A NUMERICAL METHOD OF CALCULATING THE KERNEL FUNCTION FROM SCHLUMBERGER APPARENT RESISTIVITY DATA*

A method to calculate the resistivity transform of Schlumberger VES curves has been developed. It consists in approximating the field apparent resistivity data by utilizing a linear combination of simple functions, which must satisfy the following requirements: (i) they must be suitable for fitting the resistivity data; (ii) once the fitting function has been obtained they allow the kernel to be determined in an analytic way. The fitting operation is carried out by the least mean squares method, which also accomplishes a useful smoothing of the field curve (and therefore a partial noise filtering). It gives the possibility of assigning different weights to the apparent resistivity values to be approximated according to their different reliability. For several examples (theoretical resistivity curves in order to estimate the precision of the method and with field data to verify the practicality) yield good results with short execution time independent of shape the apparent resistivity curve.     

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.     

ERROR PROPAGATION AND UNCERTAINTY IN THE INTERPRETATION OF RESISTIVITY SOUNDING DATA*

An analysis is made of the propagation of the measuring error in the different stages of the interpretation by the linear filter and reducing method. This analysis leads to an understanding of the range of possible values of the layer parameters and of the nature of the relation between them. It is shown that this relation is not always adequately described by the equivalence expressions of Maillet.     

THREE-DIMENSIONAL INTERPRETATION OF GRAVITY DATA FROM SEDIMENTARY BASINS USING AN EXPONENTIAL DENSITY-DEPTH FUNCTION*

In mapping the topography of the basement of deep sedimentary basins by gravity modelling, the accuracy can be improved by incorporating an exponential increase in density with depth. For calculating the gravity effect of a three-dimensional (3D) structure with such an exponential density-depth relation a frequency-domain forward algorithm based on series expansion is presented, the numerical evaluation of which can be performed efficiently by fast Fourier transform. The algorithm can be applied in a recursive procedure to give the inverse solution in terms of basement relief. The inversion procedure is satisfactorily tested on a 2D synthetic example and a 3D field example of gravity data from the western margin of the Pannonian Basin in eastern Austria, where up to 2.2 km of Tertiary sediments overlie an igneous or metamorphic basement. The results are confirmed by basement intersections in several wells.     

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 DIRECT INTERPRETATION METHOD OF RESISTIVITY SOUNDINGS USING THE PEKERIS MODEL*

A numerical method is presented for direct interpretation of resistivity sounding measurements. The early part of the resistivity transform curve derived from field observations by standard methods is approximated by a two-layer curve. The resistivity of the first layer is determined from the arithmetic mean of the successive computations which are carried on each of three successive discrete values of the resistivity transform curve. Using this mean value of the resistivity, the thickness of the first layer is computed from the sample values in pairs of the resistivity transform curve. After these determinations, the top layer is removed by Pekeris's reduction equation. The parameters of the second layer are obtained from the discrete values of the reduced transform curve (which corresponds to the second part of the resistivity transform curve) by the same procedure as described for the first layer. The same computational scheme is repeated until the parameters of all intermediate layers are obtained. The resistivity of the substratum is determined from the reduction equation.     

COMPUTING THE KERNEL FUNCTION IN RESISTIVITY SOUNDING WITH AN ARBITRARY ELECTRODE CONFIGURATION*

A power series expansion can be used to obtain the kernel from apparent resistivities for an arbitrary electrode configuration. Three types of function are most appropriate for this purpose. The expansion coefficients can be by a least-squares method. In this case, ortho-normalization of the functions is of great advantage. An example with the Wenner configuration is given.     

A STUDY ON THE DIRECT INTERPRETATION OF RESISTIVITY SOUNDING DATA MEASURED BY WENNER ELECTRODE CONFIGURATION*

This paper describes certain procedures for deriving from the apparent resistivity data as measured by the Wenner electrode configuration two functions, known as the kernel and the associated kernel respectively, both of which are functions dependent on the layer resistivities and thicknesses. It is shown that the solution of the integral equation for the Wenner electrode configuration leads directly to the associated kernel, from which an integral expression expressing the kernel explicitly in terms of the apparent resistivity function can be derived. The kernel is related to the associated kernel by a simple functional equation where K₁(λ) is the kernel and B₁(λ) the associated kernel. Composite numerical quadrature formulas and also integration formulas based on partial approximation of the integrand by a parabolic arc within a small interval are developed for the calculation of the kernel and the associated kernel from apparent resistivity data. Both techniques of integration require knowledge of the values of the apparent resistivity function at points lying between the input data points. It is shown that such unknown values of the apparent resistivity function can satisfactorily be obtained by interpolation using the least-squares method. The least-squares method involves the approximation of the observed set of apparent resistivity data by orthogonal polynomials generated by Forsythe's method (Forsythe 1956). Values of the kernel and of the associated kernel obtained by numerical integration compare favourably with the corresponding theoretical values of these functions.     

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.     

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.     

A SIMPLE METHOD OF MAGNETOTELLURIC INTERPRETATION *

A simple method of magnetotelluric interpretation is derived using the property that the earth currents flow in horizontal sheets. It is shown that when the depth is taken as two-thirds of the Cagniard's depth of penetration (Cagniard, 1953), the mean resistivity-over the depth is the same as the apparent resistivity of the medium. From a mean resistivity versus depth of penetration plot, resistivity can be easily computed at all depths of the sounding. This method gives satisfactory results in a short time and makes it possible to interpret the soundings over media of several layers. However, the results are inaccurate near maximum and minimum points of the plot as well as in the zone of thin layers.     

THREE-DIMENSIONAL INTERPRETATION OF MULTIPLE-SOURCE BIPOLE-DIPOLE RESISTIVITY DATA USING THE APPARENT RESISTIVITY TENSOR1

The bipole-dipole resistivity technique, which uses a single current source (bipole) to map variations in (apparent) resistivity has been much criticized in the past. A series of 3D models are used to show that the use of two distinct current bipoles in the same location but with different orientations, combined with analysis in the form of a previously defined tensor apparent resistivity, can greatly improve many aspects of bipole-dipole mapping. The model study shows that, for measurement stations more than a few bipole lengths from the current source, the apparent resistivity tensor behaves, to a close approximation, as though the current bipoles are idealized dipoles, and hence is independent of the orientation of the individual current sources used. Any pair of current bipoles (in the same location but with different orientations) can therefore be used to determine the tensor resistivity properties. The invariants of the apparent resistivity tensor have considerable advantages over the normal scalar apparent resistivities. Modelling shows that although the electric field vector corresponding to a single current bipole can be highly perturbed by a local inhomogeneity for some considerable distance beyond the inhomogeneity itself, the tensor invariants are virtually unperturbed beyond the extent of the inhomogeneity. Hence false anomalies, which are a characteristic of apparent resistivity measured using only single current bipole models, are almost completely eliminated by the use of tensor invariants. Of the possible tensor invariants, the invariant given by the square root of the determinant gives the best representation of a buried 3D body. Resistivity anomalies are localized, and occur only over the causative body. Even with complex models involving several buried bodies, the tensor invariants clearly delineate the extent of each body. Outside the bounds of perturbing bodies, the tensor data can be analysed by conventional techniques, for example, to determine layered structure.     

A FAST METHOD FOR DETERMINING THE LAYER DISTRIBUTION FROM THE RAISED KERNEL FUNCTION IN GEOELEGTRICAL SOUNDING*

In a previous publication (Koefoed 1968) a function called the “raised kernel function” has been introduced as an intermediate function in the interpretation of resistivity sounding data, and methods have been described both for the determination of the raised kernel function from the apparent resistivity function, and for the determination of the layer distribution from the raised kernel function. In the present paper a procedure is described by which the second step in this interpretation method–i.e. the determination of the layer distribution from the raised kernel function–is considerably accelerated. This gain in interpretation speed is attained by the use of a standard graph for a function which defines the reduction of the raised kernel function to a lower boundary plane.     

SIMPLE DATA PROCESSING OF TRIPOTENTIAL APPARENT RESISTIVITY MEASUREMENTS AS AN AID TO THE INTERPRETATION OF SUBSURFACE STRUCTURE*

The advantages of the Wenner tripotential method (Carpenter 1955) for apparent resistivity profiling are described and two new data processing techniques introduced as an aid to the interpretation of apparent resistivity sections (pseudo-sections). These techniques were developed from model data computed using a two-dimensional finite difference method. Oscillatory components present in anomalies on tripotential profiles and related to electrode spacing are shown to be effectively removed by linear filtering that also simplifies their form and aids recognition. Furthermore, it is shown that the ratio of the beta- and gamma-apparent resistivities is a good indicator of resistivity variation, and is particularly sensitive to lateral change. Model data indicates that, over a wide range of conditions, enough subsurface information can be obtained by inspection of tripotential resistivity and ratio profiles, and from space sections to make possible a useful—and sometimes semi-quantitative—interpretation. A rationale for the general interpretation of tripotential data is developed. Field data are described from an area of weathered granite basement in Nigeria. A model of the subsurface is developed using parameters derived from the processed observations. The observed and calculated apparent resistivity space sections are very similar.     

ON THE INTERPRETATION OF RESISTIVITY SOUNDINGS BY THE LEAST-SQUARES METHOD*

An interactive least-squares method for the interpretation of VES curves was proposed by Johansen (1977). The method permits one to select some parameters (thicknesses and/or resistivities of individual layers) and to change the rest in such a way that the interpreted model approaches the measured data. This note suggests a modification of Johansen's method, in which not only the individual parameters can be selected but also linear combinations of parameters—in particular, the sum of thicknesses of several layers.     

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.     

电阻率测深的数字解释

本文主要介绍了应用积分变换的方法和采样定理将视电阻率ρs曲线作线性滤波,得出一新的电阻率转换函数T′曲线,然后,以层参数（各层的电阻率和厚度）算得的T用最优化数值方法在DJS-6型电子计算机上与其进行自动拟合,以达到解释电测深曲线的目的。 文中简述了戈什（Ghosh）提出的对ρs作线性滤波的原理,介绍了与国外不同的取样间距和滤波系数的确定以及阻尼最小二乘法和变尺度最优化法的计算框图和应用,最后附有实例和简要的讨论。     

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.     

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.     

THE APPLICATION OF LINEAR FILTER THEORY TO THE DIRECT INTERPRETATION OF GEOELECTRICAL RESISTIVITY SOUNDING MEASUREMENTS*

Koefoed has given practical procedures of obtaining the layer parameters directly from the apparent resistivity sounding measurements by using the raised kernel function H(λ) as the intermediate step. However, it is felt that the first step of his method—namely the derivation of the H curve from the apparent resistivity curve—is relatively lengthy. In this paper a method is proposed of determining the resistivity transform T(λ), a function directly related to H(λ), from the resistivity field curve. It is shown that the apparent resistivity and the resistivity transform functions are linearily related to each other such that the principle of linear electric filter theory could be applied to obtain the latter from the former. Separate sets of filter coefficients have been worked out for the Schlumberger and the Wenner form of field procedures. The practical process of deriving the T curve simply amounts to running a weighted average of the sampled apparent resistivity field data with the pre-determined coefficients. The whole process could be graphically performed within an quarter of an hour with an accuracy of about 2%.