Velocity continuation in the downward continuation approach to seismic imaging

Abstract

S U M M A R YOne way of developing a wave-equation approach to seismic imaging is based on the concept of downward continuation of observed surface reflection data. Seismic imaging is typically based on the single scattering approximation, and assumes the knowledge of a smooth background model in which the mentioned downward continuation is carried out using the double-squareroot (DSR) equation. The downward continued data are subjected, at each depth, to an imaging condition generating an image or to an angle transform generating common-image gathers. Motivated by the problem of estimation of the background model we develop a framework for velocity continuation in the downward continuation approach to seismic imaging. Velocity continuation is a term used to describe the transformation of a common-image gather (or image) obtained in an 'initial' background velocity model into a gather (image) obtained in another 'final' model. We construct evolution equations for velocity continuation. We then introduce the concept of data-extended-image 'volume'. In the data-extended-image 'volume' we consider so-called extended images, which are functions of subsurface source-receiver midpoint, subsurface source-receiver offset and depth. We introduce and analyse velocity continuation of the individual components that make up the process of migration, imaging and the transformation to angle common-image gathers.

Evolution-equation approach to seismic image, and data, continuation

“…First we apply method 3 (Section 4.3) with phase functions given in (49). This leads to equations for velocity continuation characteristics, repeating the reasoning in (45)- (47) (with only one phase variable, τ )…”

“…The operators A we (α) are microlocally invertible under certain conditions on the ray geometry [48] (for small p). The general application of the continuation theory developed here to the downward continuation approach and associated angle transform [32,33] can be found in [49]. As the family of background models v[α], we take…”

Correlation-based seismic velocity inversion

“…Therefore, it is important that the generalized image has the same dimensionality as the data. Some interesting developments in this area are the velocity continuation of the extended images (Duchkov et al, 2008;Duchkov & de Hoop, 2009) and generalization of the principle to the non-linear case (Symes, 2008a).…”

Azimuthally anisotropic 3D velocity continuation

“…Instead of wavefronts propagating as a function of time, images propagate as a function of migration velocity. Recent work has extended the concept to heterogeneous and anisotropic velocity models (Alkhalifah and Fomel, 1997;Adler, 2002;Iversen, 2006;Schleicher and Alexio, 2007;Schleicher et al, 2008b;Duchkov and de Hoop, 2009). To account for anisotropy, the seismic velocity model must become multi-parameter.…”