Summary

This paper develops a finite element method which gives accurate numerical approximations to magnetotelluric data over two-dimensional conductivity structures. The method employs a simple finite element technique to find the field component parallel to the strike of the structure and a new numerical differentiation scheme to find the field component perpendicular to strike. Examples show that the new numerical differentiation scheme is a significant improvement over the standard finite element procedure when meshes of poor quality are used. Algorithms for incorporating the differentiation scheme into the finite element matrix equation and for computing partial derivatives of magnetotelluric data with respect to mesh parameters are derived in order to simplify the computation needed to do the inverse problem.

A Technique for Improving the Accuracy of Finite Element Solutions for Magnetotelluric Data

Three-dimensional (3D) interpretation of electromagnetic (EM) dsta is still in its infancy, due to a lack of practical numerical solutions for the forward problem. However, a number of algorithms for simulating the responses of simple 3D models have been developed over the last ten years, and they have provided important new insight. Integral equation methods have been more successful than differential equation methods, because they require calculating the electric field only in small anomalous regions, rather than throughout the earth. Utilizing a vector-scalar potential approach and incorporating symmetry through group theory improves the general 3D integral equation solution. Thin-sheet integral equation formulations have been particularly useful. Much recent research has focused on hybrid methods, which are finite element differential equation solutions within a mesh of limited extent, with boundary values determined by integrating over the interior fields. An elegant eigencurrent technique has been developed for calculating the transient response of a thin 3D sheet in free space, but general 3D time domain responses have only been calculated by Fourier transforming frequency domain results. Direct time domain calculations have been carried out only for 2D bodies.

Three-dimensional em modeling

Magnetotelluric measurements made at 10 locations in and around the Kenyan Rift Valley in 1973–74 have been processed to yield single station induction vectors and magnetotelluric principal impedances. Following a brief description of the field work and the data processing procedure, the results are presented and subjected to both qualitative and quantitative interpretations. The induction vector results indicate a concentration of current below the rift valley. Since the maximum impedances are along the rift for sites within the valley but at right angles to the rift outside it, the apparent resistivity results are qualitatively consistent with the existence of a good conductor there. Within the rift maximum apparent resistivities lie between 2 and 20 Ωm. The persistence of small induction responses and low apparent resistivities down to the shortest periods at which measurements were made suggests that the conductor lies at very shallow depths below the rift. The quantitative interpretation of the magnetotelluric results is given in terms of one-and two-dimensional conductivity models. It is shown that the data require the presence of conductive material at a depth less than 8 km below the rift floor but that the required thickness of this upper crustal conductor (> 5 km) is too great to be explained in terms of a conductive infill of the rift. The favoured explanation is in terms of high temperatures and water saturation of the crust under the rift. To satisfy the long-period data, conductive material is also required at depths of greater than 30 km, corresponding unambiguously to the upper mantle below the rift. Because of the obscuring effect of the crustal conductor, the depth to the top of and the thickness of the upper mantle conductor cannot be resolved.

/pdf/a-magnetotelluric-and-magnetovariational-study-of-the-2c0qli9qep.pdf

A magnetotelluric and magnetovariational study of the Gregory Rift Valley, Kenya

The Southern Uplands of Scotland: is at present a region of great interest to both geologists and geophysicists. The complex tectonic history associated with the closing of the proto-Atlantic ocean has yet to be determined. Previous geophysical studies have shown that the Southern Uplands is atypical of normal continent. Geomagnetic investigations have indicated a zone of anomalously high electrical conductivity underlying the Southern Uplands region at a depth of 12-30 km. In order to determine further the exact structure and spatial variation of this conductivity zone., two geomagnetic studies have been carried out in the region. A Geomagnetic Deep Sounding Array of 19 Gough-Reitzel variometers recorded the naturally varying Earth's magnetic field during December 1913 and January 1914. This was followed by • Magneto-Telluric observations on lines perpendicular and parallel to the strike of the supposed anomaly. • In this thesis, reviews of various regional MT studies, the geophysical significance of conductivity measurements and the known geology and geophysics of the Southern Uplands region are given. The aims, and relevant theory, of induction studies are also presented. The observa-. • tional procedures for both the GDS and MT techniques, and the type and form of the MT activity, are described. Techniques for analysing the • MT data are developed in some detail. Methods are proposed, and examples given, for (a) estimating, the gross structural strike direction, • (b) averaging response function estimates, (c) estimating the confidence • intervals of the response functions, (d) estimating new forms of coherence functions, which exhibit many desirable properties, and (e) a frequencytime analysis for estimating the response functions for sub-intervals of the data set. The full MT and GDS estimates from analysis of the single station data rotated major and rotated minor impedance estimates, azimuthal angles, skew factors and real and imaginary induction vectors are presented and qualitatively discussed. A one-dimensional interpretation, of two-dimensional MT data is examined and shown to be valid for 'rotated major' impedance estimates from locations sufficiently distant from gross lateral inhomogeneity. Various methods for determining an 'optimum model' that best satisfies, in some manner, the observed MT responses are reviewed. A Monte-Carlo inversion procedure is developed and.applie& to the 'rotated major? data from six of the thirteen locations. It was considered,,for various reasons, unjustifiable to interpret an the data. The GDS and MT results agree on the complexity of the conductivity variations in the Midland Valley,the Southern Uplands and Northern England. The simple 'Eskdalernuiranomaly' proposed by Edwards et al. ---(1971) cannot explain the observations.. A conductive layer is required beneath the Midland Valley at a depth of no greater thaii 11 km. The conductive zone underlying the Southern Uplands is at a depth greater than 24 km. For the Northern England response, the top layer of highly conducting sediments 'screen' the possible effects of a 'lower crustal/ upper mantle' conductive layer. The geological and geophysical implications of the acceptable MT models are discussed. In this work, the quantitative information, offered by the MT technique, is shown to be necessary fora full interpretation of the conductivity distribution. Also, estimation of the phase response, as well as the amplitude response, of the impedance tensor elements is shown to resolve the surface structure of the acceptable models. Various suggestions are made about further investigation of the region. a-

Geomagnetic induction studies in Southern Scotland

SUMMARY We describe finite-difference approximations to the equations of 2-D electromagnetic induction that permit discrete boundaries to have arbitrary geometrical relationships to the nodes. This allows finite-difference modelling with the flexibility normally ascribed to finite-element modelling. Accuracy is demonstrated by comparison with finite-element computations. We also show that related approximations lead to substantially improved accuracy in regions of steep, but not discontinuous, conductivity gradient.

/pdf/the-forward-problem-of-electromagnetic-induction-accurate-2l0ilqross.pdf

The forward problem of electromagnetic induction: accurate finite-difference approximations for two-dimensional discrete boundaries with arbitrary geometry

It is apparent that the finite difference approximation to the Laplacian obtained by Williamson, Hewlett & Tammemagi (1974) differs from that given by Jones R: Pascoe (1971). The difference is related to the manner in which first derivatives are approximated in the two cases. Jones & Pascoe assunzed a central diflerence form for first derivatives from the outset, whereas the method of Williamson et al. implies a linear cornhination of backward and forward differences for first derivatives. It is instructive to examine the two approaches in greater detail. Following Jones & Pascoe, Taylor's theorem may bc used to obtain the field values around the point '0' as: for the real part of the field values and similar equations for the imaginary part (a misplaced minus sign which appeared in the earlier paper has been correctly located above). Combining the equations for f1 and f3 gives:

A Discussion of the Finite Difference Method in Computer Modelling of Electrical Conductivity Structures. A Reply to the Discussion by Williamson, Hewlett and Tammemagi

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A Discussion of the Finite Difference Method in Computer Modelling of Electrical Conductivity Structures. A Reply to the Discussion by Williamson, Hewlett and Tammemagi