P-SV MULTIMODE SUMMATION DIFFERENTIAL SEISMOGRAMS FOR LAYERED STRUCTURES

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1 IC/97/12 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS P-SV MULTIMODE SUMMATION DIFFERENTIAL SEISMOGRAMS FOR LAYERED STRUCTURES Z.J. Du International Centre for Theoretical Physics, SAND Group, Trieste, Italy and Dipartimento di Scienze della Terra, Universita degli Studi, Trieste. Italy, A, Michelini Dipartimento di Scienze della Terra, Universita degli Studi, Trieste, Italy and G.F. Panza International Centre for Theoretical Physics, SAND Group. Trieste, Italy and Dipartimento di Scienze della Terra, Universita degli Studi, Trieste, Italy. MIRAMARE - TRIESTE February 1997

2 ABSTRACT We present a fast and accurate analytical method for the calculation of differential seismograms resulting from perturbations in the structural model. The method adopts the P-SV wave multi-mode summation formalism for laterally homogeneous layered media. The differential seismograms calculated analytically do not suffer from the numerical noise and instability affecting the numerical calculations. The frequency range for the application of the method spans from regional scale seismology to seismic engineering. The computational overhead in the calculation of the partial derivatives amounts to the calculation of just one additional full mode-spectrum. The speed and accuracy of the technique makes it extremely attractive for structural parameters waveform inversions. Key words: P-SV wave, multi-mode summation, differential seismograms, partial derivatives, waveform inversion.

3 1. Introduction Aided by the large quantity of accumulated broadband waveform data, nowadays formal waveform inversion for the medium structural parameters can be performed by adopting either global, non-linear procedures such as the "hedgehog"(valyus, 1972) and the more recent variants represented by the genetic algorithm (Sambridge and Drijkoningen, 1992) or simulated annealing (Sen and Stoffa, 1991), or through iterative, local linearization of the misfit function (e.g., Nolet, 1990). In fact, the use of these two classes of methods are not disconnected but are complementary, and through an appropriate combination they can often provide additional information about the structure of the earth. The first class of methods requires extremely fast forward modeling procedures whereas, the second class, even if it needs much fewer forward calculations, requires efficiency and accuracy for the determination of the differential seismograms. The first efforts to calculate differential seismograms for regional and local scale waveform modeling were made by Shaw and Orcutt (1985), using the "WKBJ seismogram" technique {Chapman, 1978), and by Gomberg and Masters (1988) who proposed, within the locked mode approximation, a hybrid technique in which the differentiation of the phase term is performed analytically, whereas that of the amplitude term is performed numerically. More recently, methods based on the matrix-formalism of Kennett (1983) and Luco and Apsel (1983) have been proposed by Randall (1989, 1994) and Zeng and Anderson (1995), respectively. In this paper, we present a technique to compute analytically complete differential seismograms for laterally homogeneous layered structures using the multimode summation formalism (Knopoff, 1964; Schwab, 1970; Schwab and Knopoff, 1972; Panza, 1985; Panza and Suhadolc, 1987; Urban et al., 1993, 1997).

4 2. Method Following Panza et al. (1973), for a given Rayleigh-mode the displacement for an assigned point-source double-couple can be expressed in the frequency domain as: (1) _ 1 y)j U r (u>) (2) where R(u>) is the Fourier transform of the equivalent point-force time function, x is the source radiation pattern, n is the unit vector perpendicular to the fault surface having units of length, k is the wave number, r is the epicentral distance, and $ = u*(0)/w(0) (u* = Im(u)) is the ellipticity calculated as the ratio between the radial and vertical components u(z) and w(z) of the Rayleigh-mode eigenfunctions. The factor E is given by: where c and U are the phase and group velocity, respectively. The energy integral is defined as where p{z) is the density. For a double-couple point source (Harkrider, 1970), x{@,/i), the source radiation pattern is: X(O, h) = d 0 + i(di sin 6 + d 2 cos 6) + d s sin 20 + d 4 cos 20 (5) where d 0 = -B(h) sin A sin 28 (6a) (6b) = C(h) cos A cos <5 (6c)

5 d% = A{h) cos A sin 5 (6d) di = A(h) sin X sin 26. (6e) it The source geometry is defined through the angles, 5 and A which are the azimuth of the station with respect to the fault-strike, the fault-dip and the slip angles, respectively. h is the source depth and A(h), B(h) and C(h) involve the eigenfunctions (i.e., the "motion-stress vector for Rayleigh waves") at the source depth. B(h) = U /wk(*o [ } I V(/i)j w{0) p{h)a 2 (h)w(0)/c 2 *(*»? (7h) K } where ( ) indicates time differentiation and a and r are the normal and tangential stresses. Equations (1) and (2) are equivalent to expressions (7.149) and (7.150) of Aki and Richards (1980) who, however, express the source term through the seismic moment tensor. 3. Differential seismograms An efficient and accurate technique for computing the partial derivatives of a seismogram with respect to the structural parameters can be developed using equations (1) and (2). In the following, we restrict our formulation to a double-couple point source, although the method can easily be extended to any other type of seismic source (e.g. Das and Suhadok, 1996). The terms of equations (1) and (2) which depend on the structural parameters are: eo, x(e, h), E = -?-, k l l\ exp(-zkr). (8) lcul\ In detail, the partial derivative of e 0 and x(0, h) with respect to the structural parameters can be obtained from the computation of the partial derivatives of the

6 eigenfunctions with respect to the structural parameters (Urban et al, 1997). The partial derivatives are and d apj d u*(0) opj w(0) ±A = -A W (1Oa) dpj dpj w 0 ±B = U l ^ ) d U * {h) 2 a * {h) (10b) 8p.j \ a 2 (h)j dpj w 0 p(h)a 2 (h)dpj w o /c j w o /c (1Oc) where p is the model parameter (i.e., a, j3 or p) and j indicates the layer sequential number. The computation of J-^ and ^ involves the calculation of the partial derivatives of the phase and group velocities and of the energy integral with respect to the model parameters (Urban et al ; 1993, 1997). Once all the partial derivatives of the terms in (8) are determined, the complete differential seismogram for a given P-SV mode can be constructed using the following expressions: \h)dpj ' \cdpj iu dpj h dpj J n n 2k J opj Q dpj J d TT. s tt f, \ 1 d The radial component, ^U T, in (11) involves four terms, each being controlled by the physics of the problem. The first term, m M^TTX( I ^) describes the source term changes induced by the model parameter perturbations. In fact, the seismogram

7 source term is not local (Kennett, 1995} in that the eigenfunction values at the source are affected by the perturbations in the structure. The second term accounts for the waveform changes resulting from perturbations in the medium properties. The first factor of the third term, ^jj~-, results from the adoption of a point source in a three-dimensional space (see Chapter 7.4 of Aki and Richards, 1980) whereas the factor i rj^ is the seismogram phase term change due to the structural perturbation. In WKBJ-type waveform inversion scheme {e.g., Nolet, 1990), only this term is accounted for in the calculation of the differential seismograms within the linearized inversion scheme. The last term, j-^r^a, accounts for the deformation of the elliptical particle motion induced by the structural perturbation. The vertical component, ^-U z, in (12) includes only the first three terms just described. 4. Analytic versus numerical differentiation In this section, we compare the results of the analytical calculation of seismogram derivatives with those obtained through "brute-force" numerical differentiation. The upper 250 km of the structure used to generate synthetic seismograms are shown in Figure la. While iasp91 velocity structure (Kennett and Engdahl, 1991) is used at larger depths, the density model in CAL8 ( Bullen and Bolt, 1985) is adopted for the same depths. From 250 to 640 km a constant Q value 450 is used, and a linear increase of Q from 450 to 680 is adopted to extend the structure down to a depth 1060 km. Our structure has continental properties and it is characterized by two main velocity discontinuities. The source is located in the crust at a depth of 33 km and the receiver is placed at a distance of 400 km. The azimuth of the station with respect to the fault-strike is 260. The source parameters are 5 = 37 and A = 283, and a double-couple mechanism is adopted. The seismograms have been scaled with Gusev's source spectrum (Gusev, 1983) for a moment of Nm. Modal summation is performed by summing about 200 modes. The top trace in Figure 2 is the resulting

8 seismogram. In Figure 2, we show the differential seismograms calculated using the analytical expressions (11, 12) together with those determined numerically. A shear-wave velocity increment of 0.1 km/s is used to compute the shown differential seismograms. For conciseness, we show only the differential seismograms corresponding to the grey shaded layers of Figure lb. The solid lines correspond to the analytical calculations whereas the dashed lines correspond to those obtained numerically. The numerical derivative of each seismogram is determined by adopting a first order centered numerical differentiation (13). This approach involves two seismogram evaluations, but guarantees some stability in the numerical differentiation. The adopted differentiation formula is: ds[t, Pj {z)} dp, = s[t,pj{z) +5 Pj {z)] - s[t, Pj {z) - 5p 3 {z)\ 2S Pj (z) where Spj(z) = km/s is the shear-wave velocity perturbation. As expected, the analytical differential seismograms do not suffer from the numerical instabilities affecting the purely numerical differentiation (Fig. 2). Computationally, the numerical differentiation involves 2N (N being the number of layers) complete time-consuming mode-spectrum calculations. In contrast, the proposed analytic method involves calculation and storing of the relevant partial derivatives only once and the computational cost required is of the order of just one additional mode-spectrum. It follows that the analytical differentiation requires only fast access to the partial derivative data file. Depending on the number of layers and on the type of model parameterization (i.e., the number of control points adopted in model definition) the exact analytical differentiation leads to consistent computational saving (as much as two orders of magnitude) when compared to the fully numerical calculation. In our tests, as much as 95% reduction in CPU time is obtained for models parameterized using 15 linear, local basis functions. Similar parameterizations have been used by Zielhuis and Nolet (1994).

9 5. Discussion Although a complete analysis of the contributions provided by the different terms in ( 11) and ( 12) is not the subject of this paper, it will be presented in a forthcoming paper, we describe here the main features that characterize the differential seismograms shown in Figure 2. In general, the amplitudes of the differential seismograms depend on the layer thicknesses. Because in our structure the thicknesses of the individual layers vary with depth, these should be accounted for while comparing the effects of the velocity perturbations onto the differential seismograms. Within the frequency range of our investigation, we observe that a perturbation applied to the shallow and 1.5 km thick layer 2 is comparable to that resulting from a similar perturbation applied to the 6 km thick layer 12 located between 33 and 39 km. In addition and as expected, a perturbation to the shallow layer 2 has a more pronounced effect on the tail of the seismogram. The shape of the differential seismograms resulting from perturbations in the crustal layers is, however, very similar. In contrast, a perturbation in the upper mantle (i.e., layer 14) affects only the early part of the synthetic seismogram (i.e., between 90 and 125 s). The development of this analytic technique for the calculation of differential seismograms has rather obvious implications for waveform inversion studies. For example, our differential seismogram method can easily be implemented within the "partitioned waveform inversion" scheme proposed by Nolet (1990) so that not only the phase term but also the amplitude term is included in the seismogram partial derivatives calculation required for the linearized inversion. In addition, the speed and the accuracy of the proposed technique may allow for the adoption of Monte Carlo derived global search minimization techniques for waveform fitting.

10 6. Conclusions We have presented a new analytic technique for the fast and accurate determination of seismogram partial derivatives with respect to the medium parameters. The technique adopts the multi-mode summation formalism (e.g., Panza, 1985). Calculation of the differential seismograms involves the determination of just one additional mode-spectrurn and, depending upon the parameterization of the structure, it can reduce the computational costs by as much as two orders of magnitude when compared to numerical methods. The technique can be used in waveform inversions for the structural parameters within both locally linearized, iterative and global search schemes. Acknowledgments We would like to thank Peter Suhadolc for carefully reading the manuscript and providing suggestions. This research has been partially funded through the European Union contracts EV5V-CT , ENV4-CT , ENV4-CT and through CNR ( , ), MURST 40% and 60% funding. The public domain GMT graphics softwaie developed by Wessel and Smith (1991) has been used throughout.

11 References Aki, K. and Richards, P. G., Quantitative Seismology, Vol. 1, Freeman, San Francisco. Bullen, K. E. and Bolt, B. A., An Introduction to the Theory of Seismology, Cambridge University Press, Cambridge. Chapman, C. H., A new method for computing seismograms, Geophys. J. R. Astron. Soc, 54, Das, S. and Suhadolc, P., On the inversion problem for earthquake rupture: The Haskell-type source model, J. Geophys. Res., 101, Gomberg, J. S. and Masters, T. G., Waveform modelling using locked-mode synthetic and differential seismogram: application to determination of the structure of Mexico, Geophys. J. R. Astron. Soc, 94, Gusev, A. A., Descriptive statistical model of earthquake source deviation and its application to an estimation of short period strong motions, Geophys. J. R. Astron. Soc, 74, Harkrider, D. G., Surface wave in multilayered elastic media, Part II - Higher mode spectra and spectral ratios from point source in plane layered earth models, Bull. Seismol. Soc. Am,, 60, Kennett, B. L. N., Seismic wave propagation in stratified media, Cambridge University Press, Cambridge, Kennett, B. L. N., Approximations for surface-wave propagation in laterally varying media, Geophys. J. R. Astron. Soc, 122, Kennett, B. L. N. and E. R. Engdahl, 1991, Treaveltimes for global earthquake location and phase identification, Geophys. J. Int., 105, Knopoff, L., A matrix method for elastic wave problems, Bull. Seismol. Soc. Am., 60, Luco, J., E., and R., J., Apsel, On the Green's function for a layered half-space. 10

12 Part I, Bull. Seismol. Soc Am., 73, Nolet, G., Partitioned waveform inversion and Two-Dimensional structure under the Network of Automously Recording Seismographs, J. Geophys. Res., 95, Panza, G. F., Synthetic seismograms: the Rayleigh waves modal summation, J. Geophysics, 58, Panza, G. F., Schwab, F., A., and L., Knopoff, Multimode surface waves for selected focal mechanisms I. Dip-slip sources on a vertical fault plane, Geophys. J. R. Astron. Soc, 34, Panza, G. F. and P., Suhadolc, Complete strong motion sythetics, in Seismic Strong Motion Synthetics, Vol. 4, pp , ed. by B. A. Bolt, Academic Press, Orlando, Florida. Randall, G. E., Efficient calculation of differential seismograms for lithospheric receiver functions, Geophys. J. Int., 94, Randall, G. E., Efficient calculation of complete differential seismograms for laterally homogeneous earth models, Geophys. J. Int., 118, Sambridge, M. S., and G. Drijkoningen, Genetic algorithms in seismic waveform inversion, Geophys. J. Int., 109, Schwab, F., Surface wave dispersion computations: Knopoff's method, Bull. Seism. Soc. Am., 60, Schwab, F. and Knopoff, L., Fast surface wave and free mode computation, in Methods of Computational Physics, pp , ed. B, A. Bolt, Academic Press, New York. Sen, M. K., and P. L. Stoffa, Nonlinear one-dimensional seismic waveform inversion using simulated annealing, Geophysics, 56, Shaw, P. R., and J. A. Orcutt, Waveform inversion of seismic refraction data and applications to young Pacific crust, Geophys. J. R. Astron. Soc, 82,

13 Urban, L., Cichowicz, A. and F. Vaccari, Computation of analytical partial derivatives of phase and group velocities for Rayleigh waves with respect to structural parameters, Studio, geoph. et geod., 37, Urban, L., Z. J. Du, and G. F. Panza, Computation of analytical partial derivatives of group velocities and eigenfunctions for Rayleigh waves with respect to structural parameters, in preparation, Valyus, V. P., Determining seismic profiles from a set of observations, in Computational Seismology, pp , ed. by V. I. Keilis-Borok, Consult. Bureau, New York. Wessel, P., and W. H. F. Smith, Free software helps map and display data, EOS Trans. Amer. Geophys. U., 72, pp. 441, Zeng, Y., and J., G., Anderson, A method for direct computation of the differential seismogram with respect to the velocity change in a layered elastic solid, Bull. Seism. Soc. Am., 85, Zielhuis, A. and Nolet, G., Shear-wave velocity variation in the upper mantle beneath central Europe, Geophys. J. Int., 117,

14 Figure 1. (a) Upper 250 km, of the velocity model used to generate the seismogram synthetics and the differential seismograms; (b) enlarged upper part of the shear-wave velocity model. The differential seisrnogram shown in Fig. 2 are computed for perturbations of the grey shaded layers. The whole structure used to determine the mode-spectrum extends to a depth of 1060 km. Figure 2. Seismogram and differential seismograms obtained using a perturbation of 0.1 km/s in shear wave velocity for the layers indicated in Fig. 1. The uppermost trace is the velocity seismogram (ju/s) which is determined using an instantaneous point source double-couple (S = 37, A = 283, and h = 33 km) located at a distance of 400 km and at a strike-receiver angle of 260. The seismograms are scaled with Gusev's source spectrum (Gusev, 1983) for a moment Mo = Nm. The remaining traces are the differential seismograms. The number at the top right side of each seismogram indicates the layer number (Fig. lb) the differentiation refers to. A Gaussian roll-off filter having a cut-off frequency of 0.85 Hz is applied to all traces to remove the ringing due to the truncation of the spectrum, (a) radial component; (b) vertical component. 13

15 Density(g/cm**3) I I I I I I I I I I I I I Velocity{km/s) Qs and Qp a Fig.la Fig.lb 14

16 Radial component synthetic.dat 0, analytical-02 numsricai analytical-04 numerical ,03 - analytical-06 numerical-06 5 > anaiytical-08 numerlcal-08 time(s)

17 Vertical component synttietic.dat 0.00 O.01 'in & 00s 0.00 analyttcal-06 numerlcal ,02 analytical H 1 1 h ir -H 1! time (s) Fig.2b

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