%!PS-Adobe-2.0 %%Creator: dvips 5.54 Copyright 1986, 1994 Radical Eye Software %%Title: r10lk3.dvi %%CreationDate: Sat Mar 10 20:10:36 2001 %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: C:\EMTEX\DVIPS16.EXE -O0.0mm,-8.8mm r10lk3 %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 2001.03.10:2010 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies 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y(If)21 b(paraxial)e(ra)o(y)h(propagator)f(matrix)i (\(4\))g(is)f(expressed)g(in)g(ra)o(y{cen)o(tred)f(co)q(ordinates)h Fk(q)1859 1841 y Fi(i)1876 1834 y Fm(,)0 1894 y(in)o(tro)q(duced)e(b)o (y)i(Luneburg)e(\(1944\))i(and)f(P)o(op)q(o)o(v)g(&)h(P)m(\024)-22 b(sen)o(\024)e(c)-6 b(\023)-19 b(\020k)19 b(\(1978\),)i(then)e Fk(Q)1539 1901 y Fi(i)p Fj(3)1595 1894 y Fm(=)g(0,)h Fk(Q)1751 1901 y Fj(3)p Fi(i)1807 1894 y Fm(=)f(0)0 1953 y(\(Klime)m(\024)-22 b(s)16 b(1994\))g(and)612 2016 y Fk(@)641 1998 y Fj(2)663 2016 y Fk(\034)p 574 2038 155 2 v 574 2084 a(@)s(x)631 2091 y Fi(i)648 2084 y Fk(@)s(x)705 2067 y Fj(0)705 2098 y Fi(j)734 2050 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)835 2029 y Fj(0)859 2050 y Fm(\))14 b(=)g Fh(\000)p Fk(H)1025 2057 y Fi(iK)1077 2050 y Fk(Q)1116 2028 y Ff(\000)p Fj(1)1116 2064 y Fi(K)r(L)1182 2050 y Fk(H)1227 2029 y Fj(0)1223 2062 y Fi(j)r(L)1826 2050 y Fm(\(5\))0 2193 y(\()25 2180 y(\024)19 2193 y(Cerv)o(en)q(\023)-26 b(y,)14 b(Klime)m(\024)-22 b(s)13 b(&)g(P)m(\024)-22 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y Fj(0)845 2664 y Fm(\))14 b(=)946 2631 y Fk(@)s(\034)p 937 2653 75 2 v 937 2699 a(@)s(x)994 2706 y Fi(i)1017 2664 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)1118 2644 y Fj(0)1142 2664 y Fm(\))665 b(\(7\))908 2806 y(116)p eop %%Page: 117 3 117 2 bop 0 -54 a Fm(corresp)q(onding)17 b(to)k(p)q(oin)o(t)e(source)g (at)h Fo(x)771 -72 y Fj(0)813 -54 y Fm(is)g(obtained)f(b)o(y)g (di\013eren)o(tiating)g(\(1\))h(with)g(resp)q(ect)g(to)0 6 y(\001)p Fk(x)70 13 y Fi(i)87 6 y Fm(,)99 126 y Fk(p)124 133 y Fi(i)140 126 y Fm(\()p Fo(x)12 b Fm(+)f(\001)p Fo(x)p Fk(;)d Fo(x)375 105 y Fj(0)408 126 y Fm(+)j(\001)p Fo(x)530 105 y Fj(0)552 126 y Fm(\))j(=)653 92 y Fk(@)s(\034)p 644 114 75 2 v 644 160 a(@)s(x)701 167 y Fi(i)724 126 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)825 105 y Fj(0)848 126 y Fm(\))k(+)972 92 y Fk(@)1001 74 y Fj(2)1023 92 y Fk(\034)p 935 114 153 2 v 935 160 a(@)s(x)992 167 y Fi(i)1009 160 y Fk(@)s(x)1066 167 y Fi(j)1094 126 y Fm(\()p Fo(x)p Fk(;)c Fo(x)1195 105 y Fj(0)1218 126 y Fm(\)\001)p Fk(x)1307 133 y Fi(j)1339 126 y Fm(+)1433 92 y Fk(@)1462 74 y Fj(2)1484 92 y Fk(\034)p 1395 114 155 2 v 1395 160 a(@)s(x)1452 167 y Fi(i)1469 160 y Fk(@)s(x)1526 143 y Fj(0)1526 174 y Fi(j)1555 126 y Fm(\()p Fo(x)p Fk(;)g Fo(x)1656 105 y Fj(0)1680 126 y Fm(\)\001)p Fk(x)1769 105 y Fj(0)1769 138 y Fi(j)1125 260 y Fm(+)p Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)p Fh(j)1322 239 y Fj(2)1344 260 y Fm(\))k(+)e Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1568 239 y Fj(0)1591 260 y Fh(j)1605 239 y Fj(2)1627 260 y Fm(\))50 b Fk(:)116 b Fm(\(8\))0 342 y(Di\013erence)443 402 y(\001)p Fk(p)510 409 y Fi(i)540 402 y Fm(=)13 b Fk(p)617 409 y Fi(i)634 402 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)735 381 y Fj(0)769 402 y Fm(+)825 382 y Fj(1)p 825 390 20 2 v 825 419 a(2)851 402 y Fm(\001)p Fo(x)923 381 y Fj(0)945 402 y Fm(\))k Fh(\000)e Fk(p)1050 409 y Fi(i)1067 402 y Fm(\()p Fo(x)p Fk(;)e Fo(x)1168 381 y Fj(0)1202 402 y Fh(\000)1258 382 y Fj(1)p 1258 390 V 1258 419 a(2)1284 402 y Fm(\001)p Fo(x)1356 381 y Fj(0)1378 402 y Fm(\))429 b(\(9\))0 483 y(of)17 b(the)g(slo)o(wness)d(v)o(ectors)i(corresp)q(onding)f(to)i(a)f (p)q(oin)o(t)g(source)g(shifted)g(b)o(y)g Fh(\006)1492 464 y Fj(1)p 1492 472 V 1492 501 a(2)1518 483 y Fm(\001)p Fo(x)g Fm(is)496 611 y(\001)p Fk(p)563 618 y Fi(i)593 611 y Fm(=)689 577 y Fk(@)718 559 y Fj(2)740 577 y Fk(\034)p 651 600 155 2 v 651 645 a(@)s(x)708 652 y Fi(i)726 645 y Fk(@)s(x)783 628 y Fj(0)783 659 y Fi(j)812 611 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)913 591 y Fj(0)936 611 y Fm(\)\001)p Fk(x)1025 591 y Fj(0)1025 623 y Fi(j)1059 611 y Fm(+)i Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1252 591 y Fj(0)1275 611 y Fh(j)1289 591 y Fj(3)1311 611 y Fm(\))50 b Fk(:)407 b Fm(\(10\))0 784 y Fo(3)38 b(Calculation)22 b(of)c(p)r(oin)n(t{source) h(geom)o(etrical)f(spreading)h(in)g(2-D)0 855 y Fm(Equation)d(\(10\))h (with)g(\(5\))g(in)f(2-D)g(reads)505 946 y(\001)p Fk(p)572 953 y Fi(i)602 946 y Fm(=)d Fh(\000)p Fk(H)734 953 y Fi(i)p Fj(1)771 946 y Fk(Q)810 925 y Ff(\000)p Fj(1)810 960 y(11)864 946 y Fk(H)909 926 y Fj(0)905 959 y Fi(j)r Fj(1)946 946 y Fm(\001)p Fk(x)1016 926 y Fj(0)1016 959 y Fi(j)1049 946 y Fm(+)e Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1243 926 y Fj(0)1266 946 y Fh(j)1280 926 y Fj(3)1302 946 y Fm(\))50 b Fk(;)416 b Fm(\(11\))0 1037 y(where)17 b Fk(Q)184 1044 y Fj(11)245 1037 y Fm(is)g(the)h(in{plane)f(comp)q (onen)o(t)g(of)h(matrix)f(\(3\).)26 b(Unit)19 b(basis)d(v)o(ector)i Fk(H)1596 1044 y Fi(i)p Fj(1)1651 1037 y Fm(of)g(the)g(ra)o(y{)0 1097 y(cen)o(tred)e(co)q(ordinate)g(system)g(is)g(p)q(erp)q(endicular)f (to)i(the)g(slo)o(wness)d(v)o(ector,)647 1188 y(\()p Fk(H)707 1195 y Fj(11)750 1188 y Fk(;)8 b(H)813 1195 y Fj(31)856 1188 y Fm(\))14 b(=)g Fk(v)k Fm(\()p Fh(\000)p Fk(p)1067 1195 y Fj(3)1090 1188 y Fk(;)8 b(p)1137 1195 y Fj(1)1160 1188 y Fm(\))50 b Fk(;)532 b Fm(\(12)p Fk(a)p Fm(\))636 1279 y(\()p Fk(H)700 1258 y Fj(0)696 1291 y(11)739 1279 y Fk(;)8 b(H)806 1258 y Fj(0)802 1291 y(31)845 1279 y Fm(\))14 b(=)g Fk(v)957 1258 y Fj(0)996 1279 y Fm(\()p Fh(\000)p Fk(p)1079 1258 y Fj(0)1079 1291 y(3)1101 1279 y Fk(;)8 b(p)1148 1258 y Fj(0)1148 1291 y(1)1171 1279 y Fm(\))50 b Fk(:)526 b Fm(\(12)p Fk(b)p Fm(\))0 1361 y(Slo)o(wness)15 b(v)o(ector)h Fo(p)h Fm(at)g(p)q(oin)o(t)f Fo(x)h Fm(is)f(kno)o(wn)f(or)h(ma)o(y)g(b)q(e)h(appro)o(ximated)e(b)o (y)225 1452 y Fk(p)250 1459 y Fi(i)266 1452 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)367 1431 y Fj(0)391 1452 y Fm(\))14 b(=)482 1432 y Fj(1)p 482 1440 20 2 v 482 1469 a(2)517 1412 y Fg(\002)537 1452 y Fk(p)562 1459 y Fi(i)579 1452 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)680 1431 y Fj(0)714 1452 y Fm(+)770 1432 y Fj(1)p 770 1440 V 770 1469 a(2)796 1452 y Fm(\001)p Fo(x)p Fm(\))j(+)g Fk(p)973 1459 y Fi(i)990 1452 y Fm(\()p Fo(x)p Fk(;)d Fo(x)1091 1431 y Fj(0)1125 1452 y Fh(\000)1181 1432 y Fj(1)p 1181 1440 V 1181 1469 a(2)1207 1452 y Fm(\001)p Fo(x)p Fm(\))1298 1412 y Fg(\003)1330 1452 y Fm(+)i Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1523 1431 y Fj(0)1546 1452 y Fh(j)1560 1431 y Fj(2)1582 1452 y Fm(\))50 b Fk(:)136 b Fm(\(13\))0 1543 y(On)20 b(the)g(other)g(hand,) g(initial)g(slo)o(wness)f(v)o(ector)h Fo(p)989 1525 y Fe(0)1034 1543 y Fm(at)h(p)q(oin)o(t)f Fo(x)1263 1525 y Fj(0)1306 1543 y Fm(is)g(unkno)o(wn)e(and)i(should)f(b)q(e)0 1603 y(estimated)d(from)g(the)h(tra)o(v)o(el)f(time)g(di\013erence)480 1694 y(\001)p Fk(\034)i Fm(=)c Fk(\034)6 b Fm(\()p Fo(x)p Fk(;)i Fo(x)744 1673 y Fj(0)778 1694 y Fm(+)833 1674 y Fj(1)p 833 1682 V 833 1711 a(2)859 1694 y Fm(\001)p Fo(x)p Fm(\))j Fh(\000)g Fk(\034)6 b Fm(\()p Fo(x)p Fk(;)i Fo(x)1140 1673 y Fj(0)1174 1694 y Fh(\000)1230 1674 y Fj(1)p 1230 1682 V 1230 1711 a(2)1256 1694 y Fm(\001)p Fo(x)p Fm(\))50 b Fk(:)390 b Fm(\(14\))0 1785 y(and)16 b(eik)o(onal)g(equation)785 1844 y Fk(p)810 1824 y Fj(0)810 1857 y Fi(i)832 1844 y Fk(p)857 1824 y Fj(0)857 1857 y Fi(i)893 1844 y Fm(=)e(\()p Fk(v)991 1824 y Fj(0)1014 1844 y Fm(\))1033 1824 y Fj(2)1801 1844 y Fm(\(15\))0 1926 y(at)j(the)g(initial)f(p)q(oin)o(t.)21 b(Equation)16 b(\(1\))h(yields)276 2043 y(\001)p Fk(\034)i Fm(=)429 2010 y Fk(@)s(\034)p 417 2032 81 2 v 417 2077 a(@)s(x)474 2060 y Fj(0)474 2091 y Fi(i)503 2043 y Fm(\()p Fo(x)p Fk(;)8 b Fo(x)604 2023 y Fj(0)628 2043 y Fm(\)\001)p Fk(x)717 2023 y Fj(0)717 2056 y Fi(i)750 2043 y Fm(+)j Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)944 2023 y Fj(0)967 2043 y Fh(j)981 2023 y Fj(3)1003 2043 y Fm(\))j(=)g Fh(\000)p Fk(p)1153 2023 y Fj(0)1153 2056 y Fi(j)1175 2043 y Fm(\001)p Fk(x)1245 2023 y Fj(0)1245 2056 y Fi(j)1278 2043 y Fm(+)d Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1472 2023 y Fj(0)1495 2043 y Fh(j)1509 2023 y Fj(3)1531 2043 y Fm(\))50 b Fk(:)187 b Fm(\(16\))0 2168 y(The)18 b(initial)f(slo)o(wness)f(v)o(ector)i(ma)o (y)f(b)q(e)h(decomp)q(osed)f(in)o(to)g(the)h(unit)g(v)o(ector)g(\001)p Fk(x)1575 2150 y Fj(0)1575 2181 y Fi(j)1597 2168 y Fk(=)p Fh(j)p Fm(\001)p Fo(x)1708 2150 y Fj(0)1730 2168 y Fh(j)f Fm(in)h(the)0 2228 y(direction)e(of)g(the)h(source{p)q(oin)o(t)e(shift) i(and)e(the)i(p)q(erp)q(endicular)e(v)o(ector)575 2319 y(\()p Fk(E)631 2326 y Fj(1)653 2319 y Fk(;)8 b(E)712 2326 y Fj(3)735 2319 y Fm(\))14 b(=)f(\()p Fh(\000)p Fm(\001)p Fk(x)948 2298 y Fj(0)948 2331 y(3)971 2319 y Fk(;)8 b Fm(\001)p Fk(x)1063 2298 y Fj(0)1063 2331 y(1)1085 2319 y Fm(\))p Fk(=)p Fh(j)p Fm(\001)p Fo(x)1215 2298 y Fj(0)1238 2319 y Fh(j)49 b Fk(;)486 b Fm(\(17\))640 2469 y Fk(p)665 2449 y Fj(0)665 2482 y Fi(j)701 2469 y Fm(=)14 b Fk(a)g Fm(\001)p Fk(x)864 2449 y Fj(0)864 2482 y Fi(j)886 2469 y Fk(=)p Fh(j)p Fm(\001)p Fo(x)997 2449 y Fj(0)1019 2469 y Fh(j)d Fm(+)g Fk(b)j(E)1166 2476 y Fi(j)1236 2469 y Fk(:)551 b Fm(\(18\))0 2551 y(Inserting)15 b(\(18\))j(in)o(to)e(\(16\),)h(w)o(e)f(get)h(co)q(e\016cien)o(t)598 2642 y Fk(a)d Fm(=)g Fh(\000)p Fm(\001)p Fk(\034)6 b(=)p Fh(j)p Fm(\001)p Fo(x)911 2622 y Fj(0)931 2642 y Fh(j)11 b Fm(+)g Fk(O)q Fm(\()p Fh(j)p Fm(\001)p Fo(x)1150 2622 y Fj(0)1173 2642 y Fh(j)1187 2622 y Fj(2)1209 2642 y Fm(\))50 b Fk(:)509 b Fm(\(19\))908 2806 y(117)p eop %%Page: 118 4 118 3 bop 0 1790 a @beginspecial 71 @llx 317 @lly 524 @urx 771 @ury 3780 @rwi @setspecial %%BeginDocument: vgr-q.ps %----------------------------------------------------------- % Default UNDEFINED procedure: /UNDEFINED {0.80 0.80 0.80 setrgbcolor} bind def % Default VALUEtoCOLOR procedure: /VCIRC 1.00000 def % Range of grid values: -1.41540 to 3.18380 /VREF 0.000000 def /CREF 0.166667 def /VRED VREF CREF VCIRC mul sub def /VALUEtoCOLOR{ VRED sub VCIRC div dup truncate sub dup 0 lt {1 add} if 1 1 sethsbcolor} bind def %----------------------------------------------------------- % User-defined VALUEtoCOLOR procedure may be inserted here: %----------------------------------------------------------- %----------------------------------------------------------- % Numerical values describing the image size and position: /N1 202 def /N2 202 def /BB1 70.9 def % 2.500cm /BB2 317.5 def % 11.200cm /BB3 453.5 def % 16.000cm /BB4 453.5 def % 16.000cm /PAPERSIZE 841.9 def /ROTATE 0.0 def % Scaling of grid values: /GMIN1 -1.41540 def /GMAX1 3.18380 def 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def %----------------------------------------------------------- % User-defined VALUEtoCOLOR procedure may be inserted here: %----------------------------------------------------------- %----------------------------------------------------------- % Numerical values describing the image size and position: /N1 202 def /N2 202 def /BB1 70.9 def % 2.500cm /BB2 317.5 def % 11.200cm /BB3 453.5 def % 16.000cm /BB4 453.5 def % 16.000cm /PAPERSIZE 841.9 def /ROTATE 0.0 def % Scaling of grid values: /GMIN1 0.500700E-01 def /GMAX1 3.17026 def %----------------------------------------------------------- PAPERSIZE 2 div dup translate ROTATE rotate PAPERSIZE -2 div dup translate /SCALE1 N1 BB3 div def /SCALE2 N2 BB4 div def /VSTEP1 GMAX1 GMIN1 sub 254 div def /RGB 3 string def N1 N2 8 [ SCALE1 0 0 SCALE2 BB1 SCALE1 mul neg BB2 SCALE2 mul neg ] {currentfile 1 string readhexstring pop dup 0 get VSTEP1 mul GMIN1 add exch 0 get 255 eq {pop UNDEFINED} {VALUEtoCOLOR} ifelse currentrgbcolor 255 mul .5 add 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33343435353637373838393A3A3B3B3C3D3D3E3F3F4041414243434445454647474849494A4B4C4C 4D4E4E4F505151525354545556575758595A5A5B5C5D5D5E5F6061626263646566666768696A6B6B 6C6D6E6F PAPERSIZE 2 div dup translate ROTATE neg rotate PAPERSIZE -2 div dup translate showpage %%EndDocument @endspecial 63 x Fd(Figure)17 b(2.)44 b Fc(In)o(v)o(erse)13 b(geometrical)h(spreading)e(in)o(terp)q(olated)h(within)h(ra)o(y)g (cells)g(b)o(y)g(program)f(\\m)o(tt.for".)20 b(Y)l(ello)o(w)0 1901 y(colour)15 b(corresp)q(onds)e(to)k(v)n(alues)e Fb(Q)578 1907 y Fa(11)633 1901 y Fc(=)g(0)p Fb(;)7 b Fc(1)p Fb(;)g Fc(2)p Fb(;)g Fc(3.)25 b(The)15 b(app)q(earance)g(of)h (the)g(\014gure)f(is)g(in\015uenced)e(b)o(y)j(sampling)0 1949 y Fb(Q)33 1955 y Fa(11)86 1949 y Fc(in)o(to)e(just)g(255)h(lev)o (els.)0 2065 y Fm(whic)o(h)h(ma)o(y)f(b)q(e)i(used)f(to)h(determine)e (the)i(sign)f(of)h Fk(b)p Fm(,)439 2237 y Fk(b)d Fm(=)g Fh(\000)566 2142 y Fg(s)p 615 2142 447 2 v 615 2167 a(\022)670 2203 y Fm(1)p 658 2226 49 2 v 658 2271 a 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y(putation)e(of)i(seismic)e(w)o(a)o(v)o(e\014elds)f(in)i (inhomogeneous)e(media.)h(Geoph)o(ys.)g(J.)h(R.)g(astr.)f(So)q(c.,)100 2079 y Fo(79)p Fm(,)f(89{104.)0 2138 y(Klime)m(\024)-22 b(s,)13 b(L.)g(\(1994\):)29 b(T)l(ransformations)11 b(for)i(dynamic)g (ra)o(y)g(tracing)h(in)f(anisotropic)g(media.)f(W)l(a)o(v)o(e)100 2198 y(Motion,)j Fo(20)p Fm(,)h(261{272.)0 2258 y(Klime)m(\024)-22 b(s,)15 b(L.)h(\(1996\):)33 b(Grid)15 b(tra)o(v)o(el{time)h(tracing:)f (second{order)f(metho)q(d)i(for)g(the)h(\014rst)e(arriv)m(als)100 2318 y(in)h(smo)q(oth)g(media.)f(P)l(A)o(GEOPH,)h Fo(148)p Fm(,)g(539{563.)0 2377 y(Luneburg,)d(R.K.)h(\(1944\):)30 b(Mathematical)15 b(Theory)f(of)h(Optics.)f(Lecture)h(notes,)g(Bro)o (wn)f(Univ)o(er-)100 2437 y(sit)o(y)l(,)i(Pro)o(vidence,)f(Rho)q(de)h (Island.)f(Reedition:)h(Univ)o(ersit)o(y)g(of)g(California)f(Press,)h (Berk)o(eley)100 2497 y(and)f(Los)i(Angeles,)f(1964.)0 2557 y(P)o(op)q(o)o(v,)11 b(M.M.)f(&)i(P)m(\024)-22 b(sen)o(\024)e(c)-6 b(\023)-19 b(\020k,)11 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