Three dimensional controlled source electromagnetics (CSEM) forward modelling in the frequency domain requires the solution of a large scale, complex and linear system. Such a system when derived from finite elements and finite differences formulations is sparse and ill-conditioned. The use of direct solvers is prohibitively expensive for realistic-sized problems since several million degrees of freedom are usually involved. To avoid inherent hardware limitations iterative solvers are an option, however their potential efficiency relies on the use of efficient pre-conditioning. We present an alternative pre-conditioner for the CSEM forward modelling problem, based on the Schur complement method, and will discuss some limitations and possible solutions.
During the past few years the CSEM technique has achieved recognition in the oil industry as an important and valuable tool to reduce exploration risk. The use of this technique often requires the use of sophisticated mathematical and computational methods and efficient forward modelling algorithms which are unavoidable for parameter estimation from the collected data sets. The solution of the CSEM forward modelling problem in the frequency domain requires the solution of large scale complex symmetric linear systems of equations for several sources, and the inverse problem requires multiple forward modelling solutions for sources, receivers, frequencies whether sensitivities are calculated using a back-propagation algorithm or McGillivray’s method (McGillivray and Oldenburg, 1990). A significant effort has been devoted to the development of different strategies to achieve an accurate and stable solution. Direct methods based on matrix factorization are suitable for multi-source problems in the sense that a single decomposition is sufficient to solve as many right-hand sides as necessary, nonetheless memory requirements make such methods prohibitive when 3-D geometries are considered. On the other hand iterative methods are less demanding in terms of hardware requirements but are inefficient unless efficient preconditioning is used due to the high condition numbers that are usually associated with the linear systems obtained from the numerical discretization wether finite differences or finite elements are used. We are going to present a method that results from combining both strategies (direct and iterative), consisting in the reordering of the linear system and the use of the Schur complement to achieve a divide and conquer approach. The linear system resulting from the Schur complement for the interface nodes is solved using an iterative solver and pre-conditioned with the ILU decomposition.
For the electric field calculation we consider its decomposition in primary and secondary components. On the other hand if not divided in enough subdomains, the factorization of the block matrices will take too much cpu time and consequently the performance of the solver degrades. The best trade-off between the number of MPI processes and number of sub-domains was obtained using 16 MPI processes and 10000 subdomains. Figures 2 and 3 show a good agreement between the solutions obtained using the finite element code with the presented solver and the ones obtained using Chave and Cox’s (1982) solution for 1-D resistivity models.