In this investigation we solve the problem of propagation of a compressional pulse in a solid half‐space which is overlain by a solid layer of different properties. The point source is situated at the depth ½H, H denoting the thickness of the layer. Theoretical seismograms of the vertical displacement w at the surface are evaluated out to ranges r = 20H. The solution is obtained by the exact ray theory. The displacement Wo due to the source was assumed to have a shape which at large distances reduces to a sawtooth. The once‐reflected waves from the interface, PP and PS, are strongly marked. The Rayleigh wave is already recognizable at r = 5H and is fully developed at r = 20H. The method of ‘ray theory’ was applied here far into the region where the normal mode theory converges well. The theoretical seismograms are illustrated. The properties assumed for the layer (1) and for the half‐space (2) are λ1 = μ1 μ2 = 2μ1cs2 ≡ c2 = 1.1cs1 ≡ 1.1c1cp2 > cp1 > c2 > c1
The impedance tensor corresponding to the magnetotelluric field for a nonisotropic one‐dimensional structure is given in terms of the solutions of a sixth‐order differential system. The conductivity tensor is three‐dimensional. Its components depend upon depth only in an arbitrary manner such that the corresponding matrix is positive definite. The impedance tensor components are found by a numerical integration procedure based on a set of one‐step methods and a variable step‐size to insure a given accuracy in the final result. Calculations were made for three models having sharp boundaries and also transitional layers. The first of these models has a middle layer of high conductivity, sandwiched between two layers of linearly varying conductivity, while in the second model the middle layer has a very low conductivity. In the third model the conductivity tensor is three‐dimensional and is linearly varying in one of the layers.
Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.
A geophysical inversion procedure based on the generalized inversion of a matrix is presented and applied to anisotropic magnetotelluric data for one-dimensional models. Various computational aspects of the iterative process involved are discussed together with the use of the resolution matrix, information matrix and the eigenvalues and eigenvectors for establishing the global significance of the parameters and their relationship with the data. A starting anisotropic model is built gradually, the first stage consisting of two separated isotropic problems corresponding to the off-diagonal elements of the impedance-tensor. The examples given include an anisotropic model as well as a model having two anisotropic layers and an isotropic one.
Exact synthetic seismograms are obtained for a simple layered elastic half‐space due to a buried point force and a point torque. Two models, similar to those encountered in seismic exploration of sedimentary basins, are examined in detail. The seismograms are complete to any specified time and make use of a Cagniard‐Pekeris method and a decomposition into generalized rays. The weathered layer is modeled as a thin low‐velocity layer over a half‐space. For a horizontal force in an arbitrary direction, the transverse component, in the near‐field, shows detectable first arrivals traveling with a compressional wave velocity. The radial and vertical components, at all distances, show a surface head wave (sP*) which is not generated when the source is compressive. A buried vertical force produces the same surface head wave prominently on the radial component. An example is given for a simple “Alberta” model as an aid to the interpretation of wide angle seismic reflections and head waves.
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