Publications

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2011
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.

2010
Parker, RL.  2010.  Can a 2-D MT frequency response always be interpreted as a 1-D response? Geophysical Journal International. 181:269-274.   10.1111/j.1365-246X.2010.04512.x   AbstractWebsite

Weidelt and Kaikkonen showed that in the transverse magnetic (TM) mode of magnetotellurics it is not always possible to match exactly the 2-D response at a single site with a 1-D model, although a good approximation usually seems possible. We give a new elementary example of this failure. We show for the first time that the transverse electric (TE) mode responses can also be impossible to match with a 1-D response, and that the deviations can be very large.

2007
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.

1996
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.