Simulating the Envelope of Scalar Waves in 2D Random Media Having Power-Law Spectra of Velocity Fluctuation

Transcription

1 Bulletin of the Seismological Society of America, Vol. 93, No., pp. 4 5, February 3 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation by Tatsuhiko Saito, Haruo Sato, Michael Fehler, and Masakazu Ohtake Abstract During propagation through random media, impulsive waves radiated from a point source decrease in amplitude and increase in duration with increasing travel distance. The excitation of coda waves is prominent at long lapse time. We use a finite-difference method to numerically simulate scalar waves that propagate through random media characterized by a von Kármán autocorrelation function. The power spectral density function of fractional velocity fluctuation for j-th order von Kármán-type random media obeys a power law at large wavenumbers. Such media are considered to be appropriate models for the random component of the structure of the Earth s lithosphere. The average of the square of numerically calculated wave traces over an ensemble of random media gives the reference envelope for the evaluation of envelope simulation methods. The Markov approximation method gives reliable quantitative predictions of the entire envelope for random media that are poor in short wavelength components of heterogeneity (j.), while it fails to predict the coda envelope for random media that have rich short-wavelength components (j.). The radiative-transfer theory reliably predicts the coda excitation for j. when the momentum-transfer scattering coefficient is used as the effective isotropic scattering coefficient. Replacing the direct term of the radiative-transfer solution with the envelope of the Markov approximation, we propose a new method for simulating the entire envelope from the direct arrival through the coda. The method quantitatively explains the whole envelope for j.. For the case of j.5, however, our method predicts too much coda excitation. In such a case, the method can explain whole envelopes by using the effective scattering coefficient estimated from coda excitation. Introduction High-frequency ( Hz) seismograms are not only composed of direct P and S waves but also many waves scattered from heterogeneities distributed in the earth, which cannot be readily explained by using deterministic models. Recognizing the complexity of seismograms, seismologists use statistical characterization of the random heterogeneity and focus on envelopes of band-pass filtered traces that disregard phase information (e.g., Sato and Fehler, 998). Observed S-wave envelope duration is longer than expected from the source duration time and the observed duration gets longer with increasing travel distance (Sato, 989). Frequency dependence of envelope broadening can be used to quantitatively estimate the spectra of random inhomogeneity in the lithosphere (Obara and Sato, 995; Saito et al., ). Gusev and Abubakirov (996, 999a) simulated broadening of envelopes based on the radiative transfer theory and estimated the vertical profile of scattering structure in Kamchatka (Gusev and Abubakirov, 999b). Petukhin and Gusev (3) analyzed seismogram envelope in Kamchatka and found that they are close to the shapes expected for a random medium with a power-law spectrum. The Markov approximation is a stochastic method for the parabolic-wave equation that reliably models wave diffraction or multiple-forward scattering in random media (Shishov, 974; Lee and Jokipii, 975). The split-step Fourier and the extended local Rytov Fourier methods are deterministic approaches, which are useful for calculating one-way wave propagation in inhomogeneous media and are often applied to migration imaging (Stoffa et al., 99; Huang and Fehler, 998; Huang et al., 999). Wu et al. () extended the one-way propagation method for the study of Lg wave propagation. The finite difference (FD) method successfully simulates waves in random media. Fehler et al. () recently confirmed the validity of the Markov approximation and the extended local Rytov Fourier methods for modeling wave propagation in D random media having Gaussian-type power spectral density functions (PSDF) by comparing waveforms and envelopes from these methods and those from waveforms of the FD simulations. 4

2 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation 4 Although observed envelopes near the direct arrivals of local earthquakes are different from station to station, coda envelopes have a common decay shape for all stations (Rautian and Kalturin, 978). To explain such an observed character of coda, single scattering models based on the radiative transfer theory were developed (Aki and Chouet, 975; Sato, 977). Later, Shang and Gao (988) and Zeng et al. (99) developed a multiple-isotropic scattering model. Jannaud et al. (99) used the FD method to simulate scalar-wave propagation through random media characterized by Gaussian and exponential-type autocorrelation functions (ACF). They compared the coda spectra predicted by the single scattering model with the Born approximation and those from numerically simulated waveforms and found a good coincidence between them. Frankel and Clayton (986) found a discrepancy between temporal coda decay estimated from the single isotropic scattering model and that calculated using finitedifference simulations. The quantitative coincidence of temporal coda decay of envelopes calculated from radiativetransfer theory and those from waveform simulations has not yet been reported. Gusev and Abubakirov (996) studied envelopes in scattering media calculated using Monte Carlo simulations of the radiative-transfer theory and concluded that coda envelopes can be modeled effectively by using the multiple-isotropic scattering model even in media where anisotropic scattering is important. As mentioned above, the Markov approximation method and the radiative-transfer theory are often used for interpreting high-frequency seismogram envelopes. The Markov approximation method, however, is limited because it neglects wide-angle scattering. The radiative-transfer theory governs the energy transport phenomenologically so that the relation between the radiative-transfer theory and wave propagation in random media is not obvious. If we assume isotropic scattering, the radiative-transfer theory will fail to explain the early part of envelopes, which are mainly composed of multiple forward-scattered waves. In this article, we compare envelopes simulated using the Markov approximation method and the radiative-transfer theory by using the mean-square (MS) envelopes calculated from the average of wave traces synthesized by the FD method over an ensemble of random media as references. We investigate the validity of these direct simulation methods. Then, we propose a hybrid method for synthesizing complete envelopes from the onset of direct wave through the later coda by combining the merits of the Markov approximation method and the multiple isotropic scattering model based on the radiative-transfer theory. Wave Propagation in Random Media In D inhomogeneous media, scalar waves obey the following wave equation: D u(x,t), () V(x) t where D is the Laplacian. In randomly inhomogeneous media, wave velocity is written as V(x) V ( n(x)), where the fractional fluctuation of wave velocity n(x) is a random function of location x and V is the mean wave velocity. To obtain statistical properties of the wavefield, we consider an ensemble of random media {n(x)} such that n(x), where the angular brackets mean the ensemble average. Random media are statistically characterized by the ACF of fractional fluctuation of velocity R(x) n(y)n(y x) or its Fourier transform, the PSDF. We assume that n(x) is a homogeneous and isotropic random function, so that the ACF is a function of r x. The magnitude of inhomogeneity is given by the MS fractional fluctuation, e R() n(x). Many authors have reported that the observed PSDF of velocity fluctuation in the earth has a power-law spectrum, that is, fractal in space (e.g., Shiomi et al., 997; Goff and Holliger, ). A von Kármán-type random medium is a wellknown one having such a character at large wavenumber. In D it is given by 4pje a j P(m) j (am) for am k, () ( am) where m m is the wavenumber and a is correlation distance that represents the characteristic spatial scale of inhomogeneity. Figure shows the PSDF of fractional velocity fluctuation n(x) in D for several values of j. The PSDF obeys a power law at large wavenumbers (am k ) and parameter j controls the power index: random media are richer in short wavelength components for smaller j. The value j. is nearly the same as the value for the PSDF of velocity fluctuation obtained from well-log data at depths shallower than km (e.g., Shiomi et al., 997; Goff and PSDF -4-8 Frequency [km ] Wavenumber, m m κ κ =..5. [Hz] [km - ] Figure. Power spectral density functions of von Kármán-type random media (e.5, a 5 km) in D for j.,.5, and.. Vertical broken line shows the wavenumber in the background medium for a wave with a frequency of Hz that we analyzed in this study.

3 4 T. Saito, H. Sato, M. Fehler, and M. Ohtake Holliger, ), while the value j.5 is nearly the same as the value obtained from seismogram-envelope analysis (Gusev and Abubakirov, 996; Saito et al., ). Random media whose PSDF are Gaussian-type are sometimes used to examine wave propagation in smooth random media. We use the von Kármán-type random media with j. for smooth media. In this study, we examine envelope characteristics in random media for two cases, j. and. representing smooth and rough random media, respectively. The intermediate case of j.5 will be also examined. Characteristics of the Envelopes on the Basis of the Finite Difference Simulation We use the same approach for numerical simulation of wave propagation and the same method for synthesizing RMS envelopes as those of Fehler et al. () and Fehler and Sato (unpublished manuscript). The geometry of the simulation is drawn in Figure. Waveforms are numerically simulated in D random media with dimensions of 3 km long and 5 km wide. We use Higdon absorbing boundaries (Higdon, 99). A point source is placed at 5 km from the nearest boundary and the center of a 5-km side. Eight receivers spaced 5 km apart are placed along a center line parallel to the long side. A -Hz Ricker wavelet is radiated from the source. We set the average propagation velocity at V 4 km/sec, the correlation length of inhomogeneity at a 5 km, the RMS fractional velocity fluctuation at e 5% to be consistent with the observed S-wave random inhomogeneity in the lithosphere, which is characterized by e /a of the order of 4 3 in Japan (Sato, 989; Saito et al., ). FD simulations for wave propagation through such random media are accomplished using grid spacing of 5 m and time step of 4 msec with fourth-order accuracy in space and second-order accuracy in time. We get MS envelope by averaging squared waveforms at a given location from 5 realizations of random media having the same statistical characteristics; envelopes are smoothed over a.3- sec time window. The MS envelopes may be considered as energy density and the square root of the MS envelope gives the root mean square (rms) envelope. The black solid curves in Figure 3a and b show numerically synthesized rms envelopes for j. and., respectively, where the envelopes at all receivers are plotted together. Envelope amplitudes are normalized by the maximum envelope amplitude at the receiver located 5 km from the source. We can see that the envelope width (from the onset to the time when the rms amplitude becomes the halfmaximum rms amplitude) increases with increasing propagation distance. We refer to this phenomenon as envelope broadening. Envelope broadening has been observed in S- wave envelopes of earthquakes and was attributed to diffraction or multiple forward scattering caused by large-scale random heterogeneity in the lithosphere (Sato, 989; Scherbaum and Sato, 99; Obara and Sato, 995; Saito et al., ). Note that the envelope broadening in our simulations Absorbing Boundary Source 3km 5km Receivers 5km Figure. Geometry of simulation. All edges have absorbing boundary conditions. The grid spacing is 5 m. The media in this study are characterized by the von Kármán-type random media with root mean square (rms) velocity fluctuation of 5%, correlation distance of 5 km, and the order.,.5, and.. is caused not only by diffraction and multiple forward scattering but also by travel-time fluctuation because we regard the lapse time from the origin time (not onset time) as the reference time. The envelope widths are larger for j. compared with those for j. although the envelopes for j. have longer tails. The larger envelope width for j. may be attributed to stronger excitation of traveltime fluctuation and multiple forward scattering due to the richer long-wavelength component. For the case j., more coda waves are excited than for the case of j., and the asymptotic coda decay curves at all receivers become identical at large lapse time. The simulation is consistent with the common decay curve observed for coda envelopes (e.g., Rautian and Khalturin, 978). Broken curves show geometrical decay for D, which is proportional to the reciprocal of the square root of travel distance. Scattering loss causes the maximum amplitudes for the case of j. to decrease more rapidly with propagation distance than expected from the geometrical decay. Maximum amplitudes for the case j. decrease more rapidly than those for the case of j. since there is more wide-angle scattering from small-scale inhomogeneity in media characterized by j.. The Markov Approximation Method for Simulating Envelopes The Markov approximation method is reliable for synthesizing envelopes in random media when forward scattering dominates. Shishov (974) simulated envelopes of outgoing spherical-wave propagation through 3D random media having Gaussian ACF. Lee and Jokipii (975) and Sreenivasiah et al. (976) simulated envelopes for plane waves incident upon a 3D random media having Gaussian ACF. Fehler et al. () extended Shishov s method to the D case and formulated the envelope simulation of outgoing cylindrical waves in random media having Gaussian ACF. They found a good coincidence between the envelopes directly simulated by the Markov approximation method and those from FD simulations. Recently, Saito et al. () ex-

4 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation 43 (a) κ = km km 5km km (b) κ = Figure 3. Comparisons of rms envelopes directly synthesized by the Markov approximation method (gray curves) with the FD envelopes synthesized from the finite difference simulations (black curves) for (a) j. and (b).. Envelopes are normalized by the peak amplitude at a distance of 5 km. Broken curves show D geometrical decay, which is proportional to the reciprocal of the square root of travel distance. tended Shishov s work to 3D von Kármán-type random media. Here we extend Saito et al. s method to the case of D von Kármán type random media. The Markov Approximation Method for Cylindrically Outgoing Waves in D Random Media At a source-receiver distance r x, which is much longer than the wavelength (r k /k ) and medium correlation distance (r k a), we may write D scalar waves radiated from a point source at the origin as a superposition of harmonic cylindrical waves of angular frequency x as U(h,r,x) i(krxt) u(x,t) e dx, (3) p kr where k x/v is wavenumber and h is polar angle. When k a k, substituting equation (3) into equation () and neglecting the second derivative of U(h, r, x) with respect to r, we obtain the parabolic wave equation ik U(h,r,x) U(h,r,x) r r h (4) k n(h,r)u(h,r,x). We note that wave frequency Hz corresponds to k a 6. A small segment on the circle of radius r from the origin is the transverse line for outgoing waves from the source. We define the two frequency mutual coherence function (TFMCF) at radial distance r as the correlation of the wavefield between two locations on the transverse line r rh and r rh, at different angular frequencies x and x according to Ishimaru (978), C (r,r,r, x,x ) U(r,r, x )U*(r,r, x ). (5) Angular brackets indicate the ensemble average, and the asterisk indicates complex conjugate. Because random media are assumed statistically homogeneous, C depends only on the difference between locations r and r, that is, C (r d r r, r, x, x ). For quasi-monochromatic waves,

5 44 T. Saito, H. Sato, M. Fehler, and M. Ohtake x x, we can get the master equation for C from equation (4) as kd C i r k r h C c d k d c d k [A() A(rh )]C A()C, (6) where h d h h and r d rh d. We used a forwardscattering approximation to evaluate n(h, r)u(h, r, x )U(h, r, x )* and n(h, r)u(h, r, x )U(h, r, x )*.We introduce the center of mass and difference coordinate in the wavenumber space as k c (k k )/ and k d k k (k c k k d ), respectively. Corresponding coordinates for angular frequency will also be used. The effect of inhomogeneity is represented by the longitudinal integral of the ACF: A(rh ) dzr(rh, z), (7) d where z is the radial coordinate in the ACF. We choose C to be the form, d k C d C e A()r/, (8) where the exponential term represents the travel-time fluctuation of the waves (Lee and Jokipii, 975). The master equation for C is kd C i C r kc r hd k [A() A(rh )] C. (9) c d The MS envelope at propagation distance r and time t can be written by using the TFMCF as u(r, t)u*(r, t) dx dx C (h (p) kr c d d c ix d(tr/v ), r, x, x )e () c dx I(r, ˆ t; x ). p The intensity spectral density Î is written as the integral over the difference in angular frequency as Î(r,t;x ) C (h pk r c d c ix d(tr/v ), r, x d, x c)e dx d. () c d c The MS envelope in a narrow-frequency band Dx c around the center frequency x c is written as xcdx c/ I (r,t) I(r,t;x)dx ˆ Mar p x c Dx c / c ix d(tr/v ), r, x d, x c)e dx d. Dx c C (h d () (p) kr For an isotropic source radiation with total source energy W within this angular-frequency band, W r I (r r, t) d t, (3) pv r V Mar we choose the initial condition of C as C (h d, r, x d, x c) C (h d, r Wk c, x d, x c). (4) Dx V c For the D von Kármán-type random media, integration of the ACF in equation (7) can be calculated analytically from the PSDF as in the 3D von Kármán-type case. At a large distance, we need only the behavior of A for a small transverse distance in equation (9). The approximation form of A() A(rh d ) can be written as p(j) d rh rhd A() A(rh d) e ac(j) for K, (5) a a where C and the power p and are numerically calculated (Saito et al., ) and are listed together with A() in Table. Numerically integrating equation (9) with the initial condition equation (4) following to the same procedure as Saito et al. (), we obtain the solution C. The contribution of the travel-time fluctuation term exp{k d A()r/} in equation (8) does not influence the broadening of individual Table List of C, p, and A() for Different j Values j C p A()/e a

6 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation 45 wavetraces but only shows the wandering effect, which is the travel-time fluctuation over different wavetraces for each element of ensemble (Lee and Jokipii, 975). We have chosen the origin time as a reference time for averaging squared FD waveforms, that is, the FD envelopes in this study include travel-time fluctuation. This travel-time fluctuation does increase the width of the ensemble-average envelopes. Thus, we must use C given by equation (8). Note that if we choose wave-onset as a reference time, we should use C. Taking the fast Fourier transform (FFT) ofc (see equation ) and taking the convolution integral of the MS envelope with a Ricker wavelet, we synthesized the MS envelope I Mar based on the Markov approximation method. We note that the resultant envelopes depend on wave frequency in the same manner as those in the 3D case (e.g., Fig. 8; Saito et al. ). Root-Mean-Square Envelopes Obtained by the Markov Approximation Method Figure 3 shows rms envelopes obtained by the Markov approximation method (gray curves) together with the envelopes from the FD simulations (solid curves). We used the total energy W evaluated from the integration of the FD envelope around the peak at the nearest receiver at 5 km to scale the FD and Markov approximation envelopes at all distances. The Markov envelopes for the case of j., where the medium is smooth compared with the wavelength, reliably predict the FD envelopes for all propagation distances (Figure 3a). Comparing the Markov envelopes with the FD envelopes in random media having Gaussian PSDF, which are also a smooth media, Fehler et al. () confirmed the validity of the Markov approximation when the medium is smooth relative to wavelength. Our results also support the validity of the Markov approximation method for the case of smooth random media that have power-law spectra. For the case of j., rough random media, the Markov envelopes cannot explain large coda excitation, especially at short hypocentral distances, and the peak amplitudes of the Markov envelopes are larger than those of the FD envelopes for all propagation distances (Fig. 3b). The early portions of the envelopes, however, are modeled better than the coda portions. We may explain the smaller coda excitation of the Markov envelopes by considering the angular dependence of scattering coefficient g(h) where h is scattering angle. The scattering coefficient represents the scattering power per unit area and can be estimated using the Born approximation for media with random velocity fluctuations. It can be written by using the PSDF P(m) as (e.g., Sato and Fehler, 998) 3 x x h g(h) P sin, (6) V V where angle h is chosen to be in the forward direction. Figure 4 shows a logarithmic plot of g(h) against scattering angle h. When a random medium is smooth as in the case of j., the scattering coefficient is very large in a small angle around the forward direction while wide-angle scattering is weak; thus the Markov approximation method is valid. On the other hand, when a random medium is rough as in the case of j., wide-angle scattering dominates as compared with the case of j., so that the Markov method becomes inapplicable. As a result, the Markov method cannot give the correct attenuation of direct waves and also fails to explain strong coda excitation for such a case. Radiative Transfer Theory for Simulating Coda Envelopes Multiple Isotropic Scattering Model Based on the Radiative Transfer Theory The radiative-transfer theory has been developed for describing energy propagation in inhomogeneous media. In seismology, a formulation of the multiple isotropic-scattering process was shown to explain coda envelopes by Shang and Gao (988) and Zeng et al. (99). For modeling wave propagation in random media, we assume that the energy propagates through media with homogeneous background velocity V that are filled with randomly distributed point-like scatterers. Scattering power per unit area in D media is characterized by the scattering coefficient g(h), which is a product of scattering cross-section of a scatterer and the number density of scatterers. In the case of isotropic scattering g(h) g where g is the total scattering coefficient (cf. p. 43, Sato and Fehler, 998), the energy density including the effect of multiple scattering is described as an integral equation as (e.g., Sato and Fehler, 998) Scattering Coefficient g(θ) [km - ] κ =. κ =.5 κ = [deg.] Scattering Angle θ Figure 4. Angular dependence of scattering coefficient g(h) in a D random medium based on the Born approximation for j.,.5, and.. Medium properties and wavenumber are identical to those used for Figure.

7 46 T. Saito, H. Sato, M. Fehler, and M. Ohtake rad (x, t) WG(x, t) Vg (7) I G(x x, t t)i rad (x, t)dtdx. We suppose isotropic impulsive radiation of total energy W from a source located at the origin, which is described by the delta function in time at t. G(x, t) is the propagator describing the causal propagation with a geometrical spreading factor in a D space and the effect of exponential scattering loss: r Vgt G(r, t) H(t)d t e. (8) pv r V Shang and Gao (988) and Sato (993) solved this scattering system in D media to get the analytical solution of the spatial-time distribution of energy density as follows: W r gvt I (r, t) d t e pv r V Rad Wg g VtrV t r e H t (9) p V t r V Wg for Vtk r. pv t The first delta function term represents the direct propagation of energy from the source to the receiver. The scattered energy density is given by the second term, which asymptotically decreases in proportion to the inverse of the lapse time for V t k r. Applying the Isotropic-Scattering Model to Non-Isotropic Scattering Media The scattering coefficient g(h) predicted by the Born approximation works as a key for connecting wave propagation through random media and energy propagation through scattering media. For media with smaller amounts of short-wavelength components of heterogeneity (increasing j), high-frequency scattering is increasingly dominated by forward scattering. For scalar-wave propagation in such random media, the isotropic-scattering model is not directly applicable to envelope simulation. However, when the momentum transfer scattering coefficient p g ( cosh)g(h)dh () m p is used as the effective isotropic-scattering coefficient, the multiple isotropic-scattering model well predicts the coda envelopes in the diffusion regime where multiple scattering dominates and the effects of nonisotropic scattering is smoothed out (cf., Morse and Feshbach, 953, p. 88). In this study we synthesize modified radiative-transfer envelopes as the radiative-transfer envelopes using g m instead of g in equation (9). Gusev and Abubakirov (996) numerically confirmed that the momentum transfer scattering coefficient works as the effective isotropic scattering coefficient in the diffusion regime by simulating nonisotropic scattering process by using the Monte Carlo method within the framework of the radiative transfer theory. RMS Envelopes Based on the Radiative Transfer Theory Gray curves in Figure 5 represent rms envelopes synthesized for j. by using equation (9) where g is substituted with g m, and envelopes are convoluted with the envelope of a -Hz Ricker wavelet for consistency with the FD envelopes. Modified radiative transfer envelopes reliably match coda amplitudes of FD envelopes (solid curves) for all propagation distances. In Figure 6, we plot the coda portion of the modified radiative transfer MS envelope (gray curve) at a distance of 5 km with that of the FD simulation (solid curve) using a logarithmic scale. In the coda portion, the result of the FD simulation indicates that MS envelope decays in proportion to the inverse of time. This is theoretically predicted by the radiative-transfer theory, which includes multiple isotropic scattering. As demonstrated by Figures 5 and 6, coda envelopes that are composed of wide-angle scattered waves can be modeled properly by the radiative-transfer theory by replacing the isotropic-scattering coefficient with the momentum transfer scattering coefficient. The modified radiative-transfer envelope, however, fails to model the envelope near the direct arrival because the model assumes isotropic scattering even though the small-angle scattering around the forward direction is dominant in waves near the direct arrivals. The first-arriving packet in the model is too impulsive because too little energy has been forward scattered into the portion immediately after the direct arrival. A Hybrid Method for Simulating Entire Envelopes Figure 7 shows a schematic illustration of energy-packet propagation in a scattering medium based on the concept of the radiative-transfer theory. The MS envelope can be considered as the time trace of energy density, which is composed of energy packets directly propagating from the point source and those scattered by distributed heterogeneities. Gray curves show energy-packet paths from a source to a receiver. The straight solid line between the source and the receiver shows the direct path. The gray curves around the direct path show multiple scattered energy-packet paths with small-angle scattering, which form the early part of the envelopes. The paths containing wide-angle scattering form coda excitation. For smooth random media or those that are poor in short-wavelength components of heterogeneity (j.), small-angle scattering around the forward direction is dominant. As a result, energy propagating along paths around the

8 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation 47.8 κ = Figure 5. A comparison of rms envelopes by the radiative transfer theory (gray curves) with envelopes synthesized by the finite difference method (black curves) for j.. κ =. MS Amplitude - -4 t - Source Small-angle scattering Receiver Wide-angle scattering Figure 6. Plot of the mean square (MS) envelopes at a distance of 5 km for von Kármán-type random media of j. using a logarithmic scale. The solid curve shows the MS envelope by the finite-difference method and the gray curve shows a coda portion of the MS envelope predicted by the multiple isotropicscattering model by using the momentum transfer scattering coefficient g m.7 3 km. Reflected waves from the boundary might appear in the shaded time window. The broken line shows the decay according to the reciprocal of lapse time. Small-angle scattering Effective isotropic scatterer Figure 7. Schematic illustration of energy-packet propagation in scattering media based on the concept of the radiative-transfer theory. MS envelope can be considered as the time trace of energy density, which is composed of energy packets radiated from the source. Gray curves show energy-packet paths. The straight solid line between the source and the receiver shows the direct path. The gray curves around the direct path show multiple energy-packet paths with small angle scattering, which form the early part of the envelope, while the paths containing wide-angle scattering form coda excitation. The Markov approximation method can model multiple paths due to small-angle scattering around the direct path between a source and a receiver as shown by the shaded area. The multiple isotropic-scattering model with momentum transfer scattering coefficient considers energy packet paths as the straight trajectories composed of isotropic scattering (dashed lines) for modeling the coda excitation. direct path becomes dominant over that propagating along paths containing wide-angle scattering. The Markov approximation method, which correctly accounts for multipleforward scattering and disregards wide-angle scattering, reliably models propagation around the direct path shown by the shaded area in Figure 7, so that it can quantitatively explain the entire suite of envelopes for smooth media. On the other hand, for random media that are rough, or those that are rich in short-wavelength components such as the case for j., the short-wavelength components of inhomogeneity efficiently excite wide-angle scattering. Then, scatterers distributed far from the direct path contribute more to the excitation of coda waves compared with the case for j.. The modified radiative-transfer theory predicts proper coda amplitude as is seen in Figures 5 and 6. This means that energy-packet paths (gray curves in Fig. 7)

9 48 T. Saito, H. Sato, M. Fehler, and M. Ohtake are equivalent to a connection of straight paths propagating with the background velocity (dashed lines) between isotropic scatterers (closed circles) for modeling coda excitation. The momentum transfer scattering coefficient is expected to work as the effective isotropic-scattering coefficient in such a case. From the above interpretation, we consider that the energy-packet paths are composed of two parts: the multiple paths around the direct path and the paths containing wideangle scattering. The former are composed of multiple smallangle scattering paths that form the early part of the envelope, which is well modeled by the Markov approximation. The latter form the coda part of the envelope, which is well modeled by the isotropic-scattering model using the momentum transfer scattering coefficient as an effective isotropic-scattering coefficient. Replacing the first term in the radiative-transfer solution (equation 9) with the Markov envelope (equation ), we propose a hybrid method to simulate the entire envelope in random media, I(r,t) I e lgmvt Mar Wgm g VtrV t r e m H t. () p V t r V The second term of the right-hand side represents wide-angle scattered energy density predicted by the modified radiative transfer theory. In the first term, an exponential decay term exp(lg m V t) shows scattering loss due to wide-angle scattering, where we introduce a parameter l to satisfy energy conservation. We choose the parameter l to satisfy energy conservation, that is, the spatial integral of the energy density I(r,t)dx I (r,t)dx e Mar lg V t m gmvt W( e ) () should be independent of lapse time. The wandering-effect term exp{k d A()r/} in equation (8) is excluded in the evaluation of equation () for simulating the Markov envelope I Mar. The parameter l is estimated to be l.7 for the case of j. by using numerical calculation. Using the parameter l and including the wandering effect in equation (), we get the MS envelopes. We plot the rms envelopes for j. in Figure 8. We can see that our method explains entire envelopes from the direct arrivals through the coda for all propagation distances. Neither the Markov approximation method nor the multiple isotropicscattering model alone can explain the envelope characteristics at all distances. Envelopes for the Case of j.5 We have studied wave envelopes in smooth random media (j.) and rough random media (j.). The PSDF for j.5 is also a viable model for crust and mantle inhomogeneity (Gusev and Abubakirov; 996; Yoshimoto et al., 997; Saito et al., ). Figure 9a shows FD envelopes together with the Markov envelopes for the case of j.5. We can see that the Markov results match the early part of the envelopes at least until the time when the RMS amplitude becomes the half maximum amplitude. Although the Markov envelopes cannot explain coda excitation, the discrepancy is small compared with the case of j.. The results of this simulation support that the Markov approximation method is applicable for modeling the early part of envelopes even though the coda part is not modeled well. In Figure 9b, the envelopes synthesized based on the hybrid method we proposed in this study are plotted together with FD envelopes, where we used the momentum transfer scattering coefficient g m.9 3 km and l.74, which was calculated separately from the case for j.. Theoretically derived coda envelopes have larger amplitudes than the FD envelopes at near-source receivers. The momentum transfer scattering coefficient cannot work as an effective isotropic-scattering coefficient in this case. The cause will be discussed later. Assuming that we could get proper effective scattering coefficient by some method, we apply our hybrid method to the case of j.5. We numerically estimated proper scattering coefficient g c km, so as to minimize residuals between radiative-transfer envelope and FD envelopes in a logarithmic scale for the time range between and 3 sec at a 5-km distance from the source. The radiative transfer envelope calculated using this g c value is plotted using a gray curve in Figure together with an envelope using g m. Replacing g m in equation () by g c and re-evaluating the energy conservation criteria to get l.3, we plot synthesized rms envelopes together with the FD envelopes in Figure 9c. We can see good agreement between them. It means that we can synthesize whole envelopes if we could estimate a proper effective scattering coefficient. Discussion The momentum transfer scattering coefficient g m works as an effective isotropic-scattering coefficient g c for the case of j., but fails for the case of j.5. Also for the case of j., g m is larger than g c. Numerically estimated g c and theoretically calculated g m are plotted by squares and triangles, respectively, for different j values in Figure. The effective isotropic scattering coefficient g c for the case of j. and. are estimated using the same procedure as for the case of j.5. The size of the square symbols corresponds to the estimated range of g c. The difference between g c and g m for the case of j.5 and. comes from the breakdown of the Born approximation. Forward scattering is strong for j.5 and. as shown by broken curves in Figure 4 and the total scattering coefficients are g.5 km and.39 km for j.5 and., respectively (Fig. ). The total scattering coefficients derived by using

10 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation κ = Figure 8. A comparison of rms envelopes derived from the new hybrid method (gray curves) with envelopes synthesized by the finite-difference method (solid curves) for j.. the Born approximation become g a, which indicates that most of the incident energy is scattered and the mean free path is shorter than the correlation distance. This contradicts the weak scattering assumption for the Born approximation, which requires g K a. This means that the estimation of scattering coefficient g(h) in equation (6) based on the Born approximation is improper for j.5 and., especially for the small-angle scattering range. Estimation of g m includes such an improper use of the Born approximation for small-angle scattering. For the case of j.5 and., large-angle scattering is weak because ga K, even though strong forward scattering causes the breakdown of the Born approximation. Only large-angle scattering can be evaluated correctly by the Born approximation in such a case. We plot scattering coefficient together with different kinds of scattering coefficients in Figure. The single scattering model predicts that coda energy at lapse time 5 sec at a 5-km distance from the source is composed of scattered energy with scattering angle from 5 to 8 (see Sato, 98). This means that scattering coefficient g c estimated from coda at lapse time 5 sec satisfies g c g(5). However, our simulation shows g c g(5) for j.,.5, and.. This may reflect that not only large-angle scattering but also multipleforward scattering plays an important role for the coda excitation. Although our hybrid method succeeded in explaining the whole envelopes better than the use of the Markov approximation or the multiple isotropic scattering model alone, we note that our method cannot always reliably predict envelopes. The discrepancy between FD envelopes and those of our method using g c is slight on a linear scale; however, it is magnified using a logarithmic plot in Figure, where the lapse time of each FD envelope trace is limited to the time before the arrival of reflected waves from the model boundaries. The discrepancies between the FD envelopes and our model predictions are seen around the lapse time between 5 sec and sec at a 5-km distance from the source for the case of j.5. Such discrepancy around the intermediate lapse-time can be recognized for all envelopes for both the j. and.5 cases. The discrepancy becomes larger with increasing hypocentral distance and is larger for the case of j.5 compared with the case of j.. A discrepancy can be recognized also at the tail portion of the FD envelopes at long distances from the source (r 5 km, km). Note that the FD envelopes would be expected to converge to an asymptotic curve irrespective of the distance from the source with increasing lapse time (e.g., Gusev and Abubakirov, 996). That is, the FD envelopes at r 5 km, km would converge to the hybrid model envelopes after the lapse time of our simulation. The discrepancy around the intermediate lapse time may be caused by the simple formalism that the energy packet paths are divided into two categories: paths with strong forward scattering and paths with wide-angle scattering. For more reliable modeling, we may be required to use the Markov envelope as the propagator in the radiative-transfer equation. Conclusions Numerically simulating wave propagation of a -Hz Ricker wavelet in a D von Kármán-type random media with order j(e.5, a 5 km, and V 4 km/sec) by using the FD method, we got the MS envelope from the ensemble average of squared waveforms for propagation distance out to km. We examined the validity of the Markov approximation modeling method and the multiple isotropic scattering model by using the envelopes of the FD ensemble average as references. For smooth random media of j., the Markov approximation method predicts excellent envelopes for all propagation distances. For rough random media (j.), large coda excitation cannot be explained by the Markov approximation method, which disregards

11 5 T. Saito, H. Sato, M. Fehler, and M. Ohtake (a) Markov FD MS Amplitude - -4 g m g c κ =.5 t (b) (c) Hybrid g m FD Figure. Plot of MS envelopes at a distance of 5 km for von Kármán-type random media of j.5 in logarithmic scale. Solid black curve shows the MS envelope calculated by the finite-difference method (FD envelope) and gray curves show a coda portion of MS envelope predicted by the multiple isotropic-scattering model with the momentum transfer scattering coefficient g m and that with the best fit total scattering coefficient g c appropriate for the coda envelope. Reflected waves from the boundary of the finite-difference simulations might appear in shaded area. The broken line shows the decay according to the reciprocal of lapse time Hybrid g c FD Figure 9. Comparison of rms envelopes for j.5 synthesized by the finite-difference method (solid curves) with those derived by different theoretical envelope models (gray curves): (a) the Markov approximation method, (b) new hybrid method with the momentum transfer scattering coefficient g m.9 3 km, and (c) new hybrid method with the effective total scattering coefficient g c km evaluated from the coda excitation of the FD envelope. wide-angle scattering. For the coda portion, MS envelopes decrease in proportion to the inverse of lapse time. This feature is predicted by the radiative-transfer theory, which includes multiple isotropic scattering. The coda excitation is quantitatively well explained by using the momentum transfer scattering coefficient as the effective isotropic-scattering coefficient for j.. The model, however, cannot explain Scattering Coefficient [km - ] g m g c g g(θ) a - θ = 3 O 45 O 9 O 5 O -5 8 O..5. κ Figure. Plots of various kinds of scattering coefficient for j.,.5, and.. Effective isotropic scattering coefficients g c are plotted as squares, where the size of square corresponds to estimation range. Momentum transfer scattering coefficients g m are plotted as triangles. Total scattering coefficients g are plotted as large dots. Scattering coefficient at various scattering angles are plotted as small dots. Horizontal dashed line indicates the upper limit of the Born approximation, the scattering coefficient is a.

12 Simulating the Envelope of Scalar Waves in D Random Media Having Power-Law Spectra of Velocity Fluctuation 5-5km 5km (a) 75km 5km κ =. km (b) 5km 5km 75km κ =.5 5km km Figure. Comparisons of rms envelopes derived from the hybrid method using g c (gray curves) with envelopes synthesized by the finite difference method (solid curves) in logarithmic scale for (a) j. and (b) j.5. Envelopes by the finitedifference method are plotted within the lapse time before the arrival of reflected waves from the boundaries. envelopes around the direct arrivals because of the isotropicscattering assumption. Combining the merits of the Markov approximation and the radiative-transfer theory, and considering energy conservation, we proposed a hybrid method for simulating the entire envelope, where envelopes around the direct arrivals caused by multiple forward scattering are simulated by the Markov approximation method and the coda part composed of wide-angle scattering is given by the multiple isotropic scattering model with the momentum transfer scattering coefficient. The hybrid method quantitatively well explains the entire envelopes for j.. For j.5, however, the momentum transfer scattering coefficient cannot be used as the effective isotropic-scattering coefficient where the Born approximation is not suitable. Even in such random media, the hybrid method can explain entire envelopes by using the effective isotropic-scattering coefficient estimated from coda excitation. Acknowledgments The authors are grateful to R. L. Nowack, I. R. Abubakirov, A. A. Gusev, and anonymous reviewer for their helpful review. This research was partially supported by Grant-in-Aid for JSPS Fellows (No. 53) and Grant-in-Aid for Scientific Research (C), MEXT, Japan (No. 644). Work at Los Alamos National Laboratory was supported by the United States Department of Energy Office through contract W-745-ENG-36 from the Office of Basic Energy Sciences headed by Nick Woodward. References Aki, K., and B. Chouet (975). Origin of coda waves: Source, attenuation and scattering effects, J. Geophys. Res. 8, Fehler, M., H. Sato, and L.-J. Huang (). Envelope broadening of outgoing waves in -D random media: A comparison between the Markov approximation and numerical simulations, Bull. Seism. Soc. Am. 9, Frankel, A., and R. W. Clayton (986). Finite-difference simulations of

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Res. 96, Department of Geophysics Graduate School of Science Tohoku University Aramaki-aza Aoba, Aoba-ku Sendai , Japan saito@zisin.geophys.tohoku.ac.jp (T.S., H.S., M.O.) Los Alamos Seismic Research Center Los Alamos National Laboratory Los Alamos, New Mexico (M.F.) Manuscript received April.