Monte Carlo simulation of seismogram envelopes in scattering media

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. B3, PAGES , MARCH 10, 2000 Monte Carlo simulation of seismogram envelopes in scattering media Kazuo Yoshimoto Department of Geophysics, Graduate School of Science, Tohoku University, Sendai, Japan Abstract. The analysis of the seismogram coda envelopes of local and regional earthquakes is one of the most effective strategies for investigating the heterogeneous lithospheric structure characterized by the seismic scattering and attenuation. In order to synthesize the coda envelope we introduce a numerical scheme called the direct simulation Monte Carlo method, which has been used in the field of the kinetic theory of gases. Because of the simplicity of the algorithm the method has several advantages over previous methods in terms of the flexibility of the numerical calculation to incorporate various factors required to construct realistic seismogram envelopes. On the basis of coda envelope simulations, including multiple scattering, we show that an increase of seismic velocity with depth severely affects the shape of the coda envelope. The effects of ray bending due to the velocity increase at the Moho and the reflection at the free surface are clearly found in the synthesized envelope for a shallow earthquake. Our simulation demonstrates that the amplitude of the envelope is magnified by stagnation of seismic energy at shallow depths due to the positive velocity gradient with depth. Because of this effect, for an a priori assumption of a homogeneous velocity model the measurement of the scattering coefficient by conventional methods may be overestimated. 1. Introduction of the seismic energy just beneath the free surface, as suggested by Hayakawa [1998] for the single isotropic scattering Coda waves of local and regional earthquakes are consid- model. In order to quantify this effect on seismogram enveered to be incoherent successive arrivals of the scattered waves lopes of local earthquakes it is necessary to trace the ray paths from random heterogeneoustructures [Aki end Chouet, 1975]. Aki [1980] indicated that coda waves have the same attenuation characteristics as S waves in the lithosphere. Array observations of high-frequency seismic waves reported that the coda waves consist of scattered body waves, except for the surface waves generated by local near-surface weathered layers [e.g., l/emon et el., 1998]. On the basis of these characteristics the analysis of the seismogram coda envelope has been one of the most effective tools to investigate the heterogeneity of the of seismic waves in three dimensions by using a realistic velocity structure model. The Monte Carlo method plays an important role in investigating the effect of velocity structure on coda envelopes. Adopting an analytical integral representation for the distance between scattering points, Hoshiba [1997] developed an efficient algorithm for a simple structure model and studied the characteristics of coda envelopes in layered media. In order to calculate seismogram envelopes for a realistic Earth model in lithosphere. Many methods for synthesizing coda envelopes which seismic velocity varies complexly we introduce a direct based on the energy transport approach have been proposed for this purpose [e.g., Gusev end Abubekirov, 1987, 1996; Hoshibe, 1993; Zeng et el., 1991; Nishimure et el., 1997; Yoshimoto et el., 1997a, b]. By incorporating the effect of multiple seismic scattering, the analysis of coda envelopes at large lapse simulation Monte Carlo (DSMC) method, which has been used in the field of the kinetic theory of gases [e.g., Bird, 1976, 1994; Bellomo, 1995]. The DSMC method, which utilizes a finite difference scheme for ray tracing, has an advantage over previous methods in terms of flexibility and can be applied to times made it possible to separately measure the scattering three-dimensional (3-D) (i.e., laterally varying) velocity struccoefficient and the intrinsic attenuation of the lithosphere [e.g., Meyede et el., 1992; Hoshibe, 1993; Jin et el., 1994]. However, tures. In this paper, we synthesize seismogram envelopes for a shallow local earthquake using a realistic velocity structure. there are few studies that adopt a realistic model of seismic Instead of the layered structures which were used by Hoshiba velocity structure [Hoshibe, 1997; Mergerin et el., 1998]. The use of an inappropriate model, such as a homogeneous velocity structure, for the envelope analysis may cause significant error [1997] and Margetin et al. [1998], we adopt a lithospheric model with a positive velocity gradient with depth to evaluate the effect of seismic ray bending on coda envelopes. in estimated medium parameters of the heterogeneous lithosphere (e.g., scattering coefficient). 2. Method In general, seismic velocity in the lithosphere increases with depth [e.g., Christensen end Moony, 1995]. In such a structure, We apply the DSMC method to synthesize seismogram envelopes in 3-D scattering media. The treatment of seismic not only ray paths of direct waves but also those of scattered waves is completely acoustic in this paper (i.e., only S waves waves bend upward. This causes concentration and stagnation are considered acoustically). For simplicity, let us suppose a Copyright 2000 by the American Geophysical Union. Paper number 1999JB /00/1999JB D medium in which the scattering coefficient # is spatially uniform and the seismic velocity changes only vertically. The scattering coefficient is also known as the turbidity and the 6153

2 6154 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE (a) (b) Source z So u rc Particle i x Receiver Scat,t/ering Free Surface Torus Volume x Particle time interval At the particle at depth z propagates over a distance of v(z)at, and the probability of the occurrence of scattering along this path is represented by 1 - exp[-v(z)at/ l] --- v(z)at/l, where v(z) is the seismic velocity at the depth and l -- #-1 is the mean free path of the scattering medium. This probability can be rewritten as g v(z)at by using the scattering coefficient. Under the condition of v(z)at << #- 1 the process of propagation is approximately characterized by successive independent mathematical operations that represent free propagation and scattering, respectively [Bird, 1967, 1994; Nanbu, 1980]. This is called the principle of uncoupling. In the case that the particle is scattered in this time interval its motion is calculated as the sequence that the particle first moves over the distance v(z)at without scattering and then changes its propagation direction by scattering. The occurrence of scattering along the distance of v(z)at is determined by whether the following inequality is satisfied: gv(z)at > U3, (2) where U3 is the uniform random deviate between 0.0 and 1.0. Since the probability distribution of the scattering direction is assumed to be isotropic, the propagation direction of the scattered particle is also evaluated by using (1); a newly generated set of parameters is used for the next step of calculation. Introducing the x'-z plane which includes ray path as shown in Figure la, the vector representing the propagation of the particle in time interval At is (Ax', Az) = [v(z)at sin O, v(z)at cos O], (3) where O represents the angle of particle propagation measured from the downward vertical and satisfies the following equation [Cerveny and Rayindra, 1971]' dv(z) A O = d-- -- At sin O. (4) Figure 1. Schematic illustration of the propagation of energy particle. The location of the source is taken as the origin of the Cartesian coordinate. (a) A spherical volume at a receiver to count the number of energy particles. (b) A torus volume just beneath the free surface to count the number of energy particles at a receiver at epicentral distance A. scattering cross section per unit volume [Dainty, 1981]. We assume that the source radiation and the scattering of seismic waves are isotropic, which implies a spherical symmetry of these phenomena. Random perturbation of seismic velocity in space is not considered here. We shoot an energy particle from a point source (Figure la). Because of the assumption of isotropy, takeoff angle 0 and azimuthal angle 0 of the particle are given randomly as arccos (1-2U1), 2'rrU2, (1) NO. I l SmaRT I ß I SHOO,N I '1 I,.,,, ua NO I YES (gvat> 1' J Sq.1 I where U and U 2 are the uniform random deviates between 0.0 and 1.0. Random variables U and U2 are generated independently. Equation (1) ensures uniform probability distribution of the shooting direction on the focal sphere. The energy particle, which represents a seismic wavelet of unit energy, propagates in the 3-D medium following seismic ray theory. The location of the particle at a certain lapse time is calculated on the basis of a finite difference scheme. For a small T YES Figure 2. Flow chart of the numerical calculation of seismogram envelope.

3 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE 6155 (a) lo 2o 3o 4o 5o Model 1" Uødel11' Model 2 I ' Model 1 I I I ' ' ' dia for every time step, unlike the methods of Hoshiba [1997] and Margerin et al. [1998]. This step-by-step calculation requires additional computational work. However, because of the simplicity of the algorithm the DSMC method can be easily applied to a structure with a complex spatial variation of velocity and is executable on a personal computer with small on-board memory. A computational investigation of the accuracy of numerical calculation by this method is presented in the appendix. 3. Numerical Simulations 3.1. Seismogram Envelopes in a Realistic Velocity Structure We synthesize seismogram envelopes in a scattering medium with a vertical velocity gradient by using a model of S wave velocity structure beneath the Tohoku district, Japan [Hase- $ wave Velocity (km/s) (b) Epicentral Distance (km) 10'7 _o 40.o loo 1.o u 1/ ' -... / -. ß '- G 5o Figure 3. (a) Veloci structure models of S waves used in the synthesis. (b) Schematic illustration of the free propagation of energy particle. Model 1 is used for the calculation of ray path. =50km The left-hand term represents the increment of 0 in time interval At. The initial value of 0 is set to be 0 at the location of the source. The location of the energy particle at a certain lapse time t is evaluated by iterating the calculation described above. In this computation the highest order of scattering possible is the total number of iterations because a scattering is allowed only once in each time interval At. Under the condition of #v(z)at << 1 a small time step gives a good approximation for the tracing of the particle in a scattering media. A small volume element A V is considered at a receiver. In the case where the particle is included in this volume element, we count it as an arrival of a seismic energy packet. On the basis of the Monte Carlo scheme we shoot many energy particles of number N from the source. The energy density at the receiver at a time t is evaluated as n(t)/(nav), where n(t) is the number of particles included in the volume element at the receiver. We refer to the time history of the seismic energy density at a receiver as the seismogram envelope throughout this paper. The size and shape of the volume element affect the result; however, this effect is not severe for the coda portion where the spatial and temporal variation of energy density is small. Time-space distribution of energy density is also evaluated from the same calculation for a set of spatially distributed receivers. Figure 2 shows the flow chart of the seismogram envelope synthesis. The DSMC traces particle motion in scattering me- A=lOOkm A= 150km A=200km \ 35km/s Figure 4. An example of the synthetic seismogram envelopes for model 1 in Figure 3a. Envelopes of the surface receivers at epicentral distances from 10 to 200 km are exhibited. Sampling interval of the data is 1 s. The lapse time is measured from the source origin time. The dashed line indicates apparent velocity of 4.35 km s -, which is the S wave velocity of the uppermost mantle. The source depth is 15 km. The scattering coefficient is 0.01 km -. The parameters used in the simulation are as follows: At = 0.2 s, N = 107 and AV = 1.0 x x 103 km 3 (varying with epicentral distance as A V c A).

4 , 6156 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE. '10-5 I,,,,,,,,I,,, 10-6, 10 '7 LU ß 10.8, I 10 ' 2 = 50 kn "'"" >' I 10-6 L 10.7 ß 10-8 E z lo ,J II... : '5 r 10 ' LU ß 10.8 E ' '6 10 ' A=I I i i i i i i1[ i i i i i i i i i i ij i i i A =, Figure 5. Seismogram envelopes for different velocity structures. The solid, dashed and dotted lines indicate the envelopes for the velocity models of model 1, 2, and 3 in Figure 3a, respectively. Envelopes at epicentral distances of 50, 100, 150, and 200 km are displayed on Figures 5a, 5b, 5c, and 5d, respectively. Sampling interval of the data is 1 s. The source depth is 15 km. The scattering coefficient is 0.01 km -. The parameters used in the simulation are as follows: At = 0.2 s, N = 107, and AV = 5.0 x 102, 1.0 x 103, 1.5 x 103, and 2.0 x 103 km 3 (at epicentral distances 50, 100, 150, and 200 km, for lapse times smaller than 200 s). For lapse times larger than 200 s, a receiver volume 4 times as large as the original is used to minimize the ripple of envelope. gawa et al., 1978]. The model, which is used for routine hypo- on the envelopes is a result of the finite number of energy central determination Tohoku University, has an S wave particles. It takes 4 hours of CPU time on a 270-MHz Sun velocity of 3.15 km s - at the free surface and 4.35 km s - in Ultra5 to calculate the 20 envelopes on Figure 4, where 20% of the uppermost mantle, as shown in Figure 3a. The velocity has the time is consumed by generating random numbers. The maximum gradient at the crust-mantle transition at a depth of apparent velocity of the first arrival increases with epicentral -30 km. For simplicity, a reflection from the transition struc- distance, as is expected for the direct wave. The shape of the ture which includes a discontinuity of the first derivative of envelope is impulsive at small epicentral distances broadening velocity is not modeled in the synthesis. We assume isotropic as the distance increases. At epicentral distances smaller than scattering and a constant scattering coefficient of 0.01 km km the first part of the envelope corresponds to the direct irrespective of depth. Intrinsic attenuation is not considered arrival of S waves, which propagate through the crust. In conhere. We put a point source of isotropic radiation at a depth of trast, the shape of the first arrival at larger distances affected 15 km so that the effect of velocity structure on seismogram by the velocity structure of the crust-mantle transition, and the envelopes is emphasized. Examples of free (nonscattering) effect of surface reflection dominates. The dominant impulsive propagation of energy particles are schematically shown in signal seen at epicentral distances larger than 150 km reflects Figure 3b. For the free surface we assume total reflection the arrival of the ss phase. according to ray theory. By multiple reflections at the free At large lapse time the decay of envelope amplitude is surface an energy particle that is shot in a nearly horizontal smooth irrespective of epicentral distance. This indicates that direction from the source propagates through the crust, that is, the characteristic structure of the crust-mantle transition does a crust with a positive velocity gradient behaves as a waveguide. not cause any short-time fluctuations of envelope amplitude In the numerical calculations of particle tracing, we adopt a except around the direct arrivals. We note that the small decay time step of At = 0.2 s, which is roughly one hundredth of the rate of amplitude in the coda is due to the exclusion of the mean free time l/v(z) and is small enough to ensure the intrinsic attenuation in the synthesis. The intrinsic attenuation stability of the calculation (see Figure A2). By usin geomet- can easily be incorporated into the DSMC method in a rical symmetry with respect to the epicenter we used a torus straightforward manner. volume just beneath the free surface to count the number of energy particles at a receiver (Figure lb). The size of the torus, 3.2. Effect of a Velocity Gradient on which increases with the epicentral distance, chosen to be Seismogram Envelopes large enough to stabilize ripple of the envelopes but small Figure 5 is a comparative plot of envelopes for the three enough to retain temporal resolution. velocity models: models 1, 2, and 3 in Figure 3a. Models 2 and Figure 4 exhibitsynthesized envelopes for surface receivers 3 are homogeneous velocity models and take 3.15 and 4.35 km at epicentral distances from 10 to 200 km. The amplitude s -, which correspond to the velocity at the free surface and sampled every 1 s is shown on a logarithmic scale. The ripple the uppermost mantle of model 1, respectively. Envelopes at

5 . YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE 6157 the epicentral distances of 50, 100, 150, and 200 km are calculated with the same parameters as those of Figure 4. A nonimpulsive shape of the seismogram envelope around the direct arrival is most prominent for model 1 with the velocity gradient. This is due to the free surface reflection and the ray bending caused by the velocity structure of the crustmantle transition mentioned in section 3.1. For each velocity model, envelope amplitudes at different epicentral distances are almost identical at large lapse times (see, for example, amplitudes at a lapse time of 500 s). As seen in Figures 5a-5d, the envelope shows a systematic difference in amplitude between the homogeneous velocity models (models 2 and 3). The amplitude at a fixed lapse time is consistently smaller for the high-velocity model. This is explained by the fast spatial spreading of the seismic energy in the high-velocity model. The normalization of the lapse time by using the seismic velocities of these models substantially diminishes this difference (see equation (A1) in the appendix). From just after the onset of envelope, as seen in Figure 5a, the exponent of the decay rate of amplitude is almost equal to t-.s, which is predicted by the long-time limit of the diffusion solution for the infinite medium. 10 '4 10 's 10 ' s ' '9 _ A :50km (a) 100km Model 1 L. J5Okm... I... I... I... I... I... I... I... I I... I... I... I... I... I... I... h,,,,,,,,h,,,,,,,, (b) Model 1' [... I'"' '1... I... I... I... I... I... I '4 I... I... I... I... I... I... I... I... I... I... I As depicted by the seismogram envelope of model 1 in Figure 5, the positive velocity gradient with depth increases the Model 1" amplitude of the envelope. This is the same tendency reported 10-6 by Hoshiba [1997] and Margerin et al. [1998] for layered structures. The amplitude calculated for the velocity gradient model 10'7 is roughly twice that for the homogeneous velocity models at 10-8 large lapse times. The temporal decay rate of amplitude is not identical with those of the homogeneous velocity models; how- 10 ' I ever, it is roughly characterized by t-.5 as the lapse time increases. Our result clearly indicates that upward ray bending due to the positive velocity gradient as well as the multiple reflection in the layer structure causes amplification of the seismogram envelope. We investigate how a small variation of a velocity structure Figure 6. Seismogram envelopes for different velocity structures. Envelopes at epicentral distances of 50, 100, 150, and affects the shape of seismogram envelopes. For this purpose, 200 km are shown. Sampling interval of the data is 1 s. Figures we use two models: one is a model with a low-velocity zone in 6a, 6b, and 6c are calculated using the velocity models of the lower crust (model 1') and the other has a high positive models 1, 1', and 1" in Figure 3a, respectively. The source velocity gradient near the free surface (model 1"). These mod- depth is 15 km. Set of parameters, except for At = 0.05 s, used els are constructed from model 1. Figure 6 shows seismogram in the calculation is identical with that used in Figure 4. envelopes for these three velocity models. The parameters, except for At = 0.05 s, used in the synthesis are identical with those of Figure 4. Except for the onset of envelopes, the shape of envelopes is almost identical among these models. The difference of amplitudes at a lapse time of 200 s is <10%. This implies that a small difference of velocity structure is easily hidden from the shape of coda envelopes. temporal rate of the energy leak decreases with lapse time; however, it is not identical among the velocity models. The smallest rate is found for the model with positive velocity gradient (model 1). In this model, >10% of the energy radiated from the source is seen at depths shallower than 15 km even at a lapse time of 200 s, twice the value for the homoge Spatial Energy Distribution neous velocity models. We call this phenomenon "stagnation." Figure 7, showing the depth distribution of the energy versus the lapse time, gives an intuitive insight into the dependence of the seismogram envelopes on velocity structure. The ratio of the energy included in the three layers bounded by depths 0-15, and 30-o km is plotted on the vertical axis (i.e., horizontal distribution of the energy is not reflected in Figure 7). For example, at a lapse time of 100 s, Figure 7a depicts that 21%, 13%, and 66% of energy is included in the top, middle, and bottom layers, respectively. As the lapse time increases, the seismic energy gradually leaks out of the crust for all the velocity models (Figures 7a, 7b, and 7c). The leak starts at a lapse time of -4 s when the wave front of the direct waves reaches the Moho. In general, the The dependence of energy stagnation on source depth is found in Figures 7a, 7d, and 7e. Depth of source is 15, 40, and 80 km for Figures 7a, 7d, and 7e, respectively. Since the source is located in the mantle for Figures 7d and 7e, the energy in the top and middle layer increases just after the arrival of the wave front at the lower boundary of each layer. As lapse time increases, energy propagation in the opposite direction due to the reflection of the free surface decreases the energy in these layers. The distribution of energy at short lapse times is quite different for each case; however, it is worthy to compare them at lapse times larger than several tens of seconds. The amount of energy stagnating in the crust, the top layer, and middle layer is clearly large for the shallow source: e.g., at a lapse time

6 6158 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE '?.!:!::::...:;:::.i::..:.:?....: i5.. i!?i:..: :::.i..:!????.! o ( ~ O Figure 7. Depth distribution of seismic energy versus lapse time. The ratio of seismic energy included in three different depth ranges (0-15, 15-30, 30-oo km) is shown. Figures 7a, 7b, and 7c are calculated using the velocity models of models 1, 2, and 3 in Figure 3a, respectively. Figures 7d and 7e depict the depth distribution of energy in model 1 for the sources with depths of 40 and 80 km, respectively. Set of parameters used in the calculation is identical with those used in Figure 5. of 80 s the energy of the event with a depth of 15 km is twice as large as that of 80 km. This indicates that the positive velocity gradient with depth effectively behaves as a waveguide for a shallow source Analysis of Seismogram Envelopes Our synthesis demonstrates that the effect of a positive velocity gradient with depth on the envelope amplitude is not negligible; a significant enlargement of envelope amplitude due to the stagnation of energy at shallow depths occurs for a scattering medium. It is not easy to discriminate between this phenomenon and that due to strong scattering. This implies that the analysis of observed seismogram envelopes of local and regional earthquakes might result in an overestimate of the scattering coefficient when a homogeneous velocity model is assumed. To investigate this, we carry out a numerical investigation of estimation of the scattering parameter by inversion analysis of seismogram envelopes using the theory of this paper. The first 20 s of the seismogram envelopes at epicentral distances of 10, 30, 50, 70, and 90 km in Figure 4 are used as the data of the inversion. We adopt a linearized iterative technique which is based on a singular value decomposition method. Assuming a homogeneous velocity model with an S wave velocity of 3.5 km s -, we seek a model parameter # which minimizes the squared residual between the seismogram envelopes of the data and this model. The residual of each sampling point is normalized by the amplitude of the envelope. A finite differ- ence approximation is used to calculate the differentials of seismogram envelopes by perturbing the # value 5%. Irrespective of an initial value of #, the squared residual becomes sufficiently small after several iterations. The estimated value is not sensitive to the initial value and is almost 60% larger than # km-, which is used to calculate the data envelopes. We apply the multiple lapse time window analysis [Hoshiba, 1993] to the envelopes at epicentral distances between 10 and 100 km in Figure 4. This method, which employs time windows for the analysis of seismogram envelopes, is commonly used in investigating the scattering coefficient # on the basis of the analysis of the time-space distribution of coda wave energy. In this numerical test, following Hoshiba [1993], we use three time windows of 15 s, which are applied from the onset of S waves without overlap. Let us representhe integral of seismic energy in each time window with the normalization of the envelope amplitude at lapse time of 60 s as E (r), E2(r), and E3(r), where r is the hypocentral distance. Adopting the homogenous velocity model used previously, we carried out the multiple lapse time window analysis to seek the # value which minimizes the squared log residual between data and calculation. In the grid search analysis, # value is varied from 0.0 to 2.0 x 10-2 km - by a step of 5.0 x 10-3 km -. The best fit result is shown in Figure 8. The residual takes a minimum value, which is one half of that for the true value, at# = 1.35 x 10-2 km -. Figure 8 shows good coincidence between data and calculation. However, our result indicates that the multiple lapse time

7 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE r r 2El(r) the field measurement of scattering coefficient under the assumption of homogeneous velocity structure might be overestimated as a result. Our result indicates that an appropriate structure model is indispensable for the synthesis of realistic seismogram envelopes r To evaluate the DSMC method, we investigate the accuracy of the numerical calculation by comparing it with an analytic r r2e3(r) solution, and varying the calculation parameters. Since there is at present no analytic solution, or Green function, of the multiple isotropic scattering model for a 3-D medium in which g best-' 1.35x 10-2 km'l velocity change spatially, we use the analytic solution obtained '0 4'0 6'0 ' 8'0 ' 1 )0 by Zeng et al. [1991] for a homogeneous velocity structure in an infinite volume. We synthesize seismogram envelopes for the 3-D infinite scattering medium characterized by the parameters in section 2 and compare them with those of analytic Hypocentral Distance (km) solutions. Source radiation is assumed to be temporally impul- Figure 8. Plots of normalized energy corrected for geo- sive and azimuthally isotropic. For the syntheses in this section metrical spreading by 4 rr 2. Circles, triangles, and diamonds represent energy integrals for the time windows of 0-15, 15-30, and s after the onset of S waves. The solid line shows the best fit result, which is calculated using # = 1.35 x 10-2 ' a) 0.5 >, 0.4 p =0.8 window analysis may overestimate # value under an inappropriate (homogeneous) structure model. The use of a precise velocity model is essential for envelope analysis. We believe that the analysis of seismogram envelopes is important for investigation of the global and regional structure of the lithosphere in connection with the characteristics of short-wavelengtheterogeneities. The DSMC method, which is very flexible in incorporating the effects of free-surface reflection, intrinsic attenuation, nonisotropic source radiation, and nonisotropic scattering on the seismogram envelope, would be a very effective tool for this study. 4. Conclusions A numerical algorithm called the DSMC method is introduced to synthesize seismogram envelopes in a scattering medium with nonuniform seismic velocity. We checked the capability of this method by comparing its numerical outputs and analytic solutions for a multiple isotropic scattering model (see the appendix). The result of the inspection indicates high precision and numerical stability of the DSMC method. Because of the simplicity of the algorithm it is easy to incorporate flee-surface reflection, intrinsic attenuation, nonisotropic source radiation, and nonisotropic scattering as well as the depth dependency of seismic wave velocity. The calculation can be executed on a personal computer with small on-board memory. Using the DSMC method, we investigated the characteristics of the seismogram envelope in a structure with a positive velocity gradient with depth. The velocity gradient alters not only envelope amplitude but also its temporal shape. The en- Appendix: Computational Accuracy and Stability o z ß 0.07 ß 0.06 >, 0.05 o) ' Normalized Time! i i i i i i i p u.i 0.02 O.Ol o Z ' O. 002 E O Normalized Time Normalized Time Figure A1. Normalized energy density at normalized distances of p = 0.8, 1.6, and 3.2. The result from the DSMC method (the solid line) and that from the analytic calculation velope shape of a shallow earthquake is affected by the flee- (the dashed line) are compared. The former is the average of surface reflection and by the ray bending due to the velocity 20 independent numerical simulations. Standard deviation is change at the crust-mantle transition. The stagnation of energy represented by the dotted lines. The solid and dotted lines lie at shallow depths is clearly found in the synthesis. Since this on top of the dashed line except for p = 3.2. The parameters used phenomenon is not easily discriminated from strong scattering, in the simulation are A, = 0.01, N = 106, and AV =

8 6160 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE o.o?, o.o?j I >,0.05' 0.06' I >,0.05J 0.06'J [ I A ' = ' O.04-J 0.03,. c e 0.03l 0.02l W o.02j [ O.OlJ O.01J 11 o o.oo I O0'00 'J Normalized Time Normalized Time c 0.07' ( 0.06 >,0.05 b z = e c 0.03 iil 0.02 E O.01 o 0.00 o 3 6 f 8 Normalized Time ' >,0.05- o.o4l ce o.o3l E o.oo Normalized Time Figure A2. Numerical investigation of the dependence of the result of the simulation on AT. Normalized energy density is calculated for p = 1.6. The result from the DSMC method (the solid line) and that from the analytic calculation (the dashed line) are compared. The former is the average of 20 independent numerical simulations. Standardeviation is represented by the dotted line. The solid and dotted lines lie on top of the dashed line for AT and 0.1. The set of parameters used in the calculation is identical with those used in Figure A1. we use the normalized time T and the normalized distance p following Sato [1995]: T = gvt, p = gr. (A1) The parameter r is the distance between the source and the receiver. the coarse sampling interval and is not critical. The shapes of the envelopes are identical until Av exceeds 0.1. As the value of Av increases beyond 0.1, the DSMC method exhibits slightly larger outputs compared with the reference. However, the DSMC method does not show any numerical instability. This result indicates that the upper limit of the time step interval in the calculation for accurate results is about one-tenth of the mean free time of the scattering medium. Figure A1 shows seismogram envelopes at different normalized distances: p - 0.8, 1.6, and 3.2. The average and plus or minus one standard deviation for 20 simulations are repre- Acknowledgments. The author is grateful to Haruo Sato for his sented by the solid line and the dotted line, respectively. The valuable discussions and suggestive comments. The author thanks number of particles used in each simulation is 106. A spherical Masakazu Ohtake for his thoughtful reviews and suggestions for volume with the radius of 0.1 is used to estimate the spatial greatly improving the manuscript. Thanks also to Shigeo Kinoshita for his kind encouragement for this study. Critical and helpful reviews of density of energy particles at the receiver. The amplitude of this manuscript by Anton Dainty, Stephen Gao, and Mitsuyuki seismogram envelopes calculated by the DSMC method is Hoshiba are gratefully acknowledged. For this study the author used identical with that calculated by the method of Zeng et al. the computer system of the Earthquake Research Institute, University [1991](the dashed line), except for ripples of the envelope at of Tokyo. p = 3.2. The ripples scatter around the analytic solution; however, they do not show any apparent bias within plus or minus one standard deviation. Shooting a larger number of particles References substantially diminishes the ripples. In addition, the geometric Aki, K., Scattering and attenuation of shear waves in the lithosphere, symmetry with respecto the point source may also be used for J. Geophys. Res., 85, , this purpose: the use of a concentric spherical shell with radius Aki, K., and B. Chouet, Origin of coda waves: Source, attenuation and scattering effects, J. Geophys. Res., 80, , p = 3.2 as the receiver volume element effectively reduces the Bellomo, N. (Ed.) Lecture Notes on the Mathematical Theory of the ripples. Boltzmann Equation, World Sci., River Edge, N.J., The DSMC method employs the decoupling principle, ex- Bird, G. A., The velocity distribution function within a shock wave, J. pressed by the inequality AT << 1, to calculate the motion of Fluid Mech., 30, , Bird, G. A., Molecular Gas Dynamics, Oxford Univ. Press, New York, energy particles. Next, we investigate the dependence of syn thesized envelopes on AT. Figure A2 displays the seismogram Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas envelopes at p = 1.6 for the numerical conditions AT = 0.01, Flows, Oxford Univ. Press, New York, , 0.2, and 0.4. After synthesizing 20 envelopes for each Cerveny, V., and R. Ravindra, Theory of Seismic Head Waves, Univ. of value, their average (the solid line) and standard deviation (the Toronto Press, Toronto, Ont., Christensen, N. I., and W. D. Mooney, Seismic velocity structure and dotted line) are calculated. One million particles are used in composition of the continental crust: A global view, J. Geophys. Res., the calculation. The analytic solution (the dashed line) is plot- 100, , ted for comparison. The difference around the onset is due to Dainty, A.M., A scattering model to explain seismic Q observations in

9 YOSHIMOTO: MONTE CARLO SIMULATION OF SEISMOGRAM ENVELOPE 6161 the lithosphere between 1 and 30 Hz, Geophys. Res. Lett., 8, equation, I, Monocomponent gases, J. Phys. Soc. Jpn., 49, , 1128, Gusev, A. A., and I. R. Abubakirov, Monte-Carlo simulation of record Nishimura, T., M. Fehler, W. S. Baldridge, P. Roberts, and L. Steck, envelope of a near earthquake, Phys. Earth Planet. Inter., 49, 30-36, Heterogeneoustructure around the Jemez volcanic field, New Mex ico, USA, as inferred from the envelope inversion of active- Gusev, A. A., and I. R. Abubakirov, Simulated envelopes of non- experiment seismic data, Geophys. J. Int., 131, , isotropically scattered body waves as compared to observed ones: Sato, H., Formulation of the multiple non-isotropic scattering process Another manifestation of fractal heterogeneity, Geophys. J. Int., 127, in 3-D space on the basis of energy transportheory, Geophys. J. Int., 49-60, , , Hasegawa, A., N. Umino, and A. Takagi, Double-planed structure of Vernon, F. L., G. L. Pavlis, T. J. Owens, D. E. McNamara, and P. N. the deep seismic zone in the northeastern Japan arc, Tectonophysics, Anderson, Near-surface scattering effects observed with a high- 47, 43-58, frequency phased array at Pinyon Flats, California, Bull. Seismol. Hayakawa, T., Derivation of the envelope Green function in scattering Soc. Am., 88, , media with the depth-dependent velocity structure, and its applica- Yoshimoto, K., H. Sato, and M. Ohtake, Three-component seismotion to the analysis of source rupture process (in Japanese), master's gram envelope synthesis in randomly inhomogeneous semi-infinite thesis, 94 pp., Tohoku Univ., Sendai, Japan, media based on the single scattering approximation, Phys. Earth Hoshiba, M., Separation of scattering attenuation and intrinsic absorp- Planet. Inter., 104, 37-61, 1997a. tion in Japan using the multiple lapse time window analysis of full Yoshimoto, K., H. Sato, and M. Ohtake, Short-wavelength crustal seismogram envelope, J. Geophys. Res., 98, 15,809-15,824, heterogeneities in the Nikko area, central Japan, revealed from the Hoshiba, M., Seismic coda wave envelope in depth-dependent S wave three-component seismogram envelope analysis, Phys. Earth Planet. velocity structure, Phys. Earth Planet. Inter., 104, 15-22, Inter., 104, 63-73, 1997b. Jin, A., K. Mayeda, D. Adams, and K. Aki, Separation of intrinsic and Zeng, Y., F. Su, and K. Aki, Scattering wave energy propagation in a scattering attenuation in southern California using TERRAscope random isotropic scattering medium, 1, Theory, J. Geophys. Res., 96, data, J. Geophys. Res., 99, 17,835-17,848, , Margerin, L., M. Campillo, and B. V. Tiggelen, Radiative transfer and diffusion of waves in a layered medium: New insight into coda Q, K. Yoshimoto, Department of Geophysics, Graduate School of Sci- Geophys. J. Int., 134, , ence, Tohoku University, Aoba-ku, Sendai , Japan. Mayeda, K., S. Koyanagi, M. Hoshiba, K. Aki, and Y. Zeng, A com- tohoku.ac.jp) parative study of scattering, intrinsic, and coda Q- for Hawaii, Long Valley, and central California between 1.5 and 15.0 Hz, J. Geophys. Res., 97, , (Received January 25, 1999; revised October 19, 1999; Nanbu, K., Direct simulation scheme derived from the Boltzmann accepted December 6, 1999.)