Foundations of Geophysical Electromagnetic Theory and Methods

788 Pages·2017·11.29 MB·English

by  Michael S. Zhdanov

FOUNDATIONS OF
GEOPHYSICAL
ELECTROMAGNETIC
THEORY AND METHODS

FOUNDATIONS OF
GEOPHYSICAL
ELECTROMAGNETIC
THEORY AND METHODS
Second Edition
MICHAEL S. ZHDANOV
UniversityofUtah,
SaltLakeCity,UT,UnitedStates
Elsevier
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CONTENTS
Prefatcoet heS econEdd iti.o.n.. .... . ._ . . . . . . . . . . . . . . . . . .x .v ii. . . . . . . . . . . . . . . . . ..
Prefac. .e . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. X.I X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduct. .i.o.n. .. . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . x xiii
PART1 INTRODUCTTIOOF NIE LTDH EORY
CHAPTE1R DifefrentCiaallc uolfu sV ectora nFdiD eiflfdesr enFtoiramls 3
1.1Th eB asiDci fferentRiealla tionsohfFi ipesl Tdh eor.y, . . .. . .. _ . . . . . . • . . • . 4
1.1.C1o ncepotft heP hysicFaile ld. . . . .. . . . . . . . . . . . . . . •. . • . . • . . • . . . . 4
1.1.D2o t( Scalaarnd)C ros(sV ectoPrr)o ducotfsV ector.s. . . . . . . . • . . • . . . . 6
1.1.V3e ctoDri fferenOtpiearla to.r. .s. . . . . . . . . . . . . . . . . . . . . . . . • . . • . . . . 7
1.1.4D ifferentioaftt hiePo rno ducotfsS calaarn dV ectoFri eld. .s. . . • . . . . . . . II
1.2T heB asiIcn tegRrealla tionsohfFi ipesl Tdh eory. . . . . . .. . . . . . . . . . . . • . . . . 12
1.2.C1o ncepotfW orka ndF luxo fa F iel.d . .. . . . . . . . . . . . . . . . . . • .. • . . 12
1.2.G2a ussT'hse oreamn dI tVse ctoFro rmulati.o . n. s. . . . .. •. • . . . • . •. . . 1.4
1.2.S3t okesT'hse oreamn dI tVse ctoFro rmulat.i o. .n .s . . .• . • .. . . . . .. .• . 1.6
1.2.G4r eenF'osr mula.s. . . . . . . . . .. . . . . . . . . . . . . . . . . . • . . • . . • . . . . . . . 17
1.3D ifferentFioarlm si n FiTehledo ry. . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. . 1. .8•
1.3.C1o ncepotft heD ifferenFtoiram.l. . . . . .. ... .... ... ... ... ..... . .1 8
1.3.E2x teri(oWre dgeP)r oducotft heL ineaFro rm.s .. . . . . . . .2 1. . . . . . . . . . . .
1.3.C3a nonicRaelp resentaotfit ohneDs i fferenFtoiramlsi nT hree-Dimensional
EuclideSapna ce. . ... . . . . .. . . . . . . . . . . . . . . ... . . . . . . . . . ... . .. 22
1.3.T4h eE xteriDoerr ivat.i v.. . e .. . . . . . . .. . . . .. . . . . . ... . .. . • . . . . 2.3 .
ReferenacnedsR ecommendeRde adintgoC hapteIr . . • . . . . .. . . . . . . . . . . • . . 26
CHAPTE2R FoundatoifoF nise Tlhde ory 27
. . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . .
2.1F ielGde nerat.i.o.n. ................................... ..2.8. ......
2.1.1H annoniFcu nctioLniso;u vilTlhee'osr e.m. .................. ..2.8.
2.1.2U niqueneosfDs e terminaotfit ohneS calaFri elbdy I tGsr adieanntd t he
VectoFri elbdy I tDsi vergenacnedC ur.l. ................... ...2.9
2.1.F3i elGde neratCioonnd itio.n. .s. . . . . ... . . . . . . . . . . • . . .. . . . • . . . 31
2.1.S4o urceosft heF ieladn dT heiPrh ysicMaela nin.g. 32
2.1.V5o rticoefts h eF ieladn dT heiPrh ysicMaela nin.g. . . . . . . • . . •. . • . 35
2.1.S6o urcFei eladn dV orteFxi el.d. . .. . . . . . . . . . . . . . . . . •. . .. . . . • . . . . 37
2.2S tationFaireylE dq uatioannsdM ethodosf T heiSro luti.o.n. s. . . •.. •..• . ... 38
2.2.P1o issoEnq'usa tiofnosSr c alaarn dV ectoFri eld.s. . . . . . . . . . . . . . • . . . . 38
2.2.P2o inSto urcDei;r aSci ngulFaurn cti.o.n. .................... ..4.0. .
2.2.F3u ndamentGarle enF'usn ctifoonrt heL aplaEcqeu ati.o.n. ......... ..4 1
2.3S calaarn dV ectoPro tentioaftl hseS tationFairey.l d. . . . . . . . . • . . 44
2.3.S1c alaPro tentoifat lh eS ourcFei el.d. . . . . . . . . . . . . . . . . . • . . • . . • . . . . 44
v
VI Contents
2.3.Ve2c torP otentoifta hleV ortFeixel d . . . . . . . . . . . . . . . . . . .. .• .. .. .. • 4. 5
2.3.H3e lmholTthze oreamn dC lassificoaftt ihoeVn e ctoFri. eld.s.. . . .. ... . . 46
2.4N onstationaFriye ldasn dD ifferenitalF orms . . . . . . . . . . .4 8. . . . . . . . . . . .
2.4.N1o nstatioVneacrtyFo ire ladnsd D iffereFnotrimaisln F our-Dimensional
SpacEe4 . . . . . . . . . . .. .. 4.8 . . . . . . . . . . .
2.4.D2i fferenFtoirma lE quations . . . . . . .. . . . . . . . . . . . . . . . . . . . • . . . . . 49
2.4.A3m pere·TyDpiffeer entialF ormasn da oCntinuity Equation . . . . • . . . . . . 51
2.4.F4a raday-Type DiFfofremrasen dn tthieaFo lu rP-otential . . . . . . . •. . . . . . 52
2.4.N5o nstationarVye ctFoire lEdq uations . . . . . .. .• • . . . .5 3.
Referenacneds R ecommendReeda dintgoC hapter2 54
PART2 FOUNDATIOOFNE SL ECTROMAGNTEHTEIOCR Y
CHAPTE3R ElectromagFnieetlEidqc u ations 57
. . . . . .... . . . . .. . . . . . . . .. . . . . . . . • . .
3.1M axwell'sE qutaionsa ndB oundaCroyn ditions . . . . . . . . . . . . .. .a • • a • • • • 59
3.1.B1a siEcq uatiosni nt heT heoroyf E lectromagFnieetli.dc.s . . • ., • . . . . 59
3.1.P2hy sicalI nterpretoafMt aixown eWsE quations . . . . . .. • . . • .. . . . 62
3.1.B3o undaryC onditifoonrts h eV ectFoire l.d. ....... 68
3.1.T4h eF ielidna HomogeneuosM edium 72
3.2T ime-HamloEnliecc tromagFnieetli.d.c 73
3.3E lectromagnneertgiyac n dEP oyntinTgh'eosre m . . . .. . . . . . .. . . . . ._ . • . . . 75
3.3R.a1d iatCoinodinti osn. ..... .......... 76
3.3.P2o yntinTgh'eso reimnt heT imeD omai.n. . . . . . . . . . . . . . .. • .. . .7 7. . . .
3.3.E3n ergIyn equailintt hyTe i me omDain . . . . . . . . . . . . . . . . . • .. • .. . . . . 78
3.3.P4o yntinTg'heso rem int heFr equenDcoym ain. 80
3.4El ectromagnetci GreenT'esn sors 82
3.4.G1r een's Teinnst ohreFs r equenDcoym ain 82
3.4G.r2e enTe'nssor si nt heT imeD omain 83
3.5R eciprocity Relatonis . . . ... .. 84
3.5.L1o renLtCzln. ln.a , . . .. . . . ,. . .. , .,. . .,. , . ,. . . 84
3.5R.e2ci procity Relatifoontrsh eG reenT'esn soarnsd E lectromagnetic Fields 86
3.5.E3l ectromagGnreeetn'iscT ensoRre persentationT heorem.s_ . , ., . .., ., 87
ReferenceRse caonmmde ndedR eadintgoC hapter3 , . ,. . . . ., . ,. ._ . , .....' . , 89
CHAPTE4R ModelosfE lectrgonmeatInidcuc tioni nt heE art. .h. . . • . . . . .. • . . . . • .. . . 91
4.1M odelosf E lectromagFnieetli.dc s .. .................. 91
4.2S taticle cEtormagnetk Fjelds . . , . ,. . , . , . ., . . . . . . . • . . • . . • . . . . . 93
4.2.E1le crtostaticF ieladnsd E lectrosPtotanettiilacs .................... .9 3
4.2.B2o undaryC onditifoonErsl ectrosPtotaenttiialc. ...., . .. . .. ,. .. .. ,. . 95
4.2.C3al culation oft heEl ectorstatic Fielodfa S pecifiCehda rgDei stribution9 7 ..
4.2.A4n aloyg BetweeCno nstanCtu rreFnjte lds Elaencdt rostFaietlidsc 98
4.2.D5i rect CuFrlroewAn,st s ociaMtaegdne itc Fieladn,dt heB iot-SaLvaawn 101
4.2.Po6i nta ndD ipole Sourocne sUa n iforEma lth . . ... ... ]03
4.2.D7C Potentiali na nA nisotro picE arth. .. . . . . . . . . . 1.0 9. . . . . . . . . . . . . . ..
4.3E lectromagFnieeldt Diic ffusiinoC onn ductMievdei a. . . . . . . • . . . . . . . . • . . . 11 2
Contents VII
4.3.M1o nohcromatic Quasi-StEaMt iFcie lds . . . . .. . . . . . . . . . . . . • . . . . 113
4.3.P2Ja ne ElectromagWnaevteisci na H omogeneousM edium. ..., . . . .5 . 1]
4.3.E3l ectromagneticP otenti.a.l.s. ........................1.2.1. .
4.3.Q4u asi-StatiFoinealordfya D iploeS ourciena HomogeneousM edium 122
4.3.S5ph ericalE lectromagWnaevteisc. . . . . . . . . .... . . ........... 126
4.4E lectromagnetic Waves ...................................... 129
ReferenacnedRs e commendReedad ing toC hapt4e r 131
CHAPTE5R ElcetroamgnetFiice lidnHs o rinztoalSltyr atiMfeideida . .. .... ........ .. 133
5.1P lanWea veP ropagatinia o Lna yeredE arth 134
5.1.P1l anEel cetromagnetic Wavei na HorizontSatrlatlifiyed Medium 134
5.1.L2o w-FrequencyB ehavioofWr aveI mpedance 140
5.1.D3e finitoin ofF requency Windows . . . . . . ._ 143
5.2S pectMraetlho d ofC omputingE M FielidnsH orziontally Stratified. Medi1a45
5.2.F1ou rierT ransfiontr hIeSl pla tDioamla in. . . . . . . . . . 1.4 5. . . . . . . . . . . . . ..
5.2.P2o inSto urcoef t he DC FielidnH orizontally StratifieMd edium . . . . . . . . 148
5.2.E3l ectrFiice lodf thPco inSto urciena L ayereEda rt.h. .............15.8.
5.2.E4l etcricalD ipolSeo urcteh eDo Cf F ielidna HorizonLtaaylelryeM de dium1 61
5.2.E5x pressionsf oErl ectFriiecl idnsa H orinztoalyl Layered MediumU sing
the HankeTlr ansform ...............................1.6.3. .....
5.3E lectromagnetic Fielodfa nA rbitarryS ysteomfM agnetosphCeurrirce nitnas
Horziontalyl Homogeneous Medium _ . . . . . . . ._ . . .. . . .. 165
5.3.S1p atiFarle quency-Do(mSaFiDRn)e presentoaftt ihoeEn l ectromagnetic
Fielidna Horiztaolnly LayreedM ediu.m. . . .. . .. . . . ..... ]66
5.3.L2i pskaya-VaFnoyramnu lCaosn cernIimnpge danRcaet i.o.s. ....... 168
5.3.H3o rizonPtoalla rizoaftt ihoeEn l ectFriiecli dna Horizontally
HomogeneoEuasr tahn,d th eR educedS patilaW aveN umbeSrp ectrum 170
5.4E lectromagnFejteiJGcde sn eratienLd a yered Eartbhy E lectrica ndM agnetic
DjpoleT ransmtteirs. . . . .. . .. . . . . . . . . . . . . . . . . .. .. . . . .. . . . .. . . . . 173
5.4.S1p etcralR epresentaotfti hoeFn i elodfa HorizontalC urreDnitp oloen the
Surfaocfea HorizontLaalyleyr eMde dium . . . ._ . _ ...._ . _ . 173
5.4.E2le ctormagneticF ielodfa H orizonCtuarlr eDnitp olaett heS urfaceo fa
HomogeneouHsa Jf-Space . . . . . . . .... ... . . . . . . . ....... . 178
5.4F.r3e quenDcoym ainR epresentaotfti hoeFn i elodf a eVrtcialM agnetic
DipolAeb ovea Horizontalyl StratiMfieeddi um .... .. .... . 181
5.4.T4h eM agnetic Fofiae Vled11 icalM agnetci Dipoloent heS urfaocfea
UniforHma lf-Spa.c..e 183
5.4.N5e ara ndF arF iel.ds. ........ .. 184
5.4.F6re quency DomaiMne thofdor C omputing Transient ieFlds. ....... . 187
5.4.T7ra nsientF ielodfsa DipolSeo urcOeb serviedna HomogeneoMuesd ium
ando nt heSu rfaocfea HomogeneoCuosn duticng Half-SpFaiceel;id nst he
Neara ndF arZ ones. ................................. .1.9.0. ....
ReferenacnedRs e commendReeda ditnogC hapt5e r. . . . . . . . .. . .. . . . . . .• . . . . 199
VIII Contents
CHAPTE6R ElectromaFgineelitdniIs cn homogenMeeodui.sa. . . . . . . ... 2.0 1. . . . . . . . . .
6.1In tegralE quatiMoent hod . . . . . . . . . . . . . . .2 .03 . . . . . . . . . . . . . .
6.1.1B ackgrou(nNdo nnaaln)d A nomalouPsa rtosf th e ElectromagFnieeltdi c2 03
6.1.P2o ynting'sT heoreamn dE nergIyne quailtyf ora nA nomaloFuiesld 204
6.1.I3nt egarlE quatiMoent hoidn TwDoi mensions. 205
6.1.C4a lculatoifto hneF irVsatr iat(iForne chDeetr ivatoifvt eh)e
ElectromagnFeiteilfdco r2 -DM odels. . . . . . . . . . . . . . . . . .. .• .. . • .. 208
6.51 .IntegralE quatiMoent hoidn T hreDei mensions._ ......... ..._ 210
6.1.C6o ntarctioInn tegErqalu atiMoent ho.d. ............. .. 211
6.1.C7a clulationo ft heFi rsVta ritioan (FrecDheerti vatoifvt eh)e
ElectromagneticF jelfdo 3r- DM odel.s. _ . 213
6.1.8I ntegErqalu atiMoent hoidn A nisotrMoepdici u.m....... 216
6.1.D9i screteF ormo ft heA nisortopicI EF orwarMdo deling Method. ..... 218
6.1.C1o0n tractIinotengr alE quationa sth e PreconditiCoonnevde ntional Integral
Equation . . . . . . . . . . . . . . . . . . . .2 1.9 . . . . . . . . . .
6.2In tegralE quatiMoent hoidl Mlo delWsi thIn homogeneousB ackground
Conductivity .. . . . . . . . . . .. .. .. .. .. . .2 21
6.2.M1o delW ithI nhomogeneBoaucsk grouCnodn ductiv.i t.y . . . . .2 2.1 .
6.2.A2 ccuracyC ontrooflt hem c fEM ethod. . . . . .. . . . . .. . . . . . . . . . . . . 224
6.3F amiloyf L ineaarnd Nonlinear ntIegralA pprxoimationso ft heE lectromagnetic
Field .. .. ... . .. .... . . . .............2.2.6
6.3.B1o rn andE xtended Born Approximati.o n.s . . . . . . . 22.7 . . . . . . . . . . . .
6.3.Q2u asi-LinAepaprro xmiatino andT ensoQru asi·LiEnqeuaart ion 228
6.3.Q3L ApproximatUisionng a MultigAripdpr oach ... . . . . . . 2.2 9. . . . . . .
6.3.Q4u asi-AnalyStoliutcioanls f oar3 -DE lectromagFnieletdi c 230
6.3.Q5u asi.AnalyatnidQc uaals i.BoAmp proximatifoonras M odelW iht
VariabBlaec kgrou.n.d. ......... 233
6.3.Q6u asi-AnalyStoilcuatli foonr2s - DE lectromagnetiFci elds 236
6.3.7L ocalizNeodn linearA pproximation . . . .. .. 2.3 .6 . .
6.3.L8o calizQeuda si-LinAepaprrox imation. . . . . . . .. •. . . • . .. . • • . . . .2 38
6.4D ifferenEtquiaatolin Methods . . . .. . . . . . . . . . . . . . . . . . . . .. . . 2. 4• 0
6.4.F1i eldE quatioannsdB oundaryC onditio.n s. . . . . . . ... 2.4 0. . . . . . . . . . . .
6.4.E2l ectromagPnocttekn tEiqaula tonis nad Boundary Conditions 243
6.4.F3i niDtief fereAnpcper omxiation ofB oundary-VParolbuleem 245
6.4.D4i scretizoaftM iaoxnw ellE'qsu aitonsU sing Sat aggerGerdi d 246
6.4.C5o ntractPiroenc onditfiooarn S eyrs teomf FDE quations 249
6.4.6D iscretizoaftt hieEo lne ctromagneticD iPfofteernetEniqtauiJaa tli ,o n., s25]
6.4.F7in iteE lemenSto lutoiofBn o undary-ValuPer oblems . . . . 2.5 4. . .
6.4.E8d ge·BasFeidn iteE lemeMnett hod . . . . . . . . . . .2 57. . . . . . . . . . .
ReferenacnedRs e commendReeda dintgoC hapte6r . . . . . . . . . .. • . . . . . . . . • . . . 261
Contents IX
PART3 INVERSIOANN DI MAGINOGFE LECTROMAGNFEITEILCD
DATA
CHAPTE7R PrinciopflI eIsI -PoIsnevde rPsreo blSeoml ution 269
. . . . . . . . . . . . . ... • . . . . .
7.1I LL-PosIendv erPsreo ble.m. s . . . . . . . . . . . . . . 2.7 0. . . . . . . . .
7.1.1F OimulatioonfW ell-Posaenddl I IP-osed Problem.s .. ... . . . . 2.7 0. . . ..
7.1.2C orrectSneets s . . .. . . . . . . . . . .. . . . .. . . . . . .. . • . . . . . • .. 271
7.1.3Q uasi-Solutiono fth eI II-PoPsreodb le.m. 271
7.2F oundatoinso fR egularizatoin Theory 272
7.2.1 DefinitioofMn i sfiFtu nctional 272
7.2.2R egulariOzpienrga to.r..s_ . . . 275
7.2.3S tbailziingF unctjonals 276
7.24 TikhonovP arametFruincc tional 281
7.3R egularizPaatriaomne .t.e.r. ... .... ... . ... . . ....... 282
7.3.1T ikhonovM ethoodf R egulariPzaartaimoenSte elre ct.i.o.n. . . .. • . . . 282
7.3.2L -CurvMee thoodf R egularizaPtairoanm etSeelrec toin . . . . .. • . . • . . . 285
Referencesa nd Recommended Reading toC hapter7 . . . .. . . . . . . . . . .. . . . • . . . 286
CHAPTE8R ElectromagInnevteircs ion 289
. . . . . . .. . . . . . . . . . . . . . . . . . . . . • . . . . • . . . . .
8.1L ineaIrn versi.o..n.s. . _ ._ . .. ... ....... ............. . . ...... .. . 290
8.1.1B orn Inversion . . . . . ._ . . ..... _. . . . ._ ..... 290
8.1.D2i screte LinearE M InverPsreo ble.m. . ..... .. . .. . .. . ..... . . .. 291
8.1.T3h eT ikhonoRveg ularization .Methoodf L inearI nversi.o.n., .,. . , , 2,9 2,
8.104D efiitonin oft hWee ightMiantgr ifcoerMs o dePla rametaenrdDs a ta 293
8.1.5Ap proximaRteeg ulariSzoeldu tioofLn i neaTrn verPsreo blem 295
8.61 .TheL evenberg-MarqMueathroddt . . . . . . . . . . . . . ._ . 297
8.1.C7o nductivhyI magibnygt heB ornA pporximation, ........... ... . 297
8.1.l8t eraBtoirnv el nversi.o. n. .s. . . . . . . . . . . . .... . . . . . . • . . 301
8.2N onlinear [nversion . . . . . . . .. . . . . . . . . . . . . . . . . . . . ... . 302
8.2.1Fo rmulatino oft he NonlinEeMar I nverPsroebl em .. .. . . ..... . .... .3 02
8.2.2R egularized SolutioofNn o nlinearD iscrEetMe I nverse Proble.m. .... 302
8.2.T3h eS teepeDsets cenMte thod foNro nlienarR egulariLzeeasdt -Squares
[nversion . . ._ . 303
8.2.T4h eN ewtonM ethofdo rN onlineRaegru larizedL east-SquIanrveerssio n. 304
8.2.5Nu mericSaclh emeosf t heN ewtonM ethod foonrli neNarR egularzied
Least-SquIanrveesr si, .o.n.. . . ...... , . . .... . ...... ... ... . 305
8.2.N6o nlinLeeaarst S-quares Inversbiyot nh Ceo njugaGtrea dieMnetth od 306
8.2.T7h eN umericSaclh emeo ft heR egulariCzoenjdug ateG radieMnett hofdo r
NonlineLaera st-SquIanvreerssi no .......................3.0.7
8.2.F8 rechDeetr ivaCtiavlecu lation . . . . . . . . . 308
8.2.F9r echDeetr ivatCiavlec ulatUisiongn Quasi-BAopprrno ximation 310
8.3Q uasi-LinIenavre rsion ..._ . . . . . . . . . . . . 3.1 1. . . . .
8.3.P1r inciopflQe usas i-LinearI nversi.o.n.. .. ..................3.1.1
83..2 Localized Quasi-LinearI nversion . . ._ . . ... _ . . . . . . . . . • . . • . . . . . . . 312
8.4Q uasi-AnalIyntvirescioanl . . . . .. . . . . . . . ._ . .. . . .. • . . . ._ . . . .... _ • . . . . 312