A MATRIX METHOD FOR THE INTERPRETATION OF DIRECT CURRENT RESISTIVITY DATA

Abstract

The basic aim in exploration geophysics is to add the third dimension to geological maps. By measuring physical fields near the surface of the earth the geophysicsts try to portray the three dimensional makeup of a region. This helps to identify locales of concentrations of minerals of economic value. Amongst the physical fields available for such investigations are seismic, gravity, magnetic, thermal, geoelectrical, electromagnetic and magnetotelluric. In many experiments some of these fields are excited by artificial or natural sources such as in the seismic, electrical or electromagnetic. On the other hand potential fields are passive which do not require any excitation. These potential fields are governed by the distribution of the density and magnetization in body. However the characteristics of the various other types of fields mentioned above are governed by the properties of the medium they permeate as well as the properties of their sources.lt is the characteristics imparted by the medium properties that are of interest to exploration geophysicist which they utilize for imaging the earth interior. The highly variable of all the physical properties of the earth materials is their relative resistivity (« 10-8 < P < 1CU 6Qm).Because of this large spectrum of the resistivity values of the naturally occurring rock formations, it is expedient that the geoelectrical methods should be used to exploit this large range of resistivity variation for the mapping of the subsurface geological structure and composition. In direct current resistivity method one obtains the apparent resistivity values as a function of appropriate electrode spacing. The aim of resistivity data interpretation is first to decompose the measured data in terms of resistivities and thicknesses of the subsurface layers, and subsequently these values are correlated with the geological picture of the region under investigation. Present thesis is devoted to the development of a new technique of interpretation for the direct current resistivity data. More specifically the idea is to develop a formalism that is simple and straightforward that may be used directly on the field data without previous smoothing; may be applicable on a variety of abscissa distribution without the need for extrapolation of the data set. The work in this thesis also addresses the problem of transformation of resistivity data obtained using a particular electrode configuration onto data set corresponding to another electrode configuration. The theory of the present method and examples of its application for the interpretation to a variety of geoelectrical set of data are presented in the following chapters systematically. Chapter I gives the general introduction of the subject. The previous developments in solving the resistivity interpretation problem have been summarized. The merits and demerits of different methods have been discussed. In the light

of these discussions, the research problem addressed in this thesis is identified. Chapter II presents the theory of exponential approximation of the kernel function. The error introduced in such an approximation are analysed. An error analysis has been performed by computing the standard error of the estimate of the kernel function. Consistency, stability and accuracy of the method have been shown for a variety of multilayered earth models. In Chapter III, it is shown that the apparent resistivity values for any model and for any configuration can be computed efficiently using the exponential approximation for the kernel function. The computational algorithm has been discussed for the case of (i) the symmetrical configuration, which includes Schlumberger, Wenner and two electrode configurations, (ii) all types of dipole configurations and (iii) the apparent resistivity in the presence of vertical discontinuities. Tests are also carried out over the limiting case of very high resistivity contrast, perfectly conducting and perfectly insulating basement. A number of examples show the practical applicability of the method. The procedure of the kernel function estimation from the field apparent resistivity values is presented in Chapter IV. The efficiency of the method vis-a-vis the filter method is also compared. Singular vlaue decompositions are used to demonstrate the resolution and its correlation with the eigenvlaues and condition number of the matrix for several configurations. r Numerical results are presented for a few models. The efficiency of the method in transforming the apparent resistivity data observed using a particular electrode system to the corresponding data for other electrode system is discussed in Chapter V. All types of transformations have been divided in three categories as follows: i) Inter symmetrical apparent resistivity transformation ii) Transformation from one dipole to other dipole configurations iii) Transformation from symmetrical to dipole configuration and vice-versa In numerical examples, transformed values are compared to the exact values for a number of multilayered earth models. Chapter VI deals with the inversion of apparent resistivity data in the r-domain and the \ -domain respectively. The results in both the domains are obtained using least square criterian. Synthetic examples show the reliability of either of the methods. There is no significant difference in computer time for these methods. A few field examples taken over the Saharanpur District (Uttar Pradesh,INDIA) for the case of Schlumberger configuration have been interpreted in terms of subsurface geological structure and geohydrological conditions. These results were used for the selection of well sites and give parameters for the foundation of a civil construction site. The concluding Chapter VII, discusses the merits and limitations of the new method developed here. It also discusses the possibilities of further extension of similar work for the electromagnetic data.