Theoretical calculation of the spectrum of first arrivals in layered elastic mediums

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 that the first arrival usually behaves as an imperfectly trapped mode. Mathematically, it arises from the contributions of branch line integrals and complex poles. Attempts to transform the solution into a generalization of the normal modes have been a mathematical success only. Because of the complexity of this solution, a different, less elegant approach is demanded. A practical technique is proposed. By a change of variable, the twice transformed solution is separated into a product of the form ƒ(u)e-iuR. This can be integrated with respect to the phase variable u, using standard quadrature methods, 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.