Calculation of complete theoretical seismograms in vertically varying media using collocation methods

Abstract

Summary. We present a method for calculating complete theoretical seismograms in earth models whose velocity, density and attenuation profiles are arbitrary piecewise-continuous functions of depth only. A form of attenuation valid for low loss situations is included by allowing the seismic velocities to be complex, and frequency is also allowed to be complex to avoid wraparound problems in the time-domain seismograms. Solutions for the stressdisplacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. A seismic source is applied at the Earth’s surface and a radiation condition is applied at depth. The resulting two-point boundary value problem for the expansion coefficients is solved by a collocation technique which works best for those cases that other methods, e.g. propagator matrices, work most poorly, i.e. highly evanescent solutions. Solutions for the expansion coefficients are obtained in the depth, frequency and horizontal wavenumber domain. Phase velocity filtering may be effected at this point by restricting the portion of the frequency-wavenumber plane in which solutions are sought. The transformed strain tensor at depth is formed by taking linear combinations of the solutions. This strain tensor is transformed back into the space and time domain by successive application of a Bessel transform, a fast Fourier transform, and by multiplication of the time signal by a growing exponential to remove the exponential decay introduced by the use of complex frequency. The strain tensor is contracted with a seismic moment tensor, and a reciprocity relation for Green’s functions is used to obtain displacements at the Earth’s surface caused by a buried moment tensor source.