Velocity Continuation in the Downward Continuation Approach to Seismic Imaging

One way of developing a wave-equation approach to seismic imaging is based on the concept of downward continuation of the 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. The downward continued data are subjected, at each depth, to an imaging condition generating an image or to an angle transform generating common image-point gathers. In this paper, we develop a general framework for velocity continuation in the downward continuation approach to seismic imaging, motivated by the problem of estimation of the background model. A key aspect of velocity continuation is that the downward continuation of data needs to be carried out only once, namely for an ‘initial’ background model. Velocity continuation maps these downward continued data to data, downward continued in the ‘final’ velocity model. We take a broad perspective, leading to a variety of new insights and the development of novel algorithms. To begin with, we will use an extended image, which is a function of subsurface source-receiver midpoint, subsurface source-receiver offset and depth, rather than the traditional image. We base our framework on the double-square-root (DSR) equation, and impose the DSR assumption. We then introduce the concept of data-extended-image ‘volume’. In the data-extended-image ‘volume’, we introduce and analyze velocity continuation of the individual components that make up the process of imaging, migration, and the angle transform: Data continuation nested in velocity continuation, extended image velocity continuation, and image gathers velocity continuation. Each of them can perform a different task for background model, velocity, analysis. The framework presented here extends naturally to applications in time-lapse seismics.