Apparent resistivity over a three layer earth with an inhomogeneous interstratum

Summary The theory of resistivity sounding on a three layer earth comprising an inhomogeneous interstratum has been presented. The electrical conductivity in the intermediate layer, embedded between two layers of uniform conductivity, is assumed to follow either generalised power law or exponential variation. A point source of current is placed on the surface. The electric potential functions on the ground surface have been obtained. The expressions for apparent resistivity over the above models for the Wenner and Schlumberger electrode configurations have been derived.     

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.     

APPARENT RESISTIVITY OF A MULTILAYERED EARTH WITH A LAYER HAVING EXPONENTIALITY VARYING CONDUCTIVITY*

If the conductivity of any one of the layers of a horizontally stratified earth varies exponentially with depth with or without a discontinuity at the interface, the corresponding expressions for apparent resistivity for Wenner- and Schlumberger-sounding arrays can be formulated. The general case has been broadly divided into three categories for mathematical simplicity. All previous discussions of this problem can be regarded as particular cases of the present study.     

Behavior of electric potential fields over random conductivity,layered earth models

Solutions are derived for the potential distributions over one-layer and two-layer random conductivity earth models for the direct current resistivity method. Monopole and Wenner arrays are considered. It is shown that the random potential due to a spatially stationary Gaussian random conductivity is also Gaussian but not spatially stationary. The ensemble and sample statistics of the random kernel and apparent resistivity are examined. Representative curves of these functions are presented to show the effects of a random conductivity profile on them. It is shown that the effect of the random nature of the conductivity profile on an apparent resistivity sounding curve is greatest for small electrode spacings.     

随深度指数变化的视电阻率解

从横向均匀介质满足的基本方程出发,得到视电阻率核函数的一阶非线性微分方程,通过方程求解,并利用滤波系数法容易得到电阻率随深度任意变化的视电阻率问题。当各层介质电性结构随深度呈指数变化时,还可得到各层之间核函数的递推关系,这对实际介质的正反演问题都有重要意义和应用价值。     

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.     

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.     

VERTICAL FREQUENCY EFFECT SOUNDING IN INDUCED POLARIZATION AND GALVANIC RESISTIVITY METHODS *

It has been shown that for a polarizable layer as a transmission medium there is an indirect proportionality between the frequency effect (fe) in induced polarization (IP) and the wave number of the electrical induced field. Making use of this relationship for a two layered earth a polarization transform function has been obtained. Since the mathematical expression for the polarization transform function is the same as that of the resistivity transform function, it is possible to make direct interpretation for IP frequency effect field curves. Thus, AA or QQ type resistivity sequences can be interpreted from induced polarization response of a horizontally stratified earth without resistivity extrema. A depth factor has been defined in order to obtain the true depth using the apparent depth. In this way, some electromagnetic effects between horizontal layers with different polarizabilities can partly be eliminated.     

CSAMT全区电阻率法数值模拟及应用探讨

常规的可控源电磁法理论在计算视电阻率公式上,多半采用其电磁场的渐近特征,难以直接反映全区视电阻率的值,及直观地显现地下介质的地质构造.文中采用水平偶极子激发的电磁场,提出了电场的全区精确表达式,直接计算出大地电阻率.利用汉克尔数值滤波算法和逆样条插值算法对水平层状电磁场进行正演计算,并与计算的卡尼亚视电阻率的对比和野外试验结果表明：该方法的结果在远区等价卡尼亚电阻率,在近区和过渡带则明显地改善了卡尼亚电阻率的非波场区场畸变,从而能更好地接近基底的真电阻率,更形象地反映了地下介质的垂向电性变化.     

Definitions of apparent resistivity for the presentation of magnetotelluric sounding data1

A new definition of apparent resistivity for the presentation of magnetotelluric sounding data is proposed. The new definition is based on the frequency-normalized impedance function. Both the existing and proposed definitions of apparent resistivity are analysed theoretically and are compared using model curves computed for a 1D earth model. Apparent resistivity curves computed using the proposed definition are a better approximation to the true resistivity values of the subsurface layers. In addition, the layers are more noticeable on the apparent resistivity curves, which is an advantage, especially for the ascending and descending type of apparent resistivity curve.     

电导率随深度分层线性变化的大地电磁响应

本文在电导率随深度分层线性变化的地球模型上,研究了大地电磁的正演问题.由Airy微分方程导出了电场和磁场水平分量的解、地面阻抗的递推表达式和视电阻率的计算公式;并证明了分层线性模型与均匀层状模型一样,也存在高阻薄层的H等值性和低阻薄层的S等值性;总结了电导率随深度的变化率α,对大地电磁测深曲线的影响规律.     

A Method to Remove Electromagnetic Coupling from Induced Polarization Data for an "Exponential" Earth Model

--The electromagnetic (EM) coupling effect in induced polarization (IP) data is an important problem. In many works it has been computed only considering homogeneous or layered earth models with discretely uniform conductivity. In this study, an algorithm has been developed to compute the EM coupling effect in IP data measured on the earth, whose conductivity varies (increases or decreases) exponentially with depth. The EM coupling effects for Percent Frequency Effect (PFE) and phase data are computed for a dipole-dipole array with different separations, however the method can be applied to any electrode array. The results obtained for the cases of increasing and decreasing conductivity as a function of depth indicate that the EM coupling effect strongly depends on the subsurface resistivity and the dipole length. Here an "exponential" earth model is considered to remove EM coupling from the IP data in frequency and phase domain. For this purpose, first, the region of pseudo-section is divided into segments, and within each segment a typical average apparent resistivity („_a) curve is constructed. An exponential conductivity model is fitted to average „_a data. The conductivity model is then used to compute EM responses. Next the data are corrected for the EM coupling contribution. This decoupling process is applied to field data from a galenite-pyrite mineralization area at the Dolluk site, in western Turkey. The results from the decoupling method developed here are compared with other techniques.     

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.     

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.     

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.     

水平层状介质视电阻率的高精度计算公式及其误差分析

为适应地电阻率法预报地震深入研究的需要,本文介绍了几种高精度计算水平层状介质视电阻率的公式及方法。对于MN→0的视电阻率计算,本文给出了J_1(x)滤波系数法的计算步骤,此外对二层、三层介质的传统级数求和法,给出了估算精度的公式;对于MN≠O的视电阻率计算,本文讨论了数值积分法和J_0(x)滤波系数法的计算步骤以及为提高精度所采取的技术措施,并且给出了一批用两种方法所得到的计算结果。     

INDUCED POLARIZATION RESPONSE OF A HORIZONTALLY MULTILAYERED EARTH WITH NO RESISTIVITY CONTRAST*

The induced polarization response of a horizontally multilayered earth with no resistivity contrast can rapidly be calculated on a desk calculator or minicomputer for any electrode array. The formulation is a simple series summation of the products of weighting coefficients and the true induced polarization responses for each of the layers. The coefficients are directly derivable from the corresponding resistivity model. This series approach to IP formulation was originally described by Seigel but has not been treated extensively in the present-day geophysical literature. This method can be applied to either time or frequency domain induced polarization measurements. Once the coefficients are known, apparent induced polarization response can readily be obtained by judicious substitution of known, suspected, or assumed values of the true induced polarization of each layer. Basic formulation is presented for the IP potential coefficients (pole-pole or two array) with no resistivity contrast between the layers. From these coefficients, response of any number of layers for any electrode array can be obtained by suitable differentiation. Some examples of Wenner array for a three-layered earth and dipole-dipole array for a four-layered earth are used to illustrate the application. The results of this technique are valid for many natural situations of modest resistivity contrast. However, they definitely cannot be used if there are highly contrasting resistivity layers present. Such an approach is conceptually simple and is useful for survey planning, checking or setting the “depth-of-penetration”of a given array. For field induced polarization data that fits reasonably well to the no-resistivity-contrast model, this simple approach facilitates quantitative interpretation.     

Increased resolution of subsurface parameters from 1D magnetotelluric modeling

In electric and electromagnetic techniques, it is well known that the principle of equivalence poses a problem in the interpretation of subsurface layers. This means the inversion problem can provide the conductivity-thickness product more confidently than the individual parameters — conductivity and thickness — separately. The principle of equivalence corresponds to the middle layer in a three-layer earth structure. In order to resolve this problem, we have touched upon the different formulae of apparent resistivity proposed by earlier workers considering the real and imaginary parts of the impedance tensor and designed a new formula to compute apparent resistivity for different models. We observed that the application of our new formula for apparent resistivity using the combination of real and imaginary parts of the impedance has a better resolution as compared to earlier conventional formulae of apparent resistivity. These results have been demonstrated through both forward and inverse modeling schemes.     

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.     

Study on electric variations of media in epicentral area by geomagnetic transfer functions

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