%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: VCMoserblack.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\PROGRA~1\TEXLIVE\BIN\WIN32\DVIPS.EXE VCMoserblack %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.06.05:1646 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (VCMoserblack.dvi) @start %DVIPSBitmapFont: Fa cmmi12 17.28 2 /Fa 2 60 df<027FB8FC49B912C01307131F5B5B90BA128048F0FE004848C700F0C9FC13 F8D807E0495A485A5B48C71203123E003C5D5A0070140712F05A484A5AC8FCA2151FA293 CAFC5DA4157EA315FEA25D1401A44A5AA31407A34A5AA3141FA25DA2143FA34A5AA45D92 CBFC143E423F7BBD3D>28 D59 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmr17 17.28 2 /Fb 2 42 df<150E151E153C157815F0EC01E0EC03C01407EC0F80EC1F00143EA25C5C13 015C495A13075C130F5C131F91C7FC5B133E137E137C13FCA2485AA3485AA3485AA3120F 5BA3121F5BA3123FA390C8FCA25AA5127EA312FEB3A7127EA3127FA57EA27FA3121FA37F 120FA37F1207A36C7EA36C7EA36C7EA2137C137E133E133F7F80130F8013078013036D7E 801300147C80A280EC0F80EC07C01403EC01E0EC00F01578153C151E150E1F8F73EA33> 40 D<12E07E12787E7E7E6C7E7F6C7E6C7E6C7EA2137C7F133F7F6D7E80130780130380 130180130080147C147EA280A3EC1F80A3EC0FC0A315E01407A315F01403A315F8A31401 A215FCA51400A315FEB3A715FCA31401A515F8A21403A315F0A3140715E0A3140F15C0A3 EC1F80A3EC3F00A3147EA2147C14FC5C13015C13035C13075C130F5C49C7FC5B133E5B5B A2485A485A485A5B48C8FC121E5A5A5A5A1F8F7AEA33>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmbxti10 12 15 /Fc 15 118 df<91BC12C01DE05BA26D1BC0DA007F90C86C90C7FC4C5EA203FF16FFA24C 5EA24A5EA24C5EA24A5EA24C5EA24A5EA24C5EA24A5EA24C5EA24A5EA24C5EA24A5EA293 C891C8FCA24A5EA24B5EA202FF16FFA24B5EA2495EA24B5EA2495EA24B5EA2495EA24B5E A2495EA24B5EA2495EA2007FB6D8E07FB612E0B790B7FCA36C4B6C5D5B4379C259>5 D48 D<183F4E7EA218FF5FA25FA25F5FA24D7FA25F5FA294B5FCA25E5E855E17EFEE0FC7161F 1787EE3F07A2167E16FE04FC80ED01F8A24B487E150716E0ED0FC0A2ED1F80153F040080 157EA24B7F14015D4A5A92B7FC5C5C855C91393F80000192C8FC5C147E5C13015C13034A 82495A010F167F5C131FD9FFE015FF007FD9FF8090B612FEB66C4815FFA36C028016FE48 4779C656>65 D<027FB712FC91B912C01AF0861AFE9126007FFEC76C7E191F4C16808503 FF17C0A25EA25CA25EA25C4F13805EA24A4C1300624C147F624A4C5A4E5B4C495B060F5B 4A4B90C7FC943801FFFC93B612F019C04A16F819FE4BC7380FFF80727F027F6F7FA24B6E 7FA202FF83A25DA25BA25D605B625D604960604B5E4E5B495E97C7FC4B4A5A4D485A494B 5B051F5B007FB912C0BA5A4EC8FC18F06C94C9FC4A4479C351>I<027FB712F891B97E1A F01AFC869128007FFE0001EBFF80DE001F13C04C020713E08503FF6F13F07313F84C157F 1BFC4A173FA24C16FEA25CA24C16FFA25CA25EA25CA25EF27FFE5CA293C9FC1AFF5C1BFC 5D61147F1BF85D4F13F014FFA24B4B13E0A24919C0614B1780614919004F5A5D4F5A494D 5A4E5B4B5E4E5B49040F5B4E90C7FC4BEC7FFE4E5A49030713F0057F5B007FB91280B948 C8FC18F018806C04F8C9FC504479C357>68 D<91BAFCA25BA27FDA007F90C77E4C140785 03FF81A24CEC00FEA25C1A7E5EA25CA24C017C13FE187E4A03FE13FCA24C5B0501147C4A 1700A24C485A17074A140F173F93B55AA25CA260EE007F4A141FA24B5CF101F002FFEE03 F8A24B91388007F0A249033F14E0190F4BEB1F0094C7EA1FC05BF13F805D197F49EFFF00 A24B4A5A1803494C5A180F4B141FF07FF849ED01FF051F5B007FB9FCBAFC61A26C604843 7AC24B>I<027FB600C090B6FC91B76C481580A306C01600DA007F90C8000713C009FCC7 FC4CED1FF8F23FE003FF4C5A4F48C8FC4C4A5A4F5A4A4C5AF13FE04C4A5A4F5A4ADB03FE C9FC4E5A4C495AF03FE04A4B5A4E5A4C4890CAFCEF07FC4A4A5A4D5AEEC07F17FF4A01C1 7F16C74C7F169F4A13BF93B57EA24C7F4AEBF87F04F0804C7E168091B50000804B7F4B81 834983835D854981854B7FA2496F7FA24B81844984845D86498286007FB6D8E00FB67EB7 4981A36C4B6C5D594479C359>75 D<027FB712F091B9FC1AC01AF0869128007FFE00037F 9538007FFE4C6E7E8503FF1780A24C804F13C05CA25EA25C4F13805EA25C4F13005E624A 16FF624C5D4E5B4A5D4E5B93C7481380063F90C7FC4A4AB45A93B612F819E019804A03F8 C8FC93CBFC5DA214FFA25DA25BA25DA25BA25DA25BA25DA25BA25DA25BA2007FB612E0B7 FCA36C5D4A4479C34E>80 D<933807FF80047F13FC0303B6FC031F15C092267FFC0313F0 913B01FFE0007FF84A01806D7E020F90C7EA0FFEDA1FFC6E7E4A481680DAFFF080494916 C0494980494916E04990C8FC7213F0495A495A137F4A17F813FF485BA2485BA25A4A5D5A A25C5A1AF091C95A5AA348484C13E0A4494C13C012FFA24E1380A24918006061A24E5AA2 4E5A615F007FDA3F805C9226FFE0035B4A6D5A6C6C4801F85C912707E0FC0F90C7FC913A 0FC07C1FFE6C6C48486C485A0300495A260FFF1E4A5A6C033F13C06C019E6D5B6C01DE92 C8FC6C01FEEB3FFC903A7FFF01FFF0011F90B500C01360010717F001011601D9001FEB3F E091C714039438F01FE094B5FCA261A261A296C7FC60A2705B60607013C004035BDC00FE C8FC455970C555>I<017FB912F090BAFCA25AA2DAF803EBF8014801809138F0003F0200 161F01FC4916E00007180F495D5B000F5C5B5F484818C05D485A5F90C7161F485C1A8000 7E93C7FC003E180FC84892C7FCA25EA215FFA25EA25CA25EA25CA25EA25CA25EA25CA25E A25CA25EA25CA293CAFCA25CA25DA214FFA2007FB712F083B8FCA26C5E44436FC24F>84 D<007FB66C010FB512E0B76C4914F0A36C4B16E0C66C0180C8387FF0006D18806F4BC7FC 616D4C5A6F4A5A4E5A6D4C5A6F4A5A4E5A6D7F4EC8FC18FE6D6D495A4D5A6D4B5A814D5A 6D4B5A6F495A4DC9FC6E14FEEE81FCA26EEB83F8EEC7F0EECFE06EEBFFC05F8094CAFC5E 6E5B5E5E805E5CA25EA25CA25EA25CA293CBFCA25CA25DA2147FA25DA214FF0003B612FE 4881A36C5D4C446BC356>89 D100 D112 DI117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmti12 12 45 /Fd 45 123 df12 D<137C48B4FC3803C780380703C0000F13E0120E121C13071238A21278EA700F14C0131F 00F0138012E0EA003F1400A25B137EA213FE5B12015BA212035B141E0007131C13E0A200 0F133CEBC038A21478EB807014F014E0EB81C0EA0783EBC7803803FE00EA00F8172D78AB 1E>16 D19 D<13F0EA03FC1207A2EA0FFEA4EA07FCEA 03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA0700120E5A5A5A5A 5A0F1D7A891E>44 D<120FEA3FC0127FA212FFA31380EA7F00123C0A0A76891E>46 D48 D65 D<91B712FCF0FF8019E00201903980001F F06E90C7EA07F84A6F7E727E4B81841A800203167F5DA314075D19FFA2020F17004B5C61 1803021F5E4B4A5A180F4E5A023F4B5A4BEC7F804EC7FCEF03FC027FEC0FF84BEBFFC092 B6C8FC18E0913AFF800007F892C7EA01FC717E187F49834A6F7EA30103835CA313075CA3 010F5F4A157FA24E5A131F4A4A90C7FC601703013F4B5A4A4A5A4D5A017F4B5A4D5A4A49 48C8FC01FFEC0FFEB812F817C04CC9FC41447AC345>I<91B912C0A30201902680000313 806E90C8127F4A163F191F4B150FA30203EE07005DA314074B5D190EA2140F4B1307A25F 021F020E90C7FC5DA2171E023F141C4B133C177C17FC027FEB03F892B5FCA39139FF8003 F0ED00011600A2495D5CA2160101034B13705C19F061010791C8FC4A1501611803010F5F 4A150796C7FC60131F4A151E183E183C013F167C4A15FC4D5A017F1503EF0FF04A143F01 FF913803FFE0B9FCA26042447AC342>69 D<91B91280A30201902680000713006E90C8FC 4A163FA24B81A30203160E5DA314074B151E191CA2140F5D17075F021F020E90C7FC5DA2 171E023F141C4B133CA2177C027F5CED800392B5FCA291B65AED00071601A2496E5A5CA2 160101035D5CA2160301075D4A90CAFCA3130F5CA3131F5CA3133F5CA2137FA313FFB612 E0A341447AC340>II<91B6D8803FB512E0A302010180C7387FE0006E90C86C5A4A167FA24B5E A219FF14034B93C7FCA26014074B5DA21803140F4B5DA21807141F4B5DA2180F143F4B5D A2181F147F92B75AA3DAFF80C7123F92C85BA2187F5B4A5EA218FF13034A93C8FCA25F13 074A5DA21703130F4A5DA21707131F4A5DA2170F133F4A5DA2017F151FA24A5D496C4A7E B6D8803FB512E0A34B447AC348>I<027FB512E091B6FCA20200EBE000ED7F8015FFA293 C7FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA3 5B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612E0A25D2B447BC326 >I<031FB512F05DA29239000FFC005FA35FA2161FA25FA2163FA25FA2167FA25FA216FF A294C7FCA25DA25EA21503A25EA21507A25EA2150FA25EA2151FA25EA2153FA25EA2157F A25EEA0F80D83FE013FF93C8FC127FA24A5AEAFFC04A5A1300007C495A0070495A4A5A6C 5C003C495A6C01FEC9FC380F81F83803FFE0C690CAFC344679C333>I<91B56C93387FFF C08298B5FC02014DEBC0006E614A5FA203DF4C6CC7FC1A0E63912603CFE05D038F5F1A38 1A711407030FEEE1FCA2F101C3020FEE0383020E60F107036F6C1507021E160E021C6019 1CF1380F143C023804705BA2F1E01F0278ED01C091267003F85EF003801A3F02F0ED0700 02E0030E5CA24E137F130102C04B91C8FC606201036D6C5B02805F4D5A94380380011307 0200DA07005BA2050E1303495D010E606F6C5A1907011E5D011C4B5CA27048130F133C01 384B5C017892C7FC191F01F85C486C027E5DD807FE027C4A7EB500F00178013FB512C0A2 16705A447AC357>77 D79 D<91B712F018FEF0FF800201903980007FE06E90C7EA1F F04AED07F818034B15FCF001FE1403A24B15FFA21407A25DA2140FF003FE5DA2021F16FC 18074B15F8180F023F16F0F01FE04B15C0F03F80027FED7F0018FE4BEB03FCEF0FF002FF EC7FC092B6C7FC17F892CAFC5BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2 133FA25CA2137FA25C497EB67EA340447AC342>I<91B77E18F818FE020190398001FF80 6E90C7EA3FC04AED1FE0F00FF04BEC07F8180319FC14034B15FEA314075DA3020FED07FC 5DA2F00FF8141F4B15F0F01FE0F03FC0023F16804BEC7F0018FEEF03F8027F4A5A4BEB1F C04CB4C7FC92B512F891B612E092380003F8EE00FE177F496F7E4A6E7EA28413034A140F A2171F13075CA2173F130F5CA24D5A131F5CA3013F170E5CA2017FEE801E191C4A163C49 6C1638B66C90383FC070051F13F094380FE1E0CA3803FF80943800FE003F467AC347>82 DI<48B912F85AA2913B0007FC 001FF0D807F84A130701E0010F140349160148485C90C71500A2001E021F15E05E121C12 3C0038143F4C1301007818C0127000F0147F485DA3C800FF91C7FC93C9FCA35C5DA31403 5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA21303A21307 497E007FB612C0A25E3D446FC346>I87 D97 DIIIII<15FCEC03FF91390F838380 91393E01CFC091387C00EF4A13FF4948137F010315804948133F495A131F4A1400133F91 C75A5B167E13FE16FE1201495CA215011203495CA21503A2495CA21507A25EA2150F151F 5E0001143F157F6C6C13FF913801DF8090387C039F90383E0F3FEB0FFCD903F090C7FC90 C7FC5DA2157EA215FEA25DA2001C495A127F48495A14074A5A485C023FC8FC00F8137E38 7C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE137FA3EB01FC13001301A25CA21303 A25CA21307A25CA2130FA25CA2131FA25C157F90393F83FFC091388F81F091381E00F802 387F4948137C5C4A137EA2495A91C7FCA25B484814FE5E5BA2000314015E5BA200071403 5E5B1507000F5DA249130F5E001F1678031F1370491480A2003F023F13F0EE00E090C7FC 160148023E13C01603007E1680EE070000FEEC1E0FED1F1E48EC0FF80038EC03E02D467A C432>I<143C147E14FE1301A3EB00FC14701400AE137C48B4FC3803C780380703C0000F 13E0120E121C13071238A21278EA700F14C0131F00F0138012E0EA003F1400A25B137EA2 13FE5B12015BA212035B141E0007131C13E0A2000F133CEBC038A21478EB807014F014E0 EB81C0EA0783EBC7803803FE00EA00F8174378C11E>I<14FE137FA3EB01FC13001301A2 5CA21303A25CA21307A25CA2130FA25CA2131FA25C167E013F49B4FC92380783C0913800 0E07ED3C1F491370ED603F017E13E0EC01C09026FE03801380913907000E00D9FC0E90C7 FC5C00015B5C495AEBF9C03803FB8001FFC9FCA214F03807F3FCEBF07F9038E01FC06E7E 000F130781EBC003A2001F150FA20180140EA2003F151E161C010013E0A2485DA2007E15 78167000FE01015B15F1489038007F800038021FC7FC2A467AC42D>107 DIIIII114 DI<1470EB01F8A313035CA313075CA3130F5CA3131F5CA2007FB512E0B6FC15 C0D8003FC7FCA25B137EA313FE5BA312015BA312035BA312075BA3120F5BA2EC0780001F 140013805C140E003F131EEB001C143C14385C6C13F0495A6C485AEB8780D807FEC7FCEA 01F81B3F78BD20>I<137C48B414072603C780EB1F80380703C0000F7F000E153F121C01 07150012385E1278D8700F147E5C011F14FE00F05B00E05DEA003FEC0001A2495C137E15 0313FE495CA215071201495CA2030F13380003167849ECC070A3031F13F0EE80E0153F00 011581037F13C06DEBEF8300000101148090397C03C787903A3E0F07C70090391FFE01FE 903903F000782D2D78AB34>I<017C143848B414FC3A03C78001FE380703C0000F13E012 0E001C14000107147E1238163E1278D8700F141E5C131F00F049131C12E0EA003F91C712 3C16385B137E167801FE14705BA216F0000115E05B150116C0A24848EB0380A2ED0700A2 150E12015D6D5B000014786D5B90387C01E090383F0780D90FFFC7FCEB03F8272D78AB2D >I<017CEE038048B4020EEB0FC02603C780013FEB1FE0380703C0000E7F5E001C037E13 0F01071607123804FE130300785DEA700F4A1501011F130100F001804914C012E0EA003F DA000314034C14805B137E0307140701FE1700495CA2030F5C0001170E495CA260A24848 495A60A2601201033F5C7F4B6C485A000002F713036D9039E7E0078090267E01C349C7FC 903A1F0781F81E903A0FFF007FF8D901FCEB0FE03B2D78AB41>I<02F8133FD907FEEBFF E0903A0F0F83C0F0903A1C07C780F890393803CF03017013EE01E0EBFC07120101C013F8 000316F00180EC01C000074AC7FC13001407485C120EC7FC140F5DA3141F5DA3143F92C8 FCA34AEB03C01780147EA202FEEB0700121E003F5D267F81FC130E6E5BD8FF83143CD903 BE5B26FE079E5B3A7C0F1F01E03A3C1E0F83C0271FF803FFC7FC3907E000FC2D2D7CAB2D >I<137C48B414072603C780EB1F80380703C0000F7F000E153F001C1600130712385E00 78157EEA700F5C011F14FE00F0495B12E0EA003FEC00015E5B137E150301FE5C5BA21507 00015D5BA2150F00035D5BA2151F5EA2153F12014BC7FC6D5B00005BEB7C0390383E0F7E EB1FFEEB03F090C712FE5DA214015D121F397F8003F0A24A5A4848485A5D48131F00F049 C8FC0070137E007813F8383801F0381E07C06CB4C9FCEA01FC294078AB2F>I<027C1307 49B4130F49EB800E010F141E49EBC03CEDE03890393F03F07890397C00FDF00178EB3FE0 0170EB03C001F0148049130790C7EA0F00151E5D5D5D4A5A4A5A4A5A4AC7FC141E5C5C5C 495A495A495A49C8FC011E14F04914E05B491301485A4848EB03C0D807B0130701FEEB0F 80390FCF801F3A1F07E07F00393E03FFFED83C015B486C5B00705C00F0EB7FC048011FC7 FC282D7BAB28>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmsy8 8 1 /Fe 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmex10 10 10 /Ff 10 68 df32 D<12F07E127C7E123F7E6C7E6C7E6C7E7F12016C7E7F 137E133E133F6D7E130F806D7EA26D7E80130180130080147E147F8081141F81140F8114 0781A2140381140181A2140081A2157FA36F7EA382151FA282150FA3821507A382A21503 A282A31501A282A31500A382A482A21780A7163F17C0AC161F17E0B3B3A217C0163FAC17 80167FA71700A25EA45EA31501A35EA21503A35EA21507A25EA3150F5EA3151F5EA2153F 5EA34BC7FCA315FEA25D1401A25D14035D1407A25D140F5D141F5D143F92C8FC5C147E14 FE5C13015C13035C495AA2495A5C131F49C9FC133E137E5B5B485A12035B485A485A48CA FC5A123E5A5A5A2BF87E8242>III<177C17FCEE01F8A2EE03F0EE07E0EE0FC0A2EE1F80EE3F005E167E5E1501 5E15034B5A5E150F5E151F4B5AA24BC7FCA215FEA24A5AA24A5AA24A5AA2140F5D141F5D 143F5DA2147F92C8FC5CA25C13015C1303A25C1307A3495AA3495AA3133F5CA3137F5CA3 13FF91C9FCA35A5BA31203A25BA31207A35BA3120FA45BA2121FA65BA2123FA85BA2127F AE5B12FFB3A62E95688149>48 D<12F87E127EA27E6C7E6C7EA26C7E6C7E7F12016C7E7F 137E137F6D7E131F80130F806D7EA26D7EA26D7EA26D7EA2147FA26E7EA281141F81140F 811407A281140381A2140181140081A28182A36F7EA36F7EA382150FA3821507A3821503 A3821501A382A281A31780A3167FA317C0A4163FA217E0A6161FA217F0A8160FA217F8AE 160717FCB3A62E957E8149>I64 DIII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi8 8 21 /Fg 21 121 df<90383FFFFCA2903800FE00A25CA21301A25CA21303A25CA21307A25CA2 130FA25CA2131FA25CA2133FA291C7FCA25BA2137EA213FEA25BA21201A25BA21203B512 E0A21E2D7DAC1F>73 D<91383FFFF8A29138007F00A2157EA215FE5DA314015DA314035D A314075DA3140F5DA3141F5DA3143FA292C7FCA2003C5B127E00FE137E14FE5CEAFC0100 F05B48485A386007E038781F80D81FFEC8FCEA07F0252E7BAC27>I<90263FFFFC90381F FF80A2D900FEC73803F80018E04AEC07804DC7FC0101151C5F4A14E04C5A01034A5A040E C8FC4A5B5E010714E04B5A9138E00780030EC9FC010F131F157F4A487E14C190391FC71F C014CEEC9C0F02F07F90383FE00702C07FEC0003825B6F7E137E6F7E13FE167F5B707E12 01161F4981831203B539E001FFFEA2392D7CAC3C>I77 DI81 D<000FB8FCA23B1FC003F8003F0100151F001C4A130E123C003801071406123000704A13 0EA20060010F140C12E0485CA2141FC715005DA2143FA292C8FCA25CA2147EA214FEA25C A21301A25CA21303A25CA21307A25C130F131F001FB512F0A2302D7FAC29>84 D97 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C013 30EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA2 1301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B 81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F0 03FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13 F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038 ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA21200A213 F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123E A2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E 2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01 263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A4914800081021F 130300015F491400A200034A13076049133E170F0007027EEC8080188149017C131F1801 000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB 03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B801F3870 7F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF80816 18EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E 143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C000 0713F000069038E003E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C013 3EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FF C7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<903807E03090381FF87090387C1CF0EBF80D3801F00F3903E007E0EA07C0000F130338 1F800715C0EA3F00A248130F007E1480A300FE131F481400A35C143E5A147E007C13FE5C 1301EA3E07EA1F0E380FFCF8EA03F0C7FC13015CA313035CA21307A2EBFFFEA21C2B7D9D 20>I<3807C01F390FF07FC0391CF8E0E0383879C138307B8738707F07EA607E13FC00E0 EB03804848C7FCA2128112015BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC12 0E1B1F7E9D20>II 117 D119 D<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE4813FC1218 0038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C0F1306 00FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E 9D28>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmmib10 12 12 /Fh 12 114 df13 D17 D65 D<027FB812C091B912FC1A FF1BC01BF09126007FFEC7000713F8070113FC4C80F27FFE03FF163F1BFF5EA25CA25EA2 5CF27FFE5E1BFC4A17FF1BF84C5C4F13F04A4C13E04F13C04C4A1300F17FFE4A923801FF F8063F13E093B7C7FC19FC4AEEFFC01AF04BC813FCF13FFE027F707E1B804B811BC002FF 821BE05DA25BA25D615B1BC05D61491980614B4B130062494D5A604B4A5B060F5B49043F 13C04DB55A007FB948C7FCBA12F819E096C8FC6C17F050447CC356>II<027FB87E91B912F81AFF1BC01BF09126007FFEC7001F7F07 017F4C6E6C7E747E03FF040F13807413C04C811CE04A7113F0A25E7413F85CA25EA24A19 FCA25EA25CA24C5D1CF85CA293C9FC625C1CF05D62147F1CE05D5013C014FFA24B4C1380 A2494E1300A24B5F505A4918FF634B4B5B4F5B5B4F5B4B4B5B4F90C7FC49EF7FFC4F5A4B 02035B060F5B49047F1380942607FFFEC8FC007FB912F8BA12E096C9FC18F86C94CAFC56 447BC35D>I<91B712E05BA36D16C0DA007F90C7FCA25EA215FFA25EA25CA25EA25CA25E A25CA25EA25CA25EA25CA25EA25CA293C8FCA25CA25DA214FFA25DA25BA25DA25BA25DA2 5BA25DA25BA25DA25BA2007FB612E0B77EA35E33447CC333>73 D<027FB712FE91B912E0 1AFC1AFF1B809126007FFEC7003F13E0070713F04C807313F803FF811BFC5EA24A18FEA2 5EA25C4F13FC5EA25C4F13F85E1BF04A4C13E0A24C4A13C01B804A4C13004F5A93C8B45A 06035B4A033F13E093B712804FC7FC19F04A168093CBFC5DA214FFA25DA25BA25DA25BA2 5DA25BA25DA25BA25DA25BA2007FB612E0B7FCA36C5D4F447BC348>80 DI100 D112 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmbsy10 12 1 /Fi 1 86 df85 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi12 12 33 /Fj 33 122 df13 D<1578913807FFE0021F13FC91383C7FFEEC7007EC6003 ECE0004A13381600A280A380A280147CA2147E143E143F816E7EA26E7E81140781EC3FFC 14FF903803E1FEEB07C190381F00FF133E49EB7F805B0001143F485A484814C049131F12 0F485AA248C7FC150F5A127EA300FEEC1F805AA316005A5DA2153E157E157CA26C5C127C 4A5A6C495AA26C495A6C6C485A6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I< 0203B612E0021F15F091B7FC4916E0010716C090270FF80FF8C7FC90381FC00349486C7E 017EC7FC49147E485A4848143E0007153F5B485AA2485AA2123F90C8FC5E48157E127EA2 16FE00FE5D5A15015EA24B5A007C5D15074B5A5E6C4AC8FC153E6C5C5D390F8003F03907 C007C02601F03FC9FC38007FFCEB1FE0342C7DAA37>27 D<010FB612FC013F15FE5B48B7 12FC4816F82707E001C0C7FC01805B380F0003121E121C5A4849C8FC126012E000405BC7 FC140E141EA45CA3147CA2147814F8A4495AA31303A25C1307A3130FA25C6D5A2F2C7EAA 2A>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78891B>I<1618163C167CA2167816F8A216F01501A216E015 03A216C01507A21680150FA2ED1F00A2151E153EA2153C157CA2157815F8A25D1401A24A 5AA25D1407A25D140FA292C7FC5CA2141E143EA2143C147CA25CA25C1301A25C1303A25C 1307A25C130FA291C8FC5BA2133EA2133C137CA2137813F8A25B1201A25B1203A2485AA2 5B120FA290C9FC5AA2121E123EA2123C127CA2127812F8A25A126026647BCA31>61 D64 D<1830187018F0A217011703A24D7EA2170F171FA21737A2176717E717C793380187FCA2 EE0307EE07031606160CA216181638163004607FA216C0030113011680ED0300A2150615 0E150C5D845D03707F15605DA24A5A4AB7FCA25C0206C87F5C021C157F14185CA25C14E0 5C495A8549C9FC49163F1306130E5B133C137C01FE4C7ED807FFED01FF007F01F0027FEB FFC0B5FC5C42477DC649>I<91B87E19F019FC02009039C00003FF6F480100138003FFED 3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2020717F04B151F1AE0193F 020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC 92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA3 13035CA301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7 FCEF0FFC4AEC3FF801FF913801FFE0B9128005FCC8FC17C045447CC34A>I<91B87E19F0 19FC02009039C00007FF6F489038007FC003FFED1FE0737E93C86C7E737E19014A707E5D 1A7FA20203EF3F805DA21BC014075DA3140F4B17E0A3141F4B17C0A3143F4B167FA3027F 18804B16FFA302FF180092C95A62A24917034A5F19076201034D5A5C4F5A620107173F4A 5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01FCEF0F F84AEC3FE001FF913803FF80B848C9FC17F094CAFC4B447CC351>68 D<91B912FCA3020001C0C7123F6F48EC03F803FF1501190093C91278A21A385C5DA30203 17305DA314074B1460A218E0020F4B13005DA21701021F5D4B13031707170F023F027FC8 FC92B6FCA391397FC0007E4B131EA2170E02FF140C92C7FCA2171C49031813035C611906 010392C7FC4A160E190C191C010717184A163819301970130F4A5E180161011F16034A15 074E5A013F163F4EC7FC4AEC03FF01FFED3FFEB9FCA26046447CC348>I<91B912F8A302 0001C0C7123F6F48EC07F003FF1503190193C9FCA21A705C5DA3020317605DA314075D18 C01701020F4B13005DA21703021F92C8FC4B5BA25F023F141E4B13FE92B5FCA24A5CED80 00173CA202FF141892C7FCA217384915305CA21770010315604A91C9FCA313075CA3130F 5CA3131F5CA2133FA313FFB612F8A345447CC33F>I<91B6D8E003B61280A3020001E0C7 0003EB8000DB7F806E48C7FC03FF1503A293C85BA219075C4B5EA2190F14034B5EA2191F 14074B5EA2193F140F4B5EA2197F141F4B5EA219FF143F92B8C8FCA3DA7FC0C712014B5D A2180314FF92C85BA218075B4A5EA2180F13034A5EA2181F13074A5EA2183F130F4A5EA2 187F131F4A5EA2013F16FFA24A93C9FCD9FFE002037FB6D8E003B67EA351447CC351>72 D<027FB512F8A217F09139007FF000ED3FC0157FA25EA315FF93C7FCA35C5DA314035DA3 14075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075C A3130F5CA2131FA25CEB7FF0007FB512F0B6FCA22D447DC32B>I<031FB512FC5D18F892 39000FFE00705AA35FA2160FA25FA2161FA25FA2163FA25FA2167FA25FA216FFA294C7FC A25DA25EA21503A25EA21507A25EA2150FA25EA2151FA25EA2153FA25EEA0F80D83FE013 7F5E127FA24BC8FC485A4A5A1300006C495A0060495A0070495A0030495A0038EB3F806C 49C9FC380F81FC3803FFF038007F80364679C336>I<91B600E049B512C0A3020001E0C8 383FF800DB7F80ED1FE003FF94C7FC1A3E93C9127862F101C04A4C5A4B4BC8FC191C6102 035E4B5DF003804EC9FC0207150E4B14386060020F4A5A4B0107CAFC170E5F021F14784B 13F84C7E1603023F130F4B487E163BEEE1FF91387FC1C1DB83807FED8700159CDAFFB86D 7E5D03C06D7E5D4990C7FC4A6E7EA2717E13034A811707A201076F7E5C717EA2130F4A6E 7FA2727E131F5C727E133F854A82D9FFE04B7EB600E0010FB512E05FA252447CC353>I< 91B500C0933803FFFE63630200F1FE00DB6FE0EE1BF803EF171F1B3703CFEF67F0A21BCF 0201EF018F038F60DB87F0ED030F1B1F020317060307040C5BA2F2183F02071730020661 6F6C15601B7F020E17C0020CDC018090C7FCA24F485A021C16060218606F6C5C1A010238 1618023004305BA2F16003027016C00260606F6CEB01801A0702E0ED03004A03065CA24E 130F01015E4A60047F5B1A1F01035E91C74A5CA24D48133F494BC7FC010661EE3F861A7F 010E158C010C039892C8FCA205B05C011C15E001186001386E5A190101785D01FC92C75B D803FFEF07FEB500F8011E0107B512FE161C160C5F447BC35E>77 D<91B712FEF0FFE019F802009039C0000FFE6F48EB01FF03FF9138007F80F13FC093C8EA 1FE0A24AEE0FF0A25D1AF81403A25DA21407F11FF05DA2020FEE3FE0A24B16C0197F021F 1780F1FF004B4A5A4E5A023F4B5A4E5A4BEC3FC006FFC7FC027FEC07FC92B612F0188003 80CAFC14FFA292CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133F A25CEBFFE0B612E0A345447CC33F>80 DI<91 B712F018FF19E002009039C0003FF86F48EB07FC03FFEC01FEF0007F93C8EA3F801AC0F1 1FE05C5D1AF0A214035DA30207EE3FE05DA2F17FC0020F17804B15FF1A004E5A021F4B5A 4B4A5AF00FE04E5A023F037FC7FC4BEB03FCEF1FF092B612804A4AC8FC923980007F80EF 0FC0EF07F002FF6E7E92C77F1701845B4A1400A2170113035CA2170313075CA24D5A130F 5CA3011F18185CA2013F4C13381A304A6F1370D9FFE0020314E0B600E0ED01C00501EB03 80943900FE0F00CBEA3FFEF007F045467CC34A>I<9339FF8001800307EBF003033F13FC 9239FF007E07DA01F8EB0F0FDA07E09038079F004A486DB4FC4AC77E023E804A5D187E5C 495A183C495AA213074A1538A3130F183080A295C7FC806D7E8014FF6D13E015FC6DEBFF C06D14FC6E13FF6E14C0020F80020314F8EC003F03077F9238007FFE160F1603707E8283 A283A21206A4000E163EA2120C177E001E167CA25F5F003F15014C5A6D4A5A4C5A486C4A C8FC6D143ED87CF85CD8787E495A3AF01FC00FE0D8E007B51280010149C9FC39C0003FF0 39487BC53C>I<48BA12C05AA291C7D980001380D807F092C7121F4949150F0180170748 C75B1903120E48020316005E12181238003014074C5C00701806126000E0140F485DA3C8 001F92C7FC5EA3153F5EA3157F5EA315FF93CAFCA35C5DA314035DA314075DA3140F5DA3 141F5DA3143F5DA2147FA214FF01037F001FB612FCA25E42447EC339>I100 D<141E143F5C5CA3147E143891C7FCAE133EEBFF803801C3C0380781E03806 01F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA001F5CA2133F91C7 FCA25B137E13FE5BA212015BEC03800003140013F01207495A1406140E140CEBC01C1418 14385C00035BEBE1C0C6B45A013EC7FC19437DC121>105 D<01F8D903FCEC7F80D803FE D91FFF903803FFE0D8071F903B7C0FC00F81F83E0E0F80E007E01C00FC001C9026C3C003 0178137C271807C700D9F0E0137E02CE902601F1C0133E003801DCDAFB80133F003001D8 92C7FCD90FF814FF0070495C0060495CA200E04949485CD8C01F187E4A5C1200040715FE 013F6091C75BA2040F14014960017E5D1903041F5D13FE494B130762043F160E0001060F 130C4992C713C0191F4CED801C00031A1849027E1638F2003004FE167000071A60494A16 E0F201C0030192380F0380000FF18700494AEC03FED80380D90070EC00F84F2D7DAB55> 109 D<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07C3C00300 1CD9C7007F001801CE1301003801DC80003013D8EB0FF800705B00605BA200E0491303D8 C01F5D5C12001607013F5D91C7FCA2160F495D137E161F5F13FE49143F94C7FC18700001 4B136049147E16FE4C13E0000317C049150104F81380170300071700495D170EEE781C00 0FED7C3849EC1FF0D80380EC07C0342D7DAB3A>I112 D<91380FC00391383FF007 9138F83C0F903903E00E1E90390FC0063E90381F800790393F00037E4914FC01FE130148 5AA2484814F812075B000F140316F0485AA2003F14074914E0A3007F140F4914C0A3151F 90C713805AA2153F6C1500A2127E5D007F14FE6C1301A214036C6C485A000F131E3807C0 383803E0F13901FFC1F838003F01130014035DA314075DA3140F5DA2141FA2143F011FB5 1280A21600283F7DAB2B>I115 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507121CD818035D123800 30150FA2D870075D00605B161FEAE00F00C0495CEA001F4A133FA2013F92C7FC91C7FC5E 5B017E147EA216FE13FE495CA20301EB01801703484802F81300A25F0303130616F00000 1407030F130E6D010D130C017C011D131C033913186D9038F0F838903A1F03C07870903A 07FF803FE0903A01FC000F80312D7DAB38>117 D<02FCEB07E0903A03FF801FFC903A0F 07C0781E903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848EBFF00495B00 0316FE90C71438484A130012061401000E5C120CC7FC14035DA314075DA3140F5DA3021F 143817305D1770023F1460121E003F16E0267F807FEB01C0026F148000FF01EF1303D901 CFEB070000FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803F0EB3F80302D 7EAB37>120 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507001C16 E0EA180312380030150F007016C0EA60075C161FD8E00F158000C05BEA001F4A133F1700 133F91C7FC5E49147E137EA216FE01FE5C5BA215015E485AA215035EA200001407150F6D 5C017C131F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2153F93C7 FCD807C05BD81FE0137E5DA24848485A4A5A01805B39380007C00018495A001C49C8FC6C 137C380781F83803FFE0C66CC9FC2C407DAB30>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmbx12 17.28 38 /Fk 38 122 df5 D45 D<16F04B7E1507151F153FEC01FF1407147F 010FB5FCB7FCA41487EBF007C7FCB3B3B3B3007FB91280A6395E74DD51>49 D<913801FFF8021FEBFFC091B612F8010315FF010F16C0013F8290267FFC0114F89027FF E0003F7F4890C7000F7F48486E7FD807F86E148048486E14C048486E14E048486F13F001 FC17F8486C816D17FC6E80B56C16FE8380A219FFA283A36C5BA26C5B6C90C8FCD807FC5D EA01F0CA14FEA34D13FCA219F85F19F04D13E0A294B512C019804C14004C5B604C5B4C5B 604C13804C90C7FC4C5A4C5A4B13F05F4B13804B90C8FC4B5AED1FF84B5A4B5A4B48143F 4A5B4A48C8FC4A5A4A48157E4A5A4A5AEC7F8092C9FC02FE16FE495A495A4948ED01FCD9 0FC0150749B8FC5B5B90B9FC5A4818F85A5A5A5A5ABAFCA219F0A4405E78DD51>I<92B5 FC020F14F8023F14FF49B712C04916F0010FD9C01F13FC90271FFC00077FD93FE001017F 49486D8049C86C7F484883486C6F7F14C0486D826E806E82487FA4805CA36C5E4A5E6C5B 6C5B6C495E011FC85A90C95CA294B55A614C91C7FC604C5B4C5B4C5B4C5B047F13809226 0FFFFEC8FC020FB512F817E094C9FC17F817FF91C7003F13E0040713F8040113FE707F71 7F7113E085717FA2717F85A285831A80A31AC0EA03FCEA0FFF487F487F487FA2B57EA31A 80A34D14005C7E4A5E5F6C495E49C8485BD81FF85F000F5ED807FE92B55A6C6C6C491480 6C01F0010791C7FC6C9026FF803F5B6D90B65A011F16F0010716C001014BC8FCD9001F14 F0020149C9FC426079DD51>II<01C0EE01C0D801F8160F01FF167F02F0EC07FFDAFF8090B5FC92B712 8019006060606060606095C7FC17FC5F17E0178004FCC8FC16E09026FC3FFCC9FC91CBFC ADED3FFE0203B512F0020F14FE023F6E7E91B712E001FDD9E00F7F9027FFFE00037F02F8 01007F02E06EB4FC02806E138091C8FC496F13C04917E07113F0EA00F090C914F8A219FC 83A219FEA419FFA3EA03F0EA0FFC487E487E487FA2B57EA319FEA35C4D13FC6C90C8FC5B 4917F8EA3FF001804B13F06D17E0001F5E6C6C17C06D4B1380D807FC92B512006C6C4A5B 6C6C6C01075B6C01E0011F5BD97FFE90B55A6DB712C0010F93C7FC6D15FC010115F0D900 3F1480020301F0C8FC406078DD51>III<92383FFF800203B512FC021FECFF80027F 15E049B712F849D9F0077F010F90C76C7ED91FFCEC1FFFD93FF06E7F494802037F494882 717F484980854890C9127FA24884183FA25A80A380806E157F6E5E14FE6E7E6F4A5A6C14 F003FC495B03FF495B6C1580DCE0075B6CDBF80F90C7FC9338FE1FFE6C9238FF7FF84D5A 6D16C06D5E6D4BC8FC6D6F7E6D16E00101826D16FC023F814A8149B87E01078349839026 3FFE3F8190267FFC0F819026FFF003814849C6FC48496D804849131F4890C70007801601 48486E1580003F163F49150F007F7014C0491501717E8400FF835B8484A384A21A80A27F 007F1900607F003F606D160F001F606D4C5A6C6D153F6C6D4B5A6C01F04B5A6C01FC0203 5B6C01FF021F5B6D9027F001FFFEC7FC6D90B65A010F16F001035E010093C8FC020F14F8 DA007F90C9FC426079DD51>I65 D<4DB5ED03C0057F02F014070407B600FE140F047FDBFFC0131F4BB800F013 3F030F05FC137F033F9127F8007FFE13FF92B6C73807FF814A02F0020113C3020702C091 38007FE74A91C9001FB5FC023F01FC16074A01F08291B54882490280824991CB7E494984 49498449498449865D49498490B5FC484A84A2484A84A24891CD127FA25A4A1A3F5AA348 491A1FA44899C7FCA25CA3B5FCB07EA380A27EA2F50FC0A26C7FA37E6E1A1F6C1D80A26C 801D3F6C6E1A00A26C6E616D1BFE6D7F6F4E5A7F6D6D4E5A6D6D4E5A6D6D4E5A6D6E171F 6D02E04D5A6E6DEFFF806E01FC4C90C7FC020F01FFEE07FE6E02C0ED1FF8020102F8ED7F F06E02FF913803FFE0033F02F8013F1380030F91B648C8FC030117F86F6C16E004071680 DC007F02F8C9FC050191CAFC626677E375>67 DI73 D82 DI< 913803FFFE027FEBFFF00103B612FE010F6F7E4916E090273FFE001F7FD97FE001077FD9 FFF801017F486D6D7F717E486D6E7F85717FA2717FA36C496E7FA26C5B6D5AEB1FC090C9 FCA74BB6FC157F0207B7FC147F49B61207010F14C0013FEBFE004913F048B512C04891C7 FC485B4813F85A5C485B5A5CA2B55AA45FA25F806C5E806C047D7F6EEB01F96C6DD903F1 EBFF806C01FED90FE114FF6C9027FFC07FC01580000191B5487E6C6C4B7E011F02FC130F 010302F001011400D9001F90CBFC49437CC14E>97 D<903807FF80B6FCA6C6FC7F7FB3A8 EFFFF8040FEBFF80047F14F00381B612FC038715FF038F010014C0DBBFF0011F7FDBFFC0 01077F93C76C7F4B02007F03F8824B6F7E4B6F13804B17C0851BE0A27313F0A21BF8A373 13FCA41BFEAE1BFCA44F13F8A31BF0A24F13E0A24F13C06F17804F1300816F4B5A6F4A5B 4AB402075B4A6C6C495B9126F83FE0013F13C09127F00FFC03B55A4A6CB648C7FCDAC001 15F84A6C15E091C7001F91C8FC90C8000313E04F657BE35A>I<92380FFFF04AB67E020F 15F0023F15FC91B77E01039039FE001FFF4901F8010113804901E0010713C04901804913 E0017F90C7FC49484A13F0A2485B485B5A5C5A7113E0485B7113C048701380943800FE00 95C7FC485BA4B5FCAE7EA280A27EA2806C18FCA26C6D150119F87E6C6D15036EED07F06C 18E06C6D150F6D6DEC1FC06D01E0EC7F806D6DECFF00010701FCEB03FE6D9039FFC03FFC 010091B512F0023F5D020F1580020102FCC7FCDA000F13C03E437BC148>II<92380FFFC04A B512FC020FECFF80023F15E091B712F80103D9FE037F499039F0007FFF011F01C0011F7F 49496D7F4990C76C7F49486E7F48498048844A804884485B727E5A5C48717EA35A5C7213 80A2B5FCA391B9FCA41A0002C0CBFCA67EA380A27EA27E6E160FF11F806C183F6C7FF17F 006C7F6C6D16FE6C17016D6C4B5A6D6D4A5A6D01E04A5A6D6DEC3FE0010301FC49B45A6D 9026FFC01F90C7FC6D6C90B55A021F15F8020715E0020092C8FC030713F041437CC14A> III105 D<903807FF80B6FCA6C6FC7F7FB3A90503B61280A6DD003FEB8000DE0F FCC7FCF01FF04E5AF0FFC04D5B4D90C8FCEF07FC4D5AEF3FF04D5A4D5A4C90C9FC4C5AEE 0FFC4C5A4C5AEE7FC04C7E03837F03877F158F039F7F03BF7F92B5FC838403FC804B7E03 F0804B6C7F4B6C7F1580707F707F707FA270807080717FA2717F717F717FA2717F717F83 867180727F95B57EB7D8E00FECFFF0A64C647BE355>107 D<903807FF80B6FCA6C6FC7F 7FB3B3B3B3ADB712E0A623647BE32C>I<902607FF80D91FFFEEFFF8B691B500F00207EB FF80040702FC023F14E0041F02FF91B612F84C6F488193267FE07F6D4801037F922781FE 001F9027E00FF0007FC6DA83F86D9026F01FC06D7F6DD987F06D4A487F6DD98FC0DBF87E C7804C6D027C80039FC76E488203BEEEFDF003BC6E4A8003FC04FF834B5FA24B5FA24B94 C8FCA44B5EB3B2B7D8F007B7D8803FB612FCA67E417BC087>I<902607FF80EB1FFFB691 B512F0040714FC041F14FF4C8193267FE07F7F922781FE001F7FC6DA83F86D7F6DD987F0 7F6DD98FC0814C7F039FC78015BE03BC8003FC825DA25DA25DA45DB3B2B7D8F007B71280 A651417BC05A>I<923807FFE092B6FC020715E0021F15F8027F15FE494848C66C6C7E01 0701F0010F13E04901C001037F49496D7F4990C87F49486F7E49486F7E48496F13804819 C04A814819E048496F13F0A24819F8A348496F13FCA34819FEA4B518FFAD6C19FEA46C6D 4B13FCA36C19F8A26C6D4B13F0A26C19E06C6D4B13C0A26C6D4B13806C6D4B13006D6C4B 5A6D6D495B6D6D495B010701F0010F13E06D01FE017F5B010090B7C7FC023F15FC020715 E0020092C8FC030713E048437CC151>I<902607FF80EBFFF8B6010FEBFF80047F14F003 81B612FC038715FF038F010114C09227BFF0003F7FC6DAFFC0010F7F6D91C76C7F6D496E 7F03F86E7F4B6E7F4B17804B6F13C0A27313E0A27313F0A21BF885A21BFCA3851BFEAE4F 13FCA41BF861A21BF0611BE0611BC06F92B512801B006F5C6F4A5B6F4A5B03FF4A5B7049 5B04E0017F13C09226CFFC03B55A03C7B648C7FC03C115F803C015E0041F91C8FC040313 E093CBFCB3A3B712F0A64F5D7BC05A>I114 D<913A3FFF8007800107B5EAF81F011FECFE7F017F91B5FC48B8FC48EBE0014890C7121F D80FFC1407D81FF0801600485A007F167F49153FA212FF171FA27F7F7F6D92C7FC13FF14 E014FF6C14F8EDFFC06C15FC16FF6C16C06C16F06C826C826C826C82013F1680010F16C0 1303D9007F15E0020315F0EC001F1500041F13F81607007C150100FC81177F6C163FA217 1F7EA26D16F0A27F173F6D16E06D157F6D16C001FEEDFF806D0203130002C0EB0FFE02FC EB7FFC01DFB65A010F5DD8FE0315C026F8007F49C7FC48010F13E035437BC140>II<902607FFC0ED3F FEB60207B5FCA6C6EE00076D826D82B3B3A260A360A2607F60183E6D6D147E4E7F6D6D49 48806D6DD907F0ECFF806D01FFEB3FE06D91B55A6E1500021F5C020314F8DA003F018002 F0C7FC51427BC05A>III<007FB600C0017FB512F8A6 D8001F01F8C70007EBF0006D040190C7FC6D6D5D6D6D4A5A6D6D4A5A70495A6D4C5A6E7F 6E6D495A6E6D495A7049C8FC6E4A5A6E6D485A6E6D485A6E13FFEF8FF06EEC9FE06FEBFF C06F5C6F91C9FC5F6F5B816F7F6F7F8481707F8493B57E4B805D4B80DB0FF37FDB1FE17F 04C080153F4B486C7F4B486C7F4A486D7F4A486D7F4A5A4B6D7F020F6E7F4A486D7F4A48 6D804A5A4AC86C7F49486F7F4A6F7F0107707FEB3FFFB600F049B7FCA650407EBF55>I< B700C00103B512FCA6D8003F01C0C8381FFE006FED07F0A26D6D5E190F6D6D5E191F6D6D 5E193F6D95C7FC6F5D6D177E6F15FEA26D6E495AA26E6D5C18036E6D5C18076E5E70130F 6E5E70131FA26E6D495AA26E6D91C8FC606E6D137E18FE6E5D17816F5C17C3A26FEBE7F0 A26FEBF7E017FF6F5CA26F5CA26F91C9FCA36F5BA26F5BA2705AA2705AA2705AA35FA25F 163F94CAFC5E167E16FED807E05CD81FF81301487E486C495AA2B5495AA24B5A5E151F4B 5A6C4849CBFC15FEEBFC01393FF807FC391FF03FF06CB55A6C5C6C91CCFCC613FCEB1FE0 4E5D7DBF55>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmbx12 12 43 /Fl 43 122 df5 D<023FB7FCA5DA000101E0C9FCA9D87FF8943807FFC0486C5F6D5F6D5F001F018093 387FFE006C616C6102C016FFA26C61AA6C01E04B5BA46C6102F05DA2D97FF84B5B013F95 C7FC02FC5DD91FFE4B5AD90FFF4B5A6D01814A5A6D01C14A5A6D01F901E75B6D6CB71280 6E93C8FC020715F8020015C0030F01FCC9FC030113E0AA023FB7FCA54A447AC357>9 D48 DIII<163FA25E5E 5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8 EB01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A 12FCB91280A5C8000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FF EB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C7 14C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8 FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C012 3E003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01 FCC7FC010113C02D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090 390FFC001FD93FF014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE 7F80EE1E00003F92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9 F01FFC9039FBC007FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127F A5123FA217F07F121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038 E01FFC6DB55A011F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC 90B712FEA45A17FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D1501 4B5A00F84A5A484A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2 147F5D14FFA25BA35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>III65 DIII70 D72 DI<0107B7FCA590C7001F1300B3B3A9EA1FE0487E48 7EA2487EA44B5AA26C48495A495C6C4813FF6C48485B260FFC0713C06CB65A6C4AC7FCC6 6C13F8010F138030457DC33A>II80 D<923807FFC092B512FE0207ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF0107 01C0010713C04990C700017F49486E7F49486F7E49486F7E49486F7E48496F7E48496F13 80A248496F13C0A24819E091C97E4819F0A248487013F8A3007F19FCA249177FA300FF19 FEAD007F19FCA36D17FF003F19F8A3001F19F06D5EA26C19E06E01FE5B6C912603FF8014 C06C6D486D4813804B13E06C9028E01F83F00F13006C903BF01E00F81FFE90267FF83E90 387C3FFC90263FFC3C6D485AD91FFE91381EFFF0D90FFF021F5B6D01FE5D010194C7FC6D 6D6CB45A023F90B512F8020703E0130202006F1307030713C792C7EA07F8716C130F7213 1F9538FF80FF96B5FC7114FEA3831AFCA27213F81AF0847213E07213C0721300F001FC48 587AC454>III85 DI<007FB6D8C003B61280 A5D8000F01E0C7D801F8C7FC6D4C5A6F14076D6D5D6D6D4A5A4E5A6D6D143F6E6C92C8FC 6E157E705B6EEBC0016E01E05B4D5A6E6D485A6EEBF80F6E01FC5B4D5A6E6D48C9FC6F6C 5A6F137E5F6F5B815F816F7F81836F7F707E93B5FC844B805D4B8004E77FDB0FC37FED1F 83DB3F817F04007F037E137F4B8002016E7F4B6D7F4A5A4A486D7F020F6E7F4B7F4A4881 4AC76C7F717F147E4A6F7E0101707F4A8149488349486F7F010F707FB600E00103B612FC A54E447DC355>88 D<001FB812FEA59126F8000113FC028015F801FCC75A494A13F04916 E0495C494A13C0484816805E90C84813005F003E15FF4B5B5F003C5C4B5B5F5D4B5BC85C 4B90C7FC5D5E4B5A5C5E4A5B5C5E4A5B5C5E4A90C8FC5C5D4A48140F5B5D495B5B494914 1F5D49161E495B92C8FC49163E495A5C48177E485B4A15FE481601484914034A14074816 0F4849143F91C8EAFFFC48150FB9FCA538447AC344>90 D100 DIIII<90397F8007FEB590383FFF8092B512 E0028114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA25C A35CB3A7B60083B512FEA5372D7CAC3E>110 DI<90397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF 001FFE000301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF 7FF8A318F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB5 12F002C314C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>II<90387F807FB53881FFE0028313F002 8F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7 FCA35CB3A5B612E0A5272D7DAC2E>I<90391FFC038090B51287000314FF120F381FF003 383FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF 6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143FA2 6C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8 F03F13E026E007FEC7FC232F7CAD2C>I 119 DII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr10 10 35 /Fm 35 122 df<146014E0EB01C0EB0380EB0700130E131E5B5BA25B485AA2485AA21207 5B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7EA27F12077F1203 A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD20>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA2131F7F A21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C1378A2 5BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<121C127FEAFF80A213C0 A3127F121C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>44 DI48 DIII54 D<913A01FF800180020FEBE003027F 13F8903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C812 7F01FE153F12014848151F4848150FA248481507A2485A1703123F5B007F1601A35B00FF 93C7FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D0000 1618017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007F FFFC020F13F002011380313D7BBA3C>67 DI71 D80 D82 DI85 D87 D97 D99 D101 D<147E903803FF8090380F C1E0EB1F8790383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB 487E387FFFF8A31C3B7FBA19>I104 DI108 D<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F8380 3F3D0FF3800FC7001F802603F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A348 6C496CEB1FE0B500C1B50083B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03F90 39F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C497EB500C1B51280 A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3801F 803A0FF7000FC0D803FEEB07E049EB03F04914F849130116FC150016FEA3167FAA16FEA3 ED01FCA26DEB03F816F06D13076DEB0FE001F614C09039F7803F009038F1E07E9038F0FF F8EC1FC091C8FCAB487EB512C0A328357EA42E>I<3807E01F00FFEB7FC09038E1E3E090 38E387F0380FE707EA03E613EE9038EC03E09038FC0080491300A45BB3A2487EB512F0A3 1C257EA421>114 DI<1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215 C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220>III120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmsy10 12 7 /Fn 7 86 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<0060160600F8160F6C16 1F007E163F6C167E6C6C15FC6C6CEC01F86C6CEC03F06C6CEC07E06C6CEC0FC06C6CEC1F 80017EEC3F006D147E6D6C5B6D6C485A6D6C485A6D6C485A6D6C485A6D6C485ADA7E3FC7 FCEC3F7E6E5A6E5A6E5AA24A7E4A7EEC3F7EEC7E3F4A6C7E49486C7E49486C7E49486C7E 49486C7E49486C7E49C7127E017E8049EC1F804848EC0FC04848EC07E04848EC03F04848 EC01F84848EC00FC48C9127E007E163F48161F48160F00601606303072B04D>2 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C1980CEFCB0007FBA1280 BB12C0A26C1980422C7BAE4D>17 D<1AF0A3861A78A21A7C1A3CA21A3E1A1E1A1F747EA2 747E747E87747E747E1B7E87757EF30FE0F303F8007FBC12FEBE1280A26CF3FE00CEEA03 F8F30FE0F31F8051C7FC1B7E63505A505A63505A505AA250C8FC1A1E1A3E1A3CA21A7C1A 78A21AF862A359347BB264>33 D67 D72 D<027E1718D907FF17F8011F1701017F6D150348B5EE07F00007180FEA0FC148C6EF1FE0 123CC790C913C0193FA24A17800101177FA24AEEFF001303A24A4B5A13074A150361010F 16075C011F160F4A4B5AA24948153F61017F167F91C9FC494C5A495DA248485D4D5B1203 495D00075E495FEF3F7F000F167E5B001F4C48C7FC4C5A494A5A003F5E933807E1FEEE0F C14848EC1F81EE3F0193383E03FC167C00FF15F8ED01F0923803E007ED07C0DB0F805B6D EB1F00153E15FC9026E001F0130F397FF007E09026FC0FC0153090B5C7EBFDF06C01FCED FFE04A5E6C01E093C7FC6C90C86C5AD803F016F8CAEA03C0454781C33E>85 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmbx12 14.4 20 /Fo 20 122 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmr8 8 7 /Fp 7 52 df6 D<13031307130E131C1338137013F0EA01 E013C01203EA0780A2EA0F00A2121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2 EA0780A2EA03C0120113E0EA00F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2EA00F0A21378A3133CA4131EA513 1FA2130FAB131FA2131EA5133CA41378A313F0A2EA01E0A2EA03C013801207EA0F00120E 5A5A5A5A5A10437CB11B>I48 D<130C133C137CEA03FC12FF EAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmr12 12 91 /Fq 91 127 df<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B 6C7EA292383807FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC 4A6E7E1406020E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C0101 6F7F5C0103707E91C9FC183F010683181F4983180F49831807498318034983A249707EA2 4848701380A248CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FE A2BCFC48477CC651>1 D6 D10 D<9239FFC001FC020F9038F80FFF913B3F803E3F03C0913BFC00077E07E0D903F8 90390FFC0FF0494890383FF81F4948EB7FF0495A494814E049C7FCF00FE04991393FC003 8049021F90C7FCAFB912F0A3C648C7D81FC0C7FCB3B2486CEC3FF0007FD9FC0FB512E0A3 3C467EC539>I<4AB4FC020F13E091387F80F8903901FC001C49487FD907E0130F494813 7F011FECFF80495A49C7FCA25B49EC7F00163E93C7FCACEE3F80B8FCA3C648C7FC167F16 3FB3B0486CEC7FC0007FD9FC1FB5FCA330467EC536>I<913801FFC0020FEBFB8091387F 803F903801FC00494813FFEB07E0EB1FC0A2495A49C7FC167F49143F5BAFB8FCA3C648C7 123FB3B2486CEC7FC0007FD9FC1FB5FCA330467EC536>II16 D<131F1480133F137FA2EBFF00485A485A5B485A485A138048C7FC123E123C5A12E01240 11126CC431>19 D<6C130100E013076C130F007C133E001F13F8380F81F03807E7E03803 FFC06C138038007E00133C1318180C74BF31>I23 D<1606A25E161C1618163816305EEC7F80903903FFF0C090380FC0FC90393E001F804913 0F01F0EB03C04848497E0003814848EB0CF84848147C1518001F157E48C7487E15701560 4802E01380007EECC01FEC0180A200FED9030013C0A21406140E140C141C14185CA25C00 7E16805CD87F01143F003F4914001303001F90C7123E0186147E000F157C01CC14FC0007 5DD803F8495A00014A5A00004A5A017E011FC7FC90387F807E9038C7FFF89038C07F8048 48C9FCA248CAFC5A1206120E120C5AA22A3F7DB431>28 D<001EEB03C0397F800FF000FF 131F01C013F8A201E013FCA3007F130F391E6003CC0000EB000CA401E0131C491318A300 0114384913300003147090C712604814E0000614C0000E130148EB038048EB070048130E 0060130C1E1D7DC431>34 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013 C0A312011380120313005A1206120E5A5A5A12600B1D78C41B>39 D<140C141C1438147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B1203 A2485AA3485AA348C7FCA35AA2123EA2127EA4127CA312FCB3A2127CA3127EA4123EA212 3FA27EA36C7EA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C01301EB00 E014701438141C140C166476CA26>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA2 1378137C133C133E131E131FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300 A314FCB3A214F8A31301A414F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E 133C137C13785BA2485A485AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7E B3AB007FBAFCBB1280A26C1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78891B>II<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A78891B>I<1618163C167CA2167816F8A216F01501A216E01503A2 16C01507A21680150FA2ED1F00A2151E153EA2153C157CA2157815F8A25D1401A24A5AA2 5D1407A25D140FA292C7FC5CA2141E143EA2143C147CA25CA25C1301A25C1303A25C1307 A25C130FA291C8FC5BA2133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B12 0FA290C9FC5AA2121E123EA2123C127CA2127812F8A25A126026647BCA31>I<14FF0107 13E090381F81F890383E007C01FC133F4848EB1F8049130F4848EB07C04848EB03E0A200 0F15F0491301001F15F8A2003F15FCA390C8FC4815FEA54815FFB3A46C15FEA56D130100 3F15FCA3001F15F8A26C6CEB03F0A36C6CEB07E0000315C06D130F6C6CEB1F806C6CEB3F 00013E137C90381F81F8903807FFE0010090C7FC28447CC131>I<143014F01301130313 1F13FFB5FC13E713071200B3B3B0497E497E007FB6FCA3204278C131>II<49B4FC010F13E0013F13FC9038FE01FE3A01F0007F 80D803C0EB3FC048C7EA1FE0120EED0FF0EA0FE0486C14F8A215077F5BA26C48130FEA03 C0C813F0A3ED1FE0A2ED3FC01680ED7F0015FE4A5AEC03F0EC1FC0D90FFFC7FC15F09038 0001FCEC007FED3F80ED1FC0ED0FE016F0ED07F816FC150316FEA2150116FFA3121EEA7F 80487EA416FE491303A2007EC713FC00701407003015F80038140F6C15F06CEC1FE06C6C EB3FC0D803E0EB7F803A01FE01FE0039007FFFF8010F13E0010190C7FC28447CC131>I< ED0380A21507150FA2151F153FA2157F15FFA25CEC03BF153F14071406140C141C141814 301470146014C013011480EB03005B13065B131C13185B1370136013E0485A5B120390C7 FC1206120E120C5A123812305A12E0B812C0A3C8383F8000ADEDFFE0027FEBFFC0A32A43 7DC231>I<000615C0D807C0130701FCEB7F8090B612005D5D5D15E0158026063FFCC7FC 90C9FCAE14FF010713C090381F01F090383800FC01F0137ED807C07F49EB1F8016C090C7 120F000615E0C8EA07F0A316F81503A216FCA5123E127F487EA416F890C712075A006015 F0A20070140F003015E00038EC1FC07E001EEC3F806CEC7F006C6C13FE6C6C485A3901F8 07F039007FFFE0011F90C7FCEB07F826447BC131>I I<121CA2EA1F8090B712C0A3481680A217005E0038C8120C0030151C00705D0060153016 705E5E4814014B5A4BC7FCC81206150E5D151815385D156015E04A5AA24A5A140792C8FC 5CA25C141E143EA2147E147CA214FCA21301A3495AA41307A6130FAA6D5AEB01C02A457B C231>I<14FF010713E0011F13F890387F00FE01FC133FD801F0EB1F804848EB0FC049EB 07E00007EC03F048481301A290C713F8481400A47FA26D130116F07F6C6CEB03E013FC6C 6CEB07C09039FF800F806C9038C01F006CEBF03EECF87839007FFEF090383FFFC07F0107 7F6D13F8497F90381E7FFFD97C1F1380496C13C02601E00313E048486C13F00007903800 7FF84848EB3FFC48C7120F003EEC07FE150148140016FF167F48153FA2161FA56C151E00 7C153EA2007E153C003E157C6C15F86DEB01F06C6CEB03E06C6CEB07C0D803F8EB1F80C6 B4EBFF0090383FFFFC010F13F00101138028447CC131>I<14FF010713E0011F13F89038 7F80FC9038FC007E48487F4848EB1F804848EB0FC0000FEC07E0485AED03F0485A16F800 7F140190C713FCA25AA216FE1500A516FFA46C5CA36C7E5D121F7F000F5C6C6C130E150C 6C6C131C6C6C5BD8007C5B90383F01E090390FFF80FE903801FE0090C8FC150116FCA4ED 03F8A216F0D80F801307486C14E0486C130F16C0ED1F80A249EB3F0049137E001EC75A00 1C495A000F495A3907E01FE06CB51280C649C7FCEB1FF028447CC131>I<121EEA7F80A2 EAFFC0A4EA7F80A2EA1E00C7FCB3A5121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA 1B>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121E127FEAFF80A213C0A412 7F121E1200A512011380A3120313005A1206120E120C121C5A5A12600A3E78AA1B>I<00 7FBAFCBB1280A26C1900CEFCB0007FBAFCBB1280A26C190041187BA44C>61 D64 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FE A3ED30FFA203707FED607FA203E07FEDC03FA2020180ED801FA2DA03007F160FA2020680 1607A24A6D7EA34A6D7EA34A6D7EA20270810260147FA202E08191B7FCA249820280C712 1FA249C87F170FA20106821707A2496F7EA3496F7EA3496F7EA201788313F8486C83D80F FF03037FB500E0027FEBFFC0A342477DC649>IIIIIIIII<010FB512FEA3D9000313806E130080B3B3AB 123F487E487EA44A5A13801300006C495A00705C6C13076C5C6C495A6CEB1F802603E07F C7FC3800FFFCEB1FE027467BC332>IIIIIIIII<49B41303010FEBE007013F13F89039FE00FE0FD801 F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82127C00FC81A282 A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C14FC6C14FF6C 15C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13C0167F163F16 1F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D147ED8FBC05C D8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0003FF02B487BC5 36>I<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048187C0078183CA2 0070181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA33F447DC346>I< B600C0010FB5FCA3000101E0C813F026007F80ED1F80F00F00A21806B3B3A7180E6D6C15 0CA2181C131F6E1518010F163818306D6C1570606D6C14016D6C5D6D6CEC0780027F4AC7 FC6E6C131EDA1FE0137C913907FC03F00201B55A6E6C1380DB07FCC8FC40467CC349>I< B692383FFFF0A3000301E003071300C649ED01FC4A5E017F705A6E5E133F616E1501011F 5FA26D6C4BC7FCA28001071606A26E150E0103160CA26D6C5DA2806D5EA26F1470027F15 6081023F5DA281021F4A5AA26F1303020F92C8FC8102071406A26F130E0203140CA26E6C 5BA2816E5CA2EE8070037F1360A26F6C5AA216E092381FE180A216F3030F90C9FC16FBED 07FEA36F5AA36F5AA26F5AA3166044467EC349>II 89 D<001FB81280A39126800001130001FCC7FC01F04A5A01C04A5A5B90C8485A121E4C 5A484B5AA200384B5A4C5AA24B90C7FC00304A5AA24B5AA24B5AC8485AA24B5A4B5AA24B 5A5C93C8FC4A5AA24A5A4A5AA24A5A4A5AA24A5A14FF5D4990C9FCEF0180495A495AA249 5A494814031800495AA2495A495A5F4890C8FC485A5F485A48485D5F48485D17FE484814 034848140F16FFB8FCA331447BC33C>II<01C01318000114384848137048C712E0000EEB01C0000C1480001C13030018 140000385B003013060070130E0060130CA300E0131C481318A400CFEB19E039FFC01FF8 01E013FCA3007F130FA2003F130701C013F8390F0001E01E1D71C431>II97 DII<167FED3FFFA315018182 B3EC7F80903803FFF090380FC07C90383F000E017E1307496D5AD803F87F48487F5B000F 81485AA2485AA2127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C 9038077F806C6C010E13C0013F011C13FE90380FC0F8903803FFE09026007F0013002F46 7DC436>IIIIII<143C14FFA2491380A4 6D1300A2143C91C7FCADEC7F80EB3FFFA31300147F143FB3B3AA123E127F39FF807F00A2 147EA25C6C485A383C01F06C485A3807FF80D801FEC7FC195785C21E>IIII<3901FC01FE00FF903807FFC0 91381E07F091383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA3 5BB3A8486CECFF80B5D8F83F13FEA32F2C7DAB36>II<3901FC03FC00FF90380FFF8091383C07E091387001F83A07FDE000FE000301 80137FD801FFEC3F8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE01FEABEE 03FCA3EE07F8A217F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F8 91383C0FE091381FFF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I<91387F80 03903903FFE00790380FE07890393F801C0F90387E000E496D5AD803F8EB039F0007EC01 BF4914FF48487F121F5B003F81A2485AA348C8FCAB6C7EA3123F7F121F6D5C120F6D5B12 076C6C5B6C6C497E6C6C130E013F131C90380FC0F8903803FFE09038007F0091C7FCAEEE FF80033F13FEA32F3F7DAB33>I<3903F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF38 03F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7DAB26>I< 90383FE0183901FFFC383907E01F78390F0003F8001E1301481300007C1478127800F814 38A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C14E0C614F001 1F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C147C15786C14F8 6CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I<1306A5130EA4 131EA3133E137EA213FE12011207001FB512F0B6FCA2C648C7FCB3A4150CAA017E131C01 7F1318A26D133890381F8030ECC070903807E0E0903801FFC09038007F001E3E7EBC26> IIIIII<003FB612E0A29038C0 003F90C713C0003CEC7F800038ECFF00A20030495A0070495AA24A5A0060495AA24A5A4A 5AA2C7485A4AC7FC5B5C495A13075C495A131F4A1360495A495AA249C712C0485AA2485A 485A1501485A48481303A24848EB07804848131F00FF14FF90B6FCA2232B7DAA2B>II<01F81302D803FE13073907FF800E48EBE01C391F1FF8F8393807FF F0D8700113E039E0007FC00040EB1F00200978C131>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx12 20.74 21 /Fr 21 122 df45 D<923801FFFE033FEBFFF84AB7FC020F16E0 023F16F84A16FE49B97E49DA003F80010F01F0010714F04901800101804948C880D97FF8 6F7F02E081496C834801FC6F148014FF486E6E14C08181481AE081A96C5C1BC06C4A5C6C 5C6D90C815806D5AD90FF85D90CA150062606295B55A4D5C624D5C4D5C4D91C7FC4D13FC 4D5B4CB512E0047F1480037FB548C8FC92B612F818C018F8F0FF806F15F092C7003F13FC 050713FF050114C071807213F8727F727F867214801BC07214E01BF0A27214F81BFCA372 14FEA31BFFEBFF80000313E0487F001F13FC487FA2487FA2B67EA31BFEA3601BFCA292C8 FC6C1AF84A5D4A18F06C494B14E05C6C01C04B14C06C90C915804E14006C6D4B5B6C01F0 92B55A6C01FC4A5C27007FFFC001075C6D01FE013F14C0010F90B85A6D4DC7FC010117F8 D9003F16E0020F93C8FC020015F0030749C9FC507378F061>51 D68 D82 D<92383FFFF80207B612E0027F15FC49B8 7E010717E0011F83499026F0007F13FC4948C7000F7F90B502036D7E486E6D806F6D8072 7F486E6E7F8486727FA28684A26C5C72806C5C6D90C8FC6D5AEB0FF8EB03E090CAFCA705 07B6FC041FB7FC0303B8FC157F0203B9FC021FECFE0391B612800103ECF800010F14C049 91C7FC017F13FC90B512F04814C0485C4891C8FC485B5A485B5C5A5CA2B5FC5CA360A36E 5DA26C5F6E5D187E6C6D846E4A48806C6D4A4814FC6C6ED90FF0ECFFFC6C02E090263FE0 7F14FE00019139FC03FFC06C91B6487E013F4B487E010F4B1307010303F01301D9003F02 80D9003F13FC020101F8CBFC57507ACE5E>97 D<93383FFFF00307B612C0033F15F84AB7 12FE0207707E021F17E0027F8391B526FC001F7F010302C001037F4991C7487F49495C49 5B4901F04A7F5B90B55A485CA2485C4891C8FCA248715B5C48715B725B4A6F5B48943800 7FC0071FC7FC96C8FC5AA25CA3B5FCAF7E80A47E80A27E806CF11F80F23FC06C6E167FA2 6C6EEEFF80816C606C6E17006D6D4B5A6D6D15076D6D4B5A6D6D6C4A5A6D02E0EC7FF06D 02F849485A01009126FF801F5B6E91B6C7FC021F5E020716F8020116E06E6C1580030702 FCC8FCDB003F13804A507ACE56>99 D<97380FFFE00607B6FCA8F00003190086B3AD9338 3FFF800307B512F8033F14FF4AB712C0020716F0021F16FC027F9039FE007FFE91B500F0 EB0FFF01030280010190B5FC4949C87E49498149498149498149498190B548814884484A 8192CAFC5AA2485BA25A5C5AA35A5CA4B5FCAF7EA4807EA37EA2807EA26C7F616C6E5D6C 606C80616D6D5D6D6D5D6D6D92B67E6D6D4A15FC010301FF0207EDFFFE6D02C0EB3FFE6D 6C9039FC01FFF86E90B65A020F16C002031600DA007F14FC030F14E09226007FFEC749C7 FC5F797AF76C>I<93387FFF80030FB512FC037FECFF804AB712E0020716F8021F16FE02 7FD9F8077F49B5D8C000804991C7003F13E04901FC020F7F49496E7F49498049496E7F49 496E7F90B55A48727E92C914804884485B1BC048841BE0485BA27313F05AA25C5AA21BF8 85A2B5FCA391BAFCA41BF002F8CCFCA67EA3807EA47E806CF103F0F207F86C7F1A0F6C6E 17F06C191F6F17E06C6E163F6D6DEE7FC06D6D16FF6D6D4B13806D6D4B13006D6D6CEC0F FE6D02E0EC3FFC6D02F8ECFFF86D9126FFC00F5B023F91B65A020F178002034CC7FC0200 16F8031F15E0030392C8FCDB000F13E04D507BCE58>I103 D<903801FFFCB6FCA8C67E131F7FB3AD95380FFFE095B5 12FE05036E7E050F15E0053F15F84D81932701FFF01F7F4CD900077FDC07FC6D80DC0FF0 6D80DC1FC07F4C48824CC8FC047E6F7F5EEDFDF85E03FF707F5EA25EA25EA293C9FCA45D B3B3A6B8D8E003B81280A8617879F76C>II<903801FFFCB6FCA8C67E131F 7FB3B3B3B3B3ABB812C0A82A7879F735>108 D<902601FFF891260FFFE093383FFF80B6 92B500FE0303B512F805036E6C020F14FE050F03E0023F6E7E053F03F891B712E04D6F49 82932701FFF01F6D0107D9C07F7F4CD900076D90270FFC001F7FDC07FC6D9126801FF06D 7FC66CDA0FF06D9126C03FC06D7F011FDA1FC06D4BC77E6D4A48DCE0FE834CC8ECE1FC04 7E6FD9F1F86E804CEFF3F0DBF9F8EFF7E04C6003FB7001FF6F804C6015FF4C95C9FCA24C 5FA293C95CA44B60B3B3A6B8D8E003B8D8800FB712FEA8974E79CDA2>I<902601FFF891 380FFFE0B692B512FE05036E7E050F15E0053F15F84D81932701FFF01F7F4CD900077FDC 07FC6D80C66CDA0FF06D80011FDA1FC07F6D4A48824CC8FC047E6F7F5EEDF9F85E03FB70 7F5E15FF5EA25EA293C9FCA45DB3B3A6B8D8E003B81280A8614E79CD6C>I<93381FFFE0 0303B6FC031F15E092B712FC020316FF020F17C0023FD9FC0014F091B500C0010F13FC49 91C700037F4901FC02007F010F496F13C049496F7F49496F7F4B8149496F7F90B5C96C7F 4886A24849707F481B80A248497014C0A2481BE0A348497113F0A3481BF8A5B51AFCAE6C 1BF8A46C1BF06E94B5FCA36C1BE0A26C6D4C14C0A26C1B806E5E6C1B006C6E4B5BA26C6E 4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D92B55A6D01FF02035C6D02C0010F91C7FC010002 FC90B512FC6E90B75A021F17E00207178002014CC8FCDA003F15F0030392C9FCDB001F13 E056507BCE61>I<902601FFFCEC7FFEB6020FB512F0057F14FE4CB712C0040716F0041F 82047F16FE93B5C66C7F92B500F0010F14C0C66C0380010380011F4AC76C806D4A6E8004 F06F7F4C6F7F4C6F7F4C8193C915804B7014C0861DE0A27414F0A27414F8A47513FCA575 13FEAF5113FCA598B512F8A31DF0621DE0621DC0621D806F5E701800704B5B505B704B5B 7092B55A04FC4A5C704A5C706C010F5C05E0013F49C7FC9227FE7FFC01B55A70B712F004 0F16C0040393C8FC040015F8053F14C0050301F0C9FC94CCFCB3A6B812E0A85F6F7ACD6C >I<902601FFF8EB07FEB691383FFFC094B512F00403804C14FE4C8093261FFC3F138093 263FE07F13C0DC7F80B5FCC66C5D011FDAFE0114E06DEBF9FC16F815FB16F016E015FF16 C07114C05E72138095381FFE0093C76C5AF001E095C8FCA25DA65DB3B3A2B812F8A8434E 7ACD4F>114 D<912603FFFCEB0780027F9039FFE00FC00103B6EAF83F010FEDFEFF013F 92B5FC49EB000F2601FFF01300480180143F4890C8120F4848814848814981123F83485A 187FA212FF6D163FA37F7F6DEE1F8002C092C7FC14F014FEECFFF06CECFF8016FEEEFFE0 6C16FC6C16FF18C06C836C17F86C836C836C83013F17806D17C0010717E0010117F0EB00 3F020716F8EC001F030015FC1607EE007F051F13FE1707007E82B482836D167FA2183F7F 181FA27F19FC7FA26D163F6D17F86D167F19F06D16FF6E4A13E002E04A13C06E4A138002 FE023F1300913AFFC003FFFE01E790B65A01C316F0018016C026FE003F92C7FC48010714 F80070D9007F90C8FC3F507ACE4C>I<15FFA75CA55CA45CA25CA25CA25CA25C91B5FCA2 5B5B5B131F5B90B9FC120FBAFCA6D8000791C9FCB3B3A3F01FE0AE183F7014C07F187F70 14806D16FF826D4B13006E6D485AEEFE0F6E90B55A020F5D6E5D020115C06E6C5C031F49 C7FC030113F03B6E7CEC4B>II121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 87 1 87 0 bop 40 120 a Fr(Ra)-5 b(y)65 b(propagator)f(matrices)j(in)e(3-D)f (anisotropic)628 281 y(inhomogeneous)g(la)-5 b(y)g(ered)65 b(media)0 707 y Fq(Vlastisla)m(v)459 682 y(\024)448 707 y(Cerv)m(en)q(\023)-50 b(y)863 671 y Fp(1)902 707 y Fq(\),)33 b(Tijmen)e(Jan)i(Moser)1847 671 y Fp(2)1886 707 y Fq(\))0 981 y Fp(1)39 1017 y Fq(\))g(Charles)g(Univ)m(ersit)m(y)-8 b(,)33 b(F)-8 b(acult)m(y)32 b(of)g(Mathematics)g(and)h(Ph)m(ysics,)h (Departmen)m(t)f(of)f(Geoph)m(ysics,)130 1137 y(Ke)22 b(Karlo)m(vu)f(3,)j(121)d(16)g(Praha)h(2,)h(Czec)m(h)h(Republic.)39 b(E-mail:)c(v)m(cerv)m(en)m(y@seis.k)-5 b(arlo)m(v.m\013.cuni.cz)0 1221 y Fp(2)39 1258 y Fq(\))32 b(Zeehelden)h(Geoservices,)g(v)-5 b(an)32 b(Alk)m(emadelaan)e(550A,)h(2597)g(A)-11 b(V)32 b('s-Gra)m(v)m(enhage,)h(The)g(Nether-)130 1378 y(lands.)43 b(E-mail:)e(mosertj@hotmail.com)0 2042 y Fo(Summary)146 2233 y Fq(Ra)m(y)30 b(propagator)e(matrices)h(con)m(tain)f(the)i (complete)f(solutions)f(to)h(the)g(system)i(of)d(dynamic)h(ra)m(y)0 2353 y(tracing)36 b(equations)h(connected)h(with)e(a)h(giv)m(en)f (reference)j(ra)m(y)-8 b(.)56 b(They)38 b(pla)m(y)e(an)h(imp)s(ortan)m (t)d(role)i(in)0 2474 y(studying)i(the)h(prop)s(erties)e(of)h(complete) f(four-parameteric)f(systems)j(of)f(paraxial)e(ra)m(ys)j(and)f(o\013er) 0 2594 y(man)m(y)30 b(applications)e(in)i(b)s(oth)g(n)m(umerical)f(mo)s (deling)e(and)k(practical)d(in)m(terpretational)g(problems)i(of)0 2714 y(seismic)i(ra)m(y)h(\014elds)g(in)e(the)i(high-frequency)h (asymptotic)e(appro)m(ximation.)146 2906 y(T)-8 b(raditionally)g(,)34 b(ra)m(y)i(propagator)f(matrices)h(ha)m(v)m(e)h(b)s(een)g(expressed)h (either)e(in)g(Cartesian)f(co)s(or-)0 3026 y(dinates)i(or)f(in)g(ra)m (y-cen)m(tered)i(co)s(ordinates,)g(connected)g(with)e(a)h(reference)h (ra)m(y)-8 b(.)56 b(Both)36 b(co)s(ordinate)0 3146 y(systems)46 b(ha)m(v)m(e)f(certain)f(adv)-5 b(an)m(tages.)78 b(F)-8 b(or)43 b(ra)m(y-cen)m(tered)j(co)s(ordinates,)h(the)d(dimensions)f(of) h(ra)m(y)0 3267 y(propagator)28 b(matrices)f(can)i(b)s(e)g(easily)f (reduced)i(from)e(6)14 b Fn(\002)g Fq(6)28 b(to)h(4)14 b Fn(\002)g Fq(4)28 b(\(in)g(a)h(3-D)e(medium\),)h(thereb)m(y)0 3387 y(re\015ecting)f(the)h(strictly)e(four-parameteric)f(nature)i(of)g (a)g(general)f(paraxial)f(ra)m(y)i(\014eld.)42 b(On)27 b(the)g(other)0 3508 y(hand,)32 b(in)f(Cartesian)h(co)s(ordinates,)f (the)i(computations)d(are)i(conceptually)g(simpler)e(and)h(generally)0 3628 y(v)-5 b(alid)32 b(in)h(isotropic)g(and)h(anisotropic)e(media.)46 b(In)34 b(a)g(Cartesian)g(co)s(ordinate)f(system,)i(the)f(dynamic)0 3748 y(ra)m(y)42 b(tracing)f(system)h(and)g(ra)m(y)g(propagator)f (matrices)f(are)i(w)m(ell-kno)m(wn)g(b)s(oth)f(for)g(isotropic)f(and)0 3869 y(anisotropic)35 b(la)m(y)m(ered)i(media.)55 b(In)36 b(ra)m(y-cen)m(tered)j(co)s(ordinates,)e(the)g(dynamic)f(ra)m(y)h (tracing)f(system)0 3989 y(and)26 b(ra)m(y)g(propagator)f(matrices)h (are)g(kno)m(wn)h(for)e(isotropic)g(la)m(y)m(ered)i(structures)g(and)f (for)g(anisotropic)0 4109 y(smo)s(oth)32 b(media,)f(but)i(not)f(for)g (anisotropic)f(media)h(with)g(structural)g(in)m(terfaces.)146 4301 y(W)-8 b(e)25 b(prop)s(ose)g(a)f(simple)f(and)i(in)m(v)m(ertible)e (transformation)g(b)s(et)m(w)m(een)j(ra)m(y-propagator)e(matrices)f(in) 0 4421 y(b)s(oth)28 b(co)s(ordinate)e(systems.)44 b(It)28 b(allo)m(ws)e(to)i(p)s(erform)f(con)m(v)m(en)m(tional)g(dynamic)g(ra)m (y)i(tracing)e(in)g(Carte-)0 4541 y(sian)40 b(co)s(ordinates,)i(and)f (to)f(transform)f(the)i(resulting)f(ra)m(y-propagator)f(matrix)g(to)i (ra)m(y-cen)m(tered)0 4662 y(co)s(ordinates)31 b(at)f(an)m(y)i(p)s(oin) m(t)e(of)h(the)g(ra)m(y)h(where)g(w)m(e)g(need)g(it.)42 b(This)31 b(a)m(v)m(oids)g(dynamic)g(ra)m(y)g(tracing)f(in)0 4782 y(ra)m(y-cen)m(tered)j(co)s(ordinates)e(altogether.)42 b(Vice)31 b(v)m(ersa,)i(w)m(e)g(can)e(compute,)h(at)f(an)m(y)h(p)s(oin) m(t)e(of)h(the)h(ref-)0 4903 y(erence)f(ra)m(y)-8 b(,)31 b(the)f(ra)m(y)g(propagator)f(matrix)f(in)h(Cartesian)h(co)s(ordinates) f(b)m(y)h(dynamic)f(ra)m(y)i(tracing)d(in)0 5023 y(ra)m(y-cen)m(tered) 37 b(co)s(ordinates.)49 b(W)-8 b(e)35 b(prop)s(ose)g(sev)m(eral)h (alternativ)m(e)d(v)m(ersions)j(of)e(the)i(transformation,)0 5143 y(eac)m(h)d(of)g(them)f(equally)g(v)-5 b(alid)31 b(in)g(isotropic)h(and)g(anisotropic)f(media.)p 0 5242 1512 4 v 0 5397 a Fm(Seismic)j(W)-7 b(a)n(v)n(es)32 b(in)i(Complex)f (3-D)g(Structures,)i(Rep)r(ort)e(16)g(\(Departmen)n(t)h(of)f(Geoph)n (ysics,)h(Charles)e(Univ)n(ersit)n(y)-7 b(,)0 5517 y(Praha)26 b(2006\))p Fq(,)31 b(pp.)44 b(87{114.)1841 5824 y(87)p eop %%Page: 88 2 88 1 bop 146 100 a Fq(F)-8 b(or)41 b(ra)m(ys)g(hitting)f(a)g (structural)h(in)m(terfaces,)j(the)d(ra)m(y)h(propagator)e(matrix)f (has)j(to)e(b)s(e)i(trans-)0 220 y(formed)33 b(across)h(the)g(in)m (terface.)46 b(The)35 b(relev)-5 b(an)m(t)33 b(transformation)e(matrix) h(is)h(usually)g(referred)h(to)f(as)0 340 y(the)38 b(in)m(terface)f (propagator)g(matrix.)56 b(In)38 b(Cartesian)g(co)s(ordinates,)g(the)g (6)25 b Fn(\002)h Fq(6)37 b(in)m(terface)h(propaga-)0 461 y(tor)29 b(matrix)f(has)i(b)s(een)h(published)e(b)s(efore,)i(but)f (in)e(ra)m(y-cen)m(tered)k(co)s(ordinates)d(only)g(for)g(the)h(case)h (of)0 581 y(isotropic)25 b(media.)41 b(Based)27 b(on)g(the)g (transformation)e(from)h(Cartesian)h(to)f(ra)m(y-cen)m(tered)j(co)s (ordinates,)0 702 y(w)m(e)j(presen)m(t)h(the)e(6)18 b Fn(\002)i Fq(6)30 b(and)h(4)19 b Fn(\002)g Fq(4)31 b(in)m(terface)g (propagator)f(matrices)g(in)g(ra)m(y-cen)m(tered)j(co)s(ordinates,)0 822 y(v)-5 b(alid)25 b(for)g(general)h(isotropic)f(and)h(anisotropic)f (media.)40 b(The)28 b(4)9 b Fn(\002)g Fq(4)27 b(in)m(terface)f (propagator)f(matrix)g(in)0 942 y(ra)m(y-cen)m(tered)32 b(co)s(ordinates)e(can)g(b)s(e)h(factorized)f(and)g(used)h(to)f(deriv)m (e)h(the)g(4)17 b Fn(\002)h Fq(4)30 b(surface-to-surface)0 1063 y(paraxial)i(matrices.)46 b(These)35 b(matrices)e(allo)m(w)g(to)g (relate)g(the)h(paraxial)e(ra)m(y)i(prop)s(erties)g(at)f(di\013eren)m (t)0 1183 y(surfaces)h(crossed)g(b)m(y)f(the)g(reference)i(ra)m(y)e (and)f(o\013er)h(man)m(y)f(imp)s(ortan)m(t)f(applications.)0 1492 y Fl(Keyw)m(ords)p Fq(:)40 b(Seismic)25 b(ra)m(y)i(theory)-8 b(,)28 b(seismic)d(anisotrop)m(y)-8 b(,)27 b(lateral)d (heterogeneities,)k(la)m(y)m(ered)f(media.)0 1896 y Fk(1)161 b(In)l(tro)t(duction)0 2186 y Fq(Dynamic)35 b(ra)m(y)h(tracing)f (\(abbreviated)i(DR)-8 b(T)35 b(and)h(also)g(called)e(paraxial)h(ra)m (y)h(tracing\))f(consists)i(of)0 2306 y(solving)30 b(a)h(system)i(of)d (sev)m(eral)i(linear)e(ordinary)h(di\013eren)m(tial)e(equations)j(of)f (the)h(\014rst)f(order)h(along)e(a)0 2427 y(reference)h(ra)m(y)-8 b(.)43 b(It)29 b(has)h(found)f(useful)g(applications)f(in)g(the)i(in)m (v)m(estigation)e(of)h(an)g(orthonomic)e(\(t)m(w)m(o-)0 2547 y(parameteric\))f(system)i(of)f(ra)m(ys)h(connected)h(with)e(the)h (reference)h(ra)m(y)-8 b(.)42 b(It)27 b(can)h(b)s(e)f(used)i(to)e (determine)0 2668 y(geometrical)32 b(spreading)i(\(needed)i(in)e(the)h (computation)e(of)h(amplitudes\),)f(w)m(a)m(v)m(efron)m(t)j(curv)-5 b(atures,)0 2788 y(etc.,)38 b(of)e(the)h(elemen)m(tary)f(w)m(a)m(v)m(e) j(propagating)34 b(along)h(the)i(ra)m(y)-8 b(.)56 b(It)36 b(can)h(also)e(b)s(e)i(used)g(to)f(compute)0 2908 y(the)41 b(relev)-5 b(an)m(t)41 b(propagator)f(matrix.)67 b(The)41 b(propagator)f(matrix)g(corresp)s(onding)g(to)h(the)g(dynamic)0 3029 y(ra)m(y)c(tracing)f(system)i(is)f(usually)f(referred)h(to)g(as)g (the)g(dynamic)g(ra)m(y)g(tracing)f(propagator)g(matrix,)0 3149 y(or)h(brie\015y)h(the)h(ra)m(y)f(propagator)f(matrix.)57 b(Here)39 b(w)m(e)f(refer)h(to)e(it)g(as)h(the)g(ra)m(y)g(propagator)f (matrix.)0 3269 y(The)27 b(ra)m(y)f(propagator)f(matrix)f(is)h(v)m(ery) j(con)m(v)m(enien)m(t)f(in)e(studying)h(the)g(complete)f (\(four-parameteric\))0 3390 y(system)34 b(of)e(paraxial)e(ra)m(ys)k (connected)g(with)e(the)h(reference)h(ra)m(y)f(under)h(consideration.) 146 3581 y(The)e(dynamic)e(ra)m(y)h(tracing)f(system)i(and)f(relev)-5 b(an)m(t)31 b(propagator)f(matrices)g(pla)m(y)g(a)h(v)m(ery)h(imp)s (or-)0 3701 y(tan)m(t)39 b(role)f(in)g(the)h(paraxial)e(ra)m(y)j(metho) s(ds,)g(in)e(whic)m(h)i(the)f(kinematic)e(quan)m(tities)i(are)g (studied)g(in)0 3822 y(the)g(so-called)f(paraxial)f(vicinit)m(y)h(of)h (the)g(reference)i(ra)m(y)-8 b(.)63 b(W)-8 b(e)39 b(sp)s(eak)h(of)f (paraxial)e(ra)m(ys,)k(paraxial)0 3942 y(tra)m(v)m(el)27 b(times,)f(and)h(paraxial)e(slo)m(wness)j(v)m(ector.)42 b(They)28 b(can)f(b)s(e)f(used)i(to)e(compute)h(the)f(second-order)0 4063 y(and)35 b(higher-order)g(spatial)e(deriv)-5 b(ativ)m(es)35 b(of)g(tra)m(v)m(el)g(times,)g(to)g(in)m(v)m(estigate)g(c)m(haotic)g (ra)m(ys)h(and)g(ev)-5 b(al-)0 4183 y(uate)37 b(Ly)m(apuno)m(v)h(exp)s (onen)m(ts,)h(and)e(to)f(study)h(the)g(recipro)s(cit)m(y)f(relations)f (along)g(the)i(ra)m(y)-8 b(.)55 b(Finally)-8 b(,)0 4303 y(the)39 b(ra)m(y)g(propagator)f(matrix)f(is)h(essen)m(tial)h(in)f(v)-5 b(arious)38 b(extensions)i(of)e(the)h(ra)m(y)g(metho)s(d,)h(suc)m(h)g (as)0 4424 y(the)31 b(metho)s(d)f(of)h(Gaussian)f(b)s(eams)g(and)h (Gaussian)g(pac)m(k)m(ets,)i(the)e(Maslo)m(v-Chapman)f(metho)s(d,)h (the)0 4544 y(Kirc)m(hho\013-Helmholtz)26 b(metho)s(d,)j(ra)m(y)h(p)s (erturbation)e(theories,)i(etc.)42 b(F)-8 b(or)29 b(more)f(details)g (on)g(paraxial)0 4664 y(ra)m(y)36 b(metho)s(ds)g(see)740 4639 y(\024)730 4664 y(Cerv)m(en)q(\023)-50 b(y)38 b(\(2001,)e(Chap.)55 b(4\),)36 b(and)g(Moser)h(and)2562 4639 y(\024)2552 4664 y(Cerv)m(en)q(\023)-50 b(y)38 b(\(2006\),)e(where)h(other)0 4785 y(references)43 b(are)e(giv)m(en.)69 b(See)42 b(also)e(Section)g (8.)69 b(F)-8 b(or)40 b(v)-5 b(arious)40 b(approac)m(hes)i(to)f (kinematic)e(and)i(dy-)0 4905 y(namic)g(ra)m(y)h(tracing)f(in)g (inhomogeneous)g(anisotropic)g(media)f(w)m(e)j(refer)g(the)f(reader)g (to)3441 4880 y(\024)3430 4905 y(Cerv)m(en)q(\023)-50 b(y)0 5026 y(\(1972,)32 b(2001\),)h(Han)m(yga)g(\(1982\),)f(Ga)5 b(jewski)34 b(and)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)34 b(\(1987,)f(1990\),)f(Norris)h(\(1987\),)f (Shearer)0 5146 y(and)i(Chapman)g(\(1989\),)f(F)-8 b(arra)32 b(\(1989\),)i(Kendall,)e(Guest)j(and)e(Thomson)h(\(1992\),)f(Klime)-5 b(\024)-44 b(s)32 b(\(1994,)0 5266 y(2006\),)k(F)-8 b(arra)35 b(and)g(LeB)m(\023)-46 b(egat)37 b(\(1995\),)e(Bakk)m(er)i(\(1996\),)f (Han)m(yga)g(et)g(al.)53 b(\(2001\),)35 b(Gjo)m(ystdal)g(et)i(al.)0 5387 y(\(2002\),)26 b(Iv)m(ersen)j(\(2004a\),)d(Moser)h(\(2004\),)f (and)g(Chapman)g(\(2004\),)g(where)h(man)m(y)f(other)g(references)0 5507 y(can)33 b(b)s(e)g(found.)43 b(In)33 b(sev)m(eral)g(of)g(these)g (references,)i(ra)m(y)e(propagator)f(matrices)f(are)i(also)f (discussed.)1841 5824 y(88)p eop %%Page: 89 3 89 2 bop 146 100 a Fq(The)28 b(dynamic)e(ra)m(y)h(tracing)f(system)h (and)g(ra)m(y)g(propagator)e(matrices)h(are)h(commonly)e(expressed)0 220 y(in)30 b(the)h(global)d(Cartesian)i(co)s(ordinate)f(system,)j(or)e (in)g(the)h(ra)m(y-cen)m(tered)h(co)s(ordinate)e(system,)i(con-)0 340 y(nected)27 b(to)e(a)g(suitable)f(reference)j(ra)m(y)-8 b(.)42 b(F)-8 b(or)24 b(isotropic)g(media,)i(the)g(ra)m(y-cen)m(tered)h (co)s(ordinate)e(system)0 461 y(w)m(as)34 b(\014rst)f(in)m(tro)s(duced) f(to)h(seismology)e(b)m(y)i(P)m(op)s(o)m(v)h(and)e(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)34 b(\(1978a,b\),)e(see)h(also)3288 436 y(\024)3277 461 y(Cerv)m(en)q(\023)-50 b(y,)0 581 y(Klime)-5 b(\024)-44 b(s)38 b(and)i(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)41 b(\(1988\).)65 b(F)-8 b(or)39 b(anisotropic)f(media,)j(see)g (Klime)-5 b(\024)-44 b(s)38 b(\(1994,)j(2006\).)65 b(In)40 b(global)0 702 y(Cartesian)g(co)s(ordinates,)h(the)g(ra)m(y)f (propagator)f(matrices)g(are)h(6)27 b Fn(\002)g Fq(6,)42 b(whereas)f(for)f(ra)m(y-cen)m(tered)0 822 y(co)s(ordinates)35 b(they)h(are)f(6)24 b Fn(\002)h Fq(6)35 b(or)f(4)24 b Fn(\002)h Fq(4)35 b(-)f(corresp)s(onding)i(to)f(systems)h(of)f(six)g (or)g(four)g(di\013eren)m(tial)0 942 y(equations,)27 b(resp)s(ectiv)m(ely)-8 b(.)42 b(The)26 b(propagators)e(in)g(b)s(oth)h (co)s(ordinate)f(systems)i(satisfy)f(the)g(imp)s(ortan)m(t)0 1063 y(symplectic)33 b(and)g(c)m(hain)g(prop)s(erties.)46 b(A)m(t)34 b(in)m(tersections)f(of)g(the)h(reference)h(ra)m(y)f(with)f (structural)g(in-)0 1183 y(terfaces,)g(ra)m(y)g(propagator)e(matrices)g (m)m(ust)h(b)s(e)h(c)m(hained)f(and)g(the)h(so-called)d(in)m(terface)i (propagator)0 1303 y(matrix)c(m)m(ust)h(b)s(e)h(included)f(in)g(the)h (c)m(hain.)42 b(The)30 b(problem)e(of)h(the)h(transformation)d(of)i (the)h(dynamic)0 1424 y(ra)m(y)i(tracing)f(system)i(across)g(the)f (structural)g(in)m(terface)g(has)g(b)s(een)h(solv)m(ed)f(b)m(y)h(v)-5 b(arious)31 b(approac)m(hes.)0 1544 y(F)-8 b(or)39 b(isotropic)g (media,)h(see)1085 1519 y(\024)1074 1544 y(Cerv)m(en)q(\023)-50 b(y,)45 b(Langer)40 b(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)41 b(\(1974\))e(in)g(Cartesian)h(co)s(ordinates,)0 1665 y(P)m(op)s(o)m(v)34 b(and)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)35 b(\(1978a,b\))d(and)1483 1639 y(\024)1472 1665 y(Cerv)m(en)q(\023)-50 b(y)36 b(and)d(Hron)g(\(1980\))f(in)h(ra)m (y-cen)m(tered)i(co)s(ordinates.)0 1785 y(F)-8 b(or)35 b(anisotropic)f(media,)h(see)h(Ga)5 b(jewski)36 b(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)36 b(\(1990\),)f(and)h(F)-8 b(arra)34 b(and)i(LeB)m(\023)-46 b(egat)36 b(\(1995\))0 1905 y(in)43 b(Cartesian)g(co)s(ordinates.)76 b(F)-8 b(arra)42 b(and)i(LeB)m(\023)-46 b(egat)44 b(\(1995\))e(sho)m(w) m(ed)j(that)f(some)f(previously)g(de-)0 2026 y(riv)m(ed)h (transformation)e(equations)i(do)f(not)h(satisfy)g(the)g(symplecticit)m (y)f(relation.)75 b(In)45 b(Cartesian)0 2146 y(co)s(ordinates,)c(a)e (suitable)g(form)g(of)g(the)h(in)m(terface)g(propagator)e(matrix)g (satisfying)h(symplecticit)m(y)0 2267 y(w)m(as)30 b(prop)s(osed)f(b)m (y)h(Moser)g(\(2004\),)e(and)h(for)g(some)f(other)h(co)s(ordinate)f (systems)i(b)m(y)3141 2241 y(\024)3130 2267 y(Cerv)m(en)q(\023)-50 b(y)32 b(\(2001\))0 2387 y(\(e.g.,)46 b(for)d(the)h(w)m(a)m(v)m(efron)m (t)h(orthonormal)c(co)s(ordinate)h(system\).)77 b(F)-8 b(or)43 b(ra)m(y-cen)m(tered)i(co)s(ordinates)0 2507 y(and)28 b(anisotropic)e(media,)i(ho)m(w)m(ev)m(er,)j(the)d(deriv)-5 b(ation)26 b(of)h(the)i(in)m(terface)e(propagator)g(matrix)f(has)j(not) 0 2628 y(y)m(et)34 b(b)s(een)f(rep)s(orted)g(in)f(the)h(seismological)c (literature.)146 2819 y(Let)42 b(us)f(brie\015y)h(discuss)g(the)g(adv) -5 b(an)m(tages)41 b(and)h(disadv)-5 b(an)m(tages)41 b(of)g(ra)m(y)g(propagator)f(matrices)0 2939 y(in)d(global)f(Cartesian) i(and)g(ra)m(y-cen)m(tered)i(co)s(ordinate)d(systems.)62 b(The)39 b(main)d(adv)-5 b(an)m(tage)38 b(of)g(6)25 b Fn(\002)i Fq(6)0 3060 y(ra)m(y)32 b(propagator)f(matrices)g(in)g (Cartesian)h(co)s(ordinates)f(is)g(that)h(the)g(dynamic)f(ra)m(y)i (tracing)e(system)0 3180 y(has)36 b(a)f(general)g(and)h(simple)e (structure,)k(and)e(that)f(the)h(in)m(terface)g(transformation)d (matrix)h(is)h(w)m(ell)0 3300 y(kno)m(wn.)48 b(Ho)m(w)m(ev)m(er,)36 b(from)c(the)i(six)f(linearly)f(indep)s(enden)m(t)i(solutions)f(t)m(w)m (o)h(are)f(redundan)m(t:)47 b(the)34 b(so-)0 3421 y(called)h(ra)m (y-tangen)m(t)i(and)g(non-eik)m(onal)e(solutions.)54 b(These)38 b(redundan)m(t)g(solutions)d(complicate)g(the)0 3541 y(solution)23 b(of)h(v)-5 b(arious)23 b(b)s(oundary-v)-5 b(alue)24 b(problems)g(of)g(the)g(general)g(system)i(of)d(ra)m(ys,)k (whic)m(h)e(is)f(strictly)0 3662 y(four-parameteric.)42 b(Con)m(trary)33 b(to)e(the)i(6)21 b Fn(\002)h Fq(6)32 b(propagator)g(matrix)e(in)i(Cartesian)g(co)s(ordinates,)g(the)0 3782 y(redundan)m(t)c(solutions)d(can)i(b)s(e)f(immediately)e(remo)m(v) m(ed)j(from)e(the)i(6)10 b Fn(\002)g Fq(6)26 b(propagator)f(matrix)g (in)h(ra)m(y-)0 3902 y(cen)m(tered)40 b(co)s(ordinates,)f(and)f(a)f (reduction)h(to)g(4)25 b Fn(\002)h Fq(4)38 b(form)f(is)g(straigh)m (tforw)m(ard.)59 b(F)-8 b(or)37 b(anisotropic)0 4023 y(media,)51 b(ho)m(w)m(ev)m(er,)k(the)49 b(dynamic)f(ra)m(y)h(tracing)f (system)i(in)d(ra)m(y-cen)m(tered)k(co)s(ordinates)d(is)g(more)0 4143 y(complicated)23 b(than)h(in)g(Cartesian)g(co)s(ordinates.)40 b(Among)24 b(these)i(complications)21 b(is)j(the)h(requiremen)m(t)0 4263 y(to)k(solv)m(e)g(an)g(additional)d(di\013eren)m(tial)i(equation)h (along)e(the)j(ra)m(y)f(to)g(determine)g(the)g(basis)g(v)m(ectors)i(of) 0 4384 y(the)i(ra)m(y-cen)m(tered)h(co)s(ordinate)e(system)i(\(see)f (Klime)-5 b(\024)-44 b(s,)31 b(1994,)h(2006;)2595 4359 y(\024)2584 4384 y(Cerv)m(en)q(\023)-50 b(y,)36 b(2001\).)146 4575 y(In)25 b(this)f(pap)s(er,)j(w)m(e)f(sho)m(w)f(that)g(the)g(4)6 b Fn(\002)g Fq(4)24 b(ra)m(y)h(propagator)f(matrix)f(in)g(ra)m(y-cen)m (tered)k(co)s(ordinates)0 4695 y(can)33 b(b)s(e)g(computed)g(b)m(y)h (dynamic)e(ra)m(y)h(tracing)f(in)g(Cartesian)g(co)s(ordinates.)44 b(Dynamic)31 b(ra)m(y)i(tracing)0 4816 y(in)g(ra)m(y-cen)m(tered)j(co)s (ordinates)e(is)f(not)h(required)g(at)g(all.)46 b(Moreo)m(v)m(er,)36 b(the)e(di\013eren)m(tial)e(equation)i(for)0 4936 y(the)e(basis)f(v)m (ectors)j(of)d(the)h(ra)m(y-cen)m(tered)h(co)s(ordinate)e(system)h (along)e(the)i(ra)m(y)g(need)h(not)e(b)s(e)h(solv)m(ed.)0 5057 y(Tw)m(o)43 b(basic)g(approac)m(hes)g(are)g(prop)s(osed.)74 b(In)42 b(the)h(\014rst)g(approac)m(h,)j(the)d(ra)m(y)g(propagator)e (matrix)0 5177 y(in)j(ra)m(y-cen)m(tered)j(co)s(ordinates)d(is)h (obtained)f(b)m(y)i(some)f(transformation)d(of)j(the)g(ra)m(y)g (propagator)0 5297 y(matrix)36 b(in)i(Cartesian)f(co)s(ordinates)h(at)g (the)g(initial)c(and)k(end)h(p)s(oin)m(ts)e(of)h(the)g(ra)m(y)-8 b(.)60 b(If)38 b(needed,)j(the)0 5418 y(ra)m(y)30 b(propagator)e (matrix)g(in)h(ra)m(y-cen)m(tered)j(co)s(ordinates)d(can)g(also)g(b)s (e)h(computed)f(in)g(the)h(same)f(w)m(a)m(y)0 5538 y(at)k(an)m(y)g(in)m (termediate)f(p)s(oin)m(t)g(of)h(the)g(ra)m(y)-8 b(.)45 b(In)34 b(the)f(second)h(approac)m(h,)g(the)g(ra)m(y)f(propagator)f (matrix)1841 5824 y(89)p eop %%Page: 90 4 90 3 bop 0 100 a Fq(in)41 b(ra)m(y-cen)m(tered)j(co)s(ordinates)e(is)f (obtained)h(from)f(four)g(solutions)h(of)f(the)i(dynamic)e(ra)m(y)h (tracing)0 220 y(system)30 b(in)e(Cartesian)h(co)s(ordinates,)g(with)f (strictly)g(sp)s(eci\014ed)i(initial)25 b(conditions.)41 b(Th)m(us,)32 b(only)c(four)0 340 y(solutions)d(are)g(needed)i(in)e (this)h(approac)m(h,)h(not)f(six)f(as)h(in)f(the)h(previous)g(case.)42 b(Both)26 b(approac)m(hes)g(are)0 461 y(in)m(v)m(ertible)31 b(and)h(can)h(also)e(b)s(e)h(used)h(to)f(compute)g(the)g(6)21 b Fn(\002)h Fq(6)31 b(ra)m(y)i(propagator)e(matrix)f(in)i(Cartesian)0 581 y(co)s(ordinates)g(from)f(the)i(6)22 b Fn(\002)g Fq(6)33 b(\(or)f(4)22 b Fn(\002)g Fq(4\))32 b(ra)m(y)h(propagator)f (matrix)f(in)g(ra)m(y-cen)m(tered)k(co)s(ordinates.)146 772 y(The)c(prop)s(osed)f(transformations)e(b)s(et)m(w)m(een)j(ra)m(y)f (propagator)f(matrices)g(in)f(Cartesian)i(and)g(ra)m(y-)0 893 y(cen)m(tered)d(co)s(ordinate)d(systems)i(are)f(also)g(used)h(to)e (deriv)m(e)i(the)g(6)7 b Fn(\002)g Fq(6)24 b(and)h(4)7 b Fn(\002)g Fq(4)24 b(in)m(terface)h(propagator)0 1013 y(matrices)45 b(in)f(ra)m(y-cen)m(tered)j(co)s(ordinates)e(from)f(the)i (kno)m(wn)h(in)m(terface)e(propagator)g(matrices)f(in)0 1134 y(Cartesian)36 b(co)s(ordinates.)52 b(The)37 b(in)m(terface)e (propagator)g(matrices)g(can)h(b)s(e)g(factorized)f(and)h(used)h(to)0 1254 y(deriv)m(e)j(the)f(4)26 b Fn(\002)h Fq(4)39 b(surface-to-surface) g(paraxial)f(matrices)g(in)g(ra)m(y-cen)m(tered)j(co)s(ordinates,)f (whic)m(h)0 1374 y(relate)c(the)h(paraxial)d(ra)m(y)j(quan)m(tities)g (at)f(di\013eren)m(t)h(surfaces)g(crossed)h(b)m(y)g(the)f(reference)h (ra)m(y)-8 b(.)56 b(The)0 1495 y(surfaces)34 b(ma)m(y)e(represen)m(t)i (structural)f(in)m(terfaces,)g(the)g(surface)g(of)f(the)h(Earth,)f(iso) s(c)m(hrone)h(surfaces,)0 1615 y(or)j(formal)d(surfaces.)56 b(Actually)-8 b(,)36 b(the)g(formal)e(surfaces)j(ma)m(y)f(b)s(e)g(in)m (tro)s(duced)h(at)e(an)m(y)i(p)s(oin)m(t)e(of)h(the)0 1735 y(reference)48 b(ra)m(y)f(where)g(needed.)87 b(W)-8 b(e)46 b(require)h(that)f(the)h(surfaces)h(represen)m(ting)f(the)g (structural)0 1856 y(in)m(terfaces)33 b(do)g(not)f(in)m(tersect)h(in)f (the)h(paraxial)e(vicinit)m(y)g(of)i(the)g(reference)h(ra)m(y)-8 b(.)146 2047 y(Brie\015y)35 b(to)g(the)g(con)m(ten)m(t)i(of)d(the)i (pap)s(er.)51 b(The)36 b(kinematic)d(and)i(dynamic)g(ra)m(y)g(tracing)f (and)h(ra)m(y)0 2167 y(propagator)e(matrices)h(in)f(Cartesian)h(co)s (ordinates)g(are)g(review)m(ed)h(in)f(Section)g(2.)48 b(Section)34 b(3)g(in)m(tro-)0 2288 y(duces)40 b(the)g(ra)m(y-cen)m (tered)g(co)s(ordinate)e(system)i(and)f(its)f(basis)h(v)m(ectors)h(\(b) s(oth)f(con)m(tra)m(v)-5 b(arian)m(t)38 b(and)0 2408 y(co)m(v)-5 b(arian)m(t\),)32 b(giv)m(es)h(the)g(dynamic)f(ra)m(y)h (tracing)f(system)h(in)f(ra)m(y-cen)m(tered)i(co)s(ordinates)e(and)h (the)g(ex-)0 2529 y(pression)28 b(for)f(the)h(ra)m(y)g(propagator)e (matrix)g(in)h(ra)m(y-cen)m(tered)i(co)s(ordinates.)42 b(Section)27 b(4)g(sp)s(eci\014es)i(the)0 2649 y(general)24 b(transformation)f(relation)g(b)s(et)m(w)m(een)k(the)e(ra)m(y)g (propagator)f(matrices)g(corresp)s(onding)g(to)h(t)m(w)m(o)0 2769 y(co)s(ordinate)f(systems,)j(deriv)m(ed)f(in)1304 2744 y(\024)1293 2769 y(Cerv)m(en)q(\023)-50 b(y)27 b(\(2001,)e(eq.)42 b(\(4.3.38\)\),)25 b(to)f(Cartesian)h(and)g(ra)m(y-cen)m(tered)0 2890 y(co)s(ordinate)34 b(systems.)52 b(Basic)35 b(relations)e(\(61\))h (with)h(\(59\))f(are)h(deriv)m(ed.)51 b(The)36 b(transformations)d(are) 0 3010 y(fully)23 b(in)m(v)m(ertible;)j(they)g(can)e(b)s(e)h(used)g(in) f(b)s(oth)g(directions.)40 b(In)24 b(Section)g(5,)i(w)m(e)g(discuss)f (an)f(alternativ)m(e)0 3131 y(equation)37 b(\(64\))f(for)g(the)h(ra)m (y)g(propagator)f(matrix)f(in)h(ra)m(y-cen)m(tered)j(co)s(ordinates,)e (whic)m(h)h(requires)0 3251 y(the)30 b(determination)d(of)i(only)g (four)g(solutions)f(of)h(the)g(dynamic)g(ra)m(y)h(tracing)e(equation)h (in)g(Cartesian)0 3371 y(co)s(ordinates.)42 b(In)32 b(Section)e(6,)h (simple)f(expressions)i(for)f(the)g(4)19 b Fn(\002)g Fq(4)30 b(and)h(6)19 b Fn(\002)g Fq(6)31 b(in)m(terface)f(propagator)0 3492 y(matrices)j(in)g(ra)m(y-cen)m(tered)j(co)s(ordinates)d(are)h (deriv)m(ed.)47 b(In)35 b(Section)e(7,)h(it)f(is)g(sho)m(wn)i(that)f(b) s(oth)f(the)0 3612 y(6)7 b Fn(\002)g Fq(6)24 b(and)i(4)7 b Fn(\002)g Fq(4)24 b(in)m(terface)h(propagator)f(matrices)g(in)g(ra)m (y-cen)m(tered)j(co)s(ordinates)e(ma)m(y)g(b)s(e)g(factorized)0 3732 y(and)39 b(used)i(to)e(deriv)m(e)h(the)f(surface-to-surface)h (paraxial)e(matrices)g(in)h(ra)m(y-cen)m(tered)i(co)s(ordinates.)0 3853 y(Finally)-8 b(,)30 b(Section)i(8)h(o\013ers)g(some)f(concluding)g (remarks.)146 4044 y(The)h(notations)e(in)g(the)i(pap)s(er)f(are)g(as)g (follo)m(ws:)42 b(3)20 b Fn(\002)i Fq(3)31 b(and)i(6)20 b Fn(\002)h Fq(6)32 b(matrices)f(and)h(3)21 b Fn(\002)g Fq(1)32 b(v)m(ectors)0 4164 y(are)i(denoted)i(b)m(y)f(Roman)e(b)s (oldface)h(sym)m(b)s(ols,)h(2)23 b Fn(\002)h Fq(2)34 b(and)g(4)23 b Fn(\002)h Fq(4)34 b(matrices)g(and)g(2)24 b Fn(\002)f Fq(1)35 b(v)m(ectors)h(b)m(y)0 4285 y(Italic)29 b(b)s(oldface)g(sym)m(b)s(ols.)43 b(The)31 b(sym)m(b)s(ol)f(T)h(in)f (the)g(sup)s(erscript)i(of)e(a)g(matrix)f(means)h(\\transp)s(ose",)0 4405 y(and)j(the)g(sym)m(b)s(ol)f(-T)h(in)f(the)h(sup)s(erscript)h (denotes)g(\\in)m(v)m(erse)g(transp)s(ose".)44 b(The)34 b(lo)m(w)m(ercase)g(indices)0 4526 y(i,)29 b(j,)i(k,)g(...)42 b(tak)m(e)31 b(the)f(v)-5 b(alues)30 b(1,)g(2,)g(3,)g(and)g(the)g(upp)s (ercase)i(indices)d Fj(I)8 b(;)17 b(J)n(;)g(K)r(;)g(:::)29 b Fq(the)h(v)-5 b(alues)30 b(1,)g(2.)43 b(The)0 4646 y(Einstein)32 b(summation)f(con)m(v)m(en)m(tion)j(is)e(applied)f(to)h (rep)s(eated)i(indices)e(in)g(the)h(same)f(pro)s(duct.)1841 5824 y(90)p eop %%Page: 91 5 91 4 bop 0 100 a Fk(2)161 b(Dynamic)57 b(ra)l(y)g(tracing)g(and)g(ra)l (y-propagator)f(matrices)242 282 y(in)e(Cartesian)g(co)t(ordinates)0 572 y Fq(Consider)33 b(the)g(eik)m(onal)f(equation)g(for)g(tra)m(v)m (el-time)f(\014eld)h Fj(T)14 b Fq(\()p Fl(x)p Fq(\))32 b(in)g(Hamiltonian)d(form)1607 778 y Fn(H)q Fq(\()p Fl(x)p Fj(;)17 b Fl(p)p Fq(\))28 b(=)f(0)33 b Fj(;)1482 b Fq(\(1\))0 983 y(where)42 b Fn(H)h Fq(is)d(the)i(Hamiltonian,)d Fl(x)j Fq(is)e(the)i(p)s(osition)d(v)m(ector,)45 b(and)c Fl(p)g Fq(the)h(slo)m(wness)h(v)m(ector,)h Fl(p)f Fq(=)0 1103 y Fj(@)5 b(T)11 b(=@)5 b Fl(x)p Fq(.)65 b(W)-8 b(e)39 b(consider)g(the)h(Hamiltonians)c(whic)m(h)j(are)g(homogeneous)g (functions)g(of)g(the)g(second)0 1224 y(degree)g(in)f Fl(p)g Fq(\(with)g(a)g(p)s(ossible)f(additional)e(constan)m(t\).)62 b(The)39 b(kinematic)d(ra)m(y)j(tracing)e(equations)0 1344 y(then)c(read)820 1464 y(d)p Fl(x)p Fj(=)p Fq(d)p Fj(\034)39 b Fq(=)28 b Fi(U)38 b Fq(=)27 b Fj(@)5 b Fn(H)q Fj(=@)g Fl(p)35 b Fj(;)114 b Fq(d)p Fl(p)p Fj(=)p Fq(d)p Fj(\034)40 b Fq(=)27 b Fh(\021)k Fq(=)d Fn(\000)p Fj(@)5 b Fn(H)q Fj(=@)g Fl(x)35 b Fj(:)695 b Fq(\(2\))0 1632 y(Here)33 b Fj(\034)44 b Fq(is)32 b(a)g(monotonically)d(increasing)j (sampling)e(parameter)i(along)f(the)h(ra)m(y)-8 b(,)33 b(whic)m(h)g(represen)m(ts)0 1753 y(the)39 b(tra)m(v)m(el)f(time.)59 b(The)40 b(initial)34 b(p)s(oin)m(t)k(of)f(the)i(ra)m(y)g(corresp)s (onds)g(to)f Fj(\034)49 b Fq(=)38 b Fj(\034)2861 1768 y Fp(0)2900 1753 y Fq(.)61 b(Equations)39 b(\(2\))e(also)0 1873 y(de\014ne)27 b(the)f(ra)m(y)g(v)m(elo)s(cit)m(y)f(v)m(ector)i Fi(U)36 b Fq(\(also)25 b(called)f(the)i(group)g(v)m(elo)s(cit)m(y)f(v)m (ector\))i(and)f(v)m(ector)g Fh(\021)t Fq(,)h(whic)m(h)0 1994 y(will)j(b)s(e)j(broadly)f(used)i(in)e(this)g(pap)s(er.)146 2185 y(The)40 b(dynamic)e(ra)m(y)h(tracing)f(system)i(is)e(designed)i (to)e(compute)h(t)m(w)m(o)g(3)27 b Fn(\002)f Fq(1)39 b(matrices)f Fl(Q)3565 2149 y Fp(\()p Fg(x)p Fp(\))3702 2185 y Fn(\021)0 2305 y Fq(\()p Fj(Q)115 2254 y Fp(\()p Fg(x)p Fp(\))115 2327 y(1)214 2305 y Fj(;)17 b(Q)335 2254 y Fp(\()p Fg(x)p Fp(\))335 2327 y(2)434 2305 y Fj(;)g(Q)555 2254 y Fp(\()p Fg(x)p Fp(\))555 2327 y(3)653 2305 y Fq(\))691 2269 y Fg(T)779 2305 y Fq(and)33 b Fl(P)1046 2269 y Fp(\()p Fg(x)p Fp(\))1172 2305 y Fn(\021)28 b Fq(\()p Fj(P)1392 2254 y Fp(\()p Fg(x)p Fp(\))1378 2327 y(1)1490 2305 y Fj(;)17 b(P)1611 2254 y Fp(\()p Fg(x)p Fp(\))1597 2327 y(2)1709 2305 y Fj(;)g(P)1830 2254 y Fp(\()p Fg(x)p Fp(\))1816 2327 y(3)1928 2305 y Fq(\))1966 2269 y Fg(T)2053 2305 y Fq(along)31 b(the)i(ra)m(y)-8 b(,)1183 2511 y Fl(Q)1267 2469 y Fp(\()p Fg(x)p Fp(\))1394 2511 y Fq(=)28 b Fj(@)5 b Fl(x)p Fj(=@)g(\015)39 b(;)114 b Fl(P)2026 2469 y Fp(\()p Fg(x)p Fp(\))2152 2511 y Fq(=)28 b Fj(@)5 b Fl(p)p Fj(=@)g(\015)39 b(;)1059 b Fq(\(3\))0 2716 y(where)26 b Fj(\015)j Fq(is)24 b(an)g(arbitrary)f(ra)m(y)i(parameter,)h(or)e(an)m(y)h(of)f(the)h (initial)20 b(v)-5 b(alues)25 b Fj(x)2785 2731 y Fg(i)p Fp(0)2876 2716 y Fq(=)j Fj(x)3035 2731 y Fg(i)3063 2716 y Fq(\()p Fj(\034)3143 2731 y Fp(0)3183 2716 y Fq(\),)e Fj(p)3323 2731 y Fg(i)p Fp(0)3414 2716 y Fq(=)i Fj(p)3567 2731 y Fg(i)3595 2716 y Fq(\()p Fj(\034)3675 2731 y Fp(0)3714 2716 y Fq(\).)0 2836 y(T)-8 b(aking)32 b(the)g(partial)e(deriv)-5 b(ativ)m(e)31 b(of)h(\(2\))f(with)h(resp)s(ect)h(to)f Fj(\015)5 b Fq(,)32 b(w)m(e)h(obtain)e(the)i(dynamic)e(ra)m(y)h (tracing)0 2957 y(system)i(in)d(Cartesian)i(co)s(ordinates)1081 3162 y(d)p Fl(Q)1219 3121 y Fp(\()p Fg(x)p Fp(\))1318 3162 y Fj(=)p Fq(d)p Fj(\034)95 b Fq(=)83 b Fl(A)1802 3121 y Fp(\()p Fg(x)p Fp(\))1900 3162 y Fl(Q)1984 3121 y Fp(\()p Fg(x)p Fp(\))2105 3162 y Fq(+)22 b Fl(B)2283 3121 y Fp(\()p Fg(x)p Fp(\))2382 3162 y Fl(P)2459 3121 y Fp(\()p Fg(x)p Fp(\))2590 3162 y Fj(;)1089 3307 y Fq(d)p Fl(P)1220 3266 y Fp(\()p Fg(x)p Fp(\))1318 3307 y Fj(=)p Fq(d)p Fj(\034)95 b Fq(=)83 b Fn(\000)p Fl(C)1875 3266 y Fp(\()p Fg(x)p Fp(\))1974 3307 y Fl(Q)2058 3266 y Fp(\()p Fg(x)p Fp(\))2179 3307 y Fn(\000)23 b Fl(D)2365 3266 y Fp(\()p Fg(x)p Fp(\))2463 3307 y Fl(P)2540 3266 y Fp(\()p Fg(x)p Fp(\))2671 3307 y Fj(:)957 b Fq(\(4\))0 3513 y(Here)33 b Fl(A)315 3476 y Fp(\()p Fg(x)p Fp(\))414 3513 y Fj(;)49 b Fl(B)570 3476 y Fp(\()p Fg(x)p Fp(\))668 3513 y Fj(;)h Fl(C)826 3476 y Fp(\()p Fg(x)p Fp(\))957 3513 y Fq(and)32 b Fl(D)1232 3476 y Fp(\()p Fg(x)p Fp(\))1364 3513 y Fq(are)g(3)22 b Fn(\002)h Fq(3)32 b(matrices,)g(giv)m(en)h(b)m(y)g(relations)1008 3711 y Fl(A)1093 3675 y Fp(\()p Fg(x)p Fp(\))1219 3711 y Fq(=)28 b Fj(@)1379 3675 y Fp(2)1419 3711 y Fn(H)q Fj(=@)5 b Fl(p)p Fj(@)g Fl(x)35 b Fj(;)83 b Fl(B)2011 3675 y Fp(\()p Fg(x)p Fp(\))2137 3711 y Fq(=)27 b Fj(@)2296 3675 y Fp(2)2337 3711 y Fn(H)q Fj(=@)5 b Fl(p)p Fj(@)g Fl(p)34 b Fj(;)1008 3833 y Fl(C)1089 3797 y Fp(\()p Fg(x)p Fp(\))1215 3833 y Fq(=)28 b Fj(@)1375 3797 y Fp(2)1415 3833 y Fn(H)q Fj(=@)5 b Fl(x)p Fj(@)g Fl(x)35 b Fj(;)90 b Fl(D)2017 3797 y Fp(\()p Fg(x)p Fp(\))2143 3833 y Fq(=)28 b Fj(@)2303 3797 y Fp(2)2343 3833 y Fn(H)q Fj(=@)5 b Fl(x)p Fj(@)g Fl(p)35 b Fj(:)3655 3772 y Fq(\(5\))0 4032 y(The)f(solutions)d(of)h(the)h(dynamic)f(ra)m(y)h(tracing)f(equation)g (m)m(ust)h(satisfy)f(the)h(constrain)m(t)g(relation)1403 4237 y Fi(U)1484 4195 y Fg(T)1539 4237 y Fl(P)1616 4196 y Fp(\()p Fg(x)p Fp(\))1736 4237 y Fn(\000)23 b Fh(\021)1899 4196 y Fg(T)1954 4237 y Fl(Q)2038 4196 y Fp(\()p Fg(x)p Fp(\))2165 4237 y Fq(=)28 b(0)k Fj(;)1278 b Fq(\(6\))0 4442 y(whic)m(h)39 b(follo)m(ws)f(from)f(\(1\).)62 b(The)40 b(sup)s(erscript)g(\()p Fj(x)p Fq(\))f(in)f(Equations)h(\(3\)-\(6\),)g (and)g(in)f(the)h(follo)m(wing,)0 4563 y(indicates)32 b(that)g(the)h(quan)m(tities)g(are)f(expressed)k(in)31 b(Cartesian)i(co)s(ordinates.)146 4754 y(Dynamic)28 b(ra)m(y)h(tracing) f(system)h(\(4\))g(consists)g(of)f(six)h(linear)e(ordinary)h (di\013eren)m(tial)f(equations)i(of)0 4888 y(the)e(\014rst)f(order)g (for)g Fj(Q)824 4837 y Fp(\()p Fg(x)p Fp(\))824 4911 y Fg(i)949 4888 y Fq(and)g Fj(P)1209 4837 y Fp(\()p Fg(x)p Fp(\))1195 4911 y Fg(i)1307 4888 y Fq(,)i(and)e(has)g(six)h(linearly)d (indep)s(enden)m(t)j(solutions.)40 b(Consequen)m(tly)-8 b(,)0 5008 y(w)m(e)45 b(can)e(construct)i(the)f(6)30 b Fn(\002)g Fq(6)43 b(ra)m(y)h(propagator)f(matrix)f Fl(\005)2345 4966 y Fp(\()p Fg(x)p Fp(\))2444 5008 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2616 5023 y Fp(0)2656 5008 y Fq(\),)46 b(whic)m(h)e(is)f(de\014ned)i(as)f(the)0 5129 y(fundamen)m(tal)38 b(matrix)g(of)h(\(4\),)h(sub)5 b(ject)41 b(to)e(initial)c(conditions)j Fl(\005)2525 5087 y Fp(\()p Fg(x)p Fp(\))2624 5129 y Fq(\()p Fj(\034)2704 5144 y Fp(0)2743 5129 y Fj(;)17 b(\034)2829 5144 y Fp(0)2869 5129 y Fq(\))39 b(=)g Fl(I)p Fq(,)h(where)h Fl(I)d Fq(is)h(the)0 5249 y(6)25 b Fn(\002)g Fq(6)37 b(iden)m(tit)m(y)g(matrix.)55 b(Propagator)36 b(matrix)f Fl(\005)1919 5207 y Fp(\()p Fg(x)p Fp(\))2018 5249 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2190 5264 y Fp(0)2229 5249 y Fq(\))37 b(has)g(a)g(n)m(um)m(b)s(er)g(of)f (useful)h(prop)s(erties.)0 5369 y(The)d(most)e(imp)s(ortan)m(t)e(of)i (them)h(is)f(symplecticit)m(y:)1525 5575 y Fl(\005)1612 5533 y Fp(\()p Fg(x)p Fp(\))p Fg(T)1762 5575 y Fl(J\005)1907 5533 y Fp(\()p Fg(x)p Fp(\))2034 5575 y Fq(=)27 b Fl(J)33 b Fj(;)1400 b Fq(\(7\))1841 5824 y(91)p eop %%Page: 92 6 92 5 bop 0 100 a Fq(where)34 b Fl(J)e Fq(is)g(a)h(6)22 b Fn(\002)g Fq(6)32 b(matrix)f(giv)m(en)i(b)m(y)h(the)f(form)m(ula)1521 367 y Fl(J)27 b Fq(=)1710 221 y Ff( )1817 306 y Fl(0)146 b(I)1817 427 y Fn(\000)p Fl(I)83 b(0)2117 221 y Ff(!)2232 367 y Fj(:)1396 b Fq(\(8\))0 635 y(Here)31 b Fl(0)f Fq(is)g(a)g(3)18 b Fn(\002)g Fq(3)30 b(zero)h(matrix,)e(and)h Fl(I)g Fq(the)h(3)17 b Fn(\002)i Fq(3)30 b(iden)m(tit)m(y)g(matrix.)41 b(The)31 b(con)m(tin)m(uation)f(prop)s(ert)m(y)0 755 y(of)i Fl(\005)199 713 y Fp(\()p Fg(x)p Fp(\))297 755 y Fq(\()p Fj(\034)6 b(;)17 b(\034)469 770 y Fp(0)509 755 y Fq(\))32 b(reads)1019 792 y Ff( )1126 877 y Fl(Q)1210 841 y Fp(\()p Fg(x)p Fp(\))1309 877 y Fq(\()p Fj(\034)11 b Fq(\))1126 999 y Fl(P)1203 963 y Fp(\()p Fg(x)p Fp(\))1301 999 y Fq(\()p Fj(\034)g Fq(\))1480 792 y Ff(!)1573 938 y Fq(=)28 b Fl(\005)1765 896 y Fp(\()p Fg(x)p Fp(\))1863 938 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2035 953 y Fp(0)2075 938 y Fq(\))2130 792 y Ff( )2237 877 y Fl(Q)2321 841 y Fp(\()p Fg(x)p Fp(\))2420 877 y Fq(\()p Fj(\034)2500 892 y Fp(0)2539 877 y Fq(\))2237 999 y Fl(P)2314 963 y Fp(\()p Fg(x)p Fp(\))2412 999 y Fq(\()p Fj(\034)2492 1014 y Fp(0)2532 999 y Fq(\))2619 792 y Ff(!)2734 938 y Fj(:)894 b Fq(\(9\))0 1171 y(Equation)32 b(\(9\))h(can)f(also)g(b)s(e)h(expressed)i(in)d(the) h(form)1093 1298 y Ff( )1200 1383 y Fj(\016)t Fl(x)p Fq(\()p Fj(\034)11 b Fq(\))1200 1503 y Fj(\016)t Fl(p)p Fq(\()p Fj(\034)g Fq(\))1480 1298 y Ff(!)1573 1444 y Fq(=)28 b Fl(\005)1765 1402 y Fp(\()p Fg(x)p Fp(\))1863 1444 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2035 1459 y Fp(0)2075 1444 y Fq(\))2130 1298 y Ff( )2237 1383 y Fj(\016)t Fl(x)p Fq(\()p Fj(\034)2423 1398 y Fp(0)2462 1383 y Fq(\))2237 1503 y Fj(\016)t Fl(p)p Fq(\()p Fj(\034)2426 1518 y Fp(0)2466 1503 y Fq(\))2545 1298 y Ff(!)2660 1444 y Fj(;)919 b Fq(\(10\))0 1711 y(where)35 b Fj(\016)t Fl(x)f Fq(and)f Fj(\016)t Fl(p)h Fq(are)g(small)d(p)s(erturbations)i(of)g Fl(x)h Fq(and)g Fl(p)g Fq(around)f(the)h(reference)i(ra)m(y)-8 b(.)47 b(Equation)0 1832 y(\(10\))39 b(sho)m(ws)h(w)m(ell)f(the)g(ph)m (ysical)g(meaning)f(of)h(ra)m(y)g(propagator)g(matrix)e Fl(\005)2865 1790 y Fp(\()p Fg(x)p Fp(\))2964 1832 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3136 1847 y Fp(0)3175 1832 y Fq(\))39 b(in)g(Cartesian)0 1952 y(co)s(ordinates.)k(The)34 b(c)m(hain)e(rule)g(states)i(that)1124 2163 y Fl(\005)1211 2121 y Fp(\()p Fg(x)p Fp(\))1310 2163 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1482 2178 y Fp(0)1522 2163 y Fq(\))27 b(=)h Fl(\005)1778 2121 y Fp(\()p Fg(x)p Fp(\))1877 2163 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2049 2178 y Fp(1)2089 2163 y Fq(\))p Fl(\005)2214 2121 y Fp(\()p Fg(x)p Fp(\))2313 2163 y Fq(\()p Fj(\034)2393 2178 y Fp(1)2433 2163 y Fj(;)g(\034)2519 2178 y Fp(0)2558 2163 y Fq(\))33 b Fj(;)950 b Fq(\(11\))0 2374 y(for)32 b(an)m(y)h Fj(\034)375 2389 y Fp(1)447 2374 y Fq(situated)f(on)h(the)g(ra)m(y)-8 b(,)32 b(and)h(Liouville's)d (theorem)i(yields)g(det)q(\()p Fl(\005)2880 2332 y Fp(\()p Fg(x)p Fp(\))2979 2374 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3151 2389 y Fp(0)3191 2374 y Fq(\)\))27 b(=)h(1)k(for)g(an)m(y)0 2494 y Fj(\034)11 b Fq(.)44 b(Finally)-8 b(,)30 b(the)j(symplectic)f (prop)s(ert)m(y)h(yields)g(a)f(simple)f(relation)g(for)h(the)h(in)m(v)m (erse)h(of)e Fl(\005)3377 2452 y Fp(\()p Fg(x)p Fp(\))3476 2494 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3648 2509 y Fp(0)3688 2494 y Fq(\),)1039 2638 y Ff( )1146 2723 y Fl(\005)1234 2738 y Fp(11)1392 2723 y Fl(\005)1479 2738 y Fp(12)1146 2843 y Fl(\005)1234 2858 y Fp(21)1392 2843 y Fl(\005)1479 2858 y Fp(22)1596 2638 y Ff(!)1661 2661 y Fe(\000)p Fp(1)1783 2784 y Fq(=)1887 2638 y Ff( )1994 2723 y Fl(\005)2082 2681 y Fg(T)2082 2747 y Fp(22)2317 2723 y Fn(\000)p Fl(\005)2482 2681 y Fg(T)2482 2747 y Fp(12)1994 2846 y Fn(\000)p Fl(\005)2159 2804 y Fg(T)2159 2871 y Fp(21)2317 2846 y Fl(\005)2405 2804 y Fg(T)2405 2871 y Fp(11)2598 2638 y Ff(!)2713 2784 y Fj(;)866 b Fq(\(12\))0 3074 y(where)36 b Fl(\005)372 3089 y Fp(11)447 3074 y Fj(;)17 b Fl(\005)578 3089 y Fp(12)653 3074 y Fj(;)g Fl(\005)784 3089 y Fp(21)894 3074 y Fq(and)35 b Fl(\005)1174 3089 y Fp(22)1284 3074 y Fq(are)g(3)24 b Fn(\002)g Fq(3)35 b(submatrices)g(of)g Fl(\005)2448 3032 y Fp(\()p Fg(x)p Fp(\))2547 3074 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2719 3089 y Fp(0)2759 3074 y Fq(\).)51 b(Note)35 b(that)g(the)h(partial)0 3195 y(deriv)-5 b(ativ)m(es)33 b(with)f(resp)s(ect)i(to)e Fj(\015)37 b Fq(in)32 b Fl(\005)1451 3152 y Fp(\()p Fg(x)p Fp(\))1550 3195 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1722 3210 y Fp(0)1761 3195 y Fq(\))33 b(are)f(tak)m(en)i(for)e(a)g(\014xed)i (tra)m(v)m(el)e(time)g Fj(T)41 b Fq(=)28 b Fj(\034)11 b Fq(.)146 3386 y(As)42 b(the)f(complete)f(system)h(of)f(ra)m(ys)i(is)e (four-parameteric,)h(t)m(w)m(o)g(solutions)f(in)g Fl(\005)3261 3344 y Fp(\()p Fg(x)p Fp(\))3359 3386 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3531 3401 y Fp(0)3571 3386 y Fq(\))40 b(are)0 3506 y(redundan)m(t.)54 b(The)37 b(expressions)h(for)d(these)i(t)m(w)m (o)g(solutions)e(can)h(b)s(e)g(found)g(explicitly)-8 b(.)51 b(The)37 b(\014rst)f(of)0 3627 y(them)c(is)g(the)h(so-called)f (ra)m(y-tangen)m(t)h(solution:)1374 3837 y Fl(Q)1458 3796 y Fp(\()p Fg(x)p Fp(\))1585 3837 y Fq(=)27 b Fi(U)43 b Fj(;)147 b Fl(P)2053 3796 y Fp(\()p Fg(x)p Fp(\))2179 3837 y Fq(=)27 b Fh(\021)36 b Fj(;)1201 b Fq(\(13\))0 4048 y(and)33 b(the)g(latter)e(is)h(the)h(non-eik)m(onal)f(solution,) 966 4259 y Fl(Q)1050 4218 y Fp(\()p Fg(x)p Fp(\))1177 4259 y Fq(=)27 b(\()p Fj(\034)34 b Fn(\000)23 b Fj(\034)1536 4274 y Fp(0)1575 4259 y Fq(\))p Fi(U)43 b Fj(;)114 b Fl(P)1945 4218 y Fp(\()p Fg(x)p Fp(\))2071 4259 y Fq(=)28 b Fl(p)22 b Fq(+)g(\()p Fj(\034)34 b Fn(\000)23 b Fj(\034)2613 4274 y Fp(0)2652 4259 y Fq(\))p Fh(\021)36 b Fj(:)793 b Fq(\(14\))146 4541 y(F)-8 b(or)41 b(more)g(details)g(on)h(dynamic)f (ra)m(y)h(tracing)f(and)h(ra)m(y)g(propagator)f(matrices)g(in)g (Cartesian)0 4661 y(co)s(ordinates)31 b(in)g(inhomogeneous)g (anisotropic)f(media)g(and)i(on)f(the)h(ra)m(y-tangen)m(t)g(and)g (non-eik)m(onal)0 4782 y(solutions)45 b(see)603 4756 y(\024)593 4782 y(Cerv)m(en)q(\023)-50 b(y)48 b(\(2001\),)h(and)d (Chapman)g(\(2004\),)j(where)f(man)m(y)e(other)g(references)i(are)0 4902 y(giv)m(en.)72 b(Chapman)42 b(\(2004,)i(p.152\))e(also)f(presen)m (ts)j(a)e(ph)m(ysical)g(explanation)f(of)h(the)h(non-eik)m(onal)0 5022 y(solution,)31 b(giv)m(en)i(b)m(y)g(R.)g(Burridge.)146 5214 y(The)46 b(ab)s(o)m(v)m(e)g(theory)f(w)m(ould)g(not)g(b)s(e)g (complete)f(without)g(considering)g(structural)h(in)m(terfaces)0 5334 y(in)h(the)i(mo)s(del.)85 b(Ra)m(y-tracing)46 b(equations)h (\(2\),)j(dynamic)d(ra)m(y)g(tracing)f(equations)h(\(4\))g(and)g(the)0 5454 y(expressions)g(for)e(the)h(ra)m(y)g(propagator)f(matrix)f(should) i(b)s(e)f(mo)s(di\014ed)g(when)h(the)g(ra)m(y)g(crosses)i(a)0 5575 y(structural)29 b(in)m(terface.)42 b(Under)30 b(\\crossing)f(the)h (in)m(terface")f(w)m(e)h(understand)h(not)e(only)g(transmission,)1841 5824 y(92)p eop %%Page: 93 7 93 6 bop 0 100 a Fq(but)41 b(also)f(re\015ection.)68 b(W)-8 b(e)41 b(consider)g(a)f(structural)h(in)m(terface)f(\006,)k (whic)m(h)d(separates)h(t)m(w)m(o)f(parts)g(of)0 220 y(the)29 b(medium)e(with)h(di\013eren)m(t)h(smo)s(oth)e(distributions)g (of)i(the)g(elastic)e(mo)s(duli)f(and)j(densit)m(y)-8 b(.)43 b(W)-8 b(e)28 b(use)0 340 y(the)40 b(parameteric)f(description)g (of)g(the)h(in)m(terface,)h(giv)m(en)f(b)m(y)h(relation)c Fl(x)j Fq(=)g Fl(g)r Fq(\()p Fj(u)3085 355 y Fp(1)3123 340 y Fj(;)17 b(u)3223 355 y Fp(2)3262 340 y Fq(\),)41 b(where)g Fj(u)3713 355 y Fg(I)3752 340 y Fq(,)0 461 y Fj(I)e Fq(=)31 b(1)p Fj(;)17 b Fq(2,)34 b(are)g(the)h(Gaussian)f(co)s (ordinates)g(of)g(the)h(surface.)50 b(As)35 b(a)f(sp)s(ecial)g(case)h (of)f Fj(u)3248 476 y Fp(1)3287 461 y Fj(;)17 b(u)3387 476 y Fp(2)3426 461 y Fq(,)35 b(w)m(e)g(can)0 581 y(also)26 b(consider)i(lo)s(cal)d(Cartesian)i(co)s(ordinates)f(in)h(a)g(plane)f (tangen)m(t)i(to)f(in)m(terface)g(\006)g(at)g(the)h(reference)0 702 y(ra)m(y)i(\012.)43 b(A)m(t)31 b(an)m(y)f(p)s(oin)m(t)f(of)h(\006,) h(the)f(in)m(terface)g(can)g(b)s(e)h(appro)m(ximated)e(to)g(the)i (second)g(order)f(in)f Fj(u)3607 717 y Fg(I)3677 702 y Fq(b)m(y)0 822 y(the)k(relation)1216 942 y Fl(x)p Fq(\()p Fj(u)1369 957 y Fg(I)1409 942 y Fq(\))27 b(=)h Fl(x)1637 957 y Fp(0)1698 942 y Fq(+)22 b Fl(g)1852 957 y Fg(I)1892 942 y Fj(u)1948 957 y Fg(I)2010 942 y Fq(+)2118 903 y Fp(1)p 2118 919 36 4 v 2118 977 a(2)2163 942 y Fl(g)2219 957 y Fg(I)5 b(J)2304 942 y Fj(u)2360 957 y Fg(I)2399 942 y Fj(u)2455 957 y Fg(J)2536 942 y Fj(:)1043 b Fq(\(15\))0 1101 y(Here)31 b Fl(g)284 1116 y Fg(I)351 1101 y Fq(=)d Fj(@)5 b Fl(g)r Fj(=@)g(u)730 1116 y Fg(I)800 1101 y Fq(are)30 b(v)m(ectors)i(tangen)m(t)e(to)g(\006)g(at)g Fl(x)e Fq(=)f Fl(x)2224 1116 y Fp(0)2264 1101 y Fq(,)j(and)h Fl(g)2565 1116 y Fg(I)5 b(J)2677 1101 y Fq(=)28 b Fj(@)2837 1065 y Fp(2)2877 1101 y Fl(g)r Fj(=@)5 b(u)3096 1116 y Fg(I)3136 1101 y Fj(@)g(u)3248 1116 y Fg(J)3327 1101 y Fq(are)30 b(related)0 1222 y(to)i(the)h(curv)-5 b(ature)33 b(matrix)e(of)h(the)g(in)m(terface)h(at)f Fl(x)c Fq(=)f Fl(x)2082 1237 y Fp(0)2122 1222 y Fq(,)32 b Fj(I)8 b(;)17 b(J)37 b Fq(=)27 b(1)p Fj(;)17 b Fq(2.)43 b(See)33 b(more)f(details)f (in)h(Moser)0 1342 y(and)201 1317 y(\024)190 1342 y(Cerv)m(en)q(\023) -50 b(y)35 b(\(2006\).)146 1533 y(Let)f(us)h(consider)f(a)f(ra)m(y)h (inciden)m(t)f(at)h(in)m(terface)g(\006)g(at)f Fl(x)2254 1548 y Fp(0)2294 1533 y Fq(,)h(and)f(denote)i(the)f(relev)-5 b(an)m(t)34 b(sampling)0 1654 y(parameter)40 b Fj(\034)53 b Fq(at)41 b(the)g(p)s(oin)m(t)f(of)h(incidence)g Fj(\034)1729 1669 y Fp(\006)1784 1654 y Fq(.)69 b(A)m(t)41 b Fj(\034)2071 1669 y Fp(\006)2127 1654 y Fq(,)i(v)m(ectors)f Fl(p)p Fq(,)i Fi(U)51 b Fq(and)41 b Fh(\021)j Fq(are)d(kno)m(wn)i(from)0 1774 y(kinematic)e(ra)m(y)j(tracing)e(\(2\).)73 b(Using)43 b(Snell's)f(la)m(w,)j(w)m(e)f(can)f(compute)g(these)h(quan)m(tities)f (also)f(at)0 1894 y(the)37 b(re\015ection/transmission)f(p)s(oin)m(t)f (of)i(the)g(selected)h(re\015ected/transmitted)f(w)m(a)m(v)m(e.)58 b(W)-8 b(e)37 b(denote)0 2015 y(them)31 b(with)f(a)h(tilde,)f(namely)35 b(~)-54 b Fj(\034)1175 2030 y Fp(\006)1230 2015 y Fq(,)1295 2014 y(~)1288 2015 y Fl(p)q Fq(,)1425 1989 y(~)1409 2015 y Fi(U)10 b Fq(,)1556 2014 y(~)1548 2015 y Fh(\021)t Fq(.)43 b(Although)30 b(the)h(p)s(oin)m(ts)g(of)f(incidence)h Fj(\034)3141 2030 y Fp(\006)3227 2015 y Fq(and)g(of)g(re\015ec-)0 2135 y(tion/transmission)k(~)-54 b Fj(\034)824 2150 y Fp(\006)912 2135 y Fq(coincide,)32 b(w)m(e)i(shall)d(consider)i(them)f (formally)e(as)j(t)m(w)m(o)g(di\013eren)m(t)g(p)s(oin)m(ts.)146 2326 y(The)g(matrices)754 2301 y(~)736 2326 y Fl(Q)820 2290 y Fp(\()p Fg(x)p Fp(\))919 2326 y Fj(;)993 2301 y Fq(~)980 2326 y Fl(P)1057 2290 y Fp(\()p Fg(x)p Fp(\))1187 2326 y Fq(at)e(the)i(re\015ection/transmission)d(p)s(oin)m(t)37 b(~)-55 b Fj(\034)2772 2341 y Fp(\006)2828 2326 y Fq(,)32 b(and)g Fl(Q)3160 2290 y Fp(\()p Fg(x)p Fp(\))3259 2326 y Fq(,)g Fl(P)3395 2290 y Fp(\()p Fg(x)p Fp(\))3525 2326 y Fq(at)g(the)0 2447 y(p)s(oin)m(t)g(of)g(incidence)h Fj(\034)831 2462 y Fp(\006)919 2447 y Fq(are)f(then)h(related)f(b)m(y)i (the)f(equation,)1148 2544 y Ff(")1256 2604 y Fq(~)1238 2630 y Fl(Q)1322 2593 y Fp(\()p Fg(x)p Fp(\))1252 2732 y Fq(~)1238 2757 y Fl(P)1315 2721 y Fp(\()p Fg(x)p Fp(\))1463 2544 y Ff(#)1539 2691 y Fq(=)27 b Fl(\005)1730 2648 y Fp(\()p Fg(x)p Fp(\))1829 2691 y Fq(\()5 b(~)-54 b Fj(\034)1909 2706 y Fp(\006)1964 2691 y Fj(;)17 b(\034)2050 2706 y Fp(\006)2105 2691 y Fq(\))2192 2544 y Ff(")2282 2630 y Fl(Q)2366 2593 y Fp(\()p Fg(x)p Fp(\))2282 2752 y Fl(P)2359 2715 y Fp(\()p Fg(x)p Fp(\))2507 2544 y Ff(#)2604 2691 y Fj(;)975 b Fq(\(16\))0 2957 y(where)36 b Fl(\005)372 2915 y Fp(\()p Fg(x)p Fp(\))471 2957 y Fq(\()5 b(~)-54 b Fj(\034)551 2972 y Fp(\006)606 2957 y Fj(;)17 b(\034)692 2972 y Fp(\006)747 2957 y Fq(\))35 b(is)g(the)g(so-called)f(in)m (terface)h(propagator)f(matrix)g(in)g(Cartesian)h(co)s(ordinates.)0 3077 y(A)e(particularly)e(concise)i(form)f(of)h(the)g(6)22 b Fn(\002)h Fq(6)33 b(in)m(terface)g(propagator)f(matrix)g(w)m(as)i (giv)m(en)f(b)m(y)h(Moser)0 3198 y(\(2004\),)1008 3384 y Fl(\005)1096 3342 y Fp(\()p Fg(x)p Fp(\))1195 3384 y Fq(\()5 b(~)-54 b Fj(\034)1275 3399 y Fp(\006)1330 3384 y Fj(;)17 b(\034)1416 3399 y Fp(\006)1471 3384 y Fq(\))28 b(=)1641 3238 y Ff(")1748 3297 y Fq(~)1731 3323 y Fl(X)o(X)1900 3286 y Fe(\000)p Fp(1)2271 3323 y Fl(0)1748 3425 y Fq(~)1731 3450 y Fl(X)1816 3414 y Fe(\000)p Fg(T)1925 3450 y Fl(RX)2093 3408 y Fe(\000)p Fp(1)2289 3425 y Fq(~)2271 3450 y Fl(X)2356 3414 y Fe(\000)p Fg(T)2465 3450 y Fl(X)2550 3414 y Fg(T)2646 3238 y Ff(#)2744 3384 y Fj(;)835 b Fq(\(17\))0 3625 y(where)34 b(3)22 b Fn(\002)g Fq(3)33 b(matrices)e Fl(X)i Fq(and)1249 3599 y(~)1231 3625 y Fl(X)f Fq(are)h(giv)m(en)g(b)m (y)g(relations)1122 3809 y Fl(X)27 b Fq(=)h(\()p Fl(g)1432 3824 y Fp(1)1471 3809 y Fj(;)17 b Fl(g)1571 3824 y Fp(2)1610 3809 y Fj(;)g Fi(U)10 b Fq(\))33 b Fj(;)1965 3783 y Fq(~)1947 3809 y Fl(X)27 b Fq(=)h(\()p Fl(g)2257 3824 y Fp(1)2296 3809 y Fj(;)17 b Fl(g)2396 3824 y Fp(2)2435 3809 y Fj(;)2495 3783 y Fq(~)2479 3809 y Fi(U)10 b Fq(\))33 b Fj(;)948 b Fq(\(18\))0 3993 y(and)33 b(the)g(3)22 b Fn(\002)g Fq(3)33 b(matrix)e Fl(R)h Fq(reads)1121 4222 y Fl(R)27 b Fq(=)1335 4076 y Ff(")1425 4158 y Fl(g)1483 4121 y Fg(T)1481 4182 y(I)5 b(J)1566 4158 y Fq(\()1611 4157 y(~)1604 4158 y Fl(p)22 b Fn(\000)h Fl(p)p Fq(\))83 b Fl(g)2029 4121 y Fg(T)2027 4182 y(I)2084 4158 y Fq(\()2129 4157 y(~)2122 4158 y Fh(\021)25 b Fn(\000)e Fh(\021)t Fq(\))1425 4285 y Fl(g)1483 4249 y Fg(T)1481 4310 y(J)1538 4285 y Fq(\()1583 4284 y(~)1576 4285 y Fh(\021)j Fn(\000)c Fh(\021)t Fq(\))1979 4284 y(~)1971 4285 y Fh(\021)2035 4244 y Fg(T)2106 4260 y Fq(~)2090 4285 y Fi(U)32 b Fn(\000)23 b Fh(\021)2356 4249 y Fg(T)2411 4285 y Fi(U)2534 4076 y Ff(#)2632 4222 y Fj(:)947 b Fq(\(19\))0 4486 y(The)23 b(in)m(terface)e(propagator)g(matrix)f Fl(\005)1456 4443 y Fp(\()p Fg(x)p Fp(\))1555 4486 y Fq(\()5 b(~)-54 b Fj(\034)1635 4501 y Fp(\006)1690 4486 y Fj(;)17 b(\034)1776 4501 y Fp(\006)1831 4486 y Fq(\))22 b(is)f(symplectic,)j(satis\014es)e (the)g(relation)e(det)d Fl(\005)3569 4443 y Fp(\()p Fg(x)p Fp(\))3668 4486 y Fq(\()5 b(~)-54 b Fj(\034)3748 4501 y Fp(\006)3803 4486 y Fj(;)17 b(\034)3889 4501 y Fp(\006)3944 4486 y Fq(\))28 b(=)0 4606 y(1)k(and)h(preserv)m(es)i(constrain)m(t)e (relation)e(\(6\))h(across)h(the)g(in)m(terface.)146 4797 y(F)-8 b(or)40 b(the)h(follo)m(wing)d(sections,)43 b(it)d(will)e(b)s(e)j(useful)f(to)g(express)j(the)e(matrix)e(in)m(v)m (ersions)i(in)f(\(17\))0 4918 y(explicitly)-8 b(.)42 b(W)-8 b(e)32 b(in)m(tro)s(duce)h(3)22 b Fn(\002)h Fq(3)32 b(matrices)762 5101 y Fl(U)27 b Fq(=)h Fl(X)1064 5060 y Fe(\000)p Fg(T)1201 5101 y Fq(=)g(\()p Fl(h)1405 5116 y Fp(1)1444 5101 y Fj(;)17 b Fl(h)1550 5116 y Fp(2)1590 5101 y Fj(;)g Fl(h)1696 5116 y Fp(3)1735 5101 y Fq(\))33 b Fj(;)1966 5076 y Fq(~)1947 5101 y Fl(U)28 b Fq(=)2182 5076 y(~)2164 5101 y Fl(X)2249 5060 y Fe(\000)p Fg(T)2386 5101 y Fq(=)g(\()2535 5075 y(~)2528 5101 y Fl(h)2590 5116 y Fp(1)2630 5101 y Fj(;)2680 5075 y Fq(~)2674 5101 y Fl(h)2736 5116 y Fp(2)2775 5101 y Fj(;)2825 5075 y Fq(~)2819 5101 y Fl(h)2881 5116 y Fp(3)2920 5101 y Fq(\))33 b Fj(:)588 b Fq(\(20\))0 5285 y(Here)33 b Fl(h)292 5300 y Fp(1)332 5285 y Fj(;)17 b Fl(h)438 5300 y Fp(2)477 5285 y Fj(;)g Fl(h)583 5300 y Fp(3)655 5285 y Fq(can)33 b(b)s(e)g(calculated)f(from)f Fl(g)1714 5300 y Fp(1)1753 5285 y Fj(;)17 b Fl(g)1853 5300 y Fp(2)1893 5285 y Fj(;)g Fi(U)42 b Fq(b)m(y)34 b(relations,)655 5526 y Fl(h)717 5541 y Fp(1)784 5526 y Fq(=)1011 5458 y Fl(g)1067 5473 y Fp(2)1129 5458 y Fn(\002)22 b Fi(U)p 898 5502 525 4 v 898 5600 a(U)979 5558 y Fg(T)1034 5600 y Fq(\()p Fl(g)1128 5615 y Fp(1)1190 5600 y Fn(\002)h Fl(g)1346 5615 y Fp(2)1385 5600 y Fq(\))1465 5526 y Fj(;)115 b Fl(h)1669 5541 y Fp(2)1736 5526 y Fq(=)1963 5458 y Fi(U)32 b Fn(\002)23 b Fl(g)2222 5473 y Fp(1)p 1850 5502 V 1850 5600 a Fi(U)1931 5558 y Fg(T)1986 5600 y Fq(\()p Fl(g)2080 5615 y Fp(1)2141 5600 y Fn(\002)g Fl(g)2297 5615 y Fp(2)2337 5600 y Fq(\))2417 5526 y Fj(;)114 b Fl(h)2620 5541 y Fp(3)2688 5526 y Fq(=)2869 5458 y Fl(n)2931 5422 y Fp(\006)p 2801 5502 254 4 v 2801 5600 a Fi(U)2882 5558 y Fg(T)2937 5600 y Fl(n)2999 5571 y Fp(\006)3097 5526 y Fj(;)482 b Fq(\(21\))1841 5824 y(93)p eop %%Page: 94 8 94 7 bop 0 100 a Fq(where)31 b Fl(n)341 63 y Fp(\006)426 100 y Fq(is)e(a)g(v)m(ector)i(normal)c(to)i(\006)h(at)g(a)f(p)s(oin)m (t)f(of)i(incidence)f(\(not)h(necessarily)g(unit\).)41 b(Similarly)-8 b(,)0 220 y Fl(g)56 235 y Fp(1)95 220 y Fj(;)17 b Fl(g)195 235 y Fp(2)235 220 y Fj(;)g Fl(g)335 235 y Fp(3)406 220 y Fq(can)33 b(b)s(e)g(calculated)f(from)f Fl(h)1471 235 y Fp(1)1511 220 y Fj(;)17 b Fl(h)1617 235 y Fp(2)1656 220 y Fj(;)g Fl(h)1762 235 y Fp(3)1834 220 y Fq(as)33 b(follo)m(ws:)767 447 y Fl(g)823 462 y Fp(1)890 447 y Fq(=)1100 379 y Fl(h)1162 394 y Fp(2)1224 379 y Fn(\002)23 b Fl(h)1386 394 y Fp(3)p 1004 423 519 4 v 1004 515 a Fl(h)1066 481 y Fg(T)1066 537 y Fp(3)1121 515 y Fq(\()p Fl(h)1221 530 y Fp(1)1283 515 y Fn(\002)g Fl(h)1445 530 y Fp(2)1484 515 y Fq(\))1565 447 y Fj(;)114 b Fl(g)1762 462 y Fp(2)1829 447 y Fq(=)2039 379 y Fl(h)2101 394 y Fp(3)2163 379 y Fn(\002)23 b Fl(h)2325 394 y Fp(1)p 1943 423 V 1943 515 a Fl(h)2005 481 y Fg(T)2005 537 y Fp(3)2060 515 y Fq(\()p Fl(h)2160 530 y Fp(1)2222 515 y Fn(\002)f Fl(h)2383 530 y Fp(2)2423 515 y Fq(\))2503 447 y Fj(;)115 b Fl(g)2701 462 y Fp(3)2768 447 y Fq(=)27 b Fi(U)43 b Fj(:)594 b Fq(\(22\))0 680 y(Similar)39 b(expressions)44 b(are)f(obtained)e(also)h(for)g(the)h(tilded)e(v)m(ectors;)49 b(w)m(e)43 b(only)f(need)i(to)e(tak)m(e)h(in)m(to)0 800 y(accoun)m(t)35 b(that)579 799 y(~)575 800 y Fl(g)631 815 y Fp(1)701 800 y Fq(=)30 b Fl(g)863 815 y Fp(1)902 800 y Fj(;)950 799 y Fq(~)946 800 y Fl(g)1002 815 y Fp(2)1072 800 y Fq(=)g Fl(g)1234 815 y Fp(2)1273 800 y Fq(.)49 b(The)35 b(expression)g(for)f(the)g(6)23 b Fn(\002)h Fq(6)34 b(in)m(terface)g(propagator)f(matrix)0 920 y Fl(\005)88 878 y Fp(\()p Fg(x)p Fp(\))186 920 y Fq(\()5 b(~)-54 b Fj(\034)266 935 y Fp(\006)322 920 y Fj(;)17 b(\034)408 935 y Fp(\006)463 920 y Fq(\))32 b(in)g(Cartesian)h(co)s (ordinates)f(then)h(reads)1120 1168 y Fl(\005)1208 1126 y Fp(\()p Fg(x)p Fp(\))1306 1168 y Fq(\()5 b(~)-54 b Fj(\034)1386 1183 y Fp(\006)1442 1168 y Fj(;)17 b(\034)1528 1183 y Fp(\006)1583 1168 y Fq(\))28 b(=)1752 1022 y Ff( )1877 1082 y Fq(~)1859 1107 y Fl(XU)2030 1071 y Fg(T)2250 1107 y Fl(0)1878 1210 y Fq(~)1859 1235 y Fl(U)q(R)m(U)2113 1199 y Fg(T)2269 1210 y Fq(~)2250 1235 y Fl(UX)2421 1199 y Fg(T)2518 1022 y Ff(!)2632 1168 y Fj(;)947 b Fq(\(23\))0 1416 y(see)34 b(\(17\))e(and)g(\(20\).)146 1608 y(Note)22 b(that)g(v)m(ectors)h Fl(g)948 1623 y Fp(1)987 1608 y Fj(;)17 b Fl(g)1087 1623 y Fp(2)1127 1608 y Fj(;)g Fl(g)1227 1623 y Fp(3)1294 1608 y Fq(=)27 b Fi(U)32 b Fq(represen)m(t)24 b(the)e(con)m(tra)m(v)-5 b(arian)m(t)21 b(basis)h(v)m(ectors,)j(and)d Fl(h)3471 1623 y Fp(1)3511 1608 y Fj(;)17 b Fl(h)3617 1623 y Fp(2)3656 1608 y Fj(;)g Fl(h)3762 1623 y Fp(3)3830 1608 y Fq(=)0 1728 y Fl(n)62 1692 y Fp(\006)117 1728 y Fj(=)p Fi(U)247 1686 y Fg(T)303 1728 y Fl(n)365 1692 y Fp(\006)442 1728 y Fq(the)22 b(co)m(v)-5 b(arian)m(t)21 b(basis)h(v)m(ector)g(in)f(the)i(lo)s(cal)c(nonorthogonal)h(co)s (ordinate)h(system)i Fj(u)3470 1743 y Fp(1)3509 1728 y Fj(;)17 b(u)3609 1743 y Fp(2)3647 1728 y Fj(;)g(u)3747 1743 y Fp(3)3814 1728 y Fq(=)0 1848 y Fj(\034)36 b Fq(at)24 b(the)h(p)s(oin)m(t)f(of)g(incidence.)41 b(A)m(t)25 b(the)g(p)s(oin)m (t)e(of)i(re\015ection/transmission,)f(the)h(con)m(tra)m(v)-5 b(arian)m(t)25 b(basis)0 1992 y(v)m(ectors)31 b(are)492 1991 y(~)487 1992 y Fl(g)543 2007 y Fp(1)610 1992 y Fq(=)d Fl(g)770 2007 y Fp(1)809 1992 y Fj(;)874 1991 y Fq(~)870 1992 y Fl(g)926 2007 y Fp(2)993 1992 y Fq(=)f Fl(g)1152 2007 y Fp(2)1192 1992 y Fj(;)1240 1991 y Fq(~)1236 1992 y Fl(g)1292 2007 y Fp(3)1359 1992 y Fq(=)1478 1967 y(~)1462 1992 y Fi(U)10 b Fq(,)30 b(and)g(co)m(v)-5 b(arian)m(t)2210 1966 y(~)2204 1992 y Fl(h)2266 2007 y Fp(1)2305 1992 y Fj(;)2356 1966 y Fq(~)2349 1992 y Fl(h)2411 2007 y Fp(2)2451 1992 y Fq(,)2514 1966 y(~)2508 1992 y Fl(h)2570 2007 y Fp(3)2637 1992 y Fq(=)28 b Fl(n)2803 1956 y Fp(\006)2858 1992 y Fj(=)2923 1967 y Fq(~)2907 1992 y Fi(U)2988 1927 y Fg(T)3043 1992 y Fl(n)3105 1956 y Fp(\006)3160 1992 y Fq(.)43 b(Basis)29 b(v)m(ectors)0 2113 y(satisfy)k(the)g(relations) 770 2295 y Fl(h)832 2254 y Fg(T)832 2320 y(k)888 2295 y Fl(g)944 2310 y Fg(l)998 2295 y Fq(=)27 b Fj(\016)1144 2310 y Fg(k)r(l)1241 2295 y Fj(;)1389 2269 y Fq(~)1383 2295 y Fl(h)1445 2254 y Fg(T)1445 2320 y(k)1504 2294 y Fq(~)1500 2295 y Fl(g)1556 2310 y Fg(l)1610 2295 y Fq(=)g Fj(\016)1756 2310 y Fg(k)r(l)1853 2295 y Fj(;)115 b Fl(h)2057 2310 y Fg(k)2100 2295 y Fl(g)2158 2254 y Fg(T)2156 2320 y(k)2240 2295 y Fq(=)27 b Fl(I)33 b Fj(;)2566 2269 y Fq(~)2559 2295 y Fl(h)2621 2310 y Fg(k)2668 2294 y Fq(~)2664 2295 y Fl(g)2722 2254 y Fg(T)2720 2320 y(k)2804 2295 y Fq(=)28 b Fl(I)k Fj(;)597 b Fq(\(24\))0 2478 y(where)37 b Fl(I)e Fq(is)g(the)g(3)24 b Fn(\002)h Fq(3)35 b(iden)m(tit)m(y)g (matrix.)50 b(V)-8 b(ectors)36 b Fl(g)2024 2493 y Fp(1)2099 2478 y Fq(and)g Fl(g)2348 2493 y Fp(2)2422 2478 y Fq(are)g(tangen)m(t)g (to)f(surface)h(\006,)g Fl(h)3594 2493 y Fp(1)3634 2478 y Fj(;)17 b Fl(h)3740 2493 y Fp(2)0 2598 y Fq(and)193 2572 y(~)186 2598 y Fl(h)248 2613 y Fp(1)288 2598 y Fj(;)338 2572 y Fq(~)332 2598 y Fl(h)394 2613 y Fp(2)462 2598 y Fq(are)29 b(p)s(erp)s(endicular)f(to)g(the)i(ra)m(y)-8 b(.)42 b(V)-8 b(ectors)30 b Fl(g)2111 2613 y Fp(1)2179 2598 y Fq(and)f Fl(g)2421 2613 y Fp(2)2489 2598 y Fq(need)h(not)f(b)s (e)g(unit)f(and)h(m)m(utually)0 2719 y(p)s(erp)s(endicular,)j(and)h (similarly)28 b Fl(h)1295 2734 y Fp(1)1368 2719 y Fq(and)k Fl(h)1619 2734 y Fp(2)1692 2719 y Fq(and)1888 2692 y(~)1881 2719 y Fl(h)1943 2734 y Fp(1)2015 2719 y Fq(and)2212 2692 y(~)2205 2719 y Fl(h)2267 2734 y Fp(2)2307 2719 y Fq(.)146 2910 y(In)46 b(a)f(la)m(y)m(ered)i(medium)c(with)j (structural)f(in)m(terfaces,)k(the)d(ra)m(y)g(propagator)f(matrix)f (can)h(b)s(e)0 3030 y(c)m(hained)30 b(and)g(the)g(in)m(terface)f (propagator)g(matrix)f(should)h(b)s(e)h(included)g(in)e(the)i(c)m(hain) g(at)f(an)m(y)h(p)s(oin)m(t)0 3151 y(of)40 b(incidence.)68 b(F)-8 b(or)40 b(example,)i(if)d(the)i(ra)m(y)g(crosses)i(one)e(in)m (terface)f(at)h Fj(\034)2745 3166 y Fp(\006)2841 3151 y Fq(b)s(et)m(w)m(een)h Fj(\034)3267 3166 y Fp(0)3348 3151 y Fq(and)e Fj(\034)11 b Fq(,)43 b(w)m(e)0 3271 y(obtain)858 3391 y Fl(\005)945 3349 y Fp(\()p Fg(x)p Fp(\))1044 3391 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1216 3406 y Fp(0)1255 3391 y Fq(\))28 b(=)g Fl(\005)1512 3349 y Fp(\()p Fg(x)p Fp(\))1611 3391 y Fq(\()p Fj(\034)6 b(;)22 b Fq(~)-54 b Fj(\034)1783 3406 y Fp(\006)1838 3391 y Fq(\))p Fl(\005)1964 3349 y Fp(\()p Fg(x)p Fp(\))2063 3391 y Fq(\()5 b(~)-54 b Fj(\034)2143 3406 y Fp(\006)2198 3391 y Fj(;)17 b(\034)2284 3406 y Fp(\006)2339 3391 y Fq(\))p Fl(\005)2465 3349 y Fp(\()p Fg(x)p Fp(\))2563 3391 y Fq(\()p Fj(\034)2643 3406 y Fp(\006)2699 3391 y Fj(;)g(\034)2785 3406 y Fp(0)2824 3391 y Fq(\))33 b Fj(:)684 b Fq(\(25\))0 3550 y(F)-8 b(or)32 b(a)g(more)g(thorough)g(treatmen)m(t)h(see)g(Moser)h(\(2004\),) d(where)j(a)f(general)f(sampling)e(parameter)i Fj(\034)0 3670 y Fq(along)f(the)i(ra)m(y)g(is)f(used)i(\(not)e(necessarily)h(the) g(tra)m(v)m(el)g(time\).)146 3861 y(A)42 b(terminological)37 b(note.)70 b(The)43 b(term)e Fd(interfac)-5 b(e)42 b(pr)-5 b(op)g(agator)42 b(matrix)g Fq(w)m(as)g(used)h(b)m(y)3441 3836 y(\024)3430 3861 y(Cerv)m(en)q(\023)-50 b(y)0 3982 y(\(2001\))30 b(and)g(Moser)i(\(2004\),)e(to)g(emphasize)h(the)g(fact)f (that)h(the)g(matrix)e(transforms)h(the)h(ra)m(y)g(prop-)0 4102 y(agator)d(matrix)g(across)j(the)f(in)m(terface.)42 b(Analogous)29 b(term)g Fd(wave)i(pr)-5 b(op)g(agator)32 b(for)g(the)g(interfac)-5 b(e)29 b Fq(w)m(as)0 4223 y(also)j(used)j(b)m (y)f(Kennett)h(\(1983,)d(p.)46 b(111\))33 b(in)g(the)h (stress-displacemen)m(t)g(propagator)e(formalism)e(for)0 4343 y(1-D)h(la)m(y)m(ered)j(media.)0 4741 y Fk(3)161 b(Ra)l(y-cen)l(tered)51 b(co)t(ordinate)j(systems)0 5031 y Fq(The)32 b(ra)m(y-cen)m(tered)g(co)s(ordinate)e(system)i Fl(q)27 b Fq(=)h(\()p Fj(q)1823 5046 y Fp(1)1863 5031 y Fj(;)17 b(q)1950 5046 y Fp(2)1989 5031 y Fj(;)g(q)2076 5046 y Fp(3)2115 5031 y Fq(\))31 b(connected)h(with)f(the)g(reference)h (ra)m(y)f(\012)g(in)0 5151 y(an)i(inhomogeneous)g(anisotropic)f(medium) g(can)h(b)s(e)h(in)m(tro)s(duced)f(in)g(man)m(y)g(alternativ)m(e)g(w)m (a)m(ys.)47 b(F)-8 b(or)0 5272 y(an)27 b(up-to-date)f(review)i(of)e(v) -5 b(arious)26 b(p)s(osibilities)e(see)k(Klime)-5 b(\024)-44 b(s)25 b(\(2006\).)40 b(One)28 b(of)e(the)h(simplest)f(options)0 5392 y(is)32 b(in)m(tro)s(duced)h(b)m(y)g(the)g(relation,)e(see)j (Klime)-5 b(\024)-44 b(s)30 b(\(1994,)i(2006\):)1244 5575 y Fj(x)1299 5590 y Fg(i)1327 5575 y Fq(\()p Fj(q)1408 5590 y Fg(j)1445 5575 y Fq(\))c(=)f Fj(x)1669 5590 y Fg(i)p Fp(0)1733 5575 y Fq(\()p Fj(q)1814 5590 y Fp(3)1854 5575 y Fq(\))22 b(+)g Fj(H)2093 5590 y Fg(iM)2196 5575 y Fq(\()p Fj(q)2277 5590 y Fp(3)2316 5575 y Fq(\))p Fj(q)2397 5590 y Fg(M)2509 5575 y Fj(;)1070 b Fq(\(26\))1841 5824 y(94)p eop %%Page: 95 9 95 8 bop 0 100 a Fj(i)28 b Fq(=)g(1)p Fj(;)17 b Fq(2)p Fj(;)g Fq(3,)31 b(and)i Fj(M)38 b Fq(=)28 b(1)p Fj(;)17 b Fq(2.)42 b(The)34 b(reference)g(ra)m(y)f(\012)g(is)f(sp)s(eci\014ed)h (b)m(y)h(the)f(relation)e Fj(x)3184 115 y Fg(i)3212 100 y Fq(\()p Fj(q)3293 115 y Fp(3)3333 100 y Fq(\))d(=)f Fj(x)3557 115 y Fg(i)p Fp(0)3621 100 y Fq(\()p Fj(q)3702 115 y Fp(3)3742 100 y Fq(\))0 220 y(and)c(represen)m(ts)j(the)e Fj(q)832 235 y Fp(3)871 220 y Fq(-co)s(ordinate)e(line)g(of)h(the)g (nonorthogonal)f(ra)m(y-cen)m(tered)j(co)s(ordinate)d(system,)0 340 y(and)41 b Fj(q)241 355 y Fp(3)322 340 y Fq(=)h Fj(\034)11 b Fq(,)43 b(the)e(tra)m(v)m(el-time)f(along)f(the)i(ra)m(y)-8 b(.)69 b(Co)s(ordinates)40 b Fj(q)2507 355 y Fp(1)2588 340 y Fq(and)h Fj(q)2829 355 y Fp(2)2909 340 y Fq(uniquely)g(sp)s (ecify)g(the)0 461 y(p)s(osition)30 b(of)i(a)g(p)s(oin)m(t)f(in)g(the)i (plane)f(tangen)m(t)g(to)g(the)g(w)m(a)m(v)m(efron)m(t)i(in)m (tersecting)e(the)h(reference)h(ra)m(y)e(\012)0 581 y(for)g(a)g (\014xed)i Fj(q)509 596 y Fp(3)549 581 y Fq(.)43 b(F)-8 b(or)32 b Fj(q)837 596 y Fp(1)904 581 y Fq(=)c Fj(q)1051 596 y Fp(2)1118 581 y Fq(=)g(0,)k(the)h(p)s(oin)m(t)f(is)g(situated)g (directly)g(on)h(the)g(reference)h(ra)m(y)f(\012.)146 772 y(W)-8 b(e)43 b(in)m(tro)s(duce)f(the)h(3)28 b Fn(\002)i Fq(3)41 b(transformation)g(matrix)f Fl(H)i Fq(from)f(Cartesian)h(co)s (ordinates)g Fj(x)3622 787 y Fg(i)3693 772 y Fq(to)0 893 y(ra)m(y-cen)m(tered)34 b(co)s(ordinates)f Fj(q)1117 908 y Fg(m)1183 893 y Fq(,)1531 1013 y Fj(H)1612 1028 y Fg(im)1730 1013 y Fq(=)27 b Fj(@)5 b(x)1944 1028 y Fg(i)1974 1013 y Fj(=@)g(q)2122 1028 y Fg(m)2222 1013 y Fj(;)1357 b Fq(\(27\))0 1184 y Fj(i;)17 b(m)53 b Fq(=)f(1)p Fj(;)17 b Fq(2)p Fj(;)g Fq(3.)86 b(The)48 b(con)m(tra)m(v)-5 b(arian)m(t)46 b(basis)i(v)m(ectors)g(of)f(the)g(ra)m(y-cen)m(tered)i (co)s(ordinate)e(system)0 1305 y Fl(e)51 1320 y Fp(1)91 1305 y Fj(;)17 b Fl(e)186 1320 y Fp(2)225 1305 y Fj(;)g Fl(e)320 1320 y Fp(3)392 1305 y Fq(are)32 b(giv)m(en)h(b)m(y)g(the)g (relations)1427 1517 y Fl(H)27 b Fq(=)h(\()p Fl(e)1735 1532 y Fp(1)1774 1517 y Fj(;)17 b Fl(e)1869 1532 y Fp(2)1908 1517 y Fj(;)g Fl(e)2003 1532 y Fp(3)2070 1517 y Fq(=)28 b Fi(U)10 b Fq(\))33 b Fj(:)1253 b Fq(\(28\))0 1730 y Fl(e)51 1745 y Fp(1)124 1730 y Fq(and)34 b Fl(e)366 1745 y Fp(2)439 1730 y Fq(are)g(situated)f(in)g(the)h(w)m(a)m(v)m(efron)m(t) i(tangen)m(t)e(plane,)f Fl(e)2403 1745 y Fp(3)2477 1730 y Fq(coincides)g(with)g(the)i(ra)m(y)f(v)m(elo)s(cit)m(y)0 1851 y(v)m(ector)g Fi(U)10 b Fq(,)33 b(tangen)m(t)g(to)f(the)h(ra)m(y) -8 b(.)146 2042 y(Co)m(v)j(arian)m(t)32 b(basis)h(v)m(ectors)h Fl(f)1198 2057 y Fp(1)1238 2042 y Fj(;)17 b Fl(f)1316 2057 y Fp(2)1355 2042 y Fj(;)g Fl(f)1433 2057 y Fp(3)1505 2042 y Fq(are)33 b(giv)m(en)f(b)m(y)i(the)f(relation)1310 2254 y Fl(S)28 b Fq(=)f Fl(H)1591 2213 y Fe(\000)p Fg(T)1729 2254 y Fq(=)g(\()p Fl(f)1904 2269 y Fp(1)1944 2254 y Fj(;)17 b Fl(f)2022 2269 y Fp(2)2061 2254 y Fj(;)g Fl(f)2139 2269 y Fp(3)2206 2254 y Fq(=)28 b Fl(p)p Fq(\))33 b Fj(:)1136 b Fq(\(29\))0 2467 y Fl(f)34 2482 y Fp(1)104 2467 y Fq(and)30 b Fl(f)325 2482 y Fp(2)394 2467 y Fq(are)g(p)s(erp)s(endicular)f(to)h (the)g(ra)m(y)-8 b(,)31 b Fl(f)1670 2482 y Fp(3)1739 2467 y Fq(equals)g(slo)m(wness)g(v)m(ector)g Fl(p)p Fq(.)43 b(Basis)30 b(v)m(ectors)h Fl(e)3472 2482 y Fg(i)3530 2467 y Fq(and)f Fl(f)3751 2482 y Fg(i)0 2587 y Fq(satisfy)j(the)g (relations,)1371 2708 y Fl(e)1422 2667 y Fg(T)1422 2733 y(k)1477 2708 y Fl(f)1511 2723 y Fg(l)1565 2708 y Fq(=)28 b Fj(\016)1712 2723 y Fg(k)r(l)1809 2708 y Fj(;)147 b Fl(e)2034 2723 y Fg(k)2076 2708 y Fl(f)2120 2667 y Fg(T)2110 2733 y(k)2204 2708 y Fq(=)27 b Fl(I)32 b Fj(;)1198 b Fq(\(30\))0 2879 y(where)42 b Fl(I)f Fq(is)g(the)g(3)28 b Fn(\002)h Fq(3)40 b(iden)m(tit)m(y)h(matrix.)68 b(V)-8 b(ectors)42 b Fl(f)2068 2894 y Fp(1)2108 2879 y Fj(;)17 b Fl(f)2186 2894 y Fp(2)2225 2879 y Fj(;)g Fl(f)2303 2894 y Fp(3)2384 2879 y Fq(can)41 b(b)s(e)g(calculated)g(from)e Fl(e)3471 2894 y Fp(1)3511 2879 y Fj(;)17 b Fl(e)3606 2894 y Fp(2)3645 2879 y Fj(;)g Fl(e)3740 2894 y Fp(3)0 2999 y Fq(using)32 b(relations)813 3262 y Fl(f)847 3277 y Fp(1)914 3262 y Fq(=)1100 3194 y(\()p Fl(e)1189 3209 y Fp(2)1251 3194 y Fn(\002)23 b Fi(U)10 b Fq(\))p 1027 3238 516 4 v 1027 3336 a Fi(U)1109 3294 y Fg(T)1164 3336 y Fq(\()p Fl(e)1253 3351 y Fp(1)1315 3336 y Fn(\002)22 b Fl(e)1465 3351 y Fp(2)1505 3336 y Fq(\))1585 3262 y Fj(;)114 b Fl(f)1760 3277 y Fp(2)1828 3262 y Fq(=)2014 3194 y(\()p Fi(U)32 b Fn(\002)23 b Fl(e)2306 3209 y Fp(1)2346 3194 y Fq(\))p 1941 3238 V 1941 3336 a Fi(U)2023 3294 y Fg(T)2078 3336 y Fq(\()p Fl(e)2167 3351 y Fp(1)2228 3336 y Fn(\002)g Fl(e)2379 3351 y Fp(2)2418 3336 y Fq(\))2499 3262 y Fj(;)114 b Fl(f)2674 3277 y Fp(3)2741 3262 y Fq(=)28 b Fl(p)33 b Fj(:)639 b Fq(\(31\))0 3530 y(Similarly)-8 b(,)29 b(v)m(ectors)34 b Fl(e)816 3545 y Fp(1)855 3530 y Fj(;)17 b Fl(e)950 3545 y Fp(2)990 3530 y Fj(;)g Fl(e)1085 3545 y Fp(3)1156 3530 y Fq(can)33 b(b)s(e)g(calculated)f(from)f Fl(f)2193 3545 y Fp(1)2233 3530 y Fj(;)17 b Fl(f)2311 3545 y Fp(2)2350 3530 y Fj(;)g Fl(f)2428 3545 y Fp(3)2500 3530 y Fq(as)33 b(follo)m(ws:)831 3793 y Fl(e)882 3808 y Fp(1)949 3793 y Fq(=)1127 3725 y(\()p Fl(f)1199 3740 y Fp(2)1260 3725 y Fn(\002)23 b Fl(p)p Fq(\))p 1062 3769 463 4 v 1062 3861 a Fl(p)1124 3832 y Fg(T)1180 3861 y Fq(\()p Fl(f)1252 3876 y Fp(1)1313 3861 y Fn(\002)g Fl(f)1447 3876 y Fp(2)1487 3861 y Fq(\))1567 3793 y Fj(;)114 b Fl(e)1759 3808 y Fp(2)1827 3793 y Fq(=)2005 3725 y(\()p Fl(p)22 b Fn(\002)g Fl(f)2260 3740 y Fp(1)2300 3725 y Fq(\))p 1940 3769 V 1940 3861 a Fl(p)2002 3832 y Fg(T)2057 3861 y Fq(\()p Fl(f)2129 3876 y Fp(1)2191 3861 y Fn(\002)h Fl(f)2325 3876 y Fp(2)2365 3861 y Fq(\))2445 3793 y Fj(;)114 b Fl(e)2637 3808 y Fp(3)2704 3793 y Fq(=)28 b Fi(U)43 b Fj(:)657 b Fq(\(32\))146 4158 y(Similarly)40 b(as)k(in)f(\(3\),)j(w)m (e)f(in)m(tro)s(duce)f(t)m(w)m(o)g(3)30 b Fn(\002)g Fq(1)44 b(matrices)f Fl(Q)2586 4122 y Fp(\()p Fg(q)r Fp(\))2726 4158 y Fn(\021)k Fq(\()p Fj(Q)2965 4107 y Fp(\()p Fg(q)r Fp(\))2965 4180 y(1)3058 4158 y Fj(;)17 b(Q)3179 4107 y Fp(\()p Fg(q)r Fp(\))3179 4180 y(2)3272 4158 y Fj(;)g(Q)3393 4107 y Fp(\()p Fg(q)r Fp(\))3393 4180 y(3)3486 4158 y Fq(\))3524 4122 y Fg(T)3622 4158 y Fq(and)0 4292 y Fl(P)77 4256 y Fp(\()p Fg(q)r Fp(\))197 4292 y Fn(\021)28 b Fq(\()p Fj(P)417 4241 y Fp(\()p Fg(q)r Fp(\))403 4314 y(1)510 4292 y Fj(;)17 b(P)631 4241 y Fp(\()p Fg(q)r Fp(\))617 4314 y(2)723 4292 y Fj(;)g(P)844 4241 y Fp(\()p Fg(q)r Fp(\))830 4314 y(3)936 4292 y Fq(\))974 4256 y Fg(T)1061 4292 y Fq(along)31 b(the)i(ra)m(y)-8 b(,)1126 4505 y Fl(Q)1210 4463 y Fp(\()p Fg(q)r Fp(\))1331 4505 y Fq(=)28 b Fj(@)5 b Fl(q)p Fj(=@)g(\015)39 b(;)147 b Fl(P)1996 4463 y Fp(\()p Fg(q)r Fp(\))2116 4505 y Fq(=)28 b Fj(@)5 b Fl(p)2338 4463 y Fp(\()p Fg(q)r Fp(\))2432 4505 y Fj(=@)g(\015)38 b(;)953 b Fq(\(33\))0 4717 y(where)37 b Fj(\015)42 b Fq(is)35 b(an)h(arbitrary)g(ra)m(y)g(parameter,)h(and)f Fl(p)1961 4681 y Fp(\()p Fg(q)r Fp(\))2091 4717 y Fq(is)f(the)i(slo)m (wness)h(v)m(ector)f(expressed)i(in)d(ra)m(y-)0 4838 y(cen)m(tered)e(co)s(ordinates,)1572 4958 y Fj(p)1621 4907 y Fp(\()p Fg(q)r Fp(\))1621 4981 y Fg(i)1742 4958 y Fq(=)27 b Fj(@)5 b(T)11 b(=@)5 b(q)2117 4973 y Fg(i)2180 4958 y Fj(:)1399 b Fq(\(34\))0 5143 y(As)48 b Fj(q)202 5158 y Fp(3)294 5143 y Fq(=)k Fj(\034)11 b Fq(,)52 b Fj(p)603 5092 y Fp(\()p Fg(q)r Fp(\))603 5165 y(3)748 5143 y Fq(=)g(1)47 b(along)f(the)h(reference)i(ra)m(y)-8 b(.)88 b(The)48 b(sup)s(erscript)g(\()p Fj(q)t Fq(\))f(indicates)f (that)h(the)0 5263 y(quan)m(tities)32 b(are)h(expressed)i(in)d(ra)m (y-cen)m(tered)j(co)s(ordinates.)146 5454 y(The)52 b(dynamic)d(ra)m(y)i (tracing)f(system)h(for)f(smo)s(oth)f(anisotropic)g(inhomogeneous)h (media)f(in)0 5575 y(ra)m(y-cen)m(tered)g(co)s(ordinates)d(for)g(the)h (3)31 b Fn(\002)h Fq(1)47 b(matrices)e Fl(Q)2220 5539 y Fp(\()p Fg(q)r Fp(\))2360 5575 y Fq(and)i Fl(P)2641 5539 y Fp(\()p Fg(q)r Fp(\))2780 5575 y Fq(w)m(as)g(deriv)m(ed)h(b)m(y) f(Klime)-5 b(\024)-44 b(s)1841 5824 y(95)p eop %%Page: 96 10 96 9 bop 0 100 a Fq(\(1994,2006\),)42 b(see)h(also)947 74 y(\024)936 100 y(Cerv)m(en)q(\023)-50 b(y)45 b(\(2001,)e(sec.)72 b(4.2.4/3\).)f(It)42 b(can)g(b)s(e)g(decoupled)g(in)m(to)g(t)m(w)m(o)g (sub-)0 234 y(systems.)79 b(The)45 b(\014rst)g(subsystem)h(for)e(the)g (2)30 b Fn(\002)g Fq(1)44 b(matrices)f Fh(Q)2449 192 y Fp(\()p Fg(q)r Fp(\))2589 234 y Fn(\021)48 b Fq(\()p Fj(Q)2829 183 y Fp(\()p Fg(q)r Fp(\))2829 255 y(1)2922 234 y Fj(;)17 b(Q)3043 183 y Fp(\()p Fg(q)r Fp(\))3043 255 y(2)3136 234 y Fq(\))3174 197 y Fg(T)3273 234 y Fq(and)44 b Fh(P)3562 192 y Fp(\()p Fg(q)r Fp(\))3702 234 y Fn(\021)0 368 y Fq(\()p Fj(P)115 317 y Fp(\()p Fg(q)r Fp(\))101 389 y(1)207 368 y Fj(;)17 b(P)328 317 y Fp(\()p Fg(q)r Fp(\))314 389 y(2)420 368 y Fq(\))458 331 y Fg(T)546 368 y Fq(consists)33 b(of)f(four)g(equations)1079 588 y(d)p Fh(Q)1220 545 y Fp(\()p Fg(q)r Fp(\))1313 588 y Fj(=)p Fq(d)p Fj(\034)95 b Fq(=)82 b Fh(A)1798 545 y Fp(\()p Fg(q)r Fp(\))1891 588 y Fh(Q)1978 545 y Fp(\()p Fg(q)r Fp(\))2093 588 y Fq(+)22 b Fh(B)2282 545 y Fp(\()p Fg(q)r Fp(\))2375 588 y Fh(P)2463 545 y Fp(\()p Fg(q)r Fp(\))2588 588 y Fj(;)1078 733 y Fq(d)p Fh(P)1220 691 y Fp(\()p Fg(q)r Fp(\))1313 733 y Fj(=)p Fq(d)p Fj(\034)95 b Fq(=)82 b Fn(\000)p Fh(C)1877 691 y Fp(\()p Fg(q)r Fp(\))1970 733 y Fh(Q)2057 691 y Fp(\()p Fg(q)r Fp(\))2172 733 y Fn(\000)22 b Fh(D)2368 691 y Fp(\()p Fg(q)r Fp(\))2461 733 y Fh(P)2549 691 y Fp(\()p Fg(q)r Fp(\))2674 733 y Fj(;)905 b Fq(\(35\))0 953 y(where)36 b Fh(A)370 911 y Fp(\()p Fg(q)r Fp(\))463 953 y Fj(;)51 b Fh(B)632 911 y Fp(\()p Fg(q)r Fp(\))725 953 y Fj(;)g Fh(C)891 911 y Fp(\()p Fg(q)r Fp(\))984 953 y Fq(,)35 b(and)f Fh(D)1334 911 y Fp(\()p Fg(q)r Fp(\))1461 953 y Fq(are)g(2)24 b Fn(\002)f Fq(2)34 b(matrices,)g(calculated)g(from)f(the)h(3)24 b Fn(\002)f Fq(3)34 b(matrices)0 1073 y Fl(A)85 1037 y Fp(\()p Fg(x)p Fp(\))183 1073 y Fj(;)50 b Fl(B)340 1037 y Fp(\()p Fg(x)p Fp(\))438 1073 y Fj(;)f Fl(C)595 1037 y Fp(\()p Fg(x)p Fp(\))694 1073 y Fq(,)33 b(and)f Fl(D)1029 1037 y Fp(\()p Fg(x)p Fp(\))1160 1073 y Fq(as)h(follo)m(ws:) 827 1292 y Fh(A)914 1250 y Fp(\()p Fg(q)r Fp(\))1034 1292 y Fq(=)28 b Fl(f)1182 1256 y Fg(T)1238 1292 y Fl(A)1323 1256 y Fp(\()p Fg(x)p Fp(\))1421 1292 y Fl(e)22 b Fq(+)g Fh(d)32 b Fj(;)287 b Fh(B)2090 1250 y Fp(\()p Fg(q)r Fp(\))2211 1292 y Fq(=)27 b Fl(f)2358 1256 y Fg(T)2414 1292 y Fl(B)2494 1256 y Fp(\()p Fg(x)p Fp(\))2592 1292 y Fl(f)43 b Fj(;)827 1420 y Fh(C)916 1378 y Fp(\()p Fg(q)r Fp(\))1036 1420 y Fq(=)28 b Fl(e)1191 1384 y Fg(T)1246 1420 y Fq(\()p Fl(C)1365 1384 y Fp(\()p Fg(x)p Fp(\))1486 1420 y Fn(\000)22 b Fh(\021)1649 1384 y Fg(T)1704 1420 y Fh(\021)s Fq(\))p Fl(e)33 b Fj(;)83 b Fh(D)2096 1378 y Fp(\()p Fg(q)r Fp(\))2216 1420 y Fq(=)28 b Fl(e)2371 1384 y Fg(T)2426 1420 y Fl(D)2512 1384 y Fp(\()p Fg(x)p Fp(\))2611 1420 y Fl(f)k Fn(\000)23 b Fh(d)2838 1377 y Fg(T)2925 1420 y Fj(;)3606 1353 y Fq(\(36\))0 1634 y(and)33 b Fh(d)f Fq(is)g(a)g(2)22 b Fn(\002)h Fq(2)32 b(matrix)f(giv)m(en)i(b)m(y)g(the)g(relation:)1583 1854 y Fh(d)28 b Fq(=)f Fl(f)1819 1812 y Fg(T)1875 1854 y Fq(d)p Fl(e)p Fj(=)p Fq(d)p Fj(\034)44 b(:)1410 b Fq(\(37\))0 2074 y(Here)32 b(the)f(3)18 b Fn(\002)i Fq(3)30 b(matrices)g Fl(A)1112 2037 y Fp(\()p Fg(x)p Fp(\))1211 2074 y Fq(,)h Fl(B)1349 2037 y Fp(\()p Fg(x)p Fp(\))1447 2074 y Fq(,)h Fl(C)1587 2037 y Fp(\()p Fg(x)p Fp(\))1716 2074 y Fq(and)f Fl(D)1990 2037 y Fp(\()p Fg(x)p Fp(\))2120 2074 y Fq(are)g(giv)m(en)g (b)m(y)h(\(5\),)e(and)h Fh(\021)k Fq(b)m(y)d(\(2\).)42 b(Finally)-8 b(,)0 2194 y Fl(e)33 b Fq(and)f Fl(f)43 b Fq(are)33 b(3)22 b Fn(\002)g Fq(2)33 b(matrices,)1294 2414 y Fl(e)28 b Fq(=)f(\()p Fl(e)1565 2429 y Fp(1)1605 2414 y Fj(;)17 b Fl(e)1700 2429 y Fp(2)1739 2414 y Fq(\))33 b Fj(;)146 b Fl(f)38 b Fq(=)28 b(\()p Fl(f)2231 2429 y Fp(1)2271 2414 y Fj(;)17 b Fl(f)2349 2429 y Fp(2)2388 2414 y Fq(\))32 b Fj(:)1121 b Fq(\(38\))146 2718 y(The)40 b(second)g(subsystem)g(consists)g(of)e(t)m(w)m(o)h(equations)g(for)f Fj(Q)2498 2668 y Fp(\()p Fg(q)r Fp(\))2498 2740 y(3)2630 2718 y Fq(and)h Fj(P)2903 2668 y Fp(\()p Fg(q)r Fp(\))2889 2740 y(3)2995 2718 y Fq(,)h(and)f(is)f(extremely)0 2839 y(simple,)1120 2959 y(d)p Fj(Q)1251 2908 y Fp(\()p Fg(q)r Fp(\))1251 2981 y(3)1344 2959 y Fj(=)p Fq(d)p Fj(\034)h Fq(=)28 b Fj(P)1709 2908 y Fp(\()p Fg(q)r Fp(\))1695 2981 y(3)1834 2959 y Fj(;)179 b Fq(d)p Fj(P)2171 2908 y Fp(\()p Fg(q)r Fp(\))2157 2981 y(3)2264 2959 y Fj(=)p Fq(d)p Fj(\034)39 b Fq(=)27 b(0)33 b Fj(:)946 b Fq(\(39\))0 3134 y(The)34 b(analytic)d(solution)g(of)h(\(39\))g(is)625 3367 y Fj(Q)702 3316 y Fp(\()p Fg(q)r Fp(\))702 3389 y(3)795 3367 y Fq(\()p Fj(\034)11 b Fq(\))28 b(=)g Fj(Q)1133 3316 y Fp(\()p Fg(q)r Fp(\))1133 3389 y(3)1226 3367 y Fq(\()p Fj(\034)1306 3382 y Fp(0)1346 3367 y Fq(\))22 b(+)g Fj(P)1581 3316 y Fp(\()p Fg(q)r Fp(\))1567 3389 y(3)1673 3367 y Fq(\()p Fj(\034)1753 3382 y Fp(0)1793 3367 y Fq(\)\()p Fj(\034)34 b Fn(\000)22 b Fj(\034)2086 3382 y Fp(0)2126 3367 y Fq(\))32 b Fj(;)115 b(P)2415 3316 y Fp(\()p Fg(q)r Fp(\))2401 3389 y(3)2507 3367 y Fq(\()p Fj(\034)11 b Fq(\))28 b(=)g Fj(P)2845 3316 y Fp(\()p Fg(q)r Fp(\))2831 3389 y(3)2937 3367 y Fq(\()p Fj(\034)3017 3382 y Fp(0)3057 3367 y Fq(\))k Fj(:)452 b Fq(\(40\))0 3587 y(It)30 b(should)f(b)s(e)h(noted)g(that)f(the)h (constrain)m(t)g(relation,)e(analogous)h(to)g(\(6\),)h(is)f(trivial)e (in)i(ra)m(y-cen)m(tered)0 3708 y(co)s(ordinates,)1621 3828 y Fj(P)1698 3777 y Fp(\()p Fg(q)r Fp(\))1684 3850 y(3)1790 3828 y Fq(\()p Fj(\034)11 b Fq(\))28 b(=)g(0)k Fj(;)1447 b Fq(\(41\))0 4016 y(and,)32 b(therefore,)h Fj(Q)729 3965 y Fp(\()p Fg(q)r Fp(\))729 4038 y(3)823 4016 y Fq(\()p Fj(\034)11 b Fq(\))32 b(is)g(constan)m(t)h(along)d(the)j (ra)m(y)f(in)g(an)m(y)g(smo)s(oth)g(part)g(of)f(the)i(medium.)42 b(The)0 4136 y(constrain)m(t)24 b(relation)f(\(41\))h(applies)f(to)i (the)g(subsystem)h(\(39\))e(only)-8 b(.)40 b(There)26 b(is)e(no)g(constrain)m(t)g(imp)s(osed)0 4257 y(on)32 b(the)h(subsystem)i(\(35\).)146 4448 y(Note)g(that)f(the)g(DR)-8 b(T)34 b(subsystem)i(\(35\))e(remains)f(v)-5 b(alid)32 b(ev)m(en)k(when)g Fh(Q)2858 4406 y Fp(\()p Fg(q)r Fp(\))2985 4448 y Fq(and)e Fh(P)3264 4406 y Fp(\()p Fg(q)r Fp(\))3391 4448 y Fq(represen)m(t)0 4568 y(2)22 b Fn(\002)h Fq(2)32 b(matrices)418 4846 y Fh(Q)504 4804 y Fp(\()p Fg(q)r Fp(\))625 4846 y Fq(=)728 4700 y Ff( )836 4785 y Fj(@)5 b(q)935 4800 y Fp(1)975 4785 y Fj(=@)g(\015)1131 4800 y Fp(1)1254 4785 y Fj(@)g(q)1353 4800 y Fp(1)1394 4785 y Fj(=@)g(\015)1550 4800 y Fp(2)836 4906 y Fj(@)g(q)935 4921 y Fp(2)975 4906 y Fj(=@)g(\015)1131 4921 y Fp(1)1254 4906 y Fj(@)g(q)1353 4921 y Fp(2)1394 4906 y Fj(=@)g(\015)1550 4921 y Fp(2)1631 4700 y Ff(!)1746 4846 y Fj(;)114 b Fh(P)1975 4804 y Fp(\()p Fg(q)r Fp(\))2095 4846 y Fq(=)2199 4700 y Ff( )2306 4785 y Fj(@)5 b(p)2411 4734 y Fp(\()p Fg(q)r Fp(\))2411 4807 y(1)2505 4785 y Fj(=@)g(\015)2661 4800 y Fp(1)2784 4785 y Fj(@)g(p)2889 4734 y Fp(\()p Fg(q)r Fp(\))2889 4807 y(1)2983 4785 y Fj(=@)g(\015)3139 4800 y Fp(2)2306 4922 y Fj(@)g(p)2411 4871 y Fp(\()p Fg(q)r Fp(\))2411 4944 y(2)2505 4922 y Fj(=@)g(\015)2661 4937 y Fp(1)2784 4922 y Fj(@)g(p)2889 4871 y Fp(\()p Fg(q)r Fp(\))2889 4944 y(2)2983 4922 y Fj(=@)g(\015)3139 4937 y Fp(2)3220 4700 y Ff(!)3335 4846 y Fj(;)244 b Fq(\(42\))0 5141 y(where)29 b Fj(\015)328 5156 y Fp(1)395 5141 y Fq(and)f Fj(\015)631 5156 y Fp(2)698 5141 y Fq(are)g(t)m(w)m(o)h(ra)m (y)f(parameters)g(\(corresp)s(onding)g(to)g(an)g(orthonomic)e(system)j (of)f(ra)m(ys\).)146 5332 y(In)49 b(isotropic)f(inhomogeneous)g(media,) k(the)d(arclength)f Fj(s)h Fq(along)e(the)i(reference)i(ra)m(y)e(\012)g (has)0 5452 y(b)s(een)40 b(traditionally)35 b(used)40 b(as)f(the)g Fj(q)1376 5467 y Fp(3)1416 5452 y Fq(-co)s(ordinate)e(in)h (the)i(ra)m(y-cen)m(tered)g(co)s(ordinate)e(system)i Fl(q)f Fq(=)0 5573 y(\()p Fj(q)81 5588 y Fp(1)121 5573 y Fj(;)17 b(q)208 5588 y Fp(2)247 5573 y Fj(;)g(q)334 5588 y Fp(3)373 5573 y Fq(\);)36 b(see)g(P)m(op)s(o)m(v)g(and)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)36 b(\(1978a,b\),)1961 5548 y(\024)1950 5573 y(Cerv)m(en)q(\023)-50 b(y,)38 b(Klime)-5 b(\024)-44 b(s)33 b(and)i(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)36 b(\(1988\).)49 b(This)1841 5824 y(96)p eop %%Page: 97 11 97 10 bop 0 100 a Fq(c)m(hoice)37 b(do)s(es)h(not)f(in\015uence)h(the)g (\014rst)f(subsystem)i(\(35\))d(of)h(the)h(dynamic)e(ra)m(y)i(tracing)e (system)i(at)0 220 y(all,)31 b(but)h(only)g(its)h(second)g(subsystem)i (\(39\).)43 b(F)-8 b(or)31 b(more)h(details,)g(see)i(Klime)-5 b(\024)-44 b(s)30 b(\(2006\).)146 411 y(Note)41 b(that)f(the)g (equations)h(for)f(the)g(transformation)f(of)g Fl(Q)2436 375 y Fp(\()p Fg(q)r Fp(\))2570 411 y Fq(and)h Fl(P)2844 375 y Fp(\()p Fg(q)r Fp(\))2977 411 y Fq(across)h(a)f(structural)0 532 y(in)m(terface,)53 b(analogous)47 b(to)h(\(16\),)k(i.e.)90 b(the)49 b(expressions)h(for)e(the)h(in)m(terface)g(propagator)e (matrix)0 652 y Fl(\005)88 610 y Fp(\()p Fg(q)r Fp(\))181 652 y Fq(\()5 b(~)-54 b Fj(\034)261 667 y Fp(\006)316 652 y Fj(;)17 b(\034)402 667 y Fp(\006)457 652 y Fq(\))28 b(in)g(ra)m(y-cen)m(tered)i(co)s(ordinates,)f(ha)m(v)m(e)h(not)e(y)m (et)i(b)s(een)f(published.)42 b(They)30 b(are,)f(ho)m(w)m(ev)m(er,)0 772 y(deriv)m(ed)k(here)h(as)f(a)f(b)m(y-pro)s(duct)h(of)f(our)h (treatmen)m(t,)f(see)i(Section)e(6.)146 964 y(No)m(w)i(w)m(e)g(in)m (tro)s(duce)e(the)i(6)22 b Fn(\002)h Fq(6)32 b(ra)m(y)h(propagator)f (matrix)g Fl(\005)2439 922 y Fp(\()p Fg(q)r Fp(\))2532 964 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2704 979 y Fp(0)2744 964 y Fq(\))32 b(in)g(ra)m(y-cen)m(tered)j(co)s(ordi-)0 1084 y(nates,)g(corresp)s(onding)f(to)g(the)h(DR)-8 b(T)34 b(system)h(\(35\))e(\(with)h(\(42\)\))g(and)g(\(39\).)48 b(It)34 b(has)g(trivial)e(en)m(tries)0 1204 y(in)g(the)h(third)f(and)g (sixth)h(columns)f(and)h(ro)m(ws.)44 b(Using)32 b(\(39\))g({)g(\(41\),) g(w)m(e)i(obtain)737 1619 y Fl(\005)825 1577 y Fp(\()p Fg(q)r Fp(\))918 1619 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1090 1634 y Fp(0)1130 1619 y Fq(\))27 b(=)1299 1349 y Ff(0)1299 1495 y(B)1299 1545 y(B)1299 1594 y(B)1299 1644 y(B)1299 1697 y(@)1413 1438 y Fh(Q)1499 1387 y Fp(\()p Fg(q)r Fp(\))1499 1460 y(1)1592 1438 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1764 1453 y Fp(0)1804 1438 y Fq(\))84 b Fc(0)96 b Fh(Q)2168 1387 y Fp(\()p Fg(q)r Fp(\))2168 1460 y(2)2261 1438 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2433 1453 y Fp(0)2472 1438 y Fq(\))85 b Fc(0)1413 1558 y(0)1485 1520 y Fg(T)1926 1558 y Fq(1)106 b Fc(0)2153 1520 y Fg(T)2595 1558 y Fj(\034)33 b Fn(\000)23 b Fj(\034)2812 1573 y Fp(0)1413 1695 y Fh(P)1501 1644 y Fp(\()p Fg(q)r Fp(\))1501 1717 y(1)1594 1695 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1766 1710 y Fp(0)1805 1695 y Fq(\))83 b Fc(0)96 b Fh(P)2169 1644 y Fp(\()p Fg(q)r Fp(\))2169 1717 y(2)2262 1695 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2434 1710 y Fp(0)2474 1695 y Fq(\))83 b Fc(0)1413 1816 y(0)1485 1778 y Fg(T)1926 1816 y Fq(0)106 b Fc(0)2153 1778 y Fg(T)2595 1816 y Fq(1)2893 1349 y Ff(1)2893 1495 y(C)2893 1545 y(C)2893 1594 y(C)2893 1644 y(C)2893 1697 y(A)3015 1619 y Fj(:)564 b Fq(\(43\))0 2060 y(Here)36 b Fh(Q)320 2009 y Fp(\()p Fg(q)r Fp(\))320 2082 y(1)449 2060 y Fq(and)f Fh(Q)728 2009 y Fp(\()p Fg(q)r Fp(\))728 2082 y(2)857 2060 y Fq(are)g(2)24 b Fn(\002)h Fq(2)36 b(matrices)e(corresp)s(onding)i(to)f Fh(Q)2514 2018 y Fp(\()p Fg(q)r Fp(\))2643 2060 y Fq(in)f(\(42\),)i (and)g Fh(P)3277 2009 y Fp(\()p Fg(q)r Fp(\))3277 2082 y(1)3406 2060 y Fq(and)g Fh(P)3687 2009 y Fp(\()p Fg(q)r Fp(\))3687 2082 y(2)0 2181 y Fq(are)g(2)25 b Fn(\002)g Fq(2)36 b(matrices)g(corresp)s(onding)g(to)g Fh(P)1663 2139 y Fp(\()p Fg(q)r Fp(\))1792 2181 y Fq(in)g(\(42\),)h Fc(0)47 b Fq(=)34 b(\(0)p Fj(;)17 b Fq(0\))2582 2145 y Fg(T)2636 2181 y Fq(.)55 b(F)-8 b(or)36 b(a)g(detailed)f(deriv)-5 b(ation)0 2301 y(of)36 b(\(43\))g(see)i(the)e(pioneering)g(pap)s(er)g (b)m(y)i(Klime)-5 b(\024)-44 b(s)34 b(\(1994\).)55 b(Subscripts)37 b(1)f(and)h(2)f(of)g Fh(Q)3276 2259 y Fp(\()p Fg(q)r Fp(\))3405 2301 y Fq(and)h Fh(P)3687 2259 y Fp(\()p Fg(q)r Fp(\))0 2435 y Fq(distinguish)31 b(di\013eren)m(t)i(initial)c (conditions)i(of)i Fh(Q)1825 2384 y Fp(\()p Fg(q)r Fp(\))1825 2457 y(1)1918 2435 y Fq(,)g Fh(P)2066 2384 y Fp(\()p Fg(q)r Fp(\))2066 2457 y(1)2191 2435 y Fq(and)g(of)f Fh(Q)2578 2384 y Fp(\()p Fg(q)r Fp(\))2578 2457 y(2)2671 2435 y Fq(,)h Fh(P)2819 2384 y Fp(\()p Fg(q)r Fp(\))2819 2457 y(2)2912 2435 y Fq(:)146 2637 y(a\))27 b Fh(Q)347 2586 y Fp(\()p Fg(q)r Fp(\))347 2659 y(1)440 2637 y Fq(\()p Fj(\034)6 b(;)17 b(\034)612 2652 y Fp(0)651 2637 y Fq(\))28 b(and)f Fh(P)989 2586 y Fp(\()p Fg(q)r Fp(\))989 2659 y(1)1082 2637 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1254 2652 y Fp(0)1294 2637 y Fq(\))27 b(are)g(2)12 b Fn(\002)g Fq(2)27 b(solutions)f(of)h(\(35\))g(with)g(the)h(plane-w)m(a)m(v)m (efron)m(t)g(initial)0 2757 y(conditions)k(at)g Fj(\034)39 b Fq(=)28 b Fj(\034)813 2772 y Fp(0)852 2757 y Fq(:)1380 2878 y Fh(Q)1467 2827 y Fp(\()p Fg(q)r Fp(\))1467 2899 y(1)1587 2878 y Fq(=)g Fh(I)40 b Fj(;)147 b Fh(P)2044 2827 y Fp(\()p Fg(q)r Fp(\))2044 2899 y(1)2164 2878 y Fq(=)28 b Fc(0)45 b Fj(:)1207 b Fq(\(44\))146 3136 y(b\))37 b Fh(Q)362 3085 y Fp(\()p Fg(q)r Fp(\))362 3158 y(2)455 3136 y Fq(\()p Fj(\034)6 b(;)17 b(\034)627 3151 y Fp(0)666 3136 y Fq(\))37 b(and)g Fh(P)1023 3085 y Fp(\()p Fg(q)r Fp(\))1023 3158 y(2)1116 3136 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1288 3151 y Fp(0)1328 3136 y Fq(\))36 b(are)h(2)25 b Fn(\002)h Fq(2)37 b(solutions)e(of)i(\(35\))f(with)h(the)g(p)s(oin)m (t)f(source)i(initial)0 3257 y(conditions)32 b(at)g Fj(\034)39 b Fq(=)28 b Fj(\034)813 3272 y Fp(0)852 3257 y Fq(:)1380 3377 y Fh(Q)1467 3326 y Fp(\()p Fg(q)r Fp(\))1467 3399 y(2)1587 3377 y Fq(=)g Fc(0)45 b Fj(;)147 b Fh(P)2057 3326 y Fp(\()p Fg(q)r Fp(\))2057 3399 y(2)2178 3377 y Fq(=)27 b Fh(I)40 b Fj(:)1207 b Fq(\(45\))146 3622 y(As)38 b(w)m(e)g(can)f(simply)e(see)j(from)e(\(43\),)h(the)g(4)25 b Fn(\002)h Fq(4)36 b(ra)m(y)i(propagator)e(matrix)f Fc(\005)3114 3580 y Fp(\()p Fg(q)r Fp(\))3207 3622 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3379 3637 y Fp(0)3419 3622 y Fq(\))36 b(in)h(ra)m(y-)0 3743 y(cen)m(tered)30 b(co)s(ordinates)d Fj(q)939 3758 y Fp(1)979 3743 y Fj(;)17 b(q)1066 3758 y Fp(2)1105 3743 y Fq(,)29 b(corresp)s(onding)f(to)f(the) i(DR)-8 b(T)27 b(system)i(\(35\))e(is)g(giv)m(en)h(b)m(y)h(the)f (relation)983 4037 y Fc(\005)1089 3995 y Fp(\()p Fg(q)r Fp(\))1182 4037 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1354 4052 y Fp(0)1393 4037 y Fq(\))28 b(=)1563 3891 y Ff( )1670 3976 y Fh(Q)1756 3925 y Fp(\()p Fg(q)r Fp(\))1756 3998 y(1)1849 3976 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2021 3991 y Fp(0)2061 3976 y Fq(\))84 b Fh(Q)2270 3925 y Fp(\()p Fg(q)r Fp(\))2270 3998 y(2)2363 3976 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2535 3991 y Fp(0)2574 3976 y Fq(\))1670 4113 y Fh(P)1758 4062 y Fp(\()p Fg(q)r Fp(\))1758 4135 y(1)1851 4113 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2023 4128 y Fp(0)2062 4113 y Fq(\))83 b Fh(P)2271 4062 y Fp(\()p Fg(q)r Fp(\))2271 4135 y(2)2364 4113 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2536 4128 y Fp(0)2576 4113 y Fq(\))2655 3891 y Ff(!)2770 4037 y Fj(:)809 b Fq(\(46\))0 4349 y(The)38 b(4)24 b Fn(\002)h Fq(4)37 b(ra)m(y)g(propagator)e(matrix)g Fc(\005)1568 4307 y Fp(\()p Fg(q)r Fp(\))1661 4349 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1833 4364 y Fp(0)1872 4349 y Fq(\))37 b(\(46\))f(is)g(obtained)g(from)f(the)i(6)25 b Fn(\002)g Fq(6)36 b(ra)m(y)h(propa-)0 4469 y(gator)c(matrix)g(\(43\)) h(b)m(y)h(simply)d(crossing)j(the)f(third)g(and)g(sixth)g(columns)g (and)g(ro)m(ws.)50 b(The)35 b(partial)0 4590 y(deriv)-5 b(ativ)m(es)37 b(with)f(resp)s(ect)j(to)d Fj(\015)1229 4605 y Fp(1)1305 4590 y Fq(and)h Fj(\015)1550 4605 y Fp(2)1626 4590 y Fq(in)f(the)h(2)25 b Fn(\002)h Fq(2)36 b(submatrices)h(of)f Fc(\005)2942 4548 y Fp(\()p Fg(q)r Fp(\))3035 4590 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3207 4605 y Fp(0)3246 4590 y Fq(\))37 b(in)f(\(46\))g(are)0 4710 y(tak)m(en)e(for)e(a)g(\014xed)i(tra)m(v)m(el)e(time)g Fj(T)41 b Fq(=)27 b Fj(\034)11 b Fq(.)146 4901 y(Analogously)37 b(to)h(\(7\)-\(12\),)g(the)h(6)25 b Fn(\002)i Fq(6)37 b(and)i(4)25 b Fn(\002)i Fq(4)38 b(ra)m(y)g(propagator)f(matrices)g (\(43\))h(and)g(\(46\))0 5022 y(in)c(ra)m(y-cen)m(tered)i(co)s (ordinates)e(satisfy)g(the)h(symplectic)e(prop)s(ert)m(y)-8 b(,)36 b(con)m(tin)m(uation)d(prop)s(ert)m(y)-8 b(,)36 b(c)m(hain)0 5142 y(prop)s(ert)m(y)-8 b(,)33 b(etc.)44 b(The)34 b(con)m(tin)m(uation)e(prop)s(ert)m(y)h(for)f(the)h(4)22 b Fn(\002)g Fq(4)33 b(matrix)e(\(46\))h(reads:)1014 5282 y Ff( )1122 5367 y Fh(Q)1208 5325 y Fp(\()p Fg(q)r Fp(\))1301 5367 y Fq(\()p Fj(\034)11 b Fq(\))1122 5495 y Fh(P)1210 5453 y Fp(\()p Fg(q)r Fp(\))1303 5495 y Fq(\()p Fj(\034)g Fq(\))1473 5282 y Ff(!)1567 5428 y Fq(=)27 b Fc(\005)1777 5386 y Fp(\()p Fg(q)r Fp(\))1870 5428 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2042 5443 y Fp(0)2081 5428 y Fq(\))2136 5282 y Ff( )2243 5367 y Fh(Q)2330 5325 y Fp(\()p Fg(q)r Fp(\))2423 5367 y Fq(\()p Fj(\034)2503 5382 y Fp(0)2542 5367 y Fq(\))2243 5495 y Fh(P)2331 5453 y Fp(\()p Fg(q)r Fp(\))2424 5495 y Fq(\()p Fj(\034)2504 5510 y Fp(0)2544 5495 y Fq(\))2623 5282 y Ff(!)2738 5428 y Fj(;)841 b Fq(\(47\))1841 5824 y(97)p eop %%Page: 98 12 98 11 bop 0 100 a Fq(whic)m(h)33 b(can)g(also)f(b)s(e)g(expressed)k(in) c(the)h(follo)m(wing)c(form:)989 230 y Ff( )1096 314 y Fj(\016)t Fc(q)10 b Fq(\()p Fj(\034)h Fq(\))1096 437 y Fj(\016)t Fc(p)1210 400 y Fp(\()p Fg(q)r Fp(\))1303 437 y Fq(\()p Fj(\034)g Fq(\))1473 230 y Ff(!)1567 376 y Fq(=)27 b Fc(\005)1777 334 y Fp(\()p Fg(q)r Fp(\))1870 376 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2042 391 y Fp(0)2081 376 y Fq(\))2136 230 y Ff( )2243 314 y Fj(\016)t Fc(q)10 b Fq(\()p Fj(\034)2433 329 y Fp(0)2473 314 y Fq(\))2243 437 y Fj(\016)t Fc(p)2357 400 y Fp(\()p Fg(q)r Fp(\))2450 437 y Fq(\()p Fj(\034)2530 452 y Fp(0)2569 437 y Fq(\))2649 230 y Ff(!)2764 376 y Fj(;)815 b Fq(\(48\))0 685 y(Here)38 b(the)f(2)25 b Fn(\002)h Fq(1)37 b(v)m(ectors)h Fj(\016)t Fc(q)11 b Fq(\()p Fj(\034)g Fq(\))36 b Fn(\021)f Fq(\()p Fj(\016)t(q)1521 700 y Fp(1)1561 685 y Fq(\()p Fj(\034)11 b Fq(\))p Fj(;)17 b(\016)t(q)1824 700 y Fp(2)1864 685 y Fq(\()p Fj(\034)11 b Fq(\)\))2031 649 y Fg(T)2123 685 y Fq(and)37 b Fj(\016)t Fc(p)2431 649 y Fp(\()p Fg(q)r Fp(\))2524 685 y Fq(\()p Fj(\034)11 b Fq(\))36 b Fn(\021)f Fq(\()p Fj(\016)t(p)2935 634 y Fp(\()p Fg(q)r Fp(\))2935 706 y(1)3028 685 y Fq(\()p Fj(\034)11 b Fq(\))p Fj(;)17 b(\016)t(p)3297 634 y Fp(\()p Fg(q)r Fp(\))3297 706 y(2)3390 685 y Fq(\()p Fj(\034)11 b Fq(\)\))3557 649 y Fg(T)3649 685 y Fq(are)0 805 y(small)32 b(p)s(erturbations)j(of)f Fc(q)45 b Fq(and)35 b Fc(p)1339 769 y Fp(\()p Fg(q)r Fp(\))1467 805 y Fq(around)f(the)i(reference)g(ra)m(y)f(\012,)g (measured)h(along)d(the)i(plane)0 925 y(tangen)m(t)28 b(to)g(the)g(w)m(a)m(v)m(efron)m(t)i(at)e(\012.)42 b(They)29 b(can)g(also)e(b)s(e)h(expressed)i(in)e(terms)g(of)f(2)13 b Fn(\002)g Fq(2)27 b(matrices)g Fc(Q)3687 883 y Fp(\()p Fg(q)r Fp(\))0 1046 y Fq(and)33 b Fc(P)278 1004 y Fp(\()p Fg(q)r Fp(\))403 1046 y Fq(as)g(follo)m(ws:)1187 1166 y Fj(\016)t Fc(q)38 b Fq(=)28 b Fc(Q)1523 1124 y Fp(\()p Fg(q)r Fp(\))1616 1166 y Fj(\016)t Fh(\015)38 b Fj(;)115 b(\016)t Fc(p)2015 1125 y Fp(\()p Fg(q)r Fp(\))2136 1166 y Fq(=)28 b Fc(P)2328 1124 y Fp(\()p Fg(q)r Fp(\))2421 1166 y Fj(\016)t Fh(\015)38 b Fj(:)1014 b Fq(\(49\))0 1341 y(Here)36 b Fj(\016)t Fh(\015)j Fn(\021)33 b Fq(\()p Fj(\015)577 1356 y Fp(1)641 1341 y Fn(\000)24 b Fj(\015)793 1356 y Fp(10)868 1341 y Fj(;)17 b(\015)963 1356 y Fp(2)1026 1341 y Fn(\000)24 b Fj(\015)1178 1356 y Fp(20)1253 1341 y Fq(\))1291 1304 y Fg(T)1346 1341 y Fq(,)36 b(and)g Fj(\015)1653 1356 y Fp(10)1727 1341 y Fj(;)17 b(\015)1822 1356 y Fp(20)1932 1341 y Fq(are)35 b(the)h(ra)m(y)g(parameters)g(of)f (the)h(reference)h(ra)m(y)0 1461 y(\012.)146 1652 y(In)j(the)f (solution)f(of)h(DR)-8 b(T)38 b(system)j(\(35\),)f(it)e(is)h(necessary) i(to)e(kno)m(w)h(the)g(basis)f(v)m(ectors)i Fl(e)3606 1667 y Fp(1)3645 1652 y Fj(;)17 b Fl(e)3740 1667 y Fp(2)0 1773 y Fq(and)48 b Fl(f)239 1788 y Fp(1)279 1773 y Fj(;)17 b Fl(f)357 1788 y Fp(2)445 1773 y Fq(at)48 b(an)m(y)h(p)s(oin)m(t)e(of) h(the)h(reference)h(ra)m(y)f(\012,)k(see)c(\(36\).)90 b(These)50 b(basis)f(v)m(ectors)h(can)e(b)s(e)0 1893 y(calculated)33 b(b)m(y)i(solving)e(ordinary)g(di\013eren)m(tial)f (equations)i(of)g(the)g(\014rst)h(order)f(along)e(\012.)48 b(Actually)-8 b(,)0 2013 y(it)36 b(is)h(su\016cien)m(t)i(to)e (calculate)f(only)h(one)g(pair)f(of)h(basis)h(v)m(ectors)g(in)f(this)g (w)m(a)m(y)-8 b(,)40 b(as)d(the)h(second)h(pair)0 2134 y(can)c(b)s(e)g(obtained)f(using)g(\(31\))g(or)g(\(32\).)50 b(The)35 b(relev)-5 b(an)m(t)35 b(di\013eren)m(tial)e(equations)h(for)g Fl(e)3259 2149 y Fp(1)3299 2134 y Fj(;)17 b Fl(e)3394 2149 y Fp(2)3468 2134 y Fq(con)m(tain)0 2254 y(only)38 b(\014rst-order)i(deriv)-5 b(ativ)m(es)39 b(of)f(Hamiltonian,)f(but)i (for)g Fl(f)2263 2269 y Fp(1)2302 2254 y Fj(;)17 b Fl(f)2380 2269 y Fp(2)2459 2254 y Fq(the)39 b(second-order)h(deriv)-5 b(ativ)m(es)39 b(of)0 2374 y(Hamiltonian)28 b(are)j(needed.)45 b(Consequen)m(tly)-8 b(,)34 b(it)c(is)h(simpler)f(and)h(more)g (e\016cien)m(t)h(to)f(calculate)g Fl(e)3606 2389 y Fp(1)3645 2374 y Fj(;)17 b Fl(e)3740 2389 y Fp(2)0 2495 y Fq(b)m(y)46 b(n)m(umerical)d(in)m(tegration)g(along)g(the)j(ra)m(y)-8 b(,)48 b(not)c Fl(f)1989 2510 y Fp(1)2029 2495 y Fj(;)17 b Fl(f)2107 2510 y Fp(2)2146 2495 y Fq(.)80 b(It)45 b(is)g(common)e(to) h(tak)m(e)i(the)f(ordinary)0 2615 y(di\013eren)m(tial)31 b(equation)h(of)g(the)h(\014rst)g(order)g(for)f Fl(e)1821 2630 y Fg(I)1861 2615 y Fj(;)17 b(I)35 b Fq(=)28 b(1)p Fj(;)17 b Fq(2,)32 b(as)g(follo)m(ws:)1276 2835 y(d)p Fl(e)1381 2850 y Fg(I)1421 2835 y Fj(=)p Fq(d)p Fj(\034)39 b Fq(=)28 b Fn(\000)p Fq(\()p Fl(p)1886 2794 y Fg(T)1942 2835 y Fl(p)p Fq(\))2042 2794 y Fe(\000)p Fp(1)2136 2835 y Fq(\()p Fl(e)2225 2794 y Fg(T)2225 2860 y(I)2280 2835 y Fh(\021)t Fq(\))p Fl(p)k Fj(:)1103 b Fq(\(50\))0 3055 y(where)49 b Fh(\021)58 b Fq(=)c Fn(\000)p Fj(@)5 b Fn(H)q Fj(=@)g Fl(x)p Fq(,)54 b(see)49 b(\(2\).)90 b(If)48 b(w)m(e)h(tak)m(e)g Fl(e)1973 3070 y Fp(1)2012 3055 y Fj(;)17 b Fl(e)2107 3070 y Fp(2)2194 3055 y Fq(as)49 b(t)m(w)m(o)f(v)m(ectors)i(p)s(erp)s (endicular)d(to)h(the)0 3176 y(slo)m(wness)38 b(v)m(ector)g Fl(p)f Fq(at)g(the)g(initial)c(p)s(oin)m(t)j(of)h(the)g(ra)m(y)g(\012,) h(then)g(\(50\))e(guaran)m(tees)i(that)f Fl(e)3419 3191 y Fp(1)3495 3176 y Fq(and)g Fl(e)3740 3191 y Fp(2)0 3296 y Fq(are)28 b(p)s(erp)s(endicular)g(to)g Fl(p)h Fq(along)e(the)i(whole) f(ra)m(y)-8 b(.)43 b(The)29 b(basis)f(v)m(ectors)i Fl(e)2676 3311 y Fp(1)2716 3296 y Fj(;)17 b Fl(e)2811 3311 y Fp(2)2879 3296 y Fq(need)29 b(not)g(b)s(e)f(m)m(utually)0 3416 y(p)s(erp)s(endicular)37 b(unit)f(v)m(ectors.)60 b(If)37 b(w)m(e,)i(ho)m(w)m(ev)m(er,)i(tak)m(e)d(them)g(m)m(utually)d(p)s(erp)s (endicular)i(and)g(unit)0 3537 y(at)j(the)g(initial)c(p)s(oin)m(t)k(of) f(the)h(ra)m(y)-8 b(,)43 b(then)d(\(50\))g(also)f(guaran)m(tees)i(that) f Fl(e)2753 3552 y Fp(1)2832 3537 y Fq(and)g Fl(e)3080 3552 y Fp(2)3160 3537 y Fq(ha)m(v)m(e)h(the)g(same)0 3657 y(prop)s(ect)m(y)29 b(along)e(the)h(whole)g(ra)m(y)-8 b(.)42 b(In)29 b(suc)m(h)g(case,)h(it)d(is)h(su\016cien)m(t)h(to)e (solv)m(e)i(\(50\))e(only)h(for)f(one)h(v)m(ector)0 3778 y(of)k Fl(e)162 3793 y Fp(1)234 3778 y Fq(and)h Fl(e)475 3793 y Fp(2)514 3778 y Fq(,)g(and)g(to)f(compute)h(the)g(latter)e (using)h(the)h(orthogonalit)m(y)e(condition)g(of)h Fl(e)3327 3793 y Fp(1)3367 3778 y Fq(,)g Fl(e)3477 3793 y Fp(2)3517 3778 y Fq(,)h Fl(p)p Fq(.)146 3969 y(Let)39 b(us)g(no)m(w)g(brie\015y)g (compare)f(the)h(DR)-8 b(T)38 b(systems)i(\(4\))e(in)f(Cartesian)i(co)s (ordinates)f(and)g(\(35\))0 4089 y(in)43 b(ra)m(y-cen)m(tered)i(co)s (ordinates)f(for)f(anisotropic)f(inhomogeneous)h(medium.)75 b(The)44 b(DR)-8 b(T)44 b(system)0 4210 y(\(35\))32 b(in)h(ra)m(y-cen)m (tered)i(co)s(ordinates)d(consists)i(of)f(four)f(equations)i(only)-8 b(,)32 b(but)i(the)f(DR)-8 b(T)33 b(system)h(\(4\))0 4330 y(in)41 b(Cartesian)g(co)s(ordinates)g(of)h(six)f(equations.)71 b(F)-8 b(rom)40 b(the)i(n)m(umerical)e(p)s(oin)m(t)h(of)g(view,)j(this) e(is)f(a)0 4450 y(great)32 b(adv)-5 b(an)m(tage)32 b(of)g(DR)-8 b(T)32 b(system)h(\(35\).)43 b(In)32 b(other)h(asp)s(ects,)g(the)g(DR) -8 b(T)32 b(system)h(\(4\))f(in)g(Cartesian)0 4571 y(co)s(ordinates)37 b(is)g(simpler)e(than)j(that)f(of)f(\(35\).)57 b(Actually)-8 b(,)38 b(DR)-8 b(T)37 b(system)h(\(4\))f(uses)h(only)f(the)h(3)25 b Fn(\002)h Fq(3)0 4691 y(matrices)42 b Fl(A)486 4655 y Fp(\()p Fg(x)p Fp(\))585 4691 y Fj(;)59 b Fl(B)751 4655 y Fp(\()p Fg(x)p Fp(\))850 4691 y Fj(;)g Fl(C)1017 4655 y Fp(\()p Fg(x)p Fp(\))1159 4691 y Fq(and)43 b Fl(D)1445 4655 y Fp(\()p Fg(x)p Fp(\))1544 4691 y Fq(,)j(see)e(\(5\).)74 b(These)45 b(matrices)d(are)h(needed)i(ev)m(en)g(in)d(DR)-8 b(T)0 4811 y(system)33 b(\(35\),)f(but)h(system)g(\(35\))f(also)g (requires)h(certain)f(additional)e(computations)h(at)h(eac)m(h)h(p)s (oin)m(t)0 4932 y(of)f(the)h(ra)m(y)-8 b(.)44 b(W)-8 b(e)33 b(shall)e(list)g(them)h(brie\015y)h(here:)146 5123 y(1\))39 b(In)g(the)h(solution)d(of)i(DR)-8 b(T)38 b(system)i(\(35\),)g(it)e(is)h(necessary)i(to)e(kno)m(w)h(basis)f(v)m (ectors)i Fl(e)3555 5138 y Fp(1)3594 5123 y Fq(,)g Fl(e)3713 5138 y Fp(2)3752 5123 y Fq(.)0 5243 y(These)f(basis)e(v)m(ectors)h (should)f(b)s(e)h(calculated)e(b)m(y)i(n)m(umerical)d(in)m(tegration)h (along)f(the)j(ra)m(y)f(\012,)i(e.g.)0 5364 y(\(50\).)k(The)34 b(DR)-8 b(T)32 b(system)i(in)e(Cartesian)h(co)s(ordinates)f(\(4\))h(do) s(es)g(not)g(require)g(the)h(computation)d(of)0 5484 y(an)m(y)i(basis)g(v)m(ector.)1841 5824 y(98)p eop %%Page: 99 13 99 12 bop 146 100 a Fq(2\))35 b(If)g(w)m(e)h(determine)f Fl(e)1023 115 y Fp(1)1097 100 y Fq(and)h Fl(e)1341 115 y Fp(2)1415 100 y Fq(b)m(y)g(n)m(umerical)e(in)m(tegration)f(along)h (the)h(ra)m(y)-8 b(,)36 b(e.g.)51 b(using)35 b(\(50\),)0 220 y(w)m(e)f(ha)m(v)m(e)g(to)e(compute)g Fl(f)917 235 y Fp(1)990 220 y Fq(and)g Fl(f)1213 235 y Fp(2)1285 220 y Fq(using)h(\(31\).)146 411 y(3\))i(F)-8 b(urther,)37 b(w)m(e)f(ha)m(v)m(e)h(to)e(compute)h(the)g(2)24 b Fn(\002)g Fq(2)36 b(matrix)e Fc(d)10 b Fq(.)53 b(Using)35 b(\(37\))g(and)g (\(50\),)h(w)m(e)h(obtain)0 532 y(for)32 b(the)h(comp)s(onen)m(ts)g Fj(d)905 547 y Fg(N)7 b(M)1079 532 y Fq(of)32 b Fc(d)11 b Fq(:)1253 711 y Fd(d)1303 726 y Fg(N)c(M)1473 711 y Fq(=)27 b Fn(\000)p Fq(\()p Fl(f)1735 670 y Fg(T)1725 736 y(N)1793 711 y Fl(p)p Fq(\)\()p Fl(e)1982 670 y Fg(T)1982 736 y(M)2061 711 y Fh(\021)t Fq(\))p Fj(=)p Fq(\()p Fl(p)2312 670 y Fg(T)2367 711 y Fl(p)p Fq(\))32 b Fj(:)1080 b Fq(\(51\))146 961 y(4\))22 b(Only)f(then)i(can)f(w)m(e)h(p)s(erform)e(the)i(matrix)d (m)m(ultiplications)e(in)j(\(36\))g(and)h(get)g Fh(A)3190 919 y Fp(\()p Fg(q)r Fp(\))3283 961 y Fj(;)39 b Fh(B)3440 919 y Fp(\()p Fg(q)r Fp(\))3533 961 y Fj(;)f Fh(C)3687 919 y Fp(\()p Fg(q)r Fp(\))0 1082 y Fq(and)33 b Fh(D)286 1040 y Fp(\()p Fg(q)r Fp(\))379 1082 y Fq(.)146 1273 y(Consequen)m(tly)-8 b(,)33 b(the)d(con)m(v)m(en)m(tional)g(DR)-8 b(T)30 b(system)h(\(4\))f(in)f(Cartesian)h(co)s(ordinates)f(is)h (conceptu-)0 1393 y(ally)h(simpler)g(and)i(more)f(straigh)m(tforw)m (ard)g(than)g(the)h(DR)-8 b(T)32 b(system)i(\(35\))e(in)g(ra)m(y-cen)m (tered)i(co)s(ordi-)0 1514 y(nates.)146 1705 y(In)42 b(practical)d(applications,)i(ho)m(w)m(ev)m(er,)46 b(the)41 b(4)28 b Fn(\002)h Fq(4)40 b(DR)-8 b(T)41 b(propagator)f(matrix)f Fc(\005)3315 1663 y Fp(\()p Fg(q)r Fp(\))3408 1705 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3580 1720 y Fp(0)3619 1705 y Fq(\))41 b(in)0 1825 y(ra)m(y-cen)m(tered)29 b(co)s(ordinates)e (\(46\))g(is)g(often)g(more)g(useful)g(and)h(ph)m(ysically)f(more)f (attractiv)m(e)i(than)f(the)0 1946 y(6)15 b Fn(\002)g Fq(6)28 b(DR)-8 b(T)29 b(propagator)f(matrix)f Fl(\005)1367 1904 y Fp(\()p Fg(x)p Fp(\))1465 1946 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1637 1961 y Fp(0)1677 1946 y Fq(\))29 b(in)f(Cartesian)h (co)s(ordinates.)42 b(The)30 b(reason)f(is)f(that)h(the)0 2066 y(general)36 b(system)h(of)f(ra)m(ys)i(in)d(smo)s(oth)h (inhomogeneous)g(media)f(is)h(four-parameteric,)f(so)i(that)f(the)0 2186 y(n)m(um)m(b)s(er)e(of)f(equations)h(corresp)s(onds)h(strictly)d (to)i(the)g(n)m(um)m(b)s(er)g(of)f(free)h(parameters.)46 b(In)34 b(the)g(6)22 b Fn(\002)i Fq(6)0 2307 y(DR)-8 b(T)31 b(propagator)g(matrix)g Fl(\005)1142 2265 y Fp(\()p Fg(x)p Fp(\))1241 2307 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1413 2322 y Fp(0)1452 2307 y Fq(\))32 b(in)f(Cartesian)h(co)s(ordinates,)g (t)m(w)m(o)g(columns)f(are)h(redundan)m(t)0 2427 y(\(ra)m(y-tangen)m(t) h(and)g(non-eik)m(onal)e(solutions\))g(and)i(m)m(ust)g(b)s(e)f(remo)m (v)m(ed.)146 2618 y(In)41 b(the)g(follo)m(wing)d(sections,)43 b(w)m(e)f(prop)s(ose)f(metho)s(ds)f(of)g(a)m(v)m(oiding)g(the)h(DR)-8 b(T)40 b(in)g(ra)m(y-cen)m(tered)0 2739 y(co)s(ordinates)25 b(\(35\))g(in)g(the)h(computation)e(of)i(the)g(4)8 b Fn(\002)g Fq(4)25 b(ra)m(y)i(propagator)d Fc(\005)2782 2697 y Fp(\()p Fg(q)r Fp(\))2875 2739 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3047 2754 y Fp(0)3086 2739 y Fq(\))26 b(in)f(ra)m(y-cen)m (tered)0 2859 y(co)s(ordinates.)42 b(W)-8 b(e)29 b(determine)f Fc(\005)1276 2817 y Fp(\()p Fg(q)r Fp(\))1369 2859 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1541 2874 y Fp(0)1581 2859 y Fq(\))28 b(directly)g(from)g Fl(\005)2312 2817 y Fp(\()p Fg(x)p Fp(\))2411 2859 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2583 2874 y Fp(0)2623 2859 y Fq(\),)29 b(without)f(p)s(erforming)f(DR)-8 b(T)0 2980 y(\(35\))33 b(in)f(ra)m(y-cen)m(tered)k(co)s(ordinates,)d(and)g(without)g(the)h (four)f(computational)e(steps)j(\(1\)-\(4\))e(giv)m(en)0 3100 y(ab)s(o)m(v)m(e.)41 b(W)-8 b(e)24 b(also)e(sho)m(w)i(that)f(it)f (w)m(ould)h(b)s(e)h(su\016cien)m(t)g(to)f(compute)g(only)g(four)g (columns)f(of)h Fl(\005)3431 3058 y Fp(\()p Fg(x)p Fp(\))3530 3100 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3702 3115 y Fp(0)3742 3100 y Fq(\))0 3220 y(if)31 b(w)m(e)j(wished)f(to)g(determine)f Fc(\005)1234 3178 y Fp(\()p Fg(q)r Fp(\))1327 3220 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1499 3235 y Fp(0)1539 3220 y Fq(\))32 b(from)f Fl(\005)1927 3178 y Fp(\()p Fg(x)p Fp(\))2026 3220 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2198 3235 y Fp(0)2237 3220 y Fq(\).)0 3618 y Fk(4)161 b(Relation)37 b(b)t(et)l(w)l(een)c(ra)l(y)j(propagator)f(matrices)g(\005)3311 3558 y Fq(\()p Fj(x)p Fq(\))3447 3618 y Fb(\()p Fa(\034)8 b(;)24 b(\034)3691 3640 y Fq(0)3744 3618 y Fb(\))242 3801 y Fk(and)54 b(\005)680 3741 y Fq(\()p Fj(q)t Fq(\))807 3801 y Fb(\()p Fa(\034)8 b(;)24 b(\034)1051 3822 y Fq(0)1104 3801 y Fb(\))0 4091 y Fq(A)33 b(general)g(relation)e(b)s(et)m(w)m(een) 36 b(the)d(6)23 b Fn(\002)g Fq(6)33 b(ra)m(y)h(propagator)e(matrices,)h (computed)g(along)f(the)i(same)0 4211 y(reference)46 b(ra)m(y)e(\012)g(from)f Fj(\034)998 4226 y Fp(0)1082 4211 y Fq(to)g Fj(\034)11 b Fq(,)48 b(related)43 b(to)h(t)m(w)m(o)g (arbitrary)f(co)s(ordinate)g(systems,)49 b(is)43 b(giv)m(en)h(in)11 4306 y(\024)0 4331 y(Cerv)m(en)q(\023)-50 b(y)38 b(\(2001\),)d(see)i (eq.)53 b(\(4.3.38\).)f(In)36 b(the)g(equation,)g(it)e(is)h(assumed)i (that)e(parameter)g Fj(\034)47 b Fq(along)0 4452 y(the)34 b(ra)m(y)g(is)e(the)i(same)f(in)g(b)s(oth)g(co)s(ordinate)g(systems,)i (and)e(represen)m(ts)j(the)e(tra)m(v)m(el)f(time)f(along)g(the)0 4572 y(ra)m(y)-8 b(.)146 4763 y(First)30 b(w)m(e)i(presen)m(t)h(the)e (relations)f(b)s(et)m(w)m(een)j(the)e(3)18 b Fn(\002)i Fq(1)30 b(matrices)g Fl(Q)2680 4727 y Fp(\()p Fg(x)p Fp(\))2779 4763 y Fq(,)h Fl(P)2914 4727 y Fp(\()p Fg(x)p Fp(\))3044 4763 y Fq(and)g Fl(Q)3316 4727 y Fp(\()p Fg(q)r Fp(\))3409 4763 y Fq(,)g Fl(P)3544 4727 y Fp(\()p Fg(q)r Fp(\))3637 4763 y Fq(.)42 b(A)0 4884 y(suitable)32 b(form)f(of)h(these) i(relations)d(is)h(giv)m(en)h(in)1829 4859 y(\024)1818 4884 y(Cerv)m(en)q(\023)-50 b(y)35 b(\(2001,)c(eq.)45 b(4.3.35\).)d(It)33 b(reads:)849 5063 y Fl(Q)933 5022 y Fp(\()p Fg(q)r Fp(\))1054 5063 y Fq(=)28 b Fl(H)1246 5022 y Fe(\000)p Fp(1)1340 5063 y Fl(Q)1424 5022 y Fp(\()p Fg(x)p Fp(\))1555 5063 y Fj(;)114 b Fl(P)1773 5022 y Fp(\()p Fg(q)r Fp(\))1894 5063 y Fq(=)27 b Fl(H)2085 5022 y Fg(T)2140 5063 y Fl(P)2217 5022 y Fp(\()p Fg(x)p Fp(\))2337 5063 y Fq(+)22 b Fl(FH)2593 5022 y Fe(\000)p Fp(1)2688 5063 y Fl(Q)2772 5022 y Fp(\()p Fg(x)p Fp(\))2903 5063 y Fj(:)676 b Fq(\(52\))0 5242 y(Here)41 b Fl(H)f Fq(is)g(the)h(3)27 b Fn(\002)h Fq(3)40 b(transformation)e(matrix)h(giv) m(en)i(b)m(y)g(\(27\),)h(and)e Fl(F)h Fq(is)f(a)g(3)27 b Fn(\002)h Fq(3)40 b(symmetric)0 5363 y(matrix)31 b(giv)m(en)i(b)m(y)g (the)g(relation)1503 5537 y Fj(F)1566 5552 y Fg(mn)1703 5537 y Fq(=)27 b Fj(p)1855 5552 y Fg(i)1960 5470 y Fj(@)2016 5433 y Fp(2)2057 5470 y Fj(x)2112 5485 y Fg(i)p 1893 5514 314 4 v 1893 5605 a Fj(@)5 b(q)1992 5620 y Fg(m)2060 5605 y Fj(@)g(q)2159 5620 y Fg(n)2250 5537 y Fj(:)1329 b Fq(\(53\))1841 5824 y(99)p eop %%Page: 100 14 100 13 bop 0 100 a Fq(If)33 b(w)m(e)g(tak)m(e)g(in)m(to)f(accoun)m(t)i (\(26\),)e(w)m(e)h(obtain)776 295 y Fj(F)839 310 y Fg(M)7 b(N)1009 295 y Fq(=)28 b(0)k Fj(;)114 b(F)1398 310 y Fg(M)7 b Fp(3)1540 295 y Fq(=)28 b Fj(F)1707 310 y Fp(3)p Fg(M)1849 295 y Fq(=)f Fn(\000)p Fl(e)2080 254 y Fg(T)2080 320 y(M)2160 295 y Fh(\021)36 b Fj(;)114 b(F)2460 310 y Fp(33)2563 295 y Fq(=)27 b Fn(\000)p Fi(U)2825 253 y Fg(T)2880 295 y Fh(\021)36 b Fj(:)603 b Fq(\(54\))0 491 y(Relations)31 b(\(52\))h(can)h(b)s(e)g(expressed)i(in)d(a)g(more)g (compact)g(matrix)f(form)677 603 y Ff( )784 688 y Fl(Q)868 652 y Fp(\()p Fg(q)r Fp(\))784 810 y Fl(P)861 774 y Fp(\()p Fg(q)r Fp(\))1003 603 y Ff(!)1096 749 y Fq(=)d Fl(\011)1287 764 y Fp(1)1343 603 y Ff( )1450 688 y Fl(Q)1534 652 y Fp(\()p Fg(x)p Fp(\))1450 810 y Fl(P)1527 774 y Fp(\()p Fg(x)p Fp(\))1674 603 y Ff(!)1789 749 y Fj(;)1963 603 y Ff( )2070 688 y Fl(Q)2154 652 y Fp(\()p Fg(x)p Fp(\))2070 810 y Fl(P)2147 774 y Fp(\()p Fg(x)p Fp(\))2295 603 y Ff(!)2388 749 y Fq(=)g Fl(\011)2579 764 y Fp(2)2635 603 y Ff( )2742 688 y Fl(Q)2826 652 y Fp(\()p Fg(q)r Fp(\))2742 810 y Fl(P)2819 774 y Fp(\()p Fg(q)r Fp(\))2961 603 y Ff(!)3076 749 y Fj(;)503 b Fq(\(55\))0 1002 y(where)34 b Fl(\011)369 1017 y Fp(1)441 1002 y Fq(and)f Fl(\011)718 1017 y Fp(2)790 1002 y Fq(are)f(6)22 b Fn(\002)h Fq(6)32 b(matrices,)g(giv)m(en)h(b)m(y)g(the)g(relations)716 1255 y Fl(\011)803 1270 y Fp(1)870 1255 y Fq(=)974 1108 y Ff( )1081 1194 y Fl(H)1169 1157 y Fe(\000)p Fp(1)1416 1194 y Fl(0)1081 1314 y(FH)1239 1278 y Fe(\000)p Fp(1)1416 1314 y Fl(H)1504 1278 y Fg(T)1600 1108 y Ff(!)1715 1255 y Fj(;)147 b Fl(\011)1976 1270 y Fp(2)2043 1255 y Fq(=)2147 1108 y Ff( )2254 1194 y Fl(H)341 b(0)2254 1314 y Fn(\000)p Fl(H)2419 1278 y Fe(\000)p Fg(T)2529 1314 y Fl(F)84 b(H)2771 1278 y Fe(\000)p Fg(T)2922 1108 y Ff(!)3037 1255 y Fj(;)542 b Fq(\(56\))0 1507 y(and)45 b Fl(0)f Fq(is)g(a)h(3)30 b Fn(\002)h Fq(3)44 b(n)m(ull)g(matrix.)78 b(Ob)m(viously)-8 b(,)47 b Fl(\011)1961 1522 y Fp(1)2001 1507 y Fl(\011)2088 1522 y Fp(2)2176 1507 y Fq(=)h Fl(I)p Fq(.)79 b(Both)45 b(matrices)e Fl(\011)3193 1522 y Fp(1)3277 1507 y Fq(and)h Fl(\011)3565 1522 y Fp(2)3649 1507 y Fq(are)0 1628 y(symplectic.)146 1819 y(T)-8 b(o)33 b(express)i Fl(\011)717 1834 y Fp(1)789 1819 y Fq(and)e Fl(\011)1066 1834 y Fp(2)1138 1819 y Fq(in)e(a)i(more)f(suitable)f(form,)h(w)m(e)h(in)m(tro)s(duce)g(a)f(3) 22 b Fn(\002)h Fq(3)32 b(matrix)f Fl(V)r Fq(,)1617 2015 y Fl(V)e Fq(=)f Fl(H)1923 1973 y Fe(\000)p Fg(T)2032 2015 y Fl(F)33 b Fj(:)1444 b Fq(\(57\))0 2210 y(The)34 b(columns)d(of)i(matrix)e Fl(V)i Fq(are)g(giv)m(en)g(b)m(y)g (expressions)733 2406 y Fl(V)818 2421 y Fp(1)885 2406 y Fq(=)27 b Fn(\000)p Fq(\()p Fl(e)1154 2365 y Fg(T)1154 2431 y Fp(1)1210 2406 y Fh(\021)s Fq(\))p Fl(p)33 b Fj(;)82 b Fl(V)1600 2421 y Fp(2)1666 2406 y Fq(=)28 b Fn(\000)p Fq(\()p Fl(e)1936 2365 y Fg(T)1936 2431 y Fp(2)1992 2406 y Fh(\021)s Fq(\))p Fl(p)33 b Fj(;)81 b Fl(V)2381 2421 y Fp(3)2448 2406 y Fq(=)28 b Fn(\000)p Fq(\()p Fl(e)2718 2365 y Fg(T)2718 2431 y(m)2785 2406 y Fh(\021)s Fq(\))p Fl(f)2920 2421 y Fg(m)3020 2406 y Fj(:)559 b Fq(\(58\))0 2602 y(The)34 b(6)21 b Fn(\002)i Fq(6)32 b(matrices)g Fl(\011)930 2617 y Fp(1)1002 2602 y Fq(and)h Fl(\011)1279 2617 y Fp(2)1351 2602 y Fq(then)g(read)968 2850 y Fl(\011)1055 2865 y Fp(1)1122 2850 y Fq(=)1225 2704 y Ff( )1333 2789 y Fl(f)1377 2753 y Fg(T)1557 2789 y Fl(0)1333 2910 y(V)1420 2873 y Fg(T)1557 2910 y Fl(e)1608 2873 y Fg(T)1705 2704 y Ff(!)1819 2850 y Fj(;)115 b Fl(\011)2048 2865 y Fp(2)2115 2850 y Fq(=)2219 2704 y Ff( )2326 2789 y Fl(e)195 b(0)2326 2910 y Fn(\000)p Fl(V)84 b(f)2670 2704 y Ff(!)2785 2850 y Fj(;)794 b Fq(\(59\))0 3108 y(where)34 b Fl(V)29 b Fn(\021)f Fq(\()p Fl(V)624 3123 y Fp(1)663 3108 y Fj(;)17 b Fl(V)792 3123 y Fp(2)831 3108 y Fj(;)g Fl(V)960 3123 y Fp(3)998 3108 y Fq(\).)146 3300 y(No)m(w)24 b(w)m(e)g(deriv)m(e)g (the)g(transformation)d(b)s(et)m(w)m(een)k(the)f(6)s Fn(\002)s Fq(6)f(ra)m(y)g(propagator)f(matrices)h Fl(\005)3431 3258 y Fp(\()p Fg(x)p Fp(\))3530 3300 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3702 3315 y Fp(0)3742 3300 y Fq(\))0 3420 y(and)31 b Fl(\005)276 3378 y Fp(\()p Fg(q)r Fp(\))368 3420 y Fq(\()p Fj(\034)6 b(;)17 b(\034)540 3435 y Fp(0)580 3420 y Fq(\).)43 b(Multiplying)28 b(\(9\))i(from)g(the)h(left)f(b)m(y)i Fl(\011)2157 3435 y Fp(1)2196 3420 y Fq(\()p Fj(\034)11 b Fq(\),)32 b(and)f(using)f(the)h(second)h(equation)f(of)0 3540 y(\(55\))h(for)g Fj(\034)397 3555 y Fp(0)437 3540 y Fq(,)g(w)m(e)i(obtain)755 3652 y Ff( )862 3737 y Fl(Q)946 3701 y Fp(\()p Fg(q)r Fp(\))1039 3737 y Fq(\()p Fj(\034)11 b Fq(\))862 3860 y Fl(P)939 3823 y Fp(\()p Fg(q)r Fp(\))1031 3860 y Fq(\()p Fj(\034)g Fq(\))1210 3652 y Ff(!)1303 3798 y Fq(=)28 b Fl(\011)1494 3813 y Fp(1)1533 3798 y Fq(\()p Fj(\034)11 b Fq(\))p Fl(\005)1750 3756 y Fp(\()p Fg(x)p Fp(\))1849 3798 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2021 3813 y Fp(0)2061 3798 y Fq(\))p Fl(\011)2186 3813 y Fp(2)2225 3798 y Fq(\()p Fj(\034)2305 3813 y Fp(0)2345 3798 y Fq(\))2400 3652 y Ff( )2507 3737 y Fl(Q)2591 3701 y Fp(\()p Fg(q)r Fp(\))2684 3737 y Fq(\()p Fj(\034)2764 3752 y Fp(0)2804 3737 y Fq(\))2507 3860 y Fl(P)2584 3823 y Fp(\()p Fg(q)r Fp(\))2676 3860 y Fq(\()p Fj(\034)2756 3875 y Fp(0)2796 3860 y Fq(\))2883 3652 y Ff(!)2998 3798 y Fj(:)581 b Fq(\(60\))0 4074 y(This)32 b(implies)e(that)i(the)g(6)22 b Fn(\002)f Fq(6)32 b(ra)m(y)h(propagator)e(matrix)f Fl(\005)2251 4032 y Fp(\()p Fg(q)r Fp(\))2344 4074 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2516 4089 y Fp(0)2555 4074 y Fq(\))33 b(in)e(ra)m(y-cen)m(tered)j(co)s(ordinates)0 4194 y(can)43 b(b)s(e)f(expressed)k(in)41 b(terms)i(of)f(the)g(6)29 b Fn(\002)g Fq(6)42 b(ra)m(y)h(propagator)f(matrix)f Fl(\005)2858 4152 y Fp(\()p Fg(x)p Fp(\))2957 4194 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3129 4209 y Fp(0)3169 4194 y Fq(\))42 b(in)g(Cartesian)0 4315 y(co)s(ordinates)32 b(as)h(follo)m(ws:)1091 4510 y Fl(\005)1179 4468 y Fp(\()p Fg(q)r Fp(\))1272 4510 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1444 4525 y Fp(0)1483 4510 y Fq(\))28 b(=)g Fl(\011)1740 4525 y Fp(1)1779 4510 y Fq(\()p Fj(\034)11 b Fq(\))p Fl(\005)1996 4468 y Fp(\()p Fg(x)p Fp(\))2095 4510 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2267 4525 y Fp(0)2306 4510 y Fq(\))p Fl(\011)2432 4525 y Fp(2)2471 4510 y Fq(\()p Fj(\034)2551 4525 y Fp(0)2591 4510 y Fq(\))32 b Fj(:)918 b Fq(\(61\))0 4706 y(This)51 b(is)g(the)h(\014nal)e(transformation)f(relation)h(b)s(et)m(w)m(een)j (the)f(6)34 b Fn(\002)h Fq(6)51 b(ra)m(y)h(propagator)e(matrices)0 4827 y Fl(\005)88 4785 y Fp(\()p Fg(q)r Fp(\))181 4827 y Fq(\()p Fj(\034)6 b(;)17 b(\034)353 4842 y Fp(0)392 4827 y Fq(\))38 b(and)g Fl(\005)751 4785 y Fp(\()p Fg(x)p Fp(\))850 4827 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1022 4842 y Fp(0)1061 4827 y Fq(\).)60 b(It)38 b(also)f(follo)m(ws)g(from)g (the)i(general)e(expression)i(giv)m(en)g(in)3441 4801 y(\024)3430 4827 y(Cerv)m(en)q(\023)-50 b(y)0 4947 y(\(2001,)32 b(eq.\(4.3.38\)\).)146 5138 y(Equation)25 b(\(61\))f(can)g(b)s(e)h (simply)e(mo)s(di\014ed)h(to)g(yield)g(a)g(4)6 b Fn(\002)g Fq(4)24 b(ra)m(y)h(propagator)f(matrix)f Fc(\005)3437 5096 y Fp(\()p Fg(q)r Fp(\))3530 5138 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3702 5153 y Fp(0)3742 5138 y Fq(\))0 5259 y(in)31 b(ra)m(y-cen)m(tered)i(co)s(ordinates)e Fj(q)1227 5274 y Fp(1)1267 5259 y Fj(;)17 b(q)1354 5274 y Fp(2)1393 5259 y Fq(,)32 b(corresp)s(onding)f(to)g(DR)-8 b(T)31 b(system)h(\(35\).)43 b(W)-8 b(e)32 b(simply)e(remo)m(v)m(e)0 5379 y(the)j(third)f(and)h(sixth)f(columns)g(in)g Fl(\011)1424 5394 y Fp(2)1496 5379 y Fq(and)h(third)f(and)g(sixth)h(ro)m(ws)g(in)f Fl(\011)2787 5394 y Fp(1)2827 5379 y Fq(.)43 b(W)-8 b(e)33 b(obtain)1082 5575 y Fc(\005)1188 5533 y Fp(\()p Fg(q)r Fp(\))1281 5575 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1453 5590 y Fp(0)1493 5575 y Fq(\))27 b(=)h Fl(\011)1749 5533 y Fg(r)1749 5599 y Fp(1)1789 5575 y Fq(\()p Fj(\034)11 b Fq(\))p Fl(\005)2006 5533 y Fp(\()p Fg(x)p Fp(\))2104 5575 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2276 5590 y Fp(0)2316 5575 y Fq(\))p Fl(\011)2441 5533 y Fg(r)2441 5599 y Fp(2)2480 5575 y Fq(\()p Fj(\034)2560 5590 y Fp(0)2600 5575 y Fq(\))33 b Fj(:)908 b Fq(\(62\))1817 5824 y(100)p eop %%Page: 101 15 101 14 bop 0 100 a Fq(The)38 b(\\reduced")g(matrices)e Fl(\011)1152 58 y Fg(r)1152 124 y Fp(1)1229 100 y Fq(and)h Fl(\011)1510 58 y Fg(r)1510 124 y Fp(2)1586 100 y Fq(are)g(4)25 b Fn(\002)h Fq(6)37 b(and)g(6)25 b Fn(\002)g Fq(4,)38 b(resp)s(ectiv)m(ely)-8 b(,)39 b(and)e(are)g(giv)m(en)h(b)m(y)0 220 y(relations)710 494 y Fl(\011)797 452 y Fg(r)797 519 y Fp(1)864 494 y Fq(=)968 248 y Ff(0)968 394 y(B)968 444 y(B)968 494 y(B)968 547 y(@)1082 313 y Fl(f)1126 276 y Fg(T)1116 337 y Fp(1)1306 313 y Fl(0)1362 276 y Fg(T)1082 433 y Fl(f)1126 397 y Fg(T)1116 458 y Fp(2)1306 433 y Fl(0)1362 397 y Fg(T)1082 553 y Fl(V)1169 517 y Fg(T)1167 578 y Fp(1)1306 553 y Fl(e)1357 517 y Fg(T)1357 578 y Fp(1)1082 674 y Fl(V)1169 638 y Fg(T)1167 698 y Fp(2)1306 674 y Fl(e)1357 638 y Fg(T)1357 698 y Fp(2)1459 248 y Ff(1)1459 394 y(C)1459 444 y(C)1459 494 y(C)1459 547 y(A)1580 494 y Fj(;)115 b Fl(\011)1809 452 y Fg(r)1809 519 y Fp(2)1876 494 y Fq(=)1979 348 y Ff( )2087 433 y Fl(e)2138 448 y Fp(1)2371 433 y Fl(e)2422 448 y Fp(2)2656 433 y Fl(0)101 b(0)2087 553 y Fn(\000)p Fl(V)2249 568 y Fp(1)2371 553 y Fn(\000)p Fl(V)2533 568 y Fp(2)2656 553 y Fl(f)2690 568 y Fp(1)2813 553 y Fl(f)2847 568 y Fp(2)2928 348 y Ff(!)3043 494 y Fj(:)536 b Fq(\(63\))0 851 y(Here)26 b Fl(0)i Fq(=)g(\(0)p Fj(;)17 b Fq(0)p Fj(;)g Fq(0\))722 815 y Fg(T)775 851 y Fq(.)41 b(The)27 b(reduced)g(matrices)e Fl(\011)1864 809 y Fg(r)1864 876 y Fp(1)1929 851 y Fq(and)h Fl(\011)2199 809 y Fg(r)2199 876 y Fp(2)2264 851 y Fq(again)e(satisfy)i(the)g(relation)d Fl(\011)3413 809 y Fg(r)3413 876 y Fp(1)3453 851 y Fl(\011)3540 809 y Fg(r)3540 876 y Fp(2)3607 851 y Fq(=)28 b Fl(I)p Fq(,)0 972 y(where)34 b Fl(I)e Fq(is)g(a)g(4)22 b Fn(\002)h Fq(4)32 b(iden)m(tit)m(y)h(matrix.)146 1163 y(The)i(relations)d(\(61\)) h(and)g(\(62\))g(are)h(v)-5 b(alid)31 b(for)i(arbitrarily)e(c)m(hosen)k (con)m(tra)m(v)-5 b(arian)m(t)33 b(basis)g(v)m(ector)0 1283 y Fl(e)51 1298 y Fp(1)91 1283 y Fq(\()p Fj(\034)171 1298 y Fp(0)210 1283 y Fq(\))27 b(and)f Fl(e)509 1298 y Fp(2)549 1283 y Fq(\()p Fj(\034)629 1298 y Fp(0)668 1283 y Fq(\),)i(situated)e(in)g(the)h(w)m(a)m(v)m(efron)m(t)h(tangen)m (t)e(plane)g(at)g Fj(\034)2594 1298 y Fp(0)2634 1283 y Fq(,)i(and)e(co)m(v)-5 b(arian)m(t)26 b(basis)g(v)m(ector)0 1404 y Fl(f)34 1419 y Fp(1)74 1404 y Fq(\()p Fj(\034)154 1419 y Fp(0)193 1404 y Fq(\),)42 b Fl(f)334 1419 y Fp(2)374 1404 y Fq(\()p Fj(\034)454 1419 y Fp(0)493 1404 y Fq(\),)g(situated)e (in)f(the)h(plane)f(p)s(erp)s(endicular)g(to)g(the)i(reference)g(ra)m (y)f(\012,)i(satisfying)d(the)0 1524 y(relation)29 b(\(30\).)43 b(Note)31 b(that)g Fl(e)1095 1539 y Fp(3)1135 1524 y Fq(\()p Fj(\034)1215 1539 y Fp(0)1255 1524 y Fq(\))c(=)h Fi(U)10 b Fq(\()p Fj(\034)1585 1539 y Fp(0)1625 1524 y Fq(\))31 b(and)g Fl(f)1916 1539 y Fp(3)1956 1524 y Fq(\()p Fj(\034)2036 1539 y Fp(0)2076 1524 y Fq(\))c(=)h Fl(p)p Fq(\()p Fj(\034)2387 1539 y Fp(0)2427 1524 y Fq(\).)43 b(Similarly)-8 b(,)27 b(the)32 b(c)m(hoice)g(of)e Fl(e)3583 1539 y Fp(1)3623 1524 y Fq(\()p Fj(\034)11 b Fq(\),)0 1645 y Fl(e)51 1660 y Fp(2)91 1645 y Fq(\()p Fj(\034)g Fq(\),)39 b Fl(f)320 1660 y Fp(1)360 1645 y Fq(\()p Fj(\034)11 b Fq(\),)39 b(and)f Fl(f)784 1660 y Fp(2)823 1645 y Fq(\()p Fj(\034)11 b Fq(\))38 b(is)f(also)g(arbitrary)-8 b(,)38 b(they)g(only)g(m)m(ust)f(satisfy)h(\(30\),)g(with)f Fl(e)3251 1660 y Fp(3)3291 1645 y Fq(\()p Fj(\034)11 b Fq(\))37 b(=)f Fi(U)10 b Fq(\()p Fj(\034)h Fq(\))0 1765 y(and)28 b Fl(f)219 1780 y Fp(3)259 1765 y Fq(\()p Fj(\034)11 b Fq(\))28 b(=)g Fl(p)p Fq(\()p Fj(\034)11 b Fq(\).)42 b(Th)m(us,)31 b(the)d(n)m(umerical)f(in)m(tegration)g(of)g (\(50\))h(is)f(not)h(required)h(in)f(this)f(case,)j(and)0 1885 y(relations)c(\(48\))i(with)f(\(62\))g(can)h(b)s(e)g(safely)g (used)h(to)e(compute)h Fj(\016)t Fh(q)t Fq(\()p Fj(\034)11 b Fq(\))28 b(and)g Fj(\016)t Fh(p)2857 1849 y Fp(\()p Fg(q)r Fp(\))2950 1885 y Fq(\()p Fj(\034)11 b Fq(\))28 b(from)e Fj(\016)t Fh(q)t Fq(\()p Fj(\034)3517 1900 y Fp(0)3557 1885 y Fq(\))h(and)0 2006 y Fj(\016)t Fh(p)107 1970 y Fp(\()p Fg(q)r Fp(\))200 2006 y Fq(\()p Fj(\034)280 2021 y Fp(0)319 2006 y Fq(\),)33 b(for)f Fl(e)617 2021 y Fp(1)657 2006 y Fq(\()p Fj(\034)737 2021 y Fp(0)776 2006 y Fq(\),)h Fl(e)925 2021 y Fp(2)964 2006 y Fq(\()p Fj(\034)1044 2021 y Fp(0)1084 2006 y Fq(\))g(and)f Fl(e)1395 2021 y Fp(1)1435 2006 y Fq(\()p Fj(\034)11 b Fq(\),)33 b Fl(e)1675 2021 y Fp(2)1714 2006 y Fq(\()p Fj(\034)11 b Fq(\))33 b(arbitrarily)d(and)j(indep)s(enden)m(tly)g(sp)s(eci\014ed.) 146 2197 y(Equation)c(\(62\),)g(ho)m(w)m(ev)m(er,)j(giv)m(es)d(the)g(4) 14 b Fn(\002)g Fq(4)30 b(propagator)e(matrix)f(in)h(ra)m(y-cen)m(tered) j(co)s(ordinates)0 2317 y(only)38 b(if)g Fc(\005)422 2275 y Fp(\()p Fg(q)r Fp(\))515 2317 y Fq(\()p Fj(\034)595 2332 y Fp(0)635 2317 y Fj(;)17 b(\034)721 2332 y Fp(0)761 2317 y Fq(\))38 b(=)g Fh(I)8 b Fq(,)40 b(where)g Fh(I)47 b Fq(is)38 b(a)g(4)27 b Fn(\002)g Fq(4)38 b(iden)m(tit)m(y)h(matrix.)60 b(This)39 b(is)g(guaran)m(teed)g(only)g(if)0 2438 y Fl(\011)87 2396 y Fg(r)87 2462 y Fp(1)127 2438 y Fq(\()p Fj(\034)11 b Fq(\))41 b Fn(!)g Fl(\011)525 2396 y Fg(r)525 2462 y Fp(1)565 2438 y Fq(\()p Fj(\034)645 2453 y Fp(0)684 2438 y Fq(\))g(for)f Fj(\034)52 b Fn(!)41 b Fj(\034)1197 2453 y Fp(0)1237 2438 y Fq(,)h(see)g(\(61\).)67 b(In)40 b(suc)m(h)i(case,)i(a)c(solution)f(of)h(\(50\))g(\(or)g(some)g(other)0 2558 y(analogous)31 b(equation\))i(is)f(needed.)0 2961 y Fk(5)161 b(Alternativ)l(e)54 b(computation)e(of)i(\005)2344 2901 y Fq(\()p Fj(q)t Fq(\))2471 2961 y Fb(\()p Fa(\034)8 b(;)24 b(\034)2715 2982 y Fq(0)2768 2961 y Fb(\))0 3266 y Fq(W)-8 b(e)29 b(no)m(w)h(use)g(an)f(alternativ)m(e)f(approac)m(h)i (to)e(determine)h Fl(\005)2217 3224 y Fp(\()p Fg(q)r Fp(\))2310 3266 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2482 3281 y Fp(0)2521 3266 y Fq(\),)30 b(prop)s(osed)f(b)m(y)h(Klime)-5 b(\024)-44 b(s)27 b(\(1994\).)0 3387 y(The)41 b(adv)-5 b(an)m(tage)40 b(of)f(this)g(approac)m(h)h(is)g(that)f(it)g(do)s(es)h (not)g(require)g(six)g(dynamic)f(ra)m(y)h(tracings)f(in)0 3507 y(Cartesian)33 b(co)s(ordinates,)g(but)g(only)f(four,)h(for)f (some)h(selected)h(initial)29 b(conditions.)44 b(The)34 b(remaining)0 3628 y(t)m(w)m(o)49 b(solutions)d(of)i(the)g(dynamic)f (ra)m(y)h(tracing)f(system)i(in)e(Cartesian)h(co)s(ordinates)f(are)h (kno)m(wn)0 3748 y(explicitly)31 b(\(ra)m(y)i(tangen)m(t)g(solutions,)e (non-eik)m(onal)h(solutions\).)146 3939 y(W)-8 b(e)33 b(can)g(pro)s(ceed)h(in)d(the)i(follo)m(wing)d(w)m(a)m(y)-8 b(.)45 b(W)-8 b(e)33 b(express)i(\(61\))d(as)g(follo)m(ws:)1292 4154 y Fl(\005)1380 4112 y Fp(\()p Fg(q)r Fp(\))1473 4154 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1645 4169 y Fp(0)1684 4154 y Fq(\))28 b(=)g Fl(\011)1941 4169 y Fp(1)1980 4154 y Fq(\()p Fj(\034)11 b Fq(\))p Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)2349 4169 y Fp(0)2390 4154 y Fq(\))32 b Fj(;)1119 b Fq(\(64\))0 4369 y(where)1275 4489 y Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)1515 4504 y Fp(0)1555 4489 y Fq(\))28 b(=)f Fl(\005)1812 4447 y Fp(\()p Fg(x)p Fp(\))1911 4489 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2083 4504 y Fp(0)2122 4489 y Fq(\))p Fl(\011)2247 4504 y Fp(2)2287 4489 y Fq(\()p Fj(\034)2367 4504 y Fp(0)2407 4489 y Fq(\))32 b Fj(:)1102 b Fq(\(65\))0 4661 y(The)39 b(6)26 b Fn(\002)g Fq(6)38 b(matrix)f Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)1034 4676 y Fp(0)1074 4661 y Fq(\))38 b(is)f(not)h(a)g(propagator)f(matrix,) h(as)h Fl(Z)p Fq(\()p Fj(\034)2648 4676 y Fp(0)2688 4661 y Fj(;)17 b(\034)2774 4676 y Fp(0)2813 4661 y Fq(\))37 b(=)g Fl(\011)3088 4676 y Fp(2)3128 4661 y Fq(\()p Fj(\034)3208 4676 y Fp(0)3248 4661 y Fq(\))h(is)f(di\013eren)m(t)0 4782 y(from)c(iden)m(tit)m(y)g(matrix)g Fl(I)p Fq(.)47 b(The)35 b(six)f(columns)f(of)g Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)2114 4797 y Fp(0)2154 4782 y Fq(\))34 b(are)g(the)g (solutions)f(of)g(the)i(dynamic)e(ra)m(y)0 4902 y(tracing)f(system)h (\(4\))f(in)g(Cartesian)h(co)s(ordinates,)f(with)g(initial)d (conditions)i(giv)m(en)i(b)m(y)h Fl(\011)3358 4917 y Fp(2)3397 4902 y Fq(\()p Fj(\034)3477 4917 y Fp(0)3517 4902 y Fq(\).)146 5093 y(Tw)m(o)48 b(of)e(the)h(solutions)e(of)h(the)h (dynamic)f(ra)m(y)h(tracing)e(system)j(in)d(Cartesian)i(co)s (ordinates,)0 5214 y(ho)m(w)m(ev)m(er,)35 b(are)e(kno)m(wn)i (analytically)-8 b(.)41 b(This)33 b(is)g(the)g(ra)m(y-tangen)m(t)g (solution,)f(see)i(\(13\),)e(and)h(the)h(non-)0 5334 y(eik)m(onal)e(solution)f(\(14\).)44 b(Con)m(v)m(en)m(tionally)-8 b(,)33 b(w)m(e)h(store)g(the)f(ra)m(y)g(tangen)m(t)h(solution)d(\(13\)) i(in)f(the)h(third)0 5454 y(column)f(of)g Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)693 5469 y Fp(0)733 5454 y Fq(\),)33 b(and)f(the)i(non-eik)m(onal)d(solution)g(\(14\))h(in)g(its)g(sixth)h (column.)43 b(The)34 b(remaining)0 5575 y(columns)c(represen)m(t)j (standard)e(paraxial)d(solutions)i(of)g(the)h(DR)-8 b(T)31 b(in)f(Cartesian)g(co)s(ordinates,)h(with)1817 5824 y(101)p eop %%Page: 102 16 102 15 bop 0 100 a Fq(the)33 b(initial)c(condition)j(giv)m(en)h(in)f (the)h(relev)-5 b(an)m(t)33 b(column)f(of)g Fl(\011)2325 115 y Fp(2)2365 100 y Fq(\()p Fj(\034)2445 115 y Fp(0)2484 100 y Fq(\).)44 b(The)34 b(standard)g(paraxial)c(solu-)0 220 y(tions)f(in)f Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)586 235 y Fp(0)626 220 y Fq(\))29 b(m)m(ust)g(satisfy)g(the)h(constrain)m (t)f(equation)f(\(6\).)42 b(Moreo)m(v)m(er,)32 b Fl(Q)2977 184 y Fp(\()p Fg(x)p Fp(\))3105 220 y Fq(m)m(ust)d(satisfy)g(the)0 340 y(relation)1620 461 y Fl(p)1682 420 y Fg(T)1737 461 y Fl(Q)1821 420 y Fp(\()p Fg(x)p Fp(\))1948 461 y Fq(=)e(0)33 b Fj(;)1446 b Fq(\(66\))0 620 y(expressing)34 b(the)f(fact)f(that)h(v)m (ector)g Fl(Q)1418 583 y Fp(\()p Fg(x)p Fp(\))1550 620 y Fq(is)f(tangen)m(t)h(to)f(the)h(w)m(a)m(v)m(efron)m(t)h(at)f(the)g (reference)h(ra)m(y)-8 b(.)146 811 y(No)m(w)27 b(w)m(e)g(shall)e (compute)h Fl(\005)1198 769 y Fp(\()p Fg(q)r Fp(\))1291 811 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1463 826 y Fp(0)1502 811 y Fq(\))26 b(using)g(Eq.)42 b(\(64\).)f(First)25 b(w)m(e)i(compute)f(the)h(pro)s(duct)f(of)g Fl(\011)3713 826 y Fp(1)3752 811 y Fq(,)0 931 y(giv)m(en)34 b(b)m(y)g(\(59\),)f (with)h(the)g(third)e(and)i(sixth)g(columns)f(of)g Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)2430 946 y Fp(0)2470 931 y Fq(\),)33 b(represen)m(ting)i(the)f(ra)m(y-tangen)m(t)0 1052 y(and)26 b(non-eik)m(onal)e(solutions)h(\(13\))g(and)h(\(14\).)41 b(F)-8 b(or)25 b(the)h(ra)m(y-tangen)m(t)g(solutions,)g(\(59\))f(and)h (\(13\))f(yield)805 1534 y Fl(\011)892 1549 y Fp(1)948 1388 y Ff( )1055 1473 y Fi(U)1055 1593 y Fh(\021)1178 1388 y Ff(!)1271 1534 y Fq(=)1375 1164 y Ff(0)1375 1310 y(B)1375 1360 y(B)1375 1409 y(B)1375 1459 y(B)1375 1509 y(B)1375 1559 y(B)1375 1609 y(B)1375 1659 y(B)1375 1712 y(@)1489 1232 y Fl(f)1533 1196 y Fg(T)1523 1257 y Fp(1)1588 1232 y Fi(U)1489 1353 y Fl(f)1533 1316 y Fg(T)1523 1377 y Fp(2)1588 1353 y Fi(U)1489 1473 y Fl(f)1533 1437 y Fg(T)1523 1498 y Fp(3)1588 1473 y Fi(U)1489 1593 y Fn(\000)p Fq(\()p Fl(e)1655 1557 y Fg(T)1655 1618 y Fp(1)1711 1593 y Fh(\021)s Fq(\)\()p Fl(p)1912 1557 y Fg(T)1967 1593 y Fi(U)11 b Fq(\))22 b(+)g Fl(e)2258 1557 y Fg(T)2258 1618 y Fp(1)2313 1593 y Fh(\021)1489 1714 y Fn(\000)p Fq(\()p Fl(e)1655 1678 y Fg(T)1655 1738 y Fp(2)1711 1714 y Fh(\021)s Fq(\)\()p Fl(p)1912 1678 y Fg(T)1967 1714 y Fi(U)11 b Fq(\))22 b(+)g Fl(e)2258 1678 y Fg(T)2258 1738 y Fp(2)2313 1714 y Fh(\021)1489 1834 y Fn(\000)p Fh(\021)1630 1798 y Fg(T)1685 1834 y Fi(U)32 b Fq(+)22 b Fl(e)1937 1798 y Fg(T)1937 1859 y Fp(3)1993 1834 y Fh(\021)2418 1164 y Ff(1)2418 1310 y(C)2418 1360 y(C)2418 1409 y(C)2418 1459 y(C)2418 1509 y(C)2418 1559 y(C)2418 1609 y(C)2418 1659 y(C)2418 1712 y(A)2518 1534 y Fq(=)2622 1164 y Ff(0)2622 1310 y(B)2622 1360 y(B)2622 1409 y(B)2622 1459 y(B)2622 1509 y(B)2622 1559 y(B)2622 1609 y(B)2622 1659 y(B)2622 1712 y(@)2736 1232 y Fq(0)2736 1353 y(0)2736 1473 y(1)2736 1593 y(0)2736 1714 y(0)2736 1834 y(0)2826 1164 y Ff(1)2826 1310 y(C)2826 1360 y(C)2826 1409 y(C)2826 1459 y(C)2826 1509 y(C)2826 1559 y(C)2826 1609 y(C)2826 1659 y(C)2826 1712 y(A)2948 1534 y Fj(:)631 b Fq(\(67\))0 2023 y(Here)30 b(w)m(e)f(ha)m(v)m(e)i(tak)m(en)e(in)m(to)g(accoun)m(t)g (that)g Fl(f)1652 1987 y Fg(T)1642 2048 y(I)1707 2023 y Fi(U)38 b Fq(=)28 b(0,)h Fl(p)2087 1987 y Fg(T)2143 2023 y Fi(U)38 b Fq(=)27 b(1)i(and)g Fl(f)2663 1987 y Fg(T)2653 2048 y Fp(3)2718 2023 y Fi(U)38 b Fq(=)28 b Fl(p)2993 1987 y Fg(T)3048 2023 y Fi(U)38 b Fq(=)28 b(1.)41 b(Similarly)-8 b(,)0 2144 y(for)32 b(the)h(non-eik)m(onal)e(solution,)g (\(59\))h(and)h(\(14\))f(yield)237 2626 y Fl(\011)325 2641 y Fp(1)381 2480 y Ff( )488 2565 y Fi(U)10 b Fq(\001)p Fj(\034)488 2685 y Fl(p)22 b Fq(+)g Fh(\021)t Fq(\001)p Fj(\034)910 2480 y Ff(!)1003 2626 y Fq(=)1107 2256 y Ff(0)1107 2402 y(B)1107 2452 y(B)1107 2502 y(B)1107 2551 y(B)1107 2601 y(B)1107 2651 y(B)1107 2701 y(B)1107 2751 y(B)1107 2804 y(@)1221 2324 y Fq(\001)p Fj(\034)11 b Fq(\()p Fl(f)1437 2288 y Fg(T)1427 2349 y Fp(1)1493 2324 y Fi(U)g Fq(\))1221 2445 y(\001)p Fj(\034)g Fq(\()p Fl(f)1437 2409 y Fg(T)1427 2469 y Fp(2)1493 2445 y Fi(U)g Fq(\))1221 2565 y(\001)p Fj(\034)g Fq(\()p Fl(f)1437 2529 y Fg(T)1427 2590 y Fp(3)1493 2565 y Fi(U)g Fq(\))1221 2685 y Fn(\000)p Fq(\001)p Fj(\034)g Fq(\()p Fl(e)1521 2649 y Fg(T)1521 2710 y Fp(1)1577 2685 y Fi(U)g Fq(\)\()p Fl(p)1797 2649 y Fg(T)1852 2685 y Fi(U)f Fq(\))22 b(+)g Fl(e)2142 2649 y Fg(T)2142 2710 y Fp(1)2198 2685 y Fl(p)g Fq(+)g(\001)p Fj(\034)11 b Fq(\()p Fl(e)2603 2649 y Fg(T)2603 2710 y Fp(1)2659 2685 y Fh(\021)s Fq(\))1221 2806 y Fn(\000)p Fq(\001)p Fj(\034)g Fq(\()p Fl(e)1521 2770 y Fg(T)1521 2831 y Fp(2)1577 2806 y Fi(U)g Fq(\)\()p Fl(p)1797 2770 y Fg(T)1852 2806 y Fi(U)f Fq(\))22 b(+)g Fl(e)2142 2770 y Fg(T)2142 2831 y Fp(2)2198 2806 y Fl(p)g Fq(+)g(\001)p Fj(\034)11 b Fq(\()p Fl(e)2603 2770 y Fg(T)2603 2831 y Fp(2)2659 2806 y Fh(\021)s Fq(\))1221 2926 y Fn(\000)p Fq(\001)p Fj(\034)g Fq(\()p Fh(\021)1535 2890 y Fg(T)1590 2926 y Fi(U)f Fq(\))22 b(+)g Fl(e)1880 2890 y Fg(T)1880 2951 y Fp(3)1935 2926 y Fl(p)h Fq(+)f(\001)p Fj(\034)11 b Fq(\()p Fl(e)2341 2890 y Fg(T)2341 2951 y Fp(3)2397 2926 y Fh(\021)s Fq(\))2802 2256 y Ff(1)2802 2402 y(C)2802 2452 y(C)2802 2502 y(C)2802 2551 y(C)2802 2601 y(C)2802 2651 y(C)2802 2701 y(C)2802 2751 y(C)2802 2804 y(A)2902 2626 y Fq(=)3006 2256 y Ff(0)3006 2402 y(B)3006 2452 y(B)3006 2502 y(B)3006 2551 y(B)3006 2601 y(B)3006 2651 y(B)3006 2701 y(B)3006 2751 y(B)3006 2804 y(@)3120 2324 y Fq(0)3120 2445 y(0)3120 2565 y(\001)p Fj(\034)3120 2685 y Fq(0)3120 2806 y(0)3120 2926 y(1)3296 2256 y Ff(1)3296 2402 y(C)3296 2452 y(C)3296 2502 y(C)3296 2551 y(C)3296 2601 y(C)3296 2651 y(C)3296 2701 y(C)3296 2751 y(C)3296 2804 y(A)3606 2626 y Fq(\(68\))0 3103 y(Here)33 b(\001)p Fj(\034)45 b Fq(stands)33 b(for)f Fj(\034)i Fn(\000)22 b Fj(\034)1068 3118 y Fp(0)1108 3103 y Fq(.)146 3294 y(Th)m(us,)37 b(the)e(third)e(and)i(sixth)f(columns)g(of)g Fl(\005)1855 3252 y Fp(\()p Fg(q)r Fp(\))1948 3294 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2120 3309 y Fp(0)2160 3294 y Fq(\))34 b(are)g(giv)m(en)h(b)m(y)g(simple)e(expressions)j (\(67\))0 3415 y(and)d(\(68\).)146 3606 y(W)-8 b(e)39 b(shall)d(no)m(w)j(consider)f(the)g(standard)g(paraxial)e(solutions)h (\(the)h(\014rst,)i(second,)h(fourth)c(and)0 3726 y(\014fth)c (columns\))e(of)i Fl(Z)p Fq(\()p Fj(\034)6 b(;)17 b(\034)978 3741 y Fp(0)1018 3726 y Fq(\).)43 b(W)-8 b(e)33 b(denote)g(an)m(y)h(of) e(them)1653 3826 y Ff( )1760 3911 y Fl(Q)1844 3875 y Fp(\()p Fg(x)p Fp(\))1760 4033 y Fl(P)1837 3997 y Fp(\()p Fg(x)p Fp(\))1985 3826 y Ff(!)2099 3972 y Fj(;)1480 b Fq(\(69\))0 4244 y(where)30 b Fl(Q)362 4208 y Fp(\()p Fg(x)p Fp(\))488 4244 y Fn(\021)e Fq(\()p Fj(Q)708 4193 y Fp(\()p Fg(x)p Fp(\))708 4266 y(1)807 4244 y Fj(;)17 b(Q)928 4193 y Fp(\()p Fg(x)p Fp(\))928 4266 y(2)1027 4244 y Fj(;)g(Q)1148 4193 y Fp(\()p Fg(x)p Fp(\))1148 4266 y(3)1247 4244 y Fq(\))1285 4208 y Fg(T)1368 4244 y Fq(and)29 b Fl(P)1631 4208 y Fp(\()p Fg(x)p Fp(\))1757 4244 y Fn(\021)f Fq(\()p Fj(P)1977 4193 y Fp(\()p Fg(x)p Fp(\))1963 4266 y(1)2075 4244 y Fj(;)17 b(P)2196 4193 y Fp(\()p Fg(x)p Fp(\))2182 4266 y(2)2294 4244 y Fj(;)g(P)2415 4193 y Fp(\()p Fg(x)p Fp(\))2401 4266 y(3)2513 4244 y Fq(\))2551 4208 y Fg(T)2606 4244 y Fq(.)42 b(The)29 b(pro)s(duct)g(of)f (matrix)f Fl(\011)3740 4259 y Fp(1)0 4364 y Fq(with)32 b(an)m(y)h(of)g(the)g(solutions)e(\(69\))h(reads)633 4852 y Fl(\011)720 4867 y Fp(1)776 4706 y Ff( )883 4791 y Fl(Q)967 4755 y Fp(\()p Fg(x)p Fp(\))883 4913 y Fl(P)960 4877 y Fp(\()p Fg(x)p Fp(\))1108 4706 y Ff(!)1201 4852 y Fq(=)1305 4482 y Ff(0)1305 4628 y(B)1305 4678 y(B)1305 4727 y(B)1305 4777 y(B)1305 4827 y(B)1305 4877 y(B)1305 4927 y(B)1305 4977 y(B)1305 5030 y(@)1419 4547 y Fl(f)1463 4511 y Fg(T)1453 4571 y Fp(1)1518 4547 y Fl(Q)1602 4511 y Fp(\()p Fg(x)p Fp(\))1419 4669 y Fl(f)1463 4633 y Fg(T)1453 4694 y Fp(2)1518 4669 y Fl(Q)1602 4633 y Fp(\()p Fg(x)p Fp(\))1419 4791 y Fl(f)1463 4755 y Fg(T)1453 4816 y Fp(3)1518 4791 y Fl(Q)1602 4755 y Fp(\()p Fg(x)p Fp(\))1419 4913 y Fn(\000)p Fq(\()p Fl(e)1585 4877 y Fg(T)1585 4938 y Fp(1)1640 4913 y Fh(\021)t Fq(\)\()p Fl(p)1842 4877 y Fg(T)1897 4913 y Fl(Q)1981 4877 y Fp(\()p Fg(x)p Fp(\))2080 4913 y Fq(\))22 b(+)g Fl(e)2289 4877 y Fg(T)2289 4938 y Fp(1)2344 4913 y Fl(P)2421 4877 y Fp(\()p Fg(x)p Fp(\))1419 5035 y Fn(\000)p Fq(\()p Fl(e)1585 4999 y Fg(T)1585 5060 y Fp(2)1640 5035 y Fh(\021)t Fq(\)\()p Fl(p)1842 4999 y Fg(T)1897 5035 y Fl(Q)1981 4999 y Fp(\()p Fg(x)p Fp(\))2080 5035 y Fq(\))g(+)g Fl(e)2289 4999 y Fg(T)2289 5060 y Fp(2)2344 5035 y Fl(P)2421 4999 y Fp(\()p Fg(x)p Fp(\))1419 5157 y Fn(\000)p Fh(\021)1560 5121 y Fg(T)1615 5157 y Fl(Q)1699 5121 y Fp(\()p Fg(x)p Fp(\))1820 5157 y Fq(+)g Fl(e)1969 5121 y Fg(T)1969 5182 y Fp(3)2024 5157 y Fl(P)2101 5121 y Fp(\()p Fg(x)p Fp(\))2561 4482 y Ff(1)2561 4628 y(C)2561 4678 y(C)2561 4727 y(C)2561 4777 y(C)2561 4827 y(C)2561 4877 y(C)2561 4927 y(C)2561 4977 y(C)2561 5030 y(A)2661 4852 y Fq(=)2765 4482 y Ff(0)2765 4628 y(B)2765 4678 y(B)2765 4727 y(B)2765 4777 y(B)2765 4827 y(B)2765 4877 y(B)2765 4927 y(B)2765 4977 y(B)2765 5030 y(@)2879 4550 y Fn(\002)2879 4671 y(\002)2879 4791 y Fq(0)2879 4911 y Fn(\002)2879 5032 y(\002)2879 5152 y Fq(0)2998 4482 y Ff(1)2998 4628 y(C)2998 4678 y(C)2998 4727 y(C)2998 4777 y(C)2998 4827 y(C)2998 4877 y(C)2998 4927 y(C)2998 4977 y(C)2998 5030 y(A)3120 4852 y Fj(:)459 b Fq(\(70\))0 5334 y(The)28 b(crosses)h Fn(\002)e Fq(denote)h(p)s(ossibly)f(non-zero) g(en)m(tries.)42 b(The)28 b(zero)f(in)g(the)g(third)f(ro)m(w)i(follo)m (ws)e(immedi-)0 5454 y(ately)32 b(from)f(\(66\),)h(and)g(the)h(zero)f (in)g(the)g(sixth)h(ro)m(w)f(from)f(the)i(relation)d(\()p Fl(e)2819 5418 y Fg(T)2819 5479 y(m)2886 5454 y Fh(\021)s Fq(\)\()p Fl(f)3069 5418 y Fg(T)3059 5479 y(m)3126 5454 y Fl(Q)3210 5418 y Fp(\()p Fg(x)p Fp(\))3309 5454 y Fq(\))e(=)f Fh(\021)3542 5418 y Fg(T)3597 5454 y Fl(Q)3681 5418 y Fp(\()p Fg(x)p Fp(\))0 5575 y Fq(and)33 b(from)e(the)i(constrain)m(t)f (relation)f(\(6\).)1817 5824 y(102)p eop %%Page: 103 17 103 16 bop 146 100 a Fq(Th)m(us,)44 b(the)d(6)27 b Fn(\002)h Fq(6)39 b(ra)m(y)i(propagator)e(matrix)g Fl(\005)1970 58 y Fp(\()p Fg(q)r Fp(\))2062 100 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2234 115 y Fp(0)2274 100 y Fq(\))40 b(in)f(ra)m(y-cen)m (tered)k(co)s(ordinates)c(tak)m(es)0 220 y(a)k(sp)s(eci\014c)g(form:)63 b(It)43 b(has)g(zero)h(v)-5 b(alues)42 b(in)g(the)i(third)e(and)h (sixth)g(columns)f(and)h(ro)m(ws,)k(with)42 b(the)0 340 y(exception)32 b(of)e(p)s(ositions)g(33)h(and)g(66,)g(where)h(it)e(is)h (1,)g(and)h(of)e(p)s(osition)f(36,)i(where)i(it)d(equals)h Fj(\034)g Fn(\000)20 b Fj(\034)3713 355 y Fp(0)3752 340 y Fq(.)0 461 y(Consequen)m(tly)-8 b(,)49 b(it)44 b(can)g(b)s(e)h (expressed)i(in)c(the)i(form)e(of)h(\(43\).)78 b(This)44 b(pro)m(vides)h(an)f(indep)s(enden)m(t)0 581 y(deriv)-5 b(ation)31 b(of)h(the)h(sp)s(eci\014c)g(form)f(of)g(\(43\).)146 772 y(F)-8 b(rom)34 b(the)h(6)24 b Fn(\002)g Fq(6)34 b(ra)m(y)i(propagator)e(matrix)f Fl(\005)1911 730 y Fp(\()p Fg(q)r Fp(\))2004 772 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2176 787 y Fp(0)2215 772 y Fq(\))35 b(giv)m(en)g(b)m(y)h(\(64\),)f(w)m(e)h (can)f(again)f(obtain)0 893 y(the)f(4)22 b Fn(\002)h Fq(4)32 b(ra)m(y)h(propagator)e(matrix)g Fc(\005)1510 851 y Fp(\()p Fg(q)r Fp(\))1603 893 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1775 908 y Fp(0)1814 893 y Fq(\))33 b(as)g(follo)m(ws:) 1264 1064 y Fc(\005)1370 1022 y Fp(\()p Fg(q)r Fp(\))1463 1064 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1635 1079 y Fp(0)1675 1064 y Fq(\))27 b(=)h Fl(\011)1931 1022 y Fg(r)1931 1089 y Fp(1)1971 1064 y Fq(\()p Fj(\034)11 b Fq(\))p Fl(Z)2168 1023 y Fg(r)2206 1064 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2378 1079 y Fp(0)2418 1064 y Fq(\))32 b Fj(:)1091 b Fq(\(71\))0 1235 y(Here)38 b Fl(\011)322 1193 y Fg(r)322 1260 y Fp(1)398 1235 y Fq(corresp)s(onds)g(to)f Fl(\011)1148 1250 y Fp(1)1224 1235 y Fq(giv)m(en)g(b)m(y)h(\(59\),)f(in)f(whic)m(h)i(the)f(third)f (and)h(sixth)g(ro)m(ws)h(ha)m(v)m(e)g(b)s(een)0 1356 y(remo)m(v)m(ed,)45 b(see)d(\(63\).)70 b(Similarly)-8 b(,)40 b Fl(Z)1376 1319 y Fg(r)1414 1356 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1586 1371 y Fp(0)1626 1356 y Fq(\))41 b(\()p Fl(Z)p Fq(-reduced\))i(con)m(tains)e(four)h(solutions)e(of)h(the)h(DR) -8 b(T)0 1476 y(system)38 b(in)f(Cartesian)g(co)s(ordinates)g(\(4\))g (along)f(the)h(reference)i(ra)m(y)-8 b(,)39 b(from)d(the)i(initial)33 b(p)s(oin)m(t)k Fj(\034)3616 1491 y Fp(0)3693 1476 y Fq(to)0 1596 y Fj(\034)11 b Fq(.)54 b(These)38 b(four)e(solutions)f (are)h(sp)s(eci\014ed)h(b)m(y)g(the)f(initial)c(conditions)j(giv)m(en)i (b)m(y)g(the)f(6)24 b Fn(\002)h Fq(4)36 b(matrix)0 1717 y Fl(\011)87 1675 y Fg(r)87 1741 y Fp(2)127 1717 y Fq(\()p Fj(\034)207 1732 y Fp(0)246 1717 y Fq(\))d(\()p Fl(\011)442 1732 y Fp(2)481 1717 y Fq(-reduced\),)h(see)g(\(63\),)1031 1950 y Fl(\011)1118 1908 y Fg(r)1118 1975 y Fp(2)1157 1950 y Fq(\()p Fj(\034)1237 1965 y Fp(0)1277 1950 y Fq(\))28 b(=)1446 1804 y Ff( )1553 1889 y Fl(e)1604 1904 y Fp(1)1944 1889 y Fl(e)1995 1904 y Fp(2)2335 1889 y Fl(0)101 b(0)1553 2010 y Fq(\()p Fl(e)1642 1973 y Fg(T)1642 2034 y Fp(1)1698 2010 y Fh(\021)s Fq(\))p Fl(p)83 b Fq(\()p Fl(e)2033 1973 y Fg(T)2033 2034 y Fp(2)2088 2010 y Fh(\021)t Fq(\))p Fl(p)g(f)2369 2025 y Fp(1)2492 2010 y Fl(f)2526 2025 y Fp(2)2607 1804 y Ff(!)2722 1950 y Fj(:)857 b Fq(\(72\))0 2184 y(The)34 b(\014rst)f(t)m(w)m(o)g(columns)f(in)g Fl(\011)1167 2142 y Fg(r)1167 2208 y Fp(2)1206 2184 y Fq(\()p Fj(\034)1286 2199 y Fp(0)1326 2184 y Fq(\))g(corresp)s(ond)i (to)e(the)h(plane)f(w)m(a)m(v)m(efron)m(t)j(initial)28 b(conditions)1161 2417 y Fl(\011)1248 2375 y Fg(pw)1340 2417 y Fq(\()p Fj(\034)1420 2432 y Fp(0)1460 2417 y Fq(\))g(=)1629 2271 y Ff( )1736 2356 y Fl(e)1787 2371 y Fp(1)2127 2356 y Fl(e)2178 2371 y Fp(2)1736 2476 y Fq(\()p Fl(e)1825 2440 y Fg(T)1825 2501 y Fp(1)1881 2476 y Fh(\021)s Fq(\))p Fl(p)83 b Fq(\()p Fl(e)2216 2440 y Fg(T)2216 2501 y Fp(2)2271 2476 y Fh(\021)t Fq(\))p Fl(p)2477 2271 y Ff(!)2591 2417 y Fj(;)988 b Fq(\(73\))0 2645 y(and)33 b(the)g(third)f(and)g(fourth)h (column)e(corresp)s(ond)i(to)g(the)g(p)s(oin)m(t)e(source)j(initial)29 b(condition)1405 2873 y Fl(\011)1492 2831 y Fg(ps)1565 2873 y Fq(\()p Fj(\034)1645 2888 y Fp(0)1684 2873 y Fq(\))f(=)1853 2727 y Ff( )1961 2812 y Fl(0)100 b(0)1961 2932 y(f)1995 2947 y Fp(1)2117 2932 y Fl(f)2151 2947 y Fp(2)2233 2727 y Ff(!)2347 2873 y Fj(:)1232 b Fq(\(74\))146 3190 y(Exactly)24 b(the)g(same)g(initial)c(conditions)i(as)i(\(73\))f(and)h(\(74\))f(w)m (ere)h(also)f(deriv)m(ed)i(in)3157 3165 y(\024)3146 3190 y(Cerv)m(en)q(\023)-50 b(y)26 b(\(2001,)0 3311 y(eqs.)57 b(\(4.2.50\))36 b({)h(\(4.2.52\)\),)f(in)g(relation)f(to)i(w)m(a)m(v)m (efron)m(t)h(orthonormal)d(co)s(ordinates.)55 b(T)-8 b(o)37 b(compare)0 3431 y(\(74\))j(for)f(a)h(p)s(oin)m(t)f(source)i (with)f(\(4.2.51\))f(of)1752 3406 y(\024)1741 3431 y(Cerv)m(en)q(\023) -50 b(y)43 b(\(2001\),)e(w)m(e)g(can)f(tak)m(e)h(in)m(to)f(accoun)m(t)h (that)0 3552 y Fl(f)34 3567 y Fg(I)74 3552 y Fq(,)c(giv)m(en)e(b)m(y)i (\(31\),)f(can)g(also)e(b)s(e)i(expressed)j(in)c(the)h(form:)48 b Fl(f)2324 3567 y Fg(I)2397 3552 y Fq(=)33 b Fl(e)2557 3567 y Fg(I)2621 3552 y Fn(\000)25 b Fl(p)p Fq(\()p Fl(e)2874 3515 y Fg(T)2874 3576 y(I)2929 3552 y Fi(U)11 b Fq(\),)36 b(if)e Fl(e)3255 3567 y Fp(1)3331 3552 y Fq(and)h Fl(e)3574 3567 y Fp(2)3649 3552 y Fq(are)0 3672 y(m)m(utually)i(p)s(erp)s (endicular)h(unit)h(v)m(ectors.)64 b(Equations)39 b(fully)f(analogous)g (to)g(\(74\))h(with)f(the)i(ab)s(o)m(v)m(e)0 3792 y(expression)34 b(for)e Fl(f)653 3807 y Fg(I)726 3792 y Fq(w)m(ere)i(also)d(deriv)m(ed) j(b)m(y)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)34 b(and)f(T)-8 b(eles)33 b(\(1996\).)146 3984 y(Equation)41 b(\(71\))g(represen)m(ts)j(the)e(\014nal)e(result)h(of)g (this)g(section.)70 b(It)41 b(expresses)k(the)d(4)27 b Fn(\002)i Fq(4)41 b(ra)m(y)0 4104 y(propagator)c(matrix)f Fc(\005)933 4062 y Fp(\()p Fg(q)r Fp(\))1026 4104 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1198 4119 y Fp(0)1237 4104 y Fq(\))38 b(in)f(ra)m(y-cen)m(tered)j(co)s(ordinates,)f(in)e(terms)h(of) f(the)i(6)25 b Fn(\002)h Fq(4)38 b(matrix)0 4224 y Fl(Z)68 4188 y Fg(r)106 4224 y Fq(\()p Fj(\034)6 b(;)17 b(\034)278 4239 y Fp(0)318 4224 y Fq(\),)51 b(comp)s(osed)c(of)g(four)g(solutions) f(of)h(the)g(dynamic)g(ra)m(y)h(tracing)e(system)i(in)f(Cartesian)0 4345 y(co)s(ordinates,)38 b(with)e(initial)d(conditions)j(sp)s (eci\014ed)i(b)m(y)g(matrix)e Fl(\011)2481 4303 y Fg(r)2481 4369 y Fp(2)2520 4345 y Fq(\()p Fj(\034)2600 4360 y Fp(0)2640 4345 y Fq(\),)i(giv)m(en)f(b)m(y)h(\(72\).)56 b(Dynamic)0 4465 y(ra)m(y)39 b(tracing)f(in)g(ra)m(y-cen)m(tered)i(co)s(ordinates)f (is)f(not)g(necessary)-8 b(.)65 b(Similarly)-8 b(,)36 b(the)j(non-eik)m(onal)e(and)0 4585 y(ra)m(y)c(tangen)m(t)g(solutions)f (are)g(not)h(needed.)0 4982 y Fk(6)161 b(In)l(terface)60 b(propagator)g(matrix)i(in)f(ra)l(y-cen)l(tered)d(co)t(or-)242 5165 y(dinates)0 5454 y Fq(W)-8 b(e)30 b(shall)e(no)m(w)j(apply)e (\(61\))g(to)h(compute)f(the)i(6)16 b Fn(\002)g Fq(6)30 b(in)m(terface)g(propagator)e(matrix)g Fl(\005)3261 5412 y Fp(\()p Fg(q)r Fp(\))3354 5454 y Fq(\()5 b(~)-54 b Fj(\034)3434 5469 y Fp(\006)3489 5454 y Fj(;)17 b(\034)3575 5469 y Fp(\006)3631 5454 y Fq(\))29 b(in)0 5575 y(ra)m(y-cen)m(tered)37 b(co)s(ordinates)d(from)f(the)i(6)24 b Fn(\002)g Fq(6)34 b(in)m(terface)h(propagator)f(matrix)f Fl(\005)3042 5533 y Fp(\()p Fg(x)p Fp(\))3141 5575 y Fq(\()5 b(~)-54 b Fj(\034)3221 5590 y Fp(\006)3276 5575 y Fj(;)17 b(\034)3362 5590 y Fp(\006)3418 5575 y Fq(\))34 b(\(17\))g(in)1817 5824 y(103)p eop %%Page: 104 18 104 17 bop 0 100 a Fq(Cartesian)33 b(co)s(ordinates.)43 b(W)-8 b(e)33 b(obtain)996 320 y Fl(\005)1084 278 y Fp(\()p Fg(q)r Fp(\))1177 320 y Fq(\()5 b(~)-54 b Fj(\034)1257 335 y Fp(\006)1312 320 y Fj(;)17 b(\034)1398 335 y Fp(\006)1454 320 y Fq(\))27 b(=)h Fl(\011)1710 335 y Fp(1)1749 320 y Fq(\()5 b(~)-54 b Fj(\034)1829 335 y Fp(\006)1885 320 y Fq(\))p Fl(\005)2010 278 y Fp(\()p Fg(x)p Fp(\))2109 320 y Fq(\()5 b(~)-54 b Fj(\034)2189 335 y Fp(\006)2244 320 y Fj(;)17 b(\034)2330 335 y Fp(\006)2386 320 y Fq(\))p Fl(\011)2511 335 y Fp(2)2550 320 y Fq(\()p Fj(\034)2630 335 y Fp(\006)2686 320 y Fq(\))32 b Fj(:)823 b Fq(\(75\))0 540 y(Here)37 b Fl(\011)321 555 y Fp(1)396 540 y Fq(and)f Fl(\011)676 555 y Fp(2)751 540 y Fq(are)g(giv)m(en)g(b)m(y)h(\(59\),)f (and)g Fl(\005)1831 498 y Fp(\()p Fg(x)p Fp(\))1930 540 y Fq(\()5 b(~)-54 b Fj(\034)2010 555 y Fp(\006)2066 540 y Fj(;)17 b(\034)2152 555 y Fp(\006)2207 540 y Fq(\))36 b(b)m(y)g(\(17\).)53 b(Mostly)-8 b(,)37 b(ho)m(w)m(ev)m(er,)i(only)c (the)0 660 y(2)26 b Fn(\002)h Fq(2)38 b(matrices)f Fh(Q)749 618 y Fp(\()p Fg(q)r Fp(\))880 660 y Fq(and)i Fh(P)1164 618 y Fp(\()p Fg(q)r Fp(\))1295 660 y Fq(are)g(used.)62 b(The)40 b(relev)-5 b(an)m(t)38 b(4)26 b Fn(\002)h Fq(4)38 b(in)m(terface)g(propagator)g(matrix)0 780 y Fc(\005)106 738 y Fp(\()p Fg(q)r Fp(\))199 780 y Fq(\()5 b(~)-54 b Fj(\034)279 795 y Fp(\006)335 780 y Fj(;)17 b(\034)421 795 y Fp(\006)476 780 y Fq(\))32 b(in)g(ra)m(y-cen)m(tered)j(co)s (ordinates)d(then)h(reads)987 1000 y Fc(\005)1093 958 y Fp(\()p Fg(q)r Fp(\))1186 1000 y Fq(\()5 b(~)-54 b Fj(\034)1266 1015 y Fp(\006)1322 1000 y Fj(;)17 b(\034)1408 1015 y Fp(\006)1463 1000 y Fq(\))28 b(=)f Fl(\011)1719 958 y Fg(r)1719 1025 y Fp(1)1759 1000 y Fq(\()5 b(~)-54 b Fj(\034)1839 1015 y Fp(\006)1894 1000 y Fq(\))p Fl(\005)2020 958 y Fp(\()p Fg(x)p Fp(\))2118 1000 y Fq(\()5 b(~)-54 b Fj(\034)2198 1015 y Fp(\006)2254 1000 y Fj(;)17 b(\034)2340 1015 y Fp(\006)2395 1000 y Fq(\))p Fl(\011)2520 958 y Fg(r)2520 1025 y Fp(2)2560 1000 y Fq(\()p Fj(\034)2640 1015 y Fp(\006)2695 1000 y Fq(\))32 b Fj(;)814 b Fq(\(76\))0 1220 y(where)34 b(the)f(4)22 b Fn(\002)g Fq(6)33 b(matrix)e Fl(\011)1106 1178 y Fg(r)1106 1245 y Fp(1)1178 1220 y Fq(and)h(the)h(6)22 b Fn(\002)h Fq(4)32 b(matrix)f Fl(\011)2192 1178 y Fg(r)2192 1245 y Fp(2)2263 1220 y Fq(are)i(giv)m(en)g(b)m(y)g (\(63\).)146 1412 y(Consequen)m(tly)-8 b(,)41 b(Equations)c(\(75\))g (and)h(\(76\))e(pro)m(vide)i(the)f(p)s(ossibilit)m(y)f(of)g(p)s (erforming)g(the)h(dy-)0 1532 y(namic)j(ra)m(y)h(tracing)f(\(35\))h(in) f(ra)m(y-cen)m(tered)j(co)s(ordinates)d(ev)m(en)j(for)d(anisotropic)g (inhomogeneous)0 1652 y(media)31 b(with)h(structural)h(in)m(terfaces.) 146 1844 y(W)-8 b(e)45 b(no)m(w)f(express)i(the)f(6)29 b Fn(\002)i Fq(6)43 b(in)m(terface)h(propagator)f(matrix)f Fl(\005)2682 1802 y Fp(\()p Fg(q)r Fp(\))2775 1844 y Fq(\()5 b(~)-54 b Fj(\034)2855 1859 y Fp(\006)2910 1844 y Fj(;)17 b(\034)2996 1859 y Fp(\006)3051 1844 y Fq(\))44 b(\(75\))f(in)h(a)f(more)0 1964 y(compact)32 b(form.)42 b(Using)33 b(\(17\))f(and)g(\(56\),)g(w)m(e)i(obtain)d(from)h(\(75\),) 152 2250 y Fl(\005)239 2207 y Fp(\()p Fg(q)r Fp(\))332 2250 y Fq(\()5 b(~)-54 b Fj(\034)412 2265 y Fp(\006)468 2250 y Fj(;)17 b(\034)554 2265 y Fp(\006)609 2250 y Fq(\))27 b(=)778 2103 y Ff( )905 2163 y Fq(~)885 2188 y Fl(H)973 2152 y Fe(\000)p Fp(1)1221 2188 y Fl(0)896 2291 y Fq(~)885 2316 y Fl(F)975 2291 y Fq(~)956 2316 y Fl(H)1044 2280 y Fe(\000)p Fp(1)1240 2291 y Fq(~)1221 2316 y Fl(H)1309 2280 y Fg(T)1405 2103 y Ff(!)16 b( )1612 2163 y Fq(~)1594 2188 y Fl(XX)1764 2152 y Fe(\000)p Fp(1)2135 2188 y Fl(0)1612 2291 y Fq(~)1594 2316 y Fl(X)1679 2280 y Fe(\000)p Fg(T)1789 2316 y Fl(RX)1958 2280 y Fe(\000)p Fp(1)2152 2291 y Fq(~)2135 2316 y Fl(X)2220 2280 y Fe(\000)p Fg(T)2329 2316 y Fl(X)2414 2280 y Fg(T)2510 2103 y Ff(!)h( )2700 2188 y Fl(H)285 b(0)2700 2309 y Fn(\000)p Fl(H)2865 2273 y Fg(T)2920 2309 y Fl(F)83 b(H)3161 2273 y Fe(\000)p Fg(T)3313 2103 y Ff(!)3427 2250 y Fj(:)152 b Fq(\(77\))0 2529 y(This)33 b(yields,)527 2809 y Fl(\005)614 2767 y Fp(\()p Fg(q)r Fp(\))707 2809 y Fq(\()5 b(~)-54 b Fj(\034)787 2824 y Fp(\006)843 2809 y Fj(;)17 b(\034)929 2824 y Fp(\006)984 2809 y Fq(\))27 b(=)1153 2663 y Ff( )1280 2723 y Fq(~)1260 2748 y Fl(K)1348 2712 y Fg(T)1403 2748 y Fl(K)1491 2712 y Fe(\000)p Fg(T)2800 2748 y Fl(0)1280 2851 y Fq(~)1260 2876 y Fl(K)1348 2840 y Fe(\000)p Fp(1)1442 2876 y Fl(RK)1614 2840 y Fe(\000)p Fg(T)1746 2876 y Fq(+)1855 2851 y(~)1844 2876 y Fl(F)1934 2851 y Fq(~)1914 2876 y Fl(K)2002 2840 y Fg(T)2057 2876 y Fl(K)2145 2840 y Fe(\000)p Fg(T)2277 2876 y Fn(\000)2396 2851 y Fq(~)2376 2876 y Fl(K)2464 2840 y Fe(\000)p Fp(1)2558 2876 y Fl(KF)2819 2851 y Fq(~)2800 2876 y Fl(K)2888 2840 y Fe(\000)p Fp(1)2982 2876 y Fl(K)3111 2663 y Ff(!)3226 2809 y Fj(;)353 b Fq(\(78\))0 3089 y(where)1201 3210 y Fl(K)27 b Fq(=)h Fl(X)1505 3168 y Fg(T)1559 3210 y Fl(H)1647 3168 y Fe(\000)p Fg(T)1789 3210 y Fj(;)1983 3184 y Fq(~)1963 3210 y Fl(K)g Fq(=)2200 3184 y(~)2182 3210 y Fl(X)2267 3168 y Fg(T)2341 3184 y Fq(~)2322 3210 y Fl(H)2410 3168 y Fe(\000)p Fg(T)2552 3210 y Fj(:)1027 b Fq(\(79\))0 3384 y(Equation)32 b(\(78\))g(can)h(also)f(b)s(e)h (expressed)i(in)d(a)g(more)g(symmetrical)e(form)563 3669 y Fl(\005)651 3627 y Fp(\()p Fg(q)r Fp(\))744 3669 y Fq(\()5 b(~)-54 b Fj(\034)824 3684 y Fp(\006)879 3669 y Fj(;)17 b(\034)965 3684 y Fp(\006)1020 3669 y Fq(\))28 b(=)1190 3523 y Ff( )1316 3583 y Fq(~)1297 3608 y Fl(K)1385 3572 y Fg(T)1440 3608 y Fl(K)1528 3572 y Fe(\000)p Fg(T)2763 3608 y Fl(0)1316 3711 y Fq(~)1297 3736 y Fl(K)1385 3700 y Fe(\000)p Fp(1)1479 3736 y Fq(\()p Fl(R)22 b Fq(+)1740 3711 y(~)1721 3736 y Fl(K)1819 3711 y Fq(~)1808 3736 y Fl(F)1898 3711 y Fq(~)1879 3736 y Fl(K)1967 3700 y Fg(T)2044 3736 y Fn(\000)g Fl(KFK)2389 3700 y Fg(T)2444 3736 y Fq(\))p Fl(K)2570 3700 y Fe(\000)p Fg(T)2782 3711 y Fq(~)2763 3736 y Fl(K)2851 3700 y Fe(\000)p Fp(1)2945 3736 y Fl(K)3074 3523 y Ff(!)3189 3669 y Fj(;)390 b Fq(\(80\))0 3962 y(It)37 b(ma)m(y)f(b)s(e)h(useful)f(to)g(express)j Fl(H)1300 3925 y Fe(\000)p Fg(T)1446 3962 y Fq(and)e Fl(K)1728 3925 y Fe(\000)p Fg(T)1874 3962 y Fq(explicitly)-8 b(,)35 b Fl(H)2408 3925 y Fe(\000)p Fg(T)2552 3962 y Fq(=)f Fl(S)j Fq(and)g Fl(X)3040 3925 y Fe(\000)p Fg(T)3184 3962 y Fq(=)d Fl(U)p Fq(,)j(see)h(\(20\))0 4082 y(and)33 b(\(29\).)43 b(Then)33 b(w)m(e)h(obtain)806 4299 y Fl(K)27 b Fq(=)h Fl(X)1110 4263 y Fg(T)1164 4299 y Fl(H)1252 4263 y Fe(\000)p Fg(T)1389 4299 y Fq(=)g Fl(X)1578 4263 y Fg(T)1633 4299 y Fl(S)k Fj(;)205 b Fl(K)2047 4263 y Fg(T)2129 4299 y Fq(=)28 b Fl(H)2321 4263 y Fe(\000)p Fp(1)2415 4299 y Fl(X)f Fq(=)h Fl(S)2693 4263 y Fg(T)2748 4299 y Fl(X)k Fj(;)806 4420 y Fl(K)894 4383 y Fe(\000)p Fp(1)1015 4420 y Fq(=)c Fl(H)1207 4383 y Fg(T)1262 4420 y Fl(X)1347 4383 y Fe(\000)p Fg(T)1484 4420 y Fq(=)f Fl(H)1675 4383 y Fg(T)1730 4420 y Fl(U)33 b Fj(;)83 b Fl(K)2047 4383 y Fe(\000)p Fg(T)2184 4420 y Fq(=)28 b Fl(X)2373 4383 y Fe(\000)p Fp(1)2467 4420 y Fl(H)f Fq(=)h Fl(U)2772 4383 y Fg(T)2827 4420 y Fl(H)k Fj(;)3606 4360 y Fq(\(81\))0 4645 y(and)i(similarly)c(for)j(the)h(tilded)f(quan)m (tities.)47 b(Consequen)m(tly)-8 b(,)36 b(all)c(3)23 b Fn(\002)g Fq(3)34 b(matrices)f Fl(K)p Fj(;)17 b Fl(K)3309 4609 y Fg(T)3363 4645 y Fj(;)g Fl(K)3495 4609 y Fe(\000)p Fp(1)3622 4645 y Fq(and)0 4765 y Fl(K)88 4729 y Fe(\000)p Fg(T)239 4765 y Fq(are)41 b(expressed)j(in)c(terms)h(of)g(scalar)f(pro) s(ducts)i(of)f(the)h(basis)f(v)m(ectors)h(of)f(the)h(ra)m(y-cen)m (tered)0 4886 y(co)s(ordinate)29 b(system)i(and)g(the)f(basis)g(v)m (ectors)i(of)e(the)g(lo)s(cal)e(co)s(ordinate)i(system)h(at)f(in)m (terface)g(\006.)43 b(As)0 5006 y(will)35 b(b)s(e)j(explained)g(b)s (elo)m(w,)h(this)e(giv)m(es)i(a)e(clear)g(ph)m(ysical)h(meaning)e(to)i (all)e(matrices)h Fl(K)p Fj(;)17 b Fl(K)3500 4970 y Fg(T)3554 5006 y Fj(;)g Fl(K)3686 4970 y Fe(\000)p Fp(1)0 5127 y Fq(and)33 b Fl(K)278 5090 y Fe(\000)p Fg(T)387 5127 y Fq(.)146 5318 y(W)-8 b(e)33 b(can)g(also)f(replace)g(matrix)f Fl(F)i Fq(b)m(y)h Fl(V)g Fq(using)e(\(57\),)1119 5538 y Fl(KFK)1365 5496 y Fg(T)1448 5538 y Fq(=)c Fl(X)1637 5497 y Fg(T)1691 5538 y Fl(H)1779 5497 y Fe(\000)p Fg(T)1889 5538 y Fl(FK)2047 5497 y Fg(T)2130 5538 y Fq(=)f Fl(X)2318 5497 y Fg(T)2373 5538 y Fl(VK)2546 5497 y Fg(T)2633 5538 y Fj(;)946 b Fq(\(82\))1817 5824 y(104)p eop %%Page: 105 19 105 18 bop 0 100 a Fq(and)33 b(similarly)c(for)757 74 y(~)738 100 y Fl(K)836 74 y Fq(~)825 100 y Fl(F)915 74 y Fq(~)896 100 y Fl(K)984 63 y Fg(T)1039 100 y Fq(.)146 291 y(It)46 b(is)f(simple)f(to)h(sho)m(w)i(that)e(the)h(3)31 b Fn(\002)g Fq(3)45 b(matrices)g Fl(K)p Fj(;)17 b Fl(K)2391 255 y Fe(\000)p Fp(1)2485 291 y Fj(;)g Fl(K)2617 255 y Fg(T)2717 291 y Fq(and)45 b Fl(K)3007 255 y Fe(\000)p Fg(T)3162 291 y Fq(ha)m(v)m(e)i(a)e(sp)s(ecial)0 411 y(form.)59 b(The)39 b(third)f(columns)f(of)h(matrices)g Fl(K)1728 375 y Fg(T)1821 411 y Fq(and)g Fl(K)2104 375 y Fe(\000)p Fg(T)2252 411 y Fq(are)g(\(0)p Fj(;)17 b Fq(0)p Fj(;)g Fq(1\))2731 375 y Fg(T)2785 411 y Fq(,)39 b(and)g(the)f(third)g(ro)m(ws)h(of)0 532 y Fl(K)d Fq(and)h Fl(K)406 495 y Fe(\000)p Fp(1)537 532 y Fq(are)f(\(0)p Fj(;)17 b Fq(0)p Fj(;)g Fq(1\).)54 b(This)37 b(follo)m(ws)e(from)h(the) h(prop)s(ert)m(y)g(that)g(basis)f(v)m(ectors)i Fl(f)3283 547 y Fp(1)3323 532 y Fj(;)17 b Fl(f)3401 547 y Fp(2)3440 532 y Fj(;)g Fl(h)3546 547 y Fp(1)3622 532 y Fq(and)0 652 y Fl(h)62 667 y Fp(2)140 652 y Fq(are)39 b(p)s(erp)s(endicular)f (to)h Fi(U)10 b Fq(,)41 b(and)e(that)f Fl(p)1683 616 y Fg(T)1739 652 y Fi(U)48 b Fq(=)38 b Fl(h)2034 616 y Fg(T)2034 677 y Fp(3)2090 652 y Fi(U)48 b Fq(=)39 b(1.)62 b(Analogous)38 b(prop)s(erties)g(are)h(v)-5 b(alid)0 772 y(also)41 b(for)h(the)g(tilded)f(quan)m(tities.)72 b(The)43 b(partition)d(matrices)2390 747 y(~)2370 772 y Fl(K)2458 736 y Fg(T)2513 772 y Fl(K)2601 736 y Fe(\000)p Fp(1)2737 772 y Fq(and)2956 747 y(~)2936 772 y Fl(K)3024 736 y Fe(\000)p Fp(1)3119 772 y Fl(K)h Fq(in)h(\(80\))f(ha)m(v)m(e)0 893 y(a)d(similar)d(form:)54 b(the)38 b(third)g(column)f(of)1599 867 y(~)1580 893 y Fl(K)1668 857 y Fg(T)1722 893 y Fl(K)1810 857 y Fe(\000)p Fp(1)1943 893 y Fq(is)h(\(0)p Fj(;)17 b Fq(0)p Fj(;)g Fq(1\))2358 857 y Fg(T)2411 893 y Fq(,)40 b(and)e(the)h(third)e(ro)m(w)i(of)3425 867 y(~)3406 893 y Fl(K)3494 857 y Fe(\000)p Fp(1)3588 893 y Fl(K)f Fq(is)0 1013 y(\(0)p Fj(;)17 b Fq(0)p Fj(;)g Fq(1\).)47 b(In)35 b(addition,)e(all)f(elemen)m(ts)i(of)g(the)h(third)e(ro)m(w)i(and)f (third)g(column)f(of)h(the)g(3)23 b Fn(\002)h Fq(3)34 b(matrix)0 1134 y Fl(R)22 b Fq(+)223 1108 y(~)204 1134 y Fl(K)302 1108 y Fq(~)291 1134 y Fl(F)381 1108 y Fq(~)362 1134 y Fl(K)450 1097 y Fg(T)527 1134 y Fn(\000)h Fl(KFK)873 1097 y Fg(T)927 1134 y Fq(,)33 b(and)g(consequen)m(tly)h(of)1885 1108 y(~)1866 1134 y Fl(K)1954 1097 y Fe(\000)p Fp(1)2048 1134 y Fq(\()p Fl(R)21 b Fq(+)2309 1108 y(~)2289 1134 y Fl(K)2388 1108 y Fq(~)2377 1134 y Fl(F)2467 1108 y Fq(~)2448 1134 y Fl(K)2536 1097 y Fg(T)2612 1134 y Fn(\000)i Fl(KFK)2958 1097 y Fg(T)3013 1134 y Fq(\))p Fl(K)3139 1097 y Fe(\000)p Fg(T)3249 1134 y Fq(,)32 b(v)-5 b(anish.)146 1325 y(Then)29 b(w)m(e)g(can)g(express)h(the)e(6)13 b Fn(\002)g Fq(6)28 b(in)m(terface)g(propagator)e(matrix)h(in)g(ra)m (y-cen)m(tered)j(co)s(ordinates)0 1445 y(as)j(follo)m(ws:)421 1845 y Fl(\005)508 1803 y Fp(\()p Fg(q)r Fp(\))601 1845 y Fq(\()5 b(~)-54 b Fj(\034)681 1860 y Fp(\006)737 1845 y Fj(;)17 b(\034)823 1860 y Fp(\006)878 1845 y Fq(\))27 b(=)1047 1575 y Ff(0)1047 1721 y(B)1047 1771 y(B)1047 1821 y(B)1047 1870 y(B)1047 1924 y(@)1188 1638 y Fq(~)1161 1664 y Fc(K)1264 1598 y Fg(T)1319 1664 y Fc(K)1423 1622 y Fe(\000)p Fg(T)2538 1664 y Fc(0)96 b(0)325 b(0)1161 1787 y(A)1247 1745 y Fg(T)2538 1787 y Fq(1)106 b Fc(0)2765 1749 y Fg(T)3077 1787 y Fq(0)1188 1906 y(~)1161 1932 y Fc(K)1264 1866 y Fe(\000)p Fp(1)1359 1932 y Fq([)p Fc(E)34 b Fn(\000)1613 1906 y Fq(~)1594 1932 y Fc(E)g Fn(\000)23 b Fq(\()p Fj(\033)j Fn(\000)i Fq(~)-54 b Fj(\033)t Fq(\))p Fc(D)9 b Fq(])p Fc(K)2345 1889 y Fe(\000)p Fg(T)2538 1932 y Fc(0)2721 1906 y Fq(~)2693 1932 y Fc(K)2797 1866 y Fe(\000)p Fp(1)2891 1932 y Fc(K)97 b(B)1161 2052 y(0)1233 2014 y Fg(T)2538 2052 y Fq(0)106 b Fc(0)2765 2014 y Fg(T)3077 2052 y Fq(1)3210 1575 y Ff(1)3210 1721 y(C)3210 1771 y(C)3210 1821 y(C)3210 1870 y(C)3210 1924 y(A)3332 1845 y Fj(:)247 b Fq(\(83\))0 2283 y(Here)38 b Fc(K)14 b Fj(;)j Fc(K)485 2241 y Fg(T)540 2283 y Fj(;)g Fc(K)687 2241 y Fe(\000)p Fp(1)819 2283 y Fq(and)38 b Fc(K)1117 2241 y Fe(\000)p Fg(T)1264 2283 y Fq(are)g(the)f(2)26 b Fn(\002)f Fq(2)38 b(upp)s(er)f(left)g(submatrices)g(of)g(the)h(3)25 b Fn(\002)h Fq(3)37 b(matrices)0 2404 y Fl(K)p Fj(;)17 b Fl(K)220 2368 y Fg(T)274 2404 y Fj(;)g Fl(K)406 2368 y Fe(\000)p Fp(1)540 2404 y Fq(and)39 b Fl(K)824 2368 y Fe(\000)p Fg(T)934 2404 y Fq(,)i(resp)s(ectiv)m(ely)-8 b(.)66 b(They)41 b(are)e(expressed)j(as)e(scalar)f(pro)s(ducts)h(of)g (the)g(basis)0 2524 y(v)m(ectors)34 b(as)f(follo)m(ws:)773 2720 y Fc(K)42 b Fq(=)28 b(\()p Fl(g)1102 2735 y Fp(1)1141 2720 y Fj(;)17 b Fl(g)1241 2735 y Fp(2)1280 2720 y Fq(\))1318 2683 y Fg(T)1373 2720 y Fq(\()p Fl(f)1445 2735 y Fp(1)1485 2720 y Fj(;)g Fl(f)1563 2735 y Fp(2)1602 2720 y Fq(\))33 b Fj(;)223 b Fc(K)2027 2678 y Fg(T)2110 2720 y Fq(=)27 b(\()p Fl(f)2285 2735 y Fp(1)2325 2720 y Fj(;)17 b Fl(f)2403 2735 y Fp(2)2442 2720 y Fq(\))2480 2683 y Fg(T)2535 2720 y Fq(\()p Fl(g)2629 2735 y Fp(1)2668 2720 y Fj(;)g Fl(g)2768 2735 y Fp(2)2808 2720 y Fq(\))32 b Fj(;)773 2843 y Fc(K)877 2801 y Fe(\000)p Fp(1)999 2843 y Fq(=)27 b(\()p Fl(e)1191 2858 y Fp(1)1231 2843 y Fj(;)17 b Fl(e)1326 2858 y Fp(2)1365 2843 y Fq(\))1403 2807 y Fg(T)1458 2843 y Fq(\()p Fl(h)1558 2858 y Fp(1)1597 2843 y Fj(;)g Fl(h)1703 2858 y Fp(2)1743 2843 y Fq(\))32 b Fj(;)83 b Fc(K)2027 2801 y Fe(\000)p Fg(T)2164 2843 y Fq(=)28 b(\()p Fl(h)2368 2858 y Fp(1)2408 2843 y Fj(;)17 b Fl(h)2514 2858 y Fp(2)2553 2843 y Fq(\))2591 2807 y Fg(T)2646 2843 y Fq(\()p Fl(e)2735 2858 y Fp(1)2774 2843 y Fj(;)g Fl(e)2869 2858 y Fp(2)2909 2843 y Fq(\))32 b Fj(:)3606 2781 y Fq(\(84\))0 3056 y(The)g(scalar)e(quan)m(tities)h Fj(\033)k Fq(and)i(~)-54 b Fj(\033)35 b Fq(represen)m(t)e(the)e(comp)s (onen)m(ts)h(of)e(slo)m(wness)j(v)m(ectors)f Fl(p)g Fq(and)3528 3055 y(~)3521 3056 y Fl(p)f Fq(in)m(to)0 3177 y Fl(n)62 3141 y Fp(\006)117 3177 y Fq(,)i Fj(\033)f Fq(=)27 b Fl(p)429 3141 y Fg(T)485 3177 y Fl(n)547 3141 y Fp(\006)634 3177 y Fq(and)38 b(~)-54 b Fj(\033)32 b Fq(=)1021 3176 y(~)1014 3177 y Fl(p)1076 3141 y Fg(T)1132 3177 y Fl(n)1194 3141 y Fp(\006)1249 3177 y Fq(.)44 b(The)33 b(2)22 b Fn(\002)h Fq(2)32 b Fd(curvatur)-5 b(e)35 b(matrix)d Fc(D)42 b Fq(is)32 b(giv)m(en)h(b)m(y)h(its)e(elemen)m(ts)1582 3397 y Fj(D)1663 3412 y Fg(I)5 b(J)1776 3397 y Fq(=)27 b Fl(g)1937 3356 y Fg(T)1935 3421 y(I)5 b(J)2020 3397 y Fl(n)2082 3356 y Fp(\006)2170 3397 y Fj(;)1409 b Fq(\(85\))0 3617 y(Note)40 b(that)f Fc(D)48 b Fq(=)40 b Fc(0)52 b Fq(for)39 b(plane)g(surface)h(\006.)65 b(The)40 b(2)27 b Fn(\002)g Fq(2)39 b Fd(inhomo)-5 b(geneity)40 b(matrix)f Fc(E)51 b Fq(is)39 b(giv)m(en)h(b)m(y)0 3737 y(elemen)m(ts:)1004 3858 y Fj(E)1076 3873 y Fg(I)5 b(J)1189 3858 y Fq(=)27 b(\()p Fl(g)1388 3816 y Fg(T)1386 3882 y(I)1443 3858 y Fl(p)p Fq(\)\()p Fl(e)1632 3816 y Fg(T)1632 3882 y(K)1700 3858 y Fh(\021)t Fq(\)\()p Fl(g)1898 3816 y Fg(T)1896 3882 y(J)1952 3858 y Fl(f)1986 3873 y Fg(K)2055 3858 y Fq(\))22 b(+)g(\()p Fl(g)2309 3816 y Fg(T)2307 3882 y(I)2364 3858 y Fh(\021)s Fq(\)\()p Fl(g)2561 3816 y Fg(T)2559 3882 y(J)2616 3858 y Fl(p)p Fq(\))32 b Fj(:)831 b Fq(\(86\))0 4032 y(Matrix)345 4006 y(~)326 4032 y Fc(E)46 b Fq(is)33 b(quite)h(analogous.)46 b(Note)33 b(that)h Fc(E)41 b Fq(=)30 b Fc(0)47 b Fq(for)33 b(homogeneous)h(media,)e(as)i Fh(\021)f Fq(=)d Fl(0)k Fq(there.)0 4152 y(Here)f(and)g(ab)s(o)m(v)m (e,)g Fc(0)46 b Fq(is)32 b(a)g(2)22 b Fn(\002)h Fq(1)32 b(or)g(2)22 b Fn(\002)h Fq(2)32 b(zero)h(matrix.)146 4344 y(It)42 b(remains)f(to)h(giv)m(e)g(the)g(expressions)i(for)e(the)g (2)28 b Fn(\002)i Fq(1)41 b(matrices)g Fc(A)j Fq(=)g(\()p Fj(A)3039 4359 y Fp(1)3078 4344 y Fj(;)17 b(A)3195 4359 y Fp(2)3235 4344 y Fq(\))3273 4307 y Fg(T)3370 4344 y Fq(and)42 b Fc(B)54 b Fq(=)0 4464 y(\()p Fj(B)112 4479 y Fp(1)151 4464 y Fj(;)17 b(B)269 4479 y Fp(2)309 4464 y Fq(\))347 4428 y Fg(T)402 4464 y Fq(:)549 4684 y Fj(A)622 4699 y Fg(I)690 4684 y Fq(=)27 b(\()p Fl(e)882 4643 y Fg(T)882 4709 y(I)938 4684 y Fl(h)1000 4699 y Fg(K)1068 4684 y Fq(\)\()p Fl(g)1202 4643 y Fg(T)1200 4709 y(K)1268 4684 y Fl(p)p Fq(\))c(+)f(\()p Fl(e)1578 4643 y Fg(T)1578 4709 y(I)1633 4684 y Fl(h)1695 4699 y Fp(3)1735 4684 y Fq(\))32 b Fj(;)114 b(B)2020 4699 y Fg(I)2088 4684 y Fq(=)28 b(\()2231 4683 y(~)2230 4684 y Fl(e)2281 4643 y Fg(T)2281 4709 y(I)2342 4658 y Fq(~)2336 4684 y Fl(h)2398 4699 y Fg(K)2466 4684 y Fq(\)\()p Fl(g)2600 4643 y Fg(T)2598 4709 y(K)2673 4683 y Fq(~)2667 4684 y Fl(p)p Fq(\))22 b(+)g(\()2926 4683 y(~)2925 4684 y Fl(e)2976 4643 y Fg(T)2976 4709 y(I)3038 4658 y Fq(~)3031 4684 y Fl(h)3093 4699 y Fp(3)3133 4684 y Fq(\))32 b Fj(:)376 b Fq(\(87\))0 4904 y(In)33 b(fact,)f(it)g(is)g(easy)i(to)e(sho)m(w)i(that)e Fc(A)h Fq(and)f Fc(B)43 b Fq(are)33 b Fc(0)13 b Fq(.)43 b(As)33 b Fl(g)2263 4868 y Fg(T)2261 4929 y Fp(3)2318 4904 y Fl(p)28 b Fq(=)f(1,)33 b(w)m(e)g(obtain)f(from)f(\(87\))914 5124 y Fj(A)987 5139 y Fg(I)1055 5124 y Fq(=)d Fl(e)1210 5083 y Fg(T)1210 5149 y(I)1265 5124 y Fl(h)1327 5139 y Fg(K)1395 5124 y Fl(g)1453 5083 y Fg(T)1451 5149 y(K)1520 5124 y Fl(p)22 b Fq(+)g Fl(e)1753 5083 y Fg(T)1753 5149 y(I)1808 5124 y Fl(h)1870 5139 y Fp(3)1910 5124 y Fl(g)1968 5083 y Fg(T)1966 5149 y Fp(3)2023 5124 y Fl(p)28 b Fq(=)f Fl(e)2267 5083 y Fg(T)2267 5149 y(I)2322 5124 y Fl(h)2384 5139 y Fg(k)2427 5124 y Fl(g)2485 5083 y Fg(T)2483 5149 y(k)2540 5124 y Fl(p)h Fq(=)g Fc(0)45 b Fj(:)741 b Fq(\(88\))0 5344 y(Here)25 b(w)m(e)h(ha)m(v)m(e)g(used)g Fl(h)852 5359 y Fg(k)895 5344 y Fl(g)953 5308 y Fg(T)951 5369 y(k)1035 5344 y Fq(=)i Fl(I)p Fq(,)e(see)f(\(24\),)h(and)f Fl(e)1843 5308 y Fg(T)1843 5369 y(I)1898 5344 y Fl(p)j Fq(=)f(0.)41 b(Quite)24 b(analogously)-8 b(,)25 b(w)m(e)g(obtain)f Fj(B)3518 5359 y Fg(I)3586 5344 y Fq(=)j Fd(0)14 b Fq(.)0 5464 y(Consequen)m(tly)-8 b(,)38 b(the)d(\014nal)f(expression)i(for)e (the)h(6)24 b Fn(\002)g Fq(6)34 b(in)m(terface)h(propagator)f(matrix)f Fl(\005)3372 5422 y Fp(\()p Fg(q)r Fp(\))3465 5464 y Fq(\()5 b(~)-54 b Fj(\034)3545 5479 y Fp(\006)3600 5464 y Fj(;)17 b(\034)3686 5479 y Fp(\006)3742 5464 y Fq(\))1817 5824 y(105)p eop %%Page: 106 20 106 19 bop 0 100 a Fq(in)32 b(ra)m(y-cen)m(tered)i(co)s(ordinates)e (reads:)430 517 y Fl(\005)518 475 y Fp(\()p Fg(q)r Fp(\))611 517 y Fq(\()5 b(~)-54 b Fj(\034)691 532 y Fp(\006)746 517 y Fj(;)17 b(\034)832 532 y Fp(\006)887 517 y Fq(\))28 b(=)1057 246 y Ff(0)1057 392 y(B)1057 442 y(B)1057 492 y(B)1057 542 y(B)1057 595 y(@)1198 311 y Fq(~)1171 337 y Fc(K)1274 272 y Fg(T)1329 337 y Fc(K)1432 295 y Fe(\000)p Fg(T)2548 337 y Fc(0)96 b(0)325 b(0)1171 457 y(0)1243 419 y Fg(T)2548 457 y Fq(1)106 b Fc(0)2775 419 y Fg(T)3087 457 y Fq(0)1198 576 y(~)1171 601 y Fc(K)1274 536 y Fe(\000)p Fp(1)1368 601 y Fq([)p Fc(E)34 b Fn(\000)1623 576 y Fq(~)1604 601 y Fc(E)g Fn(\000)23 b Fq(\()p Fj(\033)j Fn(\000)h Fq(~)-54 b Fj(\033)t Fq(\))p Fc(D)9 b Fq(])p Fc(K)2355 559 y Fe(\000)p Fg(T)2548 601 y Fc(0)2730 576 y Fq(~)2703 601 y Fc(K)2806 536 y Fe(\000)p Fp(1)2901 601 y Fc(K)97 b(0)1171 722 y(0)1243 684 y Fg(T)2548 722 y Fq(0)106 b Fc(0)2775 684 y Fg(T)3087 722 y Fq(1)3200 246 y Ff(1)3200 392 y(C)3200 442 y(C)3200 492 y(C)3200 542 y(C)3200 595 y(A)3322 517 y Fj(:)257 b Fq(\(89\))146 1011 y(The)34 b(6)22 b Fn(\002)h Fq(6)33 b(in)m(terface)g(propagator)f(matrix)f (\(89\))h(in)g(ra)m(y-cen)m(tered)j(co)s(ordinates)d(is)h(symplectic,)0 1131 y(and)24 b(its)f(determinan)m(t)h(equals)g(unit)m(y)-8 b(.)41 b(The)25 b(con)m(tin)m(uation)e(prop)s(ert)m(y)h(across)h(in)m (terface)f(\006)g(then)h(reads:)1115 1275 y Ff(0)1115 1421 y(B)1115 1471 y(B)1115 1521 y(B)1115 1570 y(B)1115 1620 y(B)1115 1673 y(@)1252 1347 y Fq(~)1229 1373 y Fc(Q)1323 1307 y Fp(\()p Fg(q)r Fp(\))1251 1484 y Fq(~)1229 1509 y Fj(Q)1306 1459 y Fp(\()p Fg(q)r Fp(\))1306 1531 y(3)1249 1635 y Fq(~)1229 1661 y Fc(P)1317 1595 y Fp(\()p Fg(q)r Fp(\))1251 1772 y Fq(~)1229 1798 y Fj(P)1306 1747 y Fp(\()p Fg(q)r Fp(\))1292 1819 y(3)1458 1275 y Ff(1)1458 1421 y(C)1458 1471 y(C)1458 1521 y(C)1458 1570 y(C)1458 1620 y(C)1458 1673 y(A)1558 1570 y Fq(=)i Fl(\005)1749 1528 y Fp(\()p Fg(q)r Fp(\))1842 1570 y Fq(\()5 b(~)-54 b Fj(\034)1922 1585 y Fp(\006)1977 1570 y Fj(;)17 b(\034)2063 1585 y Fp(\006)2119 1570 y Fq(\))2174 1300 y Ff(0)2173 1446 y(B)2173 1496 y(B)2173 1546 y(B)2173 1595 y(B)2173 1649 y(@)2287 1373 y Fc(Q)2381 1331 y Fp(\()p Fg(q)r Fp(\))2287 1509 y Fj(Q)2364 1459 y Fp(\()p Fg(q)r Fp(\))2364 1531 y(3)2287 1637 y Fc(P)2376 1595 y Fp(\()p Fg(q)r Fp(\))2287 1774 y Fj(P)2364 1723 y Fp(\()p Fg(q)r Fp(\))2350 1796 y(3)2516 1300 y Ff(1)2516 1446 y(C)2516 1496 y(C)2516 1546 y(C)2516 1595 y(C)2516 1649 y(A)2638 1570 y Fj(;)941 b Fq(\(90\))0 2042 y(where)48 b Fl(\005)384 1999 y Fp(\()p Fg(q)r Fp(\))477 2042 y Fq(\()5 b(~)-54 b Fj(\034)557 2057 y Fp(\006)612 2042 y Fj(;)17 b(\034)698 2057 y Fp(\006)753 2042 y Fq(\))47 b(is)g(a)f(6)32 b Fn(\002)h Fq(6)46 b(matrix)g(giv)m (en)h(b)m(y)h(\(89\),)i Fc(Q)2428 1999 y Fp(\()p Fg(q)r Fp(\))2573 2042 y Fn(\021)j Fq(\()p Fj(Q)2818 1991 y Fp(\()p Fg(q)r Fp(\))2818 2063 y(1)2911 2042 y Fj(;)17 b(Q)3032 1991 y Fp(\()p Fg(q)r Fp(\))3032 2063 y(2)3125 2042 y Fq(\))3163 2005 y Fg(T)3265 2042 y Fq(and)47 b Fc(P)3557 1999 y Fp(\()p Fg(q)r Fp(\))3702 2042 y Fn(\021)0 2175 y Fq(\()p Fj(P)115 2125 y Fp(\()p Fg(q)r Fp(\))101 2197 y(1)207 2175 y Fj(;)17 b(P)328 2125 y Fp(\()p Fg(q)r Fp(\))314 2197 y(2)420 2175 y Fq(\))458 2139 y Fg(T)513 2175 y Fq(.)62 b(The)40 b(sixth)f(line)e(of)i(\(89\))f(indicates)g (that)h(the)g(in)m(terface)g(propagator)e(matrix)h(pre-)0 2309 y(serv)m(es)g(the)e(constrain)m(t)g(relation)e Fj(P)1357 2258 y Fp(\()p Fg(q)r Fp(\))1343 2331 y(3)1483 2309 y Fq(=)f(0)j(ev)m(en)h(across)g(the)f(in)m(terface,)h(see)g(\(41\).)52 b(Similarly)-8 b(,)33 b(the)0 2458 y(third)27 b(line)f(of)i(\(89\))f (sho)m(ws)i(that)e Fj(Q)1284 2407 y Fp(\()p Fg(q)r Fp(\))1284 2479 y(3)1405 2458 y Fq(remains)g(constan)m(t)h(across)h(the)f(in)m (terface.)42 b(F)-8 b(urther,)3430 2432 y(~)3408 2458 y Fc(Q)3502 2392 y Fp(\()p Fg(q)r Fp(\))3622 2458 y Fq(and)20 2581 y(~)0 2606 y Fc(P)88 2541 y Fp(\()p Fg(q)r Fp(\))214 2606 y Fq(do)32 b(not)h(dep)s(end)g(on)g Fj(Q)1074 2555 y Fp(\()p Fg(q)r Fp(\))1074 2628 y(3)1167 2606 y Fq(.)146 2797 y(Consequen)m(tly)-8 b(,)34 b(w)m(e)f(can)f(in)m(tro)s(duce)f(the) h(4)20 b Fn(\002)g Fq(4)32 b(in)m(terface)f(propagator)g(matrix)f(in)h (ra)m(y-cen)m(tered)0 2918 y(co)s(ordinates)h Fc(\005)625 2876 y Fp(\()p Fg(q)r Fp(\))717 2918 y Fq(\()5 b(~)-54 b Fj(\034)797 2933 y Fp(\006)853 2918 y Fj(;)17 b(\034)939 2933 y Fp(\006)994 2918 y Fq(\):)565 3224 y Fc(\005)671 3182 y Fp(\()p Fg(q)r Fp(\))764 3224 y Fq(\()5 b(~)-54 b Fj(\034)844 3239 y Fp(\006)900 3224 y Fj(;)17 b(\034)986 3239 y Fp(\006)1041 3224 y Fq(\))28 b(=)1210 3053 y Ff(0)1210 3202 y(@)1352 3139 y Fq(~)1324 3164 y Fc(K)1428 3099 y Fg(T)1483 3164 y Fc(K)1586 3122 y Fe(\000)p Fg(T)2723 3164 y Fc(0)1352 3283 y Fq(~)1324 3309 y Fc(K)1428 3243 y Fe(\000)p Fp(1)1522 3309 y Fq(\()p Fc(E)34 b Fn(\000)1787 3283 y Fq(~)1768 3309 y Fc(E)g Fn(\000)23 b Fq(\()p Fj(\033)j Fn(\000)i Fq(~)-54 b Fj(\033)t Fq(\))p Fc(D)9 b Fq(\))p Fc(K)2530 3266 y Fe(\000)p Fg(T)2750 3283 y Fq(~)2723 3309 y Fc(K)2826 3243 y Fe(\000)p Fp(1)2921 3309 y Fc(K)3066 3053 y Ff(1)3066 3202 y(A)3187 3224 y Fj(:)392 b Fq(\(91\))0 3548 y(The)34 b(con)m(tin)m(uation)d(prop)s(ert)m(y)i(for)g Fc(Q)1406 3506 y Fp(\()p Fg(q)r Fp(\))1531 3548 y Fq(and)g Fc(P)1809 3506 y Fp(\()p Fg(q)r Fp(\))1935 3548 y Fq(then)g(reads:)1112 3679 y Ff(0)1112 3829 y(@)1249 3764 y Fq(~)1226 3789 y Fc(Q)1321 3724 y Fp(\()p Fg(q)r Fp(\))1246 3915 y Fq(~)1226 3940 y Fc(P)1315 3875 y Fp(\()p Fg(q)r Fp(\))1455 3679 y Ff(1)1455 3829 y(A)1555 3850 y Fq(=)28 b Fc(\005)1765 3808 y Fp(\()p Fg(q)r Fp(\))1858 3850 y Fq(\()5 b(~)-54 b Fj(\034)1938 3865 y Fp(\006)1994 3850 y Fj(;)17 b(\034)2080 3865 y Fp(\006)2135 3850 y Fq(\))2190 3704 y Ff( )2297 3789 y Fc(Q)2391 3747 y Fp(\()p Fg(q)r Fp(\))2297 3917 y Fc(P)2385 3875 y Fp(\()p Fg(q)r Fp(\))2525 3704 y Ff(!)2640 3850 y Fj(:)939 b Fq(\(92\))146 4229 y(Equations)28 b(\(91\))f(and)h (\(92\))f(represen)m(t)j(the)e(\014nal)e(results)j(of)e(this)g (section.)42 b(It)28 b(w)m(ould)f(b)s(e)h(p)s(ossible)0 4350 y(to)f(deriv)m(e)h(them)f(directly)f(from)g(\(76\).)41 b(W)-8 b(e,)29 b(ho)m(w)m(ev)m(er,)i(wished)c(to)g(explain)g(the)g(ph)m (ysical)g(argumen)m(ts)0 4470 y(leading)35 b(to)i(the)h (simpli\014cation)33 b(of)k(the)g(6)25 b Fn(\002)h Fq(6)37 b(to)g(4)24 b Fn(\002)i Fq(4)37 b(matrices.)56 b(F)-8 b(or)36 b(this)h(reason,)i(w)m(e)f(started)0 4590 y(the)33 b(deriv)-5 b(ation)31 b(from)g(Equation)i(\(75\).)146 4782 y(The)g(4)20 b Fn(\002)g Fq(4)31 b(in)m(terface)h(propagator)e (matrix)g Fc(\005)1907 4740 y Fp(\()p Fg(q)r Fp(\))2000 4782 y Fq(\()5 b(~)-54 b Fj(\034)2080 4797 y Fp(\006)2135 4782 y Fj(;)17 b(\034)2221 4797 y Fp(\006)2276 4782 y Fq(\))32 b(in)f(ra)m(y-cen)m(tered)i(co)s(ordinates,)e(giv)m(en)0 4902 y(b)m(y)j(\(91\),)e(is)g(symplectic,)g(and)h(the)g(determinan)m(t) f(of)h(\(91\))f(equals)h(unit)m(y)-8 b(.)43 b(Basis)33 b(v)m(ectors)h Fl(g)3422 4917 y Fp(1)3494 4902 y Fq(and)f Fl(g)3740 4917 y Fp(2)0 5022 y Fq(are)j(tangen)m(t)h(to)f(surface)h (\006,)g(but)g(need)g(not)f(to)g(b)s(e)g(unit)g(or)f(m)m(utually)g(p)s (erp)s(endicular.)53 b(Similarly)-8 b(,)0 5143 y Fl(n)62 5107 y Fp(\006)150 5143 y Fq(is)32 b(p)s(erp)s(endicular)g(to)g(\006,)h (but)g(need)h(not)e(b)s(e)h(unit.)146 5334 y(The)26 b(4)7 b Fn(\002)g Fq(4)25 b(in)m(terface)g(propagator)f(matrix)f(\(91\))i(in) f(ra)m(y-cen)m(tered)j(co)s(ordinates)d(is)h(expressed)j(in)c(a)0 5454 y(form)d(similar)f(to)i(the)h(4)r Fn(\002)r Fq(4)f(in)m(terface)h (propagator)e(matrix)g(in)h(w)m(a)m(v)m(efron)m(t)j(orthonormal)20 b(co)s(ordinates,)0 5575 y(deriv)m(ed)26 b(in)452 5550 y(\024)441 5575 y(Cerv)m(en)q(\023)-50 b(y)28 b(\(2001,)e(eq.)42 b(4.14.66\).)e(The)26 b(w)m(a)m(v)m(efron)m(t)h(orthonormal)c(co)s (ordinate)i(system)h(is)f(a)1817 5824 y(106)p eop %%Page: 107 21 107 20 bop 0 100 a Fq(lo)s(cal)21 b(Cartesian)j(co)s(ordinate)e (system,)27 b(connected)e(with)e(the)h(mo)m(ving)e(w)m(a)m(v)m(efron)m (t.)43 b(In)24 b(the)g(w)m(a)m(v)m(efron)m(t)0 220 y(orthonormal)35 b(co)s(ordinate)h(system)h Fj(y)1422 235 y Fp(1)1461 220 y Fj(;)17 b(y)1553 235 y Fp(2)1592 220 y Fj(;)g(y)1684 235 y Fp(3)1723 220 y Fq(,)38 b(co)s(ordinates)e Fj(y)2358 235 y Fp(1)2397 220 y Fj(;)17 b(y)2489 235 y Fp(2)2565 220 y Fq(are)37 b(Cartesian)g(co)s(ordinates)f(in)0 340 y(the)f(plane)e(tangen)m(t)i(to)e(the)i(w)m(a)m(v)m(efron)m(t,)h(with)e (the)g(origin)e(on)i(the)h(reference)h(ra)m(y)e(\012,)h(analogous)e(to) 0 461 y(ra)m(y-cen)m(tered)d(co)s(ordinates)e Fj(q)1108 476 y Fp(1)1148 461 y Fj(;)17 b(q)1235 476 y Fp(2)1275 461 y Fq(.)42 b(Co)s(ordinate)27 b Fj(y)1894 476 y Fp(3)1933 461 y Fq(,)j(ho)m(w)m(ev)m(er,)h(is)e(di\013eren)m(t)f(from)g Fj(q)3132 476 y Fp(3)3171 461 y Fq(.)42 b(A)m(t)29 b(an)m(y)g(p)s(oin)m (t)0 581 y(of)c(the)h(ra)m(y)-8 b(,)27 b Fj(y)494 596 y Fp(3)558 581 y Fq(is)e(c)m(hosen)i(along)c(slo)m(wness)k(v)m(ector)g Fl(p)e Fq(\(p)s(erp)s(endicular)g(to)f(the)i(w)m(a)m(v)m(efron)m(t\),)j (not)c(along)0 702 y(the)36 b(ra)m(y)-8 b(.)52 b(Th)m(us,)37 b(the)f(ra)m(y)g(is)e(not)i(a)f(co)s(ordinate)f(axis)h(of)g(the)g(w)m (a)m(v)m(efron)m(t)j(orthonormal)32 b(co)s(ordinate)0 822 y(system,)44 b(and)d(the)g(6)28 b Fn(\002)g Fq(6)41 b(ra)m(y)g(propagator)f(matrix)f(in)h(the)h(w)m(a)m(v)m(efron)m(t)i (orthonormal)38 b(co)s(ordinate)0 942 y(system)j(cannot)g(b)s(e)g (constructed.)68 b(Ho)m(w)m(ev)m(er,)45 b(w)m(e)d(can)e(exp)s(ect)i (that)e(the)h(4)27 b Fn(\002)h Fq(4)41 b(ra)m(y)f(propagator)0 1063 y(matrices)34 b(in)h(b)s(oth)f(co)s(ordinate)g(systems)j(are)e (the)h(same,)f(as)g(the)h(ra)m(y-cen)m(tered)h(co)s(ordinates)d Fj(q)3613 1078 y Fp(1)3653 1063 y Fj(;)17 b(q)3740 1078 y Fp(2)0 1183 y Fq(and)38 b(the)g(w)m(a)m(v)m(efron)m(t)i(orthonormal) 35 b(co)s(ordinates)j Fj(y)1953 1198 y Fp(1)1992 1183 y Fj(;)17 b(y)2084 1198 y Fp(2)2160 1183 y Fq(coincide)37 b(at)h(an)m(y)g(p)s(oin)m(t)f(of)h(the)g(reference)0 1303 y(ra)m(y)d(\012.)51 b(Analogously)-8 b(,)35 b(w)m(e)h(can)f(exp)s (ect)h(that)f(the)g(4)24 b Fn(\002)g Fq(4)35 b(in)m(terface)f (propagator)g(matrices)h(in)f(b)s(oth)0 1424 y(co)s(ordinate)25 b(systems)h(are)g(the)g(same.)41 b(In)1549 1399 y(\024)1539 1424 y(Cerv)m(en)q(\023)-50 b(y)28 b(\(2001\),)d(the)h(4)8 b Fn(\002)g Fq(4)25 b(in)m(terface)h(propagator)e(matrix)0 1544 y(in)32 b(w)m(a)m(v)m(efron)m(t)j(orthonormal)30 b(co)s(ordinates)i(w)m(as)i(deriv)m(ed)g(directly)-8 b(,)32 b(based)i(on)f(the)g(phase)h(matc)m(hing)0 1665 y(argumen)m(t,)27 b(without)e(in)m(v)m(oking)h(the)g(6)8 b Fn(\002)g Fq(6)26 b(in)m(terface)g(propagator)f(matrix)f(at)i(all.)39 b(In)26 b(the)g(deriv)-5 b(ation,)0 1785 y(it)28 b(w)m(as)j(assumed)f (that)f(basis)h(v)m(ectors)h Fl(g)1496 1800 y Fp(1)1565 1785 y Fq(and)e Fl(g)1807 1800 y Fp(2)1876 1785 y Fq(and)h Fl(n)2125 1749 y Fp(\006)2209 1785 y Fq(are)g(m)m(utually)e(p)s(erp)s (endicular)g(and)i(unit.)0 1905 y(A)47 b(2)32 b Fn(\002)g Fq(2)47 b(anisotrop)m(y)f(matrix)g Fc(A)1315 1863 y Fg(an)1446 1905 y Fq(w)m(as)i(arti\014cially)c(in)m(tro)s(duced)j(there)g(to)g (transform)f(certain)0 2026 y(\\isotropic")37 b(quan)m(tities)h(to)h (anisotropic)e(ones.)63 b(In)39 b(spite)g(of)f(these)i(di\013erences,)i (it)37 b(is)i(p)s(ossible)f(to)0 2146 y(sho)m(w)23 b(that)f(for)g(m)m (utually)e(p)s(erp)s(endicular)h(unit)h(v)m(ectors)i Fl(g)2152 2161 y Fp(1)2213 2146 y Fq(and)e Fl(g)2448 2161 y Fp(2)2510 2146 y Fq(b)s(oth)g(4)q Fn(\002)q Fq(4)f(in)m(terface) h(propagator)0 2267 y(matrices)28 b(are)h(fully)f(equiv)-5 b(alen)m(t,)29 b(although)f(deriv)m(ed)i(in)e(a)h(quite)g(di\013eren)m (t)g(w)m(a)m(y)h(and)g(corresp)s(onding)0 2387 y(to)h(t)m(w)m(o)g (di\013eren)m(t)h(co)s(ordinate)e(systems.)44 b(The)32 b(equation)f(\(91\))f(for)h(the)g(4)19 b Fn(\002)h Fq(4)30 b(in)m(terface)h(propagator)0 2507 y(matrix)40 b(in)i(ra)m(y-cen)m (tered)h(co)s(ordinate)e(system,)46 b(deriv)m(ed)c(here,)k(is)41 b(ho)m(w)m(ev)m(er)j(more)e(general,)h(as)f(it)0 2628 y(allo)m(ws)26 b(to)h(construct)i(simply)d(the)h(relev)-5 b(an)m(t)28 b(6)11 b Fn(\002)g Fq(6)28 b(in)m(terface)f(propagator)f (matrix)g(\(89\))h(from)f(it,)h(and)0 2748 y(as)35 b(it)e(uses)j(more)d (general)h(co)s(ordinate)g(systems)h(at)f(surfaces.)51 b(It)34 b(is)g(also)f(more)h(transparen)m(t,)i(as)e(it)0 2868 y(is)i(expressed)j(in)d(terms)g(of)g(scalar)f(pro)s(ducts)i(of)f (basis)h(v)m(ectors,)i(and)d(remo)m(v)m(es)i(fully)c(the)j (arti\014cial)0 2989 y(anisotrop)m(y)e(matrix)f Fc(A)886 2947 y Fg(an)970 2989 y Fq(.)51 b(Finally)-8 b(,)33 b(it)h(is)h (simpler)e(as)j(it)e(uses)i(simpler)e(expressions)j(for)d(the)i(2)23 b Fn(\002)i Fq(2)0 3109 y(matrices)32 b Fc(E)44 b Fq(and)719 3084 y(~)700 3109 y Fc(E)11 b Fq(.)0 3513 y Fk(7)161 b(Surface-to-surface)34 b(paraxial)k(matrices)e(in)g(ra)l(y-cen)l (tered)242 3695 y(co)t(ordinates)0 3985 y Fq(The)d(4)22 b Fn(\002)g Fq(4)32 b(in)m(terface)h(propagator)e(matrix)g Fc(\005)1768 3943 y Fp(\()p Fg(q)r Fp(\))1861 3985 y Fq(\()5 b(~)-54 b Fj(\034)1941 4000 y Fp(\006)1997 3985 y Fj(;)17 b(\034)2083 4000 y Fp(\006)2138 3985 y Fq(\))32 b(in)g(ra)m(y-cen)m(tered)i(co)s(ordinates,)e(giv)m(en)h(b)m(y)0 4106 y(\(91\),)h(can)g(b)s(e)g(simply)e(factorized)i(at)f(in)m(terface) h(\006.)48 b(W)-8 b(e)34 b(in)m(tro)s(duce)g(the)g(4)23 b Fn(\002)h Fq(4)33 b(matrix)g Fc(Y)20 b Fq(,)34 b(related)0 4247 y(to)e(the)h(p)s(oin)m(t)f(of)g(incidence)h(of)f(the)h(reference)h (ra)m(y)f(\012)g(on)f(\006,)h(and)g(the)g(4)22 b Fn(\002)h Fq(4)32 b(matrix)3256 4222 y(~)3227 4247 y Fc(Y)3333 4182 y Fe(\000)p Fp(1)3427 4247 y Fq(,)h(related)0 4367 y(to)f(the)h(corresp)s(onding)g(re\015ection/transmission)e(p)s(oin)m (t:)238 4675 y Fc(Y)48 b Fq(=)475 4529 y Ff( )583 4615 y Fc(K)686 4573 y Fe(\000)p Fg(T)1303 4615 y Fc(0)583 4736 y Fq(\()p Fc(E)33 b Fn(\000)23 b Fj(\033)t Fc(D)9 b Fq(\))p Fc(K)1126 4694 y Fe(\000)p Fp(1)1303 4736 y Fc(K)1448 4529 y Ff(!)1563 4675 y Fj(;)1765 4649 y Fq(~)1736 4675 y Fc(Y)1842 4609 y Fe(\000)p Fp(1)1964 4675 y Fq(=)2068 4504 y Ff(0)2068 4653 y(@)2209 4589 y Fq(~)2182 4615 y Fc(K)2285 4550 y Fg(T)2980 4615 y Fc(0)2182 4759 y Fn(\000)2287 4734 y Fq(~)2259 4759 y Fc(K)2363 4694 y Fe(\000)p Fp(1)2457 4759 y Fq(\()2514 4734 y(~)2495 4759 y Fc(E)34 b Fn(\000)28 b Fq(~)-54 b Fj(\033)t Fc(D)9 b Fq(\))3007 4734 y(~)2980 4759 y Fc(K)3083 4694 y Fe(\000)p Fp(1)3219 4504 y Ff(1)3219 4653 y(A)3341 4675 y Fj(:)238 b Fq(\(93\))0 4982 y(The)39 b(individual)d(2)26 b Fn(\002)h Fq(2)38 b(matrices)g(in)g(\(93\))g(ha)m(v)m(e)h(the)g(same)g(meaning)e (as)i(in)e(\(91\),)j(see)g(\(84\),)f(\(85\))0 5102 y(and)f(\(86\).)57 b(It)38 b(is)f(easy)h(to)f(v)m(erify)h(that)f(b)s(oth)g Fc(Y)58 b Fq(and)2104 5077 y(~)2075 5102 y Fc(Y)g Fq(are)37 b(symplectic.)58 b(T)-8 b(aking)37 b(in)m(to)g(accoun)m(t)0 5243 y(that)32 b Fl(g)267 5258 y Fg(I)335 5243 y Fq(=)443 5242 y(~)439 5243 y Fl(g)495 5258 y Fg(I)567 5243 y Fq(and)h Fl(g)813 5258 y Fg(I)5 b(J)925 5243 y Fq(=)1033 5242 y(~)1029 5243 y Fl(g)1085 5258 y Fg(I)g(J)1202 5243 y Fq(in)32 b(b)s(oth)g(matrices)g Fc(Y)52 b Fq(and)2294 5218 y(~)2265 5243 y Fc(Y)2371 5178 y Fe(\000)p Fp(1)2465 5243 y Fq(,)33 b(w)m(e)h(obtain)1384 5479 y Fc(\005)1491 5437 y Fp(\()p Fg(q)r Fp(\))1584 5479 y Fq(\()5 b(~)-54 b Fj(\034)1664 5494 y Fp(\006)1719 5479 y Fj(;)17 b(\034)1805 5494 y Fp(\006)1860 5479 y Fq(\))28 b(=)2058 5453 y(~)2029 5479 y Fc(Y)2135 5414 y Fe(\000)p Fp(1)2230 5479 y Fc(Y)52 b Fj(:)1211 b Fq(\(94\))1817 5824 y(107)p eop %%Page: 108 22 108 21 bop 0 100 a Fq(Let)31 b(us)g(brie\015y)g(discuss)g(the)g(ph)m (ysical)g(meaning)e(of)h(the)h(4)18 b Fn(\002)h Fq(4)30 b(matrix)f Fc(Y)20 b Fq(,)31 b(related)f(to)g(the)h(p)s(oin)m(t)f(of)0 220 y(incidence)j Fj(\034)465 235 y Fp(\006)520 220 y Fq(.)44 b(It)32 b(represen)m(ts)j(the)e(transformation)e(matrix)1300 330 y Ff( )1407 415 y Fj(\016)t Fc(u)1407 537 y Fj(\016)t Fc(p)1521 501 y Fp(\()p Fg(u)p Fp(\))1663 330 y Ff(!)1756 477 y Fq(=)c Fc(Y)1982 330 y Ff( )2089 415 y Fj(\016)t Fc(q)2089 537 y Fj(\016)t Fc(p)2203 501 y Fp(\()p Fg(q)r Fp(\))2337 330 y Ff(!)2452 477 y Fj(:)1127 b Fq(\(95\))0 765 y(The)32 b(meaning)e(of)g(the)i(2)19 b Fn(\002)g Fq(1)31 b(ra)m(y-cen)m(tered)i(v)m(ectors)g Fj(\016)t Fc(q)38 b Fn(\021)28 b Fq(\()p Fj(\016)t(q)2363 780 y Fp(1)2402 765 y Fj(;)17 b(\016)t(q)2536 780 y Fp(2)2576 765 y Fq(\))2614 729 y Fg(T)2700 765 y Fq(and)31 b Fj(\016)t Fc(p)3001 729 y Fp(\()p Fg(q)r Fp(\))3122 765 y Fn(\021)d Fq(\()p Fj(\016)t(p)3361 714 y Fp(\()p Fg(q)r Fp(\))3361 786 y(1)3454 765 y Fj(;)17 b(\016)t(p)3594 714 y Fp(\()p Fg(q)r Fp(\))3594 786 y(2)3687 765 y Fq(\))3725 729 y Fg(T)0 885 y Fq(is)47 b(explained)f(in)h(detail)e(under)j(Equation)f (\(48\).)87 b(Relation)45 b(\(95\))h(transforms)h(v)m(ectors)i Fj(\016)t Fc(q)57 b Fq(and)0 1005 y Fj(\016)t Fc(p)114 969 y Fp(\()p Fg(q)r Fp(\))250 1005 y Fq(\(measured)44 b(along)e(the)i(plane)f(tangen)m(t)h(to)g(the)g(w)m(a)m(v)m(efron)m (t\))h(in)m(to)e Fj(\016)t Fc(u)56 b Fn(\021)46 b Fq(\()p Fj(\016)t(u)3261 1020 y Fp(1)3300 1005 y Fj(;)17 b(\016)t(u)3447 1020 y Fp(2)3486 1005 y Fq(\))3524 969 y Fg(T)3622 1005 y Fq(and)0 1139 y Fj(\016)t Fc(p)114 1103 y Fp(\()p Fg(u)p Fp(\))263 1139 y Fn(\021)49 b Fq(\()p Fj(\016)t(p)523 1088 y Fp(\()p Fg(u)p Fp(\))523 1161 y(1)623 1139 y Fj(;)17 b(\016)t(p)763 1088 y Fp(\()p Fg(u)p Fp(\))763 1161 y(2)862 1139 y Fq(\))900 1103 y Fg(T)955 1139 y Fq(,)48 b(measured)e(along)e (surface)i(\006.)81 b(V)-8 b(ector)45 b Fj(\016)t Fc(u)55 b Fq(determines)45 b(the)g(p)s(osition)0 1260 y(of)40 b(the)i(p)s(oin)m(t)e(of)g(incidence)h(of)g(the)g(paraxial)e(ra)m(y)j (under)f(consideration)g(on)f(surface)i(\006.)69 b(V)-8 b(ector)0 1380 y Fj(\016)t Fc(p)114 1344 y Fp(\()p Fg(u)p Fp(\))241 1380 y Fn(\021)28 b Fq(\()p Fj(\016)t(p)480 1329 y Fp(\()p Fg(u)p Fp(\))480 1402 y(1)580 1380 y Fj(;)17 b(\016)t(p)720 1329 y Fp(\()p Fg(u)p Fp(\))720 1402 y(2)820 1380 y Fq(\))858 1344 y Fg(T)943 1380 y Fq(represen)m(ts)34 b(the)d(di\013erence)h(b)s(et)m(w)m(een)h(the)e(tangen)m(tial)e(comp)s (onen)m(ts)j(of)e(the)0 1501 y(slo)m(wness)i(v)m(ector)f(at)f(the)h(p)s (oin)m(t)f(of)g(incidence)g(of)g(the)h(paraxial)e(ra)m(y)h(and)h(at)f (the)h(p)s(oin)m(t)e(of)h(incidence)0 1621 y(of)i(reference)i(ra)m(y)f (\012.)146 1812 y(A)m(t)38 b(the)f(p)s(oin)m(t)f(of)h (re\015ection/transmission)e(of)i(the)g(reference)i(ra)m(y)e(\012)h(at) e(\006,)j(matrix)3438 1787 y(~)3409 1812 y Fc(Y)57 b Fq(has)38 b(a)0 1933 y(meaning)31 b(analogous)h(to)g(\(95\):)1253 2049 y Ff( )1360 2130 y Fj(\016)1414 2129 y Fq(~)1407 2130 y Fc(q)1360 2257 y Fj(\016)1416 2256 y Fq(~)1407 2257 y Fc(p)1474 2216 y Fp(\()p Fg(q)r Fp(\))1608 2049 y Ff(!)1702 2195 y Fq(=)1834 2169 y(~)1805 2195 y Fc(Y)1911 2129 y Fe(\000)p Fp(1)2022 2049 y Ff( )2129 2133 y Fj(\016)t Fc(u)2129 2255 y Fj(\016)t Fc(p)2243 2219 y Fp(\()p Fg(u)p Fp(\))2385 2049 y Ff(!)2499 2195 y Fj(:)1080 b Fq(\(96\))0 2457 y(As)27 b(the)g(tangen)m(tial)d(comp)s(onen)m(ts)j(of)f(the)h(slo) m(wness)h(v)m(ector)f(are)f(con)m(tin)m(uous)h(across)g(\006)g (\(Snell's)f(la)m(w\),)0 2577 y Fj(\016)t Fc(u)42 b Fq(and)32 b Fj(\016)t Fc(p)454 2541 y Fp(\()p Fg(u)p Fp(\))586 2577 y Fq(should)h(b)s(e)g(the)g(same)f(on)h(b)s(oth)f(sides)h(of)f (\006.)44 b(Com)m(bining)31 b(\(95\))h(with)g(\(96\),)g(w)m(e)i(get:) 662 2693 y Ff( )769 2775 y Fj(\016)823 2774 y Fq(~)816 2775 y Fc(q)769 2902 y Fj(\016)825 2901 y Fq(~)816 2902 y Fc(p)883 2861 y Fp(\()p Fg(q)r Fp(\))1018 2693 y Ff(!)1111 2839 y Fq(=)1243 2814 y(~)1214 2839 y Fc(Y)1320 2774 y Fe(\000)p Fp(1)1415 2839 y Fc(Y)1537 2693 y Ff( )1645 2777 y Fj(\016)t Fc(q)1645 2899 y Fj(\016)t Fc(p)1758 2863 y Fp(\()p Fg(q)r Fp(\))1893 2693 y Ff(!)1986 2839 y Fq(=)28 b Fc(\005)2196 2797 y Fp(\()p Fg(q)r Fp(\))2289 2839 y Fq(\()5 b(~)-54 b Fj(\034)2369 2854 y Fp(\006)2424 2839 y Fj(;)17 b(\034)2510 2854 y Fp(\006)2566 2839 y Fq(\))2621 2693 y Ff( )2727 2777 y Fj(\016)t Fc(q)2727 2899 y Fj(\016)t Fc(p)2841 2863 y Fp(\()p Fg(q)r Fp(\))2975 2693 y Ff(!)3090 2839 y Fj(:)489 b Fq(\(97\))0 3096 y(This)37 b(corresp)s(onds)g(fully)e(to)h(the)h(de\014nition)f(of)f(the)i(4)25 b Fn(\002)g Fq(4)36 b(in)m(terface)h(propagator)e(matrix)g(in)h(ra)m (y-)0 3216 y(cen)m(tered)e(co)s(ordinates,)f(see)g(\(94\).)146 3407 y(Consider)43 b(no)m(w)f(the)h(reference)g(ra)m(y)g(\012,)i(with)c (the)i(initial)38 b(p)s(oin)m(t)j(at)h Fj(S)48 b Fq(and)42 b(end)h(p)s(oin)m(t)e(at)g Fj(R)q Fq(,)0 3528 y(inciden)m(t)33 b(at)g(surface)i(\006)e(at)g(p)s(oin)m(t)g Fj(Q)p Fq(,)h(with)f(the)h (relev)-5 b(an)m(t)33 b(re\015ection/transmission)g(p)s(oin)m(t)3484 3503 y(~)3461 3528 y Fj(Q)p Fq(.)47 b(The)0 3648 y(c)m(hain)32 b(rule)h(in)e(ra)m(y-cen)m(tered)k(co)s(ordinates)d(reads:)861 3848 y Fc(\005)968 3806 y Fp(\()p Fg(q)r Fp(\))1060 3848 y Fq(\()p Fj(R)q(;)17 b(S)6 b Fq(\))28 b(=)f Fc(\005)1559 3806 y Fp(\()p Fg(q)r Fp(\))1651 3848 y Fq(\()p Fj(R)q(;)1831 3823 y Fq(~)1808 3848 y Fj(Q)p Fq(\))p Fc(\005)2030 3806 y Fp(\()p Fg(q)r Fp(\))2123 3848 y Fq(\()2183 3823 y(~)2161 3848 y Fj(Q;)17 b(Q)p Fq(\))p Fc(\005)2503 3806 y Fp(\()p Fg(q)r Fp(\))2596 3848 y Fq(\()p Fj(Q;)g(S)6 b Fq(\))32 b Fj(:)688 b Fq(\(98\))0 4048 y(Using)21 b(\(94\),)j(w)m(e)e(can)g (replace)g(the)g(4)p Fn(\002)p Fq(4)g(in)m(terface)g(propagator)f (matrix)f Fc(\005)2747 4006 y Fp(\()p Fg(q)r Fp(\))2840 4048 y Fq(\()2901 4023 y(~)2878 4048 y Fj(Q;)d(Q)p Fq(\))22 b(b)m(y)h Fc(Y)3367 4006 y Fe(\000)p Fp(1)3461 4048 y Fq(\()3521 4023 y(~)3499 4048 y Fj(Q)p Fq(\))p Fc(Y)d Fq(\()p Fj(Q)p Fq(\).)0 4168 y(Consider)36 b(further)f(that)g(the)h(p)s (oin)m(t)e Fj(S)41 b Fq(is)35 b(situated)g(on)g(the)h(an)m(terior)e (surface)i(\006)3039 4132 y Fg(a)3117 4168 y Fq(and)f(p)s(oin)m(t)f Fj(R)j Fq(on)0 4304 y(the)g(p)s(osterior)f(surface)h(\006)988 4268 y Fg(p)1028 4304 y Fq(.)56 b(Multiplying)34 b(\(98\))i(b)m(y)h Fc(Y)20 b Fq(\()p Fj(R)q Fq(\))37 b(from)e(the)i(left)f(and)h(b)m(y) 3232 4279 y(~)3203 4304 y Fc(Y)3309 4239 y Fe(\000)p Fp(1)3404 4304 y Fq(\()p Fj(S)6 b Fq(\))36 b(from)0 4424 y(the)d(righ)m(t,)f(w)m(e)h(obtain)193 4640 y Fc(Y)20 b Fq(\()p Fj(R)q Fq(\))p Fc(\005)556 4598 y Fp(\()p Fg(q)r Fp(\))649 4640 y Fq(\()p Fj(R)q(;)d(S)6 b Fq(\))938 4614 y(~)910 4640 y Fc(Y)1015 4574 y Fe(\000)p Fp(1)1110 4640 y Fq(\()p Fj(S)g Fq(\))27 b(=)h Fc(Y)20 b Fq(\()p Fj(R)q Fq(\))p Fc(\005)1746 4598 y Fp(\()p Fg(q)r Fp(\))1839 4640 y Fq(\()p Fj(R)q(;)2018 4614 y Fq(~)1996 4640 y Fj(Q)p Fq(\))2139 4614 y(~)2111 4640 y Fc(Y)2217 4574 y Fe(\000)p Fp(1)2311 4640 y Fq(\()2371 4614 y(~)2349 4640 y Fj(Q)p Fq(\))p Fc(Y)g Fq(\()p Fj(Q)p Fq(\))p Fc(\005)2830 4598 y Fp(\()p Fg(q)r Fp(\))2923 4640 y Fq(\()p Fj(Q;)d(S)6 b Fq(\))3214 4614 y(~)3186 4640 y Fc(Y)3291 4574 y Fe(\000)p Fp(1)3386 4640 y Fq(\()p Fj(S)g Fq(\))32 b Fj(:)0 4840 y Fq(W)-8 b(e)39 b(no)m(w)f(in)m(tro)s(duce)g(the)h(4)26 b Fn(\002)g Fq(4)38 b(surface-to-surface)h(paraxial)d(matrix)g Fc(T)14 b Fq(\()p Fj(R)q(;)j(S)6 b Fq(\))37 b(in)h(ra)m(y-cen)m(tered)0 4960 y(co)s(ordinates)32 b(b)m(y)i(the)f(relation)1089 5175 y Fc(T)13 b Fq(\()p Fj(R)q(;)k(S)6 b Fq(\))27 b(=)h Fc(Y)20 b Fq(\()p Fj(R)q Fq(\))p Fc(\005)1936 5133 y Fp(\()p Fg(q)r Fp(\))2029 5175 y Fq(\()p Fj(R)q(;)d(S)6 b Fq(\))2318 5150 y(~)2290 5175 y Fc(Y)2395 5110 y Fe(\000)p Fp(1)2490 5175 y Fq(\()p Fj(S)g Fq(\))32 b Fj(:)915 b Fq(\(99\))0 5375 y(The)34 b(ab)s(o)m(v)m(e)f(equation)f(then)h(yields) 1259 5575 y Fc(T)13 b Fq(\()p Fj(R)q(;)k(S)6 b Fq(\))27 b(=)g Fc(T)14 b Fq(\()p Fj(R)q(;)2014 5550 y Fq(~)1992 5575 y Fj(Q)p Fq(\))p Fc(T)f Fq(\()p Fj(Q;)k(S)6 b Fq(\))32 b Fj(:)1036 b Fq(\(100\))1817 5824 y(108)p eop %%Page: 109 23 109 22 bop 0 100 a Fq(The)30 b(requiremen)m(t)g(is)e(that)h Fj(Q)g Fq(corresp)s(onds)i(to)e(the)g(p)s(oin)m(t)g(of)f(incidence)i (of)e(the)i(reference)g(ra)m(y)g(\012)g(on)0 220 y(surface)36 b(\006)g(and)652 195 y(~)630 220 y Fj(Q)f Fq(to)g(the)h(relev)-5 b(an)m(t)35 b(re\015ection/transmission)f(p)s(oin)m(t)h(on)g(the)h (same)f(surface)i(\006,)f(so)0 340 y(that)f Fl(g)270 355 y Fg(I)310 340 y Fq(\()p Fj(Q)p Fq(\))g(and)h Fl(g)747 355 y Fg(I)5 b(J)832 340 y Fq(\()p Fj(Q)p Fq(\))35 b(are)g(the)h(same)f (as)g Fl(g)1781 355 y Fg(I)1821 340 y Fq(\()1882 315 y(~)1859 340 y Fj(Q)q Fq(\))g(and)g Fl(g)2258 355 y Fg(I)5 b(J)2343 340 y Fq(\()2403 315 y(~)2381 340 y Fj(Q)p Fq(\).)52 b(The)36 b(2)24 b Fn(\002)g Fq(2)35 b(surface-to-surface)0 461 y(paraxial)h(matrix)g Fc(T)13 b Fq(\()p Fj(R)q(;)k(S)6 b Fq(\))37 b(satis\014es)i(the)f(symplectic)f(prop)s(ert)m(y)-8 b(,)40 b(its)d(determinan)m(t)g(equals)h(unit)m(y)-8 b(,)0 581 y(and)35 b(ma)m(y)f(b)s(e)h(further)f(c)m(hained.)50 b(If)35 b(the)g(ra)m(y)f(crosses)j Fj(n)d Fq(in)m(terfaces)h(\006)2629 596 y Fp(1)2669 581 y Fj(;)17 b Fq(\006)2783 596 y Fp(2)2823 581 y Fj(;)g(:::;)g Fq(\006)3062 596 y Fg(n)3144 581 y Fq(b)s(et)m(w)m(een)36 b Fj(S)k Fq(and)0 702 y Fj(R)q Fq(,)677 822 y Fc(T)13 b Fq(\()p Fj(R)q(;)k(S)6 b Fq(\))27 b(=)h Fc(T)13 b Fq(\()p Fj(R)q(;)1432 797 y Fq(~)1410 822 y Fj(Q)1487 837 y Fg(n)1534 822 y Fq(\))p Fc(T)g Fq(\()p Fj(Q)1779 837 y Fg(n)1826 822 y Fj(;)1892 797 y Fq(~)1870 822 y Fj(Q)1947 837 y Fg(n)p Fe(\000)p Fp(1)2084 822 y Fq(\))p Fj(:::)p Fc(T)h Fq(\()p Fj(Q)2411 837 y Fp(2)2450 822 y Fj(;)2516 797 y Fq(~)2494 822 y Fj(Q)2571 837 y Fp(1)2611 822 y Fq(\))p Fc(T)f Fq(\()p Fj(Q)2856 837 y Fp(1)2895 822 y Fj(;)k(S)6 b Fq(\))32 b Fj(:)455 b Fq(\(101\))0 996 y(The)34 b(con)m(tin)m(uation)d(relation)g(reads:) 1027 1108 y Ff( )1134 1192 y Fj(\016)t Fc(u)9 b Fq(\()p Fj(R)q Fq(\))1134 1314 y Fj(\016)t Fc(p)1248 1278 y Fp(\()p Fg(u)p Fp(\))1348 1314 y Fq(\()p Fj(R)q Fq(\))1540 1108 y Ff(!)1634 1254 y Fq(=)27 b Fc(T)13 b Fq(\()p Fj(R)q(;)k(S)6 b Fq(\))2107 1108 y Ff( )2214 1192 y Fj(\016)t Fc(u)j Fq(\()p Fj(S)d Fq(\))2214 1314 y Fj(\016)t Fc(p)2327 1278 y Fp(\()p Fg(u)p Fp(\))2427 1314 y Fq(\()p Fj(S)g Fq(\))2610 1108 y Ff(!)2725 1254 y Fj(:)805 b Fq(\(102\))0 1530 y(Consequen)m(tly)-8 b(,)37 b(the)e(surface-to-surface)f(paraxial) f(matrix)f(transforms)i(the)h(paraxial)d(ra)m(y)j(quan)m(ti-)0 1651 y(ties)29 b Fj(\016)t Fc(u)38 b Fq(and)29 b Fj(\016)t Fc(p)623 1615 y Fp(\()p Fg(u)p Fp(\))752 1651 y Fq(from)f(one)h (surface)h(to)f(another.)42 b(The)30 b(surfaces)h(ma)m(y)e(b)s(e)g(of)g (an)g(arbitrary)f(shap)s(e)0 1771 y(and)38 b(ma)m(y)f(represen)m(t)i (geological)c(in)m(terfaces,)k(the)f(surface)g(of)f(the)h(Earth,)h(iso) s(c)m(hrone)f(surfaces,)i(or)0 1892 y(merely)33 b(formal)e(surfaces.)48 b(The)35 b(formal)c(surfaces)k(can)f(b)s(e)f(in)m(tro)s(duced)h(at)f (an)m(y)i(p)s(oin)m(t)d(of)i(the)g(refer-)0 2012 y(ence)f(ra)m(y)-8 b(.)44 b(It)32 b(is,)g(ho)m(w)m(ev)m(er,)i(required)f(that)f(the)g(ph)m (ysical)g(surfaces)h(\(structural)f(in)m(terfaces\))h(do)f(not)0 2132 y(in)m(tersect)h(in)f(the)h(paraxial)e(vicinit)m(y)g(of)i(the)g (reference)h(ra)m(y)-8 b(.)146 2324 y(The)39 b(surface-to-surface)g (paraxial)d(matrices)h(are)h(closely)f(link)m(ed)h(to)f(the)i (so-called)d(theory)j(of)0 2444 y(seismic)34 b(systems.)53 b(F)-8 b(or)34 b(a)h(medium)f(comp)s(osed)h(of)g(homogeneous)g(and)g (inhomogeneous)g(isotropic)0 2564 y(la)m(y)m(ers,)52 b(see)c(Bortfeld)f(\(1989\),)j(Hubral)c(et)i(al.)87 b (\(1992,1993,1995\),)48 b(Sc)m(hleic)m(her)g(et)g(al.)86 b(\(1993\),)11 2659 y(\024)0 2685 y(Cerv)m(en)q(\023)-50 b(y)31 b(\(2001,)d(sec.)43 b(4.4.7\),)28 b(Chira-Oliv)-5 b(a)25 b(et)k(al.)40 b(\(2001\),)28 b(Biloti)e(et)i(al.)41 b(\(2002\),)28 b(Sun)g(\(2004\).)41 b(F)-8 b(or)0 2805 y(anisotropic)28 b(structures,)j(a)e(most)g(detailed)f(treatmen)m(t,)h (with)g(man)m(y)g(applications)e(and)i(references,)0 2925 y(can)d(b)s(e)g(found)g(in)f(Moser)i(and)1149 2900 y(\024)1138 2925 y(Cerv)m(en)q(\023)-50 b(y)29 b(\(2006\).)40 b(The)27 b(deriv)-5 b(ations)24 b(in)h(the)h(last)f(reference)j(are)e (based)0 3046 y(on)f(the)g(factorization)e(of)h(the)i(in)m(terface)f (propagator)f(matrix)f(expressed)28 b(in)c(Cartesian)h(co)s(ordinates.) 0 3166 y(Analogously)-8 b(,)23 b(the)g(factorization)d(of)h(the)i(in)m (terface)f(propagator)g(matrix)e(in)i(w)m(a)m(v)m(efron)m(t)i (orthonormal)0 3287 y(co)s(ordinates)44 b(w)m(as)i(used)f(b)m(y)1123 3261 y(\024)1112 3287 y(Cerv)m(en)q(\023)-50 b(y)47 b(\(2001,)f(sec.)81 b(4.14.10\))44 b(to)g(deriv)m(e)h(the)g(surface-to-surface)0 3407 y(paraxial)30 b(matrices)h(in)h(these)h(co)s(ordinates.)43 b(See)33 b(also)e(Section)h(6)g(for)f(comparison)g(of)h(the)g(in)m (terface)0 3527 y(propagator)k(matrix)g(in)g(w)m(a)m(v)m(efron)m (t-orthonormal)g(and)h(ra)m(y-cen)m(tered)i(co)s(ordinates.)56 b(Similarly)33 b(as)0 3648 y(the)40 b(in)m(terface)f(propagator)f (matrices,)j(the)e(surface-to-surface)h(paraxial)e(matrices)g(in)h(w)m (a)m(v)m(efron)m(t)0 3768 y(orthonormal)30 b(co)s(ordinates)i(can)h(b)s (e)g(obtained)f(as)h(a)f(sp)s(ecial)g(case)h(of)f(the)h(equations)g (deriv)m(ed)g(here.)146 3959 y(A)25 b(terminological)c(note:)40 b(The)26 b(surface-to-surface)g(paraxial)d(matrix)g(has)j(also)e(b)s (een)i(referred)g(to)0 4080 y(as)j(the)g(surface-to-surface)g (propagator)f(matrix)f(in)h(the)h(seismological)d(literature.)40 b(This)29 b(term)g(w)m(as)0 4200 y(used)f(to)f(emphasize)h(that)f(the)g (paraxial)f(solutions)g Fj(\016)t Fc(u)36 b Fq(and)28 b Fj(\016)t Fc(p)2378 4164 y Fp(\()p Fg(u)p Fp(\))2506 4200 y Fq(can)f(b)s(e)h(\\propagated")e(from)g(one)0 4321 y(surface)41 b(to)f(another,)i(see)f(\(102\).)66 b(Mathematically)-8 b(,)40 b(ho)m(w)m(ev)m(er,)k(the)d (surface-to-surface)f(paraxial)0 4441 y(matrices)c(are)i(not)f (propagator)f(matrices,)i(see)g(Gilb)s(ert)d(and)j(Bac)m(kus)g (\(1966\),)g(Ursin)f(\(1983\).)56 b(T)-8 b(o)0 4561 y(a)m(v)m(oid)24 b(p)s(ossible)g(confusion,)i(w)m(e)g(use)f(the)g(term)f (surface-to-surface)i(paraxial)c(matrices)i(consisten)m(tly)-8 b(.)0 4682 y(Analogous)32 b(terminology)e(w)m(as)j(also)f(used)i(b)m(y) f(Moser)h(and)2246 4656 y(\024)2235 4682 y(Cerv)m(en)q(\023)-50 b(y)36 b(\(2006\).)146 4873 y(The)f(formalism)c(of)i (surface-to-surface)h(paraxial)e(matrices)h(o\013ers)i(man)m(y)f (useful)g(applications)0 4993 y(of)i(b)s(oth)g(theoretical)f(and)i (practical)d(in)m(terest.)56 b(Certain)36 b(of)g(these)h(applications)e (are)h(discussed)i(in)0 5114 y(detail)28 b(b)m(y)j(Moser)f(and)886 5088 y(\024)875 5114 y(Cerv)m(en)q(\023)-50 b(y)32 b(\(2006\),)e(where) h(also)d(appropriate)h(references)j(can)e(b)s(e)g(found.)43 b(See)0 5234 y(also)32 b(the)h(next)g(section.)1817 5824 y(109)p eop %%Page: 110 24 110 23 bop 0 100 a Fk(8)161 b(Concluding)53 b(remarks)0 390 y Fq(W)-8 b(e)29 b(ha)m(v)m(e)h(prop)s(osed)f(t)m(w)m(o)h(metho)s (ds)e(of)g(calculating)f(the)i(4)14 b Fn(\002)g Fq(4)29 b(ra)m(y)g(propagator)f(matrix)f Fc(\005)3437 347 y Fp(\()p Fg(q)r Fp(\))3530 390 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3702 405 y Fp(0)3742 390 y Fq(\))0 510 y(in)34 b(ra)m(y-cen)m(tered)i(co)s (ordinates)e(along)f(the)i(reference)i(ra)m(y)e(\012)g(from)e Fj(\034)2586 525 y Fp(0)2660 510 y Fq(to)h Fj(\034)11 b Fq(.)50 b(None)35 b(of)f(these)i(meth-)0 630 y(o)s(ds)44 b(requires)h(dynamic)e(ra)m(y)h(tracing)f(in)g(ra)m(y-cen)m(tered)j(co) s(ordinates;)j(b)s(oth)43 b(are)h(based)h(fully)e(on)0 751 y(con)m(v)m(en)m(tional)33 b(dynamic)f(ra)m(y)h(tracing)e(in)h (Cartesian)g(co)s(ordinates.)146 942 y(1\))h(In)h(the)g(\014rst)f (metho)s(d,)g Fc(\005)1249 900 y Fp(\()p Fg(q)r Fp(\))1342 942 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1514 957 y Fp(0)1553 942 y Fq(\))33 b(is)g(computed)h(from)e(the)h(6)23 b Fn(\002)g Fq(6)33 b(ra)m(y)h(propagator)e(matrix)0 1062 y Fl(\005)88 1020 y Fp(\()p Fg(x)p Fp(\))186 1062 y Fq(\()p Fj(\034)6 b(;)17 b(\034)358 1077 y Fp(0)398 1062 y Fq(\))32 b(in)g(Cartesian)h(co)s(ordinates,)f(using)g(\(62\))g(with)g(\(63\).) 146 1254 y(2\))k(In)h(the)g(second)h(metho)s(d,)f Fc(\005)1380 1211 y Fp(\()p Fg(q)r Fp(\))1472 1254 y Fq(\()p Fj(\034)6 b(;)17 b(\034)1644 1269 y Fp(0)1684 1254 y Fq(\))36 b(is)g(computed)h (using)f(\(71\).)55 b(This)37 b(metho)s(d)f(do)s(es)h(not)0 1374 y(require)d(the)g(whole)f(6)23 b Fn(\002)g Fq(6)33 b(ra)m(y)h(propagator)f(matrix)f Fl(\005)2103 1332 y Fp(\()p Fg(x)p Fp(\))2202 1374 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2374 1389 y Fp(0)2413 1374 y Fq(\))34 b(in)f(Cartesian)g(co)s (ordinates)g(to)g(b)s(e)0 1494 y(kno)m(wn,)26 b(but)e(only)f(four)f (solutions)h(of)f(the)i(dynamic)e(ra)m(y)i(tracing)e(system)i(in)f (Cartesian)g(co)s(ordinates,)0 1615 y(for)32 b(strictly)g(sp)s (eci\014ed)h(initial)c(conditions)j(\(72\).)146 1806 y(Both)k(metho)s(ds)f(should)h(yield)f(the)h(same)f(4)24 b Fn(\002)h Fq(4)35 b(ra)m(y)h(propagator)f(matrix)f Fc(\005)3117 1764 y Fp(\()p Fg(q)r Fp(\))3210 1806 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3382 1821 y Fp(0)3421 1806 y Fq(\))36 b(in)f(ra)m(y-)0 1926 y(cen)m(tered)f(co)s(ordinates;)e (the)h(di\013erences)h(are)e(only)g(in)g(the)h(n)m(umerical)e (e\016ciency)j(of)e(the)g(computa-)0 2047 y(tions.)45 b(The)34 b(second)g(metho)s(d)f(is)f(in)h(general)f(n)m(umerically)f (more)i(e\016cien)m(t)h(than)f(the)g(\014rst)h(metho)s(d,)0 2167 y(as)e(it)e(a)m(v)m(oids)i(the)g(unnecessary)i(computation)c(of)h (elemen)m(ts)h(in)f(the)h(third)e(and)i(sixth)g(columns)e(and)0 2287 y(ro)m(ws)j(of)g Fl(\005)424 2245 y Fp(\()p Fg(x)p Fp(\))523 2287 y Fq(\()p Fj(\034)6 b(;)17 b(\034)695 2302 y Fp(0)734 2287 y Fq(\),)33 b(whic)m(h)g(are)g(later)e(remo)m(v)m (ed.)146 2479 y(In)f(some)e(exceptional)h(cases,)i(the)e(\014rst)h (metho)s(d)e(ma)m(y)h(b)s(e)g(more)f(useful)h(than)g(the)h(second.)43 b(This)0 2599 y(applies)30 b(to)h(the)h(case)g(in)f(whic)m(h)h(w)m(e)g (need)h(to)e(kno)m(w)h Fc(\005)2051 2557 y Fp(\()p Fg(q)r Fp(\))2144 2599 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2316 2614 y Fp(0)2356 2599 y Fq(\))31 b(for)g(v)-5 b(arious)30 b(initial)e(conditions.)42 b(In)0 2719 y(this)31 b(case,)h(it)f(ma)m(y) g(b)s(e)g(useful)h(to)f(compute)g(the)h(whole)f(6)19 b Fn(\002)h Fq(6)31 b(ra)m(y)h(propagator)e(matrix)g Fl(\005)3404 2677 y Fp(\()p Fg(x)p Fp(\))3503 2719 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3675 2734 y Fp(0)3714 2719 y Fq(\),)0 2840 y(and)33 b(v)-5 b(ary)32 b(only)g Fl(\011)708 2798 y Fg(r)708 2865 y Fp(2)747 2840 y Fq(\()p Fj(\034)827 2855 y Fp(0)867 2840 y Fq(\).)43 b(W)-8 b(e)33 b(do)f(not)h(need)g(to)g(p)s(erform)e(new)i(dynamic)f(ra)m(y)h(tracing) f(in)g(Cartesian)0 2960 y(co)s(ordinates)g(when)i(w)m(e)f(c)m(hange)h (the)f(matrix)e(of)h(initial)d(conditions)i Fl(\011)2671 2918 y Fg(r)2671 2985 y Fp(2)2710 2960 y Fq(\()p Fj(\034)2790 2975 y Fp(0)2830 2960 y Fq(\).)146 3152 y(Analogously)e(to)g(the)h(t)m (w)m(o)g(metho)s(ds)g(to)f(determine)g(4)16 b Fn(\002)g Fq(4)30 b(propagator)f(matrices)g Fl(\005)3326 3109 y Fp(\()p Fg(q)r Fp(\))3419 3152 y Fq(\()p Fj(\034)6 b(;)17 b(\034)3591 3167 y Fp(0)3631 3152 y Fq(\))29 b(in)0 3272 y(ra)m(y-cen)m(tered)35 b(co)s(ordinates)d(describ)s(ed)i(ab)s(o)m(v)m (e,)g(w)m(e)g(can)f(also)f(compute)h(the)g(6)22 b Fn(\002)h Fq(6)32 b(ra)m(y-propagator)0 3392 y(matrix)c Fl(\005)401 3350 y Fp(\()p Fg(q)r Fp(\))494 3392 y Fq(\()p Fj(\034)6 b(;)17 b(\034)666 3407 y Fp(0)706 3392 y Fq(\))29 b(in)g(ra)m(y-cen)m (tered)i(co)s(ordinates.)42 b(In)30 b(the)f(\014rst)h(metho)s(d,)g(w)m (e)g(use)g(\(61\))f(with)g(\(59\),)0 3513 y(and)k(in)e(the)i(second)h (metho)s(d)e(\(64\).)146 3704 y(Let)46 b(us)g(no)m(w)h(consider)e(an)m (y)i(p)s(oin)m(t)d Fj(Q)i Fq(situated)g(on)f(the)h(reference)h(ra)m(y)f (\012)g(b)s(et)m(w)m(een)i Fj(\034)3537 3719 y Fp(0)3622 3704 y Fq(and)0 3824 y Fj(\034)11 b Fq(.)56 b(Assuming)36 b(the)h(6)25 b Fn(\002)g Fq(6)36 b(propagator)g(matrix)f Fl(\005)1932 3782 y Fp(\()p Fg(x)p Fp(\))2031 3824 y Fq(\()p Fj(\034)6 b(;)17 b(\034)2203 3839 y Fp(0)2243 3824 y Fq(\))36 b(is)g(kno)m(wn)i(along)d(the)i(ra)m(y)g(from)f Fj(\034)3617 3839 y Fp(0)3693 3824 y Fq(to)0 3945 y Fj(\034)11 b Fq(,)41 b(including)d(the)h(p)s(oin)m(t)f Fj(\034)1028 3960 y Fg(Q)1088 3945 y Fq(,)j(w)m(e)f(can)g(ev)-5 b(aluate)38 b(the)i(4)26 b Fn(\002)h Fq(4)39 b(propagator)f(matrix)g Fc(\005)3257 3903 y Fp(\()p Fg(q)r Fp(\))3350 3945 y Fq(\()p Fj(\034)3430 3960 y Fg(Q)3490 3945 y Fj(;)17 b(\034)3576 3960 y Fp(0)3616 3945 y Fq(\))39 b(at)0 4065 y(that)h(p)s(oin)m(t)g(using)g(\(62\),)i(where)g(w)m(e)g(replace)e Fj(\034)52 b Fq(b)m(y)42 b Fj(\034)2048 4080 y Fg(Q)2108 4065 y Fq(.)67 b(Analogously)-8 b(,)41 b(w)m(e)h(can)f(store)g Fl(Z)3438 4029 y Fg(r)3476 4065 y Fq(\()p Fj(\034)3556 4080 y Fg(Q)3616 4065 y Fj(;)17 b(\034)3702 4080 y Fp(0)3742 4065 y Fq(\))0 4185 y(and)35 b(use)i(\(71\),)e(where)h(w)m(e)h(again)c (replace)j Fj(\034)46 b Fq(b)m(y)36 b Fj(\034)1895 4200 y Fg(Q)1955 4185 y Fq(.)52 b(Actually)-8 b(,)35 b(the)g(only)g (additional)d(w)m(ork)k(is)f(the)0 4306 y(computation)c(of)h(the)h(4)22 b Fn(\002)h Fq(6)32 b(matrix)f Fl(\011)1504 4264 y Fg(r)1504 4330 y Fp(1)1544 4306 y Fq(\()p Fj(\034)1624 4321 y Fg(Q)1684 4306 y Fq(\))h(at)g(p)s(oin)m(t)g Fj(Q)p Fq(.)146 4497 y(A)h(note)f(to)g(the)g(c)m(hoice)h(of)e(basis)i(v)m(ectors)g Fl(e)1775 4512 y Fg(I)1815 4497 y Fq(\()p Fj(\034)1895 4512 y Fp(0)1935 4497 y Fq(\),)f Fl(e)2083 4512 y Fg(I)2123 4497 y Fq(\()p Fj(\034)11 b Fq(\),)33 b Fl(f)2346 4512 y Fg(I)2386 4497 y Fq(\()p Fj(\034)2466 4512 y Fp(0)2506 4497 y Fq(\))f(and)g Fl(f)2799 4512 y Fg(I)2839 4497 y Fq(\()p Fj(\034)11 b Fq(\).)44 b(Assume)33 b(\014rst)g(that)0 4617 y Fl(e)51 4632 y Fg(I)91 4617 y Fq(\()p Fj(\034)171 4632 y Fp(0)211 4617 y Fq(\))k(are)f(c)m(hosen)j(as)e(m)m(utually)e(p)s (erp)s(endicular)h(unit)g(v)m(ectors,)k(p)s(erp)s(endicular)c(to)h(the) g(slo)m(wness)0 4738 y(v)m(ector)25 b Fl(p)p Fq(\()p Fj(\034)426 4753 y Fp(0)466 4738 y Fq(\).)41 b(Actually)-8 b(,)24 b(it)f(is)h(su\016cien)m(t)h(to)f(sp)s(ecify)g(only)g Fl(e)2241 4753 y Fp(1)2280 4738 y Fq(\()p Fj(\034)2360 4753 y Fp(0)2400 4738 y Fq(\),)i(as)e Fl(e)2653 4753 y Fp(2)2693 4738 y Fq(\()p Fj(\034)2773 4753 y Fp(0)2812 4738 y Fq(\))k(=)f Fn(C)6 b Fq(\()p Fj(\034)3119 4753 y Fp(0)3159 4738 y Fq(\))p Fl(p)p Fq(\()p Fj(\034)3339 4753 y Fp(0)3379 4738 y Fq(\))f Fn(\002)g Fl(e)3555 4753 y Fp(1)3595 4738 y Fq(\()p Fj(\034)3675 4753 y Fp(0)3714 4738 y Fq(\).)0 4858 y(The)45 b(orien)m(tation)d(of)h(unit)h(v)m(ector) g Fl(e)1413 4873 y Fp(1)1453 4858 y Fq(\()p Fj(\034)1533 4873 y Fp(0)1573 4858 y Fq(\))g(\(p)s(erp)s(endicular)f(to)g Fl(p)p Fq(\()p Fj(\034)2594 4873 y Fp(0)2634 4858 y Fq(\)\),)k(ho)m(w)m (ev)m(er,)i(ma)m(y)43 b(b)s(e)h(quite)0 4979 y(arbitrary)e(in)h(the)h (plane)f(p)s(erp)s(endicular)g(to)g(slo)m(wness)i(v)m(ector)f Fl(p)p Fq(\()p Fj(\034)2604 4994 y Fp(0)2644 4979 y Fq(\).)76 b(V)-8 b(ectors)44 b Fl(f)3177 4994 y Fg(I)3218 4979 y Fq(\()p Fj(\034)3298 4994 y Fp(0)3337 4979 y Fq(\))g(m)m(ust,)i(of)0 5099 y(course,)c(strictly)c(corresp)s(ond)i(to)f(this)g(c)m(hoice)h(of) e Fl(e)1965 5114 y Fg(I)2005 5099 y Fq(\()p Fj(\034)2085 5114 y Fp(0)2125 5099 y Fq(\),)j(see)f(\(31\).)63 b(The)40 b(same)f(is)g(v)-5 b(alid)37 b(ev)m(en)k(at)0 5219 y(p)s(oin)m(t)32 b Fj(\034)11 b Fq(:)44 b(W)-8 b(e)33 b(can)f(c)m(ho)s(ose)i Fl(e)1086 5234 y Fg(I)1126 5219 y Fq(\()p Fj(\034)11 b Fq(\))33 b(arbitrarily)-8 b(.)146 5411 y(Analogous)47 b(rules)g(are)g(v)-5 b(alid)45 b(ev)m(en)k(if)d(w)m(e)i(c)m(ho)s(ose)g Fl(f)2168 5426 y Fg(I)2208 5411 y Fq(\()p Fj(\034)2288 5426 y Fp(0)2328 5411 y Fq(\))f(as)g(m)m(utually)f(p)s(erp)s(endicular) g(unit)0 5531 y(v)m(ectors,)d(p)s(erp)s(endicular)38 b(to)h(the)h(ra)m(y)g(v)m(elo)s(cit)m(y)f(v)m(ector)h Fi(U)10 b Fq(\()p Fj(\034)2294 5546 y Fp(0)2334 5531 y Fq(\).)64 b(It)39 b(is)g(again)f(su\016cien)m(t)i(to)f(sp)s(ecify) 1817 5824 y(110)p eop %%Page: 111 25 111 24 bop 0 100 a Fq(unit)27 b(v)m(ector)i Fl(f)523 115 y Fp(1)563 100 y Fq(\()p Fj(\034)643 115 y Fp(0)682 100 y Fq(\))f(arbitrarily)d(in)i(the)h(plane)g(p)s(erp)s(endicular)f (to)g Fi(U)10 b Fq(\()p Fj(\034)2628 115 y Fp(0)2668 100 y Fq(\);)29 b(v)m(ector)g Fl(f)3084 115 y Fp(2)3124 100 y Fq(\()p Fj(\034)3204 115 y Fp(0)3244 100 y Fq(\))e(=)h(\()p Fi(U)10 b Fq(\()p Fj(\034)3612 115 y Fp(0)3652 100 y Fq(\))i Fn(\002)0 220 y Fl(f)34 235 y Fp(1)74 220 y Fq(\()p Fj(\034)154 235 y Fp(0)193 220 y Fq(\)\))p Fj(=)p Fn(U)e Fq(\()p Fj(\034)470 235 y Fp(0)510 220 y Fq(\).)62 b(V)-8 b(ectors)40 b Fl(e)1042 235 y Fg(I)1082 220 y Fq(\()p Fj(\034)1162 235 y Fp(0)1202 220 y Fq(\),)g(ho)m(w)m(ev)m(er,)j(m)m (ust)c(strictly)f(corresp)s(ond)i(to)f(the)g(c)m(hoice)g(of)g Fl(f)3555 235 y Fg(I)3595 220 y Fq(\()p Fj(\034)3675 235 y Fp(0)3714 220 y Fq(\),)0 340 y(see)34 b(\(32\).)43 b(The)33 b(same)g(is)f(again)f(v)-5 b(alid)30 b(at)j(p)s(oin)m(t)e Fj(\034)11 b Fq(:)44 b(W)-8 b(e)33 b(can)g(c)m(ho)s(ose)h Fl(f)2629 355 y Fg(I)2669 340 y Fq(\()p Fj(\034)11 b Fq(\))33 b(arbitrarily)-8 b(.)146 532 y(A)37 b(\014nal)f(remark.)55 b(In)37 b(Section)f(6,)h(the)g(expressions)i(for)d(the)h(6)24 b Fn(\002)i Fq(6)36 b(in)m(terface)g(propagator)g(ma-)0 652 y(trix)f(in)g(ra)m(y-cen)m(tered)i(co)s(ordinates)e(w)m(ere)i (deriv)m(ed)f(from)e(the)i(expression)h(for)e(the)h(6)24 b Fn(\002)g Fq(6)35 b(in)m(terface)0 772 y(propagator)30 b(matrix)g(in)g(Cartesian)h(co)s(ordinates,)g(giv)m(en)h(b)m(y)g(Moser) g(\(2004\).)42 b(It)31 b(is)g(sho)m(wn)h(ho)m(w)g(this)0 893 y(expression)k(is)e(reduced)i(from)d(6)23 b Fn(\002)h Fq(6)35 b(to)f(4)23 b Fn(\002)h Fq(4)34 b(in)m(terface)h(propagator)e (matrix.)48 b(This)35 b(extends)h(the)0 1013 y(p)s(ossibilities)d(of)i (dynamic)g(ra)m(y)h(tracing)f(in)g(ra)m(y-cen)m(tered)i(co)s(ordinates) e(also)g(to)h(anisotropic)e(inho-)0 1134 y(mogeneous)43 b(media)e(with)h(structural)g(in)m(terfaces.)74 b(In)43 b(Section)g(7,)i(the)e(expression)g(for)f(in)m(terface)0 1254 y(propagator)36 b(matrix)g(is)h(factorized)g(and)h(used)g(to)f (deriv)m(e)h(the)g(4)25 b Fn(\002)h Fq(4)37 b(surface-to-surface)h (paraxial)0 1374 y(matrices)27 b(in)h(ra)m(y-cen)m(tered)i(co)s (ordinates.)41 b(Suc)m(h)30 b(4)13 b Fn(\002)g Fq(4)28 b(surface-to-surface)h(paraxial)d(matrices)i(o\013er)0 1495 y(man)m(y)k(imp)s(ortan)m(t)f(applications)f(in)h(the)i(paraxial)e (ra)m(y)h(metho)s(ds)g(for)g(inhomogeneous)g(anisotropic)0 1615 y(la)m(y)m(ered)26 b(media.)40 b(T)-8 b(o)25 b(k)m(eep)i(the)e (pap)s(er)h(to)f(an)g(admissible)e(length,)j(w)m(e)g(do)g(not)f (discuss)h(these)h(applica-)0 1735 y(tions)k(here;)i(w)m(e)g(only)e (brie\015y)h(list)f(sev)m(eral)h(of)f(them:)43 b(a\))32 b(Solution)e(of)h(b)s(oundary-v)-5 b(alue)31 b(ra)m(y)h(tracing)0 1856 y(problems)h(in)h(four-parameteric)e(system)j(of)e(paraxial)g(ra)m (ys)i(connecting)f(t)m(w)m(o)h(surfaces.)49 b(b\))34 b(Deter-)0 1976 y(mination)k(of)h(the)i(slo)m(wness)h(v)m(ectors)g(at)e (initial)c(and)41 b(end)g(p)s(oin)m(ts)e(of)h(a)g(paraxial)f(ra)m(y)h (connecting)0 2097 y(t)m(w)m(o)d(surfaces.)58 b(c\))37 b(Tw)m(o-p)s(oin)m(t)f(eik)m(onal,)h(i.e.)55 b(the)37 b(tra)m(v)m(el)g(time)f(along)f(a)h(paraxial)f(ra)m(y)i(connecting)0 2217 y(an)m(y)k(p)s(oin)m(t)f(of)g(one)h(surface)h(with)e(an)m(y)i(p)s (oin)m(t)d(of)i(other)f(surface.)69 b(d\))41 b(Determination)d(of)i (relativ)m(e)0 2337 y(geometrical)31 b(spreading,)i(needed)i(in)e(the)g (computation)f(of)h(the)g(ra)m(y-theory)h(Green's)g(function.)45 b(e\))0 2458 y(Determination)26 b(of)i(relativ)m(e)f(geometrical)g (spreading)h(from)f(the)i(tra)m(v)m(el)g(time)e(measuremen)m(ts)i (along)0 2578 y(the)f(t)m(w)m(o)g(surfaces.)43 b(f)7 b(\))27 b(Determination)e(of)i(the)g(complete)g(ra)m(y)h(propagator)e (matrix)g(from)g(the)i(mixed)0 2699 y(second)42 b(deriv)-5 b(ativ)m(es)42 b(of)e(the)i(tra)m(v)m(el)f(time)f(\014eld)h(along)f(t)m (w)m(o)h(surfaces.)71 b(g\))41 b(Determination)d(of)j(the)0 2819 y(F)-8 b(resnel)33 b(zone)i(matrix)d(and)h(F)-8 b(resnel)34 b(regions.)45 b(h\))34 b(F)-8 b(actorization)31 b(of)i(relativ)m(e)g(geometrical)e(spread-)0 2939 y(ing)42 b(using)g(the)h(F)-8 b(resnel)43 b(zone)g(matrix.)72 b(i\))42 b(Iso)s(c)m(hrons.)75 b(Iso)s(c)m(hron)44 b(ra)m(ys.)74 b(j\))43 b(P)m(oin)m(t)f(to)h(curv)m(e)h(ra)m(y)0 3060 y(tracing.)f(k\))34 b(Computation)d(of)h(Gaussian)h(b)s(eams)f(and)h (Gaussian)f(b)s(eams)h(and)g(Gaussian)f(pac)m(k)m(ets.)0 3180 y(l\))42 b(Computation)g(of)h(Maslo)m(v)h(syn)m(thetic)g (seismograms.)75 b(m\))43 b(Application)e(in)h(ra)m(y-p)s(erturbation)0 3300 y(theory)-8 b(.)50 b(n\))35 b(Application)d(in)i(v)-5 b(arious)34 b(di\013raction)f(problems.)49 b(Edge)35 b(and)g(tip)f(w)m(a)m(v)m(es.)52 b(o\))34 b(Applica-)0 3421 y(tions)c(in)f(the)i(true-amplitude)d(migration)f(and)j(in)g(the)g (theory)h(of)f(Kirc)m(hho\013-Helmholtz)d(in)m(tegrals.)0 3541 y(F)-8 b(or)28 b(more)h(details,)g(w)m(e)h(refer)g(the)g(reader)f (to)g(Hubral)g(et)h(al.)41 b(\(1992,)29 b(1993\),)g(F)-8 b(arra)28 b(\(1999\),)3441 3516 y(\024)3430 3541 y(Cerv)m(en)q(\023)-50 b(y)0 3662 y(\(2001,)30 b(Chap.)44 b(4\),)31 b(Han)m(yga)g(et)h(al.)41 b(\(2001\),)31 b(Gj\034ystdal)f(et)h(al.)42 b(\(2002\),)30 b(Chapman)h(\(2004\),)f(Iv)m(ersen)0 3782 y(\(2004b\),)46 b(Sun)f(\(2004\),)h(Moser)f(and)1465 3757 y(\024)1454 3782 y(Cerv)m(en)q(\023)-50 b(y)47 b(\(2006\),)f(where)f(man)m(y)f (other)h(references)h(can)e(b)s(e)0 3902 y(found.)f(Of)29 b(course,)j(the)e(surface-to-surface)g(paraxial)e(matrices)h(in)g(ra)m (y-cen)m(tered)j(co)s(ordinates)d(are)0 4023 y(primarily)38 b(suitable)j(for)f(implemen)m(tation)f(in)m(to)h(ra)m(y)i(tracing)e (pac)m(k)-5 b(ages,)45 b(based)d(on)g(ra)m(y-cen)m(tered)0 4143 y(co)s(ordinates.)0 4452 y Fo(Ac)l(kno)l(wledgemen)l(ts)146 4644 y Fq(The)32 b(authors)g(are)f(greatly)g(indebted)g(to)g(Chris)g (Chapman,)h(Lud)m(\024)-46 b(ek)32 b(Klime)-5 b(\024)-44 b(s)30 b(and)h(Iv)-5 b(an)31 b(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k)0 4764 y(for)28 b(helpful)g(suggestions)i(and)f(v)-5 b(aluable)27 b(discussions.)43 b(This)29 b(researc)m(h)i(has)e(b)s(een) g(supp)s(orted)h(b)m(y)g(the)0 4884 y(Gran)m(t)c(Agency)i(of)f(the)g (Czec)m(h)i(Republic)d(under)h(the)g(Con)m(tracts)h(205/01/D097)c(and)j (205/04/1104,)0 5005 y(and)37 b(b)m(y)g(the)g(Consortium)f(\\Seismic)f (W)-8 b(a)m(v)m(es)38 b(in)e(Complex)g(3-D)f(Structures".)57 b(P)m(art)36 b(of)g(this)h(w)m(ork)0 5125 y(has)h(b)s(een)g(done)h (during)e(TJM's)i(sta)m(ys)g(at)e(the)h(Charles)g(Univ)m(ersit)m(y)-8 b(,)39 b(Prague,)h(sp)s(onsored)e(b)m(y)h(the)0 5246 y(EC)33 b(pro)5 b(ject)34 b(EV)m(G3-CT-2002-80006)c(\(MA)m(GMA\).)1817 5824 y(111)p eop %%Page: 112 26 112 25 bop 0 100 a Fo(References)146 291 y Fq(Bakk)m(er,)34 b(P)-8 b(.M.)32 b(,)g(1996.)42 b(Theory)33 b(of)f(anisotropic)e (dynamic)h(ra)m(y)h(tracing)f(in)g(ra)m(y-cen)m(tered)i(co)s(or-)0 411 y(dinates.)43 b Fd(P)-7 b(A)n(GEOPH)p Fq(,)33 b Fl(148)p Fq(,)g(583)f(-)g(589.)146 603 y(Biloti,)k(R.,)j(San)m(tos,)h(L.T.,)f (and)f(T)m(ygel,)h(M.,)g(2002.)58 b(Multiparametric)35 b(tra)m(v)m(eltime)h(in)m(v)m(ersion.)0 723 y Fd(Stud.)45 b(Ge)-5 b(oph.)44 b(Ge)-5 b(o)g(d.)p Fq(,)33 b Fl(46)p Fq(,)f(177)g(-192.)146 914 y(Bortfeld,)42 b(R.,)g(1989.)65 b(Geometric)39 b(ra)m(y)h(theory:)60 b(Ra)m(ys)41 b(and)f(tra)m(v)m (eltimes)f(in)h(seismic)f(systems)0 1035 y(\(second-order)34 b(appro)m(ximation)c(of)i(the)h(tra)m(v)m(eltimes\).)43 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(54)p Fq(,)g(342)g(-)g(349.)157 1201 y(\024)146 1226 y(Cerv)m(en)q(\023)-50 b(y,)52 b(V.,)e(1972.)84 b(Seismic)45 b(ra)m(ys)i(and)f(ra)m(y)g(in)m(tensities)g(in)f (inhomogeneous)h(anisotropic)0 1346 y(media.)c Fd(Ge)-5 b(ophys.)45 b(J.)35 b(R.)f(astr.)45 b(So)-5 b(c.)p Fq(,)32 b Fl(29)p Fq(,)h(1-13.)157 1512 y(\024)146 1537 y(Cerv)m(en)q(\023)-50 b(y,)35 b(V.,)e(2001.)43 b Fd(Seismic)34 b(r)-5 b(ay)35 b(the)-5 b(ory)p Fq(.)43 b(Cam)m(bridge:)g(Cam)m(bridge)32 b(Univ.)43 b(Press.)157 1703 y(\024)146 1729 y(Cerv)m(en)q(\023)-50 b(y,)47 b(V.,)e(and)d(Hron,)j(F.,)f(1980.)71 b(The)43 b(ra)m(y)f(series)h(metho)s(d)e(and)i(dynamic)e(ra)m(y)h(tracing)0 1849 y(system)c(for)e(three-dimensional)e(inhomogeneous)i(media.)54 b Fd(Bul)5 b(l.)56 b(Seismol.)f(So)-5 b(c.)55 b(A)n(m.)p Fq(,)37 b Fl(70)p Fq(,)h(47)e(-)0 1969 y(77.)157 2135 y(\024)146 2161 y(Cerv)m(en)q(\023)-50 b(y,)33 b(V.,)e(Klime)-5 b(\024)-44 b(s,)29 b(L.,)h(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)32 b(I.,)f(1988.)42 b(Complete)29 b(seismic)g(ra)m(y)h(tracing)f(in)g(three-)0 2281 y(dimensional)34 b(structures.)55 b(In:)c Fd(Seismolo)-5 b(gic)g(al)36 b(algorithms)p Fq(,)g(ed.)54 b(D.)36 b(J.)g(Do)s(orn)m(b)s(os,)g(pp.)54 b(89)36 b(-)f(168.)0 2401 y(New)e(Y)-8 b(ork)33 b(:)44 b(Academic)32 b(Press.)157 2567 y(\024)146 2593 y(Cerv)m(en)q(\023)-50 b(y,)34 b(V.,)d(Langer,)f(J.,)h(and)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)32 b(I.,)f(1974.)42 b(Computation)29 b(of)h(geometrical)e(spreading)0 2713 y(of)g(seismic)g(b)s(o)s(dy)h(w)m(a)m(v)m(es)j(in)c(laterally)e (inhomogeneous)i(media)g(with)g(curv)m(ed)j(in)m(terfaces.)43 b Fd(Ge)-5 b(ophys.)0 2833 y(J.R.)35 b(astr.)45 b(So)-5 b(c.)p Fq(,)32 b Fl(38)p Fq(,)g(9)h(-)f(19.)146 3025 y(Chapman,)c(C.)f(H.,)h(2004.)41 b Fd(F)-7 b(undamentals)27 b(of)i(seismic)g(wave)f(pr)-5 b(op)g(agation)p Fq(.)41 b(Cam)m(bridge:)e(Cam-)0 3145 y(bridge)32 b(Univ.)43 b(Press.)146 3336 y(Chira-Oliv)-5 b(a,)35 b(P)-8 b(.,)39 b(T)m(ygel,)f(M.,)g(Zhang,)g(Y.,)g(and)f(Hubral,)g(P)-8 b(.,)39 b(2001.)55 b(Analytic)36 b(CRS)h(form)m(ula)0 3457 y(for)h(a)g(2D)g(curv)m(ed)i(measuremen)m(t)e(surface)i(and)e (\014nite-o\013set)g(re\015ections.)62 b Fd(J.)40 b(Seis.)60 b(Explor)-5 b(ation)p Fq(,)0 3577 y Fl(10)p Fq(,)33 b(245)f(-)g(262.) 146 3768 y(F)-8 b(arra,)47 b(V.,)g(1989.)79 b(Ra)m(y)44 b(p)s(erturbation)g(theory)h(for)f(heterogeneous)i(hexagonal)e (anisotropic)0 3889 y(medium.)e Fd(Ge)-5 b(ophys.)44 b(J.)35 b(Int.)p Fq(,)e Fl(99)p Fq(,)f(723)g(-)g(738.)146 4080 y(F)-8 b(arra,)32 b(V.,)h(and)f(LeB)m(\023)-46 b(egat,)33 b(S.,)g(1995.)42 b(Sensitivit)m(y)32 b(of)g(qP)h(w)m(a)m(v)m(e)i(tra)m (v)m(eltimes)c(and)i(p)s(olarization)0 4200 y(v)m(ectors)h(to)e (heterogeneit)m(y)-8 b(,)34 b(anisotrop)m(y)e(and)h(in)m(terfaces.)44 b Fd(Ge)-5 b(ophys.)45 b(J.)35 b(Int.)p Fq(,)d Fl(121)p Fq(,)g(371)g(-)g(384.)146 4392 y(Ga)5 b(jewski,)28 b(D.,)e(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)28 b(I.,)f(1987.)40 b(Computation)24 b(of)h(high-frequency)i(seismic)d(w)m (a)m(v)m(e\014elds)0 4512 y(in)35 b(3-D)f(laterally)g(inhomogeneous)h (anisotropic)f(media.)52 b Fd(Ge)-5 b(ophys.)53 b(J.)38 b(R.)g(astr.)53 b(So)-5 b(c.)p Fq(,)36 b Fl(91)p Fq(,)h(383)e(-)0 4632 y(411.)146 4824 y(Ga)5 b(jewski,)29 b(D.,)f(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)30 b(I.,)f(1990.)41 b(V)-8 b(ertical)26 b(seismic)g(pro\014le)h(syn)m (thetics)j(b)m(y)e(dynamic)f(ra)m(y)0 4944 y(tracing)33 b(in)f(laterally)f(v)-5 b(arying)33 b(la)m(y)m(ered)i(anisotropic)d (structures.)48 b Fd(J.)36 b(Ge)-5 b(ophys.)47 b(R)-5 b(es.)p Fq(,)34 b Fl(95)p Fq(,)g(11301)e(-)0 5064 y(11315.)146 5256 y(Gilb)s(ert,)24 b(F.,)i(and)e(Bac)m(kus,)k(G.E.,)e(1966.)40 b(Propagator)23 b(matrices)h(in)f(elastic)h(w)m(a)m(v)m(e)i(and)e (vibration)0 5376 y(problems.)43 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(31)p Fq(,)h(326)e(-)h(332.)146 5567 y(Gj\034ystdal,)i(H.,) g(Iv)m(ersen,)i(E.,)f(Laurain,)e(R.,)h(Lecom)m(te,)g(I.,)h(Vinje,)f (V.,)g(and)3034 5542 y(\027)3034 5567 y(Asteb)s(ol,)g(K,)f(2002.)1817 5824 y(112)p eop %%Page: 113 27 113 26 bop 0 100 a Fq(Review)27 b(of)f(ra)m(y)h(theory)h(applications)c (in)i(mo)s(delling)d(and)k(imaging)d(of)i(seismic)g(data.)41 b Fd(Stud.)i(Ge)-5 b(oph.)0 220 y(Ge)g(o)g(d.)p Fq(,)32 b Fl(46)p Fq(,)h(113)f(-)g(164.)146 411 y(Han)m(yga,)38 b(A.,)g(1982.)53 b(Dynamic)35 b(ra)m(y)i(tracing)e(in)h(anisotropic)f (medium.)53 b Fd(T)-7 b(e)i(ctonophysics)p Fq(,)35 b Fl(90)p Fq(,)0 532 y(243)d(-)g(251.)146 723 y(Han)m(yga,)37 b(A.,)f(Druzhinin,)f(A.)g(B.,)h(Dzhafaro)m(v,)g(A.)g(D.,)f(F)-8 b(r\034yland,)35 b(L.,)h(2001.)51 b(A)35 b(Hamiltonian)0 843 y(approac)m(h)i(to)g(asymptotic)e(seismic)h(re\015ection)h(and)g (di\013raction)e(mo)s(deling.)53 b Fd(Ge)-5 b(oph.)56 b(Pr)-5 b(osp.)p Fq(,)37 b Fl(49)p Fq(,)0 964 y(213)32 b(-)g(227.)146 1155 y(Hubral,)g(P)-8 b(.,)32 b(Sc)m(hleic)m(her,)h(J.,) f(and)g(T)m(ygel,)h(M.,)f(1992.)42 b(Three-dimensional)31 b(paraxial)f(ra)m(y)i(prop-)0 1275 y(erties:)50 b(P)m(art)37 b(I.)f(Basic)g(relations.)52 b Fd(J.)38 b(Seis.)54 b(Explor)-5 b(ation)p Fq(,)36 b Fl(1)p Fq(,)h(265)e(-)h(279.)53 b(P)m(art)36 b(I)s(I.)g(Applications.)0 1396 y Fd(J.)f(Seis.)44 b(Explor)-5 b(ation)p Fq(,)32 b Fl(1)p Fq(,)h(347)e(-)h(362.)146 1587 y(Hubral,)j(P)-8 b(.,)35 b(Sc)m(hleic)m(her,)h(J.,)g(and)e(T)m (ygel,)i(M.,)g(and)e(Hanitzsc)m(h,)i(C.,)g(1993.)49 b(Determination)32 b(of)0 1707 y(F)-8 b(resnel)32 b(zones)i(from)d(tra)m(v)m(eltime)h (measuremen)m(ts.)44 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(58)p Fq(,)h(703)f(-)g(712.)146 1899 y(Hubral,)24 b(P)-8 b(.,)25 b(T)m(ygel,)g(M.,)g(and)e(Sc)m(hleic)m(her,)i(J.,)g (1995.)39 b(Geometrical)20 b(spreading)i(and)h(ra)m(y-caustics)0 2019 y(decomp)s(osition)31 b(of)h(elemen)m(tary)g(seismic)g(w)m(a)m(v)m (es.)46 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(60)p Fq(,)h(1195)e(-)h (1202.)146 2210 y(Iv)m(ersen,)54 b(E.,)e(2004a.)88 b(Reform)m(ulated)47 b(kinematic)f(and)i(dynamic)f(ra)m(y)h(tracing)f(systems)i(for)0 2331 y(arbitrary)32 b(anisotropic)f(media.)42 b Fd(Stud.)j(Ge)-5 b(oph.)44 b(Ge)-5 b(o)g(d.)p Fq(,)33 b Fl(48)p Fq(,)f(1-20.)146 2522 y(Iv)m(ersen,)43 b(E.,)d(2004b.)60 b(The)40 b(iso)s(c)m(hron)e(ra) m(y)h(in)e(seismic)h(mo)s(deling)e(and)i(imaging.)58 b Fd(Ge)-5 b(ophysics)p Fq(,)0 2642 y Fl(69)p Fq(,)33 b(1053)e(-)h(1070.)146 2833 y(Kendall,)c(J-M.,)i(Guest,)g(W.,)g(S.,)g (and)e(Thomson,)i(C.)f(J.,)h(1992.)41 b(Ra)m(y)29 b(theory)g(Green's)h (function)0 2954 y(recipro)s(cit)m(y)e(and)g(ra)m(y-cen)m(tered)j(co)s (ordinates)d(in)g(anisotropic)f(media.)40 b Fd(Ge)-5 b(ophys.)44 b(J.)31 b(Int.)p Fq(,)e Fl(108)p Fq(,)g(364)0 3074 y(-)j(371.)146 3265 y(Kennett,)49 b(B.L.N.,)g(1983.)80 b Fd(Seismic)45 b(wave)h(pr)-5 b(op)g(agation)45 b(in)h(str)-5 b(ati\014e)g(d)46 b(me)-5 b(dia)p Fq(.)80 b(Cam)m(bridge:)0 3386 y(Cam)m(bridge)32 b(Univ.)43 b(Press.)146 3577 y(Klime)-5 b(\024)-44 b(s,)25 b(L.,)h(1994.)40 b(T)-8 b(ransformations)23 b(for)h(dynamic)f(ra)m(y)i(tracing)f(in)f(anisotropic)g(media.)40 b Fd(Wave)0 3697 y(Motion)p Fq(,)33 b Fl(20)p Fq(,)f(261)g(-)g(272.)146 3889 y(Klime)-5 b(\024)-44 b(s,)25 b(L.,)h(2006.)40 b(Ra)m(y-cen)m (tered)26 b(co)s(ordinate)e(systems)i(in)e(anisotropic)f(media.)39 b Fd(Stud.)k(Ge)-5 b(oph.)0 4009 y(Ge)g(o)g(d.,)p Fq(,)32 b Fl(50)p Fq(,)h(in)f(press.)146 4200 y(Moser,)j(T.J.,)g(2004.)47 b(Review)34 b(of)f(the)i(anisotropic)d(in)m(terface)i(ra)m(y)g (propagator:)45 b(symplecticit)m(y)-8 b(,)0 4321 y(eigen)m(v)j(alues,) 33 b(in)m(v)-5 b(arian)m(ts)31 b(and)i(applications.)41 b Fd(Stud.)k(Ge)-5 b(oph.)45 b(Ge)-5 b(o)g(d.)p Fq(,)32 b Fl(48)p Fq(,)h(47)f(-)g(73.)146 4512 y(Moser,)k(T.J.,)f(and)904 4487 y(\024)894 4512 y(Cerv)m(en)q(\023)-50 b(y,)37 b(V.,)e(2006.)47 b(P)m(araxial)33 b(ra)m(y)h(metho)s(ds)g(for)g(anisotropic)f(inhomoge-) 0 4632 y(neous)h(media.)42 b Fd(Ge)-5 b(ophys.)44 b(Pr)-5 b(osp)g(e)g(cting)p Fq(,)32 b(in)g(press.)146 4824 y(Norris,)f(A.)h (N.,)f(1987.)42 b(A)32 b(theory)f(of)g(pulse)h(propagation)d(in)i (anisotropic)e(elastic)i(solids.)42 b Fd(Wave)0 4944 y(Motion)p Fq(,)33 b Fl(9)p Fq(,)f(509)g(-)g(532.)146 5135 y(P)m(op)s(o)m(v,)i(M.)f(M.,)g(and)g(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)34 b(I.,)f(1978a.)43 b(Ra)m(y)33 b(amplitudes)e(in)h(inhomogeneous)g(media)g(with)0 5256 y(curv)m(ed)37 b(in)m(terfaces.)50 b(In)36 b Fd(Ge)-5 b(ofyzik\023)-50 b(aln)-10 b(\023)-40 b(\020)36 b(sb)-5 b(orn)-10 b(\023)-40 b(\020k)p Fq(,)34 b(v)m(ol.)50 b(24,)35 b(ed.)51 b(A)35 b(Z\023)-49 b(atop)s(ek,)35 b(pp.111)g(-)f(129.)49 b(Praha:)0 5376 y(Academia.)146 5567 y(P)m(op)s(o)m(v,)37 b(M.)f(M.,)g(and)f(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)38 b(I.,)e(1978b.)50 b(Computation)34 b(of)h(ra)m(y)g(amplitudes)f(in)h(inhomoge-)1817 5824 y(113)p eop %%Page: 114 28 114 27 bop 0 100 a Fq(neous)34 b(media)d(with)h(curv)m(ed)i(in)m (terfaces.)44 b Fd(Stud.)h(Ge)-5 b(oph.)45 b(Ge)-5 b(o)g(d.)p Fq(,)32 b Fl(22)p Fq(,)h(248)f(-)g(258.)146 291 y(P)-5 b(\024)-44 b(sen)m(\024)e(c)-11 b(\023)-38 b(\020k,)27 b(I.,)g(and)d(T)-8 b(eles,)26 b(T.)f(N.,)h(1996.)39 b(P)m(oin)m(t)24 b(source)h(radiation)d(in)h(inhomogeneous)h(anisotrop-)0 411 y(ic)32 b(structures.)45 b Fd(P)-7 b(A)n(GEOPH)p Fq(,)33 b Fl(148)p Fq(,)g(591)f(-)g(623.)146 603 y(Sun,)40 b(J.,)f(2004.)58 b(T)-8 b(rue)39 b(amplitude)d(w)m(eigh)m(t)i(function) f(in)g(3D)g(limited)e(ap)s(erture)j(migration)d(re-)0 723 y(visited.)43 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(60)p Fq(,)h(1025)e(-)h(1036.)146 914 y(Shearer,)42 b(P)-8 b(.)39 b(M.,)j(and)d(Chapman,)i(C.)f(H.,)h(1989.)62 b(Ra)m(y)40 b(tracing)e(in)g(azim)m(uthally)f(anisotropic)0 1035 y(media.)49 b(I.)36 b(Results)f(for)f(mo)s(dels)g(of)g(aligned)g(crac)m (ks)i(in)f(the)g(upp)s(er)h(crust.)51 b Fd(Ge)-5 b(ophys.)51 b(J.)p Fq(,)36 b Fl(96)p Fq(,)g(51)e(-)0 1155 y(64.)146 1346 y(Sc)m(hleic)m(her,)41 b(J.,)f(T)m(ygel,)g(M.,)g(and)e(Hubral,)h (P)-8 b(.,)40 b(1993.)60 b(3-D)37 b(true-amplitude)f(\014nite-o\013set) i(mi-)0 1467 y(gration.)k Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(58)p Fq(,)h(1112)e(-)h(1126.)146 1658 y(Ursin,)37 b(B.,)f(1983.)52 b(Review)36 b(of)f(elastic)g(and)h(electromagnetic)e (w)m(a)m(v)m(e)k(propagation)c(in)h(horizon-)0 1778 y(tally)c(la)m(y)m (ered)i(media.)42 b Fd(Ge)-5 b(ophysics)p Fq(,)32 b Fl(48)p Fq(,)h(1063)f(-)g(1081.)1817 5824 y(114)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF