We present a compressed implicit Jacobian scheme for the regularized Gauss-Newton inversion algorithm for reconstructing three-dimensional conductivity distribution from electromagnetic data. In this scheme, the Jacobian matrix, whose storage usually requires a large amount of memory, is decomposed in terms of electric fields excited by sources located and oriented identically to the physical sources and receivers. As a result, the memory usage for the Jacobian matrix reduces from O(NFNSNRNP) to O[NF (NS +NR)NP], in which NF is the number of frequencies, NS is the number of sources, NR is the number of receivers, and NP is the number of conductivity cells to be inverted. Moreover, we apply the adaptive cross approximation (ACA) to compress these fields to further reduce the memory requirement and to improve the efficiency of the method. This implicit Jacobian scheme provides a good balance between the memory usage and the computational time and renders the Gauss-Newton algorithm more efficient. We demonstrate the benefits of this scheme using numerical examples including both synthetic and field data for both crosswell and surface electromagnetic applications.
With rapid developments in sensor technology and survey design, many geophysical surveys acquire data in more than one straight-line direction in order to fully illuminate the survey region and acquire more information on the formation. For example, in marine controlled-source electromagnetic (CSEM) methods (Constable et al. (1986); MacGregor and Sinha (2000); Johansen et al. (2005)), data acquired by multiple tow lines together with broadside data have been inverted simultaneously. In crosswell electromagnetic surveys for reservoir characterization (Spies and Habashy (1995)), data from multiple well pairs are measured. To reconstruct the conductivity models from these data sets, both forward and inversion algorithms must be able to handle three-dimensional (3D) models. This leads to millions of unknowns, an order larger than that of twodimensional models. Moreover, the measurement data quantity also increases tremendously. Hence, this increases the demand for efficient algorithms in both computational time and memory usage. In inverting data from electromagnetic surveys, the Gauss-Newton algorithm is well-known to be robust because of its rapid convergence (Mackie et al. (2001); Chen et al. (2002); Abubakar et al. (2005); Soleimani et al. (2007); Abubakar et al. (2008b)). In this algorithm, the conductivity distributions are updated iteratively based on the difference between the measurement data and the data computed from the reconstructed model. The Gauss-Newton method relies on the Jacobian matrix that contains information about the derivative of the simulated data with respect to the conductivity. The Jacobian matrix size is equal to the number of measurement data points times the number of conductivity cells to be inverted. In 3D inversion, the size of the data set and the inversion region are usually very large. However, this method requires calculation of the forward solver at each conjugate gradient least-squares (CGLS) iteration to calculate the Gauss-Newton step, which could be large. One remedy is to terminate the CGLS iteration after a small number of iterations, but this will deteriorate the accuracy of the solution.
