Most probable seismic pulses in single realizations of two- and three-dimensional random media

We consider the time evolution of seismic primary arrivals in single realiz ations of randomly heterogeneous media. Using the Rytov approximation, we c onstruct 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 propagat ion in 1-D heterogeneous media, known also as the generalized O'Doherty-Ans tey (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 d escribe the primary arrivals in single most probable realizations of seismo grams. Similar to the attenuation coefficient and phase increment of transm issivities in one dimension, the logarithmic wavefield attributes in two an d three dimensions also demonstrate self-averaging, restricted mainly to th e weak fluctuation range, however. We show how to derive the statistical ap proximations and discuss their limitations. We also show that in the limit of long travel distances (Fraunhofer approximation) the Green's function te nds to attain the universal form of a Gaussian pulse. In addition, we compa re the outcome of finite difference experiments with the theoretically pred icted wavefield and find a good agreement: the statistical approximations p resented give a smooth version of the primary arrivals. A statistical analy sis of the simulated wavefield allows us to identify the most probable seis mograms whose primaries are well predicted by the ODA formalism. In additio n, we formulate the traveltime-corrected averaging from first principles. W e 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 gene ralized O'Doherty-Anstey formalism, however, is.