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.     

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.     

Gaussian beams in elastic 2-D laterally varying layered structures

Summary. In a paper by Červený & Pšenčik, high-frequency Gaussian beams in elastic 2-D, laterally inhomogeneous, smooth media were investigated as asymptotic high-frequency solutions of elastodynamic equations, concentrated close to rays of P - and S -waves. This paper generalizes the above results for 2-D, laterally inhomogeneous, layered structures. Gaussian beams concentrated close to any multiply-reflected, possibly converted, ray are investigated. Gaussian beams are regular everywhere, including caustic regions. The paraxial ray approximation, which allows the wavefield in the zero-order ray approximation to be evaluated not only directly on the ray, but also in its vicinity, is derived as a limiting case of the Gaussian beams.     

A method for calculating synthetic seismograms in laterally varying media

Summary An effective algorithm for computing synthetic seismograms in laterally inhomogeneous media has been developed. The method, based on zero-order asymptotic ray theory, is primarily intended for use in refraction and reflection studies and provides an economical means of seismic modelling.
A given smoothed velocity-depth-distance model is divided into small squares with constant seismic parameters and first-order interfaces are represented by an arbitrary number of dipping linear segments. The computation of ray propagation and amplitudes through such a model does not involve complicated analytic expressions and therefore minimizes computer time.
Amplitudes are determined by geometrical spreading of spherical wave-fronts and energy partitioning at interfaces. Synthetic seismograms calculated for laterally homogeneous models are in good agreement with those obtained by the Reflectivity Method.     

Finite difference seismograms for laterally varying marine models

Summary. The method of finite differences is applied to the elastic wave equation to generate synthetic seismograms for laterally varying seafloor structures. The results are compared with borehole seismic data from the Gulf of California (Deep Sea Drilling Project Site 485) in which lines were shot over flat and rough topography. The significant new phenomenon observed in both the synthetic seismograms and the field data is the generation of a 'double head wave' due to the interaction of the incident wavefront with the side of a hill and the flat seafoor adjacent to the hill.
In these models the hills are on the order of a seismic wavelength in height and steep velocity gradients occur over distances comparable to wavelengths. Ray theoretical methods would not be suitable for studying such structures. True amplitude record sections are obtained by the finite difference method, which show for these models that the head wave generated at the flat seafloor adjacent to the hill is lower in amplitude than if the hill were not present and is lower in amplitude than the head wave generated at the hill.
A second feature which is important for borehole receivers is the existence of the 'direct wave root' in the upper basement. This energy occurs below the sharp interface when the direct wave impinges on the interface from above. There is no corresponding Snell's law ray path for this energy and the energy is evanescent with depth in the lower medium.
The properties of both the double head wave and the direct wave root are clearly demonstrated in the finite difference 'snapshot' displays.     

2-Dreflectivity method and synthetic seismograms for irregularly layered structures -I. SH-wave generation

Summary. The reflectivity method for complete SH seismograms has been extended to two-dimensionally layered structures. The Aki-Larner technique is generalized to solve the integral equations for 2-D boundary conditions, and propagator matrices are enlarged to express a total SH wavefield. Synthetic seismograms in a soft basin are calculated for an incident plane-wave. They compare favourably with the results of the finite-element and finite-difference methods even in the later portion where asymptotic ray and beam theories break down. Synthetic seismograms due to a line force and a point dislocation are also presented.     

Computation of complete synthetic seismograms for laterally heterogeneous models using the Direct Solution Method

We use the Direct Solution Method (DSM) together with the modified operators derived by Geller & Takeuchi (1995) and Takeuchi, Geller & Cummins (1996) to compute complete synthetic seismograms and their partial derivatives for laterally heterogeneous models in spherical coordinates. The methods presented in this paper are well suited to conducting waveform inversion for 3-D Earth structure. No assumptions of weak perturbation are necessary, although such approximations greatly improve computational efficiency when their use is appropriate.
An example calculation is presented in which the toroidal wavefield is calculated for an axisymmetric model for which velocity is dependent on depth and latitude but not longitude. The wavefield calculated using the DSM agrees well with wavefronts calculated by tracing rays. To demonstrate that our algorithm is not limited to weak, aspherical perturbations to a spherically symmetric structure, we consider a model for which the latitude-dependent part of the velocity structure is very strong.     

Some remarks on the Gaussian beam summation method

Summary. Recently, a method using superposition of Gaussian beams has been proposed for the solution of high-frequency wave problems. The method is a potentially useful approach when the more usual techniques of ray theory fail: it gives answers which are finite at caustics, computes a nonzero field in shadow zones, and exhibits critical angle phenomena, including head waves. Subsequent tests by several authors have been encouraging, although some reported solutions show an unexplained dependence on the 'free' complex parameter ε which specifies the initial widths and phases of the Gaussian beams.
We use methods of uniform asymptotic expansions to explain the behaviour of the Gaussian beam method. We show how it computes correctly the entire caustic boundary layer of a caustic of arbitrary complexity, and computes correctly in a region of critical reflection. However, the beam solution for head waves and in edge-diffracted shadow zones are shown to have the correct asymptotic form, but with governing parameters that are explicitly ε-dependent. We also explain the mechanism by which the beam solution degrades when there are strong lateral inhomogeneities. We compare numerically our predictions for some representative, model problems, with exact solutions obtained by other means.     

Synthetic seismograms for the case of the receiver within the reflectivity zone

Synthetic seismograms are shown and discussed for the case of the receiver within the medium. Most of the discussion is on the reflectivity method with the receiver within the reflectivity zone, but results using the ray method are shown for comparison. Such synthetic seismograms can be used to interpret data from Oblique Seismic Experiments where shots generated on the surface up to large ranges are recorded in crustal boreholes.