A Rapidly Direct Algorithm of Explicit Fréchet Derivatives for MCSEM in a 3-D TI Formation Using Finite Volume of Coupled Potentials

This paper presents a rapidly direct algorithm (RDA) to calculate explicit Fréchet derivatives (EFD) for marine controlled-source electromagnetic measurement (MCSEM) in a three-dimensional (1-D) transversely isotropic (TI) formation. By discretizing the Helmholtz equations about coupled potentials on Yee’s staggered grids by the 3-D finite volume method (FVM), we obtain a complex linear system about the unknown potentials excited by a mass of moving electric current sources. To efficiently determine electromagnetic (EM) fields, we introduce an interpolation operator and projection operator per receiver by using the direct solver PARDISO and 3-D Newtown interpolation. Based on this, the perturbation in goal conductivity is expressed as a piece-wise constant function according to block or pixel model. The spatial scattered electric currents will be decomposed into a series of electric current elements distributed on Yee’s grids due to the conductivity perturbation. We then discretize the scattered electric currents by 3-D FVM and obtain the new right-hand terms about the unknown scattered potentials. This allows for fast production of the linear relationship between the changes in the EM fields and the relative conductivity perturbation per block or pixel, ultimately resulting in the EFD of MCSEM responses. Numerical results demonstrate the efficiency and accuracy of this method. The 3-D pixel sensitivities are presented to further investigate the response characteristics of MCSEM in several cases.