Exact wavefield extrapolation in 2D for v(x)
Gary F. Margrave, Michael P. Lamoureux, Peter C. Gibson, Richard Andrew Bale, Jeff P. Grossman
We pose the wavefield extrapolation problem in two spatial dimensions for arbitrary lateral velocity variation, v(x). Then, by restricting attention to the discretely sampled case, we develop an exact solution via an eigenvalue decomposition. The matrix that is decomposed is the difference between a diagonal matrix containing the squared horizontal wavenumbers and a Toeplitz matrix, scaled by frequency squared, containing the Fourier transform of the square on the inverse velocity. This solution gives an explicit decomposition of a wavefield into independent upward and downward travelling parts.
We also analyze the continuous problem via functional analytic methods and, though we fail to reach a general solution, we demonstrate the independence of upward- and downward-travelling waves. In the specific case of the step velocity function, we calculate the eigenfunctions and exhibit a formula that determines the eigenvalues, though the latter must be solved numerically.
Numerical testing shows that the exact extrapolator produces physically understandable result. When compared with three different approximate extrapolators based on Fourier integral operators, the approximate operators can be grossly wrong if a large extrapolation step is taken in the presence of strong velocity gradients. However, if the approximate operators are used in a recursion taking many small steps, they appear to approach the correct result. In a simulation of inversion with an erroneous velocity model, the three approximate extrapolators produce a very similar result to the exact extrapolator. This indicates that a very precise velocity model is required to take advantage of the exact extrapolator.