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Fi(denotes)f(a)g(phase)g(v)o(elo)q(cit)o(y)h(in)f(the) h(unp)q(erturb)q(ed)e(medium.)29 b(F)l(or)19 b Ff(m)f Fi(=)h Ff(n)g Fi(=)f(3,)i(w)o(e)g(ha)o(v)o(e)0 1137 y Ff(c)22 1144 y Fj(0)55 1137 y Fi(=)15 b Ff(\013)j Fi(and)e(for)h Ff(m)e Fi(=)g Ff(n)g Fi(=)g(1)j(or)f(2,)g(w)o(e)g(ha)o(v)o(e)g Ff(c)918 1144 y Fj(0)951 1137 y Fi(=)e Ff(\014)s Fi(.)24 b(Note)18 b(that)g(in)f (Eq.\(7\),)h(the)g(elemen)o(ts)e(of)i(the)0 1195 y(matrix)f Ff(B)201 1202 y Fe(mn)271 1195 y Fi(are)h(expressed)f(in)g(terms)h(of)g (elastic)g(parameters)e(of)i(a)g(w)o(eakly)g(anisotropic)f(medium)0 1253 y(instead)f(of)h(the)f(p)q(erturbations)f(\001)p Ff(a)694 1260 y Fe(ij)r(k)q(l)752 1253 y Fi(.)75 1459 y Fg(2.2)j(F)-5 b(orm)m(ulae)18 b(for)f(a)i(p)r(erturb)r(ed)f(phase)g(norm)o(al)75 1548 y Fi(In)11 b(man)o(y)g(situations)f(p)q(erturbations)g(of)h(phase)g(v)o (elo)q(cities)h(or)f(of)g(p)q(olarization)g(v)o(ectors)g(are)g(sough)o(t)0 1606 y(for)16 b(the)g(phase)f(normal,)g(whic)o(h)g(in)g(the)h(anisotropic)f (medium)f(is)i(di\013eren)o(t)f(from)g(the)i(phase)e(normal)0 1664 y(in)j(the)h(bac)o(kground)e(medium.)26 b(F)l(or)18 b(example,)g(w)o(e)h (often)g(seek)g(quan)o(tities)f(on)g(a)h(ra)o(y)l(.)27 b(Since)18 b(in)g(an)0 1722 y(anisotropic)c(medium,)g(the)i(ra)o(y)g(direction)e(and)i (the)g(phase)e(normal)h(generally)g(di\013er,)g(suc)o(h)f(a)i(phase)0 1780 y(normal)f(m)o(ust)g(b)q(e)i(sough)o(t)f(whic)o(h)f(yields)h(the)h (required)e(ra)o(y)h(direction.)75 1868 y(Let)g(us)f(assume)f(that)i(the)g (deviation)g(\001)p Ff(n)878 1875 y Fe(i)907 1868 y Fi(of)g(the)g(phase)e (normal)g(from)h(its)h(v)m(alue)g(in)f(the)h(bac)o(kg-)0 1926 y(round)e(isotropic)g(medium)f(is)i(a)g(\014rst-order)e(quan)o(tit)o(y)l(.)21 b(The)15 b(p)q(erturbation)f(\001\000)1533 1933 y Fe(j)r(k)1584 1926 y Fi(of)i(the)f(Christo\013el)0 1985 y(matrix)h(has,)g(in)g(this)g (case,)g(the)h(form)501 2087 y(\001\000)574 2094 y Fe(j)r(k)623 2087 y Fi(=)d(\001)p Ff(a)744 2094 y Fe(ij)r(k)q(l)802 2087 y Ff(n)832 2094 y Fe(i)846 2087 y Ff(n)876 2094 y Fe(l)899 2087 y Fi(+)d Ff(a)975 2066 y Fj(0)975 2099 y Fe(ij)r(k)q(l)1034 2087 y Fi(\001)p Ff(n)1106 2094 y Fe(i)1119 2087 y Ff(n)1149 2094 y Fe(l)1173 2087 y Fi(+)g Ff(a)1249 2066 y Fj(0)1249 2099 y Fe(ij)r(k)q(l)1307 2087 y Ff(n)1337 2094 y Fe(i)1351 2087 y Fi(\001)p Ff(n)1423 2094 y Fe(l)1435 2087 y Ff(:)437 b Fi(\(8\))0 2189 y(Let)21 b(us)f(presen)o(t)g(form)o(ulae)f(for)h(the)h(quan)o(tities)f Ff(B)990 2196 y Fe(mn)1062 2189 y Fi(corresp)q(onding)f(to)i(suc)o(h)e(a)i(p) q(erturbation)e(of)0 2248 y(the)i(Christo\013el)f(matrix.)34 b(In)20 b(order)g(to)h(distinguish)e(quan)o(tities)h(related)g(to)h(the)g(p)q (erturbation)f(of)0 2306 y(the)c(Christo\013el)f(matrix)h(\(8\))h(from)e(the) h(quan)o(tities)g(sp)q(eci\014ed)f(for)h(a)g(\014xed)f(direction)h(of)g Ff(n)1731 2313 y Fe(i)1761 2306 y Fi(and)f(th)o(us)0 2364 y(related)20 b(to)i(Eq.\(7\),)g(w)o(e)f(denote)f(the)h(former)f(quan)o(tities)g(b)o(y)h(a) g(bar.)33 b(F)l(rom)20 b(Eqs.\(5\))h(and)f(\(8\),)i(w)o(e)0 2422 y(simply)15 b(get)447 2467 y(\026)436 2480 y Ff(B)474 2487 y Fj(11)522 2480 y Fi(=)f Ff(B)613 2487 y Fj(11)648 2480 y Ff(;)732 2467 y Fi(\026)720 2480 y Ff(B)758 2487 y Fj(22)807 2480 y Fi(=)f Ff(B)897 2487 y Fj(22)932 2480 y Ff(;)1016 2467 y Fi(\026)1004 2480 y Ff(B)1042 2487 y Fj(33)1091 2480 y Fi(=)g Ff(B)1181 2487 y Fj(33)1216 2480 y Ff(;)1300 2467 y Fi(\026)1288 2480 y Ff(B)1326 2487 y Fj(12)1375 2480 y Fi(=)h Ff(B)1466 2487 y Fj(12)1501 2480 y Ff(;)293 2550 y Fi(\026)282 2563 y Ff(B)320 2570 y Fj(13)368 2563 y Fi(=)g Ff(B)459 2570 y Fj(13)505 2563 y Fi(+)d(\()p Ff(\013)606 2542 y Fj(2)636 2563 y Fc(\000)g Ff(\014)717 2542 y Fj(2)735 2563 y Fi(\)\()p Ff(e)796 2570 y Fj(1)p Fe(i)828 2563 y Fi(\001)p Ff(n)900 2570 y Fe(i)913 2563 y Fi(\))p Ff(;)1016 2550 y Fi(\026)1004 2563 y Ff(B)1042 2570 y Fj(23)1091 2563 y Fi(=)i Ff(B)1181 2570 y Fj(23)1227 2563 y Fi(+)e(\()p Ff(\013)1328 2542 y Fj(2)1358 2563 y Fc(\000)g Ff(\014)1439 2542 y Fj(2)1458 2563 y Fi(\)\()p Ff(e)1519 2570 y Fj(2)p Fe(i)1550 2563 y Fi(\001)p Ff(n)1622 2570 y Fe(i)1635 2563 y Fi(\))p Ff(;)218 b Fi(\(9\))938 2687 y(131)p eop %%Page: 132 4 132 3 bop 0 -58 a Fi(As)18 b(w)o(e)f(can)h(see,)f(the)h(form)o(ulae)e(\(3\))i (for)g(the)g(p)q(erturbation)e(of)i(phase)f(v)o(elo)q(cit)o(y)h(remain)e(unc) o(hanged)0 0 y(b)o(y)24 b(the)h(phase)f(normal)f(p)q(erturbation)h(since)g (the)h(diagonal)f(elemen)o(ts)g(of)h(the)g(matrix)f Ff(B)1807 7 y Fe(mn)1883 0 y Fi(are)0 58 y(unc)o(hanged,)13 b(see)j(\(9\).)22 b(F)l(orm)o(ulae)13 b(\(4\))j(for)f(the)g(p)q(erturbation)f(of)i(the)f(p)q (olarization)f(v)o(ector,)i(ho)o(w)o(ev)o(er,)0 116 y(are)e(a\013ected)h(b)o (y)f(the)h(p)q(erturbation)e(of)i(the)g(phase)f(normal.)19 b(F)l(or)14 b(the)h(p)q(erturbation)e(of)i(the)f Ff(q)r(P)22 b Fi(w)o(a)o(v)o(e)0 174 y(p)q(olarization)15 b(v)o(ector,)i(form)o(ulae)e (\(4a\))i(and)f(\(9\))h(yield)266 293 y(\001)r(\026)-27 b Ff(g)334 269 y Fj(\(3\))332 302 y Fe(i)392 293 y Fi(=)449 260 y Ff(B)487 267 y Fj(13)522 260 y Ff(e)545 267 y Fj(1)p Fe(i)587 260 y Fi(+)11 b Ff(B)675 267 y Fj(23)709 260 y Ff(e)732 267 y Fj(2)p Fe(i)p 449 282 314 2 v 526 328 a Ff(\013)558 313 y Fj(2)587 328 y Fc(\000)g Ff(\014)668 313 y Fj(2)779 293 y Fi(+)g(\()p Ff(e)871 300 y Fj(1)p Fe(k)908 293 y Fi(\001)p Ff(n)980 300 y Fe(k)1000 293 y Fi(\))p Ff(e)1042 300 y Fj(1)p Fe(i)1084 293 y Fi(+)g(\()p Ff(e)1176 300 y Fj(2)p Fe(k)1214 293 y Fi(\001)p Ff(n)1286 300 y Fe(k)1305 293 y Fi(\))p Ff(e)1347 300 y Fj(2)p Fe(i)1392 293 y Fi(=)j(\001)p Ff(g)1511 300 y Fe(i)1535 293 y Fi(+)d(\001)p Ff(n)1657 300 y Fe(i)1670 293 y Ff(:)151 b Fi(\(10)p Ff(a)p Fi(\))0 414 y(F)l(or)15 b(the)i(p)q(erturbation)e(of)i(the)g Ff(K)t Fi(-th)f Ff(q)r(S)j Fi(w)o(a)o(v)o(e)d(p)q(olarization)f(v)o(ector)i (w)o(e)f(get)h(from)f(\(4b\))h(and)f(\(9\))248 533 y(\001)r(\026)-27 b Ff(g)316 509 y Fj(\()p Fe(K)r Fj(\))314 542 y Fe(i)387 533 y Fi(=)460 499 y Ff(B)498 506 y Fe(K)r Fj(3)546 499 y Ff(n)576 506 y Fe(i)p 444 521 161 2 v 444 567 a Ff(\014)475 553 y Fj(2)505 567 y Fc(\000)11 b Ff(\013)587 553 y Fj(2)621 533 y Fc(\000)g Fi(\()p Ff(e)713 540 y Fe(K)r(k)764 533 y Fi(\001)p Ff(n)836 540 y Fe(k)856 533 y Fi(\))p Ff(n)905 540 y Fe(i)933 533 y Fi(=)j(\001)p Ff(g)1052 540 y Fe(i)1076 533 y Fi(+)c(\(\001)p Ff(e)1209 540 y Fe(K)r(k)1260 533 y Ff(n)1290 540 y Fe(k)1310 533 y Fi(\))p Ff(n)1359 540 y Fe(i)1387 533 y Fi(=)k(\001)p Ff(g)1506 540 y Fe(i)1530 533 y Fi(+)d(\001)p Ff(e)1645 540 y Fe(K)r(i)1688 533 y Ff(:)138 b Fi(\(10)p Ff(b)p Fi(\))0 653 y(In)16 b(deriving)g(Eq.\(10b\),)g(w)o(e)h(used)e(the)i(ob)o(vious)e(iden)o (tit)o(y)h(\001)p Ff(e)1177 660 y Fe(I)r(k)1214 653 y Ff(n)1244 660 y Fe(k)1279 653 y Fi(=)d Fc(\000)p Ff(e)1393 660 y Fe(I)r(k)1431 653 y Fi(\001)p Ff(n)1503 660 y Fe(k)1522 653 y Fi(,)k(follo)o(wing)e(from)h (the)0 711 y(p)q(erp)q(endicularit)o(y)g(of)i(p)q(erturb)q(ed)e(v)o(ectors)i Ff(n)867 718 y Fe(i)898 711 y Fi(and)f Ff(e)1019 718 y Fe(I)r(i)1050 711 y Fi(,)h(and)e(the)i(fact)g(that)g(the)g(p)q(erturbation)f(\001)p Ff(e)1920 718 y Fe(I)r(i)0 769 y Fi(has)k(only)h(nonzero)f(pro)s(jection)g (in)o(to)h(the)g(phase)f(normal,)h(see)g(App)q(endix)g(A.)g(F)l(or)f(b)q(oth) h Ff(q)r(P)29 b Fi(and)0 828 y Ff(q)r(S)20 b Fi(w)o(a)o(v)o(es)d(w)o(e)g (arriv)o(ed)f(to)i(an)f(in)o(teresting)g(result.)24 b(If)18 b(the)g(phase)e(normal)g(is)i(p)q(erturb)q(ed)e(b)o(y)i(a)f(\014rst-)0 886 y(order)d(v)o(ector)i(\001)p Ff(n)350 893 y Fe(i)363 886 y Fi(,)g(then)f(the)h(p)q(erturbations)e(of)h(the)h(corresp)q(onding)d(p)q (olarization)i(v)o(ectors)g(consist)0 944 y(of)k(terms)g(\(4\),)i(corresp)q (onding)16 b(to)k(unp)q(erturb)q(ed)e(phase)g(normal,)g(and)h(of)h(p)q (erturb)q(ed)e(bac)o(kground)0 1002 y(p)q(olarization)g(v)o(ectors)h Ff(e)473 1009 y Fe(mi)529 1002 y Fi(+)13 b(\001)p Ff(e)646 1009 y Fe(mi)689 1002 y Fi(.)30 b(In)19 b(other)f(w)o(ords,)h(in)f(case)h(of) h(the)f(p)q(erturb)q(ed)g(phase)f(normal,)0 1060 y(w)o(e)e(can)g(use)f(the)i (same)e(form)o(ulae)f(\(4\))j(as)f(for)g(the)g(case)g(of)g(the)g(\014xed)g (phase)f(normal.)21 b(W)l(e)16 b(ha)o(v)o(e)f(only)0 1118 y(to)i(substitute)g (the)g(corresp)q(onding)d(bac)o(kground)h(p)q(olarization)h(v)o(ector)h(b)o (y)g(a)g(p)q(erturb)q(ed)f(one.)23 b(This)0 1176 y(is)18 b(v)o(ery)h(imp)q (ortan)o(t)e(\014nding)g(since)i(it)f(allo)o(ws)g(an)g(easy)h(transformation) d(of)j(results)f(obtained)g(for)g(a)0 1234 y(\014xed)e(phase)g(normal)f(in)o (to)h(the)h(results)e(for)h(a)h(p)q(erturb)q(ed)f(phase)f(normal.)75 1323 y(In)21 b(the)g(next)h(section,)g(w)o(e)f(seek)g(a)g(sp)q(ecial)g(p)q (erturbation)f(\001)p Ff(n)1306 1330 y Fe(i)1340 1323 y Fi(of)i(the)f(phase)g (normal,)f(whic)o(h)0 1381 y(yields)j(the)h(ra)o(y)g(v)o(ector)f(in)h(the)g (anisotropic)e(medium)g(coinciding)h(with)g(the)h(ra)o(y)g(v)o(ector)f(in)h (the)0 1439 y(bac)o(kground)12 b(isotropic)h(medium.)20 b(T)l(o)13 b(\014nd)h(it,)g(w)o(e)g(m)o(ust)f(\014rst)g(determine)g(the)i(relation)e(b)q (et)o(w)o(een)h(the)0 1497 y(phase)k(normal)f(and)h(the)h(ra)o(y)g(v)o(ector) g(in)f(w)o(eakly)h(anisotropic)e(media.)28 b(Under)19 b(the)g(ra)o(y)f(v)o (ector,)h(w)o(e)0 1555 y(understand)c(a)h(unit)g(v)o(ector)h(in)f(the)h (direction)e(of)i(the)g(ra)o(y)f(v)o(elo)q(cit)o(y)l(.)75 1761 y Fg(3)i(Relation)j(b)r(et)n(w)n(een)g(phase)d(norm)o(al)f(and)h(ra)n(y)i(v)n (ector)75 1850 y Fi(W)l(e)e(start)f(from)g(the)h(w)o(ell-kno)o(wn)e(form)o (ula)g(for)h(the)h(ra)o(y)f(v)o(elo)q(cit)o(y)i(v)o(ector)e Ff(v)1543 1857 y Fe(i)1557 1850 y Fi(,)h(see)g(e.g.)1777 1837 y(\024)1771 1850 y(Cerv)o(en)q(\023)-26 b(y)0 1908 y(et)17 b(al.\(1977\),)g(Ga)s(jewski)f(and)g(P)m(\024)-22 b(sen)o(\024)e(c)-6 b(\023)-19 b(\020k)16 b(\(1990\):)631 2027 y Ff(v)655 2034 y Fe(i)683 2027 y Fi(=)d Ff(v)r(N)801 2034 y Fe(i)829 2027 y Fi(=)887 1994 y Ff(dx)941 2001 y Fe(i)p 887 2016 69 2 v 899 2062 a Ff(dt)974 2027 y Fi(=)g Ff(c)1048 2007 y Fb(\000)p Fj(1)1092 2027 y Ff(a)1118 2034 y Fe(ij)r(k)q(l)1177 2027 y Ff(n)1207 2034 y Fe(l)1220 2027 y Ff(g)1244 2034 y Fe(j)1261 2027 y Ff(g)1285 2034 y Fe(k)1305 2027 y Ff(:)542 b Fi(\(11\))0 2135 y(Here,)14 b Ff(v)i Fi(and)d Ff(c)g Fi(are)g(ra)o(y)g(and)g(phase)g(v)o(elo)q(cities)h (and)f Ff(g)1021 2142 y Fe(i)1048 2135 y Fi(denotes)g(a)h(p)q(olarization)e (v)o(ector)i(of)g(a)f(conside-)0 2193 y(red)j(w)o(a)o(v)o(e.)23 b(The)16 b(ra)o(y)h(v)o(ector)g Ff(N)608 2200 y Fe(i)638 2193 y Fi(and)g(the)g(phase)f(v)o(ector)h Ff(n)1141 2200 y Fe(i)1172 2193 y Fi(generally)f(di\013er.)22 b(This)16 b(can)h(b)q(e)g(simply)0 2252 y(sho)o(wn)j(if)h(w)o(e)g(consider)e(the)j(follo)o(wing)d(expression)h (for)h(the)g(phase)f(v)o(elo)q(cit)o(y)l(,)j(see)e(also)f(the)h(ab)q(o)o(v)o (e)0 2310 y(references:)787 2368 y Ff(c)809 2347 y Fj(2)841 2368 y Fi(=)13 b Ff(a)919 2375 y Fe(ij)r(k)q(l)978 2368 y Ff(n)1008 2375 y Fe(i)1021 2368 y Ff(n)1051 2375 y Fe(l)1064 2368 y Ff(g)1088 2375 y Fe(j)1105 2368 y Ff(g)1129 2375 y Fe(k)1150 2368 y Ff(:)697 b Fi(\(12\))0 2446 y(Multiplying)19 b(Eq.\(11\))h(b)o(y)g(the)g(phase)g (normal)e Ff(n)964 2453 y Fe(i)998 2446 y Fi(and)h(taking)i(in)o(to)e(accoun) o(t)h(Eq.\(12\),)h(w)o(e)f(\014nd)f(a)0 2505 y(useful)d(and)g(w)o(ell-kno)o (wn)e(iden)o(tit)o(y)883 2563 y Ff(n)913 2570 y Fe(i)927 2563 y Ff(v)951 2570 y Fe(i)979 2563 y Fi(=)f Ff(c:)794 b Fi(\(13\))938 2687 y(132)p eop %%Page: 133 5 133 4 bop 0 -58 a Fi(The)17 b(determination)f(of)i(the)f(ra)o(y)g(v)o(ector)g Ff(N)846 -51 y Fe(i)878 -58 y Fi(from)f(the)i(phase)e(normal)g Ff(n)1422 -51 y Fe(i)1453 -58 y Fi(can)h(b)q(e)h(p)q(erformed)e(with)0 0 y(the)g(use)f(of)h(relation)g(\(11\).)22 b(Determination)15 b(of)h Ff(n)950 7 y Fe(i)979 0 y Fi(from)f Ff(N)1136 7 y Fe(i)1150 0 y Fi(,)h(is)f(ho)o(w)o(ev)o(er,)g(m)o(uc)o(h)f(more)h(complicated)0 58 y(task)i(and)e(it)i(is)f(usually)f(solv)o(ed)h(b)o(y)g(an)g(iterativ)o(e)h (use)f(of)g(the)h(equation)f(\(11\),)h(see)f(e.g.)22 b(Jec)o(h)16 b(\(1991\).)0 116 y(As)k(sho)o(wn)f(later,)h(in)g(the)g(\014rst-order)e (appro)o(ximation)f(of)k(the)f(p)q(erturbation)f(theory)h(the)g(relation)0 174 y(b)q(et)o(w)o(een)c(the)h(v)o(ectors)f Ff(n)477 181 y Fe(i)508 174 y Fi(and)g Ff(N)645 181 y Fe(i)675 174 y Fi(is)g(rather)g (simple)f(and)h(can)g(b)q(e)h(written)g(explicitely)l(.)75 262 y(Let)h(us)f(consider)e(the)j(tensor)f Ff(a)688 269 y Fe(ij)r(k)q(l)764 262 y Fi(giv)o(en)g(in)g(Eq.\(1\))h(with)f(isotropic)f(bac)o(kground)g(sp)q (eci\014ed)g(b)o(y)0 321 y(Eq.\(2\).)22 b(The)14 b(ra)o(y)g(v)o(elo)q(cit)o (y)h(v)o(ector)f Ff(v)710 328 y Fe(i)738 321 y Fi(can)g(b)q(e)h(also)f (expressed)f(as)h(a)g(sum)f(of)i(the)g(ra)o(y)e(v)o(elo)q(cit)o(y)i(v)o (ector)0 379 y(in)h(the)h(bac)o(kground)d(isotropic)i(medium)f(and)h(of)g (its)h(p)q(erturbation)673 487 y Ff(v)697 494 y Fe(i)725 487 y Fi(=)c Ff(v)801 494 y Fj(0)p Fe(i)843 487 y Fi(+)e(\001)p Ff(v)959 494 y Fe(i)986 487 y Fi(=)j Ff(c)1061 494 y Fj(0)1079 487 y Ff(n)1109 494 y Fe(i)1134 487 y Fi(+)d(\001)p Ff(v)1250 494 y Fe(i)1263 487 y Ff(:)584 b Fi(\(14\))0 594 y(In)17 b(\(14\),)h(the)f (ra)o(y)f(v)o(elo)q(cit)o(y)i(v)o(ector)f Ff(v)714 601 y Fj(0)p Fe(i)762 594 y Fi(is)g(substituted)f(b)o(y)h(the)g(term)g Ff(c)1372 601 y Fj(0)1390 594 y Ff(n)1420 601 y Fe(i)1451 594 y Fi(b)q(ecause)g(in)f (an)h(isotropic)0 653 y(medium)11 b(the)i(phase)f(and)g(ra)o(y)g(v)o(elo)q (cities)h(are)g(equal)f(in)h(their)f(size)h(and)f(direction.)20 b(Indices)12 b(0)g(indicate)0 711 y(that)20 b(the)g(corresp)q(onding)e(quan)o (tities)h(are)h(related)f(to)h(the)g(bac)o(kground)e(isotropic)h(medium.)30 b(The)0 769 y(phase)13 b(normal)g Ff(n)333 776 y Fe(i)361 769 y Fi(in)h(\(14\))h(is)f(written)g(in)o(ten)o(tionally)f(without)h(suc)o(h)f (an)h(index)g(b)q(ecause)g(w)o(e)g(consider)0 827 y(it)j(\014xed)f(and)g(th)o (us)g(the)g(same)g(in)g(the)h(isotropic)f(as)g(w)o(ell)g(as)g(in)g (anisotropic)f(medium.)75 915 y(Before)23 b(pro)q(ceeding)f(to)i(the)f(sp)q (eci\014cation)g(of)g(\001)p Ff(v)1074 922 y Fe(i)1088 915 y Fi(,)i(let)e(us)g(estimate)g(the)h(size)f Ff(v)i Fi(of)e(the)h(ra)o(y)0 973 y(v)o(elo)q(cit)o(y)17 b(v)o(ector)g Ff(v)358 980 y Fe(i)388 973 y Fi(giv)o(en)f(b)o(y)g(Eq.\(14\).)23 b(T)l(o)16 b(the)h(\014rst-order,)d (w)o(e)i(get)686 1081 y Ff(v)g Fi(=)d Ff(v)802 1088 y Fj(0)832 1081 y Fi(+)e(\001)p Ff(v)k Fi(=)f Ff(c)1038 1088 y Fj(0)1067 1081 y Fi(+)d Ff(N)1157 1088 y Fe(i)1171 1081 y Fi(\001)p Ff(v)1237 1088 y Fe(i)1250 1081 y Ff(:)597 b Fi(\(15\))75 1219 y(The)11 b(p)q(erturbation)g(\001)p Ff(v)526 1226 y Fe(i)551 1219 y Fi(of)h(the)g(group)f(v)o(elo)q(cit)o(y)h(v)o(ector)g Ff(v)1169 1226 y Fe(i)1195 1219 y Fi(can)g(b)q(e)g(found)f(b)o(y)g(p)q(erturbing)f (Eq.\(11\))0 1277 y(from)19 b(an)h(isotropic)g(bac)o(kground)e(to)i(a)h(w)o (eakly)f(anisotropic)f(medium.)31 b(F)l(or)19 b(the)i(p)q(erturbation)e(of)0 1335 y(the)e(ra)o(y)f(v)o(elo)q(cit)o(y)h(of)f(the)h Ff(m)p Fi(-th)f(w)o(a)o(v)o(e,)g(\001)p Ff(v)826 1311 y Fj(\()p Fe(m)p Fj(\))824 1344 y Fe(i)883 1335 y Fi(,)g(w)o(e)h(get)447 1443 y(\001)p Ff(v)515 1419 y Fj(\()p Fe(m)p Fj(\))513 1452 y Fe(i)586 1443 y Fi(=)c Fc(\000)p Fi(\001)p Ff(c)741 1423 y Fj(\()p Fe(m)p Fj(\))798 1443 y Ff(n)828 1450 y Fe(i)852 1443 y Fi(+)e(\()p Ff(c)943 1423 y Fj(\()p Fe(m)p Fj(\))943 1456 y(0)1001 1443 y Fi(\))1020 1423 y Fb(\000)p Fj(1)1065 1443 y Fi([)p Ff(a)1105 1450 y Fe(ij)r(k)q(l)1163 1443 y Ff(n)1193 1450 y Fe(l)1206 1443 y Ff(e)1229 1450 y Fe(mj)1276 1443 y Ff(e)1299 1450 y Fe(mk)1360 1443 y Fc(\000)g Ff(\014)1441 1423 y Fj(2)1459 1443 y Ff(n)1489 1450 y Fe(i)350 1560 y Fi(+\()p Ff(\013)440 1540 y Fj(2)470 1560 y Fc(\000)g Ff(\014)551 1540 y Fj(2)569 1560 y Fi(\))588 1512 y Fa(\020)614 1560 y Fi(\()p Ff(n)663 1567 y Fe(k)683 1560 y Ff(e)706 1567 y Fe(mk)756 1560 y Fi(\)\001)p Ff(g)843 1536 y Fj(\()p Fe(m)p Fj(\))841 1569 y Fe(i)911 1560 y Fc(\000)g Fi(\()p Ff(n)1010 1567 y Fe(k)1031 1560 y Ff(e)1054 1567 y Fe(mk)1104 1560 y Fi(\))p Ff(g)1149 1536 y Fj(\()p Fe(m)p Fj(\))1147 1569 y Fe(i)1217 1560 y Fi(+)g(\()p Ff(n)1316 1567 y Fe(k)1337 1560 y Fi(\001)p Ff(g)1405 1536 y Fj(\()p Fe(m)p Fj(\))1403 1570 y Fe(k)1461 1560 y Fi(\))p Ff(e)1503 1567 y Fe(mi)1547 1512 y Fa(\021)1572 1560 y Fi(])p Ff(:)261 b Fi(\(16\))0 1654 y(The)17 b(pro)s(jection)f(of)h(the)g(p)q(erturbation)e(of)i(the)g(ra)o (y)f(v)o(elo)q(cit)o(y)i(v)o(ector)f(\(16\))g(in)o(to)f(the)h(v)o(ector)g Ff(n)1811 1661 y Fe(i)1842 1654 y Fi(leads)0 1712 y(to)842 1770 y Ff(n)872 1777 y Fe(i)885 1770 y Fi(\001)p Ff(v)951 1777 y Fe(i)979 1770 y Fi(=)c(\001)p Ff(c:)752 b Fi(\(17\))0 1855 y(Where)21 b(it)h(is)f(not)h(necessary)l(,)g(w)o(e)f(omit)g(the)h(sup)q (erscripts)e(\()p Ff(m)p Fi(\).)37 b(In)22 b(deriving)e(Eq.\(17\),)k(w)o(e)d (to)q(ok)0 1913 y(in)o(to)c(accoun)o(t)g(Eqs.\(12\))h(and)e(\(13\))j(sp)q (eci\014ed)d(for)i(the)f(bac)o(kground)f(isotropic)g(medium,)g(Eq.\(3\))i (for)0 1976 y(p)q(erturbation)12 b(of)h(the)h(corresp)q(onding)c(phase)j(v)o (elo)q(cit)o(y)g(\001)p Ff(c)g Fi(and)f(an)h(ob)o(vious)f(iden)o(tit)o(y)g Ff(e)1673 1983 y Fe(mj)1720 1976 y Fi(\001)p Ff(g)1788 1952 y Fj(\()p Fe(m)p Fj(\))1786 1985 y Fe(j)1859 1976 y Fi(=)h(0.)0 2034 y(Com)o(bining)d(Eqs.\(14\))i(and)f(\(17\),)i(w)o(e)f(can)g(th)o(us)f (express)g(the)h(group)e(v)o(elo)q(cit)o(y)j(v)o(ector)f(in)f(the)h(follo)o (wing)0 2092 y(form)707 2150 y Ff(v)731 2157 y Fe(i)759 2150 y Fi(=)h(\()p Ff(c)852 2157 y Fj(0)882 2150 y Fi(+)e(\001)p Ff(c)p Fi(\))p Ff(n)1045 2157 y Fe(i)1069 2150 y Fi(+)g Ff(A)1156 2157 y Fe(I)1175 2150 y Ff(e)1198 2157 y Fe(I)r(i)1229 2150 y Ff(:)618 b Fi(\(18\))0 2235 y(The)17 b(quan)o(tit)o(y)f Ff(A)339 2242 y Fe(I)375 2235 y Fi(represen)o(ts)f(a)i(pro)s(jection)f(of)h(the)g(ra)o (y)f(v)o(elo)q(cit)o(y)h(v)o(ector)g Ff(v)1476 2242 y Fe(i)1506 2235 y Fi(in)o(to)g(a)f(v)o(ector)h Ff(e)1822 2242 y Fe(I)r(i)1870 2235 y Fi(and)0 2293 y(is)f(giv)o(en)g(b)o(y)g(the)h(form)o(ula)e(simply)g (follo)o(wing)h(from)f(\(16\))644 2406 y Ff(A)681 2382 y Fj(\()p Fe(m)p Fj(\))681 2415 y Fe(I)753 2406 y Fi(=)e(\()p Ff(c)846 2386 y Fj(\()p Fe(m)p Fj(\))846 2418 y(0)904 2406 y Fi(\))923 2386 y Fb(\000)p Fj(1)968 2406 y Fi([)p Ff(a)1008 2413 y Fe(ij)r(k)q(l)1066 2406 y Ff(n)1096 2413 y Fe(l)1109 2406 y Ff(e)1132 2413 y Fe(mj)1179 2406 y Ff(e)1202 2413 y Fe(mk)1252 2406 y Ff(e)1275 2413 y Fe(I)r(i)229 2523 y Fi(+\()p Ff(\013)319 2502 y Fj(2)349 2523 y Fc(\000)e Ff(\014)430 2502 y Fj(2)448 2523 y Fi(\))467 2475 y Fa(\020)492 2523 y Fi(\()p Ff(n)541 2530 y Fe(k)562 2523 y Ff(e)585 2530 y Fe(mk)635 2523 y Fi(\)\(\001)p Ff(g)741 2499 y Fj(\()p Fe(m)p Fj(\))739 2532 y Fe(i)798 2523 y Ff(e)821 2530 y Fe(I)r(i)852 2523 y Fi(\))h Fc(\000)f Fi(\()p Ff(n)982 2530 y Fe(k)1002 2523 y Ff(e)1025 2530 y Fe(mk)1075 2523 y Fi(\)\()p Ff(e)1136 2530 y Fe(mi)1181 2523 y Ff(e)1204 2530 y Fe(I)r(i)1235 2523 y Fi(\))g(+)g(\()p Ff(n)1364 2530 y Fe(k)1385 2523 y Fi(\001)p Ff(g)1453 2499 y Fj(\()p Fe(m)p Fj(\))1451 2533 y Fe(k)1509 2523 y Fi(\)\()p Ff(e)1570 2530 y Fe(mi)1614 2523 y Ff(e)1637 2530 y Fe(I)r(i)1668 2475 y Fa(\021)1693 2523 y Fi(])p Ff(:)140 b Fi(\(19\))938 2687 y(133)p eop %%Page: 134 6 134 5 bop 0 -58 a Fi(W)l(e)17 b(can)f(see)h(that)g Ff(A)403 -51 y Fe(I)439 -58 y Fi(is)f(a)g(\014rst-order)e(quan)o(tit)o(y)l(.)75 30 y(T)l(aking)j(in)o(to)g(accoun)o(t)f(that)i Ff(v)661 37 y Fe(i)691 30 y Fi(=)d Ff(v)r(N)811 37 y Fe(i)824 30 y Fi(,)j(see)f (Eq.\(11\),)i(and)d(k)o(eeping)h(only)g(terms)g(upto)g(the)h(\014rst)0 88 y(order,)d(w)o(e)i(can)f(\014nd)f(from)h(\(18\))558 210 y Ff(N)598 217 y Fe(i)626 210 y Fi(=)d Ff(n)708 217 y Fe(i)733 210 y Fi(+)788 176 y Ff(n)818 183 y Fe(i)p 788 198 44 2 v 790 244 a Ff(c)812 251 y Fj(0)837 210 y Fi(\(\001)p Ff(c)d Fc(\000)h Ff(N)1020 217 y Fe(k)1040 210 y Fi(\001)p Ff(v)1106 217 y Fe(k)1127 210 y Fi(\))g(+)g Ff(c)1229 189 y Fb(\000)p Fj(1)1229 222 y Fe(o)1273 210 y Ff(A)1310 217 y Fe(i)1324 210 y Ff(e)1347 217 y Fe(I)r(i)1378 210 y Ff(:)469 b Fi(\(20\))0 343 y(Equation)20 b(\(20\))h(clearly)g(sho)o(ws)e(that)h(the)h(di\013erence)f(b)q(et)o(w)o(een) g(the)h(v)o(ectors)f Ff(n)1556 350 y Fe(i)1591 343 y Fi(and)g Ff(N)1732 350 y Fe(i)1766 343 y Fi(is)g(of)h(the)0 401 y(\014rst)c(order.)25 b(This)18 b(means)f(that)h(ev)o(erywhere,)g(where)f(a)h(\014rst-order)e(quan) o(tit)o(y)i(is)f(m)o(ultiplied)f(b)o(y)i Ff(N)1936 408 y Fe(i)0 459 y Fi(or)f Ff(n)92 466 y Fe(i)105 459 y Fi(,)g(b)q(oth)g(v)o(ectors)g(are) g(in)o(terc)o(hangeable.)k(W)l(e)c(can)g(use)g(this)f(prop)q(ert)o(y)h (directly)f(in)h(Eq.\(20\))h(and)0 517 y(rewrite)e(it)h(in)o(to)f(the)h (\014nal)e(form,)h(whic)o(h)g(reads)767 625 y Ff(N)807 632 y Fe(i)835 625 y Fi(=)e Ff(n)918 632 y Fe(i)942 625 y Fi(+)d Ff(c)1014 604 y Fb(\000)p Fj(1)1014 637 y Fe(o)1058 625 y Ff(A)1095 632 y Fe(I)1115 625 y Ff(e)1138 632 y Fe(I)r(i)1169 625 y Ff(:)678 b Fi(\(21\))0 733 y(Equation)20 b(\(21\))h(can)f(b)q(e)h(used)e(for)h(the)h (determination)e(of)h(eac)o(h)g(of)h(the)f(v)o(ectors)g Ff(n)1636 740 y Fe(i)1650 733 y Fi(,)h Ff(N)1725 740 y Fe(i)1759 733 y Fi(from)f(the)0 791 y(kno)o(wledge)14 b(of)h(the)h(other)e(one.)21 b(It)16 b(is)e(a)h(linear)f(equation)h(and)f(as)h(suc)o(h)f(it)h(can)g(yield) g(only)f(one)h(v)o(ector)0 849 y Ff(n)30 856 y Fe(i)61 849 y Fi(for)i(a)g(sp)q(eci\014ed)g Ff(N)422 856 y Fe(i)436 849 y Fi(.)24 b(Th)o(us,)16 b(Eq.\(21\))i(cannot,)f(in)g(principle,)f(giv)o(e)h (correct)g(results)f(in)h(regions)f(of)0 907 y(triplication)g(of)g(a)h Ff(q)r(S)i Fi(w)o(a)o(v)o(e)d(w)o(a)o(v)o(efron)o(t.)75 995 y(Equation)k(\(21\))i(can)f(b)q(e)g(useful)f(in)g(man)o(y)g(applications.)34 b(F)l(or)20 b(example,)h(it)g(can)g(b)q(e)g(useful)f(in)0 1053 y(ra)o(y)h(tracing)h(in)f(anisotropic)g(media,)h(where)f(it)i(can)e(serv)o(e) h(for)f(the)i(estimation)e(of)h(phase)f(v)o(ector)0 1112 y(directions,)15 b(for)i(whic)o(h)f(ra)o(ys)f(at)i(a)g(source)f(ha)o(v)o(e)g(prescrib)q(ed)f (directions.)21 b(F)l(or)16 b(suc)o(h)f(a)i(case,)f(Eq.\(21\))0 1170 y(yields)g(for)g(the)h(correction)f(of)h(the)f(phase)g(normal)785 1278 y(\001)p Ff(n)857 1285 y Fe(i)884 1278 y Fi(=)d Fc(\000)p Ff(c)997 1257 y Fb(\000)p Fj(1)997 1290 y(0)1041 1278 y Ff(A)1078 1285 y Fe(I)1098 1278 y Ff(e)1121 1285 y Fe(I)r(i)1152 1278 y Ff(:)695 b Fi(\(22\))75 1416 y(F)l(or)18 b(the)h(application)f(of)h (Eq.\(21\),)h(only)f(kno)o(wledge)f(of)i(the)f(quan)o(tities)g Ff(A)1543 1423 y Fe(I)1581 1416 y Fi(is)g(necessary)l(.)28 b(F)l(or)0 1474 y Ff(q)r(P)c Fi(w)o(a)o(v)o(es,)15 b(Eq.\(19\))i(yields)574 1582 y Ff(A)611 1589 y Fe(I)644 1582 y Fi(=)d(2)p Ff(\013)754 1561 y Fb(\000)p Fj(1)798 1582 y Ff(B)836 1589 y Fe(I)r Fj(3)872 1582 y Ff(;)58 b(B)982 1589 y Fe(I)r Fj(3)1031 1582 y Fi(=)13 b Ff(a)1109 1589 y Fe(ij)r(k)q(l)1168 1582 y Ff(n)1198 1589 y Fe(i)1212 1582 y Ff(n)1242 1589 y Fe(l)1254 1582 y Ff(n)1284 1589 y Fe(j)1302 1582 y Ff(e)1325 1589 y Fe(I)r(k)1362 1582 y Ff(:)485 b Fi(\(23\))0 1690 y(F)l(or)15 b Ff(K)t Fi(-th)i(of)f(the)h Ff(q)r(S)i Fi(w)o(a)o(v)o(es,)c(Eq.\(19\),)i(giv)o(es)251 1802 y Ff(A)288 1778 y Fj(\()p Fe(K)r Fj(\))288 1812 y Fe(I)360 1802 y Fi(=)d Ff(\014)444 1782 y Fb(\000)p Fj(1)488 1802 y Fi(\()p Ff(a)533 1809 y Fe(mj)r(k)q(l)610 1802 y Ff(n)640 1809 y Fe(l)652 1802 y Ff(e)675 1809 y Fe(K)r(j)723 1802 y Ff(e)746 1809 y Fe(K)r(k)796 1802 y Ff(e)819 1809 y Fe(I)r(m)879 1802 y Fc(\000)d Ff(B)967 1809 y Fe(K)r Fj(3)1015 1802 y Ff(e)1038 1809 y Fe(I)r(k)1076 1802 y Ff(e)1099 1809 y Fe(K)r(k)1149 1802 y Fi(\))p Ff(;)59 b(B)1279 1809 y Fe(K)r Fj(3)1341 1802 y Fi(=)13 b Ff(a)1419 1809 y Fe(ij)r(k)q(l)1478 1802 y Ff(n)1508 1809 y Fe(i)1522 1802 y Ff(n)1552 1809 y Fe(l)1564 1802 y Ff(n)1594 1809 y Fe(j)1612 1802 y Ff(e)1635 1809 y Fe(K)r(k)1685 1802 y Ff(;)162 b Fi(\(24\))0 1910 y(no)21 b(summation)e(o)o(v)o(er)h(brac)o(k)o (etted)h(indices.)35 b(It)22 b(is)e(v)o(ery)h(imp)q(ortan)o(t)f(to)i (distinguish)d(b)q(et)o(w)o(een)i(the)0 1968 y(indices)i Ff(I)29 b Fi(and)23 b Ff(K)29 b Fi(in)24 b(the)g(equations)g(\(23\))h(and)f(\(24\).) 46 b(The)24 b(index)h Ff(I)j Fi(is)c(used)f(to)i(denote)g(t)o(w)o(o)0 2026 y(m)o(utually)16 b(p)q(erp)q(endicular)f(unit)h(v)o(ectors)h Fg(arbitrarily)h Fi(c)o(hosen)e(in)h(the)g(plane)f(p)q(erp)q(endicular)f(to)i Ff(n)1922 2033 y Fe(i)1936 2026 y Fi(.)0 2085 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0 y(medium.)j(F)l(or)c Ff(q)r(P)23 b Fi(w)o(a)o(v)o(es,)15 b(the)i(v)o(ector)g(has)f(a)g(form)652 108 y Ff(v)676 115 y Fe(i)704 108 y Fi(=)d(\()p Ff(\013)f Fi(+)e(\001)p Ff(c)p Fi(\))p Ff(n)981 115 y Fe(i)1006 108 y Fi(+)h(2)p Ff(\013)1113 87 y Fb(\000)p Fj(1)1157 108 y Ff(B)1195 115 y Fe(I)r Fj(3)1230 108 y Ff(e)1253 115 y Fe(I)r(i)1284 108 y Ff(:)563 b Fi(\(26\))0 216 y(F)l(or)15 b Ff(K)t Fi(-th)i Ff(q)r(S)i Fi(w)o(a)o(v)o(e,)c(the)i(ra)o (y)f(v)o(ector)g(is)310 324 y Ff(v)336 299 y Fj(\()p Fe(K)r Fj(\))334 332 y Fe(i)408 324 y Fi(=)d(\()p Ff(\014)h Fi(+)d(\001)p Ff(c)635 303 y Fj(\()p Fe(K)r Fj(\))692 324 y Fi(\))p Ff(n)741 331 y Fe(i)766 324 y Fi(+)g Ff(\014)847 303 y Fb(\000)p Fj(1)892 324 y Fi(\()p Ff(a)937 331 y Fe(mj)r(k)q(l)1013 324 y Ff(n)1043 331 y Fe(l)1056 324 y Ff(e)1079 331 y Fe(K)r(j)1126 324 y Ff(e)1149 331 y Fe(K)r(k)1200 324 y Ff(e)1223 331 y Fe(I)r(m)1283 324 y Fc(\000)g Ff(B)1371 331 y Fe(K)r Fj(3)1419 324 y Ff(e)1442 331 y Fe(I)r(k)1479 324 y Ff(e)1502 331 y Fe(K)r(k)1553 324 y Fi(\))p Ff(e)1595 331 y Fe(I)r(i)1626 324 y Ff(;)221 b Fi(\(27\))75 462 y(The)18 b(ab)q(o)o(v)o(e)g(Eqs.\(26\))g(and)g(\(27\))h(could)f(b)q(e)g (used)g(for)g(\014nding)f(a)h(plane)g(of)g(symmetry)g(of)g(a)h(stu-)0 520 y(died)j(w)o(eakly)h(anisotropic)e(material.)39 b(According)22 b(to)h(Norris)f(\(1989\),)i Fh(ne)m(c)m(essary)i(and)e(su\016cient)0 578 y(c)m(onditions)f(that)d(the)g(dir)m(e)m(ction)i Ff(n)668 585 y Fe(i)702 578 y Fh(b)m(e)f(normal)g(to)f(a)h(plane)g(of)g(symmetry)f(ar) m(e)h(that)f(it)g(b)m(e)h(a)g(longi-)0 636 y(tudinal)h(dir)m(e)m(ction)i(and) f(that)f(the)g(ener)m(gy-\015ux)g(ve)m(ctor)h(b)m(e)f(in)g(the)g(dir)m(e)m (ction)i Ff(n)1543 643 y Fe(i)1579 636 y Fh(for)f(any)f(dir)m(e)m(ction)0 694 y(of)f(tr)m(ansverse)h(waves)g(with)f(phase)h(dir)m(e)m(ction)h Ff(n)937 701 y Fe(i)950 694 y Fi(.)32 b(The)20 b(ob)o(vious)e(conditions)g (for)i(the)g(existence)g(of)0 752 y(longitudinal)15 b(and)h(transv)o(erse)f 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b(the)i(expression)e(for)h(\001)p Ff(e)676 498 y Fj(2)p Fe(i)723 491 y Fi(to)h(the)g(RHS)f(of)h(the)f(\014rst)g(equation)h(and)e(\001)p Ff(e)1598 498 y Fj(1)p Fe(i)1645 491 y Fi(to)i(the)g(RHS)f(of)0 549 y(the)h(second)f(equation,)g(w)o(e)g(get,)h(after)g(some)f (manipulations,)414 657 y Ff(\017)434 664 y Fe(ij)r(k)482 657 y Ff(e)505 664 y Fj(2)p Fe(j)539 657 y Fi(\001)p Ff(n)611 664 y Fe(k)645 657 y Fi(=)e Ff(n)728 664 y Fe(i)741 657 y Fi(\(\001)p Ff(e)825 664 y Fj(1)p Fe(k)862 657 y Ff(n)892 664 y Fe(k)913 657 y Fi(\))p Ff(;)58 b(\017)1024 664 y Fe(ij)r(k)1072 657 y Fi(\001)p Ff(n)1144 664 y Fe(j)1161 657 y Ff(e)1184 664 y Fj(1)p Fe(k)1235 657 y Fi(=)13 b Ff(n)1317 664 y Fe(i)1331 657 y Fi(\(\001)p Ff(e)1415 664 y Fj(2)p Fe(k)1452 657 y Ff(n)1482 664 y Fe(k)1502 657 y Fi(\))p Ff(:)300 b Fi(\()p Ff(A:)p Fi(3\))0 765 y(Using)16 b(\(A.3\),)h(w)o(e)g(can)f(rewrite)g(\(A.2\))i(in)o(to)e(the)g (form)258 873 y(\001)p Ff(e)323 880 y Fj(1)p Fe(i)367 873 y Fi(=)e Ff(n)450 880 y Fe(i)464 873 y Fi(\(\001)p Ff(e)548 880 y Fj(1)p Fe(k)585 873 y Ff(n)615 880 y Fe(k)635 873 y Fi(\))d(+)g Ff(\017)735 880 y Fe(ij)r(k)783 873 y Fi(\001)p Ff(e)848 880 y Fj(2)p Fe(j)882 873 y Ff(n)912 880 y Fe(k)932 873 y Ff(;)58 b Fi(\001)p Ff(e)1069 880 y Fj(2)p Fe(i)1113 873 y Fi(=)14 b Ff(n)1196 880 y Fe(i)1209 873 y Fi(\(\001)p Ff(e)1293 880 y Fj(2)p Fe(k)1330 873 y Ff(n)1360 880 y Fe(k)1381 873 y Fi(\))d(+)g Ff(\017)1481 880 y Fe(ij)r(k)1529 873 y Ff(n)1559 880 y Fe(j)1576 873 y Fi(\001)p Ff(e)1641 880 y Fj(1)p Fe(k)1678 873 y Ff(:)143 b Fi(\()p Ff(A:)p Fi(4\))0 981 y(Multiplying)15 b(b)q(oth)i(equations)f(b)o (y)g Ff(n)711 988 y Fe(i)725 981 y Fi(,)g(w)o(e)h(get)624 1088 y Ff(\017)644 1095 y Fe(ij)r(k)692 1088 y Fi(\001)p Ff(e)757 1095 y Fj(2)p Fe(j)790 1088 y Ff(n)820 1095 y Fe(k)855 1088 y Fi(=)c(0)p Ff(;)58 b(\017)1024 1095 y Fe(ij)r(k)1072 1088 y Ff(n)1102 1095 y Fe(j)1119 1088 y Fi(\001)p Ff(e)1184 1095 y Fj(1)p Fe(k)1235 1088 y Fi(=)13 b(0)p Ff(:)509 b Fi(\()p Ff(A:)p Fi(5\))0 1196 y(Th)o(us)12 b(the)h(p)q(erturbations)e(of)i(the)g(v)o (ectors)g Ff(e)837 1203 y Fj(1)p Fe(i)880 1196 y Fi(and)f Ff(e)996 1203 y Fj(2)p Fe(i)1040 1196 y Fi(ha)o(v)o(e)g(only)h(pro)s(jections)e(to)j (the)f(phase)f(normal:)536 1304 y(\001)p Ff(e)601 1311 y Fj(1)p Fe(i)645 1304 y Fi(=)i Ff(n)728 1311 y Fe(i)741 1304 y Fi(\(\001)p Ff(e)825 1311 y Fj(1)p Fe(k)862 1304 y Ff(n)892 1311 y Fe(k)913 1304 y Fi(\))p Ff(;)58 b Fi(\001)p Ff(e)1069 1311 y Fj(2)p Fe(i)1113 1304 y Fi(=)14 b Ff(n)1196 1311 y Fe(i)1209 1304 y Fi(\(\001)p Ff(e)1293 1311 y Fj(2)p Fe(k)1330 1304 y Ff(n)1360 1311 y Fe(k)1381 1304 y Fi(\))p Ff(:)421 b Fi(\()p Ff(A:)p Fi(6\))75 1442 y(Seismic)23 b(W)l(a)o(v)o(es)g(in)h(Complex)g(3-D)g (Structures,)h(Rep)q(ort)f(4,)i(pp.)44 b(129-138,)26 b(Dep.Geoph)o(ys.,)0 1500 y(Charles)15 b(Univ.,)i(Prague)e(1996)938 2687 y(138)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF