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b(et)e(al.,)g(1988)6 b Fp(\))17 b(is)g(prop)q(osed.)26 b(The)17 b(prop)q(osed)h(algorithm)c(has)i(b)q(een)i(co)q(ded,)f(and) 231 291 y(t)o(w)o(o{p)q(oin)o(t)10 b(anisotropic)f(common)f(S{w)o(a)o (v)o(e)h(tra)o(v)o(el)h(times)g(ha)o(v)o(e)g(b)q(een)h(n)o(umerically)d (calculated)231 340 y(b)o(y)k(anisotropic{commo)o(n{ra)o(y)d(tracing)i (in)g(a)g(test)i(example)d(b)o(y)i Fo(Bulant)h(and)g(Klime)n(\024)-19 b(s)14 b(\(2006\))r Fp(,)231 390 y(in)g(whic)o(h)f(the)h(t)o(w)o(o{p)q (oin)o(t)f(tra)o(v)o(el)g(times)g(calculated)g(along)g(anisotropic)g (common)e(ra)o(ys)i(ha)o(v)o(e)231 439 y(b)q(een)g(compared)e(with)g (the)h(second{order)h(tra)o(v)o(el{time)d(p)q(erturbations)j(from)c (isotropic)j(ra)o(ys,)231 489 y(and)19 b(vice)h(v)o(ersa.)33 b Fo(Bulant)20 b(and)h(Klime)n(\024)-19 b(s)21 b(\(2006\))h Fp(also)c(estimated)g(the)i(tra)o(v)o(el{time)d(errors)231 539 y(due)g(to)f(the)h(anisotropic{comm)o(on{ra)o(y)c(appro)o(ximation) g(using)i(the)i(second{order)h(tra)o(v)o(el{)231 588 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b(or)231 986 y Fk(A;)7 b(B)r(;)g(C)q(;)g(:::)16 b Fp(=)k(1)p Fk(;)7 b Fp(2)17 b(is)h(used)h(throughout)g(this)f(pap)q (er.)32 b(A)19 b(subscript)g(follo)o(wing)d(a)i(comma)231 1035 y(denotes)f(the)f(partial)e(spatial)g(deriv)n(ativ)o(e)g(and)h(a)g (sup)q(erscript)i(follo)o(wing)c(a)i(comma)d(denotes)231 1085 y(the)22 b(partial)e(deriv)n(ativ)o(e)h(with)g(resp)q(ect)i(to)e (the)h(slo)o(wness)g(v)o(ector,)h(e.g.,)f Fk(H)1487 1091 y Fj(;i)1533 1085 y Fi(\021)i Fk(@)r(H)q(=@)r(x)1718 1070 y Fj(i)1732 1085 y Fp(,)231 1135 y Fk(H)269 1120 y Fj(;i)305 1135 y Fi(\021)11 b Fk(@)r(H)q(=@)r(p)474 1141 y Fj(i)488 1135 y Fp(.)314 1259 y(2.)20 b(P)m(AR)m(TIAL)14 b(PHASE{SP)m(A)o(CE)h(DERIV)-5 b(A)m(TIVES)15 b(OF)g(THE)g(EIGENV)-5 b(ALUES)650 1309 y(OF)18 b(THE)g(CHRISTOFFEL)g(MA)m(TRIX)291 1408 y(The)c(Christo\013el)g(matrix)e(reads)704 1479 y(\000)730 1485 y Fj(ij)759 1479 y Fp(\()p Fk(x)799 1461 y Fj(m)830 1479 y Fk(;)7 b(p)870 1485 y Fj(n)892 1479 y Fp(\))12 b(=)g Fk(a)986 1485 y Fj(ik)q(j)r(l)1044 1479 y Fp(\()p Fk(x)1084 1461 y Fj(m)1115 1479 y Fp(\))g Fk(p)1164 1485 y Fj(k)1184 1479 y Fk(p)1205 1485 y Fj(l)1259 1479 y Fk(;)419 b Fp(\(1\))231 1549 y(where)19 b Fk(x)379 1534 y Fj(m)428 1549 y Fp(are)f(the)g(Cartesian)g(co)q(ordinates,)h Fk(a)1026 1555 y Fj(ij)r(k)q(l)1084 1549 y Fp(\()p Fk(x)1124 1534 y Fj(m)1155 1549 y Fp(\))f(the)g(densit)o(y{normalized)e(elastic) 231 1599 y(mo)q(duli,)c(and)h Fk(p)484 1605 y Fj(i)512 1599 y Fp(the)h(comp)q(onen)o(ts)g(of)f(the)i(slo)o(wness)f(v)o(ector.) 291 1648 y(As)f(w)o(e)g(need)h(to)f(handle)f(b)q(oth)i(the)f(deriv)n (ativ)o(es)g(with)f(resp)q(ect)k(to)c Fk(x)1371 1633 y Fj(m)1415 1648 y Fp(and)h Fk(p)1516 1654 y Fj(n)1539 1648 y Fp(,)f(w)o(e)h(denote)231 1698 y(an)o(y)j(partial)g(phase{space) i(deriv)n(ativ)o(e)d(b)o(y)937 1683 y Fh(0)965 1698 y Fp(or)1019 1683 y Fj(?)1038 1698 y Fp(.)25 b(Both)17 b(\000)1207 1683 y Fh(0)1207 1709 y Fj(ij)1252 1698 y Fp(and)f(\000)1361 1683 y Fj(?)1361 1709 y(ij)1407 1698 y Fp(then)h(stand)g(for)f(the)231 1748 y(\014rst{order)g(partial)d (phase{space)i(deriv)n(ativ)o(es)710 1838 y(\000)736 1844 y Fj(ij;k)803 1838 y Fi(\021)852 1810 y Fk(@)r Fp(\000)902 1816 y Fj(ij)p 852 1828 80 2 v 858 1866 a Fk(@)r(x)906 1854 y Fj(k)948 1838 y Fp(=)997 1810 y Fk(@)r(a)1043 1816 y Fj(imj)r(n)p 997 1828 126 2 v 1025 1866 a Fk(@)r(x)1073 1854 y Fj(k)1127 1838 y Fk(p)1148 1844 y Fj(m)1180 1838 y Fk(p)1201 1844 y Fj(n)1690 1838 y Fp(\(2\))231 1908 y(or)666 1969 y(\000)692 1949 y Fj(;k)692 1981 y(ij)734 1969 y Fi(\021)783 1941 y Fk(@)r Fp(\000)833 1947 y Fj(ij)p 783 1960 80 2 v 789 1998 a Fk(@)r(p)834 2004 y Fj(k)879 1969 y Fp(=)c(\()p Fk(a)960 1975 y Fj(ik)q(j)r(m)1047 1969 y Fp(+)e Fk(a)1110 1975 y Fj(imj)r(k)1187 1969 y Fp(\))p Fk(p)1224 1975 y Fj(m)1297 1969 y Fk(:)381 b Fp(\(3\))231 2055 y(Analogously)m(,)12 b(\000)501 2040 y Fh(0)p Fj(?)501 2066 y(ij)544 2055 y Fp(stands)j(for)e(the)i (second{order)g(partial)e(phase{space)i(deriv)n(ativ)o(es)671 2158 y(\000)697 2164 y Fj(ij;k)q(l)774 2158 y Fi(\021)839 2130 y Fk(@)863 2115 y Fg(2)882 2130 y Fp(\000)908 2136 y Fj(ij)p 823 2149 130 2 v 823 2187 a Fk(@)r(x)871 2175 y Fj(k)892 2187 y Fk(@)r(x)940 2175 y Fj(l)969 2158 y Fp(=)1018 2130 y Fk(@)1042 2115 y Fg(2)1061 2130 y Fk(a)1083 2136 y Fj(imj)r(n)p 1018 2149 145 2 v 1025 2187 a Fk(@)r(x)1073 2175 y Fj(k)1094 2187 y Fk(@)r(x)1142 2175 y Fj(l)1167 2158 y Fk(p)1188 2164 y Fj(m)1219 2158 y Fk(p)1240 2164 y Fj(n)1690 2158 y Fp(\(4\))231 2229 y(or)626 2295 y(\000)652 2275 y Fj(;l)652 2308 y(ij;k)719 2295 y Fi(\021)782 2267 y Fk(@)806 2252 y Fg(2)825 2267 y Fp(\000)851 2273 y Fj(ij)p 768 2285 127 2 v 768 2323 a Fk(@)r(x)816 2312 y Fj(k)837 2323 y Fk(@)r(p)882 2329 y Fj(l)911 2295 y Fp(=)960 2267 y Fk(@)r Fp(\()p Fk(a)1022 2273 y Fj(ilj)r(m)1101 2267 y Fp(+)9 b Fk(a)1164 2273 y Fj(imj)r(l)1234 2267 y Fp(\))p 960 2285 290 2 v 1071 2323 a Fk(@)r(x)1119 2312 y Fj(k)1255 2295 y Fk(p)1276 2301 y Fj(m)1690 2295 y Fp(\(5\))231 2409 y 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Fg(\))902 1313 y Fp(=)12 b(0)41 b Fk(:)555 b Fp(\(12\))137 1386 y(Multiplying)12 b(\(7\))i(b)o(y)g Fk(g)504 1393 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))564 1386 y Fp(and)14 b(considering)g(\(8\),)709 1458 y Fk(G)736 1465 y Fg(\()-6 b Fj(a)l Fg(\))784 1458 y Fp(=)12 b Fk(g)844 1465 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))891 1458 y Fp(\000)917 1464 y Fj(ij)957 1458 y Fk(g)973 1465 y Fj(j)p Fg(\()g Fj(a)l Fg(\))1065 1458 y Fk(:)498 b Fp(\(13\))137 1530 y(Multiplying)12 b(\(10\))i(b)o(y)f Fk(g)524 1537 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))585 1530 y Fp(and)13 b(considering)h(\(7\),)g(\(8\),)715 1603 y Fk(G)748 1585 y Fh(0)742 1614 y Fg(\()-6 b Fj(a)l Fg(\))790 1603 y Fp(=)12 b Fk(g)850 1610 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))896 1603 y Fp(\000)922 1585 y Fh(0)922 1613 y Fj(ij)952 1603 y Fk(g)968 1610 y Fj(j)p Fg(\()g Fj(a)l Fg(\))1059 1603 y Fk(:)504 b Fp(\(14\))137 1675 y(Multiplying)12 b(\(11\))i(b)o(y)f Fk(g)524 1682 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))585 1675 y Fp(and)13 b(considering)h(\(7\),)g(\(8\),)f 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1892 y Fk(G)688 1899 y Fg(\()g Fj(a)l Fg(\))736 1892 y Fp(=)12 b Fk(g)796 1899 y Fj(i)n Fg(\()-6 b Fj(b)l Fg(\))839 1892 y Fp(\000)865 1875 y Fh(0)865 1902 y Fj(ij)894 1892 y Fk(g)910 1899 y Fj(j)p Fg(\()g Fj(a)l Fg(\))970 1892 y Fp(+)9 b Fk(g)1027 1899 y Fj(i)n Fg(\()-6 b Fj(b)l Fg(\))1070 1892 y Fk(g)1091 1875 y Fh(0)1086 1903 y Fj(i)n Fg(\()g Fj(a)l Fg(\))1133 1892 y Fk(G)1160 1899 y Fg(\()g Fj(b)l Fg(\))1234 1892 y Fk(:)329 b Fp(\(16\))137 1974 y(Equations)14 b(\(8\),)f(\(9\),)h (\(12\))f(and)h(\(16\))f(yield)h(\()869 1963 y Fo(\024)862 1974 y(Cerven)o(\023)-21 b(y,)16 b(2001,)f(e)n(qs.)g(4.14.9)g(and)g (4.14.10)6 b Fp(\))643 2075 y Fk(g)664 2058 y Fh(0)659 2087 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))717 2075 y Fp(=)761 2036 y Ff(X)762 2125 y Fj(b)p Fh(6)p Fg(=)p Fj(a)827 2075 y Fk(g)843 2082 y Fj(i)n Fg(\()g Fj(b)l Fg(\))891 2042 y Fk(g)907 2049 y Fj(j)p Fg(\()g Fj(b)l Fg(\))954 2042 y Fp(\000)980 2027 y Fh(0)980 2054 y Fj(j)r(k)1016 2042 y Fk(g)1032 2049 y Fj(k)o Fg(\()g Fj(a)l Fg(\))p 891 2066 194 2 v 901 2104 a Fk(G)928 2111 y Fg(\()g Fj(a)l Fg(\))974 2104 y Fi(\000)9 b Fk(G)1042 2111 y Fg(\()-6 b Fj(b)l Fg(\))1132 2075 y Fk(:)431 b Fp(\(17\))137 2193 y(W)m(e)13 b(transform)f(Christo\013el)h(matrix)e(\(1\),)i(its)g (\014rst{order)h(phase{space)h(deriv)n(ativ)o(es)e(\(2\),)f(\(3\),)137 2243 y(and)i(second{order)h(phase{space)h(deriv)n(ativ)o(es)d(\(4\),)h (\(5\),)f(\(6\))h(in)o(to)f(the)i(eigen)o(v)o(ectors,)713 2315 y(\000)733 2322 y Fg(\()-6 b Fj(ab)l Fg(\))796 2315 y Fp(=)12 b Fk(g)856 2322 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))902 2315 y Fp(\000)928 2321 y Fj(ij)957 2315 y Fk(g)973 2322 y Fj(j)p Fg(\()g Fj(b)l Fg(\))1061 2315 y Fk(;)502 b Fp(\(18\))137 2409 y Fl(452)931 b Fn(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h(\(2006\))p eop %%Page: 453 5 453 4 bop 437 32 a Fq(Common{R)n(ay)13 b(T)m(r)n(acing)g(and)f(Dynamic) g(R)n(ay)h(T)m(r)n(acing)g(for)g(S)g(Waves)g(...)807 142 y Fp(\000)833 124 y Fh(0)827 153 y Fg(\()-6 b Fj(ab)l Fg(\))890 142 y Fp(=)12 b Fk(g)950 149 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))997 142 y Fp(\000)1023 124 y Fh(0)1023 152 y Fj(ij)1052 142 y Fk(g)1068 149 y Fj(j)p Fg(\()g Fj(b)l Fg(\))1156 142 y Fk(;)502 b Fp(\(19\))807 215 y(\000)833 198 y Fh(0)p Fj(?)827 226 y Fg(\()-6 b Fj(ab)l Fg(\))890 215 y Fp(=)12 b Fk(g)950 222 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))997 215 y Fp(\000)1023 198 y Fh(0)p Fj(?)1023 225 y(ij)1052 215 y Fk(g)1068 222 y Fj(j)p Fg(\()g Fj(b)l Fg(\))1156 215 y Fk(:)502 b Fp(\(20\))231 282 y(Equation)14 b(\(13\))f(ma)o(y)f(then)j(b)q(e)f(expressed)j(as)864 357 y Fk(G)891 364 y Fg(\()-6 b Fj(a)l Fg(\))939 357 y Fp(=)12 b(\000)1003 364 y Fg(\()-6 b Fj(aa)l Fg(\))1099 357 y Fk(;)559 b Fp(\(21\))231 432 y(equation)14 b(\(14\))g(as)864 483 y Fk(G)897 465 y Fh(0)891 494 y Fg(\()-6 b Fj(a)l Fg(\))939 483 y Fp(=)12 b(\000)1009 465 y Fh(0)1003 494 y Fg(\()-6 b Fj(aa)l Fg(\))1099 483 y Fk(;)559 b Fp(\(22\))231 550 y(and)14 b(equation)g(\(15\))f(with)h(\(17\))g(as)699 657 y Fk(G)732 640 y Fh(0)p Fj(?)726 669 y Fg(\()-6 b Fj(a)l Fg(\))774 657 y Fp(=)12 b(\000)844 640 y Fh(0)p Fj(?)838 669 y Fg(\()-6 b Fj(aa)l Fg(\))902 657 y Fp(+)10 b(2)972 618 y Ff(X)972 707 y Fj(b)p Fh(6)p Fg(=)p Fj(a)1059 621 y Fp(\000)1085 605 y Fh(0)1079 634 y Fg(\()-6 b Fj(ab)l Fg(\))1130 621 y Fp(\000)1156 605 y Fj(?)1150 634 y Fg(\()g Fj(ab)l Fg(\))p 1043 648 175 2 v 1043 686 a Fk(G)1070 693 y Fg(\()g Fj(a)l Fg(\))1116 686 y Fi(\000)9 b Fk(G)1184 693 y Fg(\()-6 b Fj(b)l Fg(\))1264 657 y Fk(:)394 b Fp(\(23\))231 778 y(Equations)11 b(\(2\){\(6\),)g(\(19\),)f(\(20\),)h(\(22\))g(and)f (\(23\))h(are)g(suitable)f(for)h(the)g(n)o(umerical)e(calculation)231 828 y(of)i(the)g(\014rst{order)h(and)e(second{order)j(partial)c (phase{space)k(deriv)n(ativ)o(es)d(of)g(the)h(eigen)o(v)n(alues)231 877 y(of)j(the)g(Christo\013el)g(matrix.)304 1002 y(3.)20 b(HAMIL)m(TONIAN)15 b(AND)f(ITS)h(P)m(AR)m(TIAL)e(PHASE{SP)m(A)o(CE)j (DERIV)-5 b(A)m(TIVES)291 1102 y(F)m(or)15 b(arithmetic)g(a)o(v)o (eraging)g(of)h(the)g(Hamiltonians)e(of)h(b)q(oth)h(S{w)o(a)o(v)o(e)g (p)q(olarizations,)f(w)o(e)231 1152 y(consider)d(homogeneous)d (Hamiltonians)f(of)i(arbitrary)g(degree)i Fk(N)5 b Fp(.)16 b(Note)11 b(that)f Fo(Bakker)15 b(\(2002\))231 1202 y Fp(considered)g Fk(N)i Fp(=)12 b(2,)h(whereas)h Fo(Klime)n(\024)-19 b(s)14 b(and)i(Bulant)i(\(2004\))e Fp(considered)f Fk(N)h Fp(=)c Fi(\000)p Fp(1,)h(whic)o(h)g(is)231 1252 y(b)q(etter)k(suited)f (for)e(tra)o(v)o(el{time)g(p)q(erturbations)i(as)f(n)o(umerically)e (demonstrated)i(b)o(y)g Fo(Bulant)231 1302 y(and)d(Klime)n(\024)-19 b(s)12 b(\(2006,)g(tables)e(7{12\))r Fp(.)17 b(The)10 b(a)o(v)o(eraged)f(Hamiltonian)d(is)k(then)g(also)e(homogeneous)231 1352 y(and)20 b(of)f(degree)i Fk(N)5 b Fp(.)36 b(According)20 b(to)g(Euler's)g(theorem)f(on)h(homogeneous)f(functions,)i(the)231 1402 y(parameter)h(along)e(ra)o(ys)i(is)g(then)g(prop)q(ortional)f(to)g (the)i(tra)o(v)o(el)e(time.)40 b(W)m(e)21 b(th)o(us)h(c)o(ho)q(ose)231 1451 y(the)d(v)n(alue)e(of)g(the)h(Hamiltonian)d(in)i(the)h(stationary) f(Hamilton{Jacobi)e(equation)i(so)h(that)231 1501 y(the)d(parameter)f (along)e(ra)o(ys)i(is)g(equal)g(to)f(the)i(tra)o(v)o(el)e(time,)815 1594 y Fk(H)s Fp(\()p Fk(x)893 1577 y Fj(m)924 1594 y Fk(;)7 b(\034)955 1600 y Fj(;n)988 1594 y Fp(\))k(=)1073 1566 y(1)p 1064 1584 38 2 v 1064 1622 a Fk(N)1148 1594 y(;)510 b Fp(\(24\))231 1683 y(where)22 b Fk(N)j Fp(is)20 b(the)h(degree)g(of)f(Hamiltonian)d Fk(H)s Fp(\()p Fk(x)1054 1668 y Fj(m)1085 1683 y Fk(;)7 b(p)1125 1689 y Fj(n)1147 1683 y Fp(\),)22 b(homogeneous)d(with)h(resp)q(ect)i(to)231 1733 y(slo)o(wness)15 b(v)o(ector)g Fk(p)543 1739 y Fj(i)556 1733 y Fp(.)291 1783 y(The)e(homogeneous)g(Hamiltonia)o(n)e(of)h (degree)j Fk(N)j Fp(for)13 b(standard)h(anisotropic{ra)o(y{theory)231 1833 y(ra)o(y)g(tracing)g(is)815 1888 y Fk(H)844 1895 y Fg(\()-6 b Fj(a)l Fg(\))892 1888 y Fp(=)947 1871 y Fg(1)p 940 1878 30 2 v 940 1902 a Fj(N)975 1888 y Fp(\()p Fk(G)1018 1895 y Fg(\()g Fj(a)l Fg(\))1054 1888 y Fp(\))1075 1859 y Fe(N)p 1076 1864 25 2 v 1081 1881 a Fd(2)1149 1888 y Fk(;)509 b Fp(\(25\))231 1955 y(its)14 b(\014rst{order)i (partial)c(phase{space)k(deriv)n(ativ)o(es)e(are)768 2035 y Fk(H)806 2018 y Fh(0)797 2046 y Fg(\()-6 b Fj(a)l Fg(\))845 2035 y Fp(=)894 2018 y Fg(1)p 894 2025 17 2 v 894 2049 a(2)915 2035 y Fk(G)948 2018 y Fh(0)942 2046 y Fg(\()g Fj(a)l Fg(\))979 2035 y Fp(\()p Fk(G)1022 2042 y Fg(\()g Fj(a)l Fg(\))1059 2035 y Fp(\))1080 2006 y Fe(N)p 1080 2011 25 2 v 1085 2028 a Fd(2)1109 2018 y Fh(\000)p Fg(1)1195 2035 y Fk(;)463 b Fp(\(26\))231 2110 y(and)14 b(its)g(second{order)i(partial)c(phase{space)k(deriv)n(ativ)o (es)e(are)550 2190 y Fk(H)588 2173 y Fh(0)p Fj(?)579 2201 y Fg(\()-6 b Fj(a)l Fg(\))628 2190 y Fp(=)677 2174 y Fg(1)p 677 2181 17 2 v 677 2204 a(2)699 2190 y Fk(G)732 2173 y Fh(0)p Fj(?)726 2201 y Fg(\()g Fj(a)l Fg(\))762 2190 y Fp(\()p Fk(G)805 2197 y Fg(\()g Fj(a)l Fg(\))842 2190 y Fp(\))863 2162 y Fe(N)p 863 2167 25 2 v 868 2183 a Fd(2)892 2173 y Fh(\000)p Fg(1)946 2190 y Fp(+)993 2174 y Fj(N)s Fh(\000)p Fg(2)p 993 2181 72 2 v 1020 2204 a(4)1070 2190 y Fk(G)1103 2173 y Fh(0)1097 2201 y Fg(\()g Fj(a)l Fg(\))1133 2190 y Fk(G)1166 2173 y Fj(?)1160 2201 y Fg(\()g Fj(a)l Fg(\))1197 2190 y Fp(\()p Fk(G)1240 2197 y Fg(\()g Fj(a)l Fg(\))1276 2190 y Fp(\))1297 2162 y Fe(N)p 1298 2167 25 2 v 1303 2183 a Fd(2)1327 2173 y Fh(\000)p Fg(2)1413 2190 y Fk(:)245 b Fp(\(27\))231 2265 y(In)15 b(the)f(coupling)f(ra)o(y)h(theory)m(,)g(b)q(oth)g(S{w)o (a)o(v)o(e)f(p)q(olarizations)g(are)i(coupled.)k(It)14 b(is)g(th)o(us)g(useful)231 2315 y(to)h(ha)o(v)o(e)f(common)e (reference)17 b(ra)o(ys)e(equally)f(suitable)g(for)h(b)q(oth)f(S{w)o(a) o(v)o(e)g(p)q(olarizations.)20 b(F)m(or)231 2409 y Fn(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h(\(2006\))930 b Fl(453)p eop %%Page: 454 6 454 5 bop 810 32 a Fq(L.)14 b(Klime)n(\024)-18 b(s)137 142 y Fp(common{ra)o(y)13 b(tracing,)k(w)o(e)g(shall)f(th)o(us)i (consider)f(the)h(a)o(v)o(eraged)e(Hamiltonian)e(of)i(b)q(oth)h(S{)137 192 y(w)o(a)o(v)o(e)c(p)q(olarizations,)618 249 y Fk(H)i Fp(=)731 233 y Fg(1)p 716 240 46 2 v 716 264 a(2)p Fj(N)767 216 y Ff(\002)785 249 y Fp(\()p Fk(G)828 256 y Fg(\()-6 b(1)l(\))863 249 y Fp(\))884 221 y Fe(N)p 884 226 25 2 v 889 242 a Fd(2)925 249 y Fp(+)9 b(\()p Fk(G)1009 256 y Fg(\()-6 b(2)l(\))1045 249 y Fp(\))1066 221 y Fe(N)p 1066 226 V 1071 242 a Fd(2)1097 216 y Ff(\003)1156 249 y Fk(:)407 b Fp(\(28\))137 319 y(Note)11 b(that)f(w)o(e)g(obtain)f (di\013eren)o(t)j(anisotropic)d(common)f(ra)o(ys)i(and)g(di\013eren)o (t)h(reference)h(tra)o(v)o(el{)137 370 y(time)h(\014elds)h(for)f (di\013eren)o(t)i Fk(N)5 b Fp(.)18 b(The)c(\014rst{order)i(partial)d (phase{space)i(deriv)n(ativ)o(es)f(are)522 452 y Fk(H)560 435 y Fh(0)584 452 y Fp(=)632 436 y Fg(1)p 632 443 17 2 v 632 466 a(4)654 418 y Ff(\002)671 452 y Fk(G)704 435 y Fh(0)698 463 y Fg(\()-6 b(1)l(\))733 452 y Fp(\()p Fk(G)776 459 y Fg(\()g(1)l(\))811 452 y Fp(\))832 424 y Fe(N)p 833 429 25 2 v 838 445 a Fd(2)862 435 y Fh(\000)p Fg(1)916 452 y Fp(+)9 b Fk(G)990 435 y Fh(0)984 463 y Fg(\()-6 b(2)l(\))1020 452 y Fp(\()p Fk(G)1063 459 y Fg(\()g(2)l(\))1098 452 y Fp(\))1119 424 y Fe(N)p 1119 429 V 1124 445 a Fd(2)1148 435 y Fh(\000)p Fg(1)1193 418 y Ff(\003)1252 452 y Fk(;)311 b Fp(\(29\))137 530 y(and)14 b(the)g(second{order)i(partial)d(phase{space)i(deriv)n(ativ)o (es)f(are)137 613 y Fk(H)175 596 y Fh(0)p Fj(?)215 613 y Fp(=)264 596 y Fg(1)p 264 603 17 2 v 264 627 a(4)285 579 y Ff(\002)303 613 y Fk(G)336 596 y Fh(0)p Fj(?)330 624 y Fg(\()-6 b(1)l(\))365 613 y Fp(\()p Fk(G)408 620 y Fg(\()g(1)l(\))443 613 y Fp(\))464 584 y Fe(N)p 464 589 25 2 v 469 606 a Fd(2)494 596 y Fh(\000)p Fg(1)540 613 y Fp(+)r Fk(G)607 596 y Fh(0)p Fj(?)601 624 y Fg(\()g(2)l(\))636 613 y Fp(\()p Fk(G)679 620 y Fg(\()g(2)l(\))714 613 y Fp(\))735 584 y Fe(N)p 735 589 V 741 606 a Fd(2)765 596 y Fh(\000)p Fg(1)810 579 y Ff(\003)835 613 y Fp(+)881 596 y Fj(N)s Fh(\000)p Fg(2)p 881 603 72 2 v 909 627 a(8)958 579 y Ff(\002)975 613 y Fk(G)1008 596 y Fh(0)1002 624 y Fg(\()g(1)l(\))1038 613 y Fk(G)1071 596 y Fj(?)1065 624 y Fg(\()g(1)l(\))1100 613 y Fp(\()p Fk(G)1143 620 y Fg(\()g(1)l(\))1178 613 y Fp(\))1199 584 y Fe(N)p 1199 589 25 2 v 1204 606 a Fd(2)1229 596 y Fh(\000)p Fg(2)1275 613 y Fp(+)r Fk(G)1342 596 y Fh(0)1336 624 y Fg(\()g(2)l(\))1371 613 y Fk(G)1404 596 y Fj(?)1398 624 y Fg(\()g(2)l(\))1433 613 y Fp(\()p Fk(G)1476 620 y Fg(\()g(2)l(\))1511 613 y Fp(\))1532 584 y Fe(N)p 1532 589 V 1538 606 a Fd(2)1562 596 y Fh(\000)p Fg(2)1606 579 y Ff(\003)1637 613 y Fk(:)1575 663 y Fp(\(30\))137 713 y(W)m(e)20 b(express)i(the)f(second{order)h (deriv)n(ativ)o(es)e(of)g(the)h(common)c(S{w)o(a)o(v)o(e)j(Hamiltonian) d(in)137 764 y(terms)d(of)f(the)h(deriv)n(ativ)o(es)g(of)f(the)i (Christo\013el)f(matrix)e(using)i(equations)g(\(22\))f(and)h(\(23\),) 155 883 y Fk(H)193 866 y Fh(0)p Fj(?)233 883 y Fp(=)282 855 y(1)p 282 874 21 2 v 282 912 a(4)308 837 y Ff(h)327 883 y Fp(\000)353 866 y Fh(0)p Fj(?)347 895 y Fg(\()-6 b(11)l(\))399 883 y Fp(\()p Fk(G)442 890 y Fg(\()g(1)l(\))477 883 y Fp(\))498 855 y Fe(N)p 499 860 25 2 v 504 876 a Fd(2)528 866 y Fh(\000)p Fg(1)575 883 y Fp(+)r(\000)635 866 y Fh(0)p Fj(?)629 895 y Fg(\()g(22)l(\))682 883 y Fp(\()p Fk(G)725 890 y Fg(\()g(2)l(\))760 883 y Fp(\))781 855 y Fe(N)p 781 860 V 786 876 a Fd(2)810 866 y Fh(\000)p Fg(1)855 837 y Ff(i)884 883 y Fp(+)930 846 y(\000)956 831 y Fh(0)950 860 y Fg(\()g(13)l(\))1002 846 y Fp(\000)1028 831 y Fj(?)1022 860 y Fg(\()g(13)l(\))1074 846 y Fp(\()p Fk(G)1117 853 y Fg(\()g(1)l(\))1152 846 y Fp(\))1173 820 y Fe(N)p 1174 825 V 1179 841 a Fd(2)1203 831 y Fh(\000)p Fg(1)p 930 874 318 2 v 969 912 a Fp(2)12 b(\()p Fk(G)1045 919 y Fg(\()-6 b(1)l(\))1089 912 y Fi(\000)10 b Fk(G)1158 919 y Fg(\()-6 b(3)l(\))1193 912 y Fp(\))1262 883 y(+)1308 846 y(\000)1334 831 y Fh(0)1328 860 y Fg(\()g(23)l(\))1380 846 y Fp(\000)1406 831 y Fj(?)1400 860 y Fg(\()g(23)l(\))1452 846 y Fp(\()p Fk(G)1495 853 y Fg(\()g(2)l(\))1531 846 y Fp(\))1552 820 y Fe(N)p 1552 825 25 2 v 1557 841 a Fd(2)1581 831 y Fh(\000)p Fg(1)p 1308 874 318 2 v 1347 912 a Fp(2)12 b(\()p Fk(G)1423 919 y Fg(\()-6 b(2)l(\))1467 912 y Fi(\000)10 b Fk(G)1536 919 y Fg(\()-6 b(3)l(\))1571 912 y Fp(\))155 1033 y(+\000)213 1015 y Fh(0)207 1044 y Fg(\()g(12)l(\))259 1033 y Fp(\000)285 1015 y Fj(?)279 1044 y Fg(\()g(12)l(\))336 1002 y Fp(\()p Fk(G)379 1009 y Fg(\()g(1)l(\))414 1002 y Fp(\))435 975 y Fe(N)p 435 980 25 2 v 441 997 a Fd(2)465 987 y Fh(\000)p Fg(1)512 1002 y Fi(\000)r Fp(\()p Fk(G)589 1009 y Fg(\()g(2)l(\))625 1002 y Fp(\))646 975 y Fe(N)p 646 980 V 651 997 a Fd(2)675 987 y Fh(\000)p Fg(1)p 336 1023 384 2 v 408 1061 a Fp(2)11 b(\()p Fk(G)483 1068 y Fg(\()-6 b(1)l(\))528 1061 y Fi(\000)9 b Fk(G)596 1068 y Fg(\()-6 b(2)l(\))631 1061 y Fp(\))734 1033 y(+)780 1005 y Fk(N)7 b Fi(\000)r Fp(2)p 780 1023 96 2 v 818 1061 a(8)881 987 y Ff(h)901 1033 y Fp(\000)927 1015 y Fh(0)921 1044 y Fg(\()-6 b(11)l(\))973 1033 y Fp(\000)999 1015 y Fj(?)993 1044 y Fg(\()g(11)l(\))1044 1033 y Fp(\()p Fk(G)1087 1040 y Fg(\()g(1)l(\))1123 1033 y Fp(\))1144 1004 y Fe(N)p 1144 1009 25 2 v 1149 1025 a Fd(2)1173 1015 y Fh(\000)p Fg(2)1227 1033 y Fp(+)10 b(\000)1295 1015 y Fh(0)1289 1044 y Fg(\()-6 b(22)l(\))1341 1033 y Fp(\000)1367 1015 y Fj(?)1361 1044 y Fg(\()g(22)l(\))1413 1033 y Fp(\()p Fk(G)1456 1040 y Fg(\()g(2)l(\))1491 1033 y Fp(\))1512 1004 y Fe(N)p 1512 1009 V 1517 1025 a Fd(2)1542 1015 y Fh(\000)p Fg(2)1586 987 y Ff(i)1620 1033 y Fk(:)1575 1111 y Fp(\(31\))137 1161 y(The)12 b(term)g(with)g Fk(G)438 1168 y Fg(\()-6 b(1)l(\))478 1161 y Fi(\000)5 b Fk(G)542 1168 y Fg(\()-6 b(2)l(\))590 1161 y Fp(in)12 b(the)g(denominator)f(is)g (p)q(ossibly)h(singular.)17 b(It)12 b(is)g(th)o(us)g(desirable)137 1212 y(to)i(carry)g(out)g(the)h(division)d(b)o(y)i Fk(G)679 1219 y Fg(\()-6 b(1)l(\))723 1212 y Fi(\000)10 b Fk(G)792 1219 y Fg(\()-6 b(2)l(\))840 1212 y Fp(analytically)m(.)16 b(F)m(or)d Fk(N)d Fp(=)5 b(2,)636 1299 y(\()p Fk(G)679 1306 y Fg(\()-6 b(1)l(\))715 1299 y Fp(\))736 1273 y Fe(N)p 736 1278 V 741 1294 a Fd(2)765 1284 y Fh(\000)p Fg(1)812 1299 y Fi(\000)r Fp(\()p Fk(G)889 1306 y Fg(\()g(2)l(\))925 1299 y Fp(\))946 1273 y Fe(N)p 946 1278 V 951 1294 a Fd(2)976 1284 y Fh(\000)p Fg(1)p 636 1320 384 2 v 741 1358 a Fk(G)768 1365 y Fg(\()g(1)l(\))812 1358 y Fi(\000)10 b Fk(G)881 1365 y Fg(\()-6 b(2)l(\))1037 1330 y Fp(=)12 b(0)41 b Fk(:)420 b Fp(\(32\))137 1433 y(F)m(or)13 b Fk(N)d Fp(=)5 b(1,)348 1514 y(\()p Fk(G)391 1521 y Fg(\()-6 b(1)l(\))426 1514 y Fp(\))447 1488 y Fe(N)p 447 1493 25 2 v 452 1509 a Fd(2)477 1499 y Fh(\000)p Fg(1)523 1514 y Fi(\000)r Fp(\()p Fk(G)600 1521 y Fg(\()g(2)l(\))636 1514 y Fp(\))657 1488 y Fe(N)p 657 1493 V 663 1509 a Fd(2)687 1499 y Fh(\000)p Fg(1)p 348 1535 384 2 v 452 1573 a Fk(G)479 1580 y Fg(\()g(1)l(\))523 1573 y Fi(\000)10 b Fk(G)592 1580 y Fg(\()-6 b(2)l(\))748 1545 y Fp(=)12 b Fi(\000)1097 1517 y Fp(1)p 829 1535 556 2 v 829 1579 a(\()p Fk(G)872 1586 y Fg(\()-6 b(1)l(\))907 1579 y Fp(\))928 1554 y Fd(1)p 929 1559 15 2 v 929 1575 a(2)950 1579 y Fp([\()p Fk(G)1005 1586 y Fg(\()g(1)l(\))1039 1579 y Fp(\))1060 1554 y Fd(1)p 1061 1559 V 1061 1575 a(2)1091 1579 y Fp(+)10 b(\()p Fk(G)1176 1586 y Fg(\()-6 b(2)l(\))1211 1579 y Fp(\))1232 1554 y Fd(1)p 1232 1559 V 1232 1575 a(2)1253 1579 y Fp(]\()p Fk(G)1308 1586 y Fg(\()g(2)l(\))1343 1579 y Fp(\))1364 1554 y Fd(1)p 1364 1559 V 1364 1575 a(2)1432 1545 y Fk(:)131 b Fp(\(33\))137 1654 y(F)m(or)13 b Fk(N)d Fp(=)5 b Fi(\000)p Fp(1,)348 1735 y(\()p Fk(G)391 1742 y Fg(\()-6 b(1)l(\))426 1735 y Fp(\))447 1708 y Fe(N)p 447 1713 25 2 v 452 1730 a Fd(2)477 1720 y Fh(\000)p Fg(1)523 1735 y Fi(\000)r Fp(\()p Fk(G)600 1742 y Fg(\()g(2)l(\))636 1735 y Fp(\))657 1708 y Fe(N)p 657 1713 V 663 1730 a Fd(2)687 1720 y Fh(\000)p Fg(1)p 348 1756 384 2 v 452 1794 a Fk(G)479 1801 y Fg(\()g(1)l(\))523 1794 y Fi(\000)10 b Fk(G)592 1801 y Fg(\()-6 b(2)l(\))748 1766 y Fp(=)12 b Fi(\000)903 1735 y Fk(G)930 1742 y Fg(\()-6 b(1)l(\))974 1735 y Fp(+)10 b(\()p Fk(G)1059 1742 y Fg(\()-6 b(1)l(\))1094 1735 y Fk(G)1121 1742 y Fg(\()g(2)l(\))1156 1735 y Fp(\))1177 1708 y Fd(1)p 1177 1713 15 2 v 1177 1730 a(2)1208 1735 y Fp(+)9 b Fk(G)1276 1742 y Fg(\()-6 b(2)l(\))p 829 1756 556 2 v 829 1800 a Fp(\()p Fk(G)872 1807 y Fg(\()g(1)l(\))907 1800 y Fp(\))928 1774 y Fd(3)p 929 1779 15 2 v 929 1796 a(2)950 1800 y Fp([\()p Fk(G)1005 1807 y Fg(\()g(1)l(\))1039 1800 y Fp(\))1060 1774 y Fd(1)p 1061 1779 V 1061 1796 a(2)1091 1800 y Fp(+)10 b(\()p Fk(G)1176 1807 y Fg(\()-6 b(2)l(\))1211 1800 y Fp(\))1232 1774 y Fd(1)p 1232 1779 V 1232 1796 a(2)1253 1800 y Fp(]\()p Fk(G)1308 1807 y Fg(\()g(2)l(\))1343 1800 y Fp(\))1364 1774 y Fd(3)p 1364 1779 V 1364 1796 a(2)1432 1766 y Fk(:)131 b Fp(\(34\))137 1875 y(F)m(or)13 b Fk(N)d Fp(=)5 b Fi(\000)p Fp(2,)513 1931 y(\()p Fk(G)556 1938 y Fg(\()-6 b(1)l(\))591 1931 y Fp(\))612 1904 y Fe(N)p 612 1909 25 2 v 617 1925 a Fd(2)642 1915 y Fh(\000)p Fg(1)688 1931 y Fi(\000)r Fp(\()p Fk(G)765 1938 y Fg(\()g(2)l(\))801 1931 y Fp(\))822 1904 y Fe(N)p 822 1909 V 828 1925 a Fd(2)852 1915 y Fh(\000)p Fg(1)p 513 1952 384 2 v 617 1990 a Fk(G)644 1997 y Fg(\()g(1)l(\))688 1990 y Fi(\000)10 b Fk(G)757 1997 y Fg(\()-6 b(2)l(\))913 1962 y Fp(=)12 b Fi(\000)1020 1931 y Fk(G)1047 1938 y Fg(\()-6 b(1)l(\))1091 1931 y Fp(+)10 b Fk(G)1160 1938 y Fg(\()-6 b(2)l(\))p 994 1952 226 2 v 994 1990 a Fp(\()p Fk(G)1037 1997 y Fg(\()g(1)l(\))1072 1990 y Fp(\))1088 1978 y Fg(2)1107 1990 y Fp(\()p Fk(G)1150 1997 y Fg(\()g(2)l(\))1185 1990 y Fp(\))1201 1978 y Fg(2)1267 1962 y Fk(:)296 b Fp(\(35\))196 2059 y(Dynamic)14 b(ra)o(y)h(tracing)h(is)f(often)h(p)q (erformed)g(in)f(ra)o(y)g(cen)o(tred)j(co)q(ordinates.)24 b(The)16 b(trans-)137 2109 y(formation)j(of)i(the)i(second{order)g (phase{space)h(deriv)n(ativ)o(es)d(of)h(the)g(Hamiltonian)d(from)137 2159 y(Cartesian)c(to)f(ra)o(y)g(cen)o(tred)i(co)q(ordinates)f(dep)q (ends)h(on)e(the)h(degree)h Fk(N)j Fp(of)14 b(the)h(homogeneous)137 2210 y(Hamiltonian)f(\()p Fo(Klime)n(\024)-19 b(s,)17 b(2002)6 b Fp(\).)27 b(T)m(o)17 b(simplify)d(the)j(transformation,)e (it)i(ma)o(y)e(b)q(e)i(useful)g(to)137 2260 y(con)o(v)o(ert)h (homogeneous)e(Hamiltonian)e Fk(H)s Fp(\()p Fk(x)864 2245 y Fj(m)895 2260 y Fk(;)7 b(p)935 2266 y Fj(n)957 2260 y Fp(\))18 b(of)e(arbitrary)h(degree)i Fk(N)j Fp(in)o(to)17 b(homoge-)137 2315 y(neous)f(Hamiltonian)503 2304 y Ff(e)494 2315 y Fk(H)s Fp(\()p Fk(x)572 2300 y Fj(m)603 2315 y Fk(;)7 b(p)643 2321 y Fj(n)665 2315 y Fp(\))15 b(of)g(a)f(\014xed)i (degree.)23 b(Here)16 b(w)o(e)g(c)o(ho)q(ose)f(the)h(second)g(degree) 137 2409 y Fl(454)931 b Fn(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h (\(2006\))p eop %%Page: 455 7 455 6 bop 437 32 a Fq(Common{R)n(ay)13 b(T)m(r)n(acing)g(and)f(Dynamic) g(R)n(ay)h(T)m(r)n(acing)g(for)g(S)g(Waves)g(...)231 147 y Fp(for)308 136 y Ff(e)298 147 y Fk(H)s Fp(\()p Fk(x)376 132 y Fj(m)407 147 y Fk(;)7 b(p)447 153 y Fj(n)469 147 y Fp(\),)18 b(similarly)c(as)j Fo(Klime)n(\024)-19 b(s)20 b(\(1994\))r Fp(.)28 b(The)17 b(homogeneous)f(Hamiltonian)e(of)j (the)231 198 y(second)e(degree)h(is)833 249 y Ff(e)823 259 y Fk(H)e Fp(=)921 243 y Fg(1)p 921 250 17 2 v 921 274 a(2)954 259 y Fp(\()p Fk(N)5 b(H)s Fp(\))1072 231 y Fd(2)p 1067 236 25 2 v 1067 252 a Fe(N)1140 259 y Fk(;)518 b Fp(\(36\))231 337 y(its)14 b(\014rst{order)i(partial)c(phase{space)k (deriv)n(ativ)o(es)e(are)794 412 y Ff(e)784 423 y Fk(H)822 406 y Fh(0)845 423 y Fp(=)e Fk(H)927 406 y Fh(0)950 423 y Fp(\()p Fk(N)5 b(H)s Fp(\))1069 394 y Fd(2)p 1063 399 V 1063 416 a Fe(N)1093 406 y Fh(\000)p Fg(1)1179 423 y Fk(;)479 b Fp(\(37\))231 509 y(and)14 b(its)g(second{order)i(partial) c(phase{space)k(deriv)n(ativ)o(es)e(are)529 584 y Ff(e)520 595 y Fk(H)558 577 y Fh(0)p Fj(?)598 595 y Fp(=)e Fk(H)680 577 y Fh(0)p Fj(?)720 595 y Fp(\()p Fk(N)5 b(H)s Fp(\))838 566 y Fd(2)p 833 571 V 833 587 a Fe(N)862 577 y Fh(\000)p Fg(1)916 595 y Fp(+)10 b(\(2)r Fi(\000)r Fk(N)5 b Fp(\))12 b Fk(H)1135 577 y Fh(0)1146 595 y Fk(H)1184 577 y Fj(?)1215 595 y Fp(\()p Fk(N)5 b(H)s Fp(\))1333 566 y Fd(2)p 1328 571 V 1328 587 a Fe(N)1358 577 y Fh(\000)p Fg(2)1444 595 y Fk(:)214 b Fp(\(38\))231 684 y(F)m(or)14 b(a)g(P)g(w)o(a)o(v)o (e,)f(w)o(e)h(ma)o(y)e(directly)i(put)884 673 y Ff(e)874 684 y Fk(H)g Fp(=)972 668 y Fg(1)p 972 675 17 2 v 972 698 a(2)994 684 y Fk(G)1027 691 y Fg(\(3\))1071 684 y Fp(,)1106 673 y Ff(e)1096 684 y Fk(H)1134 669 y Fh(0)1157 684 y Fp(=)1206 668 y Fg(1)p 1206 675 V 1206 698 a(2)1228 684 y Fk(G)1261 669 y Fh(0)1261 697 y Fg(\(3\))1319 684 y Fp(and)1409 673 y Ff(e)1399 684 y Fk(H)1437 669 y Fh(0)p Fj(?)1478 684 y Fp(=)1526 668 y Fg(1)p 1526 675 V 1526 698 a(2)1548 684 y Fk(G)1581 669 y Fh(0)p Fj(?)1581 697 y Fg(\(3\))1625 684 y Fp(.)298 815 y(4.)20 b(RA)m(Y)13 b(TRA)o(CING)h(SYSTEM)h(AND)f(D)o(YNAMIC)h(RA)m(Y)f(TRA)o(CING)f (SYSTEM)667 866 y(IN)18 b(CAR)m(TESIAN)g(COORDINA)m(TES)291 972 y(The)13 b(ra)o(y)f(tracing)h(equations)g(ma)o(y)e(b)q(e)j (expressed)h(in)d(the)i(form)d(of)h(Hamilton's)e(equations)884 1058 y(_)-19 b Fk(x)901 1041 y Fj(k)933 1058 y Fp(=)12 b Fk(H)1015 1041 y Fj(;k)1086 1058 y Fk(;)572 b Fp(\(39\))872 1154 y(_)-20 b Fk(p)885 1160 y Fj(k)917 1154 y Fp(=)12 b Fi(\000)p Fk(H)1028 1160 y Fj(;k)1099 1154 y Fk(:)559 b Fp(\(40\))231 1231 y(The)23 b(dot)f(stands)h(for)f(the)g(deriv)n (ativ)o(e)g Fk(@)r(=@)r(\015)985 1237 y Fg(3)1027 1231 y Fp(along)f(the)h(ra)o(y)m(.)42 b(Since)23 b(w)o(e)f(ha)o(v)o(e)g(c)o (hosen)231 1283 y(a)13 b(homogeneous)f(Hamiltonia)o(n)e(and)j(ha)o(v)o (e)f(c)o(hosen)i(its)e(constan)o(t)i(v)n(alue)e(to)g(b)q(e)h(giv)o(en)g (b)o(y)f(\(24\),)231 1334 y(w)o(e)j(ha)o(v)o(e)836 1395 y(_)-19 b Fk(\034)16 b Fp(=)c Fk(H)945 1378 y Fj(;k)975 1395 y Fk(p)996 1401 y Fj(k)1028 1395 y Fp(=)g(1)41 b Fk(:)524 b Fp(\(41\))231 1473 y(The)15 b(dot)f(then)g(represen)o(ts)j (the)d(deriv)n(ativ)o(e)g(with)f(resp)q(ect)j(to)e(tra)o(v)o(el)g(time) e Fk(\034)19 b Fp(along)13 b(the)h(ra)o(y)m(.)291 1529 y(The)i(dynamic)e(ra)o(y)h(tracing)h(system)g(in)f(Cartesian)h(co)q (ordinates)h(\()1388 1519 y Fo(\024)1381 1529 y(Cerven)o(\023)-21 b(y,)18 b(1972)6 b Fp(\))16 b(can)231 1581 y(b)q(e)i(obtained)e(b)o(y)g (di\013eren)o(tiating)g(ra)o(y)h(tracing)f(equations)h(\(39\))f(and)g (\(40\))g(with)h(resp)q(ect)h(to)231 1632 y(ra)o(y)c(parameter)g Fk(\015)r Fp(,)780 1683 y(_)766 1693 y Fk(Q)799 1676 y Fj(i)825 1693 y Fp(=)d Fk(H)906 1673 y Fj(;i)903 1705 y(;j)930 1693 y Fk(Q)966 1676 y Fj(j)993 1693 y Fp(+)e Fk(H)1072 1676 y Fj(;ij)1111 1693 y Fk(P)1138 1699 y Fj(j)1197 1693 y Fk(;)461 b Fp(\(42\))767 1779 y(_)753 1789 y Fk(P)780 1795 y Fj(i)805 1789 y Fp(=)12 b Fi(\000)p Fk(H)916 1795 y Fj(;ij)955 1789 y Fk(Q)991 1772 y Fj(j)1018 1789 y Fi(\000)d Fk(H)1097 1769 y Fj(;j)1094 1801 y(;i)1124 1789 y Fk(P)1151 1795 y Fj(j)1210 1789 y Fk(;)448 b Fp(\(43\))231 1867 y(where)874 1943 y Fk(Q)907 1926 y Fj(i)932 1943 y Fp(=)981 1915 y Fk(@)r(x)1029 1900 y Fj(i)p 981 1933 62 2 v 988 1971 a Fk(@)r(\015)1089 1943 y(;)569 b Fp(\(44\))878 2076 y Fk(P)905 2082 y Fj(i)930 2076 y Fp(=)979 2048 y Fk(@)r(p)1024 2054 y Fj(i)p 979 2066 60 2 v 985 2104 a Fk(@)r(\015)1085 2076 y(:)k Fp(\(45\))231 2409 y Fn(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h(\(2006\))930 b Fl(455)p eop %%Page: 456 8 456 7 bop 810 32 a Fq(L.)14 b(Klime)n(\024)-18 b(s)152 142 y Fp(5.)18 b(D)o(YNAMIC)c(RA)m(Y)f(TRA)o(CING)g(SYSTEM)i(IN)f(RA)m (Y{CENTRED)f(COORDINA)m(TES)196 241 y(Ra)o(y{cen)o(tred)23 b(co)q(ordinates)g(w)o(ere)h(applied)e(to)g(the)h(p)q(olarization)e(of) h(electromagnetic)137 291 y(w)o(a)o(v)o(es)14 b(b)o(y)g Fo(Lunebur)n(g)k(\(1944\))e Fp(and)e(to)g(dynamic)e(ra)o(y)i(tracing)g (in)f(isotropic)h(media)e(b)o(y)i Fo(Pop)n(ov)137 340 y(and)i(P)n(\024)-19 b(sen)o(\024)f(c)l(\023)j(\020k)19 b(\(1978a,)c(1978b\))r Fp(,)f(and)g(then)h(to)e(dynamic)g(ra)o(y)g (tracing)h(in)g(anisotropic)f(media)137 390 y(b)o(y)h Fo(Hanyga)k(\(1982\))r Fp(,)13 b Fo(Kendal)r(l)i(et)g(al.)j(\(1992\))e Fp(and)e Fo(Klime)n(\024)-19 b(s)16 b(\(1994\))r Fp(.)196 439 y(The)10 b Fk(q)297 424 y Fg(3)310 439 y Fp({co)q(ordinate)g(line)f (of)f(ra)o(y{cen)o(tred)j(co)q(ordinates)f Fk(q)1104 424 y Fj(m)1144 439 y Fp(is)g(the)f(axial)f(ra)o(y)h(parametrized)137 489 y(b)o(y)14 b(parameter)703 560 y Fk(q)723 543 y Fg(3)753 560 y Fp(=)e Fk(q)817 543 y Fg(3)816 570 y(0)845 560 y Fp(+)886 504 y Ff(Z)928 514 y Fj(\034)909 598 y(\034)924 602 y Fd(0)942 560 y Fk(w)g Fp(d)p Fk(\034)46 b(:)492 b Fp(\(46\))137 653 y(F)m(or)13 b(example,)e(the)i(parameter)g(ma)o(y)e (b)q(e)j(tra)o(v)o(el)e(time)g Fk(q)1026 638 y Fg(3)1056 653 y Fp(=)g Fk(\034)17 b Fp(if)c Fk(w)f Fp(=)g(1,)g(or)h(arclength)g Fk(q)1555 638 y Fg(3)1586 653 y Fp(=)e Fk(s)137 703 y Fp(if)j Fk(w)i Fp(is)f(the)h(ra)o(y)f(v)o(elo)q(cit)o(y)m(,)f Fk(w)h Fp(=)665 666 y Fi(p)p 699 666 124 2 v 699 703 a Fk(H)737 691 y Fj(;i)761 703 y Fk(H)799 691 y Fj(;i)838 703 y Fp(.)22 b(In)15 b(this)g(pap)q(er,)h(dynamic)d(ra)o(y)i(tracing)h (in)e(ra)o(y{)137 752 y(cen)o(tred)f(co)q(ordinates)f(is)g(restricted)h (to)f(co)q(ordinates)g Fk(q)1008 737 y Fg(1)1038 752 y Fp(and)f Fk(q)1136 737 y Fg(2)1166 752 y Fp(only)m(.)16 b(The)c(c)o(hoice)g(of)f Fk(q)1538 737 y Fg(3)1568 752 y Fp(then)137 802 y(do)q(es)k(not)e(in\015uence)i(dynamic)e(ra)o(y)g (tracing.)18 b(W)m(e)c(shall)f(th)o(us)h(put)g Fk(w)f Fp(=)e(1.)196 852 y(The)f Fk(q)297 837 y Fj(A)333 852 y Fp(co)q(ordinates)g(lie,)f(for)g(\014xed)h Fk(q)787 837 y Fg(3)806 852 y Fp(,)f(in)g(the)h(w)o(a)o(v)o(efron)o(t)f(tangen)o (t)g(plane.)17 b(The)10 b(Cartesian)137 901 y(co)q(ordinates)15 b(corresp)q(onding)f(to)g Fk(q)695 886 y Fj(m)741 901 y Fp(are)638 971 y Fk(x)662 954 y Fj(i)687 971 y Fp(=)e Fk(x)755 954 y Fj(i)755 981 y Fg(0)773 971 y Fp(\()p Fk(q)809 954 y Fg(3)828 971 y Fp(\))d(+)h Fk(h)919 954 y Fj(i)919 981 y(M)956 971 y Fp(\()p Fk(q)992 954 y Fg(3)1010 971 y Fp(\))i Fk(q)1058 954 y Fj(M)1136 971 y Fk(;)427 b Fp(\(47\))137 1040 y(where)11 b Fk(x)277 1025 y Fj(i)277 1051 y Fg(0)296 1040 y Fp(\()p Fk(q)332 1025 y Fg(3)350 1040 y Fp(\))g(are)f(p)q(oin)o(ts)g(of)g(the)h(axial)d(ra)o(y)m(.)16 b(This)10 b(is)g(b)q(oth)h(the)f(de\014nition)g(of)g(the)h(ra)o(y{cen)o (tred)137 1090 y(co)q(ordinates,)22 b(and)e(the)g(transformation)e (equation)i(from)e(the)j(ra)o(y{cen)o(tred)g(to)f(Cartesian)137 1139 y(co)q(ordinates.)f(The)14 b(transformation)e(matrices,)h(tak)o (en)h(at)g(the)g(cen)o(tral)h(ra)o(y)m(,)d(are)598 1234 y Fk(h)622 1217 y Fj(i)622 1244 y(m)665 1234 y Fp(=)721 1206 y Fk(@)r(x)769 1191 y Fj(i)p 714 1224 76 2 v 714 1262 a Fk(@)r(q)758 1250 y Fj(m)836 1234 y Fk(;)938 1223 y Fp(^)938 1234 y Fk(h)962 1217 y Fj(m)962 1244 y(i)1005 1234 y Fp(=)1054 1206 y Fk(@)r(q)1098 1191 y Fj(m)p 1054 1224 V 1061 1262 a Fk(@)r(x)1109 1250 y Fj(i)1176 1234 y Fk(:)387 b Fp(\(48\))137 1334 y(The)16 b(\\con)o(tra)o(v)n(arian)o(t) f(basis)h(v)o(ectors")h Fk(h)793 1319 y Fj(i)793 1345 y Fg(1)812 1334 y Fp(,)f Fk(h)864 1319 y Fj(i)864 1345 y Fg(2)882 1334 y Fp(,)g Fk(h)934 1319 y Fj(i)934 1345 y Fg(3)969 1334 y Fp(and)g(the)g(\\co)o(v)n(arian)o(t)f(basis)h(v)o (ectors")1595 1323 y(^)1595 1334 y Fk(h)1619 1319 y Fg(1)1619 1345 y Fj(i)1637 1334 y Fp(,)137 1378 y(^)137 1389 y Fk(h)161 1374 y Fg(2)161 1400 y Fj(i)180 1389 y Fp(,)204 1378 y(^)203 1389 y Fk(h)227 1374 y Fg(3)227 1400 y Fj(i)258 1389 y Fp(of)c(the)g(ra)o(y{cen)o(tred)i(co)q(ordinate)e(system)g(are)g (de\014ned)i(b)o(y)d(equations)i(\()p Fo(Klime)n(\024)-19 b(s,)12 b(1994,)137 1439 y(e)n(q.)j(23)6 b Fp(\))787 1488 y Fk(h)811 1471 y Fj(i)811 1499 y Fg(3)841 1488 y Fp(=)12 b Fk(H)923 1471 y Fj(;i)988 1488 y Fk(;)575 b Fp(\(49\))801 1545 y(^)800 1556 y Fk(h)824 1539 y Fg(3)824 1566 y Fj(i)854 1556 y Fp(=)12 b Fk(p)919 1562 y Fj(i)974 1556 y Fk(;)589 b Fp(\(50\))777 1623 y Fk(h)801 1606 y Fj(i)801 1634 y(a)822 1612 y Fp(^)821 1623 y Fk(h)845 1606 y Fj(b)845 1634 y(i)873 1623 y Fp(=)12 b Fk(\016)937 1606 y Fj(b)935 1634 y(a)997 1623 y Fk(:)566 b Fp(\(51\))137 1686 y(The)12 b(\014rst)g(t)o(w)o(o)f(con)o(tra)o(v)n(arian)o(t)f (basis)h(v)o(ectors)i Fk(h)892 1671 y Fj(i)892 1697 y Fg(1)910 1686 y Fp(,)f Fk(h)958 1671 y Fj(i)958 1697 y Fg(2)987 1686 y Fp(of)f(the)h(ra)o(y{cen)o(tred)g(co)q(ordinate)g (system)137 1736 y(are)f(tangen)o(t)g(to)f(the)i(w)o(a)o(v)o(efron)o (t,)e(the)h(third)g(con)o(tra)o(v)n(arian)o(t)e(basis)i(v)o(ector)g Fk(h)1328 1721 y Fj(i)1328 1746 y Fg(3)1357 1736 y Fp(is)g(tangen)o(t)g (to)f(the)137 1790 y(ra)o(y)m(.)16 b(The)11 b(\014rst)h(t)o(w)o(o)e(co) o(v)n(arian)o(t)f(basis)i(v)o(ectors)872 1779 y(^)872 1790 y Fk(h)896 1775 y Fg(1)896 1801 y Fj(i)914 1790 y Fp(,)937 1779 y(^)937 1790 y Fk(h)961 1775 y Fg(2)961 1801 y Fj(i)990 1790 y Fp(of)f(the)h(ra)o(y{cen)o(tred)h(co)q(ordinate) f(system)137 1845 y(are)18 b(p)q(erp)q(endicular)h(to)f(the)h(ra)o(y)m (,)e(the)i(third)f(co)o(v)n(arian)o(t)f(basis)h(v)o(ector)1294 1834 y(^)1293 1845 y Fk(h)1317 1830 y Fg(3)1317 1856 y Fj(i)1354 1845 y Fp(is)g(p)q(erp)q(endicular)137 1900 y(to)h(the)i(w)o(a)o(v)o(efron)o(t.)34 b(Whereas)21 b(basis)e(v)o (ectors)i Fk(h)947 1885 y Fj(i)947 1910 y Fg(3)985 1900 y Fp(and)1072 1889 y(^)1071 1900 y Fk(h)1095 1885 y Fg(3)1095 1910 y Fj(i)1133 1900 y Fp(are)f(uniquely)f(determined)h(b)o(y)137 1949 y(equations)e(\(49\))f(and)h(\(50\),)g(w)o(e)g(ma)o(y)d(either)k (c)o(ho)q(ose)g(the)f(con)o(tra)o(v)n(arian)o(t)f(basis)g(v)o(ectors)i Fk(h)1630 1934 y Fj(i)1630 1960 y Fg(1)137 1999 y Fp(and)h Fk(h)248 1984 y Fj(i)248 2009 y Fg(2)286 1999 y Fp(arbitrarily)e(in)i (the)g(w)o(a)o(v)o(efron)o(t)f(tangen)o(t)h(plane,)h(or)e(c)o(ho)q(ose) i(the)f(co)o(v)n(arian)o(t)f(basis)137 2053 y(v)o(ectors)280 2042 y(^)280 2053 y Fk(h)304 2038 y Fg(1)304 2064 y Fj(i)338 2053 y Fp(and)421 2042 y(^)421 2053 y Fk(h)445 2038 y Fg(2)445 2064 y Fj(i)479 2053 y Fp(arbitrarily)c(in)h(the)g(plane)g(p)q (erp)q(endicular)h(to)f(the)g(ra)o(y)g(v)o(elo)q(cit)o(y)f(v)o(ector) 137 2103 y Fk(h)161 2088 y Fj(i)161 2113 y Fg(3)180 2103 y Fp(.)196 2152 y(If)10 b(the)h(con)o(tra)o(v)n(arian)o(t)e(basis)h(v)o (ectors)h Fc(h)814 2158 y Fg(1)843 2152 y Fp(and)f Fc(h)947 2158 y Fg(2)975 2152 y Fp(are)g(kno)o(wn,)h(the)f(co)o(v)n(arian)o(t)f (basis)i(v)o(ectors)140 2191 y(^)137 2202 y Fc(h)164 2187 y Fg(1)196 2202 y Fp(and)280 2191 y(^)277 2202 y Fc(h)304 2187 y Fg(2)336 2202 y Fp(are)j(giv)o(en)f(b)o(y)h(relations) 698 2280 y(^)695 2291 y Fc(h)722 2274 y Fg(1)752 2291 y Fp(=)846 2263 y Fc(h)873 2269 y Fg(3)900 2263 y Fi(\002)c Fc(h)969 2269 y Fg(2)p 800 2281 233 2 v 800 2320 a Fc(h)827 2306 y Fg(T)827 2332 y(1)860 2320 y Fp(\()p Fc(h)903 2326 y Fg(3)930 2320 y Fi(\002)g Fc(h)999 2326 y Fg(2)1017 2320 y Fp(\))1079 2291 y Fk(;)484 b Fp(\(52\))137 2409 y Fl(456)931 b Fn(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h(\(2006\))p eop %%Page: 457 9 457 8 bop 437 32 a Fq(Common{R)n(ay)13 b(T)m(r)n(acing)g(and)f(Dynamic) g(R)n(ay)h(T)m(r)n(acing)g(for)g(S)g(Waves)g(...)792 152 y Fp(^)789 163 y Fc(h)816 146 y Fg(2)846 163 y Fp(=)941 135 y Fc(h)968 141 y Fg(3)995 135 y Fi(\002)c Fc(h)1063 141 y Fg(1)p 895 154 233 2 v 895 193 a Fc(h)922 178 y Fg(T)922 204 y(2)954 193 y Fp(\()p Fc(h)997 199 y Fg(3)1025 193 y Fi(\002)g Fc(h)1093 199 y Fg(1)1111 193 y Fp(\))1174 163 y Fk(:)484 b Fp(\(53\))231 264 y(If)15 b(the)g(co)o(v)n(arian)o(t)f (basis)h(v)o(ectors)774 253 y(^)771 264 y Fc(h)798 249 y Fg(1)830 264 y Fp(and)915 253 y(^)912 264 y Fc(h)939 249 y Fg(2)972 264 y Fp(are)g(kno)o(wn,)f(the)i(con)o(tra)o(v)n(arian)o (t)d(basis)i(v)o(ectors)231 314 y Fc(h)258 320 y Fg(1)290 314 y Fp(and)f Fc(h)398 320 y Fg(1)430 314 y Fp(are)g(giv)o(en)g(b)o(y) g(relations)799 419 y Fc(h)826 425 y Fg(1)855 419 y Fp(=)950 391 y Fc(p)9 b Fi(\002)1030 380 y Fp(^)1027 391 y Fc(h)1054 376 y Fg(2)p 904 410 214 2 v 907 443 a Fp(^)904 454 y Fc(h)931 440 y Fg(T)931 465 y(1)963 454 y Fp(\()p Fc(p)g Fi(\002)1060 443 y Fp(^)1057 454 y Fc(h)1084 442 y Fg(2)1102 454 y Fp(\))1165 419 y Fk(;)493 b Fp(\(54\))799 562 y Fc(h)826 568 y Fg(2)855 562 y Fp(=)950 534 y Fc(p)9 b Fi(\002)1030 523 y Fp(^)1027 534 y Fc(h)1054 519 y Fg(1)p 904 553 V 907 587 a Fp(^)904 598 y Fc(h)931 583 y Fg(T)931 609 y(2)963 598 y Fp(\()p Fc(p)g Fi(\002)1060 587 y Fp(^)1057 598 y Fc(h)1084 586 y Fg(1)1102 598 y Fp(\))1165 562 y Fk(:)493 b Fp(\(55\))231 661 y(The)20 b(4)6 b Fi(\002)g Fp(4)19 b(dynamic)e(ra)o(y)i(tracing)g(system)g(in)g(ra)o(y{cen)o(tred) h(co)q(ordinates)g(reads)g(\()p Fo(Klime)n(\024)-19 b(s,)231 710 y(1994,)16 b(e)n(q.)f(60)6 b Fp(\))686 758 y(_)672 768 y Fk(Q)705 751 y Fj(M)705 780 y Fg(\()p Fj(q)q Fg(\))761 768 y Fp(=)11 b Fk(H)842 747 y Fg(\()p Fj(q)q Fg(\))p Fj(;M)839 780 y(;N)931 768 y Fk(Q)964 751 y Fj(N)964 780 y Fg(\()p Fj(q)q Fg(\))1018 768 y Fp(+)e Fk(H)1097 748 y Fj(;M)s(N)1094 782 y Fg(\()p Fj(q)q Fg(\))1173 768 y Fk(P)1206 747 y Fg(\()p Fj(q)q Fg(\))1200 780 y Fj(N)1291 768 y Fk(;)367 b Fp(\(56\))674 843 y(_)660 854 y Fk(P)693 832 y Fg(\()p Fj(q)q Fg(\))687 866 y Fj(M)749 854 y Fp(=)11 b Fi(\000)p Fk(H)862 832 y Fg(\()p Fj(q)q Fg(\))859 866 y Fj(;M)s(N)935 854 y Fk(Q)968 837 y Fj(N)968 865 y Fg(\()p Fj(q)q Fg(\))1022 854 y Fi(\000)e Fk(H)1101 832 y Fg(\()p Fj(q)q Fg(\))p Fj(;N)1098 866 y(;M)1185 854 y Fk(P)1218 832 y Fg(\()p Fj(q)q Fg(\))1212 866 y Fj(N)1303 854 y Fk(;)355 b Fp(\(57\))231 920 y(where)849 987 y Fk(Q)882 970 y Fj(M)882 999 y Fg(\()p Fj(q)q Fg(\))938 987 y Fp(=)986 959 y Fk(@)r(q)1030 944 y Fj(M)p 986 978 82 2 v 1003 1016 a Fk(@)r(\015)1114 987 y(;)544 b Fp(\(58\))790 1105 y Fk(P)823 1083 y Fg(\()p Fj(q)q Fg(\))817 1117 y Fj(M)878 1105 y Fp(=)939 1077 y Fk(@)p 927 1095 49 2 v 927 1133 a(@)r(\015)980 1046 y Ff(\022)1032 1077 y Fk(@)r(\034)p 1015 1095 82 2 v 1015 1133 a(@)r(q)1059 1121 y Fj(M)1102 1046 y Ff(\023)1174 1105 y Fk(:)484 b Fp(\(59\))231 1194 y(The)16 b(second{order)i(phase{space)f(deriv)n (ativ)o(es)e(of)g(the)h(Hamiltonian)c(in)j(ra)o(y{cen)o(tred)i(co)q (or-)231 1244 y(dinates)i(are)g(giv)o(en)f(b)o(y)g(relations)g(\()p Fo(Klime)n(\024)-19 b(s,)20 b(1994,)g(e)n(qs.)f(51a,)i(51b,)f(51c)r Fp(;)1472 1233 y Fo(\024)1465 1244 y(Cerven)o(\023)-21 b(y,)21 b(2001,)231 1294 y(e)n(q.)15 b(4.2.78)6 b Fp(;)13 b Fo(Klime)n(\024)-19 b(s,)14 b(2002,)i(e)n(qs.)e(51a,)i(51b,)f(51c)r Fp(\))621 1376 y Fk(H)659 1355 y Fg(\()p Fj(q)q Fg(\))656 1388 y Fj(;AB)731 1376 y Fp(=)d Fk(h)799 1356 y Fj(j)799 1388 y(A)826 1376 y Fp([)p Fk(H)873 1382 y Fj(;j)r(k)927 1376 y Fi(\000)d Fp(\()p Fk(N)14 b Fi(\000)c Fp(1\))p Fk(H)1145 1382 y Fj(;j)1172 1376 y Fk(H)1207 1382 y Fj(;k)1236 1376 y Fp(])p Fk(h)1272 1359 y Fj(k)1272 1387 y(B)1342 1376 y Fk(;)316 b Fp(\(60\))726 1467 y Fk(H)764 1445 y Fg(\()p Fj(q)q Fg(\))p Fj(;N)761 1479 y(;M)859 1467 y Fp(=)12 b Fk(h)927 1447 y Fj(j)927 1479 y(M)963 1467 y Fk(H)1001 1447 y Fj(;k)998 1478 y(;j)1032 1456 y Fp(^)1031 1467 y Fk(h)1055 1449 y Fj(N)1055 1477 y(k)1096 1467 y Fi(\000)e Fk(d)1160 1449 y Fj(N)1160 1477 y(M)1238 1467 y Fk(;)420 b Fp(\(61\))776 1545 y Fk(H)814 1528 y Fj(;M)s(N)902 1545 y Fp(=)946 1534 y(^)946 1545 y Fk(h)970 1528 y Fj(M)970 1555 y(j)1006 1545 y Fk(H)1044 1528 y Fj(;j)r(k)1090 1534 y Fp(^)1090 1545 y Fk(h)1114 1528 y Fj(N)1114 1555 y(k)1187 1545 y Fk(:)471 b Fp(\(62\))231 1611 y(The)15 b(four)e(co)q(e\016cien)o(ts)750 1661 y Fk(d)772 1644 y Fj(N)772 1671 y(M)820 1661 y Fp(=)864 1650 y(^)864 1661 y Fk(h)888 1644 y Fj(N)888 1671 y(i)924 1650 y Fp(_)919 1661 y Fk(h)943 1644 y Fj(i)943 1671 y(M)991 1661 y Fp(=)f Fi(\000)p Fk(h)1091 1644 y Fj(i)1091 1671 y(M)1134 1639 y Fp(_)1129 1650 y(^)1128 1661 y Fk(h)1152 1644 y Fj(N)1152 1671 y(i)1670 1661 y Fp(\(63\))231 1732 y(in)j(equation)g(\(61\))g(dep)q(end)h(on)f(the)g(c)o(hoice)h(of)e (basis)h(v)o(ectors)i Fk(h)1260 1717 y Fj(i)1260 1742 y Fg(1)1278 1732 y Fp(,)e Fk(h)1329 1717 y Fj(i)1329 1742 y Fg(2)1363 1732 y Fp(\(or)1431 1721 y(^)1431 1732 y Fk(h)1455 1717 y Fg(1)1455 1743 y Fj(i)1473 1732 y Fp(,)1501 1721 y(^)1500 1732 y Fk(h)1524 1717 y Fg(2)1524 1743 y Fj(i)1543 1732 y Fp(\))g(along)f(the)231 1782 y(ra)o(y)m(,)i(whic)o(h)f(is)h(discussed)h(in)f(detail)f(b)o(y)g Fo(Klime)n(\024)-19 b(s)19 b(\(2006\))r Fp(.)24 b(On)16 b(the)g(other)h(hand,)e(the)i(c)o(hoice)231 1832 y(of)d(the)g(four)g (co)q(e\016cien)o(ts)h Fk(d)669 1817 y Fj(N)669 1843 y(M)719 1832 y Fp(uniquely)e(determines)h(the)h(ev)o(olution)d(of)i (the)g(basis)g(v)o(ectors)h(of)231 1882 y(the)f(ra)o(y{cen)o(tred)f(co) q(ordinate)g(system)g(along)e(the)i(ra)o(y)f(if)g(the)h(basis)g(v)o (ectors)g(are)g(giv)o(en)f(at)h(the)231 1931 y(initial)f(p)q(oin)o(t)i (of)f(the)i(ra)o(y)e(\()p Fo(Klime)n(\024)-19 b(s,)14 b(2006)6 b Fp(\).)231 2409 y Fn(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h (50)h(\(2006\))930 b Fl(457)p eop %%Page: 458 10 458 9 bop 810 32 a Fq(L.)14 b(Klime)n(\024)-18 b(s)525 142 y Fp(6.)29 b(NUMERICAL)17 b(IMPLEMENT)m(A)m(TION)262 243 y(6)8 b(.)g(1)g(.)38 b(P)8 b(h)h(a)f(s)h(e)f({)g(s)h(p)f(a)h(c)f(e) 32 b(d)9 b(e)f(r)h(i)f(v)e(a)i(t)g(i)g(v)f(e)i(s)32 b(o)8 b(f)31 b(t)9 b(h)f(e)32 b(H)8 b(a)g(m)f(i)h(l)g(t)g(o)h(n)f(i)g(a)g(n) 196 344 y(When)19 b(calculating)f(the)i(homogeneous)e(Hamiltonian)e(of) i(the)i(second)g(degree)g(and)f(its)137 395 y(\014rst{order)f(and)e (second{order)i(phase{space)g(deriv)n(ativ)o(es,)e(w)o(e)h(\014rst)g (calculate)g(Christo\013el)137 445 y(matrix)12 b(\(1\),)h(its)g(eigen)o (v)n(alues)h Fk(G)649 452 y Fg(\()-6 b Fj(a)l Fg(\))699 445 y Fp(and)13 b(eigen)o(v)o(ectors)i Fk(g)1027 452 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))1073 445 y Fp(.)18 b(W)m(e)13 b(then)h(calculate)g(the)g(P{w)o(a)o(v)o(e)137 495 y(Hamiltonian)387 484 y Ff(e)377 495 y Fk(H)j Fp(=)e Fk(G)503 502 y Fg(\()-6 b(3)l(\))553 495 y Fp(or)15 b(common)e(S{w)o(a)o(v)o(e)i(Hamiltonian)d (\(36\))j(with)g(\(28\),)g(and)h(rescale)137 545 y(the)e(slo)o(wness)h (v)o(ector)g(according)f(to)f(the)i(v)n(alue)e(of)g(the)i(Hamiltonian.) 196 596 y(T)m(o)21 b(calculate)h(the)g(\014rst{order)h(and)f (second{order)h(phase{space)g(deriv)n(ativ)o(es)f(of)f(the)137 646 y(Christo\013el)e(matrix,)e(transformed)g(in)o(to)h(the)h(eigen)o (v)o(ectors,)h(w)o(e)f(insert)g(equations)f(\(2\){\(6\))137 696 y(in)o(to)13 b(equations)i(\(19\))e(and)h(\(20\).)19 b(The)c(v)n(alues)f(of)f(the)i(densit)o(y{reduced)h(elastic)e(mo)q (duli)e(and)137 746 y(the)h(v)n(alues)g(of)f(their)h(\014rst{order)h (and)f(second{order)h(spatial)e(deriv)n(ativ)o(es)h(are)g(m)o (ultiplied)d(b)o(y)137 796 y(factors)15 b Fk(g)289 803 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))336 796 y Fk(g)352 803 y Fj(j)p Fg(\()g Fj(b)l Fg(\))399 796 y Fk(p)420 802 y Fj(m)451 796 y Fk(p)472 802 y Fj(n)495 796 y Fp(,)14 b Fk(g)537 803 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))584 796 y Fk(g)600 803 y Fj(j)p Fg(\()g Fj(b)l Fg(\))647 796 y Fk(p)668 802 y Fj(m)714 796 y Fp(or)15 b Fk(g)782 803 y Fj(i)n Fg(\()-6 b Fj(a)l Fg(\))829 796 y Fk(g)845 803 y Fj(j)p Fg(\()g Fj(b)l Fg(\))891 796 y Fp(,)15 b(whic)o(h)g(is)g(n)o (umerically)e(more)h(e\016cien)o(t)i(than)137 847 y(to)i(co)q(de)i (equations)e(\(2\){\(6\))g(follo)o(w)o(ed)f(b)o(y)h(transforms)g (\(19\))g(and)g(\(20\).)32 b(In)18 b(this)h(w)o(a)o(y)m(,)f(w)o(e)137 897 y(calculate)c(the)h(second{order)h(phase{space)g(deriv)n(ativ)o(es) e(\000)1093 904 y Fg(\()-6 b Fj(aa)l Fg(\))1147 903 y Fj(;ij)1187 897 y Fp(,)13 b(\000)1232 904 y Fg(\()-6 b Fj(aa)l Fg(\))1287 877 y Fj(;j)1287 908 y(;i)1329 897 y Fp(and)14 b(\000)1436 877 y Fj(;ij)1430 911 y Fg(\()-6 b Fj(aa)l Fg(\))1484 897 y Fp(,)14 b(and)g(the)137 947 y(\014rst{order)k(phase{space)g(deriv)n(ativ)o(es)e(\000)806 954 y Fg(\()-6 b Fj(ab)l Fg(\))858 953 y Fj(;i)898 947 y Fp(and)16 b(\000)1007 927 y Fj(;i)1001 961 y Fg(\()-6 b Fj(ab)l Fg(\))1069 947 y Fp(with)17 b(\()p Fk(a)p Fp(\))f Fi(6)p Fp(=)g(\()p Fk(b)p Fp(\).)26 b(The)17 b(\014rst{order)137 997 y(phase{space)e(deriv)n(ativ)o(es)f(\000)597 1004 y Fg(\()-6 b Fj(aa)l Fg(\))652 1003 y Fj(;i)690 997 y Fp(and)13 b(\000)796 977 y Fj(;i)790 1011 y Fg(\()-6 b Fj(aa)l Fg(\))859 997 y Fp(can)14 b(b)q(e)h(calculated)f(from)e(\000) 1306 1004 y Fg(\()-6 b Fj(aa)l Fg(\))1361 977 y Fj(;j)1361 1009 y(;i)1402 997 y Fp(and)13 b(\000)1508 977 y Fj(;ij)1502 1011 y Fg(\()-6 b Fj(aa)l Fg(\))1557 997 y Fp(,)706 1074 y(\000)726 1081 y Fg(\()g Fj(aa)l Fg(\))781 1080 y Fj(;i)816 1074 y Fp(=)865 1057 y Fg(1)p 865 1064 17 2 v 865 1088 a(2)886 1074 y Fp(\000)906 1081 y Fg(\()g Fj(aa)l Fg(\))961 1054 y Fj(;j)961 1085 y(;i)989 1074 y Fk(p)1010 1080 y Fj(j)1068 1074 y Fk(;)495 b Fp(\(64\))745 1163 y(\000)771 1143 y Fj(;i)765 1177 y Fg(\()-6 b Fj(aa)l Fg(\))831 1163 y Fp(=)12 b(\000)901 1143 y Fj(;ij)895 1177 y Fg(\()-6 b Fj(aa)l Fg(\))950 1163 y Fk(p)971 1169 y Fj(j)1030 1163 y Fk(:)533 b Fp(\(65\))137 1235 y(The)18 b(\014rst{order)h(and)e (second{order)j(phase{space)f(deriv)n(ativ)o(es)e(of)g(the)h (homogeneous)f(P{)137 1286 y(w)o(a)o(v)o(e)c(Hamiltonian)d(of)j(the)h (second)g(degree)h(are)f(simply)d(giv)o(en)i(b)o(y)g(equations)h (\(22\))f(and)g(\(23\))137 1336 y(with)18 b(\()p Fk(a)p Fp(\))h(=)g(\(3\).)31 b(The)19 b(\014rst{order)h(and)e(second{order)i (phase{space)f(deriv)n(ativ)o(es)g(of)e(the)137 1386 y(homogeneous)12 b(common)d(S{w)o(a)o(v)o(e)j(Hamiltonian)d(of)j (degree)i Fk(N)j Fp(are)c(giv)o(en)f(b)o(y)h(equations)f(\(29\))137 1436 y(and)18 b(\(31\),)g(with)g(one)g(of)g(the)g(equations)g (\(32\){\(35\).)30 b(The)19 b(\014rst{order)g(and)f(second{order)137 1486 y(phase{space)i(deriv)n(ativ)o(es)f(of)f(the)h(homogeneous)f (common)e(S{w)o(a)o(v)o(e)i(Hamiltonian)e(of)i(the)137 1536 y(second)c(degree)h(are)f(then)f(obtained)g(from)f(the)i (\014rst{order)g(and)f(second{order)i(phase{space)137 1586 y(deriv)n(ativ)o(es)20 b(of)f(the)h(homogeneous)f(common)e(S{w)o (a)o(v)o(e)i(Hamiltonian)e(of)i(degree)i Fk(N)k Fp(using)137 1636 y(equations)14 b(\(37\))g(and)f(\(38\).)196 1687 y(The)h(calculation)e(of)h(the)g(homogeneous)g(Hamiltonia)o(n)e(of)h (the)i(second)g(degree)h(and)e(of)g(its)137 1738 y(\014rst{order)j(and) e(second{order)i(phase{space)g(deriv)n(ativ)o(es)e(has)h(b)q(een)g(co)q (ded)g(and)g(placed)f(in)137 1788 y(\014le)g(\\)p Fb(hder.for)p Fp(")e(of)h(the)h(F)m(ortran)g(77)f(pac)o(k)n(age)h(MODEL)g(\()p Fo(Bucha)i(and)g(Bulant,)f(2003)6 b Fp(\).)163 1913 y(6)i(.)g(2)g(.)36 b(M)8 b(o)h(d)f(i)g(f)f(i)h(c)h(a)f(t)g(i)g(o)g(n)31 b(o)8 b(f)30 b(t)9 b(h)f(e)31 b(i)8 b(s)h(o)f(t)g(r)h(o)f(p)g(i)g(c)31 b(r)9 b(a)e(y)30 b(t)9 b(r)f(a)g(c)h(i)f(n)g(g)31 b(a)8 b(l)g(g)g(o)g(r)g(i)g(t)h(h)f(m)196 2014 y(F)m(or)i(the)g(mo)q (di\014cation)e(of)h(the)i(isotropic)f(\\complete)f(ra)o(y)g(tracing")h (algorithm)d(b)o(y)1508 2004 y Fo(\024)1501 2014 y(Cerven)o(\023)-21 b(y)137 2064 y(et)11 b(al.)16 b(\(1988\))r Fp(,)11 b(w)o(e)f(ha)o(v)o (e)f(selected)j(options)d(whic)o(h)h(are)g(most)e(similar)f(to)j(the)g (original)e(isotropic)137 2114 y(algorithm.)26 b(In)18 b(this)f(w)o(a)o(y)m(,)g(the)h(4)5 b Fi(\002)g Fp(4)17 b(paraxial)f(ra)o(y)h(propagator)g(matrix)f(is)h(calculated)h(b)o(y)137 2165 y(dynamic)g(ra)o(y)i(tracing)g(in)g(ra)o(y{cen)o(tred)i(co)q (ordinates)e(according)h(to)f(Section)g(5,)h(and)f(the)137 2215 y(ra)o(y{cen)o(tred)14 b(co)q(ordinates)f(are)g(de\014ned)h (according)e(to)h Fo(Klime)n(\024)-19 b(s)15 b(\(2006,)f(se)n(c.)g (5.4\))r Fp(.)j(This)12 b(ra)o(y{)137 2265 y(cen)o(tred)18 b(co)q(ordinate)e(system)f(represen)o(ts)k(one)d(of)f(the)i(v)n(arious) e(p)q(ossible)h(generalizations)f(of)137 2315 y(the)c(ra)o(y{cen)o (tred)h(co)q(ordinate)f(system)f(for)h(isotropic)f(media)f(b)o(y)h Fo(Pop)n(ov)j(and)g(P)n(\024)-19 b(sen)o(\024)f(c)l(\023)j(\020k)15 b(\(1978a,)137 2409 y Fl(458)931 b Fn(Stud.)10 b(Geoph)o(ys.)g(Geo)q (d.,)g(50)h(\(2006\))p eop %%Page: 459 11 459 10 bop 437 32 a Fq(Common{R)n(ay)13 b(T)m(r)n(acing)g(and)f (Dynamic)g(R)n(ay)h(T)m(r)n(acing)g(for)g(S)g(Waves)g(...)231 142 y Fo(1978b\))r Fp(.)19 b(The)13 b(con)o(tra)o(v)n(arian)o(t)f (basis)h(v)o(ectors)h Fc(h)989 148 y Fg(1)1020 142 y Fp(and)e Fc(h)1126 148 y Fg(2)1157 142 y Fp(are)i(used)f(in)g(place)g (of)f(the)h(isotropic)231 192 y(basis)20 b(v)o(ectors,)h(b)q(ecause)g (they)f(ha)o(v)o(e)f(similar)e(prop)q(erties:)31 b Fc(h)1249 198 y Fg(1)1286 192 y Fp(and)19 b Fc(h)1399 198 y Fg(2)1437 192 y Fp(are)g(unit)h(v)o(ectors,)231 243 y(p)q(erp)q(endicular)c(to)f (the)g(slo)o(wness)h(v)o(ector,)f(and)g(m)o(utually)d(p)q(erp)q (endicular)k(\(all)e(with)g(resp)q(ect)231 293 y(to)g(the)h(Cartesian)f (metric\).)291 345 y(Anisotropic{ra)o(y{theory)20 b(ra)o(y)g(tracing)h (for)f(P)h(w)o(a)o(v)o(es)g(and)f(anisotropic{common{ra)n(y)231 396 y(tracing)12 b(for)f(S)h(w)o(a)o(v)o(es)f(in)g(smo)q(oth)f (anisotropic)i(mo)q(dels)e(without)h(in)o(terfaces)h(and)g(corresp)q (ond-)231 446 y(ing)j(dynamic)e(ra)o(y)i(tracing)g(in)g(ra)o(y{cen)o (tred)h(co)q(ordinates)g(ha)o(v)o(e)e(b)q(een)j(co)q(ded,)e(added)h(to) f(the)231 497 y(F)m(ortran)10 b(77)e(CR)m(T)h(pac)o(k)n(age,)g (debugged)h(and)f(n)o(umerically)e(tested)k(\()p Fo(Bucha)h(and)g (Bulant,)f(2004)6 b Fp(;)231 547 y Fo(Bulant)17 b(and)f(Klime)n(\024) -19 b(s,)15 b(2006)6 b Fp(\).)21 b(F)m(or)15 b(the)g(n)o(umerical)e (example)h(of)g(anisotropic{common{ra)n(y)231 598 y(tracing)e(and)f (for)g(the)i(application)d(of)h(the)h(n)o(umerical)e(algorithm)f(of)i (the)h(coupling)e(ra)o(y)i(theory)231 648 y(b)o(y)g Fo(Bulant)h(and)g (Klime)n(\024)-19 b(s)14 b(\(2002\))g Fp(along)d(anisotropic)g(common)d (S{w)o(a)o(v)o(e)j(ra)o(ys)h(refer)g(to)g Fo(Bulant)231 699 y(and)k(Klime)n(\024)-19 b(s)17 b(\(2006\))r Fp(.)802 826 y(7.)33 b(CONCLUSIONS)291 928 y(The)21 b(coupling)f(ra)o(y)g (theory)i(for)e(S)h(w)o(a)o(v)o(es)g(in)f(anisotropic)g(media)f(has)i (recen)o(tly)h(b)q(een)231 978 y(applied)c(to)f(isotropic)h(reference)i (ra)o(ys)e(\()p Fo(P)n(\024)-19 b(sen)o(\024)f(c)l(\023)j(\020k,)20 b(1998)6 b Fp(;)19 b Fo(P)n(\024)-19 b(sen)o(\024)f(c)l(\023)j(\020k)19 b(and)h(Del)r(linger,)e(2001)6 b Fp(\).)231 1029 y(The)22 b(equations)f(presen)o(ted)i(b)o(y)e Fo(Bakker)26 b(\(2002\))d Fp(and)e(the)h(equations)f(presen)o(ted)j(in)c(this)231 1079 y(pap)q(er)h(allo)o(w)c(anisotropic)j(common)c(S{w)o(a)o(v)o(e)j (reference)j(ra)o(ys)e(to)f(b)q(e)h(traced.)36 b(Commo)o(n{)231 1130 y(ra)o(y)14 b(tracing)g(for)g(S)g(w)o(a)o(v)o(es)g(and)f(corresp)q (onding)i(dynamic)e(ra)o(y)g(tracing)h(enable)h(the)f(coupling)231 1180 y(ra)o(y)20 b(theory)h(to)f(b)q(e)h(made)e(more)g(accurate,)k (while)d(preserving)h(the)g(comfort)d(of)i(common)231 1231 y(reference)i(ra)o(ys)d(for)f(b)q(oth)h(S{w)o(a)o(v)o(e)f(p)q (olarizations.)32 b(The)20 b(anisotropic)e(common)e(ra)o(ys)j(also)231 1281 y(eliminate)d(problems)g(with)h(anisotropic{ra)o(y{theory)f(ra)o (y)h(tracing)g(through)h(some)e(S{w)o(a)o(v)o(e)231 1332 y(slo)o(wness{surface)g(singularities.)291 1384 y(The)c(equations)g(of) g(this)g(pap)q(er)h(represen)o(t)h(a)e(complete)g(algorithm)d(suitable) j(for)g(anisotro-)231 1434 y(pic{common{ra)o(y)h(tracing)k(for)f(S)g(w) o(a)o(v)o(es)g(in)g(a)g(smo)q(oth)f(elastic)i(anisotropic)f(medium,)d (and)231 1485 y(for)g(corresp)q(onding)h(dynamic)e(ra)o(y)h(tracing)g (in)f(Cartesian)i(or)f(ra)o(y{cen)o(tred)h(co)q(ordinates.)19 b(F)m(or)231 1535 y(the)i(comparison)d(of)g(tra)o(v)o(el{time)g(errors) j(of)d(the)j(coupling)d(ra)o(y)h(theory)h(due)g(to)f(isotropic)231 1586 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Fq(Common{R)n(ay)13 b(T)m(r)n(acing)g(and)f (Dynamic)g(R)n(ay)h(T)m(r)n(acing)g(for)g(S)g(Waves)g(...)231 142 y Fr(Klime)n(\024)-17 b(s)14 b(L.)f(and)g(Bulan)o(t)h(P)m(.,)e (2006.)17 b(Errors)c(due)h(to)e(the)h(anisotropic{commo)q(n{ra)o(y)j (appro)o(ximation)291 183 y(of)c(the)h(coupling)j(ra)o(y)d(theory)m(.) 18 b Fq(Stud.)12 b(Ge)n(ophys.)f(Ge)n(o)n(d.)p Fr(,)f Fa(50)p Fr(,)j(463{477.)231 250 y(Luneburg)k(R.K.,)d(1944.)24 b Fq(Mathematic)n(al)13 b(The)n(ory)i(of)g(Optics.)22 b Fr(Lecture)15 b(notes,)h(Bro)o(wn)f(Univ)o(ersit)o(y)m(,)291 291 y(Pro)o(vidence,)g(Rho)q(de)g(Island.)f(Reedition:)i(Univ)o(ersit)o (y)g(of)d(California)j(Press,)e(Berk)o(eley)h(and)f(Los)291 333 y(Angeles,)f(1964.)231 399 y(P)o(op)q(o)o(v)i(M.M.)d(and)i(P)n (\024)-17 b(sen)o(\024)f(c)l(\023)j(\020k)15 b(I.,)d(1978a.)19 b(Ra)o(y)13 b(amplitudes)j(in)e(inhomogeneous)i(media)f(with)e(curv)o (ed)291 440 y(in)o(terfaces.)k Fq(T)m(r)n(avaux)12 b(Instit.)e(G)o (\023)-19 b(eophys.)10 b(A)n(c)n(ad.)h(Tch)o(\023)-19 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