Inverse Theory Applied to Multi‐source Cross‐hole Tomography.

Abstract

Frequency‐domain methods are well suited to the imaging of wide‐aperture cross‐hole data. However, although the combination of the frequency domain with the wavenumber domain has facilitated the development of rapid algorithms, such as diffraction tomography, this has also required linearization with respect to homogeneous reference media. This restriction, and association restrictions on source‐receiver geometries, are overcome by applying inverse techniques that operate in the frequency‐space domain. In order to incorporate the rigorous modelling technique of finite differences into the inverse procedure a nonlinear approach is used. To reduce computational costs the method of finite differences is applied directly to the frequency‐domain wave equation. The use of high speed, high capacity vector computers allow the resultant finite‐difference equations to be factored in‐place. In this way wavefields can be computed for additional source positions at minimal extra cost, allowing inversions to be generated using data from a very large number of source positions. Synthetic studies show that where weak scatter approximations are valid, diffraction tomography performs slightly better than a single iteration of non‐linear inversion. However, if the background velocities increase systematically with depth, diffraction tomography is ineffective whereas non‐linear inversion yields useful images from one frequency component of the data after a single iteration. Further synthetic studies indicate the efficacy of the method in the time‐lapse monitoring of injection fluids in tertiary hydrocarbon recovery projects.

Efficient full waveform inversion using the excitation representation of the source wavefield

“…Although seismic full waveform inversion (FWI) is a powerful tool to estimate the unknown subsurface parameters quantitatively with high resolution (Lailly 1983;Tarantola 1984;Gauthier et al 1986;Pratt 1990;Pratt et al 1996), it is computationally very expensive. The cost of FWI in the framework of local nonlinear-optimization problem is incurred mostly by the computation of the gradient at each iteration.…”

A parallel evolution strategy for an earth imaging problem in geophysics

“…Two formulations (either time-domain or frequency-domain based) are traditionally used for finding the solution of this inverse problem. Relevant details on both approaches can be found in, e.g., [10,34,45,52]. Since the frequencydomain approach is regarded as more advantageous when solving the full-waveform inversion in the multiple frequency case [52], we will exclusively consider this approach in our paper.…”

“…More recent methods such as stereo tomography [27] and inversion in the Laplace domain [41] are being investigated in academia. It must be also mentioned here that the use of multiscale strategies can mitigate the nonlinearity and reduce the dependence on the starting velocity model for FWI [34,45].…”

Algorithms for EOR imaging using crosshole seismic data: an experiment with scale model data

“…However, the inverse GRT is a linearized inversion method and converges in a single iteration for linear problems. Frequency-domain wave-equation imaging is a non-linear inversion method requiring multiple iterations to achieve convergence, even for linear problems (for a full discussion and an illustration of this point, see Pratt and Worthington, 1990 ). In this paper we apply only a single iteration of each of these methods.…”

“…Scale crosshole surveys were carried out in both models, and the resultant data were processed using three different imaging algorithms. The three algorithms used were traveltime tomography, the inverse generalized Radon transform (inverse GRT) (Miller et al, 1987) and frequency-domain wave-equation imaging (Pratt and Worthington, 1990).…”