Theoretical Calculation of the Spectrum of First Arrivals in Layered Elastic Media

摘要

A practical method for the calculation of the spectral parameters of first-arriving signals in seismology has been the object of much theoretical work in recent years. The difficulty has been in the fact that the first arrival usually behaves as an imperfectly trapped mode. A practical technique is proposed. By a change of variable, the twice transformed solution is separated into a product of the form f(u).e raised to the -iuR power. This can be integrated with respect to the phase variable u, using standard quadrature methods, with the real part of u changing most rapidly along the integration path. By making the frequency complex, it is possible to displace any singularities away from the vicinity of the contour. This gives the spectrum of the signal as viewed through an exponentially decaying time window, making it possible to work with the first arrival by itself. (Author)