The Application of Linear Filter Theory to the Direct Interpretation of Geoelectrical Resistivity Sounding Measurements*

GHOSH, D. I’., 1971, The Application of Linear Filter Theory to the Direct Interpretation of Geoelectrical Resistivity Sounding Measurements, Geophysical Prospecting 19,192.217. Koefoed has given practical procedures of obtaining the layer parameters directly from the apparent resistivity sounding measurements by using the raised kernel function H(A) as the intermediate step. However, it is felt that the first step of his methodnamely 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(h), a function directly related to H(A), from the resistivity field curve. It is shown that the apparent resistivity and the re,<istivity 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. Sepz,rate 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 z %. INTRODUCTION Direct interpretation of geoelectrical sounding resistivity measureme:zts is carried out in two steps. In the first step the kernel function B(A) (see eq. (I)) or some modified version of this function is evaluated from the field measurements, and in the second step the layer parameters (resistivites pi and thicknesses dg) are derived from it. The logic behind this approach is that the kernel function is dependent only on the layer parameters, and an expression relating it to the field measurements can be obtained by mathematical processes. These suggestions (Slichter 1933, Pekeris rgdo)-although made about four decades ago-found relatively little application, mainly because of the nonavailability of a simple procedure to compute the kernel. Consequently inter* Paper read at the 3znd Meeting of the European Association of Exploration Geophysicists at Edinburgh, May 1970. ** Geophysical Laboratory of the Technological University at Delft, now with Benares Hindu University, Varanasi U.P., India. APPLICATION OF LINEAR FILTER THEORY TO RESISTIVITY SOUNDING 193 preters resorted to indirect techniques like the comparison of observed apparent resistivity (p,) curves with precalculated curves for known earth models. Almost a quarter of a century elapsed before interest in direct interpretational methods was revived. The credit goes to Koefoed (1968) who gave practical procedures to directly interpret the apparent resistivity field curves, along the lines of interpretation suggested earlier. For the intermediate step, however, he chose to use the raised kernel function N(h) instead of B(A). He showed that relative variations in the apparent resistivity were not of the same order of magnitude in the corresponding kernel curve, and thus any method based on the determination of this function as the intermediary step would lead to loss of information and hence to incorrect interpretation. Koefoed (1970) subsequently modified the second step of his method by introducing standard graphs to accelerate the derivation of the layer distribution from the resistivity transform T(h), a function directly related to H(h). It is, however, felt that Koefoed’s first step is relatively lengthy which could be of disadvantage to the application of direct methods in resistivity interpretation. In this paper therefore an attempt is made to devise a simple and quick procedure to obtain the resistivity transform from the apparent resistivity field curve. An analysis of the properties of the T function given below demonstrates the suitability of this function as the intermediary step in direct interpretation : I. It is solely determined by the layer distribution. 2. It is an unambiguous representation of the pa function. 3. For small and large values of I/A it approaches the pa curve. It will be shown that the pa and T functions are linearily related to each other such that the principle of electric filter theory could be applied to derive the T curve from the pa curve. Swartz and Sokoloff (rg54), Dean (rg58), Robinson and Treitel (1964) have all demonstrated the applicability of electric filter theory to solve physical problems which are linear in nature. One essential difference that might be stressed here is that electric or “real” time filters depend on the excitation of energy at t = o to produce an output and thus can have only finite responses for t 3 o, whereas in filters used for data processing or to handle physical ,problems there can be responses of the filter for both positive and negative values of the independent variable. The implication of this is that in electric filters the output depends only on the present and past values of the input, whereas in the latter case the output in addition depends also on the future values of the input (Dean 1958, Robinson and Treitel 1964).