Migration with implicit solvers for the time-harmonic Helmholtz equation

From the measured seismic data, the location and the amplitude of reflectors can be determined via a migration algorithm. Classically, following Claerbout’s imaging principle [2], a reflector is located at the position where the source’s forward-propagated wavefield correlates with the backward-propagated wavefield of the receiver data. Lailly and Tarantola later showed that this imaging principle is an instance of inverse problems, with the associated migration operator formulated via a least-squares functional; see [6, 12, 13]. Furthermore, they showed that the migrated image is associated with the gradient of this functional with respect to the image. If the solution of the least-squares functional is done iteratively, the correlation-based image coincides up to a constant with the first iteration of a gradient method. In practice, this migration is done either in the time domain or in the frequency domain. In the frequency-domain migration, the main bottleneck thus far, which renders its full implementation to large scale problems, is the lack of efficient solvers for computing wavefields. Robust direct methods easily run into excessive memory requirements as the size of the problem increases. On the other hand, iterative methods, which are less demanding in terms of memory, suffered from lack of convergence. During the past years, however, progress has been made in the development of an efficient iterative method [4, 3] for the frequency-domain wavefield computations. In this paper, we will show the significance of this method (called MKMG) in the context of the frequency-domain migration, where multi-shot-frequency wavefields (of order of 10,000 related wavefields) need to be computed.

https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2009/Erlang...