A pulse in a binary sediment

Abstract

SUMMARY Synthetic seismograms are generated by an exact method that automatically separates stratigraphic effects from intrinsic effects. Then three different methods are used to derive rather simple expressions for the effective sound speed of a binary sediment for vertical propagation at wavelengths greater than the average layer thickness. A single-scatter theory (SST) is based on the notion that transmitted energy is reflected once less than absorbed energy. A path sum theory (PST) is used to explicitly sum the contribution of rays corresponding to multiple reflections. An averaged-multiple imbedding theory (AMI) is developed that explicitly averages multiple reflections over the distribution of layer thicknesses. Although each of these theories has a different ansatz they all agree remarkably well with numerical simulation and with each other. Unlike some earlier theories, they do not require that the impedance contrast of the layers decrease as the number of layers increases. They take account of the effect of intrinsic attenuation on multiples as well as the primary wave. They use the layer-thickness probability distributions directly rather than the autocorrelation of the reflectivity or the impedance variation. SST is shown to agree with earlier static theories of effective sound-speed in the limit as m ω 0. PST gives a stratigraphic (scattering) Q that vanishes as ω 0. Both PST and AM1 permit scattering losses due to spatial irregularities of the sand-shale interfaces, and they are more accurate than SST if reflection coefficients are large.