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1.
I. Introduction In this section the problem is stated, its physical and mathematical difficulties are indicated, and the way the authors try to overcome them are briefly outlined. Made up of a few measurements of limited accuracy, an electrical sounding does not define a unique solution for the variation of the earth resistivities, even in the case of an isotropic horizontal layering. Interpretation (i.e. the determination of the true resistivities and thicknesses of the ground-layers) requires, therefore, additional information drawn from various more or less reliable geological or other geophysical sources. The introduction of such information into an automatic processing is rather difficult; hence the authors developped a two-stage procedure:
- a) the field measurements are automatically processed, without loss of information, into more easily usable data;
- b) some additional information is then introduced, permitting the determination of several geologically conceivable solutions.
2.
D. P. GHOSH 《Geophysical Prospecting》1971,19(2):192-217
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%. 相似文献
3.
D. PATELLA 《Geophysical Prospecting》1972,20(3):561-579
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
4.
L. MANSINHA 《Geophysical Prospecting》1984,32(6):1155-1166
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. 相似文献
5.
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
6.
AN INTERACTIVE COMPUTER/GRAPHIC-DISPLAY-TERMINAL SYSTEM FOR INTERPRETATION OF RESISTIVITY SOUNDINGS*
H. K. JOHANSEN 《Geophysical Prospecting》1975,23(3):449-458
A fast computer-procedure giving the apparent resistivity curve as well as the partial derivatives with respect to the layer-parameters is presented. It is based on the linear filter method developed by D. P. Ghosh in 1971. The sampling frequency is 10 points per decade, and 3 decades are covered. The maximum relative error is less than 10?3, and in most cases orders of magnitude smaller. The computation time on a CDC 6400 for one curve given in 30 points ranges linearly from .17s for a two-layer case to .36s for a ten-layer case. The procedure is used to plot master curves interactively on a graphic display terminal (Tektronix 4010) connected to the CDC 6400. By trial-and-error adjustments a set of layer-parameters is found, giving essentially the measured curve. 相似文献
7.
Abstract Geoelectric resistivity measurements by means of direct current for solving hydrogeological problems have become increasingly significant in recent years. Measurements on the surface according to the four-point-method (SCHLUMBERGER or WENNER arrangement) result in so-called “apparent” resistivities ? α as a function of the electrode distance L. The evaluation of these measuring data, here in form of sounding graphs ? α(L/2), consists of the determination of true resistivities as a function of the depth z. Since a direct computation of ? (z) from α (L/2) is not possible in practice, theoretically computed master curves constitute the essential auxiliary means for the evaluation. New simplified calculation techniques allow to establish accurately computed master curves for an underground consisting of more than three layers. By means of such standard graphs special problems of hydrogeology can quantitatively be solved by applying geoelectrical methods. The procedure is demonstrated on hand of complicated cases of aquifers devided into several storeys. 相似文献
8.
In this paper an idealized valley of a semi-elliptic cross-section is considered. For a Schlumberger configuration on the axis, sets of master curves are calculated for the ratio of semi-axis a/b= 1, 2, 3, 4, 5 in corresponding to various resistivity ratios of surrounding rocks and valley sediments. For small resistivity ratios, these model curves have the shape of three-layer curves for horizontal bedding and are often equivalent to them within the accuracy of measurements. The axial ratio a/b considerably affects the depth determination of valley sediments. In the special case of a circular cross-section (a/b= 1) the influence of the position of the electrodes on the sounding curve is studied in more detail. The application of the master curves in practice shows that the influence of the specific shape of the valley on soundings should not be neglected. In general, the valleys have a greater “true” depth than can be seen from the interpretation of a sounding by master curves corresponding to the horizontal bedding. 相似文献
9.
E. MUNDRY 《Geophysical Prospecting》1984,32(1):124-131
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. 相似文献
10.
P. HUBRAL 《Geophysical Prospecting》1974,22(4):722-735
The implementation of a stacking filter involves the filtering of each trace with an individual filter and the subsequent summing of all outputs. The actual position of a trace in space as well as certain simultaneous shifts of traces and filter components in time do not influence the process. The resulting output is consequently invariant to various arbitrary coordinate transformations. For a certain useful class of ensembles of non-linear moveout arrival times for signals a particular transformation can be found which transforms a given ensemble into one consisting only of straight lines. It is thus possible to reduce, for instance, the analysis of a stacking filter designed for hyperbola-like moveout curves to the analysis of a velocity filter with linear moveout curves. As the (f—k) transform is a very useful concept to describe a velocity filter, it can consequently be applied to characterize a stacking filter in regard to its performance on input signals with non-linear moveout. 相似文献
11.
The authors generalize a method expounded in a previous paper (1971, Geoph. Prosp. 18, 786-799) to the case of a local conductivity σ(M) of the infinite medium satisfying the relation where the Ri's are the distances from the point M to n fixed points Si (i= 1,. n), k is a positive real constant and Ci, Cii are constants ensuring the condition α > O. The sub-surface conductivity distributions (half-spaces) complying with (1) provide a wide variety of conducting structures, which can fit quite successfully the rather complicated distributions of conductivity occurring in natural ore bodies. An exact algebraic calculation of the apparent resistivity for these grounds, valid for any dc electrical prospecting devices (Wenner, Schlumberger, dipole, etc.) leads to a set of simultaneous linear equations, with a matrix which is invariant with respect to the position of the quadrupole being used. This greatly simplifies the numerical computation. We also present some examples of cross sections for the real and apparent resistivity obtained by this method. 相似文献
12.
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
13.
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
14.
The MT method has proved to be a useful tool for the resolution of some geophysical problems not well adapted to seismic prospecting. To meet the accuracy needed by such cases, the original field and processing techniques has to be somewhat improved. The processing of the natural EM field (with two electrical and three magnetic components recorded) is made in the frequency domain, and the usual MT parameters are calculated. However, the dispersion of the results is given a great importance and the final plots (apparent resistivity vs. frequency, azimuths of the main directions vs. frequency, etc.) illustrate the distribution of the calculated values. Average curves can then be selected and studied by means of the known master curves. However, if the earth can be assumed to consist of parallel layers, a computer program derives from the apparent resistivity diagram an automatic solution consisting of a great number of layers with the same ratio of thickness to square root of resistivity. This comprehensive solution is then simplified through an auxiliary diagram called the MT Dar Zarrouk curve and a set of geologically consistent solutions can be found. 相似文献
15.
A. T. BAOKUR 《Geophysical Prospecting》1984,32(1):132-138
Different sets of filter coefficients for the linear filter technique for the computations of resistivity and EM sounding curves are evaluated for several electrode and coil configurations. Instead of this procedure, the two-electrode filter can be used for computations of Wenner, Schlumberger, and dipole—dipole apparent resistivity model curves by defining convolutional expressions which contain the new input functions in terms of the resistivity transform function. Similarly, the Schlumberger filter performs the computations of dipole—dipole apparent resistivity model curves. The Wenner, Schlumberger, and dipole—dipole filter functions are defined in terms of the two-electrode filter using the new convolutional expressions. A relationship between the Schlumberger and dipole—dipole filter functions is given. The above arguments are adopted for the computations of EM sounding curves. It is shown that the EM filter for the horizontal coplanar loop system (which is identical to the two-electrode filter) performs the computations of the mutual coupling ratios for perpendicular, vertical coplanar, and vertical coaxial loop systems. In the same way, the Schlumberger filter can be used to compute vertical coaxial sounding curves. The corresponding input functions are defined in terms of the EM kernel for all convolutional expressions presented. After these considerations, integral expressions of the mutual coupling ratios involving zero-order Bessel function are derived. The mutual coupling ratio for the vertical coaxial loop system is given in the same form as the mutual coupling ratio for the vertical coplanar loop system. 相似文献
16.
17.
S. A. Greenhalgh L. Marescot B. Zhou M. Greenhalgh T. Wiese 《Pure and Applied Geophysics》2009,166(4):673-699
In this paper we develop analytic solutions for the electric potential, current density and Fréchet derivatives at any interior
point within a 3-D transversely isotropic medium having a tilted axis of symmetry. The current electrode is assumed to be
on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any
azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are
strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity
direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken
perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse
resistivity and dip angle of the symmetry axis (dG/dσl, dG/dσt, dG/dθ0) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt
angle. Such sensitivity patterns are useful in real-time experimental design as well as in the fast inversion of resistivity
data collected over an anisotropic earth. 相似文献
18.
This paper highlights the use of a conceptual method for separating runoff components in daily hydrographs, contrary to the traditionally used graphical method of separation. In the conceptual method, the components, viz. surface flow, interflow and baseflow, are regarded as high, medium and low frequency signals and their separation is done using the principle of a recursive digital filter commonly used in signal analysis and processing. It requires estimates of the direct runoff (βd) and surface runoff (βs) filter parameters which are obtained by a least‐squares procedure involving baseflow and interflow indices based on graphical and recursive digital filter estimation techniques. The method thus circumvents the subjective element associated with the graphical procedure of hydrograph separation, in which case the eye approximation and/or one's skill at plotting is the prime basis for the whole analysis. The analysis based on three forest catchments in Kimakia, Kenya, East Africa, revealed that βd=Kb and βs=Ki , where Kb and Ki are the baseflow and interflow recession constants. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
19.
Lszl Szarka 《Geophysical Prospecting》1997,45(5):763-774
In this paper the locations where ρapp = ρ1 and ? = π/4 and where these parameters reach an extreme value in two-layer magnetotelluric (MT) sounding curves are summarized in an extremely compact form. The key parameters over two-layer models with conductivities σ1, σ2 and upper layer thickness h are the real S and α, where S is the conductivity contrast and α is the distance between the observation site and the conductivity interface, normalized to the half skindepth in the first layer. If the impedance components, various resistivity definitions ( ρRe Z, ρIm Z and ρ|Z|, based on different parts of the complex impedance Z ) and the magnetotelluric phase ? are derived as a function of S and α, then the conditions for the apparent resistivity ρapp and the phase ? are that they either satisfy ρapp = ρ1 and ? = π/4 or attain extreme values which can be given in terms of simple algebraic equations between S and α. All equations are valid for observation sites at any depth 0 ≤ z ≤ h in the first layer. The set of equations, presented in a tabular form, may make it possible to determine a layer boundary from the short period part of the sounding curves, in particular the ρRe Z and the ?MT curves. 相似文献
20.
E. SZARANIEC 《Geophysical Prospecting》1976,24(3):528-548
The potential distribution and the wave propagation in a horizontally stratified earth is considered and the analogy of the mathematical expression for seismic transfer function, electromagnetic and electric kernel functions, and magnetotelluric input impedance is discussed. Although these specific functions are conveniently treated by a separate expression in each method, it is indicated that the function for seismic and electromagnetic methods is mathematically the same with a change in the physical meaning of the variables from one method to the other. Similarly, the identity of the mathematical expressions of the resistivity kernel function and magnetotelluric input impedance is noticed. In each method a specific geophysical function depends on the thickness and the physical properties of the various layers. Every specific function involves two interdependent fundamental functions, that is Pn and Qn, or Pn and P*n, having different physical meaning for different methods. Specific functions are expressible as a ratio Pn/Qn or P*n/Pn. Fundamental functions may be reduced to polynomials. The fundamental polynomials Q*n and P*n describing the horizontally stratified media are a system of polynomials orthogonal on the unit circle, of first and second order, respectively. The interpretation of geophysical problems corresponds to the identification of the parameters of a system of fundamental orthogonal polynomials. The theorems of orthogonal polynomials are applied to the solution of identification problems. A formula for calculating theoretical curves and direct resistivity interpretation is proposed for the case of arbitrary resistivity of the substratum. The basic equation for synthetic seismograms is reformulated in appendix A. In appendix B a method is indicated for the conversion of the seismic transfer function from arbitrary to perfectly reflective substratum. 相似文献