USE OF MONTE CARLO BASED MATRIX SOLVER TO ENHANCE EFFICIENCY OF DC RESISTIVITY MODELLING ALGORITHM

Abstract

We present Markov Chain Monte Carlo (MCMC) method to solve forward problem in geophysics. Usually, geophysical models are discretized using Finite Difference Method (FDM) and a banded coefficient matrix is obtained. Solving this matrix would yield the computed response to be compared with observed data. Observations involved in Controlled Source Electromagnetic (CSEM), Magnetotelluric (MT) and Electromagnetic (EM) surveys are mostly surface based; thereby discretizing our model domain and solving System of Linear Algebraic Equations (SLAE) using MCMC only for the nodes lying on the surface would be more efficient against computing entire solution vector using deterministic methods. For large sparse matrices, we find MC methods are more efficient than other deterministic methods. We have tested MCMC algorithm on a banded matrix for varying sizes and for different number of Markov chains. The results show interesting characteristics: 1) number of computations for one component of solution vector is independent of size of the matrix and 2) number of Markov chains required for a given accuracy is independent of size of the matrix; given the structure and norm of matrices is same. These results emphasize on benefits of using MCMC for extremely large sparse matrices. While performing inverse modelling of CSEM and MT data, like forward modelling, this method can be used to compute Jacobian matrices associated with observation nodes only, so selective estimate of those matrices itself can be performed using MCMC, making inversion more efficient. The technique can be extended to all modelling formulations where only a small percentage of solution vector needs to be obtained, which would increase computational efficiency significantly.