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Effects of weighting functions on Maslov uniform seismograms: A robust weighting method

Department of Physics University of Toronto, Toronto, Ontario Canada M5S 1A7

Abstract

In principle, Maslov theory provides a uniform asymptotic solution for modeling wave fields in a generally inhomogeneous medium by blending ray theory and Maslov transform theory. However, in a real calculation, it is still not clear how choice of the weighting functions that select the ray and Maslov transform solutions will affect the accuracy of Maslov seismograms.

To answer this question, we have investigated two weighting functions: One is the hyperbolic function suggested by Brown (1994), and the other is a trigonometric function we propose. The former relates the weights to the local slope of the Lagrangian manifold cross section, reflecting the opportunity for the ray solution or the Maslov solution to contribute to the wave-field construction. But the transition level of the hyperbolic function is controlled by an artificial coefficient. We found that the hyperbolic function performs accurately, particularly when a wave front is only smoothly bent, but there exists a tolerance range for the value of the coefficient. Outside this range, errors of caustic type or pseudocaustic type arise. Because the range is limited, and because it changes for different modelings, use of the hyperbolic function can become troublesome in intensive modelings where it is impractical to have human supervision.

The trigonometric function that we propose works differently. Caustic and pseudocaustic points are located by a root-finding algorithm, and trigonometric weighting functions control switching of the two solutions in a symmetric way between pairs of caustics and pseudocaustics. In most cases, it will provide an accurate result automatically, and it should be considered first for intensive modelings such as those needed in controlled source seismology.