THEORY FOR THE BIPOLE-DIPOLE METHOD OF RESISTIVITY SOUNDING*

A theory for the bipole-dipole method of resistivity sounding is developed. Bipole-dipole apparent resistivities are related to Schlumberger apparent resistivities at two spacings. The theory can also be used to compute exact dipole-dipole apparent resistivity curves providing an improvement over the existing techniques which involve far field approximations. A comparison of bipole-dipole and dipole-dipole systems reveals the similarity between the two. However, the resolution of the bipole-dipole system depends on the azimuth angle. The flexibility of the theoretical expressions lead to a generalized field scheme independent of the bipolar or dipolar nature of the current source.     

ZERO—PHASE FORWARD FILTERS FOR RESISTIVITY SOUNDING*

Forward filters to transform the apparent resistivity function over a layered half-space into the resistivity transform have been derived for a number of sample intervals. The filters have no apparent Gibbs' oscillations and hence require no phase shift. In addition, the end points of the filter were modified to compensate for truncation. The filters were tested on simulated ascending and descending two-layer cases. As expected, “dense” filters with sample spacing of In (10)/6 or smaller performed very well. However, even “sparse” filters with spacing of In (10)/2 and a total of nine coefficients have peak errors of less than 5% for p₁:p₂ ratios of 10^–6 to 10⁶. If a peak error of 5.5% is acceptable, then an even sparser filter with only seven coefficients at a spacing of 3 In (10)/5 may be used.     

RESISTIVITY SOUNDING ON A THREE-LAYER TRANSITIONAL MODEL*

The expression for the potential due to a point source of current, placed on the surface of a horizontal three-layer earth, has been derived when conductivity in the second layer varies linearly with depth and changes abruptly at the boundaries. Master curves for Wenner and Schlumberger configurations have been presented for an insulating basement and for one value of conductivity gradient in the second layer.     

RESISTIVITY SOUNDING ON A LAYERED TRANSITIONAL EARTH*

Master curves are presented for three-layer earth-sections, the electrical resistivity in the intermediate layer of which has a linear variation with depth. A new approach is proposed to interpret the sounding data with the help of RMS difference in apparent resistivity values.     

UNCERTAIN RESISTIVITY SOUNDING AND EQUIVALENT MODELS*

Problems of the resistivity sounding method may be reparametrized by means of Dar Zarrouk—or generalized D.Z.—transformations. A complete set of equivalent models is subdivided into three classes: initial models, intermediate models, and all other models. Equivalent reparametrized models are subdivided into three analogous classes. There is no one-to-one mapping of intermediate primary and intermediate reparametrized models. Non-trivial equivalent models are generated from initial models via intermediate reparametrized models. A method is developed to find equivalent models relative to the uncertainty of resistivity soundings. There are different forms of approximate equivalence laws, apart from those commonly accepted. The equivalence depends considerably on the “flatness” of the sounding curve. For mean flatness, the electric models exhibit the (hp^±2)-equivalence rather than the (hp^±1)-equivalence.     

RESISTIVITY SOUNDING ON A HORIZONTALLY STRATIFIED MULTI-LAYERED EARTH*

The electrical potential due to a point source of current placed on the ground surface is studied for a multi-layered earth consisting of homogeneous overburden of constant conductivity over a stack of transition layers where conductivity varies with depth according to power and exponential laws in even and odd layers, respectively. The general recursion relations are derived and are used to obtain expressions for the apparent resistivities for Schlumberger and Wenner electrode arrays. Their asymptotic behavior has been studied. The solutions for some particular cases are given: (i) odd layers with conductivities exponentially varying with depth while all even layers (and the first) have constant conductivities; (ii) even layer conductivities varying as a power law while odd number layers are of constant conductivity; and (iii) any two successive layers as transition layers and all others having constant conductivities. Further it is shown that Patella's theory is a particular case of the present study. It is concluded, therefore, that the present treatment is more general as all earlier models consisting of trasition layers can be derived from this study.     

RESISTIVITY SOUNDING ON A MULTILAYERED EARTH CONTAINING TRANSITION LAYERS*

For curves over a horizontally stratified earth where any of the layers has a conductivity variation proportional to (1 +β·z)^N, where β and N are arbitrary constants and z is the depth to the layer, expressions for apparent resistivity for Wenner and Schlumberger sounding are derived. No assumption has been made about the continuity of conductivity at the interface. It is shown that most of the previous investigations in this connection can be regarded as particular cases of the present study.     

THEORETICAL RESISTIVITY SOUNDING RESULTS OVER A TRANSITION LAYER MODEL *

An earth model with a transition layer (anisotropic inhomogeneous) is considered. The inhomogeneity in σ_v (vertical conductivity) of the transition layer is represented by a power law variation. Expressions for potential distribution in the upper layer, transition layer and bottom layer are obtained by solving appropriate differential equation for each layer. By utilizing the boundary conditions, expressions of apparent resistivity for Wenner and Schlumberger configurations are derived. Numerical analysis is performed for linear and quadratic variation of σ_v. The results are presented in the form of theoretical apparent resistivity curves for both configurations. Negative apparent resistivities are the interesting feature of this analysis.     

RESISTIVITY SOUNDING ON A TWO-LAYER EARTH WITH TRANSITIONAL BOUNDARY*

A theoretical solution is obtained for the problem of a two-layer earth with transitional boundary. In practice, the transition layer can stand for the weathered zone in hard rock areas where the degree of weathering diminishes with depth. Master curves and tables of data are presented for the case when the lower half-space is infinitely resistive.     

AN ANALYSIS OF EQUIVALENCE IN RESISTIVITY SOUNDING*

A mathematical analysis is given of the phenomenon of equivalence in resistivity sounding, which is based upon the properties of the raised kernel function. Analysis of this function instead of the apparent resistivity function is justified because, as has been shown in a previous publication, variations in the apparent resistivity function lead to variations in the raised kernel function with relative values of the same order of magnitude The expression for the raised kernel function is expanded into a Mac Laurin series. Equivalence can occur only if the second order term of this series is negligible. The coefficient of the first order term depends on the resistivity and the thickness of the layer under consideration. There is an infinite set of combinations of values for these two quantities, for which the coefficient of the first order term has the same value. All these combinations represent equivalent layer distributions.     

AN APPROXIMATE METHOD OF RESISTIVITY SOUNDING INTERPRETATION*

An approximate method of interpretation of resistivity sounding is presented, which may be described as a very crude manner of application of the exact direct interpretation method. The accuracy of this method is fairly low, the errors being in the order of 25%. The method is very fast in application and well suited for application to multilayer cases. The main advantage of the method is that it is in close and clear relation to the exact theory.     

RESISTIVITY SOUNDING ON A MULTI-LAYERED EARTH WITH TRANSITIONAL LAYERS. PART II: THEORETICAL AND FIELD EXAMPLES *

This final part of our study of the resistivity sounding method in the presence of transitional layers deals with the direct application of the theoretical developments, previously reported, to both theoretical and field examples. The former are used to explain in more detail all steps of the direct procedure of interpreting resistivity measurements, when transitional layers with different dispositions are present in the assumed earth section. An alternate sequence of uniform and transitional layers is at first considered, then the case of a random sequence of these layers, and finally a sequence of only transitional layers. Then we give some logical considerations about the best fields of application of the new interpretation theory, with particular reference to hydrogeological, geological- structural, and mining exploration problems. Representative field examples are finally presented and discussed in detail for each of the exploration fields above listed. The comparison of the geophysical models obtained with the old interpretative scheme and the new system shows the greater resolving power of the new theory, provided that preliminary geological considerations, and where existing–borehole evidence justify the use of the new approach.     

A DIGITAL LINEAR FILTER FOR RESISTIVITY SOUNDING WITH A GENERALIZED ELECTRODE ARRAY*

This paper presents a digital linear filter which maps composite resistivity transforms to apparent resistivities for any four—electrode array over a horizontally layered earth. A filter is provided for each of three sampling rates; the choice of filter will depend on resistivity contrasts and computational facilities. Two methods of filter design are compared. The Wiener-Hopf least-squares method is preferable for low sampling rate filters. The Fourier transform method is more successful in producing a filter with a high sampling rate which can handle resistivity contrasts of 100 000: 1.     

COMPARISON OF FIVE LEAST-SQUARES INVERSION TECHNIQUES IN RESISTIVITY SOUNDING*

A brief history of the development of the inverse problem in resistivity sounding is presented with the development of the equations governing the least-squares inverse. Five algorithms for finding the minimum of the least-square problem are described and their speed of convergence is compared on data from two planar earth models. Of the five algorithms studied, the ridge-regression algorithm required the fewest numbers of forward problem evaluations to reach a desired minimum. Solution space statistics, including (1) parameter-standard errors, (2) parameter correlation coefficients, (3) model parameter eigenvectors, and (4) data eigenvectors are discussed. The type of weighting applied to the data affects these statistical parameters. Weighting the data by taking log₁₀ of the observed and calculated values is comparable to weighting by the inverse of a constant data error. The most reliable parameter standard errors are obtained by weighting by the inverse of observed data errors. All other solution statistics, such as dataparameter eigenvector pairs, have more physical significance when inverse data error weighting is used.     

THE AUTOMATIC FITTING OF A RESISTIVITY SOUNDING BY A GEOMETRICAL PROGRESSION OF DEPTHS *

The problem of interpretation of geoelectrical resistivity soundings has been studied and a compromise method of interpretation developed. A simple depth sequence of increasing bed thickness is taken and resistivities assigned to the various layers so that a good fit is obtained to the observed apparent resistivity curve. An initial estimate of the resistivities is made by a direct method but the model is then altered to give an improved fit by the indirect method of curve matching. The whole method is computerized and input data to the computer program consists of spacings and apparent resistivities. The method is extremely rapid, about one third of the computation time is devoted to obtaining the resistivity transform by the method of convolution. A normal type of solution, incorporating additional information if necessary, can also be obtained.     

A NOTE ON THE LINEAR FILTER METHOD OF INTERPRETING RESISTIVITY SOUNDING DATA*

In the linear filter method of interpreting resistivity sounding data, as developed by Ghosh (1971), it appears that the filter function in the x-domain approaches an oscillating function for both large positive and large negative abscissa values. In the present note the reason for this oscillating behaviour is derived, and a possible practical application is indicated.     

A NUMERICAL METHOD TO COMPUTE THE RESISTIVITY TRANSFORM FROM WENNER SOUNDING DATA*

A numerical technique to compute the resistivity transform directly from the observed Wenner sounding data has been developed. In principle, the procedure is based on a decomposition method and consists of two steps: the first step determines a function that approximates the apparent resistivity data and the second step transforms this function into the corresponding kernel by an analytical operation. The proposed method is tested on some theoretical master curves. A high degree of precision is achieved with very little computer time. The applicability is shown on two field examples.     

MATHEMATICAL ANALYSIS OF D.C. RESISTIVITY SOUNDING OVER A PARABOLOID*

A complete mathematical analysis is proposed for direct current resistivity prospecting over the surface of a layered paraboloid. The analysis evaluates the Green's function in parabolic coordinates for a current source at the vertex. The general solution is obtained as a Fourier-Bessel integral involving those curvilinear coordinates that have a kernel function which is similar to that of a half-plane containing inhomogeneous layers. This similarity permits the computation of a class of sounding curves over such an oval surface providing a way to analyse field data over hilly terrain.     

DETERMINATION OF RESISTIVITY SOUNDING FILTERS BY THE WIENER-HOPF LEAST-SQUARES METHOD*

It has been found that the Wiener-Hopf least-squares method is a very successful tool for the determination of resistivity sounding filters. The values of the individual filter coefficients differ quite appreciably from those obtained by the Ghosh procedure. These differences in the filter coefficients, however, have only a negligible effect on the output of the filter. It seems that these differences in the coefficients correspond to a filter function of a rather narrow frequency band around the Nyquist frequency, which is only very weakly present in the input and output functions.