A TIME-DOMAIN FINITE-DIFFERENCE METHOD WITH ATTENUATION BY A RECURSIVE ALGORITHM

Transcription

1 A TIME-DOMAIN FINITE-DIFFERENCE METHOD WITH ATTENUATION BY A RECURSIVE ALGORITHM Ningya Cheng Arthur 'C. H."Cheng and M. Nafi Tksz Earth Resurces Labratry Department f Earth Atmspheric and Planetary Sciences Massachusetts Institute f Technlgy Cambridge MA ABSTRACT A recursive algrithm t incrprate attenuatin int a time-dmain finite-difference calculatin is develped. First a rhelgical mdel f the generalized Maxwell bdy is chsen. The discrete relaxatin frequency and the peak strength f these Maxwell bdies are jintly determined by fitting t an arbitrary Q law in the frequency band f the interest. A cnjugate gradient technique and a randmly chsen starting mdel are used t determine ptimum fitting. Examples f cnstant and frequency dependent Q mdels are shwn. Secnd in rder t include the attenuatin int a finite-difference staggered-grid scheme the cnvlutin integral f stress and strain is evaluated directly. The cnvlutin integral can be expressed recursively. This is pssible because the timedmain viscelastic mdulus functin is expnential. The implementatin f 1-D wave prpagatin in a cnstant Q medium is shwn as a example. At the distance f 50 wavelengths and with three relaxatin frequencies the finite-difference results are in very gd agreement with the analytic slutins. INTRODUCTION Time-dmain finite-difference is a widely used methd in seismic wave prpagatin simulatin fr prblems with cmplex gemetry. One majr disadvantage f the time-dmain finite-difference methd is that it is hard t include attenuatin in the calculatin. Fr frequency dmain methds attenuatin can be incrprated int the calculatin simply by making the elastic mduli a cmplex functin f frequency. On the ther hand in the time-dmain stress is expressed as a cnvlutin integral f the histry f strain. 12-1

2 Cheng et al. Direct evaluatin f this cnvlutin integral requires immense strage as well as the number f peratins. Day and Minster (1984) presented a methd t incrprate attenuatin int timedmain calculatins. Their apprach is t transfrm the cnvlutin integral relating stress t strain histry int a sequence f differential peratrs. The Pade apprximatin is used in this transfrmatin. Fr a frequency independent Q the peratr cefficients can be btained in clsed frm in terms f Legendre plynmials. Emmerich and Krn (1987) imprved this apprach by using the rhelgic.al mdel fthe generalized Maxwell bdy which has a mdulus f the desired ratinal frm. This differential peratr apprach is als used in viscelastic finite-difference mdeling (Rbertssn et ai. 1994) and the pseudspectral methd (Carcine 1993). The purpse f this paper is t develp an algrithm t incrprate the attenuatin int time-dmain finite-difference calculatins. We chse the rhelgical mdel f the generalized Maxwell bdy. The discrete relaxatin frequencies and their peaks f strength are jintly determined by fitting a specific attenuatin law. The cnjugate gradient methd is used t d the ptimizatin. We_ directly evaluate the cnvlutin integral by a recursive algrithm. This is pssible because the time-dmain viscelastic mdulus functin is expnential. Finally a 1-D finite-difference wave prpagatin example is shwn. APPROXIMATION OF ATTENUATION LAW The relatinship between stress (J and strain E: be expressed in the frequency dmain as: in the case f linear viscelastisity can (J(W) = M(w)E:(w) (1) where M is the cmplex viscelastic mdulus. M beys a Kramers-Krnig relatin t make the relatinship between stress and strain causal. The quality factr Q is defined as: Q-l( ) = Im(M) W Re(M) (2) Here we apprximate the cmplex mdulus M by MN in which the relaxatin spectra cnsists f N single peaks f strength aj at discrete relaxatin frequencies Wj. This gives the cmplex mdulus as: (3) where M. is the unrelaxed mdulus. IlM = M. - Mr where Mr is the relaxed mdulus. The cmplex mdulus MN can als be expressed by using Mr. This allws us t interpret the mdel as a sum f Maxwell bdies with viscsity ajllm/wj and elastic 12-2

3 Time-Dmain Finite Difference Methd mdulus ajb.m plus an extra elastic element M r be determined by (Emmerich and Krn 1987). Q can (4) where Yj = ~Z' aj' Fr a given N the cefficients Yj and Wj have t be determined by fitting t a Q law in which we are interested. Seismic data suggest that in the seismic wave bandq-l is nearly independent f frequency. This can be characterized by a brad relaxatin spectrum (Liu et al. 1976). Fr a frequency independent Q Emmerich and Krn (1987) set the discrete relaxatin frequencies Wj equidistant n a lgarithmic scale and then determined Yj by least squares fitting. Blanch et al. (1995) apprximate Q by the summatin f Debye functins and slved the prblem analytically by the T methd but nce again the relaxatin time was pre-determined. In this paper we determine Yj and Wj by a purely numerical methd. The prblem is psed as minimizatin f expressin: with respect t Yj and Wj; [w a Wb] is the frequency band we are interested in. This methd is used fr this minimizatin purpse (Press et ai. 1992). The cnjugate gradient methd requires nt nly the value f functin but als its gradient. The gradient is cmputed by the finite-difference apprximatin. The gradient infrmatin is als updated when the mdel is updated. T avid being trapped in a lcal minimum the initial Yj is randmly chsen between zer and Q-l and Wj is randmly chsen between Wa and Wb. In the calculatin there are 1000 randm starting mdels tested and Yj and Wj are determined frm the minimum 0 f all randm starting mdels. As numerical examples we cnsider a frequency independent Q = 20 in the frequency range frm t Hz. The cmparisn f Q 0 1 and Q-;/ are shwn in Figures 1 and 2 fr the cases f N = 3 and 5 respectively. The discrete relaxatin spectra are als shwn n the plts. With N =5 Q"il prvided a very gd apprximatin t the cnstant Q mdel in the frequency band. Fr N :':: 4 the discrete relaxatin spectra are apprximately equally distanced n a lgarithmic scale. Hwver when N = 5 the spacing n lgarithmic scale is n lnger equal. The unevenly distributed relaxatin spectrum has a much smaller amplitude than an evenly distributed relaxatin spectra (Figure 2a). Next we cnsider a frequency dependent Q. It is given by Q = Q..fl. This kind f frequency dependence is due t fluid flw abve a critical frequency (Tksiiz et al. 1987). The result is pltted in Figure 3 fr N = 3. The spacing n a lgarithmic scale is nt unifrm fr discrete rela;{atin spectra distributins in the frequency dependent Q case. Even with three relaxatin spectra the apprximatin is already quite gd (5)

4 Cheng et al. FINITE-DIFFERENCE METHOD WITH ATTENUATION The wave equatin in a 1-D medium is: 8v 8u P8t=8x' where P is the density and v is the velcity. The time-dmain viscelastic mdulus MN can be btained by an '.nverse Furier transfrm f Equatin (3) MN(t)=Mu (O(t)- ~YjWje-Wj'H(t)) where H(t) is a unit step functin. In rder t use a velcity-stress frmulatin we need the time derivative f the stress. Thus we multiply by iw n bth sides f equatin (1) substituting in MN(w) and ding an inverse Furier transfrm t a time-dmain we btain: 8u 8t (6) (7) (8) and 8E: 8v = 8t 8x We let t = k6.t and assume ~~ Xj(k6.t) = xj in a discrete manner: xj = = k-l l(m+lll:>t 8 ( ) I: e-wj('-tly_~dt m=o ml:>t 8x (9) (10) t be cnstant ver each time interval 6.t. We evaluate ewjl:>t - 1 -kw.l:>t ~ mw L:>t8v(m6.t) ----e ' ~e '. Wj m=o 8x At first glance it seems that cmputing the cnvlutin summatin (Eq. 11) requires immense strage and a large number f peratins. But since it is a summatin f expnential functins it can be updated recursively (Luebbers et al. 1990). We derive the frmula t cmpute xj+l in terms f Xj. With sme algebra we btain (11) X k+l _ X k -w.l:>t (1 -e-wjl:>') 8v(k6.t). - e ' + J J w 8x J (12) 12-4

5 Time-Dmain Finite Difference Methd This is the basic equatin fr cmputing xj+1 recursively with initial cnditin XJ = O. T check its cnsistency with earlier wrk we derive the differential equatin which Xj satisfies. With straightfrward differentiatin f Equatin (9) we have X j OE: at + WjXj = at' (13) This is exactly equivalent t the first-rder differential equatin which the memry variables satisfied [Day and Minster 1984 Eq.(27)]. STAGGERED GRID ALGORITHM Equatins (6) (8) and (10) were used fr the finite-difference calculatins. Derivatives can be discretized using centered finite-difference n a staggered grid. The finitedifference scheme can be written as: and where k+l/2 _ k-i/2 6.t k k v i + I / 2 - v i +1/2 + p6.x (O'i+1 - O'i) (14) N k+l k M6.t (k+i/2 k+1/2 '" k+l O'i = O'i +~ v i +1/2 - v i _ I / 2 ) - M6.t L.. YjWjXji J:=1 -w.!!.t) k+l/2 _ k+i/2 Xk+1 = X k.e-wj!!.t e' vi+ 1/ 2 vi_ I / 2 ]t J't ( w'..6..x' J where i is the index fr the x axis discretizatin. T test ur recursive algrithm we cnsider a hmgeneus medium with velcity V p = V~" = 2000mjs density p = 1 gjcc and frequency-independent Q = 20. The discretized grid size is 6.x = 100m and 6.t = The surce signal is fed int the stress grid and a Kelly wavelet with a center frequency f 1 Hz is used. A very simple absrbing bundary cnditin is used at bth ends f the grid. The ptimal discrete relaxatin frequency and peak strength fr attenuatin are determined in the frequency bandwidth between t 10 5 Hz. The analytic slutin in the hmgeneus medium is btained by a substitute velcity via the rule: ( 1 Wi) V p = VI 1 + -QIn(-) - - 1r 21r 2Q where VI is the velcity at frequency 1 Hz. The cmparisns f the finite-difference slutin and the analytic slutin are pltted in Figure 4 fr the distance f 50 wavelengths. The finite-difference slutin with ne relaxatin frequency experienced a big errr. The slutin drifted int the lwer frequencies. With three relaxatin frequencies they prvide a fairly gd apprximatin t the cnstant Q. The finite-difference wavefrm matches the analytic slutin very well (15) (16) (17)

6 Cheng et al. CONCLUSIONS We develped a recursive algrithm t incrprate attenuatin int a time-dmain finitedifference methd. Unlike previus studies we directly evaluated the cnvlutin integral f strain and stress by a recursive algrithm. This methd is easy t understand and implement and cmputatinally efficient. In rder t apprximate the particular attenuatin law the discrete relaxatin frequency and strength are determined jintly by fitting in a least-squares sense. The system is slved by the cnjugate gradient methd with a randmized starting mdel t avid -pssible lcal minima. The numerical example shws very gd agreement between analytic slutin and the results f finite-difference with attenuatin ACKNOWLEDGMENTS One authr (N.C.) was partially supprted by Ls Alams Natinal Labratry as a Psdctral Assciate. We thank Dr. Steven Day and_ annymus reviewers fr valuable cmments n the manuscript. This wrk is supprted by the Brehle Acustics and Lgging and the Reservir Delineatin Cnsrtia at MIT. 12-6

7 Time-Dmain Finite Difference Methd REFERENCES Blanch J.O. Rbertssn J.O.A. and Symes W.W Mdeling f a cnstant Q: Methdlgy and algrithm fr an efficient and ptimally inexpensive viscelastic technique: Gephysics Carcine J.M Seismic mdeling in viscelastic media: Gephysics 58 1l0-l20. Day S.M and Minster J.B Numerical simulatin f attenuated wavefields using a Pade apprximatin methd: Gephys. l.r. Astr. Sc Emmerich H. and Krn M Incrpratin f attenuatin int time-dmain cmputatins f seismic wave field: Gephysics Liu H.P. Andersn D.L. and Kanamri H Velcity dispersin due t anelasticity: implicatins fr seismlgy and mantle cmpsitin: Gephys. l.r. astr. Sc Luebbers R. Hunsberger F.P. Kunz KS. Standler R.B. and Schneider M A frequency-dependent finite difference time dmain frmulatin fr dispersive materials: IEEE Trans. Electrmagn. Cmpat Press W.H. Teuklsky S.A. Vetterling W~T. and Flannery B.P Numerical Recipes: Cambridge University Press. Rbertssn J.O.A. Blanch J.O. and Symes W.W Viscelastic finite-difference mdeling: Gephysics 5g Tksiiz M.N. Mandai B. and Dainty A.M Frequency dependent attenuatin in the crust: Gephys. Res. Let

8 Cheng et al. 0.2 " (a) " " " Frequency (Hz) I (b)?! ' Frequency (Hz) Figure 1: Apprximatin f frequency independent Q-l with discrete relaxatin spectra with N=3. (a) Discrete relaxatin frequency and its peak strength. (b) Slid line fr exact Q-l and dashed line fr apprximated Q-l. 12-8

9 " Time-Dmain Finite Difference Methd (a). > 'I " Frequency (Hz) 10 I 100 (b) e' Frequency (Hz) Figure 2: Apprximatin f frequency independent Q-l with discrete relaxatin spectra with N=5. (a) Discrete relaxatin frequency and its peak strength. (b)slid line fr exact Q-l and dashed line fr apprximated Q-l. 12-9

10 Cheng et al f- - I 0 'r 'I 'I Freqnency (Hz) (b) CI Frequency (Hz) Figure 3: Apprximatin f frequency dependent Q-l with discrete relaxatin spectra with N=3. (a) Discrete relaxatin frequency and its peak strength. (b) Slid line fr exact Q-l and dashed line fr apprximated Q-l

11 Time-Dmain Finite Difference Methd N=l "' T XN (sec) N=2 ~ - - ~ T XN (sec) N=3 ~ T X/V (sec) 6 8 Figure 4: Cmparisn f analytic slutin with finite difference result at distance 50 wavelength fr Q=20. Discrete relaxatin spectra N = 123 are cnsidered

12 Cheng et al