3-D Adaptive Regularization Nonlinear Inversion of LWD Ultradeep Resistivity in Anisotropic Formation Based on Finite Volume Method of Secondary Field Coupled Potentials and Explicit Fréchet Derivative

The article advances a 3-D adaptive regularization nonlinear inversion of the logging while drilling (LWD) ultradeep multicomponent resistivity (LWD-UDMCR) by the Gauss-Newton (GN) method. We manage to reconstruct the pixel-based horizontal and vertical conductivities simultaneously in a goal domain outside an arbitrary dipping borehole. The piecewise constant functions are used to describe the spatial distribution of the block-based and the pixel-based conductivity. The background formation is assumed as the horizontally layered transversely isotropic (TI) media, and the background electromagnetic (EM) fields are determined analytically by the transmission line method (TLM). We then use the 3-D finite volume method (FVM) of secondary field coupled potentials and parallel direct sparse solver (PARDISO) to simulate the tool responses and Fréchet derivatives simultaneously. Through the projection operator and OpenMP parallel technique, we further enhance the computational efficiency of the pixel-based explicit Fréchet derivatives and set up a complete normalization linearized response. After that, the large normal equation from the quadratic objective function is solved by the preconditioned conjugate gradient (PCG) to determine the gradient of the objective function. By properly controlling the maximum component of the gradient per iteration step, we acquire an adaptive regularization factor so that the stabilization of the inversion solution is assured as well as the realization of the best fit of the input data with the modeling logs. Finally, numerical tests validate the algorithm and antinoise ability.