Composite memory variables for viscoelastic synthetic seismograms

Transcription

1 Geophys. J. Int. (1995) 121, RESEARCH NOTE Composite memory variables for viscoelastic synthetic seismograms Tong Xu George A. McMechan Center for Lithospheric Studies, The University of Texas at Dallas, PO Box , Richardson, TX , USA Accepted 1994 October 8. Received 1994 September 2; in original form 1994 June 4 INTRODUCTION Recent developments in linear viscoelastic theory, based on the superposition of relaxation mechanisms (Liu, Anderson & Kanamori 1976; Day & Minster 1984), produced formulations that allow the computation of synthetic seismograms for models with arbitrary spatial distributions of quality factors (Q, Q5). The key concepts are given by Emmerich & Korn (1987), Carcione, Kosloff & Kosloff (1988a, b, c), Carcione (1990) Tal-Ezer, Carcione & Kosloff (1990). The main numerical feature of this approach is replacement of a time convolution between stress strain by memory variables. The memory variables reduce the computer memory requirements to the point where viscoelastic computations are practical, even in three dimensions (e.g. Dong & McMechan 1995). Memory variables were originally defined to be associated with individual compressional shear relaxation mechanisms. Because Q effects are physically more naturally associated with particle motions than with wave types, because various wave types (P, S, Rayleigh, Love) are not confined to a single response component, it makes more sense to redefine the memory variables, simply by rearranging the equations, to give one memory variable per displacement component. This also provides significant savings in the computer memory needed for computation of synthetic seismograms; the amount of computation remains unchanged as the new formulation is equivalent to the original one. SUMMARY By rearranging terms in the viscoelastic wave equations for the stard linear solid, new composite memory variables are defined. Instead of having one memory variable for compressional relaxation one for each of the six independent components of quasi-shear stress relaxation, as in the stard relaxation mechanism formulation, the new grouping contains only one for each displacement response component. Thus, only two (rather than three) memory variables are required for 2-D computations, only three (rather than seven) are required for 3-D computations. The amount of computation is not changed as the new formulation is equivalent to the stard one, but there is a significant saving in computer memory. Key words: memory variables, synthetic seismograms, viscoelasticity. THE 3-D VISCOELASTIC EQUATIONS The most common version of the viscoelastic equations used for seismogram synthesis includes memory variables that replace convolutions between stress strain (Carcione et al. 1988~). There is one memory variable for each compressional relaxation mechanism, six for each quasi-shear relaxation mechanism (corresponding to the six independent components of the quasi-shear stress tensor). In the usual 3-D formulation, different combinations of four of these seven appear in the equation for each displacement component. By grouping these together, only one composite memory variable needs to be computed stored for each displacement component, thereby significantly reducing the computer memory requirements. The equations of motion in a 3-D isotropic linear viscoelastic medium are 634

2 Viscoelastic synthetic seismograms 635 av au aw L1 p-=- (A+2p)-+A -+- +: c el,+ c ez2,] at2 a2v a [ ay (ax a,) /=I I= 1 ay L? a2w - atz a az a au aw L2 +- p -+- +Eel,, ax [ (a, ax) [=, 1 where au av ay az ' the response functions a av aw L2 +- p -+- +Ce,,,, ay I (a, ay) /=I 1 where u, v w are the displacements; x, y z are the space coordinates; t is time; p is density; L, L, are the numbers of compressional shear relaxation mechanisms present (usually 2 or 3 are sufficient): el,, el,/, ez2/, e33,, ei2,, el,, ez3, are the memory variables (defined below); A p are the unrelaxed Lam6 parameters (defined below). Carcione (1993) presents a similar formulation in terms of stress rather than displacement, but omits e,,, in the vertical component, replacing it by a combination of el,, e,,,: this trick works in 3-D, but does not allow direct reduction to 2-D, because e,,, is not zero in 2-D. Let Me Mu be the unrelaxed compressional shear moduli. Then, where MpR MR are relaxed compressional shear moduli, respectively; z,,,, z, are the stress strain relaxation times for shear waves for the 1 th relaxation mechanism;,, z T ~ are ~, the stress strain relaxation times for the compressional waves. The unrelaxed Lam6 parameters are A = 3 qm e--3 = f Ms U. The memory variable equations are el, e,, = - ~ %pr + +ll(o)@, (3a) &/(O) = ( : ::::I In this form, seven memory variables need to be solved for each mechanism. C 0 M P 0 SITE ME M 0 R'Y V A R I A B L E S If it is assumed that there are an equal number of relaxation mechanisms for both P S waves, L = L, = L,, the relaxation mechanism terms in eqs l(a)-(c) may be grouped together to give where l a a a a?i, =--el, +-ell, + -el,/ + -el,,, 3ax ax ay az a l a a a rcrr =-~lzl+--ee,,+-~22, +-e2,,, ax 3ay ay az a a i a a 6 = e el/ + - e331. ax ay 3az az So that the individual original memory variables (e,,, el,,, e,,,, e331, el,/, e,31 ez3,) do not have to be saved, we now

3 636 T. Xu G. A. McMechan reparametrize the relaxation mechanisms in terms of relaxation frequencies (q = 7;; = ;~ z; for each of the L mechanisms) as described by Emmerich (1992). Since the choice of wi values is arbitrary, its only function is to define a desired Q behaviour, it in no way limits the model definition. Q,(w) Q,(w) can have different shapes if desired; for convenience in the examples below, we take Q(w) to be constant for both. With this change, the equations for the memory variables (v/ I+%/) associated with the horizontal displacements, (5,) associated with the vertical displacement, become vertical displacements become, respectively, 3 /I (7a) Evaluation of eqs 6(a)-(c) with 7(a)-(c) involves the same amount of computation as for eqs l(a)-(c) with 4(a)-(g). However, the number of memory variables is reduced from seven to three. Reduction to 2-D In the 2-D case, all the terms related to the derivatives with respect to y the displacement component u are omitted, the divisions by three of the terms in e,, in eq. (1) 0 in eqs (4) (7) become divisions by two. Then, from eqs 4(b) (d), e331 = -ell1. So the 3-D equations of motion may be simplified to the following for 2-D: au au aw +- a:[ p (a, -+-,)I +L TI, EXAMPLES To test the viability of the parametrization proposed above, to illustrate the equivalence to the stard formulation (e.g. Kang & McMechan 1993), results of some representative 2-D numerical computations are presented in this section. Figure 1 shows the recording geometry parameters for a simple test model. To simulate constant Q functions, three relaxation mechanisms with relaxation frequencies of 2, 10 50Hz were summed (Emmerich 1992). The resulting Q(w) behaviour is shown in Fig. 2. The source time function was the first derivative of the Gaussian function with a dominant frequency 10 Hz; the spatial X-POSITION (KM) ~ l x x x x x x x 2 RECEIVERS It SOURCE Vp = 3.0 KMIS VS = 1.6 KM/S p = 1.25 G/CM3 Qp = 80 QS = 50 (8b) The memory variables associated with the horizontal Figure 1. Recording geometry model parameters for the fixed-time snapshots seismograms in Figs 3 4. The grid size is 128 X 128, with an increment of 0.02 km in both the horizontal vertical directions.

4 Viscoelastic synthetic seismograms CT P 0 2 z a 3 B FREQUENCY (HZ) FREQUENCY (HZ) Figure 2. Q behaviour as a function of frequency for the example in Fig. 1. Superposition of three relaxation mechanisms (dashed lines) gives nearly constant Q, (a) Q, (b) from 2 to 50Hz (solid lines). geometry is a vertical impulse at (x, z) = (1.28, 1.00) km. Computations used eighth-order finite-differencing in space, second-order in time. Fixed-time snapshots of both horizontal vertical displacement components at 0.3 s (Fig. 3) show virtually identical results for the stard new formulations. Only a small residual difference that is consistent with X-POSITION (KM) numerical round-off error is present. Similar observations are made for the corresponding seismograms in Fig. 4. DISCUSSION AND SYNOPSIS a By grouping the relaxation memory variables in the equation for each displacement component, the number of HORIZONTAL DISPLACEMENT (U) VERTICAL DISPLACE-NT (w) STANDARD FORMULATION NEW FORMULATION DIFFERENCE Figure 3. Fixed-time snapshots for the configuration in Fig. 1.

5 638 T. Xu G. A. McMechan X-POSITION (KM) g h) 0 STANDARD FORMULATION NEW FORMULATION DIFFERENCE Figure 4. Synthetic viscoelastic seismograms for the configuration in Fig. 1. The time sample increment is s there is a total of 1200 time steps. memory variables can be reduced from three to two in 2-D viscoelastic computations. The reduction for the corresponding 3-D wave equation would be from seven to three, which is much more significant. For the viscoacoustic equation, there is no saving as there is only one memory variable in both parametrizations. Although the number of computations is not different in the new formulation, it was found to be 10 per cent faster than the stard formulation in 2-D it is estimated to be 25 per cent faster in 3-D (assuming L, = L, = 3 for both). This improvement is attributed to the fact that less input output of memory variables is involved in the new one. There are also additional memory savings involved in using W, instead of,, z z,; only a few values of wi are usually needed, these can be the same for all grid points (Emmerich 1992). For the example in Fig. 3, the new formulation needed 435 s of CPU time on one Convex C-3 processor 493 s on a four-processor Intel ipsc860. Figures 3 4 confirm the equivalence of the stard new algorithms. Since the new one is more efficient in both CPU time space, it appears advantageous to use it rather than the stard formulation. ACKNOWLEDGMENTS The research leading to this paper was funded by the National Science Foundation under grant EAR , by the Petroleum Research Fund of the American Chemical Society under grant AC2, by the sponsors of the UT-Dallas Parallel Computing Consortium. Computations were performed on a Convex C-3200 on an InteliPSC860. This paper is Contribution No. 789 from the Programs in Geosciences at the University of Texas at Dallas. REFERENCES Carcione, J.M., Wave propagation in anisotropic linear viscoelastic media, Geophys. J. Inf., 101, Carcione, J.M., Seismic modeling in viscoelastic media, Geophysics, 58, Carcione, J.M., Kosloff, D. & Kosloff, R., 1988a. Wave propagation in a linear viscoacoustic medium, Geophys. J., 93, Carcione. J.M., Kosloff, D. & Kosloff, R., 1988b. Viscoacoustic wave propagation simulation in the Earth, Geophysics, 53, Carcione, J.M., Kosloff, D. & Kosloff. R., 1988c. Wave propagation in a linear viscoelastic medium, Geophys. J., 95, Day, S.M. & Minster, J.B., Numerical simulation of attenuated wavefield using a Pade approximant method, Geophys. J. R. astr. Soc., 78, Dong, Z. & McMechan, G.A., D viscoelastic anisotropic modeling of data from a multi-component, multi-azimuth experiment in northeast Texas, Geophysics, in press.

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