A first-order scattering solution for modelling elastic wave codas — I. The acoustic case

Summary A corrected, first-order solution for modelling acoustic wave scattering in layered halfspaces containing random inhomogeneities is derived. Energy lost to higher order scattering and intrinsic attenuation is included in the correction, which is constructed so that energy is conserved to first order. The complex propagation effects of the layering are overcome by representing the motion as a sum of normal modes. This approach renders the kinematic description of the scattering two dimensional, with the wave vectors of incident and scattered modes lying parallel to the layering. At each level in the halfspace, the inhomogeneities are resolved into two-dimensional Fourier spectra also parallel to the layering. The root mean square (rms) motion of a scattered mode depends on the correlation between spectra at different levels and the group velocity of the mode. To simplify the solution, it is assumed that the inhomogeneity spectra are piecewise constant and that the energy of a normal model propagates only at its group velocity. The final step of the theory establishes a criterion for the source—receiver separations over which the results are accurate. Numerical calculations have been carried out for a single layer of inhomogeneities over a halfspace. The spectra of the inhomogeneities were assumed band limited, and several different spectra were examined. The results suggest the existence of a diagonal selection rule whereby a wavelet of mode order n scatters mostly to wavelets of the same order. Moreover, a resonant frequency of scattering occurs, causing the rms signal to appear monochromatic. The frequency of the resonance is controlled by the inhomogeneity spectra band limits. With the aid of the diagonal selection rule, the simplified solution allows for both rapid computation of synthetic signals and inversion of data for scattering cross-section. Existing data suggest the theory may be applied to obtain approximate models of local earthquake codas. The synthetic signals of the single layer cases, for example, have codas similar to published observations. To illustrate the inversion of data with the theory, a preliminary scattering cross-section for the lunar crust is presented.