Modeling space-time domain acoustic wave fields in media with attenuation: The symbolic manipulation approach

Abstract

A method is presented for the determination of the space-time domain acoustic wave field, and in particular the space-time domain Green’s function, in homogeneous, isotropic, lossy equivalent fluid media that represent solids with a complicated viscoelastic behavior. The loss properties of the equivalent fluids are modeled with the aid of arbitrarily intricate compliance memory functions and, eventually, inertia memory functions. The presented integral transformation-type method consists of three steps. First, a temporal Laplace transformation and a horizontal spatial Fourier transformation are performed. Due to the application of the temporal Laplace transformation, causality of the acoustic wave field is automatically dealt with. Second, the resulting transform domain problem is solved using a convergent Neumann series solution. Analytical expressions for the terms of this Neumann series solution are obtained by means of a recurrence scheme that can ideally be evaluated with the aid of symbolic manipulation. Third, the transformation back to the space-time domain is performed analytically using the Cagniard–De Hoop method. No numerically imposed limitation of the bandwidth of the wave field quantities or the Green’s function shows up. In principle, the method offers the possibility of generating space-time domain results, both for the acoustic wave field and the Green’s function, for any time instant with any desired degree of accuracy. After deriving the general theory, numerical results for several lossy equivalent fluids with an almost constant-Q behavior, as shown by many types of rock, are presented.