A review of the finite-element method in seismic wave modelling

Transcription

1 Revew of he fe-eleme meho A revew of he fe-eleme meho sesmc wave moellg Faraak ahmoa a Gary F. argrave ABSTRACT mercal solos of he scalar a elasc wave eqaos have grealy ae geophyscss boh he forwar moellg a mgrao of sesmc wavefels complcae geologc mea. I P- a S-wave propagao, he fe-eleme meho s a powerfl ool for eermg he effec of srcral rreglares o wave propagao. Depeece of he wave eqao o boh spaal a emporal ffereals reqres solvg boh spaal a emporal screzao. I he spaal screzao sep D a D, pecewse lear bass fcos a he Galerk meho are he mos commoly se ools. Afer solvg spaal screzao wh he fe-eleme meho, he wave eqao reces o a ODE orary ffereal eqao. I hs regar, ffere ahors se ffere ODE solver clg Rge-Ka meho a fe-fferece meho. Ths paper wll famlarze he reaer wh he verse approaches of solvg emporal screzao. A applcao of fe-eleme mehos o solve sesmc wave moo lear vscoelasc mea where elasc sras evelopme epes o oly o he crre sae of he sress a sra b o he fll hsory of her evelopme, sg memory varable formalsm spaal screzao sep, s oe of he revewe secos. ITRODUCTIO mercal solos of he scalar a elasc wave eqaos have grealy ae geophyscss boh forwar moellg a mgrao of sesmc wavefels complcae geologc mea, a hey promse o be valable solvg he fll verse problem. Oe of he mercal mehos ha ca be apple o he problem of wave propagao s he fe-eleme meho. The fe-eleme meho has become he mos wely accepe geeral-prpose echqe for mercal smlaos egeerg a apple mahemacs. Prcpal applcaos arse com mechacs, fl flow, hermoyamcs, a fel heory. I hese areas, compaoal mehos are esseal a beef srogly from he eormos avaces comper echology. The fe-eleme meho s a powerfl ool for he mercal moellg of sesmc boy-wave propagao a heerogeeos elasc mea. The fe-eleme resls agree wh fe-fferece resls Smh, 975 b spe of hs, he meho has ever become poplar he geophyscal lerare. Perhaps hs s becase mplemeao s more ffcl ha oher mehos Kay, 996. The fe-eleme meho s a geeral echqe for cosrcg appromae solos o boary-vale problems o solve physcal problems. I he fe-eleme aalyss, a boy s cosere o be a assemblage of scree fe elemes ercoece a oal pos o eleme boares. Oe-, wo-, a hree-mesoal elemes have a varable mber of oes. Fgre shows some ypcal com elemes. Bahe 996 cosers he followg seps o solve a problem wh he feeleme meho: CREWES Research Repor Volme 5 3

2 ahmoa a argrave The ealzao of he physcal problem o a mahemacal moel reqres cera assmpos ha ogeher lea o ffereal eqaos goverg he mahemacal moel. The efcao of he kow splacemes ha compleely efe he splaceme respose of he srcral ealzao. 3 The formlao force-balace eqaos correspog o he kow splaceme a he solo of hese eqaos. 4 The erpreao of he splaceme prece by he solo of he srcral ealzao Base o he assmpo se. a Trss a cable elemes b Two-mesoal elemes c Three-mesoal elemes FIG.. Typcal elemes for fe-eleme meho afer Bahe 998. Sce he fe-eleme solo echqe s a mercal procere, s ecessary o assess he solo accracy. If he crera are o me, he fe-eleme solo has o be repeae wh refe solo parameers sch as fer mesh l sffce accracy s reache. Appe A shows he fe-eleme process brefly. Depeece of wave eqao o boh spaal a emporal ervaves reqres boh spaal a emporal screzao. I spaal screzao D a D pecewse lear bass fcos, he Galerk meho varaoal form s oe of he fe-eleme mehos ha s se by may ahors. CREWES Research Repor Volme 5 3

3 WAVE PROPAGATIO I ELASTIC EDIA D Spaal Dscrezao Revew of he fe-eleme meho Coser a oe mesoal D oma, X, a elasc mem. The eqao of moo for oe egree of freeom elasc wave s wh smpler form as, ρ, ρ ',, ρ,,, where s he approprae mols relae o he esy ρ hrogh he phase velocy. Here he fe-eleme solo of smple wave eqao wh he boary coos, l, a al coo, s revewe. The PDE s or fe-eleme moel problem. The ffcly s he reqreme ha a solo, o eqao sasfes he ffereal eqao a every po, < < l, s oo large. To overcome hs ffcly, we shall reformlae he boary-vale problem a way ha wll am weaker coos he solo a s ervaves. Sch reformlaos are calle weak or varaoal formlaos of he problems. Oe weak saeme of he moel problem s gve as follow: f he fco, sch ha he ffereal eqao, ogeher wh boary coos, are sasfe he sese of weghe averages Becker, 98. By he sasfaco of all weghe averages of he ffereal eqao, as Becker 98 efes, we reqre l ρ,, v, for all v H. 3 v belogs o class H Lear Hlber space a has zero vales a a l he boares. The frs ervave of sch Hlber fcos v s sqare-egrable v <. ay ahors calle hs v he weghg fco. Takg egrao by pars for eqao 3 yels l l l v ρ, v, v,. 4 I s clear ha f eqao 3 s re, here ca be o poro of fe eleme legh of he erval < < l wh whch he ffereal eqao fals o be sasfe. If fco, sasfes eqao 3, wll sasfy eqao as well. Becker 98 saes ha for havg a symmerc weak formlao we assme ha he solo fcos, also belog o class H. ow we choose a se of bass orhogoal fcos for H sch ha every fco H ca be epresse as a lear combao of sch bass fcos. Becker 98 represes v he form CREWES Research Repor Volme 5 3 3

4 ahmoa a argrave where he coeffces a are gve by v a, 5 a l v. 6 If we ake oly a fe mber of erms he seres 5, he we wll oba a appromao of v of v: v a. 7 The bass fcos {,,, } efe a -mesoal sbspace H. Afer seg he weghg fcos as a lear combao of bass fcos; we are ow reay o coser he Galerk meho for cosrcg a appromae solo o he varaoal boary-vale problem eqao 3. Becker 98 says ha he Galerk meho cosss of seekg a appromae solo, o eqao 3 sbspace H raher ha a whole space H of he form,. 8 The coeffces ake o he vale of he splaceme a a scree mber of oal pos pos,,. For a beer ersag of v, sppose he vale of v a oes,, a 3 are.9,.7, a., respecvely, wh bass fcos as Fgre b. Sbsg hese vales o eqao 7 gves v , so ha hese hree compoes combe o gve he coos pecewse-lear fco show Fgre 3a a he pecewse-lear fco v has he form of Fgre 3b. I he Galerk meho he solo fco, s appromae by pecewse-lear fcos v, wh vales cocg wh hose of, a he oes, he resl s a polygoal fco whch closely resembles,. Ths s he pecewse-lear erpolao of he eac solo,. As he mesh s refe.e., as he mber of elemes s crease, he fe-eleme erpola becomes progressvely closer o, see Fgre 3c. H of 4 CREWES Research Repor Volme 5 3

5 Revew of he fe-eleme meho Elemes: Ω Ω Ω 3 Ω 4 h h h h a l 3 4 h l l 3 4 b FIG.. a A smple D oma mesh b Eample of fe-eleme bass fcos boh, afer Becker, 98. CREWES Research Repor Volme 5 3 5

6 ahmoa a argrave There s o ee for weghe fco v o be epresse by -mesoal bass fcos{,,, }. ay ahors coser hem as separae ses Bahe, 996; Hghes, 987; a arfr, 984. arfr 984 saes ha choosg weghg fcos as lear combao of bass fcos sch ha he weghg a bass are ecal, geeraes he Galerk formlao of he fe-eleme meho. arfr 984 also meos whle he Galerk meho of seg he weghg fco v eqal o ha of he bass fcos has bee amog he mos poplar fe-eleme echqes, here s o compellg reaso o o so. Frhermore, here s o compellg reaso o se he same evalag he mass, ampg, sffess, a loa marces. Sbso of he epaso of, a v, eqaos 7 a 8, o eqao 4 resls a [ ρ ] + [ m ]. 9 Ths s he same resl as Kay a Krebes, 999. oe ha geg hs resl he boary vales are alreay apple. Sce he fcos v, a hs he coeffces a, are arbrary, he epresso he brackes of eqaos 9 ms eqal zero. The srcre of eqao 9 ca be wre more compac form as whose a K marces are a + K, K ρ. l l 6 CREWES Research Repor Volme 5 3

7 Revew of he fe-eleme meho v a 3 4 b c FIG. 3. a v , b pecewse-lear fe-eleme represeao of he fco v boh, afer Becker, 98. c Local lear erpolao of smooh fco. CREWES Research Repor Volme 5 3 7

8 ahmoa a argrave The reslg orary ffereal eqaos me for he kow splacemes a he oal pos ca be wre as + K. The marces a K are calle mass a sffess marces, respecvely. Eqao s he correspoece fe-eleme eqao. The erms mass a sffess mar are msomers for he acosc wave eqao where he recprocal of he esy s acally cle he sffess mar a he recprocal of sffess he mass mar arfr, 984. Calclag he mass mar, from eqao, leas o a sparse mar as Fgre 4. FIG. 4. Caroo of mass mar assembly D Afer Kay, 996. Coserg a sorce F a a ampg erm C wave eqao, he correspog fe-eleme eqao becomes + C + K F. 3 Eqao 3 s se for he fe-eleme moel of wave eqao by Smh 975, Jal 994, Kay 999, Hghes 987, Cohe, Sllva 983, a arfr 984, amog ohers. The al-vale problem for eqao 3 cosss of fg a splaceme,, sasfyg eqao 3 a he gve al aa:, v. 4 Here we ry o prese a fe-eleme solo of eqao. I hs regar, we ee o efe he bass fcos he cosrc he sffess a mass marces. As 8 CREWES Research Repor Volme 5 3

9 Revew of he fe-eleme meho meoe above, fe eleme mehos assme ha he solo ca be represee wh bass fcos emerae as. These fcos are chose o be zero ose some fe erval. I ao, f he has he vale of y a oe oe efe laer he compaoal mesh a zero a every oher oe, a oe-o-oe correspoece bewee bass fcos a he gr oes resls. Kay 996 also meos ha he are chose o be sffcely smooh sch ha all egrals wll es. The ma ea as Becker 98 says, s ha he bass fcos ca be efe pecewse over sbregos of he oma calle fe elemes a ca be chose o be very smple fcos sch as polyomals of low egree. Cosrcg sch a se of pecewse bass fcos, as Becker 98 saes, we frs paro he oma X he erval < < l of or problem o a fe mber of elemes. Fgre a shows, for eample, he oma of or moel problem. Wh each eleme, cera pos are efe, calle oes or oal pos, whch play a mpora role he fe-eleme cosrcos. The colleco of elemes a he oal pos makg p he oma of he appromae problem s somemes referre o as a fe-eleme mesh. The bass fcos shol be smooh eogh o be members of a Hlber space. Oe very smple sample se of bass fcos for he oma showe Fgre a, s show Fgre b. If he cooraes of he oes are eoe,,, 3, 4, show for,, 3are gve by he he bass fcos for h + for +, 5 h+ for < a > + where h s he legh of eleme Ω. The frs ervaves of bass fcos also ca be calclae very easly. Coserg eqao he ass a Sffess marces ca be compe. Solvg he wave eqao a fg mass a sffess marces by cosrcg he bass fcos, eqao may be cas o he form K. 6 The evelopme of boary-vale problems escrbg physcal pheomea wo mesos follows closely he oe-mesoal reame gve above, fferg oly aspecs cae by he hgher mesoaly. CREWES Research Repor Volme 5 3 9

10 ahmoa a argrave D Spaal Dscrezao I wo mesos, he oma s screze o eleme omas smlar o he D case. I wo mesos he eleme omas mgh be smply ragles a qarlaerals Fgre 5. oal pos may es aywhere o he oma b mos freqely appear a he eleme verces a ereleme boares a less ofe he erors Hghes, 987. FIG. 5. D oma a D mesh for fe-eleme spaal screzao afer Hghes, 987. I he D oma eqao 4, we roce a se of bass fcos as lear fcos a + a. For geeralzg he cocep o D, Becker 98 cosers he lear fco, y a + a + a y. 7 3 Ths fco eermes a plae srface. The se of sch a bass fco o a ragle eleme wll resl he appromao of a smooh fco v,y ha we ha before as weghg fcos by a plaar fco, f, y 8 f for, y as coorae of each oe fe-eleme mesh. The bass fcos, y,,, are cosrce he same maer as escrbe for D. The bass fcos correspog o aace elemes he mesh are smply pache ogeher o proce a pyram oal po he mesh Becker, 98. There are may mechacal egeerg sofware packages ha solve boary vale problems by fe-eleme mehos. Iclg SC.Para a ISA II/Dsplay III. Fgre 6 s a D seco of a D fe-eleme solo o a sample wave propagao eece by ISA II. The geeralzao of he fe-eleme meho scsse o hree mesos s heorecally smple, hogh compaoally more epesve arfr, 984. CREWES Research Repor Volme 5 3

11 Revew of he fe-eleme meho 5.E- 4.E- 3.E-.E-.E-.E+ -.E-.E+.E- 4.E- 6.E- 8.E-.E+.E+.4E+.6E+.8E+.E+ -.E- -3.E- -4.E- a+ T/ me a.e+.e+.e+ -7.8E-.E-.E- -.4E-.E-.35E E- 3.E- 7.3E- 4.6E- 4.E E-.7E- 5.E- -.64E-.34E- 6.E- -.73E- 3.77E- 7.E E- -.9E- 8.E-.66E- -.88E- 9.E- 3.7E E-.E+ 4.57E- -.88E-.E+ 3.7E- -.9E-.E+.66E- 3.77E-.3E E-.34E-.4E+ -.73E-.7E-.5E+ -.64E- 4.6E-.6E E E-.7E+ 7.3E- -.4E-.8E+.35E- -7.8E-.9E+.E- FIG. 6. D fe-eleme solo o wave eqao a elasc Isoropc maeral. The ble plo caes he al wavele a he re plo caes he wavele afer half-pero of al wavelegh he cere po a sc efe as al problem. Boary a al vale boh were se o zero. Ths s D seco of D fe-eleme solo of a sample wave propagao, Eece by ISA II/Dsplay III sofware. Dsplaye aa able shows D mercal vales. CREWES Research Repor Volme 5 3

12 ahmoa a argrave Temporal Dscrezao There are ffere ways o solvg he me-sep eqao 5. ay ahors se ffere mercal mehos for solvg hs ODE Orary Dffereal Eqao. Smh 975 ses a mercal solo sg he Rge-Ka algorhm, b here are o eals of hs solo hs paper. Rge-Ka s a mercal meho for solvg ODE ha oes o se he eplc evalao of he ervaves Ambramowz a Seg, 97. A bref smmery of he Rge-Ka meho s fo Appe B. Becker 98 preses a fe fferece solo for he me-epee hea eqao. The hea eqao s very smlar o wave eqao. The hea eqao a s fe-eleme moel are, c, c k ',, 9 C + K. The me-sep solo for he hea eqao wll hol for he wave eqao wh some slgh mofcao. Becker 98 saes ha by sg he fe-eleme meho, we have scceee recg he gve al-vale problems 9 o he sysem of orary ffereal eqao. Sce we have o ye screze he behavor of me, eqao s referre o a sem-scree fe-eleme appromao. To oba a flly scree appromao, we ms ow roce a appromao of he behavor of me. Becker 98 oles oe of he smples mehos, forwar fe-fferece appromao. The me oma T s ve o k eqal ervals of legh T / k. A, he solo s kow: ˆ.To avace he solo me from o + he forwar fe-fferece operaor s se, he eqao leas o he algorhm ; C I C K K. Sce s kow, ca be calclae sg eqao. I hs way we egrae he solo me from o k for ay esre mber of me seps. I shol be oe ha for a gve mesh sze, a lmao o he me-sep sze s eee orer for hs scheme o be mercally sable. Takg he same seps as Becker 98 for wave eqao, a sg he ceere seco fe-fferece operaor CREWES Research Repor Volme 5 3

13 Revew of he fe-eleme meho +, 3 eqao 6 becomes K. 4 So he solo me for ay mber of me seps s acheve. Srage a F 973, solvg he hea eqao wh fe-eleme mehos, also ses Galerk meho; a for solvg he me-epee eqao, hey sae ha: I s aral o ask why fe-elemes are o se also he me reco. Ths has ceraly bee aempe, b o wh grea sccess a fac a sraghforwar applcao of he Galerk prcple may cople all he me levels, a esroy he crcal propery of propagao forwar me. I he me-sep solo for eqao, Srage a F 973 aalyze he Crak- cholso scheme, whch s cere a + a herefore acheves seco-orer accracy me: Rewre, he appromao + C + K. 5 s eerme by K K +. 6 Srage a F 973 say ha a acal compao, he mar o he lef ca be T facore by Gass elmao o LL, where L s Cholesky s lower raglar mar, a he wol be compe a each sep by back sbsos, K / T / L, L. 7 Srage a F 973 sae ha becase he coeffce of eqao s meepee, he he src Galerk heory he mass a sffess marces ms be compe a each me sep. ow becomes clear ha why he mplemeao of feeleme meho wave eqao s so har. Thoro 98 akes he same fe-eleme procere for solvg he hea eqao 9 a ges he fe-eleme eqao. He also ses he fe-fferece operaor for solvg he me-epee par of he hea eqao. CREWES Research Repor Volme 5 3 3

14 ahmoa a argrave + C + K F v oel Decomposo Temporal Dscrezao, ν ν a + Cν + ν + gve ν + oel Decomposo + K F { β a + βa } { γ a + γa } h ω h + ξω + v F Temporal Dscrezao a h + ξω ν + + ν + ν ν + h ω F { β a + βa } { γ a + γa }, ν gve FIG. 7. The eplc fe-eleme meho for wave eqao afer Hghes, 987. Hghes 987 solvg he wave eqao by fe-eleme also rves eqao 3 a saes ha he mos wely se famly of rec mehos for solvg eqao 3 wh al coos 4, s he ewmark famly, whch cosss of he followg eqaos: v a + C + K F + v v + Cv + + K + F 8 [ β a + βa ] 9 [ γ a + γa ], 3 where, v, a a are he appromaos of,, a, respecvely. Eqaos 9 a 3 are fe-fferece formlas escrbg he evolo of he appromae solo. The parameers β a γ eerme he sably a accracy characerscs of he algorhm er coserao. Eqaos 8 o 3 may be hogh of as hree eqaos for eermg he hree kows, v, a a. I s beg assme ha, v, a a are kow from he prevos sep s calclaos. Hghes 987 smmarzes he eplc fe-eleme meho for he wave eqao Fgre 7 where he ω h s he weghg fco as efe prevos pars. Hghes 987 saes whe he me sep resrco s o oo severe, as s ofe he case elasc 4 CREWES Research Repor Volme 5 3

15 Revew of he fe-eleme meho wave-propagao problems, he ceral fe-fferece meho s geerally he mos ecoomcal rec egrao procere a s hs wely se. Hghes 987 scsses alerave approaches from he Galerk a ewmark mehos, for solvg he mesep of he wave eqao sg he fe-eleme meho, clg: he Lear lseps LS, a Hobol mehos; Collocao Schemes; he α-eho, he Wlso-θ meho, he Fo-Goow meho, a he precor-correcor meho, whch all are all weghe resal me egrao schemes. All meoe weghe resal me egrao schemes le a grey area bewee fe-eleme a fe-ffereces mehos Bahe, 996; Zekewcz, 997; a arfr, 984. arfr 984 ses a hree-po scheme for screzg he homogeeos me meso a saes oe ca apply he emporal bass fcos for ODE eqao 3 Zekewcz, 977. WAVE PROPAGATIO I LIEAR VISCOELASTIC EDIA Vscoelascy All we have show p l ow has bee elasc mea, b here s also he feeleme solo for wave propagao vscoelasc a asoropc mea. Kay a Krebes 999 apply he fe-eleme meho o a solo of sesmc wave moo lear vscoelasc mea. Frs we ll scss vscoelasc mea. Zekewcz characerzes vscoelasc pheomea by he rae a whch elasc sras evelop epes o oly o he crre sae of he sress a sra b, geeral, o he fll hsory of her evelopme. Ths, o eerme he creme of elasc sra over a gve me erval or me-sep, s ecessary o kow he sae of sress a sra a all preceg mes. Kay 996 escrbes he sress-sra relao vscoelasc mea as: σ ε ' R ',. 3 where σ s sress, ε s sra a he esor R s he sra relaao fco we wll efe shorly. As sae by Ak a Rchars, he R s he characersc relaao me for sra er a apple sep sress. If a he sysem goes from a sae of o sra o a sae wh some small b o-zero sra, he he sress for ay me afer, s gve by hs covolo of he sra hsory wh he relao fco. Kay 996 shows ha a egral form of he sress-sra behavor a vscoelasc mea ca be wre a covolo form as: ε ' σ R ' '. 3 ' Kay 996 saes ha here s a oe-o-oe correspoece bewee he sffess mar of elasc heory a he relaao fco here. Bla 96 saes ha he Laplace rasform of he vscoelasc problem wll formally have he same form as he elasc problem. The erms ha correspo o he elasc cosas appear as fcos of he Laplace rasform parameer,. Bla 96 also shows ha he famlar ecoplg CREWES Research Repor Volme 5 3 5

16 ahmoa a argrave of SH a P-SV waves for elasc mea hols for lear vscoelasc mea oo. Kay a Krebes 999 say he erm vscoelasc ca apply o problems ay mber of mesos, wheher acosc, aplae-sra SH waves, or wh mlple egrees of freeom a each po a ehbg cople P-SV waves. Kay 996 saes ha a geeralze awell moel compose of compoes has a scree relaao specrm wh m / τ R r + δ a e H, 33 where r s he relae mols a eqao s eqal o he sm r + δ, a H s he Heavse sep fco Bla, 96; Ch.. For hs form of R eqao 33, he sress eqao 3 ca be epresse as σ X, 34 where he X are he memory varables. Kay 996 efes memory varables erms of he sra as X a δ τ τ / τ e τ τ. 35 Ths epresso s eqvale o he ffereal eqao Spaal screzao for vscoelascy a δ X + X. 36 τ τ Kay a Krebes 999 se he followg wave eqao D heerogeeos mem ρ, σ,, 37 where σ, s sress. I s he same form as we sae eqao for elasc mea, becase for a sgle egree-of-freeom, sra s eqal o,. oe ha for elasc mea δ, he eqao of moo has he same form as eqao. So all he resls for vscoelasc mea ca also be apple o elasc mea. Kay a Krebes 999 coser he eqao 34 for sress a sbse ha o eqao 37. Repeag he Galerk aalyss, mlplyg by he wegh fco v, as show above elasc mea yels o smlar eqao as eqao 4, wh he form 6 CREWES Research Repor Volme 5 3

17 Revew of he fe-eleme meho CREWES Research Repor Volme X v v v l l l,,, ρ. 38 Usg he memory varable X, Kay a Krebes 999 presee wo mehos of spaal screzao sg he memory varable X. Here we s scss oe of hose mehos. Ths screzao scheme s aalogos o eqao 7 appromag he memory varable X as X ξ. 39 These epasos are sbse o eqao 36 a a smlar Galerk aalyss s apple o eqao 36. The seco-orer orary ffereal eqao eqao 3 s cople o wo frs-orer ffereal eqaos a v f v /. Repeag he Galerk aalyss sg hs epaso for X wll resl a ODE for. The he sysem of eqaos ca be smmarze as, v C C C C K v T T I I ξ ξ ξ ξ τ δ τ δ 4 where C K T a C I τ δ τ ρ δ τ. 4 Deals ca be fo Kay 996. I ao o he spaal screzao presee above, some meho of avacg he solo me s reqre, a he reslg scheme ms be show o appromae he solo of he coos eqaos. As meoe before applyg he spaal screzao 39 o wave eqao 37 resle a ODE me oma. Kay a Krebes 999 se a cere fefferece operaor o appromae. They screze me eqal seps of, a a sperscrp k eoes evalao a me as k. Wh hs oao, Kay a Krebes

18 ahmoa a argrave 999 propose fe-eleme space a fe-fferece me. Ther FE/FD scheme s. k + k k k k + / k /. + K. + Kδ. ξ + ξ. 4 Solvg he ODE reslg from eqao 4, Kay a Krebes 999 also se a sh-orer Rge-Ka scheme wh aapve sep-sze corol hey also meo he meho of Karasso 997 ca be se. They compare he resls from eqao 4 solve by he Karasso meho o hose from Rge-Ka meho base o applcao of he meho of les Ames, 99. Agreeme bewee he wo ffere solo mehos creases cofece he resls. I hs way, Kay a Krebes 999 showe he accracy of her meho a also showe her solo s also epee of he memory varable formlao. Resls homogeeos mem also agree wh he freqecy oma solos of Karasso s cosa-q meho for more eal see Kay 996. COCLUSIOS ay of he coceps assocae wh fe-eleme mehos are more ve ha hose fe-fferece or specral mehos, b he mplemeao s more complcae for smple oe-mesoal problems. However, for hgher mesoaly a comple geomery, fe-eleme pays for era he work. Irreglar geomeres a homogeeos mea ha represe realsc geologcal srcres ca be hale by he fe-eleme meho beer ha oher mehos lke fe-fferece O he bass of he cocep of he prcple of Galerk varaoal meho of alboary-vale problems, he fe-eleme solos for acosc a elasoyamc rase problems have bee sccessflly formlae by may researchers. Ko 98 saes he fe-eleme meho as evelope proves avaages over he more coveoal fe-fferece mehos whe apple o eplorao problems for: smple a accrae moellg of arbrary sesmc sorces a sorce arrays; ease of applyg homogeeos a homogeeos boary coos of ay ype; 3 grea flebly moellg arges of ay rreglar shape as well as he effecs of rreglar opography a weaherg zoes; a 4, perhaps mos mporaly, errors are average over he elemes hrogho he oma qeso. ACKOWLEDGEETS We wsh o hak o r. Ha ohamma for rg he package ISA II a s. Feresheh Roozbeh for los of scsso abo fe-eleme mehos; also haks o r. Kev Hall a r. Vafa Ab for los of help. 8 CREWES Research Repor Volme 5 3

19 Revew of he fe-eleme meho REFERECES Ames, W. F., 99, mercal ehos for Paral Dffereal Eqaos, 3r e.: Acaemc Press, Ic. Ambramowz,. a Seg, I. A., 97, Habook of ahemacal Fcos: Dover Pblcaos Ic. Bahe, K. J., 996, Fe-Eleme Proceres: Prece Hall, Ic. Becker E. B., Carey G. F., a Oe J. T., 98, Fe-Eleme: A roco: Prece Hall. Bla, D. R., 96, The Theory of Lear Vscoelascy: Ieraoal Seres of oographs o Pre a Apple ahemacs, Pergamo Press Cohe, G. a Faqe, S.,, Effce me fe eleme for he acoscs eqao: 7h A. Iera. g: Soc. of Eplorao Geophyscs: 5-8. Hghes, T. J. R., 987, The Fe-Eleme eho; Lear Sac a Dyamc Fe-Eleme Aalyss: Prece Hall Ic. Jal, Z., 994,.5-D fe eleme sesmc moellg: 64h A. Iera. g: Soc. of Eplorao Geophyscs: Kay, I., 996, Sesmc moellg o-eal mea wh fe eleme mehos: Ph.D. Thess, Uv. of Calgary. Kay, I. a Krebes, E.S., 999, Applyg fe-eleme aalyss o he memory varable formlao of wave propagao aelasc mea: Geophyscs, 64, Kelly, K. R. a arfr, K. J., 99, mercal moellg of sesmc wave propagao: Geophyscs Repr Seres, Socey of Eplorao Geophyscss. Karasso, E., 979, Cosa Q-wave propagao a aeao: J Geoph. Res., 84, Karasso, E., 997, Cosa Q-wave propagao a aeao: J. of Geoph. Res. 84, Ko, J. T. a Teg, Y. C., 98, Three-mesoal fe-eleme moellg of acosc a elasc waves: 5 A. Iera. g: Soc. of Eplorao Geophyscs, Sesso: RW.6. arfr, K. J., 984, Accracy of fe-fferece a fe-eleme moellg of he scalar a elasc wave eqaos: Geophyscs, 49, p arfr, K. J., 99, Aalyss of Hgher Orer Fe-Eleme ehos. mercal oellg of Sesmc Wave Propagao, Socey of Eplorao Geophyscs. Smh, W. D., 975, The applcao of fe eleme aalyss o boy wave propagao problems: Geophyscal Joral of he Royal Asroomcal Socey, 4, Smh, W. D., 98, Free oscllaos of a laerally heerogeeos earh: A fe eleme approach o realsc moellg: Physcs of he Earh a Plaeary Ierors,, Srage, G. a F, G. J., 973. A Aalyss of he Fe-Eleme eho: Prece-Hall, Ic. Sllva,. F. a Yog, T. K., 983, Desg coseraos fe-eleme program evelopme: 53r A. Iera. g: Soc. of Eplorao. Geophyscs., Sesso:S7.6. Zekewcz, O. C., 977, The Fe-Eleme eho, 3r e. ew York, cgraw-hll Book Co. CREWES Research Repor Volme 5 3 9

20 ahmoa a argrave FIG. A.The process of fe-eleme aalyss afer Bahe, 996. CREWES Research Repor Volme 5 3

21 Revew of he fe-eleme meho Rge-Ka meho APPEDIX B The Rge-Ka meho s he mos accrae mercal algorhm o solve Orary Dffereal Eqaos. Ths meho fs a polyomal o he crve, whch ses o esmae he ew y vale. Ths s a eeso o Eler's mprove meho. The Eler meho ca be cosere a frs-orer Rge-Ka meho. The forh-orer Rge- Ka meho works by evalag he fco a for separae pos a akes he average of hose for pos. The Rge-Ka meho proves mprove accracy wh larger sep-szes a who he ee of evalae hgher ffereals beyo he frs ervave of he fco of eres. Hgher orer Rge-Ka mehos evalae he solo a more pos.e., sh-orer Rge-Ka ses s solos, a herefore become more accrae. The am of meho s geerag a mercal solo o a al vale problem of he form: Here s a smmary of he meho: where, y' f, y y y o h y y + k k k3 k k f, y k f + h/, y + hk / k3 f + h/, y + hk / k f + h, y + hk 4 3 oe ha he case where f,y oes o epe po y, he above reces o whch s Smpso's rle. h y y + + [ f 4 f h/ f h] o CREWES Research Repor Volme 5 3