Amplitude‐variation‐with‐angle behavior of self‐similar interfaces

Abstract

Amplitude‐variation‐with‐angle (AVA) analysis is generally based on the assumption that the medium parameters behave as step functions of the depth coordinate z, at least in a finite region around the interface. However, outliers observed in well logs often behave quite differently from step functions. In this paper, outliers in the acoustic propagation velocity are parameterized by functions of the form [Formula: see text]. The wavelet transform of this function reveals properties similar to those of several outliers in real well logs. Moreover, this function is self‐similar, according to [Formula: see text], for β > 0. Analytical expressions are derived for the acoustic normal incidence reflection and transmission coefficients for this type of velocity function. For oblique incidence, no explicit solutions are available. However, by exploiting the self‐similarity property of the velocity function, it turns out that the acoustic angle‐dependent and frequency‐dependent reflection and transmission coefficients are self‐similar as well. To be more specific, these coefficients appear to be constant along curves described by [Formula: see text], where p is the raypath parameter and ω the angular frequency. The singularity exponent α that is reflected in these curves may prove to be a useful indicator in seismic characterization.