MOST PROBABLE SEISMIC PULSES IN SINGLE REALIZATIONS OF TWO-AND THREE-DIMENSIONAL RANDOM MEDIA

Abstract

We consider the time evolution of seismic primary arrivals in single realizations of randomly heterogeneous media. Using the Rytov approximation, we construct the Green's function of an initially plane wave propagating in 2-D and 3-D weakly heterogeneous fluids and solids. Our approach is a 2-D and 3-D extension of the dynamic-equivalent medium description of wave propagation in 1-D heterogeneous media, known also as the generalized O’Doherty–Anstey (ODA) formalism. The Green's function is constructed by using averaged logarithmic wavefield attributes and depends on the second-order statistics of the medium heterogeneities. Green's functions constructed in this way describe the primary arrivals in single most probable realizations of seismograms. Similar to the attenuation coefficient and phase increment of transmissivities in one dimension, the logarithmic wavefield attributes in two and three dimensions also demonstrate self-averaging, restricted mainly to the weak fluctuation range, however. We show how to derive the statistical approximations and discuss their limitations. We also show that in the limit of long travel distances (Fraunhofer approximation) the Green's function tends to attain the universal form of a Gaussian pulse. In addition, we compare the outcome of finite difference experiments with the theoretically predicted wavefield and find a good agreement: the statistical approximations presented give a smooth version of the primary arrivals. A statistical analysis of the simulated wavefield allows us to identify the most probable seismograms whose primaries are well predicted by the ODA formalism. In addition, we formulate the traveltime-corrected averaging from first principles. We discuss the relationship between our approach and approaches based on the traveltime-corrected formalism. Strictly speaking, such approaches are not appropriate to describe wavefields in most probable realizations; the generalized O’Doherty–Anstey formalism, however, is.