Transient acoustic wave fields in continuously layered media with depth-dependent attenuation: An analysis based on higher-order asymptotics

A method is presented for the analysis of the space-time domain acoustic wave field in a continuously layered, lossy, isotropic fluid or equivalent fluid. Application of vertically varying compliance and inertia memory functions enables the modeling of a large class of depth-dependent loss properties. The method is based on integral transformations and consists of three steps. First, a temporal Laplace transformation with a real and positive transform parameter is applied, which is followed by horizontal spatial Fourier transformations. Second, higher-order, WKBJ-like asymptotic representations are derived that form approximate solutions of the resulting transform domain problem. With the chosen forward transformation process, problems caused by a zero vertical slowness are avoided, and the need for more intricate types of asymptotics is absent. Third, the Cagniard–De Hoop method is employed for the transformation back to the space-time domain. The form of the transform domain expressions allows for a very efficient use of the inversion process, while many steps of the inverse transformation may be performed analytically. Since the method does not impose a numerical limit on the bandwidth of the result, it may also be employed to generate a Green’s function. Numerical results are presented for reflections from half-spaces filled with a continuously layered, equivalent fluid with depth-dependent losses. At each level these losses show an almost “constant-Q” behavior in the frequency domain. The results show that there is an interval, beginning with the arrival time, on which the improvement due to the use of higher-order asymptotics is significant.