Gaussian beam synthetic seismograms in a vertically varying medium

Summary We develop the Gaussian beam summation method for a medium in which velocity is only a function of depth. We show that in such a medium the kinematic and dynamic ray tracing equations, i.e. trajectories and amplitudes, may be solved in closed form for any initial wavefield specified at the source. The solution for an individual Gaussian beam is written in terms of the usual functions of ray theory: distance, travel time, intercept time and geometrical spreading. An important result of this analysis, confirmed by numerical experiments, is that one of the base functions selected by Červený, Popov & Pšenčik to solve the dynamic ray tracing equations should be modified to avoid causality problems. Finally, by means of a simple canonical transformation we rewrite all the equations in a geographical coordinate system independent of the particular ray trajectories. We then show that Gaussian beam summation is an analytical continuation to complex values of position and slowness of the WKB method proposed by Chapman. A simple computational method is developed in which it is not necessary to determine the coordinates of the observer in ray centred coordinates. This simplifies the computational effort so that Gaussian beam calculation becomes only slightly more expensive than WKB. With respect to the latter method Gaussian beam summation has the advantage that it is possible to control the amplitudes of the cut‐off phases due to a finite range of slowness integration.