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Van Beusekom, AE, Parker RL, Bank RE, Gill PE, Constable S.  2011.  The 2-D magnetotelluric inverse problem solved with optimization. Geophysical Journal International. 184:639-650.   10.1111/j.1365-246X.2010.04895.x   AbstractWebsite

P>The practical 2-D magnetotelluric inverse problem seeks to determine the shallow-Earth conductivity structure using finite and uncertain data collected on the ground surface. We present an approach based on using PLTMG (Piecewise Linear Triangular MultiGrid), a special-purpose code for optimization with second-order partial differential equation (PDE) constraints. At each frequency, the electromagnetic field and conductivity are treated as unknowns in an optimization problem in which the data misfit is minimized subject to constraints that include Maxwell's equations and the boundary conditions. Within this framework it is straightforward to accommodate upper and lower bounds or other conditions on the conductivity. In addition, as the underlying inverse problem is ill-posed, constraints may be used to apply various kinds of regularization. We discuss some of the advantages and difficulties associated with using PDE-constrained optimization as the basis for solving large-scale nonlinear geophysical inverse problems. Combined transverse electric and transverse magnetic complex admittances from the COPROD2 data are inverted. First, we invert penalizing size and roughness giving solutions that are similar to those found previously. In a second example, conventional regularization is replaced by a technique that imposes upper and lower bounds on the model. In both examples the data misfit is better than that obtained previously, without any increase in model complexity.

Stark, PB, Parker RL.  1987.  Smooth profiles from τ(p) and X(p) data. Geophysical Journal of the Royal Astronomical Society. 89:997-1010.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1987.tb05205.x   AbstractWebsite

Summary. We reduce the problem of constructing a smooth, 1-D, monotoni-cally increasing velocity profile consistent with discrete, inexact τ (p) and X(p) data to a quadratic programming problem with linear inequality constraints. For a finite-dimensional realization of the problem it is possible to find a smooth velocity profile consistent with the data whenever such a profile exists. We introduce an unusual functional measure of roughness equivalent to the second central moment or ‘Variance’ of the derivative of depth with respect to velocity for smooth profiles, and we prove that its minimal value is unique. In our experience, solutions minimizing this functional are very smooth in the sense of the two-norm of the second derivative and can be constructed inexpensively by solving one quadratic programming problem. Still smoother models (in more traditional measures) may be generated iteratively with additional quadratic programs. All the resulting models satisfy the τ (p) and X(p) data and reproduce travel-time data remarkably well, although sometimes τ (p) data alone are insufficient to ensure arrivals at large X; then an X(p) datum must be included.