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b(times)g(due)h(to)g(the)g(p)q(erturbation.)24 b(These)17 b(tra)o(v)o(el{time)d(errors)j(can)f(deteriorate)h(the)137 1657 y(coupling{ra)o(y{theory)c(solution)g(at)g(high)g(frequencies.)20 b(It)14 b(is)f(th)o(us)i(of)e(principal)g(imp)q(ortance)137 1708 y(for)e(n)o(umerical)f(applications)g(to)h(estimate)g(the)g(tra)o (v)o(el{time)f(errors)i(due)g(to)f(the)h(common{ra)o(y)137 1758 y(appro)o(ximation,)e(and)k(then)h(the)f(related)h(error)f(of)g (the)g(w)o(a)o(v)o(e\014eld.)196 1810 y(In)f(the)h(common{ra)n(y)c (appro)o(ximation,)g(the)j(S{w)o(a)o(v)o(e)g(tra)o(v)o(el)f(times)g (are)i(usually)e(appro)o(xi-)137 1860 y(mated)c(b)o(y)h(the)h (\014rst{order)h(p)q(erturbation)f(expansion)f(from)e(the)j(common)d (reference)12 b(ra)o(y)m(.)j(The)137 1911 y(errors)f(of)f(S{w)o(a)o(v)o (e)f(tra)o(v)o(el)h(times)f(ma)o(y)f(then)i(b)q(e)h(appro)o(ximated)d (b)o(y)i(the)g(second{order)i(terms)137 1961 y(in)f(the)i(p)q (erturbation)f(expansion.)21 b(Note)15 b(that)g(the)g(p)q(erturbation)h 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y(ed)j(using)f(the)g(a)o(v)o(eraged)h(Hamiltonia)o(n)d(of)h(b)q (oth)i(anisotropic{ra)o(y{theory)e(S)i(w)o(a)o(v)o(es)f(\()p Fm(Bakker,)137 1441 y(2002)6 b Fn(;)15 b Fm(Klime)n(\024)-19 b(s,)15 b(2006b)s Fn(\).)20 b(This)15 b(is)f(probably)g(the)h(b)q(est)h (common{ra)o(y)11 b(appro)o(ximation.)17 b(The)137 1490 y(errors)c(due)f(to)g(the)g(anisotropic{common{ra)o(y)c(appro)o (ximation)h(of)i(the)i(coupling)e(ra)o(y)g(theory)137 1540 y(are)i(usually)f(considerably)g(smaller)f(than)i(the)g(errors)h (due)f(to)f(the)i(routinely)e(used)h(isotropic{)137 1590 y(common{ra)o(y)g(appro)o(ximation)g(of)i(the)i(coupling)e(ra)o(y)g (theory)i(\()p Fm(Klime)n(\024)-19 b(s)16 b(and)i(Bulant,)f(2004;)137 1640 y(Bulant)e(and)h(Klime)n(\024)-19 b(s,)14 b(2006)6 b Fn(\).)196 1691 y(In)18 b(the)h(less)g(accurate)g Fm(isotr)n(opic{c)n (ommon{r)n(ay)g(appr)n(oximation)p Fn(,)g(the)g(reference)h(ra)o(y)e (is)137 1740 y(traced)i(in)f(the)h(reference)h(isotropic)e(mo)q(del.)32 b(Moreo)o(v)o(er,)21 b(the)f(reference)h(isotropic)e(mo)q(del)137 1790 y(ma)o(y)8 b(b)q(e)i(selected)h(in)f(di\013eren)o(t)g(w)o(a)o(ys,) g(yielding)e(quasi{isotropic)h(appro)o(ximations)e(of)i(di\013ering)137 1840 y(accuracies.)595 1965 y(3.)31 b(COUPLING)18 b(RA)m(Y)f(THEOR)m(Y) 196 2065 y(F)m(or)10 b(the)h(deriv)n(ation)f(of)f(the)i(coupling)f(ra)o (y)g(theory)h(refer)g(to)g Fm(Co)n(ates)g(and)i(Chapman)h(\(1990\))137 2115 y Fn(and)21 b(to)290 2105 y Fm(\024)283 2115 y(Cerven)o(\023)-21 b(y)26 b(\(2001\))r Fn(.)40 b(F)m(or)21 b(the)h(description)g(of)e(the) i(n)o(umerical)e(algorithm)f(of)h(the)137 2165 y(coupling)12 b(ra)o(y)h(theory)g(refer)h(to)f Fm(Bulant)i(and)g(Klime)n(\024)-19 b(s)15 b(\(2002\))r Fn(.)j(The)c(n)o(umerical)d(algorithm)f(of)137 2215 y(the)i(coupling)e(ra)o(y)h(theory)h(b)o(y)f Fm(Bulant)i(and)g (Klime)n(\024)-19 b(s)14 b(\(2002\))f Fn(is)e(indep)q(enden)o(t)i(of)d (the)i(reference)137 2265 y(ra)o(y)m(.)38 b(The)21 b(algorithm)d(is)i 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y(ra)o(y)g(theory)m(.)291 389 y(Here)j(w)o(e)f(shall)g(concen)o(trate)i(only)d(on)h(estimating)e (the)j(tra)o(v)o(el{time)d(errors)k(due)f(to)f(the)231 438 y(anisotropic{common{ra)o(y)h(\(and)j(also)g(isotropic{commo)o (n{ra)o(y\))d(appro)o(ximation)h(b)o(y)i(n)o(u-)231 488 y(merical)12 b(quadrature)i(along)e(the)h(anisotropic)g(common)d(ra)o (ys.)18 b(The)c(estimates)e(are)i(based)f(on)231 537 y(the)18 b(equations)f(for)f(the)i(second{order)g(p)q(erturbation)g (deriv)n(ativ)o(es)e(of)h(tra)o(v)o(el)f(time)g(deriv)o(ed)231 587 y(b)o(y)e Fm(Klime)n(\024)-19 b(s)17 b(\(2002\))r Fn(.)443 711 y(4.)23 b(TRA)-5 b(VEL)17 b(TIMES)f(IN)g(THE)g(COUPLING)g (RA)m(Y)f(THEOR)m(Y)291 810 y(Let)22 b Fi(a)395 816 y Fh(ij)r(k)q(l)477 810 y Fn(=)j Fi(a)556 816 y Fh(ij)r(k)q(l)614 810 y Fn(\()p Fi(x)654 816 y Fh(m)685 810 y Fn(\))d(b)q(e)g(the)g (densit)o(y{normalized)f(elastic)g(mo)q(duli)e(describing)k(a)231 860 y Fm(smo)n(oth)12 b(anisotr)n(opic)g(mo)n(del)p Fn(,)e(in)g(whic)o (h)h(the)g Fm(anisotr)n(opic)g(c)n(ommon)h(r)n(ays)e Fn(are)h(traced)h(using)e(the)231 909 y(a)o(v)o(eraged)h(Hamiltonian)d (of)i(b)q(oth)h(anisotropic{ra)o(y{theory)f(S)g(w)o(a)o(v)o(es.)17 b(F)m(or)11 b(completeness,)g(w)o(e)231 958 y(also)i(in)o(tro)q(duce)h (S{w)o(a)o(v)o(e)e(v)o(elo)q(cit)o(y)h Fi(v)815 964 y Fg(0)845 958 y Fn(=)f Fi(v)909 964 y Fg(0)928 958 y Fn(\()p Fi(x)968 964 y Fh(m)999 958 y Fn(\))h(in)g(the)g(smo)q(oth)f(reference) k(isotropic)d(mo)q(del)231 1008 y(in)h(whic)o(h)g(the)g(isotropic)g (common)d(ra)o(ys)j(could)g(b)q(e)g(traced.)291 1057 y(Assume)k(a)g(phase{space)i Fm(r)n(efer)n(enc)n(e)e(r)n(ay)p Fn(,)g(parametrized)g(b)o(y)g(reference)j(tra)o(v)o(el)d(time)f Fi(\034)5 b Fn(,)231 1107 y(with)15 b(reference)i(slo)o(wness)f(v)o (ectors)g Fi(p)832 1113 y Fh(i)846 1107 y Fn(\()p Fi(\034)5 b Fn(\))15 b(kno)o(wn)f(at)h(all)e(its)i(p)q(oin)o(ts)g Fi(x)1367 1113 y Fh(j)1384 1107 y Fn(\()p Fi(\034)5 b Fn(\).)21 b(In)15 b(this)g(section,)231 1156 y(the)20 b(reference)i(ra)o(y)d(is)g(general,)i(including)d(the)i(anisotropic)f (common)d(ra)o(y)j(and)g(isotropic)231 1206 y(common)11 b(ra)o(y)h(as)i(sp)q(ecial)f(cases.)19 b(Using)13 b(the)h(reference)h (slo)o(wness)f(v)o(ectors,)g(w)o(e)f(can)g(calculate)231 1255 y(the)i(reference)h(Christo\013el)e(matrix)668 1321 y(\000)694 1327 y Fh(j)r(k)730 1321 y Fn(\()p Fi(\034)5 b Fn(\))12 b(=)g Fi(p)862 1327 y Fh(i)875 1321 y Fn(\()p Fi(\034)5 b Fn(\))12 b Fi(a)964 1327 y Fh(ij)r(k)q(l)1022 1321 y Fn(\()p Fi(x)1062 1327 y Fh(m)1093 1321 y Fn(\()p Fi(\034)5 b Fn(\)\))12 b Fi(p)1197 1327 y Fh(l)1210 1321 y Fn(\()p Fi(\034)5 b Fn(\))425 b(\(1\))231 1386 y(and)17 b(its)g(eigen)o(v)o(ectors)i Fi(g)632 1392 y Fh(i\013)667 1386 y Fn(\()p Fi(\034)5 b Fn(\),)17 b Fi(\013)g Fn(=)g(1)p Fi(;)7 b Fn(2)p Fi(;)g Fn(3)15 b(along)g(the)j(reference)h(ra)o(y)m(.) 27 b(Whereas)18 b(the)g(Ein-)231 1435 y(stein)j(summation)c(o)o(v)o(er) i(the)i(pairs)f(of)f(iden)o(tical)g(Roman)f(indices)i(\(b)q(oth)g (subscripts)h(and)231 1485 y(sup)q(erscripts\))g Fi(i;)7 b(j;)g(k)q(;)g(:::)15 b Fn(=)j(1)p Fi(;)7 b Fn(2)p Fi(;)g Fn(3)15 b(or)j Fi(I)s(;)7 b(J)o(;)g(K)q(;)g(:::)15 b Fn(=)j(1)p Fi(;)7 b Fn(2)17 b(is)g(used)i(throughout)e(this)h(pap)q (er,)231 1534 y(no)13 b(implicit)d(summation)f(applies)k(to)f(Greek)h (subscripts)h Fi(\013;)7 b(\014)r(;)g(:::)k Fn(indexing)h(the)h(eigen)o (v)o(ectors)231 1584 y(of)g(the)i(Christo\013el)e(matrix)f(or)i(the)g (p)q(erturbation)g(parameters.)k(Assume)13 b(that)h(eigen)o(v)o(ectors) 231 1633 y Fi(g)251 1639 y Fh(i)p Fg(1)282 1633 y Fn(\()p Fi(\034)5 b Fn(\))17 b(and)h Fi(g)459 1639 y Fh(i)p Fg(2)489 1633 y Fn(\()p Fi(\034)5 b Fn(\))18 b(corresp)q(ond)h(to)f(the)g(S)g(w) o(a)o(v)o(es.)29 b(F)m(or)17 b(application)f(of)h(the)i(coupling)e(ra)o (y)231 1683 y(theory)m(,)i(the)f(eigen)o(v)o(ectors)h(should)f(v)n(ary) f(con)o(tin)o(uously)g(along)g(the)h(reference)i(ra)o(y)e(\()p Fm(Bulant)231 1732 y(and)e(Klime)n(\024)-19 b(s,)13 b(2002)6 b Fn(\).)19 b(This)13 b(condition)g(is)g(not)h(required)g(in)f(regions) h(where)h(the)f(t)o(w)o(o)f(S{w)o(a)o(v)o(e)231 1781 y(eigen)o(v)n(alues)h(of)f(the)i(Christo\013el)f(matrix)e(are)i(appro)o (ximately)e(equal.)291 1831 y(Let)j(us)g(denote)g(b)o(y)g Fi(\034)631 1837 y Fh(\013)655 1831 y Fn(\()p Fi(\034)5 b Fn(\))14 b(the)i(anisotropic{ra)o(y{theory)e(tra)o(v)o(el)g(time)g (corresp)q(onding)h(to)231 1880 y(the)j(selected)g(eigen)o(v)o(ector)f Fi(g)701 1886 y Fh(i\013)736 1880 y Fn(\()p Fi(\034)5 b Fn(\))17 b(of)f(the)h(Christo\013el)f(matrix.)24 b(It)17 b(ma)o(y)d(b)q(e)j(appro)o(ximated)231 1930 y(b)o(y)d(a)g(quadrature)g (along)f(the)h(unp)q(erturb)q(ed)j(reference)f(ra)o(y)m(,)772 1987 y(d)p Fi(\034)813 1993 y Fh(\013)p 772 2005 65 2 v 781 2043 a Fn(d)p Fi(\034)853 2015 y Fn(=)c([\000)935 2021 y Fh(j)r(k)970 2015 y Fi(g)990 2021 y Fh(j)r(\013)1029 2015 y Fi(g)1049 2021 y Fh(k)q(\013)1091 2015 y Fn(])1103 1998 y Ff(\000)1134 1987 y Fe(1)p 1134 1992 15 2 v 1134 2008 a(2)1196 2015 y Fi(:)482 b Fn(\(2\))231 2095 y(T)m(ra)o(v)o (el{time)18 b(appro)o(ximation)e(\(2\),)21 b(suggested)g(for)f(the)g (coupling)f(ra)o(y)g(theory)h(b)o(y)g Fm(Bulant)231 2144 y(and)14 b(Klime)n(\024)-19 b(s)15 b(\(2002\))r Fn(,)d(w)o(ould)f(b)q (ecome)g(exact)i(for)e(a)h(reference)i(ra)o(y)d(follo)o(wing)e(the)k (path)e(of)h(the)231 2193 y(resp)q(ectiv)o(e)g(anisotropic{ra)o (y{theory)e(ra)o(y)m(.)15 b(T)m(ra)o(v)o(el{time)8 b(appro)o(ximation)f (\(2\))i(can)h(b)q(e)h(deriv)o(ed)231 2243 y(as)h(the)f(\014rst{order)i (part)e(of)g(p)q(erturbation)g(expansion)g(\(19\),)g(corresp)q(onding)h (to)f(Hamiltonian)673 2315 y Fi(H)708 2321 y Fh(\013)731 2315 y Fn(\()p Fi(x)771 2321 y Fh(m)802 2315 y Fi(;)c(p)842 2321 y Fh(n)864 2315 y Fn(\))12 b(=)g Fd(\000)p Fn([)p Fi(G)1013 2321 y Fh(\013)1036 2315 y Fn(\()p Fi(x)1076 2321 y Fh(m)1107 2315 y Fi(;)7 b(p)1147 2321 y Fh(n)1169 2315 y Fn(\)])1197 2298 y Ff(\000)1228 2287 y Fe(1)p 1228 2292 V 1228 2308 a(2)1290 2315 y Fi(;)388 b Fn(\(3\))231 2409 y Fl(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h(\(2006\))930 b Fj(467)p eop %%Page: 468 6 468 5 bop 685 32 a Fo(L.)13 b(Klime)n(\024)-18 b(s)14 b(and)e(P.)i(Bulant)137 142 y Fn(see)21 b(Section)g(7.)36 b(Here)21 b Fi(G)568 148 y Fh(\013)591 142 y Fn(\()p Fi(x)631 148 y Fh(m)663 142 y Fi(;)7 b(p)703 148 y Fh(n)725 142 y Fn(\))20 b(is)g(the)g(eigen)o(v)n(alue)g(of)f(Christo\013el)h (matrix)e(\(1\),)k(cor-)137 191 y(resp)q(onding)c(to)f(eigen)o(v)o (ector)h Fi(g)645 197 y Fh(i\013)681 191 y Fn(.)28 b(Hamiltonian)14 b(\(3\))k(is)f(a)g(homogeneous)f(function)h(of)g(the)137 241 y(min)o(us)12 b(\014rst)j(degree)g(with)f(resp)q(ect)i(to)e(the)g (slo)o(wness)h(v)o(ector.)256 366 y(5.)21 b(REFERENCE)15 b(HAMIL)m(TONIAN,)g(ANISOTR)o(OPIC{RA)m(Y{THEOR)m(Y)225 416 y(HAMIL)m(TONIANS,)g(AND)g(ISOTR)o(OPIC{RA)m(Y{THEOR)m(Y)e(HAMIL)m (TONIAN)196 516 y(W)m(e)h(denote)i(b)o(y)e Fi(H)496 522 y Fg(0)514 516 y Fn(\()p Fi(x)554 522 y Fh(m)585 516 y Fi(;)7 b(p)625 522 y Fh(n)647 516 y Fn(\))15 b(the)g Fm(r)n(efer)n(enc)n(e)g(Hamiltonian)f Fn(used)h(to)g(trace)g(the)g (reference)137 566 y(ra)o(y)m(.)196 616 y(F)m(or)g(the)h(anisotropic)g (common)c(ra)o(y)m(,)j Fm(Klime)n(\024)-19 b(s)18 b(\(2006b\))g Fn(assumed)d(the)h(reference)i(Hamil-)137 666 y(tonian)13 b(to)h(b)q(e)g(equal)g(to)g(the)g(a)o(v)o(eraged)g(Hamiltonian)p 359 733 38 2 v 359 766 a Fi(H)397 772 y Fg(0)415 766 y Fn(\()p Fi(x)455 772 y Fh(m)487 766 y Fi(;)7 b(p)527 772 y Fh(n)549 766 y Fn(\))k(=)625 750 y Fg(1)p 625 757 17 2 v 625 780 a(2)654 720 y Fc(n)686 750 y Fg(1)p 686 757 V 686 780 a(2)708 766 y Fd(f)p Fn([)p Fi(G)774 772 y Fg(1)791 766 y Fn(\()p Fi(x)831 772 y Fh(m)863 766 y Fi(;)c(p)903 772 y Fh(n)925 766 y Fn(\)])958 738 y Fb(N)p 957 743 25 2 v 963 759 a Fe(2)998 766 y Fn(+)j([)p Fi(G)1085 772 y Fg(2)1103 766 y Fn(\()p Fi(x)1143 772 y Fh(m)1174 766 y Fi(;)d(p)1214 772 y Fh(n)1236 766 y Fn(\)])1269 738 y Fb(N)p 1269 743 V 1274 759 a Fe(2)1300 766 y Fd(g)1321 720 y Fc(o)1359 717 y Fe(2)p 1354 722 V 1354 738 a Fb(N)1596 766 y Fn(\(4\))137 854 y(of)16 b(b)q(oth)i(S)f(w)o(a)o(v)o(es)g(\()p Fm(Klime)n(\024)-19 b(s,)17 b(2006b,)j(e)n(qs.)d(28)h(and)h(36)6 b Fn(\),)18 b(where)g(constan)o(t)f Fi(N)22 b Fn(sp)q(eci\014es)d(the)137 904 y(w)o(a)o(y)c(of)g(a)o(v)o(eraging.)21 b(Note)16 b(that)g Fm(Bakker)k(\(2002\))e Fn(considered)f Fi(N)i Fn(=)14 b(2.)23 b(Hamiltonian)12 b(\(4\))k(is)137 953 y(a)g(homogeneous)g(function)g(of)g(the)h(second)h(degree)g(with)e (resp)q(ect)j(to)d(the)h(slo)o(wness)g(v)o(ector.)137 1003 y(The)d(v)n(alue)g(of)f(Hamiltonian)e(\(4\))i(along)g(the)i (reference)h(ra)o(y)e(is)p 1152 970 38 2 v 13 w Fi(H)1190 1009 y Fg(0)1209 1003 y Fn(\()p Fi(x)1249 1009 y Fh(m)1280 1003 y Fi(;)7 b(p)1320 1009 y Fh(n)1342 1003 y Fn(\))12 b(=)1419 987 y Fg(1)p 1419 994 17 2 v 1419 1018 a(2)1440 1003 y Fn(.)196 1054 y(The)19 b(\014rst{order)h(phase{space)g(deriv)n (ativ)o(es)f(of)f(Hamiltonian)e(\(4\))i(are)h(determined)g(b)o(y)137 1103 y(equations)c(\(29\))h(and)f(\(37\))g(of)g Fm(Klime)n(\024)-19 b(s)18 b(\(2006b\))r Fn(.)23 b(The)16 b(second{order)h(phase{space)g (deriv)n(a-)137 1153 y(tiv)o(es)23 b(of)f(Hamiltonia)o(n)e(\(4\))i(are) h(determined)g(b)o(y)f(equations)h(\(31\))f(and)g(\(38\))h(of)f Fm(Klime)n(\024)-19 b(s)137 1203 y(\(2006b\))r Fn(.)196 1254 y(Since)16 b(the)g(equations)f(for)g(the)h(p)q(erturbation)g (deriv)n(ativ)o(es)f(of)g(tra)o(v)o(el)g(time)f(w)o(ere)i(deriv)o(ed) 137 1303 y(b)o(y)e Fm(Klime)n(\024)-19 b(s)18 b(\(2002\))f Fn(on)d(the)h(condition)f(that)h(the)g(Hamiltonian)c(has)k(an)g(equal)f (v)n(alue)g(for)g(all)137 1353 y(orthonomic)f(systems)h(of)g(ra)o(ys)h (under)g(consideration,)f(reference)j(Hamiltonian)11 b Fi(H)1481 1359 y Fg(0)1499 1353 y Fn(\()p Fi(x)1539 1359 y Fh(m)1571 1353 y Fi(;)c(p)1611 1359 y Fh(n)1633 1353 y Fn(\))137 1403 y(m)o(ust)16 b(b)q(e)h(a)g(homogeneous)f (function)g(of)h(the)g(same)f(degree)i(with)f(resp)q(ect)i(to)e(the)g (slo)o(wness)137 1453 y(v)o(ector)e(as)f(Hamiltonia)o(n)d(\(3\).)18 b(W)m(e)c(th)o(us)g(select)h(reference)i(Hamiltonian)570 1535 y Fi(H)605 1541 y Fg(0)623 1535 y Fn(\()p Fi(x)663 1541 y Fh(m)695 1535 y Fi(;)7 b(p)735 1541 y Fh(n)757 1535 y Fn(\))k(=)h Fd(\000)p Fn([2)p 893 1501 38 2 v Fi(H)931 1541 y Fg(0)949 1535 y Fn(\()p Fi(x)989 1541 y Fh(m)1021 1535 y Fi(;)7 b(p)1061 1541 y Fh(n)1083 1535 y Fn(\)])1111 1518 y Ff(\000)1141 1507 y Fe(1)p 1141 1512 15 2 v 1141 1528 a(2)1204 1535 y Fi(;)380 b Fn(\(5\))137 1610 y(whic)o(h)21 b(is)h(a)f(homogeneous)g(function)g(of)g(the)h(min)o (us)e(\014rst)j(degree)g(with)e(resp)q(ect)j(to)d(the)137 1660 y(slo)o(wness)15 b(v)o(ector,)f(and)f(whic)o(h)h(yields)g(equal)f (orthonomic)g(systems)h(of)f(ra)o(ys)h(as)g(Hamiltonian)137 1710 y(\(4\).)k(Along)13 b(the)i(reference)h(ra)o(y)m(,)d(the)h(v)n (alue)f(of)h(Hamiltonia)o(n)d(\(5\))j(is)711 1785 y Fi(H)746 1791 y Fg(0)764 1785 y Fn(\()p Fi(x)804 1791 y Fh(m)835 1785 y Fi(;)7 b(p)875 1791 y Fh(n)897 1785 y Fn(\))12 b(=)g Fd(\000)p Fn(1)41 b Fi(;)521 b Fn(\(6\))137 1861 y(the)11 b(v)n(alues)f(of)f(the)i(\014rst{order)h(partial)d(deriv)n (ativ)o(es)h(with)g(resp)q(ect)j(to)d(the)h(spatial)e(co)q(ordinates) 137 1911 y(\(denoted)15 b(b)o(y)f(Roman)d(subscripts)16 b(follo)o(wing)11 b(a)j(comma\))d(of)i(Hamiltonian)e(\(5\))j(are)p 137 1950 1512 2 v 137 2043 2 93 v 758 2012 a Fi(H)793 2018 y Fg(0)p Fh(;i)844 2012 y Fn(=)p 888 1979 38 2 v 12 w Fi(H)926 2018 y Fg(0)p Fh(;i)1008 2012 y Fi(;)566 b Fn(\(7\))p 1647 2043 2 93 v 137 2045 1512 2 v 137 2101 a(the)12 b(v)n(alues)f(of)f(the)i(\014rst{order)g(partial)e(deriv)n (ativ)o(es)h(with)g(resp)q(ect)i(to)e(the)h(comp)q(onen)o(ts)f(of)f (the)137 2151 y(slo)o(wness)i(v)o(ector)g(\(denoted)h(b)o(y)e(Roman)e (sup)q(erscripts)14 b(follo)o(wing)9 b(a)i(comma\))e(of)i(Hamiltonian) 137 2201 y(\(5\))j(are)p 137 2240 V 137 2332 2 92 v 773 2302 a Fi(H)811 2282 y Fh(;i)808 2313 y Fg(0)846 2302 y Fn(=)p 890 2269 38 2 v 12 w Fi(H)928 2276 y Fh(;i)928 2312 y Fg(0)993 2302 y Fi(;)581 b Fn(\(8\))p 1647 2332 2 92 v 137 2334 1512 2 v 137 2409 a Fj(468)931 b Fl(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h(\(2006\))p eop %%Page: 469 7 469 6 bop 243 32 a Fo(Err)n(ors)13 b(due)g(to)g(the)f(A)o(nisotr)n (opic{Common{)o(R)n(ay)e(Appr)n(oximation)h(of)i(the)f(Coupling)g(R)n (ay)h(The)n(ory)231 142 y Fn(the)g(v)n(alues)f(of)g(the)h(second{order) h(partial)d(deriv)n(ativ)o(es)i(of)e(Hamiltonian)e(\(5\))k(with)f(resp) q(ect)i(to)231 192 y(the)h(spatial)e(co)q(ordinates)i(are)p 231 230 1512 2 v 231 322 2 93 v 721 291 a Fi(H)756 297 y Fg(0)p Fh(;ij)823 291 y Fn(=)p 867 258 38 2 v 12 w Fi(H)905 297 y Fg(0)p Fh(;ij)970 291 y Fd(\000)9 b Fn(3)p 1032 258 V Fi(H)1070 297 y Fg(0)p Fh(;i)p 1110 258 V 1110 291 a Fi(H)1148 297 y Fg(0)p Fh(;j)1233 291 y Fi(;)435 b Fn(\(9\))p 1742 322 2 93 v 231 324 1512 2 v 231 381 a(and)10 b(the)h(v)n(alues)e(of)g(the)i(second{order)g(partial)e(deriv) n(ativ)o(es)h(of)f(Hamiltonian)e(\(5\))i(with)h(resp)q(ect)231 432 y(to)k(the)h(comp)q(onen)o(ts)e(of)h(the)g(slo)o(wness)h(v)o(ector) f(are)p 231 470 V 231 561 2 92 v 753 531 a Fi(H)791 511 y Fh(;ij)788 542 y Fg(0)841 531 y Fn(=)p 885 498 38 2 v 12 w Fi(H)923 505 y Fh(;ij)923 542 y Fg(0)971 531 y Fd(\000)c Fn(3)p 1034 498 V Fi(H)1072 505 y Fh(;i)1072 542 y Fg(0)p 1095 498 V 1095 531 a Fi(H)1133 505 y Fh(;j)1133 542 y Fg(0)1202 531 y Fi(:)445 b Fn(\(10\))p 1742 561 2 92 v 231 563 1512 2 v 291 621 a(The)20 b(v)n(alue)f(of)h(Hamiltonian) d(\(3\),)k(corresp)q(onding)g(to)f(unit)f(S{w)o(a)o(v)o(e)h(eigen)o(v)o (ector)h Fi(g)1697 627 y Fh(i\013)1732 621 y Fn(,)231 671 y Fi(\013)12 b Fn(=)g(1)p Fi(;)7 b Fn(2)12 b(of)i(the)g (Christo\013el)g(matrix,)e(at)i(the)g(reference)i(ra)o(y)m(,)d(is)p 231 709 V 231 802 2 93 v 707 771 a Fi(H)742 777 y Fh(\013)777 771 y Fn(=)e Fd(\000)p Fn(\()p Fi(a)890 777 y Fh(ij)r(k)q(l)949 771 y Fi(g)969 777 y Fh(i\013)1004 771 y Fi(p)1025 777 y Fh(j)1043 771 y Fi(g)1063 777 y Fh(k)q(\013)1104 771 y Fi(p)1125 777 y Fh(l)1138 771 y Fn(\))1154 754 y Ff(\000)1185 743 y Fe(1)p 1185 748 15 2 v 1185 764 a(2)1248 771 y Fi(:)399 b Fn(\(11\))p 1742 802 2 93 v 231 804 1512 2 v 231 861 a(The)16 b(partial)e(deriv)n(ativ)o(es)h(of)f(Hamiltonian)e (\(11\))j(with)g(resp)q(ect)i(to)e(the)g(spatial)g(co)q(ordinates)231 911 y(are)p 231 941 V 231 1036 2 95 v 642 1003 a Fi(H)677 1009 y Fh(\013;i)733 1003 y Fn(=)d Fd(\000)814 987 y Fg(1)p 814 994 17 2 v 814 1017 a(2)836 1003 y Fn(\()p Fi(H)887 1009 y Fh(\013)910 1003 y Fn(\))926 986 y Fg(3)945 1003 y Fi(a)967 1009 y Fh(mj)r(k)q(l;i)1065 1003 y Fi(g)1085 1009 y Fh(m\013)1137 1003 y Fi(p)1158 1009 y Fh(j)1176 1003 y Fi(g)1196 1009 y Fh(k)q(\013)1238 1003 y Fi(p)1259 1009 y Fh(l)1313 1003 y Fi(:)334 b Fn(\(12\))p 1742 1036 2 95 v 231 1038 1512 2 v 231 1095 a(The)16 b(partial)f(deriv)n(ativ)o (es)g(of)g(Hamiltonian)e(\(11\))i(with)g(resp)q(ect)j(to)d(the)i(comp)q (onen)o(ts)e(of)g(the)231 1145 y(slo)o(wness)g(v)o(ector)g(are)p 231 1175 V 231 1266 2 91 v 696 1237 a Fi(H)734 1220 y Fh(;i)731 1247 y(\013)769 1237 y Fn(=)d Fd(\000)p Fn(\()p Fi(H)896 1243 y Fh(\013)920 1237 y Fn(\))936 1220 y Fg(3)954 1237 y Fi(a)976 1243 y Fh(mik)q(l)1049 1237 y Fi(g)1069 1243 y Fh(m\013)1122 1237 y Fi(g)1142 1243 y Fh(k)q(\013)1183 1237 y Fi(p)1204 1243 y Fh(l)1259 1237 y Fi(:)388 b Fn(\(13\))p 1742 1266 V 231 1268 1512 2 v 231 1326 a(The)16 b(v)n(alues)e(of)h (phase{space)h(deriv)n(ativ)o(es)f(\(12\))f(and)h(\(13\))f(are)i (calculated)f(at)f(the)i(reference)231 1376 y(ra)o(y)e(in)g(this)f(pap) q(er.)19 b(P)o(artial)13 b(deriv)n(ativ)o(es)h(\(13\))f(satisfy)h (Euler's)g(relation)846 1452 y Fi(H)884 1435 y Fh(;i)881 1462 y(\013)907 1452 y Fi(p)928 1458 y Fh(i)953 1452 y Fn(=)e Fd(\000)p Fi(H)1064 1458 y Fh(\013)1670 1452 y Fn(\(14\))231 1528 y(for)i(a)g(homogeneous)f(function)g(of)g(degree)j Fd(\000)p Fn(1)e(with)f(resp)q(ect)j(to)e Fi(p)1298 1534 y Fh(i)1312 1528 y Fn(.)291 1579 y(The)j(quan)o(tities)g(enclosed)h(ab) q(o)o(v)o(e)f(in)g(b)q(o)o(xes)g(are)h(necessary)h(for)e(estimating)e (the)j(errors)231 1629 y(due)d(to)f(the)g(anisotropic{common{ra)n(y)d (appro)o(ximation)g(of)i(the)i(coupling)e(ra)o(y)g(theory)m(.)291 1680 y(T)m(o)g(b)q(e)i(able)f(to)g(calculate)h(the)f(tra)o(v)o(el{time) f(p)q(erturbations)i(from)e(the)i(anisotropic)f(com-)231 1730 y(mon)g(ra)o(y)g(to)o(w)o(ards)h(the)g(isotropic{ra)o(y{theory)g (ra)o(y)m(,)e(w)o(e)i(also)f(in)o(tro)q(duce)i(homogeneous)e(Ha-)231 1780 y(miltonian)648 1832 y Fi(H)683 1838 y Fg(3)701 1832 y Fn(\()p Fi(x)741 1838 y Fh(m)772 1832 y Fi(;)7 b(p)812 1838 y Fh(n)834 1832 y Fn(\))12 b(=)g Fd(\000)p Fn([)p Fi(v)970 1838 y Fg(0)988 1832 y Fn(\()p Fi(x)1028 1838 y Fh(m)1060 1832 y Fn(\)])1088 1814 y Ff(\000)p Fg(1)1132 1832 y Fn(\()p Fi(p)1169 1838 y Fh(i)1183 1832 y Fi(p)1204 1838 y Fh(i)1217 1832 y Fn(\))1233 1814 y Ff(\000)1264 1803 y Fe(1)p 1264 1808 15 2 v 1264 1824 a(2)1670 1832 y Fn(\(15\))231 1900 y(for)i(the)h(reference)i(isotropic) d(medium.)j(Hamiltonian)11 b(\(15\))j(is)g(a)g(homogeneous)g(function)g (of)231 1950 y(the)j(min)o(us)d(\014rst)j(degree)g(with)e(resp)q(ect)j (to)e(the)g(slo)o(wness)h(v)o(ector.)24 b(The)17 b(partial)d(deriv)n (ativ)o(es)231 2000 y(of)g(Hamiltonian)c(\(15\))k(with)g(resp)q(ect)i (to)e(spatial)f(co)q(ordinates)h Fi(x)1271 2006 y Fh(i)1299 2000 y Fn(are)721 2082 y Fi(H)756 2088 y Fg(3)p Fh(;i)807 2082 y Fn(=)e Fi(v)871 2088 y Fg(0)p Fh(;i)918 2082 y Fn(\()p Fi(v)954 2088 y Fg(0)973 2082 y Fn(\))989 2065 y Ff(\000)p Fg(2)1034 2082 y Fn(\()p Fi(p)1071 2088 y Fh(k)1091 2082 y Fi(p)1112 2088 y Fh(k)1133 2082 y Fn(\))1149 2065 y Ff(\000)1180 2054 y Fe(1)p 1180 2059 V 1180 2075 a(2)1242 2082 y Fi(:)416 b Fn(\(16\))231 2159 y(The)16 b(partial)f(deriv)n(ativ)o(es)g(of)g(Hamiltonian)e(\(15\))i(with)g (resp)q(ect)j(to)d(the)i(comp)q(onen)o(ts)e(of)g(the)231 2209 y(slo)o(wness)g(v)o(ector)g(are)744 2260 y Fi(H)782 2240 y Fh(;i)779 2271 y Fg(3)817 2260 y Fn(=)d(\()p Fi(v)897 2266 y Fg(0)916 2260 y Fn(\))932 2243 y Ff(\000)p Fg(1)976 2260 y Fn(\()p Fi(p)1013 2266 y Fh(k)1034 2260 y Fi(p)1055 2266 y Fh(k)1075 2260 y Fn(\))1091 2243 y Ff(\000)1122 2232 y Fe(3)p 1122 2237 V 1122 2253 a(2)1143 2260 y Fi(p)1164 2266 y Fh(i)1220 2260 y Fi(:)438 b Fn(\(17\))231 2409 y Fl(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h(\(2006\))930 b Fj(469)p eop %%Page: 470 8 470 7 bop 685 32 a Fo(L.)13 b(Klime)n(\024)-18 b(s)14 b(and)e(P.)i(Bulant)361 142 y Fn(6.)24 b(P)m(ARAMETRIC)15 b(SYSTEM)i(OF)f(THE)g(HAMIL)m(TONIANS)241 192 y(AND)f(ERR)o(ORS)f(DUE)h (TO)g(THE)g(COMMON{RA)m(Y)g(APPR)o(O)o(XIMA)m(TIONS)196 294 y(W)m(e)d(consider)i(the)f(parametric)f(set)h Fi(H)8 b Fn(=)d Fi(H)s Fn(\()p Fi(x)927 300 y Fh(m)958 294 y Fi(;)i(p)998 300 y Fh(n)1020 294 y Fi(;)g(f)1059 300 y Fh(\013)1082 294 y Fn(\))13 b(of)f(Hamiltonians)e(parametrized)137 345 y(b)o(y)k Fi(f)215 351 y Fh(\013)239 345 y Fn(,)294 429 y Fi(H)s Fn(\()p Fi(x)372 435 y Fh(m)404 429 y Fi(;)7 b(p)444 435 y Fh(n)466 429 y Fi(;)g(f)505 435 y Fh(\013)528 429 y Fn(\))12 b(=)f Fi(H)634 435 y Fg(0)653 429 y Fn(\()p Fi(x)693 435 y Fh(m)724 429 y Fi(;)c(p)764 435 y Fh(n)786 429 y Fn(\))i(+)853 390 y Fc(X)872 477 y Fh(\013)913 429 y Fn([)p Fi(H)960 435 y Fh(\013)982 429 y Fn(\()p Fi(x)1022 435 y Fh(m)1054 429 y Fi(;)e(p)1094 435 y Fh(n)1116 429 y Fn(\))r Fd(\000)r Fi(H)1203 435 y Fg(0)1222 429 y Fn(\()p Fi(x)1262 435 y Fh(m)1293 429 y Fi(;)g(p)1333 435 y Fh(n)1355 429 y Fn(\)])k Fi(f)1414 435 y Fh(\013)1480 429 y Fi(;)83 b Fn(\(18\))137 542 y(where)16 b Fi(\013)6 b Fn(=)g(1)p Fi(;)h Fn(2)12 b(for)i(p)q(erturbations)i(to)o(w)o(ards)e (the)i(anisotropic{ra)o(y{theory)e(ra)o(ys.)20 b(If)14 b(w)o(e)g(are)137 593 y(also)g(in)o(terested)j(in)d(the)h(p)q (erturbations)h(to)o(w)o(ards)f(a)f(reference)j(isotropic)e(mo)q(del,)e Fi(\013)6 b Fn(=)g(1)p Fi(;)h Fn(2)p Fi(;)g Fn(3.)137 643 y(P)o(arameters)22 b Fi(f)384 649 y Fh(\013)430 643 y Fn(are)h(called)e Fm(p)n(erturb)n(ation)h(p)n(ar)n(ameters)g Fn(\()p Fm(Baumg\177)-21 b(artel,)24 b(1985)6 b Fn(;)26 b Fm(Mur)n(do)n(ck,)137 694 y(1999)6 b Fn(;)20 b Fm(Klime)n(\024)-19 b(s)18 b(and)h(Bulant,)h(2004)6 b Fn(\))19 b(or)e(mo)q(del)g (parameters)h(\()p Fm(T)m(ar)n(antola,)h(1987)6 b Fn(;)20 b Fm(Klime)n(\024)-19 b(s,)137 744 y(2002)6 b Fn(\).)20 b(P)o(arametric)14 b(system)g(\(18\))g(of)g(the)h(Hamiltonians)d(then)j (generates)h(parametric)d(sys-)137 795 y(tem)d Fi(\034)5 b Fn(\()p Fi(x)280 801 y Fh(m)312 795 y Fi(;)i(f)351 801 y Fh(\013)374 795 y Fn(\))k(of)g(the)h(tra)o(v)o(el{time)d (\014elds)j(corresp)q(onding)g(to)f(the)h(individual)e(Hamiltonia)o(ns) 137 845 y Fi(H)s Fn(\()p Fi(x)215 851 y Fh(m)246 845 y Fi(;)d(p)286 851 y Fh(n)308 845 y Fi(;)g(f)347 851 y Fh(\013)371 845 y Fn(\).)196 897 y(W)m(e)16 b(obtain)f(the)h (reference)j(Hamiltonia)o(n)13 b(for)j(\()p Fi(f)993 903 y Fg(1)1012 897 y Fi(;)7 b(f)1051 903 y Fg(2)1069 897 y Fi(;)g(f)1108 903 y Fg(3)1127 897 y Fn(\))h(=)g(\(0)p Fi(;)f Fn(0)p Fi(;)g Fn(0\),)14 b(the)j(Hamiltonian)137 948 y(corresp)q(onding)h(to)f(the)h(\014rst)g(anisotropic{ra)o (y{theory)e(S)h(w)o(a)o(v)o(e)g(for)g(\()p Fi(f)1304 954 y Fg(1)1323 948 y Fi(;)7 b(f)1362 954 y Fg(2)1380 948 y Fi(;)g(f)1419 954 y Fg(3)1438 948 y Fn(\))j(=)g(\(1)p Fi(;)d Fn(0)p Fi(;)g Fn(0\),)137 998 y(the)21 b(Hamiltonian)d(corresp)q (onding)k(to)e(the)i(second)g(anisotropic{ra)o(y{theory)e(S)h(w)o(a)o (v)o(e)f(for)137 1049 y(\()p Fi(f)173 1055 y Fg(1)192 1049 y Fi(;)7 b(f)231 1055 y Fg(2)249 1049 y Fi(;)g(f)288 1055 y Fg(3)307 1049 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(1)p Fi(;)g Fn(0\),)i(and)h(the)h(Hamiltonian)c(corresp)q(onding)12 b(to)e(the)h(isotropic{ra)o(y{theory)137 1099 y(ra)o(y)j(traced)h(in)e (the)h(reference)j(isotropic)d(mo)q(del)e(for)h(\()p Fi(f)1021 1105 y Fg(1)1041 1099 y Fi(;)7 b(f)1080 1105 y Fg(2)1098 1099 y Fi(;)g(f)1137 1105 y Fg(3)1155 1099 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(0)p Fi(;)g Fn(1\).)196 1152 y(The)15 b(T)m(a)o(ylor)e(expansion)h(with)g(resp)q(ect)j(to)d(p)q (erturbation)h(parameters)f Fi(f)1374 1158 y Fh(\013)1412 1152 y Fn(is)h(called)f(here)137 1202 y(the)f Fm(p)n(erturb)n(ation)f (exp)n(ansion)p Fn(.)19 b(The)13 b(second{order)g(p)q(erturbation)g (expansion)f(of)f(tra)o(v)o(el)h(time)137 1253 y(is)315 1307 y Fi(\034)5 b Fn(\()p Fi(x)378 1313 y Fh(m)409 1307 y Fi(;)i(f)448 1313 y Fh(\013)471 1307 y Fn(\))12 b Fd(\031)g Fi(\034)5 b Fn(\()p Fi(x)606 1313 y Fh(m)637 1307 y Fn(\))k(+)704 1267 y Fc(X)723 1354 y Fh(\013)771 1307 y Fi(\034)789 1313 y Fh(;\013)822 1307 y Fn(\()p Fi(x)862 1313 y Fh(m)894 1307 y Fn(\))p Fi(f)930 1313 y Fh(\013)963 1307 y Fn(+)1010 1290 y Fg(1)p 1010 1297 17 2 v 1010 1321 a(2)1038 1267 y Fc(X)1057 1354 y Fh(\013)1105 1267 y Fc(X)1125 1357 y Fh(\014)1172 1307 y Fi(\034)1190 1313 y Fh(;\013\014)1244 1307 y Fn(\()p Fi(x)1284 1313 y Fh(m)1315 1307 y Fn(\))p Fi(f)1351 1313 y Fh(\013)1375 1307 y Fi(f)1395 1313 y Fh(\014)1460 1307 y Fi(;)103 b Fn(\(19\))137 1420 y(where)11 b(the)f(Greek)h(subscripts)g(follo)o(wing)c(a)i(comma)e(denote)k (partial)e(deriv)n(ativ)o(es)g(with)g(resp)q(ect)137 1470 y(to)i(p)q(erturbation)g(parameters)g Fi(f)657 1476 y Fh(\013)681 1470 y Fn(,)f(hereinafter)i(called)f Fm(p)n(erturb)n (ation)g(derivatives)p Fn(.)17 b(Note)11 b(that)137 1521 y Fm(Klime)n(\024)-19 b(s)17 b(\(2002\))f Fn(refers)f(to)f(the)h(p)q (erturbation)f(deriv)n(ativ)o(es)g(brie\015y)g(as)g(the)h(\\p)q (erturbations".)137 1571 y(Here,)20 b(tra)o(v)o(el)e(time)f Fi(\034)5 b Fn(\()p Fi(x)535 1577 y Fh(m)566 1571 y Fn(\))19 b(and)f(its)g(p)q(erturbation)h(deriv)n(ativ)o(es)g Fi(\034)1226 1577 y Fh(;\013)1259 1571 y Fn(\()p Fi(x)1299 1577 y Fh(m)1331 1571 y Fn(\),)g Fi(\034)1396 1577 y Fh(;\013\014)1450 1571 y Fn(\()p Fi(x)1490 1577 y Fh(m)1521 1571 y Fn(\),)g(with)137 1622 y(argumen)o(ts)h Fi(f)366 1628 y Fh(\013)411 1622 y Fn(omitted,)g(corresp)q(ond)j(to)d(the)i(reference)h(system)d(of)h (ra)o(ys)f(\()p Fi(f)1450 1628 y Fg(1)1470 1622 y Fi(;)7 b(f)1509 1628 y Fg(2)1527 1622 y Fi(;)g(f)1566 1628 y Fg(3)1584 1622 y Fn(\))17 b(=)137 1672 y(\(0)p Fi(;)7 b Fn(0)p Fi(;)g Fn(0\).)196 1724 y(T)m(ra)o(v)o(el{time)g(appro)o (ximation)g(\(2\))j(corresp)q(onds)h(to)f(the)g(\014rst{order)h(part)f (of)f(p)q(erturbation)137 1775 y(expansion)20 b(\(19\),)h(see)h (Section)e(7.)37 b(The)21 b(error)g(of)f(tra)o(v)o(el{time)e(appro)o (ximation)f(\(2\))k(ma)o(y)137 1825 y(th)o(us)c(b)q(e)h(appro)o (ximated)d(b)o(y)i(the)g(quadratic)g(term)f(in)h(p)q(erturbation)h (expansion)e(\(19\))h(with)137 1876 y(Hamiltonians)11 b(\(18\).)196 1928 y(W)m(e)i(ma)o(y)e(no)o(w)i(express)h(the)g (estimates)f(of)g(the)g(tra)o(v)o(el{time)e(errors)k(due)e(to)g(the)h Fm(anisotr)n(o-)137 1978 y(pic{c)n(ommon{r)n(ay)c Fn(\(A)o(CR\))f Fm(appr)n(oximation)h Fn(\()o Fm(Bakker,)i(2002)6 b Fn(;)k Fm(Klime)n(\024)-19 b(s,)11 b(2006b)s Fn(\))f(of)e(anisotropic)137 2029 y(tra)o(v)o(el)i(times)f(in)g(terms)h(of)f(the)i(second{order)g(p) q(erturbation)g(deriv)n(ativ)o(es)f(of)f(tra)o(v)o(el)h(time.)15 b(The)137 2079 y(errors)h(of)f(tra)o(v)o(el{time)e(appro)o(ximation)f (\(2\))k(from)d(anisotropic)i(common)d(ra)o(y)j(\()p Fi(f)1460 2085 y Fg(1)1479 2079 y Fi(;)7 b(f)1518 2085 y Fg(2)1536 2079 y Fi(;)g(f)1575 2085 y Fg(3)1594 2079 y Fn(\))g(=)137 2130 y(\(0)p Fi(;)g Fn(0)p Fi(;)g Fn(0\))14 b(to)h(anisotropic{ra)o(y{theory)f(ra)o(y)h(\()p Fi(f)874 2136 y Fg(1)893 2130 y Fi(;)7 b(f)932 2136 y Fg(2)951 2130 y Fi(;)g(f)990 2136 y Fg(3)1008 2130 y Fn(\))g(=)g(\(1)p Fi(;)g Fn(0)p Fi(;)g Fn(0\))14 b(and)h(to)g(anisotropic{ra)o(y{)137 2180 y(theory)f(ra)o(y)g(\()p Fi(f)375 2186 y Fg(1)394 2180 y Fi(;)7 b(f)433 2186 y Fg(2)452 2180 y Fi(;)g(f)491 2186 y Fg(3)509 2180 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(1)p Fi(;)g Fn(0\))12 b(are)i(appro)o(ximately)p 137 2223 1512 2 v 137 2332 2 109 v 735 2285 a Fi(\016)r(\034)778 2267 y Fg(A)o(CR)773 2295 y Fh(\013)863 2285 y Fn(=)911 2257 y Fi(\034)929 2263 y Fh(;\013\013)p 911 2275 74 2 v 938 2313 a Fn(2)1031 2285 y Fi(;)522 b Fn(\(20\))p 1647 2332 2 109 v 137 2334 1512 2 v 137 2409 a Fj(470)931 b Fl(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h(\(2006\))p eop %%Page: 471 9 471 8 bop 243 32 a Fo(Err)n(ors)13 b(due)g(to)g(the)f(A)o(nisotr)n (opic{Common{)o(R)n(ay)e(Appr)n(oximation)h(of)i(the)f(Coupling)g(R)n (ay)h(The)n(ory)231 142 y Fn(where)k Fi(\013)7 b Fn(=)f(1)p Fi(;)h Fn(2.)20 b(Equation)14 b(\(20\))h(with)g Fi(\013)7 b Fn(=)f(3)15 b(analogously)e(yields)i(the)g(second{order)i(term)231 192 y(in)g(the)h(p)q(erturbation)g(expansion)g(from)d(the)j (anisotropic)f(common)e(ra)o(y)i(to)g(the)h(isotropic{)231 242 y(ra)o(y{theory)c(ra)o(y)g(traced)h(in)e(the)i(reference)h (isotropic)e(mo)q(del.)291 292 y(W)m(e)e(ma)o(y)f(also)h(use)i(the)f (second{order)i(p)q(erturbation)e(deriv)n(ativ)o(es)g(of)f(tra)o(v)o (el)h(time,)e(calcu-)231 342 y(lated)k(along)f(the)i(anisotropic)e (common)e(ra)o(y)m(,)i(to)h(estimate)f(the)i(accuracy)g(of)e(the)i Fm(isotr)n(opic{)231 392 y(c)n(ommon{r)n(ay)h Fn(\(ICR\))f Fm(appr)n(oximation)p Fn(.)24 b(The)16 b(estimates)g(of)f(errors)i(of)f (tra)o(v)o(el{time)e(appro)o(x-)231 442 y(imation)k(\(2\))h(from)f (isotropic)i(common)d(ra)o(y)i(\()p Fi(f)1026 448 y Fg(1)1045 442 y Fi(;)7 b(f)1084 448 y Fg(2)1103 442 y Fi(;)g(f)1142 448 y Fg(3)1160 442 y Fn(\))15 b(=)g(\(0)p Fi(;)7 b Fn(0)p Fi(;)g Fn(1\))18 b(to)h(anisotropic{ra)o(y{)231 493 y(theory)j(ra)o(y)e (\()p Fi(f)483 499 y Fg(1)502 493 y Fi(;)7 b(f)541 499 y Fg(2)560 493 y Fi(;)g(f)599 499 y Fg(3)617 493 y Fn(\))16 b(=)h(\(1)p Fi(;)7 b Fn(0)p Fi(;)g Fn(0\))19 b(and)i(to)f (anisotropic{ra)o(y{theory)g(ra)o(y)h(\()p Fi(f)1545 499 y Fg(1)1564 493 y Fi(;)7 b(f)1603 499 y Fg(2)1621 493 y Fi(;)g(f)1660 499 y Fg(3)1679 493 y Fn(\))16 b(=)231 543 y(\(0)p Fi(;)7 b Fn(1)p Fi(;)g Fn(0\))13 b(are)701 594 y Fi(\016)r(\034)744 577 y Fg(ICR)739 604 y Fh(\013)818 594 y Fn(=)866 578 y Fg(1)p 866 585 17 2 v 866 608 a(2)888 594 y Fi(\034)906 600 y Fh(;\013\013)970 594 y Fd(\000)d Fi(\034)1030 600 y Fh(;\013)p Fg(3)1089 594 y Fn(+)1136 578 y Fg(1)p 1136 585 V 1136 608 a(2)1157 594 y Fi(\034)1175 600 y Fh(;)p Fg(33)1262 594 y Fi(;)396 b Fn(\(21\))231 662 y(where)19 b Fi(\013)11 b Fn(=)g(1)p Fi(;)c Fn(2.)28 b(Error)19 b(estimates)e(\(21\))g(represen)o(t)j(the)e(con)o (tributions)g(of)f(the)h(quadratic)231 712 y(terms)e(in)f(p)q (erturbation)h(expansion)f(\(19\))h(from)d(the)j(isotropic)g(common)c (ra)o(y)k(to)f(the)h(aniso-)231 762 y(tropic{ra)o(y{theory)e(S{w)o(a)o (v)o(e)g(ra)o(ys.)291 813 y(T)m(o)i(calculate)i(the)g(\014rst{order)h (and)e(second{order)j(p)q(erturbation)e(deriv)n(ativ)o(es)f Fi(\034)1625 819 y Fh(;\013)1676 813 y Fn(and)231 863 y Fi(\034)249 869 y Fh(;\013\014)326 863 y Fn(of)k(tra)o(v)o(el)h (time,)h(w)o(e)g(need)g(the)g(\014rst{order)h(and)e(second{order)i (phase{space)f(and)231 913 y(p)q(erturbation)e(deriv)n(ativ)o(es)f(of)f (the)h(Hamiltonian.)33 b(The)20 b(phase{space)h(and)f(p)q(erturbation) 231 963 y(deriv)n(ativ)o(es)14 b(of)g(Hamiltonia)o(n)d(\(18\))j(in)f (the)i(reference)h(mo)q(del)c Fi(f)1249 969 y Fh(\013)1278 963 y Fn(=)5 b(0)14 b(are)867 1039 y Fi(H)902 1045 y Fh(;i)936 1039 y Fn(=)e Fi(H)1015 1045 y Fg(0)p Fh(;i)1097 1039 y Fi(;)561 b Fn(\(22\))872 1121 y Fi(H)910 1104 y Fh(;i)945 1121 y Fn(=)12 b Fi(H)1027 1102 y Fh(;i)1024 1133 y Fg(0)1092 1121 y Fi(;)566 b Fn(\(23\))851 1191 y Fi(H)886 1197 y Fh(;ij)936 1191 y Fn(=)12 b Fi(H)1015 1197 y Fg(0)p Fh(;ij)1112 1191 y Fi(;)546 b Fn(\(24\))856 1265 y Fi(H)894 1248 y Fh(;ij)945 1265 y Fn(=)12 b Fi(H)1027 1245 y Fh(;ij)1024 1276 y Fg(0)1107 1265 y Fi(;)551 b Fn(\(25\))818 1334 y Fi(H)853 1340 y Fh(;\013)898 1334 y Fn(=)12 b Fi(H)977 1340 y Fh(\013)1009 1334 y Fd(\000)d Fi(H)1085 1340 y Fg(0)1145 1334 y Fi(;)513 b Fn(\(26\))791 1403 y Fi(H)826 1409 y Fh(;i\013)882 1403 y Fn(=)12 b Fi(H)961 1409 y Fh(\013;i)1015 1403 y Fd(\000)d Fi(H)1091 1409 y Fg(0)p Fh(;i)1173 1403 y Fi(;)485 b Fn(\(27\))812 1478 y Fi(H)850 1461 y Fh(;i)847 1488 y(;\013)892 1478 y Fn(=)12 b Fi(H)974 1461 y Fh(;i)971 1488 y(\013)1006 1478 y Fd(\000)e Fi(H)1086 1458 y Fh(;i)1083 1489 y Fg(0)1151 1478 y Fi(;)507 b Fn(\(28\))879 1547 y Fi(H)914 1553 y Fh(;\013\014)979 1547 y Fn(=)11 b(0)42 b Fi(;)573 b Fn(\(29\))231 1615 y(where)18 b(the)f(Greek)f(subscripts)i(follo)o (wing)c(a)h(comma)f(denote)j(the)f(partial)g(deriv)n(ativ)o(es)g(with) 231 1665 y(resp)q(ect)h(to)c(p)q(erturbation)i(parameters)f Fi(f)903 1671 y Fh(\013)927 1665 y Fn(,)f(analogously)f(as)i(for)g(tra) o(v)o(el)f(time.)309 1790 y(7.)20 b(FIRST{ORDER)13 b(PER)m(TURBA)m (TION)h(EXP)m(ANSION)i(OF)f(TRA)-5 b(VEL)15 b(TIME)291 1891 y(The)20 b(\014rst{order)i(p)q(erturbation)f(deriv)n(ativ)o(es)f (of)g(tra)o(v)o(el)g(time)f(can)h(b)q(e)h(determined)f(b)o(y)231 1941 y(n)o(umerical)13 b(quadrature)h(\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)i(e)n(qs.)e(19)i(and)f(25)6 b Fn(\))717 2040 y Fi(\034)735 2046 y Fh(;\013)769 2040 y Fn(\()p Fi(\034)f Fn(\))12 b(=)g Fi(\034)898 2046 y Fh(;\013)931 2040 y Fn(\()p Fi(\034)970 2023 y Fg(0)989 2040 y Fn(\))d Fd(\000)1041 1984 y Fc(Z)1083 1994 y Fh(\034)1059 2078 y(\034)1077 2070 y Fe(0)1090 2040 y Fn(d)p Fi(\034)17 b(H)1183 2046 y Fh(;\013)1670 2040 y Fn(\(30\))231 2142 y(along)c(the)i(reference)h(ra)o(y)m(,)d(with)g(zero)i(initial)d (conditions)h Fi(\034)1186 2148 y Fh(;\013)1220 2142 y Fn(\()p Fi(\034)1259 2127 y Fg(0)1278 2142 y Fn(\))e(=)h(0)i(at)g (the)g(p)q(oin)o(t)f(source.)291 2193 y(Inserting)h(\(26\))g(for)f Fi(H)651 2199 y Fh(;\013)698 2193 y Fn(in)o(to)g(equation)h(\(30\))f (and)h(considering)g(\(6\),)g(w)o(e)g(obtain)579 2294 y Fi(\034)5 b Fn(\()p Fi(\034)g Fn(\))k(+)g Fi(\034)725 2300 y Fh(;\013)759 2294 y Fn(\()p Fi(\034)c Fn(\))12 b(=)f Fi(\034)5 b Fn(\()p Fi(\034)931 2277 y Fg(0)950 2294 y Fn(\))k(+)h Fi(\034)1035 2300 y Fh(;\013)1068 2294 y Fn(\()p Fi(\034)1107 2277 y Fg(0)1126 2294 y Fn(\))f Fd(\000)1178 2237 y Fc(Z)1221 2248 y Fh(\034)1197 2332 y(\034)1215 2323 y Fe(0)1228 2294 y Fn(d)p Fi(\034)16 b(H)1320 2300 y Fh(\013)1385 2294 y Fi(:)273 b Fn(\(31\))231 2409 y Fl(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h(\(2006\))930 b Fj(471)p eop %%Page: 472 10 472 9 bop 685 32 a Fo(L.)13 b(Klime)n(\024)-18 b(s)14 b(and)e(P.)i(Bulant)137 142 y Fn(This)e(is)g(the)h(\014rst{order)h (part)e(of)g(p)q(erturbation)g(expansion)h(\(19\))f(of)f(tra)o(v)o(el)h (time)f(from)g(aniso-)137 192 y(tropic)i(common)d(ra)o(y)i(\()p Fi(f)527 198 y Fg(1)546 192 y Fi(;)7 b(f)585 198 y Fg(2)603 192 y Fi(;)g(f)642 198 y Fg(3)661 192 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(0)p Fi(;)g Fn(0\))k(to)h(anisotropic{ra)o(y{theory)g(ra)o(ys) h(\()p Fi(f)1462 198 y Fg(1)1481 192 y Fi(;)7 b(f)1520 198 y Fg(2)1538 192 y Fi(;)g(f)1577 198 y Fg(3)1596 192 y Fn(\))e(=)137 242 y(\(1)p Fi(;)i Fn(0)p Fi(;)g Fn(0\))12 b(and)i(\()p Fi(f)399 248 y Fg(1)418 242 y Fi(;)7 b(f)457 248 y Fg(2)475 242 y Fi(;)g(f)514 248 y Fg(3)533 242 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(1)p Fi(;)g Fn(0\),)k(or)j(to)g(isotropic) g(common)d(ra)o(y)i(\()p Fi(f)1291 248 y Fg(1)1310 242 y Fi(;)7 b(f)1349 248 y Fg(2)1368 242 y Fi(;)g(f)1407 248 y Fg(3)1425 242 y Fn(\))e(=)g(\(0)p Fi(;)i Fn(0)p Fi(;)g Fn(1\).)196 293 y(Equation)k(\(31\))h(with)f(\(11\))h(for)f Fi(\013)g Fn(=)h(1)p Fi(;)7 b Fn(2)k(yields)g(tra)o(v)o(el{time)f (appro)o(ximation)f(\(2\))j(used)g(in)137 344 y(the)g(coupling)f(ra)o (y)h(theory)m(.)17 b(Equation)11 b(\(31\))h(with)f(\(15\))h(for)f Fi(\013)g Fn(=)h(3)g(represen)o(ts)i(the)e(\014rst{order)137 394 y(p)q(erturbation)k(expansion)f(from)e(the)j(anisotropic)f(common)d (ra)o(y)j(to)g(the)h(isotropic)f(common)137 444 y(ra)o(y)m(.)219 570 y(8.)20 b(D)o(YNAMIC)15 b(RA)m(Y)f(TRA)o(CING)g(AND)g(RA)m (Y{CENTRED)g(COORDINA)m(TES)196 671 y(W)m(e)j(de\014ne)i(the)f (matrices)f(of)g(the)h(partial)f(deriv)n(ativ)o(es)g(of)g(spatial)g(co) q(ordinates)h Fi(x)1551 677 y Fh(i)1582 671 y Fn(and)137 722 y(of)e(the)h(slo)o(wness{v)o(ector)h(comp)q(onen)o(ts)e Fi(p)813 728 y Fh(i)842 722 y Fn(=)g Fi(\034)908 728 y Fh(;i)949 722 y Fn(with)g(resp)q(ect)i(to)f(3)f Fm(r)n(ay)h(c)n(o)n (or)n(dinates)p Fn(:)23 b(ra)o(y)137 772 y(tak)o(e{o\013)14 b(parameters)g Fi(\015)529 778 y Fg(1)548 772 y Fn(,)f Fi(\015)594 778 y Fg(2)613 772 y Fn(,)h(and)f(indep)q(enden)o(t)j (parameter)d Fi(\015)1174 778 y Fg(3)1205 772 y Fn(=)f Fi(\034)18 b Fn(along)13 b(ra)o(ys,)602 868 y Fi(Q)635 874 y Fh(ia)678 868 y Fn(=)729 840 y Fi(@)r(x)777 846 y Fh(i)p 727 859 66 2 v 727 897 a Fi(@)r(\015)772 903 y Fh(a)839 868 y Fi(;)90 b(P)968 874 y Fh(ia)1011 868 y Fn(=)1060 840 y Fi(@)r(\034)1102 846 y Fh(;i)p 1060 859 67 2 v 1060 897 a Fi(@)r(\015)1105 903 y Fh(a)1172 868 y Fi(:)391 b Fn(\(32\))137 965 y(Matrices)11 b Fi(Q)336 971 y Fh(ia)377 965 y Fn(and)f Fi(P)481 971 y Fh(ia)523 965 y Fn(can)g(b)q(e)g(calculated)g(b)o(y)g(n)o(umerically)e(solving)h (a)h(set)g(of)g(linear)f(ordinary)137 1015 y(di\013eren)o(tial)20 b(equations)h(called)f(the)i Fm(dynamic)f(r)n(ay)g(tr)n(acing)g(e)n (quations)g Fn(\()1354 1005 y Fm(\024)1347 1015 y(Cerven)o(\023)-21 b(y,)24 b(1972)6 b Fn(\).)137 1065 y(Matrices)11 b Fi(Q)336 1071 y Fh(ia)378 1065 y Fn(and)f Fi(P)482 1071 y Fh(ia)523 1065 y Fn(describ)q(e,)i(b)o(y)e(de\014nition,)g(the)h(prop)q(erties)g (of)f(the)h(orthonomic)d(system)137 1115 y(of)i(ra)o(ys)h(corresp)q (onding)g(to)g(the)g(tra)o(v)o(el)f(time)f(under)j(consideration.)17 b(They)11 b(ma)o(y)d(b)q(e)j(expressed)137 1166 y(in)18 b(terms)f(of)h(their)g(initial)e(v)n(alues)i(and)g(the)g(paraxial)f(ra) o(y)g(propagator)h(matrix)e(\()1495 1155 y Fm(\024)1488 1166 y(Cerven)o(\023)-21 b(y,)137 1216 y(2001)6 b Fn(\).)196 1267 y(The)13 b(second{order)g(spatial)f(deriv)n(ativ)o(es)g Fi(\034)883 1273 y Fh(;ij)934 1267 y Fn(of)f(tra)o(v)o(el)h(time)f(can) h(b)q(e)h(expressed)h(in)e(terms)137 1317 y(of)h(matrices)h Fi(Q)384 1323 y Fh(ia)429 1317 y Fn(and)g Fi(P)537 1323 y Fh(ia)568 1317 y Fn(.)k(Equation)747 1395 y Fi(P)774 1401 y Fh(ia)817 1395 y Fn(=)12 b Fi(\034)879 1401 y Fh(;ij)929 1395 y Fi(Q)962 1401 y Fh(j)r(a)1575 1395 y Fn(\(33\))137 1472 y(is)i(a)f(direct)i(consequence)h(of)e(the)g(ab)q (o)o(v)o(e)g(de\014nitions)g(\(32\).)196 1528 y(Let)e(us)f(denote)h(b)o (y)f Fi(h)528 1513 y Fh(i)528 1538 y Fg(1)546 1528 y Fi(;)c(h)589 1513 y Fh(i)589 1538 y Fg(2)607 1528 y Fi(;)g(h)650 1513 y Fh(i)650 1538 y Fg(3)679 1528 y Fn(the)12 b(con)o(tra)o(v)n (arian)o(t)e(basis)h(v)o(ectors,)h(and)f(b)o(y)1381 1517 y(^)1380 1528 y Fi(h)1404 1534 y Fh(i)p Fg(1)1435 1528 y Fi(;)1454 1517 y Fn(^)1454 1528 y Fi(h)1478 1534 y Fh(i)p Fg(2)1507 1528 y Fi(;)1526 1517 y Fn(^)1526 1528 y Fi(h)1550 1534 y Fh(i)p Fg(3)1591 1528 y Fn(the)137 1578 y(co)o(v)n(arian)o(t)f(basis)h(v)o(ectors)h(of)f(the)h Fm(r)n(ay{c)n(entr)n(e)n(d)g(c)n(o)n(or)n(dinate)g(system)p Fn(.)17 b(The)11 b(basis)g(v)o(ectors)i(of)d(the)137 1628 y(ra)o(y{cen)o(tred)15 b(co)q(ordinate)g(system)f(are)g (calculated)h(along)d(the)j(reference)i(ra)o(y)m(.)h(The)c(\014rst)h(t) o(w)o(o)137 1678 y(con)o(tra)o(v)n(arian)o(t)e(basis)g(v)o(ectors)i Fi(h)657 1663 y Fh(i)657 1690 y(M)708 1678 y Fn(of)e(the)h(ra)o(y{cen)o (tred)h(co)q(ordinate)f(system)f(are)h(tangen)o(t)g(to)137 1733 y(the)i(w)o(a)o(v)o(efron)o(t,)g(the)g(third)g(con)o(tra)o(v)n (arian)o(t)f(basis)g(v)o(ector)i Fi(h)1103 1718 y Fh(i)1103 1743 y Fg(3)1136 1733 y Fn(=)e Fi(H)1221 1713 y Fh(;i)1218 1744 y Fg(0)1260 1733 y Fn(is)h(tangen)o(t)g(to)g(the)g(ra)o(y)m(.)137 1788 y(The)f(\014rst)h(t)o(w)o(o)f(co)o(v)n(arian)o(t)f(basis)h(v)o (ectors)815 1777 y(^)815 1788 y Fi(h)839 1794 y Fh(iM)902 1788 y Fn(of)g(the)g(ra)o(y{cen)o(tred)h(co)q(ordinate)g(system)e(are) 137 1838 y(p)q(erp)q(endicular)i(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)1194 1827 y(^)1193 1838 y Fi(h)1217 1844 y Fh(i)p Fg(3)1261 1838 y Fn(=)f Fi(p)1328 1844 y Fh(i)1357 1838 y Fn(is)h(p)q(erp)q(endicular)137 1889 y(to)i(the)h(w)o(a)o(v)o(efron)o(t.)28 b(The)17 b(con)o(tra)o(v)n(arian)o(t)g(basis)g(v)o(ectors)h(and)f(co)o(v)n (arian)o(t)g(basis)g(v)o(ectors)h(are)137 1939 y(m)o(utually)11 b(biorthonormal,)764 1991 y Fi(h)788 1974 y Fh(i)788 2002 y(a)808 1980 y Fn(^)808 1991 y Fi(h)832 1997 y Fh(ib)872 1991 y Fn(=)h Fi(\016)934 1997 y Fh(ab)1010 1991 y Fi(:)553 b Fn(\(34\))137 2060 y(The)13 b(equations)f(for)g(calculating)f(the)h (basis)h(v)o(ectors)g(of)e(the)i(ra)o(y{cen)o(tred)h(co)q(ordinate)e (system)137 2111 y(can)i(b)q(e)h(found,)e(e.g.,)f(in)i Fm(Klime)n(\024)-19 b(s)16 b(\(2006a\))r Fn(.)196 2162 y(The)g(quan)o(tities)g(co)o(v)n(arian)o(tly)f(transformed)g(from)f (spatial)h(co)q(ordinates)i Fi(x)1409 2168 y Fh(i)1438 2162 y Fn(to)f(the)h(lo)q(cal)137 2212 y(Cartesian)h(basis)f(of)g(the)h (ra)o(y{cen)o(tred)g(co)q(ordinate)g(system)f(are)g(mark)o(ed)g(b)o(y)g (a)g(tilde)12 b Fc(e)l Fn(.)28 b(In)137 2262 y(particular,)740 2304 y Fc(e)732 2315 y Fi(Q)765 2321 y Fh(ia)808 2315 y Fn(=)852 2304 y(^)852 2315 y Fi(h)876 2321 y Fh(mi)919 2315 y Fi(Q)952 2321 y Fh(ma)1043 2315 y Fi(;)520 b Fn(\(35\))137 2409 y Fj(472)931 b Fl(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h (\(2006\))p eop %%Page: 473 11 473 10 bop 243 32 a Fo(Err)n(ors)13 b(due)g(to)g(the)f(A)o(nisotr)n (opic{Common{)o(R)n(ay)e(Appr)n(oximation)h(of)i(the)f(Coupling)g(R)n (ay)h(The)n(ory)846 136 y Fc(e)838 147 y Fi(P)865 153 y Fh(ia)908 147 y Fn(=)f Fi(h)976 130 y Fh(m)976 157 y(i)1007 147 y Fi(P)1034 153 y Fh(ma)1125 147 y Fi(;)533 b Fn(\(36\))815 213 y Fc(e)-24 b Fi(\034)832 219 y Fh(;ij)882 213 y Fn(=)12 b Fi(h)950 196 y Fh(m)950 223 y(i)982 213 y Fi(\034)1000 219 y Fh(;mn)1062 213 y Fi(h)1086 196 y Fh(n)1086 223 y(j)1150 213 y Fi(;)508 b Fn(\(37\))834 279 y Fc(e)-24 b Fi(\034)851 285 y Fh(;i\013)908 279 y Fn(=)12 b Fi(h)976 262 y Fh(m)976 290 y(i)1007 279 y Fi(\034)1025 285 y Fh(;m\013)1130 279 y Fi(;)528 b Fn(\(38\))p 231 311 1512 2 v 231 755 2 444 v 827 362 a Fc(e)818 373 y Fi(H)853 379 y Fg(0)p Fh(;i)904 373 y Fn(=)12 b Fi(H)983 379 y Fg(0)p Fh(;m)1040 373 y Fi(h)1064 356 y Fh(m)1064 383 y(i)1137 373 y Fi(;)510 b Fn(\(39\))786 450 y Fc(e)776 460 y Fi(H)811 466 y Fg(0)p Fh(;ij)878 460 y Fn(=)12 b Fi(H)957 466 y Fg(0)p Fh(;mn)1035 460 y Fi(h)1059 443 y Fh(m)1059 470 y(i)1090 460 y Fi(h)1114 443 y Fh(n)1114 470 y(j)1178 460 y Fi(;)469 b Fn(\(40\))786 541 y Fc(e)776 551 y Fi(H)814 531 y Fh(;ij)811 562 y Fg(0)865 551 y Fn(=)11 b Fi(H)946 531 y Fh(;mn)943 562 y Fg(0)1009 540 y Fn(^)1008 551 y Fi(h)1032 557 y Fh(mi)1076 540 y Fn(^)1075 551 y Fi(h)1099 557 y Fh(nj)1179 551 y Fi(;)468 b Fn(\(41\))822 628 y Fc(e)813 639 y Fi(H)848 645 y Fh(\013;i)904 639 y Fn(=)12 b Fi(H)983 645 y Fh(\013;m)1045 639 y Fi(h)1069 621 y Fh(m)1069 649 y(i)1142 639 y Fi(;)505 b Fn(\(42\))835 715 y Fc(e)825 726 y Fi(H)863 709 y Fh(;i)860 736 y(\013)898 726 y Fn(=)12 b Fi(H)980 709 y Fh(;m)977 736 y(\013)1022 715 y Fn(^)1021 726 y Fi(h)1045 732 y Fh(mi)1130 726 y Fi(:)517 b Fn(\(43\))p 1742 755 V 231 757 1512 2 v 231 804 a(Since)15 b(w)o(e)f(consider)h(matrices)e Fi(Q)762 810 y Fh(ia)808 804 y Fn(and)g Fi(P)915 810 y Fh(ia)961 804 y Fn(de\014ned)i(for)e Fi(\015)1188 810 y Fg(3)1219 804 y Fn(=)f Fi(\034)5 b Fn(,)895 868 y Fc(e)887 879 y Fi(Q)920 885 y Fg(3)p Fh(J)971 879 y Fn(=)11 b(0)42 b Fi(:)581 b Fn(\(44\))231 958 y(The)10 b(2)r Fd(\002)r Fn(2)g(submatrices)633 948 y Fc(e)624 958 y Fi(Q)657 964 y Fh(I)r(J)707 958 y Fn(and)791 948 y Fc(e)783 958 y Fi(P)810 964 y Fh(I)r(J)859 958 y Fn(of)f(matrices)f(\(35\))h(and)h (\(36\))f(are)h(usually)e(calculated)h(b)o(y)231 1008 y(the)14 b Fm(dynamic)g(r)n(ay)f(tr)n(acing)g(e)n(quations)h(in)g(r)n (ay{c)n(entr)n(e)n(d)f(c)n(o)n(or)n(dinates)g Fn(\()p Fm(Klime)n(\024)-19 b(s,)12 b(1994,)j(e)n(q.)e(60)6 b Fn(\))674 1076 y(d)p 662 1095 46 2 v 662 1133 a(d)p Fi(\034)721 1094 y Fc(e)713 1104 y Fi(Q)746 1110 y Fh(I)r(J)798 1104 y Fn(=)851 1088 y Fc(e)851 1094 y(e)841 1104 y Fi(H)879 1084 y Fh(;I)879 1116 y Fg(0)p Fh(;K)953 1094 y Fc(e)945 1104 y Fi(Q)978 1110 y Fh(K)r(J)1040 1104 y Fn(+)1091 1094 y Fc(e)1081 1104 y Fi(H)1119 1084 y Fh(;I)r(K)1116 1115 y Fg(0)1193 1094 y Fc(e)1185 1104 y Fi(P)1212 1110 y Fh(K)r(J)1306 1104 y Fi(;)352 b Fn(\(45\))659 1190 y(d)p 647 1209 V 647 1247 a(d)p Fi(\034)706 1208 y Fc(e)698 1218 y Fi(P)725 1224 y Fh(I)r(J)777 1218 y Fn(=)11 b Fd(\000)862 1202 y Fc(e)862 1208 y(e)852 1218 y Fi(H)890 1224 y Fg(0)p Fh(;I)r(K)981 1208 y Fc(e)973 1218 y Fi(Q)1006 1224 y Fh(K)r(J)1068 1218 y Fd(\000)1119 1202 y Fc(e)1119 1208 y(e)1109 1218 y Fi(H)1147 1198 y Fh(;K)1147 1231 y Fg(0)p Fh(;I)1208 1208 y Fc(e)1200 1218 y Fi(P)1227 1224 y Fh(K)r(J)1321 1218 y Fi(:)337 b Fn(\(46\))231 1296 y(F)m(or)14 b(the)g(second{order)i(phase{space)f(deriv)n(ativ)o (es)709 1354 y Fc(e)709 1360 y(e)699 1371 y Fi(H)737 1377 y Fg(0)p Fh(;I)r(K)824 1371 y Fn(=)878 1360 y Fc(e)868 1371 y Fi(H)903 1377 y Fg(0)p Fh(;I)r(K)987 1371 y Fn(+)10 b(2)1059 1360 y Fc(e)1050 1371 y Fi(H)1085 1377 y Fg(0)p Fh(;I)1139 1360 y Fc(e)1129 1371 y Fi(H)1164 1377 y Fg(0)p Fh(;K)1264 1371 y Fi(;)394 b Fn(\(47\))241 1435 y Fc(e)241 1441 y(e)231 1452 y Fi(H)269 1432 y Fh(;I)269 1464 y Fg(0)p Fh(;K)345 1452 y Fn(and)439 1435 y Fc(e)440 1441 y(e)430 1452 y Fi(H)468 1432 y Fh(;I)r(K)468 1463 y Fg(0)544 1452 y Fn(=)604 1441 y Fc(e)594 1452 y Fi(H)632 1432 y Fh(;I)r(K)629 1463 y Fg(0)709 1452 y Fn(of)17 b(the)h(reference)i (Hamiltonian)15 b(in)i(ra)o(y{cen)o(tred)i(co)q(ordinates)231 1502 y(refer)c(to)f Fm(Klime)n(\024)-19 b(s)17 b(\(1994,)e(e)n(qs.)g (51a,)g(51b,)g(51c\))h Fn(or)e(to)g Fm(Klime)n(\024)-19 b(s)16 b(\(2006b,)g(e)n(qs.)e(60{62\))r Fn(.)291 1552 y(W)m(e)f(shall)g(th)o(us)i(express)h(the)e(equations)g(for)g (estimating)e(the)j(tra)o(v)o(el{time)d(errors)j(due)f(to)231 1601 y(the)19 b(common{ra)o(y)c(appro)o(ximations)g(of)i(the)i (coupling)e(ra)o(y)h(theory)g(in)g(terms)g(of)f(the)h(2)5 b Fd(\002)g Fn(2)231 1651 y(matrices)406 1641 y Fc(e)398 1651 y Fi(Q)431 1657 y Fh(I)r(J)485 1651 y Fn(and)574 1641 y Fc(e)565 1651 y Fi(P)592 1657 y Fh(I)r(J)646 1651 y Fn(in)13 b(ra)o(y{cen)o(tred)j(co)q(ordinates.)479 1776 y(9.)24 b(FIRST{ORDER)15 b(PER)m(TURBA)m(TION)h(DERIV)-5 b(A)m(TIVES)620 1825 y(OF)17 b(THE)h(TRA)-5 b(VEL{TIME)18 b(GRADIENT)291 1925 y(T)m(o)11 b(calculate)h(the)h(second{order)h(p)q (erturbation)f(deriv)n(ativ)o(es)f(of)g(tra)o(v)o(el)g(time,)f(w)o(e)h (need)h(to)231 1975 y(kno)o(w)h(at)g(least)g(the)g(ra)o(y{normal)e(pro) r(jection)834 2054 y Fi(\034)857 2035 y Ff(?)852 2065 y Fh(;i\013)909 2054 y Fn(=)g Fi(E)986 2035 y Ff(?)984 2065 y Fh(i)1014 2031 y(j)1032 2054 y Fi(\034)1050 2060 y Fh(;j)r(\013)1670 2054 y Fn(\(48\))231 2129 y(of)j(the)g (\014rst{order)i(p)q(erturbation)e(deriv)n(ativ)o(es)g Fi(\034)1025 2135 y Fh(;i\013)1085 2129 y Fn(of)g(the)g(tra)o(v)o (el{time)e(gradien)o(t)i(\()p Fm(Klime)n(\024)-19 b(s,)231 2186 y(2002,)16 b(e)n(qs.)f(54)h(and)f(55)6 b Fn(\).)19 b(The)c(ra)o(y{normal)d(pro)r(jection)i(matrix)e Fi(E)1329 2169 y Ff(?)1327 2196 y Fh(i)1358 2164 y(j)1389 2186 y Fn(can)i(b)q(e)h(expressed)i(in)231 2235 y(terms)d(of)f(the)i (\014rst)g(t)o(w)o(o)e(basis)h(v)o(ectors)h(of)e(the)i(ra)o(y{cen)o (tred)g(co)q(ordinate)f(system,)827 2315 y Fi(E)860 2296 y Ff(?)858 2325 y Fh(i)888 2291 y(j)917 2315 y Fn(=)962 2304 y(^)961 2315 y Fi(h)985 2321 y Fh(iM)1034 2315 y Fi(h)1058 2295 y Fh(j)1058 2327 y(M)1136 2315 y Fi(:)522 b Fn(\(49\))231 2409 y Fl(Stud.)11 b(Geoph)o(ys.)e(Geo)q(d.,)h(50)h (\(2006\))930 b Fj(473)p eop %%Page: 474 12 474 11 bop 685 32 a Fo(L.)13 b(Klime)n(\024)-18 b(s)14 b(and)e(P.)i(Bulant)137 142 y Fn(The)g(co)o(v)n(arian)o(t)f(transform)g (\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)h(e)n(q.)g(69)6 b Fn(\))741 206 y Fi(T)771 187 y Ff(?)765 216 y Fh(J)s(\013)821 206 y Fn(=)12 b Fi(\034)888 187 y Ff(?)883 216 y Fh(;i\013)936 206 y Fi(Q)969 212 y Fh(iJ)1575 206 y Fn(\(50\))137 271 y(of)g(ra)o(y{normal)f(deriv)n(ativ)o(es)i(\(48\))g(in)o(to)f(ra)o(y)h (co)q(ordinates)g(can)h(b)q(e)f(determined)g(b)o(y)g(n)o(umerical)137 320 y(quadrature)i(\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)h(e)n(q.)g (75)6 b Fn(\))p 137 350 1512 2 v 137 481 2 132 v 569 424 a Fi(T)599 405 y Ff(?)593 435 y Fh(J)s(\013)638 424 y Fn(\()p Fi(\034)f Fn(\))11 b(=)h Fi(T)778 405 y Ff(?)772 435 y Fh(J)s(\013)817 424 y Fn(\()p Fi(\034)856 407 y Fg(0)875 424 y Fn(\))d(+)927 368 y Fc(Z)969 378 y Fh(\034)945 462 y(\034)963 454 y Fe(0)976 424 y Fn(d)p Fi(\034)j(K)1067 405 y Ff(?)1064 435 y Fh(i\013)1100 424 y Fi(Q)1133 430 y Fh(iJ)1565 424 y Fn(\(51\))p 1647 481 V 137 483 1512 2 v 137 529 a(along)g(the)h(reference)i(ra)o(y)m(,)d(with)g(zero)i (initial)d(conditions)h Fi(T)1095 512 y Ff(?)1089 540 y Fh(J)s(\013)1134 529 y Fn(\()p Fi(\034)1173 514 y Fg(0)1192 529 y Fn(\))f(=)h(0)h(at)f(the)i(p)q(oin)o(t)e(source.)196 578 y(The)h(ra)o(y{normal)d(in)o(tegration)i(k)o(ernel)h(for)f(the)h(p) q(erturbation)h(deriv)n(ativ)o(es)e(of)g(the)h(tra)o(v)o(el{)137 627 y(time)g(gradien)o(t)g(is)h(\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)h(e)n(q.)g(80)6 b Fn(\))474 692 y Fi(K)512 673 y Ff(?)509 702 y Fh(i\013)556 692 y Fn(=)12 b Fi(E)633 673 y Ff(?)631 702 y Fh(i)661 669 y(a)688 692 y Fn(\()p Fd(\000)p Fi(H)771 698 y Fh(;a\013)832 692 y Fn(+)d Fi(H)908 698 y Fh(;a)945 692 y Fi(H)983 675 y Fh(;r)980 702 y(;\013)1020 692 y Fi(p)1041 698 y Fh(r)1059 692 y Fn(\))g Fd(\000)h Fi(H)1164 675 y Fh(;r)1161 702 y(;\013)1201 692 y Fi(\034)1224 673 y Ff(?)1219 702 y Fh(;r)q(i)1300 692 y Fi(;)263 b Fn(\(52\))137 757 y(where)705 806 y Fi(\034)728 787 y Ff(?)723 816 y Fh(;ij)774 806 y Fn(=)12 b Fi(E)851 787 y Ff(?)849 816 y Fh(i)879 783 y(a)899 806 y Fi(E)932 787 y Ff(?)930 816 y Fh(j)960 783 y(b)977 806 y Fi(\034)995 812 y Fh(;ab)1575 806 y Fn(\(53\))137 866 y(is)18 b(the)i(ra)o (y{normal)c(pro)r(jection)j(of)f(the)h(second{order)h(spatial)e(deriv)n (ativ)o(es)g Fi(\034)1436 872 y Fh(;ij)1494 866 y Fn(of)g(tra)o(v)o(el) 137 915 y(time.)196 965 y(Inserting)g(equations)f(\(22\),)g(\(27\),)g (\(28\))g(with)g(\(14\))g(and)g(\(34\),)g(and)g(\(53\))g(in)o(to)f (\(52\),)i(w)o(e)137 1014 y(arriv)o(e)c(at)420 1063 y Fi(K)458 1044 y Ff(?)455 1074 y Fh(i\013)502 1063 y Fn(=)e Fd(\000)p Fi(E)611 1044 y Ff(?)609 1074 y Fh(i)639 1040 y(a)666 1063 y Fn(\()p Fi(H)717 1069 y Fh(\013;a)778 1063 y Fn(+)d Fi(H)854 1069 y Fh(\013)877 1063 y Fi(H)912 1069 y Fg(0)p Fh(;a)958 1063 y Fn(\))h Fd(\000)f Fi(H)1063 1046 y Fh(;r)1060 1074 y(\013)1091 1063 y Fi(E)1124 1044 y Ff(?)1122 1074 y Fh(r)1152 1040 y(a)1172 1063 y Fi(\034)1190 1069 y Fh(;ab)1235 1063 y Fi(E)1268 1044 y Ff(?)1266 1074 y Fh(i)1296 1040 y(b)1354 1063 y Fi(:)209 b Fn(\(54\))137 1123 y(Multiplying)11 b(equation)h(\(54\))g(b)o(y)h(the)g(\014rst)g(t)o (w)o(o)f(columns)g Fi(Q)1098 1129 y Fh(iJ)1145 1123 y Fn(of)g(the)h(matrix)e(of)h(geometrical)137 1172 y(spreading,)j (inserting)h(equation)f(\(49\),)g(and)g(considering)g(transforms)g (\(35\),)g(\(36\),)g(\(37\))g(and)137 1222 y(relations)f(\(33\),)f (\(34\))h(and)f(\(44\),)g(w)o(e)h(arriv)o(e)g(at)p 137 1251 V 137 1344 2 93 v 464 1313 a Fi(K)502 1294 y Ff(?)499 1323 y Fh(i\013)535 1313 y Fi(Q)568 1319 y Fh(iJ)614 1313 y Fn(=)e Fd(\000)p Fn(\()716 1302 y Fc(e)706 1313 y Fi(H)741 1319 y Fh(\013;I)800 1313 y Fn(+)e Fi(H)877 1319 y Fh(\013)910 1302 y Fc(e)900 1313 y Fi(H)935 1319 y Fg(0)p Fh(;I)980 1313 y Fn(\))1011 1302 y Fc(e)1003 1313 y Fi(Q)1036 1319 y Fh(I)r(J)1085 1313 y Fd(\000)1136 1302 y Fc(e)1127 1313 y Fi(H)1165 1296 y Fh(;I)1162 1323 y(\013)1202 1302 y Fc(e)1194 1313 y Fi(P)1221 1319 y Fh(I)r(J)1302 1313 y Fi(;)251 b Fn(\(55\))p 1647 1344 V 137 1346 1512 2 v 137 1391 a(where)267 1381 y Fc(e)259 1391 y Fi(Q)292 1397 y Fh(I)r(J)347 1391 y Fn(and)438 1381 y Fc(e)430 1391 y Fi(P)457 1397 y Fh(I)r(J)512 1391 y Fn(are)16 b(the)g(2)s Fd(\002)s Fn(2)h(submatrices)e(of)g(matrices)g (\(35\))h(and)f(\(36\).)23 b(The)16 b(2)s Fd(\002)s Fn(2)137 1446 y(matrices)312 1436 y Fc(e)304 1446 y Fi(Q)337 1452 y Fh(I)r(J)392 1446 y Fn(and)481 1436 y Fc(e)473 1446 y Fi(P)500 1452 y Fh(I)r(J)555 1446 y Fn(are)f(usually)e(calculated)i (b)o(y)g(the)g(dynamic)e(ra)o(y)h(tracing)h(equations)137 1496 y(\(45\))f(and)f(\(46\))h(in)f(ra)o(y{cen)o(tred)j(co)q (ordinates.)196 1545 y(Inserting)c(equations)g(\(48\))g(and)g(\(49\))f (in)o(to)g(\(50\))h(while)f(considering)h(transforms)f(\(35\))h(and)137 1594 y(\(38\),)h(w)o(e)h(arriv)o(e)g(at)730 1649 y Fi(T)760 1630 y Ff(?)754 1660 y Fh(J)s(\013)811 1649 y Fn(=)e Fc(e)-24 b Fi(\034)872 1655 y Fh(;I)r(\013)938 1639 y Fc(e)930 1649 y Fi(Q)963 1655 y Fh(I)r(J)1044 1649 y Fi(:)519 b Fn(\(56\))137 1709 y(The)20 b(\014rst{order)i(p)q (erturbation)e(deriv)n(ativ)o(es)g(of)f(the)i(tra)o(v)o(el{time)d (gradien)o(t)i(expressed)i(in)137 1758 y(ra)o(y{cen)o(tred)15 b(co)q(ordinates)g(are)f(then)p 137 1785 V 137 1878 2 93 v 728 1847 a Fc(e)-24 b Fi(\034)745 1853 y Fh(;I)r(\013)807 1847 y Fn(=)12 b Fi(T)881 1828 y Ff(?)875 1857 y Fh(J)s(\013)928 1837 y Fc(e)920 1847 y Fi(Q)953 1829 y Ff(\000)p Fg(1)953 1859 y Fh(J)s(I)1039 1847 y Fi(;)514 b Fn(\(57\))p 1647 1878 V 137 1880 1512 2 v 137 1926 a(where)265 1915 y Fc(e)257 1926 y Fi(Q)290 1908 y Ff(\000)p Fg(1)290 1938 y Fh(J)s(I)347 1926 y Fn(are)14 b(the)g(comp)q(onen)o(ts)f(of)g(the)h (matrix)e(in)o(v)o(erse)i(to)f(2)q Fd(\002)q Fn(2)i(matrix)1389 1915 y Fc(e)1380 1926 y Fi(Q)1413 1932 y Fh(I)r(J)1467 1926 y Fn(calculated)137 1975 y(b)o(y)f(dynamic)e(ra)o(y)h(tracing)h (\(45\))g(and)g(\(46\))f(in)h(ra)o(y{cen)o(tred)h(co)q(ordinates.)196 2030 y(The)g(v)n(alue)f(of)i Fc(e)-24 b Fi(\034)457 2036 y Fh(;I)r(\013)522 2030 y Fn(at)15 b(the)g(p)q(oin)o(t)f(source,)i (where)g Fi(T)1046 2013 y Ff(?)1040 2042 y Fh(J)s(\013)1085 2030 y Fn(\()p Fi(\034)1124 2015 y Fg(0)1143 2030 y Fn(\))d(=)g(0)i (and)1343 2020 y Fc(e)1334 2030 y Fi(Q)1367 2036 y Fh(I)r(J)1407 2030 y Fn(\()p Fi(\034)1446 2015 y Fg(0)1465 2030 y Fn(\))e(=)g(0,)i (can)137 2080 y(b)q(e)k(obtained)f(from)f(equation)h(\(57\))g(b)o(y)g (the)h(l'Hospital)e(rule)h(with)g(resp)q(ect)j(to)d Fi(\034)24 b Fd(!)19 b Fi(\034)1587 2065 y Fg(0)1605 2080 y Fn(+.)137 2135 y(Equation)13 b(\(45\))h(and)g(deriv)n(ativ)o(e)f(\(55\))h(of)f (equation)g(\(51\))h(with)1170 2124 y Fc(e)1162 2135 y Fi(Q)1195 2141 y Fh(I)r(J)1246 2135 y Fn(=)e(0)i(then)g(yield)p 137 2164 V 137 2256 2 93 v 676 2226 a Fc(e)-24 b Fi(\034)693 2232 y Fh(;I)r(\013)744 2226 y Fn(\()p Fi(\034)783 2208 y Fg(0)802 2226 y Fn(\))11 b(=)h Fd(\000)914 2215 y Fc(e)905 2226 y Fi(G)938 2232 y Fh(I)r(J)988 2215 y Fc(e)978 2226 y Fi(H)1016 2208 y Fh(;J)1013 2236 y(\013)1091 2226 y Fi(;)462 b Fn(\(58\))p 1647 2256 V 137 2258 1512 2 v 137 2304 a(where)265 2293 y Fc(e)257 2304 y Fi(G)290 2310 y Fh(I)r(J)344 2304 y Fn(is)13 b(the)i(in)o(v)o(erse)f(matrix)e (to)788 2293 y Fc(e)780 2304 y Fi(G)813 2289 y Fh(I)r(J)865 2304 y Fn(=)918 2293 y Fc(e)908 2304 y Fi(H)946 2284 y Fh(;I)r(J)943 2315 y Fg(0)996 2304 y Fn(.)137 2409 y Fj(474)931 b Fl(Stud.)10 b(Geoph)o(ys.)g(Geo)q(d.,)g(50)h(\(2006\))p eop %%Page: 475 13 475 12 bop 243 32 a Fo(Err)n(ors)13 b(due)g(to)g(the)f(A)o(nisotr)n (opic{Common{)o(R)n(ay)e(Appr)n(oximation)h(of)i(the)f(Coupling)g(R)n (ay)h(The)n(ory)255 142 y Fn(10.)19 b(SECOND{ORDER)13 b(PER)m(TURBA)m(TION)h(DERIV)-5 b(A)m(TIVES)14 b(OF)h(TRA)-5 b(VEL)14 b(TIME)291 241 y(The)h(second{order)j(p)q(erturbation)e(deriv) n(ativ)o(es)f(of)g(tra)o(v)o(el)g(time)f(can)i(b)q(e)g(determined)f(b)o (y)231 291 y(n)o(umerical)e(quadrature)h(\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)i(e)n(qs.)e(19)i(and)f(20)6 b Fn(\))p 231 329 1512 2 v 231 461 2 132 v 687 404 a Fi(\034)705 410 y Fh(;\013\014)759 404 y Fn(\()p Fi(\034)f Fn(\))12 b(=)g Fi(\034)888 410 y Fh(;\013\014)942 404 y Fn(\()p Fi(\034)981 387 y Fg(0)999 404 y Fn(\))e(+)1052 347 y Fc(Z)1094 358 y Fh(\034)1070 442 y(\034)1088 433 y Fe(0)1101 404 y Fn(d)p Fi(\034)16 b(K)1193 410 y Fh(\013\014)1659 404 y Fn(\(59\))p 1742 461 V 231 463 1512 2 v 231 517 a(along)c(the)h(reference)i(ra)o(y)m(,)c(with)i(zero)g(initial)e (conditions)h Fi(\034)1176 523 y Fh(;\013\014)1230 517 y Fn(\()p Fi(\034)1269 502 y Fg(0)1287 517 y Fn(\))g(=)g(0)g(at)g(the)h (p)q(oin)o(t)f(source.)291 567 y(F)m(or)19 b(homogeneous)h(Hamiltonia)o (ns)e(of)h(the)i(min)o(us)e(\014rst)i(degree)g(with)f(resp)q(ect)j(to)d (the)231 617 y(slo)o(wness)13 b(v)o(ector,)g(the)g(in)o(tegration)e(k)o (ernel)i(for)e(the)i(second{order)h(p)q(erturbation)e(deriv)n(ativ)o (es)231 667 y(of)i(tra)o(v)o(el)f(time)g(can)h(b)q(e)h(expressed)h(as)e (\()p Fm(Klime)n(\024)-19 b(s,)14 b(2002,)h(e)n(q.)g(64)6 b Fn(\))524 740 y Fi(K)559 746 y 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Fi(\034)1216 919 y Ff(?)1211 948 y Fh(;i\013)1257 938 y Fi(\034)1280 919 y Ff(?)1275 948 y Fh(;j)r(\014)1364 938 y Fi(:)294 b Fn(\(61\))231 1012 y(Inserting)19 b(relations)e(\(48\))h(and)f (\(49\),)h(and)f(considering)h(transforms)f(\(38\),)h(\(41\))f(and)h (\(43\),)231 1061 y(equation)c(\(61\))g(can)g(b)q(e)g(expressed)i(in)e (ra)o(y{cen)o(tred)h(co)q(ordinates)g(as)p 231 1100 V 231 1198 2 99 v 574 1161 a Fi(K)609 1167 y Fh(\013\014)664 1161 y Fn(=)d Fd(\000)750 1151 y Fc(e)740 1161 y Fi(H)778 1144 y Fh(;I)775 1172 y(\013)808 1161 y Fc(e)-24 b Fi(\034)825 1167 y Fh(;I)r(\014)884 1161 y Fd(\000)935 1151 y Fc(e)925 1161 y Fi(H)963 1141 y Fh(;I)960 1174 y(\014)993 1161 y Fc(e)g Fi(\034)1010 1167 y Fh(;I)r(\013)1070 1161 y Fd(\000)1121 1151 y Fc(e)1112 1161 y Fi(H)1150 1141 y Fh(;I)r(J)1147 1172 y Fg(0)1201 1161 y Fc(e)f Fi(\034)1217 1167 y Fh(;I)r(\013)1269 1161 y Fc(e)h Fi(\034)1286 1167 y Fh(;J)s(\014)1381 1161 y Fi(:)266 b Fn(\(62\))p 1742 1198 V 231 1200 1512 2 v 291 1255 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