SUMMARY
The Jacobian of the non-linear mapping from model parameters to observations is a key component in all gradient-based inversion methods, including variants on Gauss–Newton and non-linear conjugate gradients. Here, we develop a general mathematical framework for Jacobian computations arising in electromagnetic (EM) geophysical inverse problems. Our analysis, which is based on the discrete formulation of the forward problem, divides computations into components (data functionals, forward and adjoint solvers, model parameter mappings), and clarifies dependencies among these elements within realistic numerical inversion codes. To be concrete, we focus much of the specific discussion on 2-D and 3-D magnetotelluric (MT) inverse problems, but our analysis is applicable to a wide range of active and passive source EM methods. The general theory developed here provides the basis for development of a modular system of computer codes for inversion of EM geophysical data, which we summarize at the end of the paper.

Computational recipes for electromagnetic inverse problems

ModEM: A modular system for inversion of electromagnetic geophysical data

Tethys tectonic evolution and its bearing on the distribution of important mineral deposits in the Sanjiang region, SW China

https://www.eri.u-tokyo.ac.jp/induction/data/SiripunvarapornPEPI05.pdf

Three-dimensional magnetotelluric inversion : data-space method

There are currently three types of algorithms in use for regularized 2-D inversion of magnetotelluric (MT) data. All seek to minimize some functional which penalizes data misfit and model structure. With the most straight‐forward approach (exemplified by OCCAM), the minimization is accomplished using some variant on a linearized Gauss‐Newton approach. A second approach is to use a descent method [e.g., nonlinear conjugate gradients (NLCG)] to avoid the expense of constructing large matrices (e.g., the sensitivity matrix). Finally, approximate methods [e.g., rapid relaxation inversion (RRI)] have been developed which use cheaply computed approximations to the sensitivity matrix to search for a minimum of the penalty functional. Approximate approaches can be very fast, but in practice often fail to converge without significant expert user intervention. On the other hand, the more straightforward methods can be prohibitively expensive to use for even moderate‐size data sets. Here, we present a new and much m...

An efficient data-subspace inversion method for 2-D magnetotelluric data

WSINV3DMT: Vertical magnetic field transfer function inversion and parallel implementation

Traditional methods for interpretation of magnetotelluric (MT) profile data are based on 2-D inversion, under the assumption that 3-D complications in the data can be treated as ‘geological noise’. We show with synthetic models that fitting 3-D data with a 2-D inversion can result in spurious features, especially if transverse electric (TE) data are used. Inversion of a single profile of MT data with a 3-D algorithm results in significantly more realistic images of structure beneath the data profile, and also allows some resolution of nearby off-profile structure. We also consider the importance of including the on-diagonal impedance tensor terms, Z
 xx and Z
 yy 
, in the inversion. In synthetic test cases, fitting these diagonals improves the accuracy of images of off-profile structure, particularly near the edge of a conductive feature.

/pdf/interpretation-of-two-dimensional-magnetotelluric-profile-4l5qn8hj8b.pdf

Interpretation of two-dimensional magnetotelluric profile data with three-dimensional inversion: synthetic examples

Magnetotelluric exploration has been used to image along strike variations in the electrical resistivity structure of the San Andreas Fault at Parkfield, California. A low resistivity wedge extending to a depth of several kilometers is continuous over a horizontal distance of 8 km. The base of the wedge is coincident with the shallowest microearthquakes. A change in the electrical and fluid connection of the San Andreas Fault with a low resistivity zone in the Franciscan formation is observed along the Parkfield segment.

/pdf/along-strike-variations-in-the-electrical-structure-of-the-494wng2z78.pdf

Weerachai Siripunvaraporn

Papers

Three-dimensional magnetotelluric inversion : data-space method

An efficient data-subspace inversion method for 2-D magnetotelluric data

WSINV3DMT: Vertical magnetic field transfer function inversion and parallel implementation

Interpretation of two-dimensional magnetotelluric profile data with three-dimensional inversion: synthetic examples

Along strike variations in the electrical structure of the San Andreas fault at Parkfield, California