On the automatic interpretation of direct current resistivity soundings

A method for the automatic inversion of resistivity soundings is presented. The procedure consists of two main stages. First, application of linear filters which transforms the apparent resistivity curve into the kernel function, and vice versa. In the second stage the first and second derivatives of the kernel function are calculated and used in a second-order modified Newton-Raphson iterative fitting procedure. The model obtained is optimal in the least squares sense. The method has been tried on some field examples and produced resistivity models which show a good agreement with the geological well logs.     

A GENERAL TECHNIQUE FOR THE DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

The indirect method of interpreting resistivity data is capable only of limited success because of the difficulty of calculating type curves for complex structures. Consequently a need arises for a direct method of interpretation for complex generalized structures. Such a technique for the direct interpretation of apparent resistivity data, obtained by electrical soundings carried out over two-dimensional structures, has been developed from an examination of the Hankel transform of such data. The method is based on the observation that the asymptotic expansion of the Hankel transform is critically dependent upon the minimum distance between the measuring device and the surface of discontinuity in resistivity. The variation in the across strike direction may be mapped by a sequence of depth soundings made parallel to each other and separated in the across strike direction.     

THE QUANTITATIVE INTERPRETATION OF DIPOLE SOUNDINGS BY MEANS OF THE RESISTIVITY TRANSFORM FUNCTION*

In this paper a theorem is demonstrated which allows—after the introduction of a suitable dipole kernel function or dipole resistivity transform function—to write the apparent resistivity function as an Hankel transformable integral expression. As a practical application of the theorem a procedure of quantitative interpretation of dipole soundings is suggested in which the dipole resistivity transform function obtained after inversion of the original dipole apparent resistivity data is used to control the goodness of the set of layering parameters which have been derived with our previous method of transformation of dipole sounding curves into equivalent Schlumberger diagrams.     

DIRECT INTERPRETATION OF TWO-ELECTRODE RESISTIVITY SOUNDINGS *

This paper describes the procedure for interpreting the apparent resistivity data measured with the two-electrode array directly with the help of kernel function. The calculation of kernel function from the observed resistivity curve is done by the method of decomposition. In the method of decomposition the resistivity curve is approximated by a sum of certain functions, whose choice is only restricted by the requirement that the contribution to the kernel function corresponding to them should be easily computable. A few such functions are classified. These, and the standard curves for corresponding kernel functions obtained by utilising an integral expression for two-electrode array expressing the kernel explicitly in terms of the apparent resistivity functions, are plotted on log-log scale. The determination of layer parameters, that is, the layer resistivities and thicknesses from the kernel function can be carried out by a method proposed by Pekeris (1940).     

THE USE OF FILTERED BESSEL FUNCTIONS IN DIRECT INTERPRETATION OF GEOELECTRICAL SOUNDINGS *

We start from the Hankel transform of Stefanescu's integral written in the convolutionintegral form suggested by Ghosh (1971). In this way it is possible to obtain the kernel function by the linear electric filter theory. Ghosh worked out the sets of filter coefficients in frequency domain and showed the very low content of high frequencies of apparent resistivity curves. Vertical soundings in the field measure a series of apparent resistivity values at a constant increment Δx of the logarithm of electrode spacing. Without loss of information we obtain the filter coefficient series by digital convolution of the Bessel function of exponential argument with sine function of the appropriate argument. With a series of forty-one values we obtain the kernel functions from the resistivity curves to an accuracy of better than 0.5%. With the digital method it is possible to calculate easily the filter coefficients for any electrode arrangement and any cut-off frequency.     

TRANSFORMATION OF DIPOLE RESISTIVITY SOUNDING MEASUREMENTS OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

The technique of linear digital filtering as developed for the direct interpretation of Schlumberger and Wenner soundings (Ghosh 1971) has been applied here for the derivation of the resistivity transform function from the field dipole measurements as the first step in directly interpreting dipole data. Filter coefficients for this transformation have been worked out for the radial-polar, perpendicular and parallel (30°) arrays of dipole sounding. The procedure combines speed with accuracy.     

A STUDY ON THE DIRECT INTERPRETATION OF DIPOLE SOUNDING RESISTIVITY MEASUREMENTS OVER LAYERED EARTH *

Dipole sounding resistivity measurements over layered earth can be interpreted directly by adapting the procedure given by Koefoed (1968) for Schlumberger system. To carry out the first step of the interpretation leading to the determination of the raised kernel function, partial resistivity functions for the dipole method are derived and given in the form of standard curves. The second step involving the derivation of layering parameters from the kernel being independent of the electrode configurations remains unaltered. The applicability and limitations of the method are discussed.     

INTERPRÉTATION DIRECTE DES COURBES DE SONDAGE ÉLECTRIQUE ET LE PROBLÈME DE DIFFÉRENTS DISPOSITIFS DE MESURE *

Direct interpretation methods of resistivity curves are discussed, which use the kernel function of the apparent resistivity. This function results from the consideration of the problem of diverse electrodes configurations. Several expressions for the determination of the kernel function of the potential from the kernel function of the apparent resistivity are given.     

A SIMPLE METHOD OF INTERPRETATION OF RESISTIVITY SOUNDING DATA USING EXPONENTIAL APPROXIMATION OF THE KERNEL FUNCTION*

A simple unified equation of apparent resistivity for a general four-electrode array is developed. The main idea is the analytical integration of the Stefanescu expression for potential over a layered earth by writing an exponential approximation for the kernel function. Finally a matrix equation is developed to estimate the kernel function from observed apparent resistivity values. The general equation automatically reduces to the particular configuration once the electrode separations are modified suitably. Examples for Schlumberger and Wenner configurations are numerically calculated to estimate the precision of the method. Good results in a short execution time are obtained, irrespective of the shape of the apparent resistivity curve. Finally, the full interpretation of one theoretical resistivity curve and two field resistivity curves is demonstrated. The more stable ridge-regression estimation method is used in the identification of layer parameters from the kernel function.     

电偶源频率电磁测深二维阻抗视电阻率计算的源效应校正法

研讨了频率域电磁法中不同源装置的大地电磁测深、线源频率电磁测深和偶极源频率电磁测深阻抗视电阻率的源效应影响特征。在唯象分析的基础上，提出了几种电磁测深法阻抗视电阻率的相互换算法──源效应校正法（大地电磁测深二维ＴＥ极化视电阻率和其它两种电磁法的赤道装置二维阻抗视电阻车）。模型试验表明，利用这一源效应校正法可以由大地电磁二维视电阻率近似地计算出线源频率电磁二维阻抗视电阻率。这一方法被尝试应用于由线源频率电磁二维阻抗视电阻率估算偶极源频率电磁二维阻抗视电阻率。     

COMPUTATION OF THE ELECTRIC POTENTIAL IN THREE-DIMENSIONAL STRUCTURES*

A method has been developed for computation of the electrical DC potential of an arbitrary three-dimensional resistivity structure using a finite difference method. The threedimensionality is necessary for interpretation of geoelectrical soundings with controlled point sources over a laterally inhomogeneous medium. Lateral inhomogeneities should be considered in resistivity soundings with large layouts. The results obtained with the described method permit a more realistic representation of geological features. The resolution of the method is determined by the number of elements in the resistivity network. The problem of core memory space has been resolved by using random access disc files. The results computed using a Fortran program are in good agreement with analytically obtained solutions.     

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.     

Gaussian character of the distribution of magnetotelluric results working in log-space

The distribution of individual values of the apparent resistivities given by two M-T soundings made over two different kinds of subsurface geology is described and computed. This distribution is Gaussian only if the logarithm of the resistivity values are used. Consequently, a statistical stability is obtained when the mean and standard deviation are calculated in log-space. Log-space is a mathematical space that may have one or more logarithmical dimensions; in this case two.     

THE DETERMINATION OF FILTER COEFFICIENTS FOR THE COMPUTATION OF STANDARD CURVES FOR DIPOLE RESISTIVITY SOUNDING OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

The technique of linear digital filtering developed for the computation of standard curves for conventional resistivity and electromagnetic depth soundings is applied to the determination of filter coefficients for the computation of dipole curves from the resistivity transform function by convolution. In designing the filter function from which the coefficients are derived, a sampling interval shorter than the one used in the earlier work on resistivity sounding is found to be necessary. The performance of the filter sets is tested and found to be highly accurate. The method is also simple and very fast in application.     

Research Note: The mean sensitivity depth of the electrical resistivity method

Until now, a simple formula to estimate the depth of investigation of the electrical resistivity method that takes into account the positions of all of the electrodes for a general four‐electrode array has not been available. While the depth sensitivity function of the method for a homogeneous infinite half‐space is well known, previous attempts to use it to characterize the depth of investigation have involved calculating its peak and median, both of which must be determined numerically for a general four‐electrode array. I will show that the mean of the sensitivity function, which has not been considered previously, does admit a very simple mathematical formula. I compare the mean depth with the median and peak sensitivity depths for some common arrays. The mean is always greater than or equal to the median that is always greater than the peak. All three measures give reasonable estimates to the depths of actual structures for most circumstances. I will further show that, for 1D soundings, the use of the mean sensitivity depth as the pseudo‐depth assigns an apparent resistivity to a given pseudo‐depth that is consistent between different arrays. One consequence of this is that smoother depth soundings are obtained as “clutches,” caused by a change in the depth sensitivity due to moving the potential electrodes, are effectively removed. I expect that the mean depth formula will be a useful “rule of thumb” for estimating the depth of investigation before the resistivity structure of the ground is known.     

LES COURBES DAR ZARROUK ET LES SONDAGES DIFFÉRENTIELS DE RÉSISTIVÉ*

Differential and difference soundings of resistivity for real, imaginary and complex values of the newly introduced parameters ν and μ are considered. The transverse and longitudinal differential soundings, known up to the present, correspond with the values ν= 1 and ν=— 1. The point of departure is the generalization of the concept of Dar Zarrouk parameters and the dimensional analogy between D.Z. resistivity and transverse and longitudinal differential resistivities. Properties of generalized D.Z. curves as well as the method of their construction are given. The examples of differential and difference curves are presented and the properties of these curves are discussed; particulary the effective spacings are determined. The problems of the realization of differential and difference soundings are considered; the bisymmetrical and asymmetrical arrays are proposed as a practical ones.     

RESISTIVITY TYPE CURVES OVER OUTCROPPING VERTICAL DYKE—I *

The theoretical horizontal resistivity profiles over an outcropping vertical dyke for various parameters-electrode spacing, vein-width and resistivity contrast—with inline alpha, beta and gamma-Wenner electrode systems are described. The resistivity profiles present a most bewildering variety of shapes as compared to those in resistivity soundings. The analysis of type curves suggests suitable electrode configuration for detection of wide, moderately wide, and thin veins. The negative apparent resistivity values on the gamma anomalies for resistive vein over certain vein-widths of higher positive values of resistivity reflection factor contradict the concept of apparent resistivity hitherto held.     

水平多层大地上垂直磁偶极频率测深的全波视电阻率

本文根据视电阻率定义的原则,以及用不同的场量定义的视电阻率效果不同这一事实,提出一种新的全波视电阻率定义.在全区同时用均匀大地上电磁场的三个分量来分区定义祝电阻率.在远区视电阻率由磁场的水平分量求出,在近区由磁场的垂直分量或其实分量定义,而在过渡区则由电场的水平分量确定.用这种方法定义的视电阻率为电磁响应的单值函数,它随频率变化的曲线显著改善,能直观地反映地层电阻率随深度的变化,数值比较接近地层的真电阻率值,假极值效应明显压低.在计算中用切比雪夫多项式分段拟合均匀大地电磁响应的反函数,并给出一套系数,由此算出的视电阻率误差小于1％.     

GEOELECTRICAL MODEL CALCULATIONS FOR TWO-DIMENSIONAL RESISTIVITY DISTRIBUTIONS*

For the calculation of geoelectrical model curves for a two-dimensional resistivity distribution, the potential equation is transformed by means of a Fourier cosine transform into a two-dimensional Helmholtz equation containing the separation parameter λ. The numerical solution of this equation for different values of λ for an irregular grid is obtained using the method of finite differences combined with the method of overrelaxation. The method by which derivatives are replaced by finite differences turned out to be very important, especially for high resistivity contrasts. After testing several methods designed to deal with any type of resistivity distribution, a method of discretization similar to that used by Brewitt—Taylor and Weaver (1976) for magnetotelluric modeling for H polarization was found the best. Examples are given of model curves for Schlumberger soundings over a vertical fault covered by overburden. The incorrect use of horizontal-layer models leads to erroneous interpretations that are more complex than the real subsurface situations.     

RESISTIVITY SOUNDING ON A MULTI-LAYERED EARTH WITH TRANSITIONAL LAYERS. PART I: THEORY*

The electrical potential generated by a point source of current on the ground surface is studied for a multi-layered earth formed by layers alternatively characterized by a constant conductivity value and by conductivity varying linearly with depth. The problem is accounted for by solving a Laplace's differential equation for the uniform layers and a Poisson's differential equation for the transitional layers. Then, by a simple algorithm and by the introduction of a suitable kernel function, the general expression of the apparent resistivity for a Schlumberger array placed on the surface is obtained. Moreover some details are given for the solution of particular cases as 1) the presence of a infinitely resistive basement, 2) the absence of any one or more uniform layers, and 3) the absence of any one or more transitional layers. The new theory proves to be rather general, as it includes that for uniform layers with sharp boundaries as a particular case. Some mathematical properties of the kernel function are studied in view of the application of a direct system of quantitative interpretation. Two steps are considered for the solution of the direct problem: (i) The determination of the kernel function from the field measurements of the apparent resistivity. Owing to the identical mathematical formalism of the old with this new resistivity theory, the procedures there developed for the execution of the first step are here as well applicable without any change. Thus, some graphical and numerical procedures, already published, are recalled. (ii) The determination of the layer distribution from the kernel function. A recurrent procedure is proposed and studied in detail. This recurrent procedure follows the principle of the reduction to a lower boundary plane, as originally suggested by Koefoed for the old geoelectrical theory. Here the method differs mainly for the presence of reduction coefficients, which must be calculated each time when passing to a reduced earth section.