CONSTRAINED INVERSION OF POTENTIAL FIELD DATA*

For constrained inversion of potential field data within the framework of generalized inversion an analysis of data error variances leads to confidence limits for the model parameters. For that purpose Pseudo-hyper-ellipsoids can be used to describe the nonlinear behaviour of the given inverse problem, and upper and lower bounds can be added to those parameters of which some independent knowledge is available. A gravity example is treated to show the application of the method.     

POTENTIAL FIELD CONTINUATION BETWEEN GENERAL SURFACES*

The theory of potential field continuation is studied with a view of extending it to include continuation between non-linear surfaces. This theoretical extension is regarded as useful in reducing observed potential fields along one general surface to another general surface. It is demonstrated that the continuation operators considered in past geophysical literature are special cases of the generalized operators. In view of the tremendous growth in the art of high-speed computing it is possible to consider applications of the generalized operators. In worked examples upward continuation between a general surface and a datum is considered for the gravity field due to geometrically simple sources; this in order to test the accuracy of the digital applications. It is indicated that for observations on an undulating surface it is possible to account for considerable errors when, during interpretation, the observation locations are taken to be along a datum. For aids to profile interpretation in areas of considerable topographic relief some simple convolution operators for continuation between linear, non-parallel surfaces are proposed.     

AUTOMATED STATIC CORRECTIONS*

The widespread use of common depth point techniques has emphasized the need for accurate static corrections. Manual interpretation methods can give excellent results, but a computer technique is desirable because of the great volumn of data recorded in common depth point shooting. The redundancy inherent in common depth point data may be used to compute a statistical estimate of the static corrections. The corrections are assumed to be time-invarient, surface-consistent, and independent of frequency. Surface consistency implies that all traces from a particular shot will receive the same shot static correction and all traces from a particular receiver position will receive the same receiver correction. Time shifts are computed for all input traces using crosscorrelation functions between common depth point traces. The time shift for each trace is composed of a shot static, a receiver static, residual normal moveout if present, and noise. Estimates of the shot and receiver static corrections are obtained by averaging different sets of the measured time shifts. Time shifts which are greatly in error are detected and removed from the computations. The method is useful for data which has a moderate to good signal to noise ratio. Residual normal moveout should be corrected before estimating the statics. The program estimates the statics for correctly stacking common depth point traces but it is not sensitive to constant or very slowly changing static errors.     

INTERPRETATION OF POTENTIAL FIELD DATA A GENERALIZED INVERSE APPROACH*

The interpretation of potential field data from two-dimensional structures with a single interface of density or susceptibility contrasts is solved in terms of generalized matrix inversion. The model equations are derived, and important features of generalized matrix inversion are treated. The method is subsequently used to solve two gravity problems, an artificial one where the solution is known, and a geophysical one related to the crust-mantle interface. The solution is shown to compare well with the FFT results of Oldenburg. The method is also used to solve two magnetic problems, an artificial one with the solution known, and a geophysical one from the continental shelf of Greenland. The advantages and limitations of the method are finally discussed.     

THE STABILITY OF THE SOLUTIONS OF THE POTENTIAL FIELD TRANSFORMATIONS *

The quantitative analysis of the potential fields leads to the solution of some operational equations which sometimes have unstable solutions representing fictitious anomalies. A general method to find numerically stable solutions of such problems is presented in this paper. The subject of the downward analytical continuation is also discussed. The method has been checked on a theoretical model and applied to a gravity map.     

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.     

A SPATIAL ANALYSIS OF UPWARD CONTINUATION OF POTENTIAL FIELD DATA *

A spatial analysis of both continuous and discrete operators for upward continuation help us realize the problems and limitations which have been encountered before (Henderson 1960, Kontis 1971) but remained unsolved in practical application of upward continuation computation due to the finite length of data and operator in spatial domain. Various numerical examples show that an improvement of accuracy of continuation computations can be achieved through proper sampling and sufficient length of data.     

TWO DIMENSIONAL STRAKHOV'S FILTER FOR EXTRACTION OF POTENTIAL FIELD SIGNAL*

A two dimensional linear filter operator based on Strakhov's method for the extraction of a potential field signal (gravity or magnetic) from a background of homogeneous random noise has been developed. An algorithm to solve for the coefficients of the operator is given together with the corresponding computer program in FORTRAN language. It may be used on any machine using FORTRAN IV. A few examples of the operator are also included.     

SEISMIC MODELING IN THE IMPLICIT MODE*

Finite-difference seismic models are often saddled with huge memory requirements for data manipulation, a prohibitive amount of CPU time and even approximate results. At least part of these costs may be due to the fact that most of the work reported on this subject is devoted to the development of explicit models which suffer from severe limitations of stability and necessitate extremely fine time sampling of the wavefield. A new method of seismic modeling which works in the implicit mode and is unconditionally stable is put forward. It is based on the self-adjoint version of the acoustic wave equation. The evaluation of the wavefield is done by using a highly efficient splitting algorithm which does not require transposing the field data at the various time steps. Moreover, it can accommodate anisotropic media as well as three-dimensional structures. Computational efficiency is achieved by introducing an unconventional procedure which yields the sum of the values of the wavefield corresponding to a new time step and a previous time step. The new value can be obtained from this sum by a simple subtraction. A number of numerical examples, some obtained by using time steps about five times larger than the largest permissible time step in the corresponding explicit model, are presented. The effect of topographic changes on the growth of the wavefield due to a source located near the surface of the ground is investigated.     

FORWARD MODELING IN THE FREQUENCY-SPACE DOMAIN*

Forward modeling of zero-offset data is performed in the frequency-space domain using a one-way extrapolation equation. The use of the frequency domain offers several advantages over conventional time domain methods. The greatest advantage of the frequency domain is that all time derivatives are evaluated exactly by a simple multiplication. Synthetic zero-offset sections are computed with a high degree of accuracy for arbitrary velocity and reflectivity structures. Examples are shown for realistic complicated models and compared with results from physical modeling.     

TOPOGRAPHIC AND ISOSTATIC CORRECTIONS TO GRAVITY SURVEYS IN MOUNTAINOUS AREAS*

In mountainous areas it is an inadequate procedure to reduce gravity observations by merely subtracting the effect of an infinite flat slab of material between the station and sea-level, and adding a terrain correction. A programme is described which directly computes the effect of masses above sea-level, and mass-deficiencies below it; as well as the effect of compensating masses under the Pratt and Airy isostatic systems. As an example, the method has been applied to a regional gravity survey of Papua and New Guinea where it is seen to remove the usually high correlation of the Bouguer anomaly with local topography.     

ON THE UPWARD CONTINUATION OF POTENTIAL FIELD DATA BETWEEN IRREGULAR SURFACES*

The potential field and its derivatives at points above an irregular surface can be approximately obtained from the sampled potential field data acquired on that surface. A method of minimizing the truncation effect, which appears when gravity and magnetic maps are processed with the aid of surface integrals, is derived. The results are compared with those of the most relevant similar methods by using a theoretical, but realistic, model.     

CONDUCTING SPHERE IN ELECTROMAGNETIC INPUT FIELD*

A frequency-domain analysis is outlined for a conducting sphere in a uniform Input field: inequispaced alternating half-sine wave pulses. The Barringer Input air-borne electromagnetic exploration system uses such source fields. Theoretical profiles of Hρ(t), the horizontal magnetic component over the sphere from different elevations and for various conductivity and geometrical factors are presented. Based on these results some useful features such as penetration and detectibility are discussed.     

CORRECTIONS FOR CONDUCTIVITY ESTIMATES IN INDUCTION PROSPECTING OF SULPHIDE DYKES IN A LAYERED ENVIRONMENT*

In a weathered environment estimates of depth and conductance of metallic sulphide dykes from conventional anomaly index diagrams for a vertical half-plane in air have to be corrected, besides the usual corrections, for: 1. moderate conductivity of the host rocks, and 2. finiteness of strike length S and depth extent D. Model experiments have been carried out to evaluate the response variation of a vertical planar conductor with varying depth extent and strike length for both insulating and conductive surroundings. The results indicate: 1. A conductor with finite depth extent (D/L < 2.5) or strike length (S/L < 5.0) in an insulating medium yields a lower estimate of conductance (mineralization) and a greater depth. 2. A moderately-conductive host rock enhances the anomaly and rotates the phase so that the conductor appears to be more resistive (less mineralized) and shallower. The results have practical significance since in weathered surroundings a highly-mineralized body of finite size could be missed, or misjudged, because of low estimates of conductivity and depth.     

COMPUTATION OF THE ELECTRIC POTENTIAL IN THREE-DIMENSIONAL STRUCTURES*

A method has been developed for computation of the electrical DC potential of an arbitrary three-dimensional resistivity structure using a finite difference method. The threedimensionality is necessary for interpretation of geoelectrical soundings with controlled point sources over a laterally inhomogeneous medium. Lateral inhomogeneities should be considered in resistivity soundings with large layouts. The results obtained with the described method permit a more realistic representation of geological features. The resolution of the method is determined by the number of elements in the resistivity network. The problem of core memory space has been resolved by using random access disc files. The results computed using a Fortran program are in good agreement with analytically obtained solutions.     

STATIC CORRECTIONS FOR SHEAR WAVE SECTIONS*

The computation of static corrections requires information about subsurface velocities. This information can be obtained by different methods: surface wave analysis, short refraction lines, downhole times, uphole times and first arrivals from seismograms. For pure shear waves generated by SH sources the analysis of first arrivals from seismograms combined, if necessary, with short refraction lines has proved to be most accurate and economic. A comparison of first-arrival plots from P- and S-wave surveys of the same line measured in areas of unconsolidated sediments in northern Germany illustrates the characteristic differences between the two velocity models. P-waves show a marked velocity increase at the water table from about 600 to 1800 m/s. S-wave velocities of the same strata increase gradually from about 100 to 400 m/s. As a consequence, S-wave models are vertically and laterally more complex and, in general, show no significant velocity increase at a defined boundary as P-wave models do. Therefore, other suitable correction levels with specific velocities must be chosen. A comparison of “tgd-corrections” (correction time between geophone position and datum level) for P- and S-waves in areas of unconsolidated sediments shows that their ratio is different from the P-/S-velocity ratio for the respective correction level because of the greater depth of the S-wave refractor. Therefore, P- and S-waves are influenced by different near-surface anomalies, and time corrections calculated for both wave types are largely independent.     

TRANSIENTS AND FIELD BEHAVIOUR IN INDUCED POLARIZATION*

This paper deals with electromagnetic fields in the so called “Transient Induced Polarization Technique”. Field equations are integrated in the case of a polarizable sphere inbedded in a sterile overburden. The existence of a remanent polarization vector P which slowly decays once the charging current has been cut off is taken as the initial condition. When the surface between the air and overburden is (as a first step) disregarded, the Laplace transform of the EM fields is given. The integral of the electric field (which is fairly often the result of prospection work) is independent of the various time constants related to electrochemical processes, but it stands as a good measurement of the total electrostatic dipole created by the charging current. We investigate the geometrical circumstances that can bring negative values of I.P. signal. Such negative values can be found in two cases: 1. The discharge currents are. distorted by the ground surface. 2. The I.P. signal is picked up by electrodes inside the polarizable material. The last part is dedicated to an account of experimental work performed on models and of an actual case history. The mathematical derivations are included in an appendix.     

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.     

CONDUCTING INFINITE HORIZONTAL CYLINDER IN ELECTROMAGNETIC INPUT FIELD *

With the help of the principle of superposition the Input response of a conducting infinite horizontal homogeneous cylinder is synthesised from its step-function response. The variation of transient peaks and the time lag between the secondary and primary fields with the cylinder constitution parameter are illustrated.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.