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 Volume 41, Issue 6, 1993
Geophysical Prospecting  Volume 41, Issue 6, 1993
Volume 41, Issue 6, 1993


MODEL‐BASED STACK: A METHOD FOR CONSTRUCTING AN ACCURATE ZERO‐OFFSET SECTION FOR COMPLEX OVERBURDENS^{1}
More LessAbstractThe interpretation of stacked time sections can produce a correct geological image of the earth in cases when the stack represents a true zero‐offset section. This assumption is not valid in the presence of conflicting dips or strong lateral velocity variations. We present a method for constructing a relatively accurate zero‐offset section. We refer to this method as model‐based stack (MBS), and it is based on the idea of stacking traces within CMP gathers along actual traveltime curves, and not along hyperbolic trajectories as it is done in a conventional stacking process. These theoretical curves are calculated for each CMP gather by tracing rays through a velocity‐depth model. The last can be obtained using one of the methods for macromodel estimation. In this study we use the coherence inversion method for the estimation of the macromodel since it has the advantage of not requiring prestack traveltime picking. The MBS represents an accurate zero‐offset section in cases where the estimated macromodel is correct. Using the velocity–depth macromodel, the structural inversion can be completed by post‐stack depth migration of the MBS.



DIELECTRIC PROPERTIES OF WET SOILS IN THE FREQUENCY RANGE 1–3000 MHz^{1}
By W.A. WENSINKAbstractThe effective relative dielectric constant ɛe, r and the effective conductivity σe have each been determined as a function of frequency in the range 1–3000 MHz at volumetric water contents of up to approximately 0.74 for clays, 0.83 for a peat and 0.56 for a silt.
At frequencies above about 25 MHz (depending on soil type), ɛe, rincreases with water content for all samples. However, at lower frequencies, ɛe, ronly increases with water content as long as the wet density also increases, which is the case for water contents up to a critical value lying between 0.35 and 0.48. At higher water contents, ɛe, rand the wet density decrease with increasing water content. Consequently, curves of ɛe, rversus frequency for two wet samples with different water contents, at least one of them higher than the critical value, are seen to cross at about 25 MHz. Below the critical value the curve of the sample with the lower water content is below the other curve at all freqencies applied. At a given frequency, σe has a maximum as a function of water content. This is tentatively explained by assuming that σe is the sum of pore water conductivity (increasing with water content until all salts in the soil are dissolved into the water and then decreasing) and surface water conductivity (increasing with wet density and therefore increasing with water content up to the critical value and then decreasing).
At frequencies higher than 1000 MHz, ɛe, rdepends only weakly on salinity (which is represented by the measured conductivity). It shows an increasing dependence if the frequency is decreased towards 1 MHz.
The highest values of ɛe, rand σe, measured in this work, occur for a sample of wet, nearly saturated silt originating from a location below sea‐level near to the Dead Sea, Israel: ɛe, rdecreases continuously from a value of about 10^{4} at 3 MHz to about 10^{2} at 200 MHz, while σe rises from about 4 S/m to 5 S/m at these respective frequencies. The dependence of the wavelength on the loss‐tangent is strong and the wavelength is considerably smaller than it would be in a dielectric. This is the only sample for which σe increases with water content, even if the latter is above its critical value. Therefore it is assumed that the pore water conductivity is greater than the surface water conductivity if the volumetric water content is lower than 0.564, the maximum value applied. The measurements give evidence for the presence of a relaxation at about 3 MHz for all samples examined.



THREE‐DIMENSIONAL INTERPRETATION OF MULTIPLE‐SOURCE BIPOLE‐DIPOLE RESISTIVITY DATA USING THE APPARENT RESISTIVITY TENSOR^{1}
Authors H.M. BIBBY and G.W. HOHMANNAbstractThe bipole‐dipole resistivity technique, which uses a single current source (bipole) to map variations in (apparent) resistivity has been much criticized in the past. A series of 3D models are used to show that the use of two distinct current bipoles in the same location but with different orientations, combined with analysis in the form of a previously defined tensor apparent resistivity, can greatly improve many aspects of bipole‐dipole mapping.
The model study shows that, for measurement stations more than a few bipole lengths from the current source, the apparent resistivity tensor behaves, to a close approximation, as though the current bipoles are idealized dipoles, and hence is independent of the orientation of the individual current sources used. Any pair of current bipoles (in the same location but with different orientations) can therefore be used to determine the tensor resistivity properties.
The invariants of the apparent resistivity tensor have considerable advantages over the normal scalar apparent resistivities. Modelling shows that although the electric field vector corresponding to a single current bipole can be highly perturbed by a local inhomogeneity for some considerable distance beyond the inhomogeneity itself, the tensor invariants are virtually unperturbed beyond the extent of the inhomogeneity. Hence false anomalies, which are a characteristic of apparent resistivity measured using only single current bipole models, are almost completely eliminated by the use of tensor invariants. Of the possible tensor invariants, the invariant given by the square root of the determinant gives the best representation of a buried 3D body. Resistivity anomalies are localized, and occur only over the causative body. Even with complex models involving several buried bodies, the tensor invariants clearly delineate the extent of each body. Outside the bounds of perturbing bodies, the tensor data can be analysed by conventional techniques, for example, to determine layered structure.



WAVE‐EQUATION MULTIPLE SUPPRESSION USING CONSTRAINED GROSS‐EQUALIZATION^{1}
More LessAbstractA method for improving the attenuation of water‐layer multiple energy is suggested. The improvement is achieved using wave‐equation extrapolation to generate an initial model of the multiple energy, and then constraining the way in which this model is modified to fit the observed multiple energy. Reconciling the initial multiple model with the input data is a critical part of this process and several techniques have been suggested previously by other authors. The approach used here is to fit the time, amplitude and phase of the wavelets by adapting the initial model trace using a weighted sum of four traces which can each be derived from the initial multiple model trace.
Results on real data suggest that attenuation of primary energy is minimized using this technique, without diminishing the level of multiple attenuation.



TEMPERATURE EFFECTS ON AIRGUN SIGNATURES^{1}
Authors JAN LANGHAMMER and MARTIN LANDRØAbstractExperiments in an 850 litre water tank were performed in order to study temperature effects on airgun signatures, and to achieve a better understanding of the physical processes that influence an airgun signature. The source was a bolt airgun with a chamber volume of 1.6 cu.in. The pressure used was 100 bar and the gun depth was 0.5 m. The water temperature in the tank was varied between 5°C and 45°C. Near‐field signatures were recorded at different water temperatures. Typical signature characteristics such as the primary‐to‐bubble ratio and the bubble time period increased with increasing water temperature. For comparison and in order to check whether this is valid for larger guns, computer modelling of airguns with chamber volumes of 1.6 and 40 cu.in. was performed. In the modelling the same behaviour of the signatures with increasing water temperature can be observed. The increase in the primary‐to‐bubble ratio and the bubble time period with increasing water temperature can be explained by an increased mass transfer across the bubble wall.



NUMERICAL SEISMOGRAM COMPUTATIONS FOR INHOMOGENEOUS MEDIA USING A SHORT, VARIABLE LENGTH CONVOLUTIONAL DIFFERENTIATOR^{1}
Authors B. ZHOU, S.A. GREENHALGH and J. ZHEAbstractA short convolutional differentiator (CD) for computing second spatial derivatives in the acoustic wave equation is presented. This differentiator is obtained by tapering the inverse Fourier transform of the band‐limited Fourier spectrum of the second‐derivative operator. This new filter has been applied to seismogram computations for inhomogeneous media and results are compared with the conventional high‐order finite‐difference (FD) and Fourier schemes. The operator can be progressively shortened at the model edges to reduce boundary artefacts. The CD method is superior to the conventional FD operator and comparable with the Fourier method in accuracy but faster to run. A strategy to reduce computation time by 20%, which exploits the localized nature of the operator, is given. The method is illustrated using simple 2D models.



ON THE COMPUTATION OF THE MAGNETIC FIELD DUE TO A D.C. POINT ELECTRODE AT THE VERTICAL BOUNDARY BETWEEN TWO QUARTER‐SPACES^{1}
By M. HVOŽDARAAbstractNew formulae are presented for the calculation of the horizontal magnetic field due to a point electrode situated at the vertical boundary between two quarter‐spaces of different electrical conductivities. Previously the solution was obtained using a double Fourier transform of the expression for the vertical component of the magnetic field. The new formulae are given in the form of single integrals.



A RECONCILIATION OF MATHEMATICAL MODELS FOR SPONTANEOUS MINERALIZATION POTENTIALS^{1}
More LessAbstractTwo methods for computing spontaneous mineralization potentials in the region external to the source body are reviewed. The first of these is a long‐established technique in which the causation is assumed to be a distribution of simple current source on the boundary of the mineralization. The second is a more recent technique which assumes a surface distribution of current dipole moment (double layer) along the boundary of the source body.
The former technique is a special case of a more general spontaneous potential (SP) model in which the source is a density of current dipole moment (current polarization) distributed throughout the mineralization. As far as the potentials in the region external to the source body are concerned this current polarization can be simply related to an equivalent double layer source function, i.e. the potential discontinuity produced over the boundary of the mineralization by an equivalent double layer model.
This simple relationship suggests an integral equation technique for the exact numerical solution of boundary value problems appropriate to the polarization model for spontaneous mineralization potentials. The technique is applied to exploring the justification of interpreting mineralization self‐potentials by the traditional approach.



ON THE COMPLETENESS OF DATA SETS WITH MULTIELECTRODE SYSTEMS FOR ELECTRICAL RESISTIVITY SURVEY
More LessAbstractThis paper describes how, using a surface linear array of equally spaced electrodes, potential data can be obtained for use in electrical resistivity imaging. The aim is to collect a complete data set which contains all linearly independent measurements of apparent resistivity on such an array using two‐, three‐ or four‐electrode configurations. From this primary data set, it is shown that any other value for apparent resistivity on the array can be synthesized through a process of superposition. Numerical tests show that such transformations are exact within the machine error for calculated data but that their use with real field data may lead to noise amplification.

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