Abstract

Multi-mode, surface-wave dispersion—eigenvalue—computations are studied in the range of frequencies from 1 to 10,000 Hz, for structures composed of a sequence of homogeneous layers. When the original version of the Thomson-Haskell technique is applied to such structures, a loss-of-precision problem is encountered with Rayleigh-wave computations. Into the range of high frequencies, we now want to know whether the Love-wave formulation and the alternate Rayleigh-wave formulations are completely free of this problem. At low, intermediate, high, and very high frequencies, our numerical testing shows no evidence whatsoever of loss-of-precision problems; not for Love waves, for Rayleigh waves when Knopoff's method is used, nor for Rayleigh waves when the deltamatrix extension of the original Thomson-Haskell formulation is employed. In these three cases, when P digits are used in the computations, the resulting phase velocities can be obtained to full. P digit accuracy. All overflow/underflow problems can be controlled by simple modifications of the usual computational methods; these modifications do not affect the accuracy of the calculated dispersion.

The computation of displacement-depth and stress-depth functions—eigenfunctions—is also studied up to extremely high frequencies. Once again, the original Rayleigh-wave formulation encounters loss-of-precision difficulties in these eigenfunction evaluations. The use of Knopoff's method completely solves this aspect of the problem with precision loss. As with dispersion calculations, all overflow/underflow problems are easily controlled by means that do not affect the accuracy of the computed eigenfunctions.

In addition to precision loss and overflow/underflow problems, three further details affect efficiency and accuracy when synthesizing high-frequency seismograms by multi-mode summation of dispersed surface waves. Avoid branch-line integrals and spherical Bessel functions of nonintegral order by using spherical structures combined with transformations permitting replacement of Bessel functions with circular or exponential functions. The number of higher modes treated explicitly, at each of the higher frequencies, should be limited by selective sampling combined with summation procedures approximating inclusion of all modes. As frequency increases, the structural specification must be monitored to ensure the homogeneous layers used to approximate the true structure do not become much thicker than the vertical extent of the eigenfunction lobes.

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