Seismic depth migration with the Dirac equation

We demonstrate the applicability of the Dirac equation in seismic wavefield extrapolation by presenting a new explicit one-way prestack depth migratio n scheme. The method is in principle accurate up to 90 degrees from the ver tical, and it tolerates lateral velocity variations. This is achieved by pe rforming the extrapolation step of migration with the Dirac equation, imple mented in the space-frequency domain. The Dirac equation is an ex act linea rization of the square-root wave equation and is equivalent to keeping infi nitely many terms in a Taylor series or continued-fraction expansion of the squareroot operator. An important property of the new method is that the l ocal velocity and the spatial derivatives decouple in separate terms within the extrapolation operator. Therefore, we do not need to precompute and st ore large tables of convolutional extrapolator coefficients depending on ve locity. The main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numerical stability. We have tested two numerical implementations of the migration scheme. In th e first implementation, we perform depth stepping using the Taylor series a pproximation and compute spatial derivatives with high-order finite differe nce operators. In the second implementation, we perform depth stepping with the Rapid expansion method and numerical differentiation with the pseudosp ectral method. The imaging condition is a generalization of Claerbout's U/D principle. For both implementations, the impulse response is accurate up to 80 degrees from the vertical. Using synthetic data from a simple fault model, we test the depth migration scheme in the presence of lateral velocity variations. The results show that the proposed migration scheme images dipping reflect ors and the fault plane in the correct positions.