Image gather reconstruction using StOMP

Introduction

Reverse time migration (RTM) is quickly becoming the standard high-end seismic imaging technique. Significant work has been done on speeding up the kernel (Nemeth et al., 2008; Micikevicius, 2009; Clapp et al., 2010; Nguyen et al., 2010) but far less on constructing image gathers(Sava and Fomel, 2003,2006) needed for rock property analysis or velocity updates. Image gather construction is a much less tractable problem because it is memory rather than compute intensive. The volume size of the domain increases by one to three orders of magnitude making the dominant cost reading/writing to distant memories (from main memory rather than a cache, across the PCI Bus, or from disk).

Donoho (2006) offers an approach termed compressive sensing potential solution to this computation and storage problem. In compressive sensing, a random subset of the desired measurements is made. An inversion problem is then set up to estimate in an $\ell_1$ , or preferably $\ell_0$ , sense, a sparse basis function that fully characterizes the desired signal. For compressive sensing to work, a signal must be highly compressible. For compressive sensing to be worthwhile, the cost of inverting for the basis function must be significantly less than the cost of acquiring the full signal. Clapp (2011) showed that correlation gather construction fit the first criteria for a successful compressive sensing problem. Multi-dimensional correlation gathers/angle gathers are compressible at nearly a 100:1 ratio. The challenge became finding an inversion scheme that could accurately enough recover the full model. Donoho et al. (2006) proposed a solution to the second problem, an $\ell_1$ version methodology that works for a large number of unknowns. In this paper, I apply the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm to correlation gather reconstruction. I show that the angle domain representation of the sparsely acquired gathers is similar to the representation of the full data. I then apply a phase encoding technique, combining many different correlations to every data point to further improve the inverted model.

Image gather reconstruction using StOMP