Compressed imaging

In 1998 Grimbergen et. al. introduced a new method for computing wavefield propagation which improved on the previously employed local explicit operator method in that it exhibited no dip limitation, accurately handled laterally varying background ground velocity models, and is unconditionally stable. These desirable properties are mainly attributed to bringing the propagation problem into an eigenvector basis that diagonalizes the propagation operators. This modal-transform method, however, requires at each depth-level the solution of a large-scale sparse eigenvalue problem to compute the square-root of the Helmholtz operator. By using recent results from compressed sensing, we hope to reduce these computational costs that typically involve the synthesizes of the imaging operators and the cost of matrix-vector products. To reduce these costs, we compress the extrapolation operators by using only a fraction of the positive eigenvalues and temporal frequencies. This reduction not only leads to smaller matrices but also to reduced synthesis costs. These reductions go at the expense of solving a recovery problem from incomplete data. During the presentation, we show that wavefields can accurately be extrapolated with a compressed operators and competitive costs.