### The 2-D magnetotelluric inverse problem solved with
optimization

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 Maxwellâ€™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.

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