An efficient algorithm of three-dimensional explicit electromagnetic sensitivity matrix in marine controlled source electromagnetic measurements

Abstract

In this paper, an efficient algorithm of three-dimensional (3D) explicit sensitivity (or called Fréchet derivatives) matrix for marine controlled source electromagnetic measurements is established by combining an electric coupled potential finite volume method with a direct solver PARDISO direct method. Firstly, on the Yee’s staggered grids, the coupled potential Helmholtz equations are discretized to form a large, sparse and complex linear system which is excited by mobile transmitters. Secondly, through the inversion of the discrete matrix and 3D linear interpolation formula, the interpolation operators and projection operators are established for each receiver at different positions. Because these interpolation operators and projection operators are unrelated to transmitters, they can be computed in advance according to the positions of all receivers. Then the multiple projection operators with discrete vector of each transmitter source can efficiently produce the electromagnetic (EM) responses. On the basis, the goal conductivity of block model and pixel model is expressed as a piece-wise constant function. By perturbing the goal conductivity, the scattered electric current density can be decomposed into a series of electric current elements distributed on Yee’s grids. Each scattered current element is equal to the product of relative perturbation of conductivity and the electric intensity on the grid. The discrete vector on the right-hand side is computed by integrating each scattered current element on the Yee’s grid and then being multiplied with the project operator. Then the linear relationship between the changes in EM field and the relative conductivity perturbation on each block or pixel can be fast produced to obtain the explicit sensitivity matrix about EM responses. Finally, numerical results demonstrate the efficiency and accuracy of this method. The characteristics of 3D sensitivity in three different cases are further investigated.