Synthetic body wave seismograms for three-dimensional laterally varying media

Summary. Two methods of computing body wave synthetic seismograms in three-dimensional laterally varying media are discussed. Both these methods are based on the summation of Gaussian beams. In the first, the initial beam parameters are chosen at the source, in the second at the beam endpoints. Both these variants eliminate the ray method singularities. The expansion of the wavefield into plane waves may be considered as the limiting case of the first approach and the Chapman–Maslov method as the limiting case of the second approach. Computer algorithms are briefly described and numerical examples presented. In the first numerical example, the comparisons of the two approaches, based on summing Gaussian beams, with the reflectivity method indicate that the computed synthetic seismograms are satisfactorily accurate even in the caustic region. The next example suggests that the two methods discussed can be simply and effectively applied to 3-D laterally inhomogeneous structures.     

Synthetic seismograms for complex three-dimensional geometries using an analytical–numerical algorithm

Summary. An algorithm which is part analytical and part numerical is suggested for the computation of complete synthetic seismograms for complex three-dimensional geological structures with radial symmetry. A partial separation of variables based on the combination of a finite Fourier integral transform with respect to the spatial coordinate z together with the finite difference method is the essence of the algorithm. Upon application of the finite transform the problem reduces to solving a system of equations containing only partial derivatives with respect to one spatial coordinate ( r ) and time. As radial symmetry is assumed, there is no functional dependence on φ in the cylindrical system of coordinates ( r , φ, z ). The coefficients of the transformed equations may contain finite Fourier integrals of the z dependence of the elastic parameters. Several examples of synthetic seismograms computed for both SH - and P – SV -waves propagating in complex subsurface geometries are presented and their interpretation discussed.     

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.     

Synthetic SH seismograms in a laterally varying medium by the discrete wavenumber method

Summary. We present a new method to calculate the SH wavefield produced by a seismic source in a half-space with an irregular buried interface. The diffracting interface is represented by a distribution of body forces. The Green's functions needed to solve the boundary conditions are evaluated using the discrete wavenumber method. Our approach relies on the introduction of a periodicity in the source-medium configuration and on the discretization of the interface at regular spacing. The technique developed is applicable to boundaries of arbitrary shapes and is valid at all frequencies. Some examples of calculation in simple configurations are presented showing the capabilities of the method.     

Determination of Lg-wave attenuation using single-station seismograms: a case study for Zimbabwe

In this paper we show how the quality factor Q may be calculated using a single seismograph station and a number of events recorded on analogue seismograms. We followed Nuttli's (1973) method and extended it to one seismograph station. Using the single station Bulawayo (BUL), we determined a mean Q value of 650 for Zimbabwe. Furthermore, we considered different propagation paths over Zimbabwe as the seismic waves travelled to BUL and found a low Q value of 350 for the Deka fault zone. the Q value of 650 obtained in this study agrees well within error with that of 603 reported by Chow et al. (1980) using the multistation, multi-event method, and lies within the range of Q values (454–759) reported by Xie & Mitchell (1990) using a 'back-project'method to image large-scale lateral variations of Lg coda Q . the results obtained are important as the Q value constitutes part of the input data in seismic hazard calculations. the method may be used to determine Q in regions where there may be only one station with reliable analogue seismogram data.     

Generation of complete theoretical seismograms for SH-III

Summary. Earlier efforts to generate the entire theoretical seismograms, including both body and surface waves for realistic sources buried in a radially heterogeneous anelastic, spherical earth, are extended to include the summation of 16 modes. The comparison between a real seismogram and theoretical time series, relative to different attenuation models in the upper mantle, yields information concerning the anelasticity under the Pacific Ocean.     

A modified first-motion approximation for the synthesis of body-wave seismograms

Summary. Modified first-motion approximations have been developed for the generation of synthetic body-wave seismograms using the Cagniard-de Hoop method. Comparisons are presented between classical first motion, modified first motion and full Cagniard treatments for problems involving a homogeneous sphere and a triplication in a realistic earth model. Results of these comparisons show that the modified first-motion approximations may be used for a wide variety of geophysically interesting problems with little loss of accuracy compared to the full Cagniard method.     

Generation of complete theoretical seismograms for SH—

Summary. We report the initial results of our attempts to obtain theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram, including both body waves and surface waves, can be generated for a spherical, anelastic earth by simple inverse Fourier transformation of the sum of the propagating fundamental and higher-mode surface waves. The key to success in reproducing the WWSSN records involves the number of modes, and the minimum period used in these computations; here we use eight modes and a minimum period of 2 s. Efficient computational algorithms make it possible to handle up to 2000 frequency points for each mode; approximately 200 layers are used to model the radial heterogeneity of the earth; attenuation is treated exactly. Examples are given of the SH theoretical seismograms resulting from dislocation sources buried at various depths in the Earth.     

Thickness estimates of the upper-mantle transition zone from bootstrapped velocity spectrum stacks of receiver functions

We modify the receiver-functions stacking technique known as velocity spectrum stacking (VSS) so as to estimate combinations of velocity model ( V_P and V_S ) and depth that stack the Ps conversion from upper-mantle discontinuities most coherently. We find that by estimating the differences in the depths to the 660 and 410 km discontinuities using velocities that maximize the stacked amplitudes of P410s and P660s phases we can estimate the thickness of the transition zone more accurately than the depths to either of these discontinuities. We present two examples indicating that the transition zone beneath Obninsk, Russia, is 252±6 km thick and that beneath Pasadena, California, is only 220±6 km thick.     

Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media

Summary. This paper discusses several aspects of the calculation of theoretical seismograms for two-dimensional inhomogeneous media with the method of Gaussian beams. The most important steps of this method, kinematic and dynamic ray tracing, can be performed very efficiently, if the model cross-section is subdivided into triangles with linear velocity laws. Each Gaussian beam is characterized by a complex beam constant ε which determines its width and phase-front curvature. Various possibilities to choose ε are discussed, including cases where beam properties at the beam endpoint (and not at the beginning) are prescribed; for instance, the beam width at the endpoint can be specified. In such cases the beam constant is a function of the radiation angle at the source, and the decomposition of a cylindrical wave into beams has to take this into account by weighting the beams differently, at least in principle. The exact weight function is derived and shown to be reasonably well approximated by the weight function, corresponding to angle-independent ε Theoretical seismograms are presented for a laterally heterogeneous model of the crust–mantle transition which is characterized by complications in the reflection from the transition and in the refraction from below. These complications are modelled by and large with success. The seismograms, however, depend to a certain extent on the choice of the beam constant. Moreover, according to the reciprocity principle calculations with source and receiver interchanged should have the same results as calculations for the original configuration. In practice this is not so, and the difference increases with the strength of lateral heterogeneities. Hence, for a successful application of Gaussian beams the model should not vary too strongly in lateral direction.