split.ps

弹性波动方程正演程序

803F00EBC03EEBE0FCEB3FF8EB0FC090C8FCA8EAFFE0A21A207E951F>I<EBFE033807FF
87380FC1C7381F006F48133F127E80127C12FCA7127EA2003E5B6C5B380FC1DF3807FF9F
3801FC1FC7FCA8ECFFE0A21B207E951E>I<EAFE1FEB3FC0381E67E013C71387A2381F83
C090C7FCADEAFFF0A213167E9517>I<EA0FF3EA3FFFEA781FEA6007EAE003A212F000FC
C7FCEA7FE013F8EA3FFEEA0FFF120138000F80EAC007130312E0A238F00700EAFC0EEAEF
FCEAC7F011167E9516>I<487EA41203A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB
8300EA07C3EA03FEEA00F811207F9F16>I<38FF01FEA2381F003EAF147E14FE380F81BE
3907FF3FC0EA01FC1A167E951F>I<39FFE01FE0A2390F800600A2EBC00E0007130CEBE0
1C00031318A26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA21B167F951E>
I E /Fj 66 123 df<90381F83E09038706E309038C07C78380180F8000313F039070070
00A9B612C03907007000B21478397FE3FF801D2380A21C>11 D<EB1FC0EB7020EBC01038
01803800031378EA0700143091C7FCA7B512F8380700781438B2397FE1FF80192380A21B
>I<EB1FD8EB7038EBC078EA0180120338070038A9B512F838070038B3397FF3FF801923
80A21B>I<90380FC07F90397031C0809039E00B00402601801E13E00003EB3E01380700
3C91381C00C01600A7B712E03907001C011500B23A7FF1FFCFFE272380A229>I<EA7038
EAF87CEAFC7EA2EA743AEA0402A4EA0804A2EA1008A2EA2010EA40200F0F7EA218>34
D<132013401380EA01005A12061204120CA25AA25AA312701260A312E0AE1260A3127012
30A37EA27EA2120412067E7EEA0080134013200B327CA413>40 D<7E12407E7E12187E12
041206A27EA2EA0180A313C01200A313E0AE13C0A312011380A3EA0300A21206A2120412
0C5A12105A5A5A0B327DA413>I<497EB0B612FEA23900018000B01F227D9C26>43
D<127012F812FCA212741204A41208A21210A212201240060F7C840E>I<EAFFE0A20B02
7F8B10>I<127012F8A3127005057C840E>I<EA01F0EA071CEA0C06487E00381380A23870
01C0A400F013E0AE007013C0A3EA780300381380A2381C0700EA0C06EA071CEA01F01322
7EA018>48 D<13801203120F12F31203B3A9EA07C0EAFFFE0F217CA018>I<EA03F0EA0C
1CEA100700201380384003C0A2008013E012F0EAF801A3EA2003120014C0A2EB07801400
130E5B13185B5B5B485A90C7FC000213205A5A00181360481340383FFFC05AB5FC13217E
A018>I<EA03F8EA0C1EEA100700201380004013C0127813031307123800001380A21400
5B130E1338EA03F0EA001C130FEB0780A2EB03C0A214E01220127012F8A200F013C01240
EB0780122038100F00EA0C1CEA03F013227EA018>I<13021306130EA2131EA2132E134E
A2138EA2EA010E1202A21204A212081210A21220A212401280B512F838000E00A7131F38
01FFF015217FA018>I<00101380381E0700EA1FFF5B13F8EA13E00010C7FCA613F8EA13
0EEA1407381803801210380001C0A214E0A4127012F0A200E013C01280EA400314803820
0700EA1006EA0C1CEA03F013227EA018>I<137EEA01C138030080380601C0EA0E03121C
381801800038C7FCA212781270A2EAF0F8EAF30CEAF4067F00F81380EB01C012F014E0A5
1270A3003813C0A238180380001C1300EA0C06EA070CEA01F013227EA018>I<EA01F0EA
060C487EEA1807383803801270A238F001C0A314E0A5127013031238EA1805120CEA0619
EA03E1380001C0A3EB0380A200301300EA78071306EA700CEA20186C5AEA0FC013227EA0
18>57 D<127012F8A312701200AB127012F8A3127005157C940E>I<127012F8A3127012
00AB127012F8A312781208A41210A312201240A2051F7C940E>I<B612FEA2C9FCA8B612
FEA21F0C7D9126>61 D<497EA3497EA3EB05E0A2EB0DF01308A2497E1478A2497EA3497E
A3497EA290B5FC3901000780A24814C000021303A24814E01401A2000CEB00F0A2003EEB
01F839FF800FFF20237EA225>65 D<903807E0109038381830EBE0063901C00170390380
00F048C7FC000E1470121E001C1430123CA2007C14101278A200F81400A812781510127C
123CA2001C1420121E000E14407E6C6C13803901C001003800E002EB381CEB07E01C247D
A223>67 D<B512F0380F801E00071307EC03C01401EC00E015F01578A2157C153CA3153E
A9153CA2157C1578A215F015E01401EC03C0EC0700000F131EB512F81F227EA125>I<B6
12C0380F80070007130114001540A215601520A314201500A3146014E013FF1380146014
20A315081400A21510A31530A2157015E0000F1303B6FC1D227EA121>I<B612C0380F80
070007130114001540A215601520A314201500A3146014E013FF138014601420A491C7FC
A9487EEAFFFE1B227EA120>I<39FFFC3FFF390FC003F039078001E0AE90B5FCEB8001AF
390FC003F039FFFC3FFF20227EA125>72 D<EAFFFCEA0FC0EA0780B3ACEA0FC0EAFFFC0E
227EA112>I<39FF8007FF3907C000F81570D805E01320EA04F0A21378137C133C7F131F
7FEB0780A2EB03C0EB01E0A2EB00F014F81478143C143E141E140FA2EC07A0EC03E0A214
01A21400000E1460121FD8FFE0132020227EA125>78 D<EB0FE0EB783CEBE00E3903C007
8039078003C0390F0001E0000E1300001E14F0481478A2007C147CA20078143CA200F814
3EA90078143C007C147CA2003C1478003E14F8001E14F06CEB01E0A239078003C03903C0
07803900E00E00EB783CEB0FE01F247DA226>I<B512F0380F803C0007130FEC078015C0
140315E0A615C014071580EC0F00143CEBFFF00180C7FCAE487EEAFFFC1B227EA121>I<
B512E0380F803C0007130E6E7E81140381A55D14075D020EC7FC143CEBFFE0EB80708080
141E140E140FA481A3168015C014073A0FC003C10039FFFC01E2C8127C21237EA124>82
D<3803F020380C0C60EA1802383001E0EA70000060136012E0A21420A36C1300A2127812
7FEA3FF0EA1FFE6C7E0003138038003FC0EB07E01301EB00F0A214707EA46C1360A26C13
C07E38C8018038C60700EA81FC14247DA21B>I<007FB512F83978078078006014180040
1408A300C0140C00801404A400001400B3A3497E0003B5FC1E227EA123>I<39FFFC07FF
390FC000F86C4813701520B3A5000314407FA2000114806C7E9038600100EB3006EB1C08
EB03F020237EA125>I<3BFFF03FFC03FE3B1F8007E000F86C486C4813701720A26C6C6C
6C1340A32703C002F01380A33B01E004780100A33A00F0083C02A39039F8183E06903978
101E04A2137C90393C200F08A390391E400790A390390F8003E0A36D486C5AA36D5C0102
13002F237FA132>87 D<12FEA212C0B3B3A912FEA207317BA40E>91
D<EA0804EA1008EA2010A2EA4020A2EA8040A4EAB85CEAFC7EA2EA7C3EEA381C0F0F7AA2
18>I<12FEA21206B3B3A912FEA207317FA40E>I<EA0FE0EA1838EA3C0C130EEA18071200
A313FFEA07C7EA1E07123C1278127000F01308A3130FEA7817383C2390380FC1E015157E
9418>97 D<120E12FE121E120EAB131FEB61C0EB8060380F0030000E1338143C141C141E
A7141C143C1438000F1370380C8060EB41C038083F0017237FA21B>I<EA01FEEA070338
0C0780121C383803000078C7FC127012F0A712700078134012386C1380380C0100EA0706
EA01F812157E9416>I<14E0130F13011300ABEA01F8EA0704EA0C02EA1C01EA38001278
127012F0A7127012781238EA1801EA0C0238070CF03801F0FE17237EA21B>I<EA01FCEA
0707380C0380381C01C01238007813E0EA700012F0B5FC00F0C7FCA51270007813201238
6C13406C138038070300EA00FC13157F9416>I<133C13C6EA018F1203130FEA0700A9EA
FFF8EA0700B21380EA7FF8102380A20F>I<14703801F19838071E18EA0E0E381C0700A2
003C1380A4001C1300A2EA0E0EEA0F1CEA19F00010C7FCA21218A2EA1FFE380FFFC014E0
383800F0006013300040131812C0A300601330A2003813E0380E03803803FE0015217F95
18>I<120E12FE121E120EABEB1F80EB60C0EB80E0380F0070A2120EAF38FFE7FF18237F
A21B>I<121C123EA3121CC7FCA8120E12FE121E120EB1EAFFC00A227FA10E>I<EA01C0EA
03E0A3EA01C0C7FCA8EA01E0120F12011200B3A51260EAF0C0EAF1801261EA3E000B2C82
A10F>I<120E12FE121E120EABEB03FCEB01F014C01480EB02005B5B5B133813F8EA0F1C
EA0E1E130E7F1480EB03C0130114E0EB00F014F838FFE3FE17237FA21A>I<120E12FE12
1E120EB3ADEAFFE00B237FA20E>I<390E1FC07F3AFE60E183803A1E807201C03A0F003C
00E0A2000E1338AF3AFFE3FF8FFE27157F942A>I<380E1F8038FE60C0381E80E0380F00
70A2120EAF38FFE7FF18157F941B>I<EA01FCEA0707380C0180381800C0003813E04813
70A200F01378A700701370007813F0003813E0381C01C0380E038038070700EA01FC1515
7F9418>I<EA0E1F38FE61C0380E8060380F0070000E1338143C141C141EA7143CA21438
000F1370380E80E0EB41C0EB3F0090C7FCA9EAFFE0171F7F941B>I<3801F82038070460
EA0E02EA1C01003813E0EA7800A25AA712701278EA3801121CEA0C02EA070CEA01F0C7FC
A9EB0FFE171F7E941A>I<EA0E3CEAFE46EA1E8FEA0F0F13061300120EAD120FEAFFF010
157F9413>I<EA0F88EA3078EA601812C01308A212E0EAF000EA7F80EA3FE0EA0FF0EA01
F8EA003CEA801C130CA212C01308EAE018EAD030EA8FC00E157E9413>I<1202A41206A3
120E121E123EEAFFF8EA0E00AB1304A6EA07081203EA01F00E1F7F9E13>I<000E137038
FE07F0EA1E00000E1370AD14F0A238060170380382783800FC7F18157F941B>I<38FFC1
FE381E0078000E13301420A26C1340A238038080A33801C100A2EA00E2A31374A21338A3
131017157F941A>I<39FF8FF8FF391E01E03C001CEBC018120EECE010A2390702602014
70A239038430401438A23901C81880141CA23900F00D00140FA2EB6006A320157F9423>
I<38FF83FE381F01F0380E00C06C1380380381001383EA01C2EA00E41378A21338133C13
4E138EEA0187EB0380380201C0000413E0EA0C00383E01F038FF03FE17157F941A>I<38
FFC1FE381E0078000E13301420A26C1340A238038080A33801C100A2EA00E2A31374A213
38A31310A25BA35B12F05B12F10043C7FC123C171F7F941A>I<383FFFC038380380EA30
0700201300EA600EEA401C133C1338C65A5B12015B38038040EA07005A000E13C0481380
5AEA7801EA7007B5FC12157F9416>I E /Fk 17 120 df<121FEA3F80EA7FC0EAFFE0A5
EA7FC0EA3F80EA1F000B0B7A8A17>46 D<EB0FFC90387FFF8048B512E03903FC0FF03907
F003F848486C7EA248486C7E003F80A2497F007F1580A400FF15C0B0007F1580A4003F15
006D5BA2001F5CEBE001000F5C3907F807F86C6C485A6CB55A6C6C1380D90FFCC7FC222E
7DAD29>48 D<EB01E01303130F137FEA1FFFB5FC13BFEAE03F1200B3B1007FB512F0A31C
2E7AAD29>I<90391FF0018090B51203000314C73907F00FFF380F800148C7127F48141F
003E140F127E150712FEA215037EA26D90C7FC13E0EA7FFCEBFFE014FE6CEBFFC06C14F0
816C800003806C806C6C1480131F010014C01407020013E0153FA2151F126000E0140FA3
16C07EA26CEC1F807EB4EC3F0001C0137E9038FC01FC00F1B55AD8E03F13E0D8C00790C7
FC23317BB02E>83 D<EB7FF80003B5FC4814C0390FE01FE0391FF007F06E7E816E7EEA0F
E0EA07C0EA0100C7FCA2EB01FF133F3801FFF13807FE01EA0FF0EA1FC0EA3F80127FEAFF
00A41403EA7F80EBC00E393FF03CFF3A0FFFF87FF86CEBF03FC6EB800F25207E9F28>97
D<EB07FC90383FFF8090B512E03903FC07F03907F801F8390FE000FC001F147E485A8112
7F5B168012FFA290B6FCA30180C8FCA3127FA26C7EED0380121F6D13076C6CEB0F006C6C
131ED803FE13FCC6B55A013F13E0010390C7FC21207E9F26>101
D<90391FF007E09039FFFE3FF0489038FF7FF83907F83FF1390FE00FE1EDE0F03A1FC007
F0601600003F80A5001F5CA26C6C485AA23907F83FC090B5C7FC00065B380E1FF090C9FC
121EA2121F7F90B512C06C14F815FE6C806C15804815C0001F15E048C7127F007EEC0FF0
4814071503A4007EEC07E06CEC0FC0D81FC0EB3F803A0FF801FF006CB55AC614F0011F13
80252F7E9F29>103 D<EA01F812FFA3120F1207ADEC07F8EC3FFF4A7F9138F03FC09039
F9C01FE09038FB000F01FE80A25BA25BB3B500C1B51280A329327DB12E>I<EA01C0EA07
F0487EA2487EA36C5AA26C5AEA01C0C8FCA8EA01F812FFA3120F1207B3A6B51280A31133
7DB217>I<EA01F812FFA3120F1207B3B3A6B512C0A312327DB117>108
D<3903F007F800FFEB3FFF4A7F9138F03FC03A0FF1C01FE03907F3000F01F68013FE5BA2
5BB3B500C1B51280A329207D9F2E>110 D<EB07FE90383FFFC090B512F03903FC03FC39
07F000FE4848137F4848EB3F80003F15C0A24848EB1FE0A300FF15F0A8007F15E0A36C6C
EB3FC0A26C6CEB7F80000F15003907F801FE3903FE07FC6CB55AD8003F13C0D907FEC7FC
24207E9F29>I<3901F80FF000FFEB7FFE01F9B512809039FFE07FC0000F9038001FE06C
48EB0FF001F8EB07F816FC150316FEA2150116FFA816FE1503A216FC15076D14F86DEB0F
F06DEB1FE09039FBE07FC001F9B512009038F87FFEEC1FE091C8FCABB512C0A3282E7E9F
2E>I<3803F03F00FFEB7FC09038F1FFE09038F3C7F0000FEB8FF83807F70F13F613FE90
38FC07F0EC03E0EC0080491300B2B512E0A31D207E9F22>114 D<1338A51378A313F8A2
120112031207121FB512FEA33807F800B01407A70003130E13FC3801FE1C3800FFF8EB7F
F0EB0FE0182E7EAD20>116 D<B538801FFEA33A07F80003E0ED01C06D1303000315806D
1307000115006D5B6C140EEC801E017F131CECC03C013F133814E0011F5BA26D6C5AA214
F901075B14FF6D5BA26D90C7FCA26D5AA2147CA2143827207E9F2C>118
D<B53A0FFFF01FFEA33C0FF0007E0001F06D017F14E0000717C092383F80036C6C1680ED
7FC06D1507000102FF14006D9038EFE00F6C160E913801C7F0D97F815C14839139C383F8
3C013F15389138C701FCD91FE75C14EF9139FE00FEF0010F5D4A137F01075DA24A133F01
035D4A131F010192C7FC4A7F0100140E37207E9F3C>I E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin

%%EndSetup
%%Page: 1 1
1 0 bop 78 154 a Fk(0.1)83 b(Splitting)27 b(the)g(w)n(a)n(v)n(e)f(op)r
(erator)78 285 y Fj(When)d(one)g(applies)f(the)h(elastic)e(w)o(a)o(v)o
(e)h(equation)h(op)q(erator)h(for)f(spatially)f(v)m(arying)h(media,)78
345 y(sev)o(eral)13 b(\014rst)h(order)f(deriv)m(ativ)o(es)g(ha)o(v)o(e)
f(to)i(b)q(e)g(calculated)f(in)g(the)h(pro)q(cess.)21
b(The)13 b(w)o(a)o(v)o(e)g(equation)78 405 y(can)j(formally)f(b)q(e)h
(split)g(in)o(to)g(t)o(w)o(o)g(comp)q(onen)o(ts)f(so)i(that)g(deriv)m
(ativ)o(es)d(are)j(tak)o(en)e(with)h(resp)q(ect)78 466
y(to)23 b(medium)d(parameters)i(and)h(with)g(resp)q(ect)f(to)h(the)g
(propagating)i(w)o(a)o(v)o(e)c(\014eld.)41 b(Splitting)78
526 y(the)17 b(w)o(a)o(v)o(e)g(equation)g(op)q(erator)i(allo)o(ws)e
(one)h(to)g(adapt)g(deriv)m(ativ)o(e)e(op)q(erators)j(to)f(the)f(ph)o
(ysical)78 586 y(quan)o(tities)h(to)h(b)q(e)g(di\013eren)o(tiated.)27
b(In)19 b(particular)f(the)h(adaption)h(can)f(b)q(e)f(guided)h(b)o(y)f
(sp)q(ecial)78 646 y(prop)q(erties)c(of)g(the)g(quan)o(tit)o(y)l(.)19
b(Practical)14 b(problems)e(can)j(arise)f(since)f(the)h(deriv)m(ated)f
(quan)o(tities)78 706 y(in)j(general)g(complex)e(media)h(ha)o(v)o(e)g
(to)i(b)q(e)f(in)o(terp)q(olated)g(bac)o(k)f(to)i(collo)q(cation)f(p)q
(oin)o(ts.)151 787 y(In)k(seismic)f(mo)q(deling)g(adaptiv)o(e)h(tec)o
(hniques)f(are)i(w)o(ell)e(kno)o(wn.)35 b(Usually)20
b(the)g(adaption)78 847 y(of)g(a)h(quan)o(tit)o(y)d(is)i(in)g(the)g
(direction)e(of)j(a)f(ph)o(ysical)f(dimension)f(suc)o(h)i(as)h(time)d
(or)i(space.)32 b(In)78 908 y(general,)14 b(adaption)h(is)f(guided)g(b)
o(y)f(criteria)g(to)i(impro)o(v)o(e)c(n)o(umerical)g(accuracy)j(and)h
(stabilit)o(y)e(or)78 968 y(to)18 b(increase)e(the)h(sp)q(eed)h(of)f
(the)g(calculation.)24 b(Cunha)18 b(\()p Fi(?)p Fj(\))25
b(discussed)18 b(the)f(notion)g(of)h(treating)78 1028
y(the)g(propagating)h(isotropic)f(elastic)e(w)o(a)o(v)o(e)h(\014eld)g
(di\013eren)o(tly)g(from)f(the)i(underlying)f(medium)o(.)78
1088 y(F)l(or)i(anisotropic)g(media,)e(Igel)h(et.)29
b(al)18 b(\()p Fi(?)p Fj(\))30 b(sho)o(w)o(ed)19 b(an)g(in)o(terp)q
(olation)f(error)h(analysis)g(of)g(FD)78 1148 y(op)q(erators)c(and)g
(their)e(spatial)i(distribution)e(in)h(the)g(case)g(of)g(a)g(staggered)
h(grid.)21 b(In)14 b(this)f(section)78 1209 y(I)j(com)o(bine)e(b)q(oth)
j(notions)g(for)g(anisotropic)f(media.)78 1376 y Fh(0.1.1)70
b(F)-6 b(ormal)23 b(splitting)d(of)k(the)e(w)n(a)n(v)n(e)h(op)r(erator)
78 1490 y Fj(The)15 b(general)g(w)o(a)o(v)o(e)e(equation)i(\()p
Fi(??)p Fj(\))22 b(in)o(v)o(olv)o(es)13 b(calculating)h(tensor)i(pro)q
(ducts)f(and)h(deriv)m(ativ)o(es,)78 1550 y(where)g(the)g(w)o(a)o(v)o
(e)f(op)q(erator)j Fi(L)p Fj(,)d(in)o(tro)q(duced)h(in)g(a)g(previous)h
(section,)e(is)h(de\014ned)g(as)801 1662 y Fi(Lu)e Fj(=)19
b Fi(a)932 1687 y Fg(\030)970 1662 y(r)p Fj(\()t Fi(b)1031
1687 y Fg(\030)1079 1662 y Ff(\017)1070 1687 y Fg(\030)1108
1662 y Fj(\))14 b Ff(:)623 b Fj(\(0)p Ff(:)p Fj(1\))78
1784 y Fg(r)14 b Fj(is)g(a)h(partial)g(deriv)m(ativ)o(e)d(op)q(erator)k
(matrix)d(\()p Fi(??)p Fj(\))22 b(acting)14 b(on)h(its)g(argumen)o(t,)i
Fi(b)1568 1809 y Fg(\030)1621 1784 y Fj(is)d(a)h(medium)78
1852 y(prop)q(ert)o(y)k(tensor)h(and)g Fi(u)g Fj(is)f(the)g
(displacemen)o(t)e(\014eld.)30 b(Using)19 b(the)g(c)o(hain)g(rule)g
(for)g(co)o(v)m(arian)o(t)78 1912 y(deriv)m(ativ)o(es)c(and)i(applying)
f(it)g(to)g(the)g(tensor)h(pro)q(duct,)f(one)h(can)f(rewrite)f
(equation)i(\()p @beginspecial @setspecial @endspecial(0.1\))f(to)124
2048 y Ff(l)139 2055 y Fe(i)166 2048 y Fj(=)e Ff(a)244
2055 y Fe(j)262 2048 y Fg(f)287 2007 y Fd(X)294 2099
y Fe(j)r(k)q(l)347 2048 y Fj([)382 2015 y(1)p 382 2037
25 2 v 382 2082 a(2)411 2048 y(\()472 2015 y Ff(@)p 451
2037 71 2 v 451 2082 a(@)s(x)508 2089 y Fe(i)538 2048
y Fj(+)615 2015 y Ff(@)p 592 2037 75 2 v 592 2082 a(@)s(x)649
2089 y Fe(j)671 2048 y Fj(\))i Fi(b)737 2057 y Fc(ijkl)817
2048 y Fj(])g Ff(\017)867 2057 y Fc(kl)932 2048 y Fj(+)997
2007 y Fd(X)1004 2099 y Fe(j)r(k)q(l)1081 2048 y Fi(b)1112
2057 y Fc(ijkl)1192 2048 y Fj([)1227 2015 y(1)p 1227
2037 25 2 v 1227 2082 a(2)1256 2048 y(\()1317 2015 y
Ff(@)p 1297 2037 71 2 v 1297 2082 a(@)s(x)1354 2089 y
Fe(i)1383 2048 y Fj(+)1476 2015 y Ff(@)p 1453 2037 75
2 v 1453 2082 a(@)s(x)1510 2089 y Fe(j)1549 2048 y Fj(\))g
Ff(\017)1604 2057 y Fc(kl)1658 2048 y Fj(])p Fg(g)d Ff(;)54
b Fj(\(0)p Ff(:)p Fj(2\))78 2202 y(where)20 b([)p Ff(:::)p
Fj(])f(indicates)h(the)g(scop)q(e)h(of)g(the)f(partial)h(deriv)m(ativ)o
(es)e(and)i(where)f(i,j,k,l)14 b(=)i(1,2,3)21 b(is)78
2263 y(the)i(range)g(of)g(tensor)h(comp)q(onen)o(ts.)40
b(Equation)23 b(\()p @beginspecial @setspecial @endspecial(0.2)q(\))g
(is)f(written)h(in)f(terms)g(of)h(its)g(tensor)78 2323
y(elemen)o(ts)8 b(and)j(is)g(v)m(alid)g(for)g(arbitrary)g(anisotropic)g
(media.)17 b(In)11 b(the)g(\014rst)g(term)e(of)i(equation)g(\()p
@beginspecial @setspecial @endspecial(0.2\))78 2383 y(deriv)m(ativ)o
(es)i(of)j(the)e(elastic)g(constan)o(ts)i(are)f(computed;)e(in)i(the)g
(second)g(term)e(only)h(deriv)m(ativ)o(es)78 2443 y(of)k(the)f(strain)h
(tensor)f(are)h(computed.)23 b(The)18 b(sti\013ness)g(comp)q(onen)o(ts)
e(in)h(the)h(second)f(term)f(act)78 2503 y(as)j(constan)o(t)f(co)q
(e\016cien)o(ts)f(for)i(the)e(deriv)m(ated)h(matrix)e(elemen)o(ts.)25
b(The)18 b(deriv)m(ativ)o(e)e(op)q(eration)78 2564 y(in)i(equation)g
(\()p @beginspecial @setspecial @endspecial(0.2)q(\))g(app)q(ears)i(in)
e(b)q(oth)h(terms.)26 b(It)18 b(acts,)h(ho)o(w)o(ev)o(er,)e(on)i(t)o(w)
o(o)g(di\013eren)o(t)e(en)o(tities.)78 2624 y(In)23 b(the)g(\014rst)h
(term)e(the)h(deriv)m(ativ)o(e)e(of)j(medium)c(prop)q(erties)k(is)f
(tak)o(en)g(and)h(in)f(the)g(second)78 2684 y(term)18
b(the)i(deriv)m(ativ)o(e)e(of)j(the)e(w)o(a)o(v)o(e)g(\014eld)h(is)g
(computed.)31 b(In)20 b(the)f(end,)i(summation)d(of)i(b)q(oth)78
2744 y(terms)e(pro)q(duces)j(the)e(complete)f(deriv)m(ativ)o(e)g
(without)i(taking)g(a)g(deriv)m(ativ)o(e)f(of)h(the)f(medium)78
2804 y(and)14 b(w)o(a)o(v)o(e)f(\014eld)g(pro)q(duct,)i(as)f(equation)g
(\()p @beginspecial @setspecial @endspecial(0.1\))f(w)o(ould)h
(suggest.)22 b(F)l(or)13 b(a)i(medium)10 b(with)k(constan)o(t)p
eop
%%Page: 2 2
2 1 bop 78 -46 a Fb(Karr)n(enb)n(ach)645 b Fj(2)319 b
Fb(Splitting)19 b(the)f(wave)h(op)n(er)n(ator)78 146
y Fj(prop)q(erties)f(the)h(\014rst)f(term)f(v)m(anishes,)i(lea)o(ving)e
(the)i(second)f(term)f(in)h(equation)h(\()p @beginspecial
@setspecial @endspecial(0.2\),)f(whose)78 206 y(F)l(ourier)j(transform)
h(represen)o(ts)f(the)h(Christo\013el)g(equation)g(\()p
Fi(?)p Fj(;)j Fi(?)p Fj(\).)40 b(F)l(rom)20 b(an)j(economical)78
266 y(p)q(oin)o(t)f(of)g(view)f(of)h(equation)g(\()p
@beginspecial @setspecial @endspecial(0.2\),)h(some)e(exp)q(ensiv)o(e)f
(computations)i(can)g(b)q(e)g(sa)o(v)o(ed)f(if)g(the)78
326 y(medium)c(deriv)m(ativ)o(es)i(are)i(calculated)f(once)g(and)h(are)
g(then)f(reused)g(through)i(the)e(course)h(of)78 387
y(computation,)i(but)g(the)f(real)g(opp)q(ortunit)o(y)h(lies)e(in)i
(computing)e(the)h(solution)h(to)g(equation)78 447 y(\()p
@beginspecial @setspecial @endspecial(0.1\))15 b(more)f(accurately)g
(and)h(more)f(realistically)e(for)j(certain)f(t)o(yp)q(es)h(of)g
(media.)k(In)c(particular,)78 507 y(the)h(deriv)m(ativ)o(e)f(prop)q
(erties)h(can)g(b)q(e)h(adjusted)f(to)h(matc)o(h)e(the)h(prop)q(erties)
g(the)g(quan)o(tities.)78 821 y Fh(0.1.2)70 b(Adaption)23
b(of)g(the)g(deriv)l(ativ)n(e)e(op)r(erator)78 963 y
Fj(Tw)o(o)d(basic)f(observ)m(ables)h(in)f(the)g(w)o(a)o(v)o(e)f
(equation)h(are)h(the)f(w)o(a)o(v)o(e)f(\014eld)h(\(e.g.)24
b(stress,)17 b(displace-)78 1024 y(men)o(t\))h(and)j(the)f(medium)c
(prop)q(erties)k(\(sti\013nesses,)h(v)o(elo)q(cities\).)31
b(If)19 b(the)h(frequency)f(con)o(ten)o(t)78 1084 y(of)e(the)f(w)o(a)o
(v)o(e)f(\014eld)h(and)h(the)f(material)e(prop)q(erties)j(are)f
(similar,)e(the)i(same)f(t)o(yp)q(e)h(of)h(deriv)m(ativ)o(e)78
1144 y(op)q(erators)k(can)e(b)q(e)g(used)h(for)f(b)q(oth.)31
b(Then)19 b(\()p @beginspecial @setspecial @endspecial(0.1\))h(should)f
(b)q(e)h(used)f(rather)g(than)h(\()p @beginspecial @setspecial
@endspecial(0.2\).)30 b(The)78 1204 y(necessary)14 b(condition)g(is)g
(that)h(mo)q(del)e(and)i(w)o(a)o(v)o(e)f(\014eld)g(roughness)h(are)g
(of)f(the)g(same)g(scale.)20 b(F)l(or)78 1264 y(example,)15
b(if)i(the)g(medium)d(prop)q(erties)j(v)m(ary)g(smo)q(othly)g(in)g
(space)g(then)g(a)h(fairly)e(long)i(deriv)m(a-)78 1325
y(tiv)o(e)f(op)q(erator)i(can)f(b)q(e)g(adequate)g(to)g(appro)o(ximate)
f(the)g(analytical)h(deriv)m(ativ)o(e)e(n)o(umerically)-5
b(.)78 1385 y(Ho)o(w)o(ev)o(er,)21 b(if)h(the)g(medium)d(is)j(v)m
(arying)g(rapidly)l(,)g(the)g(same)f(deriv)m(ativ)o(e)g(op)q(erator)i
(ma)o(y)e(not)78 1445 y(b)q(e)f(adequate.)34 b(As)20
b(an)h(extreme)d(example,)h(realistic)f(mo)q(deling)i(of)g(w)o(a)o(v)o
(e)g(propagation)i(in)e(a)78 1505 y(random)g(medium)d(is)k(a\013ected)f
(b)o(y)g(the)g(length)h(of)g(the)f(\014nite)g(di\013erence)f(op)q
(erator.)35 b(A)21 b(long)78 1565 y(op)q(erator)i(implici)o(tly)18
b(a)o(v)o(erages)j(medium)e(prop)q(erties.)37 b(T)l(o)22
b(describ)q(e)f(lo)q(cal)g(e\013ects,)h(a)g(m)o(uc)o(h)78
1626 y(shorter)16 b(op)q(erator)h(w)o(ould)f(b)q(e)g(necessary)l(,)g
(but)g(for)g(the)g(same)f(prescrib)q(ed)g(spacing,)h(that)g(op)q(er-)78
1686 y(ator)i(migh)o(t)d(giv)o(e)h(a)i(less)f(accurate)g(deriv)m(ativ)o
(e.)22 b(A)17 b(la)o(y)o(ered)e(mo)q(del)h(is)h(an)h(extreme)c(case.)24
b(The)78 1746 y(spatial)d(v)m(ariabilit)o(y)d(in)i(the)g(horizon)o(tal)
h(direction)e(is)h(zero,)h(while)e(the)h(v)m(ariabilit)o(y)f(in)h
(depth)78 1806 y(o)q(ccurs)15 b(in)g(discon)o(tin)o(uous)g(steps.)21
b(Applying)14 b(equation)h(\()p @beginspecial @setspecial
@endspecial(0.1\))g(along)h(with)f(con)o(v)o(en)o(tional)f(\014nite)78
1866 y(di\013erence)f(op)q(erators)k(to)e(suc)o(h)f(a)h(mo)q(del)e(w)o
(ould)i(implicit)o(ly)c(assume)j(a)h(con)o(tin)o(uous)g(and)g(band-)78
1927 y(limited)g(medium)o(.)23 b(Equation)18 b(\()p @beginspecial
@setspecial @endspecial(0.2\))g(allo)o(ws)f(us)h(to)g(\\separate")h
(the)f(medium)c(from)j(the)g(w)o(a)o(v)o(e)78 1987 y(\014eld.)35
b(One)21 b(can)h(adapt)g(the)f(deriv)m(ativ)o(e)e(op)q(erators)k(to)e
(the)g(prop)q(erties)g(of)h(the)f(observ)m(ables)78 2047
y(they)16 b(are)g(op)q(erating)h(on.)78 2361 y Fh(0.1.3)70
b(Practical)21 b(issues)78 2503 y Fj(In)14 b(practical)f(implem)o(e)o
(n)o(tations,)f(problems)g(arise)i(when)g(one)g(calculates)f(deriv)m
(ativ)o(es)f(of)i(quan-)78 2564 y(tities)i(on)h(a)g(staggered)h(grid.)k
(A)17 b(staggered)g(grid)g(\(cf.)22 b(Figure)17 b @beginspecial
@setspecial @endspecial(0.1\))g(can)g(b)q(e)f(c)o(hosen)h(suc)o(h)g
(that)78 2624 y(the)g(outcome)e(of)i(a)h(deriv)m(ativ)o(e)d
(calculation)h(falls)h(on)o(to)g(the)f(grid)h(p)q(oin)o(t)g(of)g(the)g
(quan)o(tit)o(y)l(,)e(and)78 2684 y(is)j(m)o(ultipli)o(ed)d(together)j
(in)f(the)g(next)h(step)f(of)i(the)e(algorithm.)25 b(In)17
b(suc)o(h)h(a)g(sc)o(heme)d(no)j(quan-)78 2744 y(tit)o(y)f(needs)i(to)g
(b)q(e)g(in)o(terp)q(olated,)f(and)h(th)o(us)g(there)e(is)i(no)g(loss)g
(in)f(accuracy)l(.)28 b(Examining)17 b(the)78 2804 y(partitioned)h
(constitutiv)o(e)f(equation,)h(one)h(can)f(see)g(ho)o(w)g(a)h(commonly)
d(used)i(staggered)h(grid)p eop