THE ANALYSIS AND DESIGN OF TWO-DIMENSIONAL FILTERS FOR TWO-DIMENSIONAL DATA *

Some of the methods such as regional removal and second derivative calculations which can be used to outline anomalies on potential data maps can be thought of as a filtering operation. The analysis and design of such two-dimensional filters by means of direct and inverse two-dimensional Fourier transforms have been considered. An analysis of several published sets of second derivative coefficient sets indicates that, in general, they are not a good approximation to the theoretical second derivative filter. Alternate methods of designing regional removal and second derivative filters are discussed. The properties of various two-dimensional filters are further illustrated by means of maps obtained from the convolution of several of these filters with a set of observed field data. These maps show the large changes in anomaly shape which can result from the inclusion or rejection of various wavelength components.     

INVERSION OF SURFACE AND BOREHOLE ELECTROMAGNETIC DATA FOR TWO-DIMENSIONAL ELECTRICAL CONDUCTIVITY MODELS*

A method for inverting electromagnetic fields induced by a line source in an earth of two-dimensional conductivity structure is developed. Certain unique features of the finite element method are used to construct an efficient algorithm for the accurate calculation of the Jacobian matrix of partial derivatives, and the resulting linearized equations are solved using the damped least squares method. Case studies of theoretical data generated from a simple model of interest in geophysical prospecting show that, in general, it is impossible to obtain, from surface data alone, accurate estimates of the conductivity of structures buried deeper than 0.2 skin depths under a conducting overburden. The addition of borehole data to the surface data is found to increase the resolving power of the electromagnetic method dramatically. In particular, the borehole data appear to stabilize the inverse when only a poor initial estimate of the likely structure is given.     

A GENERAL TECHNIQUE FOR THE DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

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.     

A NOTE ON THE LINEAR FILTER METHOD OF INTERPRETING RESISTIVITY SOUNDING DATA*

In the linear filter method of interpreting resistivity sounding data, as developed by Ghosh (1971), it appears that the filter function in the x-domain approaches an oscillating function for both large positive and large negative abscissa values. In the present note the reason for this oscillating behaviour is derived, and a possible practical application is indicated.     

A NOTE ON THE USE OF A NOMOGRAM FOR SELF-POTENTIAL ANOMALIES*

A nomogram has been devised for situations, in which the source of a self-potential anomaly can be approximated by an obliquely polarized sphere or horizontal cylinder embedded in a homogeneous half space. The nomogram can be used for rapid determination of three parameters of the target: (1) depth to the centre, (2) angle between the axis of polarization and the horizontal, (3) shift of the point vertically above the centre of the body from zero potential value. The nomogram has been tested and the parameters determined for SP results obtained over ore bodies Weiss and Süleymanköy in the Ergani Copper district, Turkey. The curves computed for the estimated parameters match the field curves well.     

A NOTE ON MAGNETIZED SPHERES*

The interpretation of total field anomalies becomes somewhat complicated, especially when an arbitrarily magnetized spherical ore mass happens to be the causative body. Even though some attempts have been made to analyze total field anomaly maps, they are often too complicated and their underlying assumptions in respect of permanent and induced components of magnetism are far from realistic. In this note, an attempt has been made to show that vertical magnetic anomalies are capable of yielding interpretation with ease and precision as far as magnetized spheres are concerned. An empirical method has been outlined for computing the magnetization inclination in the plane of the profile using the measured distances between principal maximum, principal minimum, and zero anomaly positions on a magnetic anomaly profile.     

A NOTE ON THE USE OF NOMOGRAMS FOR APPARENT RESISTIVITY COMPUTATION*

Nomograms are presented for the purpose of rapid computation of the apparent resistivity functions from the field data for symmetrical and unsymmetrical linear electrode arrays.     

A METHOD FOR INTERPOLATION AND EXTRAPOLATION OF POTENTIAL FIELD DATA*

The application of automatic methods to the processing of potential field survey results has led to the need for the development of special procedures for the interpolation and extrapolation of such data. A method is proposed which utilizes the mathematical properties of potential field data to achieve a minimum distortion solution to the interpolation/extrapolation problem. The proposed method is suitable for use with any data point distribution. Digital computer programs have been prepared using the method; results obtained through the use of the programs are presented.     

DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

A direct interpretation scheme is developed which is capable of determining most of the geological features of a ground which can be assumed to be two dimensional in structure. This scheme extends the earlier work of Pekeris (1940) and Koefoed (1968) to the case where the basal layer of a ground is undulating. It also has a limited use for finding the parameters of a dipping dyke in the lower medium. Though the top and dip of the dyke can be determined, this is not true for the thickness.     

RESISTIVITY MODELLING FOR ARBITRARILY SHAPED TWO-DIMENSIONAL STRUCTURES*

A numerical technique is developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations by using point- as well as area-discretization of the subsurface. Potential distributions at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct explicit matrix inversion techniques. An empirical mixed boundary condition is used at the “infinitely distant” edges of the lower half-space. Accurate solutions using area-discretization method are obtained with significantly less attendant computational costs than with the relaxation, finite-element, or network solution techniques for models of comparable dimensions.     

GEOELECTRICAL MODEL CALCULATIONS FOR TWO-DIMENSIONAL RESISTIVITY DISTRIBUTIONS*

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.     

ON THE INTERPRETATION OF AIRBORNE ELECTROMAGNETIC DATA FOR THE TWO-LAYER CASE*

For helicopter-borne electromagnetic systems, the distance between the transmitting and the receiving coils is small compared with the altitude above ground. For this case, a major simplification can be made for the calculation of model curves. Some two-layer curves for the interpretation of frequency measurements are presented. A very simple procedure is demonstrated for the conversion of the relative secondary field into apparent resistivity and apparent distance for the mapping of airborne electromagnetic data. Furthermore, an approximation is described for the determination of the thickness and the resistivity of a layer lying on a perfectly conducting half-space.     

GEOELECTRIC AXIAL DIPOLE SOUNDING CURVES FOR A CLASS OF TWO-DIMENSIONAL EARTH STRUCTURES*

With the aim of studying the behaviour of geoelectric axial dipole vertical soundings over complex geology, a systematic theoretical approach is presented for a class of earth structures characterized by horizontal and vertical parallel boundary planes. The two-dimensional cylindrical bodies of infinite length and rectangular cross-section are constrained to have resistivities satisfying Alfano's condition at every intersection line of the graticule, in order to adopt the image-point theory. A detailed analysis is performed for models with any number of horizontal boundaries and two vertical discontinuities. The apparent resistivity formulas are obtained and selected apparent resistivity curves are drawn for different parameter combinations and various directions of the sounding expansion axis. The class under consideration contains as a particular case the HVC model elaborated in Alpin's monograph, where only a small collection of master curves is available for the axial array. The reconstruction of those curves by the present formulation shows the existence of large discrepancies. A test based on the transformation to equivalent half-Schlumberger sounding curves supports the conclusion that an unidentified error must exist in some part of the theoretical approach of the Russian researchers. Finally, some field sounding curves based on geothermal and volcanological surveys are presented and interpreted by complete curve matching, essentially to show the applicability of the theoretical solutions.     

ELECTROMAGNETIC RESPONSE OF TWO-DIMENSIONAL INHOMOGENEITIES IN A DISSIPATIVE HALF-SPACE FOR TURAM INTERPRETATION *

An integral equation technique based on the integral formulation of the electric field of a harmonically excited line-source in the presence of a two-dimensional scattering in-homogeneity is outlined. In this treatment the inhomogeneity is assumed to have an arbitrary cross-sectional shape and to be located in a layered, dissipative half-space. The scattering responses in terms of the horizontal and the vertical magnetic field components observed on the ground surface are studied for a variety of geologic models involving overburden layers, vertical or inclined veins and multiple inhomogeneities. The horizontal component of the total magnetic field appears to be the most diagnostic response parameter. A thin vein shaped target with a cross sectional area of 0.05 δ₂× 0.50 δ₂ (δ₂ being the skin-depth in the lower half-space) located 0.50 δ₂ away from the line-source and under a highly conductive overburden layer (σ₁/σ₂= 300) of thickness 0.02 δ₂ is easily detected with a moderate resolution. The scattering responses are also sensitive to changes in depths of burial, inclination and conductivity contrast with the surrounding half-space of the target inhomogeneity. An interpretation scheme as well as a number of characteristic detectability parameters are developed for the conventional Turam method used in mineral exploration.     

METHODS FOR CONTOURING IRREGULARLY SPACED DATA*

The sampling theorem in two dimensions univocally defines a surface, provided that its values are known at points disposed on a regular lattice. If the data are irregularly spaced, the usual procedure is first to interpolate the surface on a regular grid and then to contour the interpolated data: however, the resulting surface will not necessarily assume the prescribed values on the irregular grid. One way to obtain this result is to introduce a transformation of the coordinates such that all the original data points are transferred into part of the nodes of a regular grid. The surface is then interpolated in the points correspondent to the other crosspoints of the regular grid; the contour lines are determined in the transformed plane and then, using the inverse coordinate transformation, are transferred back to the original plane where they will certainly be congruent with the original data points. Nonetheless, the resulting surface is very sensitive to the interpolation method used: two algorithms for that are analyzed. The first (harmonization) corresponds to the determination of the potential of an electrical field whose contour conditions are those defined by the data points. The second method consists in two dimensional statistical estimation (krigeing); in particular, the effects of different choices for the data auto-covariance function are discussed. The solutions are compared and some practical results are shown.     

FRACTURES AS CONDUCTING DYKES AND CORRESPONDING TWO-DIMENSIONAL MODELS*

A statistical treatment of MT data from the Pannonian Basin, Hungary, based on the distortion theory of the S-effect suggests that the local increase in crustal conductivity is connected with deep fractures. Field model measurements have recently been carried out in a shallow, quasi two-dimensional basin of well known tectonics using MT station distances of 1–3 km. The results of the investigation and numerical modelling of near surface distortions, support the idea that deep fractures (faults) contain the conducting formations. According to 2-D model computations, several conducting dykes at considerable depths can cause anomalies which are indiscernible from an anomaly due to a conducting layer. The significance of these results is discussed from the point of view of geothermal energy exploitation.     

山西阳高──河北容城剖面大地电磁资料的二维反演解释

介绍了一种快速二维大地电磁自动反演方法及其在阳高－容城剖面大地电磁资料解释中的应用．二维反演结果显示，研究区浅层电性结构与地表地质有较好的对应性；在中、下地壳范围内发育着断断续续的低阻带，并沿剖面大致分为三段；上地幔低阻带由北西向南东方向逐渐变浅，上地幔低阻带深度陡变带与巨型重力异常高梯度带对应．与一此较大型的断裂带对应的区段在电性上有较清楚的显示．     

山西阳高──河北容城剖面大地电磁资料的二维反演解释

介绍了一种快速二维大地电磁自动反演方法及其在阳高－容城剖面大地电磁资料解释中的应用．二维反演结果显示,研究区浅层电性结构与地表地质有较好的对应性;在中、下地壳范围内发育着断断续续的低阻带,并沿剖面大致分为三段;上地幔低阻带由北西向南东方向逐渐变浅,上地幔低阻带深度陡变带与巨型重力异常高梯度带对应．与一此较大型的断裂带对应的区段在电性上有较清楚的显示．     

A NOTE ON POSSIBLE USE OF MICROCOMPUTERS FOR AQUIFER STUDIES

Abstract. Microcomputers can be a major tool for the hydrologist. Small and portable, microcomputers can be useful for the acquisition and modeling of an aquifer while on site. This would result in very low cost for computer usage and fast processing time. Their low cost makes the availability almost universal. Tests were run on the TRS-80 system using the Prickett finite difference model and the Theis equation. The results indicate that with the exception of memory storage no problems should arise. With newer generations of microcomputers core memory storage will no longer be a problem.