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 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.     

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.     

A method for the computation of finite frequency body wave synthetic seismograms in laterally varying media

Summary. Body wave synthetic siesmograms for laterally varying media are computed by means of a slowness implementation of the extended WKBJ (EWKBJ) theory of Frazer & Phinney. An EWKBJ seismogram is computed by first tracing rays through a particular model to obtain conventional ray information (travel time, ray end point, ray slowness) and then using these data in the finite frequency integral expression for the EWKBJ seismogram. The EWKBJ seismograms compare favourably to geometrical ray theory (GRT) seismograms but are significantly better because of the finite frequency nature of the EWKBJ calculation. More realistic behaviour is obtained with EWKBJ seismograms at normal seismic frequencies near caustics, where the GRT amplitude is infinite, and within geometrical shadow zones where GRT predicts zero amplitudes. In addition the EWKBJ calculation is more sensitive than GRT to focuses and defocuses in the ray field. The major disadvantage of the EWKBJ calculation is the additional computer time over that of GRT, necessary to calculate one seismogram although an EWKBJ seismogram costs much less to compute than a reflectivity seismogram. Another disadvantage of EWKBJ theory is the generation of spurious, non-geometrical phases that are associated with rapidly varying lateral inhomogeneities. Fortunately the amplitudes of these spurious phases are usually much lower than that of neighbouring geometrical phases so that the spurious phases can usually be ignored. When this observation is combined with the moderately increased computational time of the EWKBJ calculation then the gain in finite frequency character significantly outweighs any disadvantages.