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%.     

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.     

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.     

THE DIRECT INTERPRETATION OF RESISTIVITY OBSERVATIONS MADE WITH A WENNER ELECTRODE CONFIGURATION*

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.     

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.     

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.     

Models of electric conductivity in the upper mantle

Summary The accessible data of the apparent resistivity _T(T) were collected and supplemented by the spectral analysis of daily means of geomagnetic components observed at world geomagnetic observatories. The data were least-squares fitted; four models of the resistivity in the upper mantle which agree with the experimental data were obtained by solving the inverse problem using Marquardt's gradient-expansion algorithm.     

MATHEMATICAL MODELS OF CONDUCTING ORE BODIES FOR DIRECT CURRENT ELECTRICAL PROSPECTING *

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 R_i's are the distances from the point M to n fixed points S_i (i= 1,. n), k is a positive real constant and C_i, C_ii 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.     

TRANSFORMATION OF RESISTIVITY SOUNDING MEASUREMENTS OBTAINED IN ONE ELECTRODE CONFIGURATION TO ANOTHER CONFIGURATION BY MEANS OF DIGITAL LINEAR FILTERING*

The technique of digital linear filtering is used for transformation of apparent resistivity data from one electrode configuration into another. Usually filter spectra are determined via the discrete Fourier transforms of input and output functions: the filter characteristic is the quotient of the spectra of the output function and input function. In this paper, the transformation of the apparent resistivities is presented for four electrode configurations (Wenner, the two-electrode, Schlumberger, and dipole configurations). In our method, there is no need to use the discrete Fourier transform of the input and output functions in order to determine the filter spectrum for converting apparent resistivity in one electrode configuration to any other configuration. Sine responses for determination of the derivative of apparent resistivities are given in analytical form. If the filter spectrum for converting the apparent resistivity to the resistivity transform for one electrode configuration is known, the filter spectra for transforming the apparent resistivity to the resistivity transform for any electrode configurations can be calculated by using newly derived expressions.     

AN ELECTRICAL RESISTIVITY PROFILE IN HAWAII WITH NOVEL ELEVATION CORRECTION*

In order to locate relatively optimum sites for drilling exploratory holes for fresh water, an electrical resistivity survey was conducted along the new Mahukona-Kawaihae Road on the west flank of the Kohala Mountain. Two resistivity soundings made at the same stations, using the Schlumberger electrode configuration, determined an a spacing of 275 feet for horizontal profiling with the Wenner array. The correlation coefficient of the elevation to profile data was 0.41. A procedure for removing elevation effect from observed apparent resistivity was developed. Based on the reduced resistivity profile, four relatively optimum sites for additional exploration, such as by drilling, are specified. There is no specific interpretation of the data that can definitely indicate the occurrence of large underground reservoirs of fresh water anywhere along the profile. This is because the interpretation of horizontal profiling data is essentially relative and not absolute.     

MODEL STUDIES ON SOME ASPECTS OF RESISTIVITY AND MEMBRANE POLARIZATION BEHAVIOUR OVER A LAYERED EARTH*

Electrolytic model tank experiments to study resistivity and time domain induced polarization (IP) response over layered earth models were initiated primarily to facilitate the understanding of field results. Alternate layers of clay and sand (or clay-coated sand) with, in some cases, a surficial layer of water were assembled in the tank and resistivity and IP measurements made for a range of electrode spacings using the Wenner configuration. Graphite and silver-silver chloride electrodes were used as current and potential electrodes respectively. Clay-coated (3% by weight) sand was found to generate stronger polarization than either clay or sand alone. Apparent chargeability m_a was observed to be positive for a nonpolarizable surface layer. For a polarizable surface layer, the sign of IP was controlled by the polarizability, the thickness of the second layer, and the spacing of the electrode spreads. The apparent chargeability m_a can theoretically change sign from positive to negative and vice versa with a gradual increase in electrode spacing, and such negative IP effects were obtained in a few observations. A simultaneous decrease in IP and an increase in resistivity, which is a qualitative diagnostic feature for the occurrence of clean freshwater sand aquifers, could also be generated in the model tank experiment. Combined resistivity and IP soundings were carried out near Fredericton Junction and Tracy, New Brunswick, Canada. Field curves are presented along with the model curves for qualitative comparison and understanding of IP behaviour over a layered earth. Twenty-five out of twenty-seven soundings show only positive apparent chargeabilities, whereas two show chargeability sign changes (positive/negative/positive). The model study gives reason to believe that surface soils and Quaternary gravel boulder deposits near Fredericton Junction are relatively non-polarizable. As an auxiliary experiment, sand and clay were taken in different proportions by weight and mixed thoroughly with water in a cement mixer. The mixtures were then compressed with a suitable die and plunger under 3.6 Pa pressure to prepare cylindrical samples of height 18 cm and diameter 15.5 cm. IP measurements were done on the flat faces using the Wenner configuration with a= 2 cm. Chargeability was found to be negative for 100 and 90% clay mixtures. It reached a positive maximum for an 80% clay-20% sand mixture and then decreased gradually with increasing sand and decreasing clay content.     

Focused imaging of electrical resistivity data in archaeological prospecting

Electrical resistivity mapping and electrical resistivity profiling are powerful instruments for investigating archaeological structures. Interpretation of geoelectrical data is complicated by near-surface anomalies and the characteristics of the applied electrode arrays. Averaging Wenner α and Wenner β data as an alternative method of focused imaging is presented to overcome these problems. The mechanism of focused imaging is explained using the sensitivity distribution of the combined arrays. Various methods of imaging geoelectrical data are examined with synthetic and field data. In electrical resistivity mapping, inversion of the data is unnecessary when using focused imaging. In electrical resistivity profiling, focused imaging gives a first idea about the subsurface resistivity distribution without achieving the quality obtainable by inversion.     

THE USE OF TWO-ELECTRODE AND SCHLUMBERGER FILTERS FOR COMPUTING RESISTIVITY AND EM SOUNDING CURVES*

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.     

Subsurface DC resistivity mapping: approximate 1-D interpretation

Resistivity prospecting is the main tool used to investigate the shallow structure of the ground. A series of new techniques for determining the 2-D and 3-D geometry of the ground is now finding increasing use, but the light and simple Wenner prospecting technique remains a practical and efficient tool for rapidly mapping lateral variations in resistivity. When the resistivity changes are smooth, 1-D modelling can be used to interpret the data, and the criteria governing this approximation can be defined from synthetic data generated by a 3-D slab-model. For a Wenner array, two quadripole configurations can be used, Normal and Dipole-Dipole. For these two configurations the width of the transition zone, the apparent anisotropy effect and the precision of the resistivity values recovered from 1-D inversion differ. However the simultaneous inversion of both sets of data gives better results than for either configuration by itself. Two examples illustrate that in geological contexts where the thickness of the weathered zone causes the changes in the apparent resistivity value, this parameter can be recovered from 1-D inversion.     

A numerical comparison of 2D resistivity imaging with 10 electrode arrays

Numerical simulations are used to compare the resolution and efficiency of 2D resistivity imaging surveys for 10 electrode arrays. The arrays analysed include pole‐pole (PP), pole‐dipole (PD), half‐Wenner (HW), Wenner‐α (WN), Schlumberger (SC), dipole‐dipole (DD), Wenner‐β (WB), γ‐array (GM), multiple or moving gradient array (GD) and midpoint‐potential‐referred measurement (MPR) arrays. Five synthetic geological models, simulating a buried channel, a narrow conductive dike, a narrow resistive dike, dipping blocks and covered waste ponds, were used to examine the surveying efficiency (anomaly effects, signal‐to‐noise ratios) and the imaging capabilities of these arrays. The responses to variations in the data density and noise sensitivities of these electrode configurations were also investigated using robust (L₁‐norm) inversion and smoothness‐constrained least‐squares (L₂‐norm) inversion for the five synthetic models. The results show the following. (i) GM and WN are less contaminated by noise than the other electrode arrays. (ii) The relative anomaly effects for the different arrays vary with the geological models. However, the relatively high anomaly effects of PP, GM and WB surveys do not always give a high‐resolution image. PD, DD and GD can yield better resolution images than GM, PP, WN and WB, although they are more susceptible to noise contamination. SC is also a strong candidate but is expected to give more edge effects. (iii) The imaging quality of these arrays is relatively robust with respect to reductions in the data density of a multi‐electrode layout within the tested ranges. (iv) The robust inversion generally gives better imaging results than the L₂‐norm inversion, especially with noisy data, except for the dipping block structure presented here. (v) GD and MPR are well suited to multichannel surveying and GD may produce images that are comparable to those obtained with DD and PD. Accordingly, the GD, PD, DD and SC arrays are strongly recommended for 2D resistivity imaging, where the final choice will be determined by the expected geology, the purpose of the survey and logistical considerations.     

Regional study of the anomalous change in apparent resistivity before the Tangshan earthquake (M=7.8, 1976) in China

Electrical resistivity measurements have been conducted as a possible means for obtaining precursory earthquake information. Before five great earthquakes (M>7,h<25 km) in China, the apparent resistivity _a showed systematic variations within a region 200 km from the epicenters. In particular, 9 stations in the Tangshan-Tianjin-Beijing region prior to the Tangshan earthquake (M=7.8,h=11 km, 27 July 1976) showed a consistent decrease of apparent resistivity around the epicenter, with a maximum resistivity change of 6% and a period of variation of 2–3 years. Simultaneous water table observations in this region showed a declining water table, and ground surface observations indicated a slight (5 mm) uplift in the epicenter region relative to its surroundings.In order to develop an explanation for the observed change of apparent resistivity associated with these great earthquakes, we have used Archie's Law to explore the effects of changes in rock porosity, water content and electrolyte resistivity on measured resistivity.Tentative conclusions of this study are as follows: (1) the apparent resistivity change is opposite to the effect expected from the simultaneous water table trend; (2) the dilatancy needed to give such resistivity variations (assuming Archie's Law holds) is much larger than that needed to explain the observed uplift (by 2–3 orders of magnitude); (3) salinity change in the pore electrolyte is a possible explanation for the variation in resistivity: an increase in the salinity would cause a proportional decrease in resistivity; the data needed to test this hypothesis, however, are lacking; and (4) the effect of changing geometry of rock pores or cracks due to pressure solution may provide an explanation for the decrease in apparent resistivity; it is different in nature from the effect of a volume change in response to stress although the geometry change is also closely related to the stress change.     

NUMERICAL INTERPRETATION OF RESISTIVITY SOUNDINGS OVER HORIZONTAL BEDS*

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.     

Su unaVexata quaestio di gerarchia nei dispositivi dei sondaggi elettrici

Riassunto L'A. mostra corne dal metodo più semplice di misura di rcsistività d'un suolo del polo singolo, si possano dedurrc tutti gli altri (Wenner, Schlumberger, ecc). CiÒ dà la possibilità di risalire immediatamente dai grafici _s di un dispositivo a quelli di un altro. L'A., dopo rettificato recenti valutazioni del metodo dei due punti, rimuove la pretesa da altri formulata di porre gerarchie nei dispositivi di misure _s .

---

Summary The Author shows how departing from simplest method of the ground resistivity measurement with the single pole can be deduced all another methods (Wenner, Schlumberger, etc.). That gives the practical possibility to pass immediately the _s graphics of one arrangement to these of other types of arrangement. After having rectified recently mades evaluations about the two points method, the Author removes the requirements made from other side to graduate the _s measuring arrangements.

---

Zusammenfassung Der Verfasser zeigt wie man von der einfachsten Methode der Widerstandsmessung eines Bodens mit Einzelpol alle anderen Methoden (Wenner, Schlumberger, usw.) ableiten kann. Dadurch hat man die praktische Möglichkeit, von den _s Diagrammen einer Anordnung sofort auf jene anderer Anordnungen überzugehen. Nach Richtigstellung von kürzlich geäusserten Ansichten über die Zweipunktmethode, beseitigt der Verfasser die von Anderen gestellten Ansprüche, die _s Messanordnungen vermeintlichen Vorzügen nach staffeln zu wollen.

---

---

Relazion e presentata il 5 Aprile 1956 alla Quarta Assemblea Generale dcllaSocietà Italiana di Geofisica e Meteorologia (Genova: 5–8 Aprile 1956).     

Approximate Conditions for the Off-Axis Triplication in Transversely Isotropic Media

The presented approximate formulas yield a critical value of anisotropy parameter , for which an incipient off-axis SV-wave triplication occurs in transversely isotropic media. The formulas are simple but approximate the exact solution with a high accuracy. The best results are obtained using the third-order approximation, which yields accuracy at least 30 times higher than the formulas presented by Thomsen and Dellinger (2003). The formula works safely for parameters  = a ₃₃/a ₄₄ > 2 and 0.2 >  = (a ₁₁ – a ₃₃)/2a ₃₃ > –0.2, and yields critical values of from 0.1 to 0.7. Outside this interval, it is recommended to use an exact solution.     

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。