Stability condition of finite difference solution for viscoelastic wave equations

Transcription

1 Erthq Sci (009): Doi: 0007/s Stbiity condition of finite difference soution for viscoestic wve equtions Chengyu Sun, Yunfei Xio Xingyo Yin nd Hongcho Peng Chin University of Petroeum, Qingdo 66555, Chin BGP, Chin tion Petroeum Corportion, Zhuozhou 0775, Chin Abstrct The stbiity probem is very importnt spect in seismic wve numeric modeing Bsed on the theory of seismic wves nd constitutive equtions of viscoestic modes, the stbiity probems of finite difference scheme for Kevin- Voigt nd Mxwe modes with rectngur grids re nyzed Expressions of stbiity conditions with rbitrry spti ccurcies for two viscoestic modes re derived With pproximtion of quity fctor Q 5, simpified expressions re deveoped nd some numeric modes re given to verify the vidity of the corresponding theoretic resuts Then this pper summrizes the infuences of seismic wve veocity, frequency, size of grid nd difference coefficients, s we s quity fctor on stbiity condition Finy the prerequisite conditions of the simpified stbiity equtions re given with error nysis Key words: finite difference; stbiity; viscoestic mode; wve eqution; quity fctor CLC number: P350 Document code: A Introduction The finite difference (FD) method pys one of the most importnt numeric roes for soving wve equtions Stbiity nd numeric dispersion re two inherent nd significnt probems in finite-different numeric soution Discussion of the two probems usuy ims t estic medium As resut, the obtined stbiity condition nd the nytic resut of dispersion re so bsed on estic ssumption However, the ctu Erth is imperfecty estic medium nd its viscoestic properties wi mke the stbiity of finite difference soution worse When we sove viscoestic wve equtions with FD method, the former stbiity condition for estic equtions does not gurntee the stbiity of the soution On the reserch of stbiity probems, Aki nd Richrds (980) gve exmpes of finite-difference stbiity gorithm, then the stbiity condition expressions of the first-order veocity-stress eqution for P-wve nd SV-wve were given respectivey by Virieux (986) Some reserchers, such s Levnder (988), Crcione et (00), Moczo et (000), de Bsbe nd Sen (007) so nyzed the stbiity of finite difference scheme of Received 3 Mrch 009; ccepted in revised form 5 June 009; pubished 0 October 009 Corresponding uthor e-mi: suncy@upceducn estic wve equtions successivey Mnning nd Mrgrve (000) introduced correction method for instbiity probem in -D estic finite-difference forwrd modeing Long-time stbe modeing of estic wve with free surfce ws discussed by Hesthoim (003) The effects of boundry condition on the instbiity of finite difference soution in estic medium hve been eborted by In nd Loewenth (976), whie Vide nd Cyton (986) discussed the stbiity of finite difference soutions in estic medium with the free surfce boundry In Chin, Dong et (000) progressed quntittive nysis on stbiity of stggered grid finite difference in estic medium And Mu nd Pei (005) presented the stbiity condition of high-order difference scheme of estic wve eqution For viscoestic cse, Robertsson et (99) expored the FD method of viscoestic medium, nd they introduced the highest owbe formtion veocity in stbe difference scheme under utr high frequency, but tht for ow frequency cse which invoves in exportion seismoogy nd erthquke reserch is bsent Senger nd Bohen (00) expored the rotted stggered-grid finite difference forwrd modeing methods of viscoestic nisotropic medium, nd they described the stbiity condition of difference scheme Rotted stggered-grid difference scheme is superior to convention rectngur grid in

2 80 Erthq Sci (009): suppressing numeric dispersion, but its stbiity is weker thn tht of convention grid to some extent It is evident tht the sttus quo on stbiity study mosty focuses on the estic medium In the finite difference soution of viscoestic wve eqution, the visco-esticity woud increse the numeric soution s instbiity Bsed on the viscoestic theory, the stbiity conditions of the popury used Kevin-Voigt mode nd Mxwe mode in wve fied modeing with convention rectngur grid re expored nd further retionships between stbiity conditions nd seismic wve veocity, frequency, quity fctor, grid size nd difference coefficients re investigted Derivtion of the stbiity conditions of bsic viscoestic modes Studies of nestic medium mosty invove two bsic modes, nmey Kevin-Voigt mode nd Mxwe mode The former suggests tht medium stress under certin strin consists of two prts, strin reted nd strin rte reted And it cn be regrded s pre connection of n estic eement (spring) nd ewtonin viscous eement (dmper) Whie the tter indictes tht medium strin under certin deformtion is composed of two prts, nmey, stress reted nd stress rte reted It cn be seen s series connection of n estic eement (spring) nd ewtonin viscous eement (dmper), it is so known s the estic viscous body Both of the stress-strin constitutive retion expressions re respectivey given by: ε σ M ε M () for Kevin-Voigt mode nd σ M ε σ M (b) M for Mxwe mode, where σ is the stress, ε is the strin nd M, M re estic coefficient nd viscous coefficient of the medium, respectivey With the hrmonic stress σ Pe it nd strin ε Ee j t, the compex moduus of the mode cn be expressed s MP/Eke jδ, the quity fctors Q re stisfied, respectivey M tnδ () Q M for Kevin-Voigt mode, nd Q M tnδ (b) M for Mxwe mode, where δ is the rgument of compex moduus It cn be drwn tht the former Q is inversey proportion to frequency, nd the tter one is proportion to frequency As the retionships of Q versus for the two modes re different, the impcts of quity fctor on the stbiity so vry According to constitutive retions, the stbiity conditions of finite difference soutions for the two modes re derived in the foowing Kevin-Voigt viscoestic mode In 3D medium, the buk strin of estic body is θ u, (3) where u is the dispcement vector Substitute eqution (3) into eqution (), suppose stress is P, we cn get θ P Mθ M () The wve eqution cn be rewritten s: u P ρ (5) We simutneousy ccute second-order prti derivtives with respect to t in eqution (), nd first-order prti derivtives to x, y, z in (5) s foowing: P M M P P (6) ρ ρ Then we do spce Fourier trnsform to stress P of eqution (6) The expression in wve number domin is ~ ~ P M ~ M P k P k, (7) ρ ρ where P ~ is the spce Fourier trnsform of P nd k is the wve number Do difference pproximtion to the time derivtive in eqution (7), we get ~ n M M ~ n P k k P ρ ρ (8) M ~ n k P, ρ ~ n ~ n ~ n where P, P, P re vues of P ~ t time n, n, nd n Assuming (M /ρ)k, b (M /ρ)k, then ~ n ~ n P b b P ~ ~ 0 (9) n n P P

3 Erthq Sci (009): Aso, we hve retion s M [( Q ) ρ ( Q / Q] Q v ) (Sun nd Yin, 007), where v is wve veocity in this medium From eqution (), we obtin M M /(Q) Thus [( Q ) ( Q / Q] Q v k ) / [( Q ) Q] b ( Q ) v k Obviousy, the stbiity condition of recursive eqution (9) is the moduus of eigenvues λ of stte trnsition mtrix A must be ess thn, where b A b 0 Suppose c b, then the eigenvues λ of the mtrix A is c ± c λ (0) And the stbiity condition of Kevin-Voigt mode becomes Δ t, () k Q Q where is time smping interv nd [(Q ) / Q]Qv /[(Q )] In the cse of -order spti difference ccurcy, ssuming tht the grid spcing in x, y, z directions re Δx,, nd wve-number components re k x, k y, k z, respectivey It is simpe to obtin (Mu nd Pei, 005), ( k x ) mx Δx ( k z ) mx, ( k y ) mx, where is the difference coefficient of -order ccurcy, specific vues re shown in Tbe nd they stisfy As resut, k Δx () By substituting eqution () into eqution (), the expression of the stbiity condition becomes (3) Q Q x y z Δ Δ Δ Speciy, when Δx, stbiity condition is simpified s h () Q h Q 3 Genery, difference scheme is often crried out in D cse, nd the stbiity condition cn be further simpified s h (5) Q h Q When M 0, Q, the bove expression deteriortes to (h/v)(/ ) /, which coincides we with the stbiity condition in estic medium, Tbe Weighting coefficients of the second derivtive with -order ccurcy (from Sun, 007) Mxwe viscoestic mode For the Mxwe viscoestic mode, the derivtion is simir to Kevin-Voigt cse Min difference ies in tht the Mxwe mode is originted from eqution (b), then M M Q/, tht is M /M /Q, so we hve pproximtion s M /ρ [(Q ) / Q] v /(Q) After comprbe derivtion, the stbiity conditions re s foows:

4 8 Erthq Sci (009): k Q k Q k, (6) where is time smping interv, nd [(Q ) / Q] v /(Q) In the owbe rnge of the wve number, stbiity expression is given by Q Δx Q Δx Δx / Speciy, when Δxh, it cn be simpified s 9Q h 3Q For the D cse, there is h Q h Q h h h 3 Simiry, when M, Q, the bove eqution cn be rewritten s (h/v)(/ ) /, which is the perfecty estic cse (7) (8) 3 Low quity fctor effect Concusions in equtions (3) (5) nd (7) (8) re derived under the ssumption of Q 5 which is described fter eqution (9) When quity fctor Q is smer thn five, M /ρ v becomes uncceptbe, the bove conditions cn not gurntee the stbiity of numeric resut Actuy, for rbitrry quity fctor Q, the wve veocities stisfy ( Q ) vk V v (9) / [( Q ) Q] Q for Kevin-Voigt mode, nd Q v M v / ( Q ) Q (9b) for Mxwe mode, respectivey (Sun nd Yin, 007), where v M /ρ In such cse, it needs to repce v K-V for the veocity v in the concusions (3) (5), nd substitute v M for the veocity v in (7) (8) However, quity fctors of most rocks nd yers cn stisfy the pproximte condition of Q 5, so in ctu ccution they cn sti mintin the stbiity with bove pproximte condition The foowing numeric exmpes re crried out using pproximted condition; the resuts give proof to this concusion 3 umeric nysis of stbiity of bsic viscoestic modes Assume tht Δx5 m, the nytic stbiity conditions for 3D Kevin-Voigt viscoestic wve eqution with sixth-order spti ccurcy re shown in Figure Figure shows the vrition of requested time smping intervs with the formtion veocity in the cse of Q 5, 50, 00 nd 000 For certin vue of Q, s veocity increses, the time smping interv becomes smer nd smer, tht is to sy the stbiity worsens; when the veocity is constnt, with the decrese of the medi quity fctor Q, the stbiity worsen, too; nd with the increse of the quity fctor, the more tendency to estic cse, the better the stbiity Difference between the two curves tht Q 00 nd Q 000 is very sm, whie the interv widens between the two curves of Q 50 nd Q 00, nd the intervs between the curve of Q 5 nd other curves re much remrkbe It is demonstrbe tht s the medi quity fctor decreses, the stbiity condition requested by difference soution wi shrpy be tougher Figure b shows the retionships between the requested time smping interv nd quity fctors under different veocities for stbe scheme From this figure, the sme concusions s Figure cn be drwn Figure shows the comprison of stbiities under different order spti difference ccurcy For the sme veocity mode, the higher precision the spti difference hs, the smer time smping

5 Erthq Sci (009): interv is needed; tht is to sy, higher spti difference precision mens worse stbiity Under such circumstnce, high-density time smping interv is necessry to keep stbiity The convention method usuy dopted is improving difference ccurcy to suppress the difference spti dispersion, but this wy of work so eicits higher requirements on time stbiity, the two re contrdiction Figure The retionship between stbiity, quity fctor nd veocity of Kevin-Voigt mode where veocity is in unit of km/s Figure The retionship between stbiity nd spti ccurcy when Q equs to 00 To be convenient for mode tri, D cse is considered Grid size nd spti difference ccurcy re the sme s bove mentioned With Assumption of wve veocity is v3 km/s nd frequency is 35 Hz, the ccuted stbiity condition re 0 ms, 09 ms nd ms when Q equs to 5, 00 nd, respectivey We conduct the finite difference numeric test ccording to the resuts, nd the trends with the number of itertions re shown in Figure 3, where bsciss is the cyce number n of itertions nd vertic coordinte is the numeric resuts For Q 5 (Figure 3), it is stbe when 03 ms nd 0 ms, but unstbe s 05 ms For Q 00 (Figure 3b), it is stbe when 05 ms nd 09 ms, but unstbe when 09 ms A of the bove verify the vidity of the resuts Figure shows the 3-D nytic soution of stbiity for Mxwe mode with sixth-order spti differenti ccurcy The bsic chrcteristics re sme s tht of Kevin-Voigt mode With the increse of veocity nd the improvement of spti difference ccurcy, the stbiity tends to be weker nd the time smping interv shoud be smer However, the infuence of quity fctor on stbiity is ess significnt thn tht in the former mode Considering sixth-order scheme, supposing the rtio of grid size to veocity is 0003, when quity fctor chnges from 5 to 000, the chnges of requested time smping interv is ony ess thn % The infuence is so sm tht cn be negected Figure 5 shows the retionships between stbiity of two modes nd frequency of seismic wve On condition tht the other prmeters re sme, with the increse of frequency, the stbiity of difference soution of Kevin-Voigt mode gets better, whie tht of Mxwe mode becomes worse The bove numeric resuts re bsed on the pproximte conditions (3) (5) nd (7) (8) From eqution (9) we cn see tht the re veocity of Kevin- Voigt mode is greter thn pproximte veocity, whie tht of Mxwe mode is smer To iustrte the error sce cused by the veocity of pproximtion, Figure 6 shows the chnges of retive errors between re veocities nd their pproximtions with quity fctor for the two viscoestic modes With the increse of quity fctor, the retive errors tend to be zero, nd the veocities of viscoestic modes pproch to the veocity of estic medi The retive errors of the two modes re ess thn 8% nd 09% when Q 5, respectivey

6 8 Erthq Sci (009): Figure 3 The convergence of Kevin-Voigt viscoestic mode by finite difference for Q 5 () nd Q 00 (b) where n denotes the cyce number of ccution Figure The vrition of stbiity with quity fctor, spti ccurcy nd veocity (in unit of km/s) of Mxwe mode Figure 5 The retionship between stbiity nd frequency for Kevin-Voigt mode () nd Mxwe mode (b)

7 Erthq Sci (009): deteriortes drmticy, for which ccurte veocity re needed to define the stbiity conditions From the nysis we cn see tht the stbiity of difference scheme of Mxwe mode is superior to tht of Kevin-Voigt mode to some extent But Kevin-Voigt mode is popury used in soving viscoestic probems t present becuse there re certin gps between inherent Q~ retionship of Mxwe mode nd the current observtion resuts References Figure 6 The retive errors between ccurte veocities nd their pproximtion of two viscoestic modes When Q is too sm, equtions (3) (5) cn not gurntee the stbiity of difference scheme becuse the re veocity of Kevin-Voigt mode is much rger thn the pproximte vue, whie the equtions (7) (8) cn gurntee the stbiity due to smer veocity of Mxwe mode thn the pproximte vue Concusions From some theory nysis, stbiity conditions of Kevin-Voigt mode nd Mxwe mode with vried difference ccurcy re nyzed Combined with nysis on numeric tests, the foowing concusions cn be drwn ) The greter the quity fctor Q nd the ower the veocity, the better the stbiity of difference scheme of the two modes Quity fctor hs ess infuence on stbiity of Mxwe mode thn tht of Kevin-Voigt mode ) On condition tht vue of Q is fixed, with the increse of the frequency, the stbiity of difference scheme of Kevin-Voigt mode gets better, whie tht of Mxwe mode becomes worse with higher frequency 3) The stbiity conditions obtined from this pper re suitbe for quity fctor Q 5; when Q is too sm, the stbiity of difference scheme of Kevin-Voigt mode Aki K nd Richrds P G (980) Quntittive Seismoogy: Theory nd Method W H Freemn nd Compny, ew York, Crcione J M, Hermn G C nd ten Kroode A P E (00) Seismic modeing Geophysics 67(): de Bsbe J D nd Sen K (007) Grid dispersion nd stbiity criteri of some common finite difference nd finite eement methods for coustic nd estic wve propgtion Expnded Abstrcts of SEG 77th Annu Meeting Sn Antonio, USA, Dong L G, M Z T nd Co J Z (000) A study on stbiity of the stggered-grid high-order difference method of first-order estic wve eqution Chinese J Geophys 3(6): (in Chinese with Engish bstrct) Hesthoim S (003) Estic wve modeing with free surfces: Stbiity of ong simutions Geophysics 68(): 3 3 In A nd Loewenth D (976) Instbiity of finite-difference schemes due to boundry conditions in estic medi Geophysic Prospecting : 3 53 Levnder A R (988) Fourth-order finite-difference P-SV seismogrms Geophysics 53(): 5 36 Mnning P M nd Mrgrve G F (000) Finite difference modeing nysis, dispersion nd stbiity Expnded Abstrcts of SEG 70th Annu Meeting Cgry, Cnd, Moczo P, Kristek J nd Bystricky E (000) Stbiity nd grid dispersion of the P-SV th-order stggered-grid finite-difference schemes Geophysics : 38 0 Mu Y G nd Pei Z L (005) Seismic umeric Modeing for 3D Compex Medi Petroeum Industry Press, Beijing, 33 3 (in Chinese) Robertsson J O A, Bnch J O nd Symes W W (99) Viscoestic finitedifference modeing Geophysics 59(9): 56 Senger E H nd Bohen T (00) Finite-difference modeing of viscoestic nd nisotropic wve propgtion using the rotted stggered grid Geophysics 69(): Sun C Y (007) Theory nd Methods of Seismic Wves Chin University of Petroeum Press, Dongying, 9 (in Chinese) Sun C Y nd Yin X Y (007) Construction of constnt-q viscoestic mode with three prmeters Act Seismoogic Sinic 8(): Vide J E nd Cyton R W (986) A stbe free-surfce boundry condition for two-dimension estic finite-difference wve simution Geophysics 5(): 7 9 Virieux J (986) P-SV wve propgtion in heterogeneous medi: Veocity-stress finite-difference method Geophysics 5():