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1.
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.  相似文献   

2.
In this paper a numerical method of direct interpretation of geoelectrical soundings is described. It has similarities with already existing direct methods, but owing to its simplicity and, in particular, to the possibility of applying it also without digital computers, it proves useful mostly in the field, where very often an accurate method for the interpretation of multi-layer curves is required. The direct interpretation system splits up into three steps: i) the evaluation of the resistivity transform after application of Hankel's inversion theorem; ii) the determination of the layer distribution after application of Koefoed's recurrent procedure; iii) the control of the solution. Each step is considered and the practical procedures suggested. Finally two field examples are presented and discussed.  相似文献   

3.
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.  相似文献   

4.
Two-layer type curves of apparent frequency effect for the Wenner configuration are presented. The formulation is based on the normal definition of frequency effect in terms of resistivities measured at different frequencies plus the definition of apparent resistivity over two horizontal layers as a function of first and second layer resistivities. The use of these type curves in the interpretation of multilayer apparent frequency-effect curves is described and some field examples are given.  相似文献   

5.
Following up our recent study of an indirect procedure for the practical determination of the maximum frequency-effect, defined as fe = 1 ? pρ∞/ρdc with ρ∞ the resistivity at infinite frequency, we show at first how, through the Laplace transform theory, ρ∞ can be related to stationary field vectors in the simple form of Ohm's law. Then applying the equation of continuity for stationary currents with a suitable set of boundary conditions, we derive the integral expression of the apparent resistivity at infinite frequency ρ∞,a in the case of a horizontally layered earth. Finally, from the definition of the maximum apparent frequency-effect, analytical expressions of feα are obtained for both Schlumberger and dipole arrays placed on the surface of the multi-layered earth section in the most general situation of vertical changes in induced polarization together with dc resistivity variations not at the same interfaces. Direct interpretation procedures are suggested for obtaining the layering parameters directly from the analysis of the sounding curves.  相似文献   

6.
The digital computer technique described for interpreting resistivity soundings over a horizontally stratified earth requires two steps. First, the kernel function is evaluated numerically from the inverse Hankel transform of the observed apparent resistivity curve. Special attention is given to the inversion of resistivity data recorded over a section with a resistant basement. The second step consists in the least-squares estimation of layer resistivities and thicknesses from the kernel function. For the case of S or T-equivalent beds only one layer-parameter can be obtained, either the longitudinal conductance, or the transverse resistance respectively. Two examples given in the paper show that a wide tolerance is permitted for Choosing the starting values of the layering parameters in the successive approximation procedure. Another important feature for practical applications is good convergence of the iterations. The method is probably best suited for interpreting profiles of electrical soundings with the purpose of mapping approximately horizontal interfaces at depth.  相似文献   

7.
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.  相似文献   

8.
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.  相似文献   

9.
The interpretation of vertical electrical sounding data can be facilitated by the application of the reciprocal geoelectric section. If an apparent resistivity field curve has a descending right end, the apparent resistivity curve of the reciprocal geoelectric section can be obtained by the application of linear filter theory; from this the total transverse resistance of the geoelectric section can be calculated without having to interpret the field curve. In addition, Orellana's auxiliary point method can now be extended to interpret three and four layer apparent resistivity curves of all types. This paper summarizes the properties of the resistivity transform curve, the apparent resistivity curve, and the apparent resistivity curve of the reciprocal geoelectric section, with several new applications.  相似文献   

10.
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.  相似文献   

11.
It is shown how to interpret, without curve-matching, Schlumberger resistivity soundings carried out with the array parallel to the surface trace of a vertical boundary plane separating two media with different resistivities (the vertical fault problem). To this end, it is demonstrated that the apparent resistivity function can be expressed as an Hankel-transformable integral that allows to invert the apparent resistivity curve into an associated resistivity transform curve. The study of the asymptotical properties of this function and of some mathematical properties of a related reduced transform function allows to realize a simple procedure for deriving the parameters of the model. In practice, this procedure consists in fitting a straight-line to a semi-logarithmic plot of the reduced transform function and in evaluating the intercept along the vertical axis and the slope.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
In a previous paper it has been shown that we can relate the transient IP electric field Ep , existing in a rock after a step wave of polarizing current, with the steady-state current density Jss during the current step wave as follows: Ep =ρ' Jss . This relation may be interpreted as a generalized Ohm's law, valid in linear cases, in which ρ’(fictitious resistivity) is defined as the product of the true resistivity ρ with the chargeability m. Supposing E p=— grad Up and applying the divergence condition div Jss = o, one can, for a layered earth, obtain a general expression for the depolarization potential Up as a solution of Laplace's equation ?2Up= o. Since the mathematical procedure for the solution of this last equation is identical to that used in resistivity problems, we propose now the introduction of an apparent fictitious resistivity ρ'a (defined as the product of the apparent resistivity ρa with the apparent chargeability ma) as a new parameter for the interpretations of IP soundings carried out over layered structures with a common electrode array. The most general expression of ρ'a as a function of the electrode distance turns out to be mathematically identical to the general expression of ρ'a. Therefore it is possible to interpret a ρ'a field curve using the same standard graphs for resistivity prospecting with the usual method of complete curve matching. In this manner a great deal of work is saved since there is no need to construct proper ma graphs for the interpretation of IP soundings, as it has been done up to now. Finally some field examples are reported.  相似文献   

15.
The polarization content of a medium, in both the time and frequency domains, can be described by parameters which differ in inherent physical meaning and their practical significance. For real situations, general expressions for the apparent parameters, previously determined for both domains, exist for the general case of soundings on a horizontally multi-layered earth. The comparative analysis of these expressions, here restricted to the simple case of a two-layered earth, shows that the theoretical sounding curves of the frequency-domain are different from those of the time-domain. In particular, for every resistivity or chargeability contrasts examined, the apparent frequency-effect curve lies always over the corresponding apparent chargeability curve, but both curves reach the same asymptotical values for shortest and largest spacings. The important conclusions which can be drawn from this result is that both techniques are suitable to investigate subsoil polarizability anomalies. However, from a practical point of view, it is more convenient to adopt the frequency-domain technique when the polarizability increases with depth, while, on the contrary, the time-domain technique is more efficacious when the polarizability decreases with depth.  相似文献   

16.
In this paper it is shown how one may obtain a generalized Ohm's law which relates the induced polarization electric field to the steady-state current density through the introduction of a fictitious resistivity defined as the product of the chargeability and the resistivity of a given medium. The potential generated by the induced polarization is calculated at any point in a layered earth by the same procedure as used for calculating the potential due to a point source of direct current. On the basis of the definition of the apparent chargeability ma, the expressions of ma for different stratigraphie situations are obtained, provided the IP measurements are carried out on surface with an appropriate AMNB array. These expressions may be used to plot master curves for IP vertical soundings. Finally some field experiments over sedimentary formations and the quantitative interpretation procedure are reported.  相似文献   

17.
In a previous paper by the present author a method was developed for direct interpretation of resistivity observations made with a Schlumberger electrode configuration. This method consisted of two steps. The first of these was to derive the kernel function in the integral expression for the apparent resistivity from the observed data; the second step was to derive the resistivity stratification from this kernel function. The first of these two steps depends on the electrode configuration that has been used. In the present paper the above mentioned method is modified so as to make it apply to a Wenner electrode configuration. The procedure is indicated by which the method may be adapted to any other electrode configuration in which the distances between the electrodes are finite. The second step in the interpretation, i.e. the derivation of the resistivity stratification from the kernel function, is independent of the electrode configuration used, and therefore needs no further discussion in the present context.  相似文献   

18.
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.  相似文献   

19.
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.  相似文献   

20.
For the two and three layer cases geo-electrical sounding graphs can be rapidly and accurately evaluated by comparing them with an adequate set of standard model graphs. The variety of model graphs required is reasonably limited and the use of a computer is unnecessary for this type of interpretation. For more than three layers a compilation of model graphs is not possible, because the variety of curves required in practice increases immensely. To evaluate a measured graph under these conditions, a model graph is calculated by computer for an approximately calculated resistivity profile which is determined, for example, by means of the auxiliary point methods. This model graph is then compared with the measured curve, and from the deviation between the curves a new resistivity profile is derived, the model graph of which is calculated for another comparison procedure, etc. This type of interpretation, although exact, is very inconvenient and time-consuming, because there is no simple method by which an improved resistivity profile can be derived from the deviations between a model graph and a measured graph. The aim of this paper is, on the one hand, to give a simple interpretation method, suitable for use during field work, for multi-layer geo-electrical sounding graphs, and, on the other hand, to indicate an automatic evaluation procedure based on these principles, suitable for use by digital computer. This interpretation system is based on the resolution of the kernel function of Stefanescu's integral into partial fractions. The system consists of a calculation method for an arbitrary multi-layer case and a highly accurate approximation method for determining those partial fractions which are important for interpretation. The partial fractions are found by fitting three-layer graphs to a measured curve. Using the roots and coefficients of these partial fractions and simple equations derived from the kernel function of Stefanescu's integral, the thicknesses and resistivities of layers may be directly calculated for successively increasing depths. The system also provides a simple method for the approximative construction of model graphs.  相似文献   

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