Transient acoustic wave modeling: Higher-order Wentzel–Kramers–Brillouin–Jeffreys asymptotics and symbolic manipulation

Abstract

A combination of Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) asymptotics and the Cagniard–De Hoop method is used to determine the space-time domain acoustic wave field in a continuously horizontally layered configuration. First, a Laplace transformation with real and positive transform parameter is applied with respect to the time coordinate, followed by a two-dimensional Fourier transformation with respect to the horizontal coordinates. Using the integral equations of the resulting transform domain problem, a recurrence scheme is derived for the coefficients of the WKBJ asymptotic expansions of the transform domain solutions. This scheme is well-suited for implementation in a symbolical manipulation program. The transformation of the resulting higher-order WKBJ asymptotic representations back to the space-time domain is performed with the aid of the Cagniard–De Hoop method, which is particularly efficient in this case. If both the source and the receiver are located in one homogeneous half-space of the configuration, the Cagniard–De Hoop method allows for an almost completely analytical transformation back to the space-time domain. For various configurations numerical results are presented, which show that the method is valid for instants ranging up to several times the arrival time.