Coupling in amplitude variation with offset and the Wiggins approximation

Abstract

Linear amplitude-variation-with-offset (AVO) approximations, which experience a reduction in accuracy as elastic parameter contrasts become large, may be adjusted with second- and higher-order corrections. Corrective terms can be expressed in many ways, but they only serve a meaningful purpose if they provide the same qualitative interpretability as did the linearization. Some aspects of nonlinear AVO can be understood, quantitatively and qualitatively, in terms of coupling — the interdependence of elastic parameter contrasts amongst themselves in their determination of reflection strengths. Coupling, for instance, explains the weak but nonnegligible dependence of the converted wave reflection coefficient on the lower half-space P-wave velocity. This fact can be exposed by expanding the solutions of the Zoeppritz equations in a particular hierarchy of series. Also explainable through this approach is the mathematical importance of what is sometimes referred to as the “Wiggins approximation,” under which [Formula: see text]. This special number is seen to coincide with a full decoupling of density contrasts from [Formula: see text] and [Formula: see text] contrasts at the second order. The decoupling persists across several variations of the nonlinear AVO approximations, including both expressions in terms of the relative changes [Formula: see text], [Formula: see text], and [Formula: see text], and expressions in terms of single-parameter reflectivities.