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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.

Medin, AE, Parker RL, Constable S.  2007.  Making sound inferences from geomagnetic sounding. Physics of the Earth and Planetary Interiors. 160:51-59.   10.1016/j.pepi.2006.09.001   AbstractWebsite

We examine the nonlinear inverse problem of electromagnetic induction to recover electrical conductivity. As this is an ill-posed problem based on inaccurate data, there is a critical need to find the reliable features of the models of electrical conductivity. We present a method for obtaining bounds on Earth's average conductivity that all conductivity profiles must obey. Our method is based completely on optimization theory for an all-at-once approach to inverting frequency-domain electromagnetic data. The forward modeling equations are constraints in an optimization problem solving for the electric fields and the conductivity simultaneously. There is no regularization required to solve the problem. The computational framework easily allows additional inequality constraints to be imposed, allowing us to further narrow the bounds. We draw conclusions from a global geomagnetic depth sounding data set and compare with laboratory results, inferring temperature and water content through published Boltzmann-Arrhenius conductivity models. If the upper mantle is assumed to be volatile free we find it has an average temperature of 1409-1539 degrees C. For the top 1000 km of the lower mantle, we find an average temperature of 1849-2008 degrees C. These are in agreement with generally accepted mantle temperatures. Our conclusions about water content of the transition zone disagree with previous research. With our bounds on conductivity, we calculate a transition zone consisting entirely of Wadsleyite has < 0.27 wt.% water and as we add in a fraction of Ringwoodite, the upper bound on water content decreases proportionally. This water content is less than the 0.4 wt.% water required for melt or pooling at the 410 km seismic discontinuity. Published by Elsevier B.V.

O'Brien, MS, Constable CG, Parker RL.  1997.  Frozen-flux modelling for epochs 1915 and 1980. Geophysical Journal International. 128:434-450.   10.1111/j.1365-246X.1997.tb01566.x   AbstractWebsite

The frozen-flux hypothesis for the Earth's liquid core assumes that convective terms dominate diffusive terms in the induction equation governing the behaviour of the magnetic field at the surface of the core. While highly plausible on the basis of estimates of physical parameters, the hypothesis has been questioned in recent work by Bloxham, Gubbins & Jackson (1989) who find it to be inconsistent with their field models for most of the century. To study this question we improve the method of Constable, Parker & Stark (1993), which tests the consistency of magnetic observations with the hypothesis by constructing simple, flux-conserving core-field models fitting the data at pairs of epochs. We introduce a new approach that fixes the patch configurations at each of the two epochs before inversion, so that each configuration is consistent with its respective data set but possesses the same patch topology. We expand upon the inversion algorithm, using quadratic programming to maintain the proper flux sign within patches; the modelling calculations are also extended to include data types that depend non-linearly on the model. Every test of a hypothesis depends on the characterization of the observational uncertainties; we undertake a thorough review of this question. For main-field models, the primary source of uncertainty comes from the crustal field. We base our analysis on one of Jackson's (1994) statistical models of the crustal magnetization, adjusted to bring it into better conformity with our data set. The noise model permits us to take into account the correlations between the measurements and requires that a different weighting be given to horizontal and vertical components. It also indicates that the observations should be fit more closely than has been the practice heretofore. We apply the revised method to Magsat data from 1980 and survey and observatory data from 1915.5, two data sets believed to be particularly difficult to reconcile with the frozen-flux hypothesis. We compute a pair of simple, flux-conserving models that fit the averaged data from each epoch. We therefore conclude that present knowledge of the geomagnetic fields of 1980 and 1915.5 is consistent with the frozen-flux hypothesis.

Parker, RL, Booker JR.  1996.  Optimal one-dimensional inversion and bounding of magnetotelluric apparent resistivity and phase measurements. Physics of the Earth and Planetary Interiors. 98:269-282.   10.1016/s0031-9201(96)03191-3   AbstractWebsite

The properties of the log of the admittance in the complex frequency plane lead to an integral representation for one-dimensional magnetotelluric (MT) apparent resistivity and impedance phase similar to that found previously for complex admittance. The inverse problem of finding a one-dimensional model for MT data can then be solved using the same techniques as for complex admittance, with similar results. For instance, the one-dimensional conductivity model that minimizes the chi(2) misfit statistic for noisy apparent resistivity and phase is a series of delta functions. One of the most important applications of the delta function solution to the inverse problem for complex admittance has been answering the question of whether or not a given set of measurements is consistent with the modeling assumption of one-dimensionality. The new solution allows this test to be performed directly on standard MT data, Recently, it has been shown that induction data must pass the same one-dimensional consistency test if they correspond to the polarization in which the electric field is perpendicular to the strike of two-dimensional structure, This greatly magnifies the utility of the consistency test. The new solution also allows one to compute the upper and lower bounds permitted on phase or apparent resistivity at any frequency given a collection of MT data, Applications include testing the mutual consistency of apparent resistivity and phase data and placing bounds on missing phase or resistivity data, Examples presented demonstrate detection and correction of equipment and processing problems and verification of compatibility with two-dimensional B-polarization for MT data after impedance tensor decomposition and for continuous electromagnetic profiling data.