TRANSFORMATION OF DIPOLE RESISTIVITY SOUNDING MEASUREMENTS OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

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.     

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.     

INTERPRETATION GRAPH FOR VERTICAL TABULAR CONDUCTORS IN E.M. IN-LINE DIP-ANGLE MEASUREMENTS*

A graphical interpretation method is presented by which separately the depth and the product of conductivity and thickness of a vertical plate-shaped conducting orebody can be determined. The method is based on the result of model experiments.     

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%.     

PRINCIPLES OF DIGITAL WIENER FILTERING*

The theory of statistical communication provides an invaluable framework within which it is possible to formulate design criteria and actually obtain solutions for digital filters. These are then applicable in a wide range of geophysical problems. The basic model for the filtering process considered here consists of an input signal, a desired output signal, and an actual output signal. If one minimizes the energy or power existing in the difference between desired and actual filter outputs, it becomes possible to solve for the so-called optimum, or least squares filter, commonly known as the “Wiener” filter. In this paper we derive from basic principles the theory leading to such filters. The analysis is carried out in the time domain in discrete form. We propose a model of a seismic trace in terms of a statistical communication system. This model trace is the sum of a signal time series plus a noise time series. If we assume that estimates of the signal shape and of the noise autocorrelation are available, we may calculate Wiener filters which will attenuate the noise and sharpen the signal. The net result of these operations can then in general be expected to increase seismic resolution. We show a few numerical examples to illustrate the model's applicability to situations one might find in practice.     

BASIC PRINCIPLES OF TWO-DIMENSIONAL DIGITAL FILTERING *

The geophysicist involved in the analysis of two-dimensional data should have an understanding of the two-dimensional finite Fourier transform and the mechanics of two-dimensional filtering. Frequency aliasing must be considered when working with sampled data. In two dimensions it is advantageous to consider aliasing in terms of the overlap of the repeating spectra inherent in the finite Fourier transform. Two-dimensional filtering can be performed as a transient convolution in the space domain, as cyclic convolution utilizing the frequency domain or as the multiplication of polynomials using the z-transform. If the “edge” effects are removed, the results of the three methods are identical.     

TIME-VARYING PREDICTION FILTERING BY MEANS OF UPDATING*

In contrast to the conventional deconvolution technique (Wiener-Levinson), the spike-, predictive-, and gap-deconvolution is realized with the help of an adaptive updating technique of the prediction operator. As the prediction operator will be updated from sample to sample, this procedure can be used for time variant deconvolution. Updating formulae discussed are the adaptive updating formula and the sequential algorithm for the sequential estimation technique. This updating technique is illustrated using both synthetic and real seismic data.     

NUMERICAL VLF MODELING*

The VLF response of laterally inhomogeneous and anisotropic models is calculated numerically using the finite element method. Some results are presented for a slab model in terms both of the polarization parameters, i.e., the tilt angle and ellipticity of the magnetic polarization ellipse, and the amplitude ratio |H_z/H_x|. On the basis of both the ellipticity and the tilt angle, it is possible to discriminate between a poor conductor and a good one. The direction of the dip can be determined from the anomaly profiles of all diagnostic parameters. The effect of the conductive overburden is most noticeable on the ellipticity profile: one observes attenuation for a poor conductor and “negative attenuation” for a good conductor. The anomaly profiles for anisotropic cases are consistent with the ones of the isotropic cases.     

COMPUTATION OF SIGNATURES OF LINEAR AIRGUN ARRAYS*

Far-field signatures from an airgun array are usually obtained by carrying out extensive field measurements. In order to decrease the need for such measurements, we have developed a method for computing signatures from linear airgun arrays where the distances between the airguns are such that the non-linear interaction among the airguns is negligible. The signature from a single airgun of a given type is computed from the following airgun parameters: airgun chamber volume, chamber pressure, airgun depth and position of the waveshape plate within the chamber. For calibration purposes, a recorded signature for one set of airgun parameters has to be provided for each type of airgun. The signatures are computed by using empirical relations between signature properties and the airgun parameters, and by treating the primary and bubble pulses separately. The far-field signature from a linear airgun array can now be computed by summation of the delayed signatures from the airguns in the array. Practical results are shown for an array with different PAR (Bolt) 1500 C airguns.     

单分量地面记录地震波场分离的非线性滤波方法研究

单分量反射地震勘探地面记录的垂直位移分量uz应表示为纵波位移分量uPz和横波位移分量uSz之和.对反射地震数据进行精确处理必须从垂直位移分量uz中分离出独立的纵波位移分量uPz和横波位移分量uSz或者其他的波场分量,例如瑞雷波.现有的地面记录的纵横波分离方法建立在多分量记录基础上,而分离瑞雷波是基于波传播视速度差异的特性,其中隐含了数据观测面是弹性自由边界的假设,这和实际的地面观测条件是有差别的.文中设地面为一非弹性自由边界,利用非弹性自由边界面上的应力条件,给出了从单分量垂直位移记录uz中分离纵波垂直位移分量uPz和横波垂直位移分量uSz的非线性滤波方法.当边界为自由弹性时,该非线性滤波退化为线性视速度滤波.最后讨论了该滤波方法用于反射纵波和频散瑞雷波场分离的可能性.     

THE DETERMINATION OF FILTER COEFFICIENTS FOR THE COMPUTATION OF STANDARD CURVES FOR DIPOLE RESISTIVITY SOUNDING OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

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.     

VLF SURVEY OF THE WEATHERED LAYER IN SOUTHERN INDIA*

The electrical properties of the weathered layer were investigated by means of Geonics VLF-EM 16/16R equipment in two areas of the Andhra Pradesh State. The resistivity and thickness of the weathered layer were found to be variable even over a small survey area. In the area underlain by Precambrian granite-gneiss, most of the recorded VLF-EM anomalies were caused by variations in the resistivity of the weathered layer. Changes in thickness were well reflected in the VLF-EMR curves. The second area was underlain by Cretaceous basalts and dolerites. Quantitative interpretation of the VLF-EMR data with a simple one-dimensional model yielded considerable detail about the weathered layer. For the granitic area, a prior estimate of at least one resistivity parameter of the ground is required. If this is not already available, a limited amount of direct-current resistivity surveying can provide the required information. A study of the EMR data from the basaltic area revealed the presence of a thin, highly conductive layer between the weathered layer and the bedrock. The parameters of this layer were found to be variable, making it necessary to use a set of diagrams for quantitative interpretation. Due to the presence of this highly conductive layer, the EMR data contain little information on the bedrock resistivity. Our field studies suggest that the VLF-EMR method can be used as a fast and inexpensive tool for mapping of the weathered layer in tropical regions with hard rock geology. Such mapping is of considerable importance because the weathered layer is an important source of groundwater.     

COMPUTATION OF VLF RESPONSE OVER HALF-PLANE AND WEDGE MODELS*

A theoretical solution is presented to the problem where a VLF anomaly is generated by a conducting half-plane or a perfectly conducting wedge below a stratified overburden. The solution is obtained by the use of a scattering matrix for plane-wave eigenfunctions. VLF anomalies have been computed for different values of the conductance and dip of the half-plane. The phase of the VLF anomaly due to a conducting half-plane depends on the conductance and the distance to the half-plane. Close to the half-plane the tilt angle and ellipticity are of opposite sign for a perfect conductor, but the ellipticity will change sign for a poor conductor. The VLF anomaly for a perfectly conducting wedge is essentially determined by the position of the upper surface of the wedge, i.e. the anomaly will closely resemble the anomaly of a perfectly conducting half-plane in the same position as the upper surface of the wedge.     

THE APPLICATION OF VHF MEASUREMENTS TO EARTHQUAKE PREDICTION*

The use of Very High Frequency measurements applied to the detection of the variation in dielectric constant as a precursor to earthquakes is examined. It is concluded that such a technique is feasible and that the installation of source and sensor in separate deep boreholes provides a suitable environment for this type of study. The boreholes could also be used for a considerable range of other types of instrumentation.     

LINEAR TRANSFORMATIONS OF GRAVITY AND MAGNETIC FIELDS*

The spectral representation of gravity and magnetic fields shows that the mathematical expressions describing these fields are the result of convolution of factors which depend on the geometry of the causative body, the physical properties of the body and the type of field being observed. If a field is known, it is possible to remove or alter these factors to map other fields or physical parameters which are linearly related to the observed field. The transformations possible are: continuation, reduction to the pole, converting between gravity and magnetic fields, converting between components of measurement, calculation of derivatives, and mapping magnetization and density distribution, relief on interfaces, and vertical thicknesses of layers.     

SIGNATURE AND AMPLITUDE OF LINEAR AIRGUN ARRAYS *

During the last few years many airgun arrays have been designed with the objective of generating a short signature of high amplitude. For linear arrays of non-interacting airguns two rules have been derived that may help in the design or evaluation of airgun arrays. To achieve a short pressure pulse, the total available air volume has to be distributed over the individual guns in such a way that the tail of the signal, owing to the added bubble signals, becomes as flat as possible. When we think of ordering airguns according to volume, this flat signal tail can be achieved by designing the volumes such that the difference in bubble times of two adjacent guns is proportional to their volume to the power 2/3. The amplitude expected from a linear array of non-interacting airguns is limited by the physical length of the array. A graph of measured values tends to confirm this relation. No relation has been found between the total volume of an array and its amplitude. The graph also detects inefficient use of available array length of existing arrays.     

OPTIMUM DIGITAL FILTERING AND INVERSE FILTERING IN THE FREQUENCY DOMAIN*

Two distinct filters are developed in the frequency domain which represent an attempt to increase the resolution of fine structure contained in the signal whilst keeping the expected filtered noise energy within reasonable bounds. A parameter termed the White Noise Amplification is defined and used together with a measure of the deconvolved pulse width in order to provide a more complete characterisation of the filters. Each of the two main types of frequency domain filters discussed varies in properties with respect to a single adjustable parameter. This may be contrasted with a time domain Wiener filter which in general has three variables: length, delay and an adjustable noise parameter or weight. The direct frequency domain analogue of the Wiener filter is termed a gamma-Fourier filter, and is shown to have properties which span the range from those of a spiking filter with zero least square error at one extreme, to those of a matched filter at the other extreme of its variable parameter's range. The second type of filter considered—termed the modulated Gaussian filter—is similarly shown to be a perfect spiking filter at one extreme of its parameter range, but adopts the properties of an output energy filter at the other extreme.     

THE EFFECTS OF ANISOTROPY ON SQUARE ARRAY RESISTIVITY MEASUREMENTS*

An account is given of the use of the square array technique in investigating the surface effects of rotational anisotropy when the axes of anisotropy are inclined to the surface. It is shown that, as with other arrays, two anisotropy parameters and n can be derived by varying the array orientation. On the basis of these considerations, the effects of such anisotropy on the values of the mean apparent resistivity and azimuthal inhomogeneity ratio normally obtained in square array measurements is reviewed. Particular attention is paid to the variation of resistivity with orientation and it is noted that, in areas of moderate anisotropy, this variation is lower for the square than for the Schlumberger array. In addition to this advantage, the azimuthal inhomogeneity ratio obtained from square array measurements may be used to indicate the severity of anisotropy in an area and two field examples of this use are given. Where anisotropy is severe, gross variations of apparent resistivity with orientation are obtained with either square or collinear arrays. In these circumstances, the use of crossed measurements is considered and the particular stability of the crossed square array demonstrated.     

TOPOGRAPHIC REDUCTION OF GRAVITY MEASUREMENTS BY NUMERICAL INTEGRATION OF BOUNDARY INTEGRALS*

A numerical method for calculating the topographic reduction of gravity measurements is developed which follows the approximation of the topography by a single valued function. The method involves the conversion of the volume integral for the gravity effect into a two-dimensional definite integral. The definite integral is partly solved by explicit, and partly by numerical, integration, using the Gauss-Legendre quadrature formula. This method is well suited to calculating the topographic reduction of 50 to approximately 1000 m from the station–especially for microgravimetric surveys in areas of steeply sloping terrain. To test the method in practice it was applied in an area of rough relief in Keban (East Turkey).