Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion

Abstract

We investigate a new algorithm for computing regularized solutions of the 2-D magnetotelluric inverse problem. The algorithm employs a nonlinear conjugate gradients (NLCG) scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity. We compare this algorithm theoretically and numerically to two previous algorithms for constructing such “minimum-structure” models: the Gauss-Newton method, which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics, and an algorithm due to Mackie and Madden, which solves a sequence of linearized inverse problems incompletely using a (linear) conjugate gradients technique. Numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients (NLCG and Mackie-Madden) are more efficient than the Gauss-Newton algorithm in terms of both computer memory requirements and CPU time needed to find accurate solutions to problems of realistic size. This owes largely to the fact that the conjugate gradients-based algorithms avoid two computationally intensive tasks that are performed at each step of a Gauss-Newton iteration: calculation of the full Jacobian matrix of the forward modeling operator, and complete solution of a linear system on the model space. The numerical tests also show that the Mackie-Madden algorithm reduces the objective function more quickly than our new NLCG algorithm in the early stages of minimization, but NLCG is more effective in the later computations. To help understand these results, we describe the Mackie-Madden and new NLCG algorithms in detail and couch each as a special case of a more general conjugate gradients scheme for nonlinear inversion.