Nonlinear conjugate gradients algorithm for 2D magnetotelluric inversion
Geophysics, 66 (1):
174--187 (Jan 1, 2001)DOI: 10.1190/1.1444893
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.