Comparison of the WKBJ and truncated asymptotic methods for an acoustic medium

Abstract

A new time-domain method for solving the 1-D acoustic wave equation with smoothly varying density and bulk modulus is used to analyse the propagation of waves in a smoothly varying medium. The new 'truncated asymptotic' method is exact for a large class of inhomogeneities and hence does not require the usual restriction of geometrical optics, that the wavelength be much less than the material stratification length.The truncated asymptotic solution is used as a benchmark to analyse a two-term WKBJ approximation for one of four classes of velocity functions. The velocity functions are such that the truncated asymptotic solutions are exact. For the case of small and intermediate velocity gradients, the WKBJ solution performs well when the length scale of the transition zone is of the same order, or larger, than the length of the applied wavelet. For steeper velocity gradients, the WKBJ solutions differs significantly from the exact truncated asymptotic solution.

Theory and Observations: Body Waves, Ray Methods, and Finite-Frequency Effects

“…Truncation errors in the solution (Yedlin et al, 1990) especially in the presence of strong velocity gradients as well as numerical effects coming from smoothing boxcar windows or from convolution operators (Dey-Sarkar and Chapman, 1978) require specific attention when one wants to reconstruct the velocity structure. Reverberations from a stack of layered structures, which can be observed in the reflecting method, are well separated as WKBJ contributions (Shaw, 1986) (see also Section 1.05.4 of this book).…”

P-SV coupling in inhomogeneous elastic media of Epstein type

“…Wave propagation studies in inhomogeneous media described by the Epstein pro¢les constitute one of the wellknown canonical problems in inhomogeneous media. In spite of the remarkable progress in the design of e¤cient and versatile computer codes for layered media, evidence of interest in canonical studies can be found in the literature (Drijkoningen 1991;Yedlin et al 1990). Amongst the studies which made use of Epstein's original solution to the scalar wave equation are those of Phinney (1970) and Hron & Chapman (1974a,b).…”