ALBERT

Notes: This is the first of a series of papers giving the solution of the inverse problem in seismic exploration. The acoustic approximation is used together with the assumption that the velocity field has the form 〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR998:GPR_998_mu1"/〉 . The forward problem is then linearized (thus neglecting multiple reflected waves) and the inverse problem of estimating δ is set up. Its rigorous solution can be obtained using an iterative algorithm, each step consisting of a classical Kirchhoff migration (hyperbola summation) plus a classical forward modeling step (circle summation).

Notes: The response of a seismic model to excitation by a source can be represented in terms of the action of reflection and transmission operators for portions of the structure. This approach provides a flexible framework for both modeling and processing problems.The operator development provides a physical description of the wave propagation process and, via the expansion of reverberation operators, gives a mechanism for assessing the accuracy of approximate developments. The representation suggests new ways of developing modeling algorithms by balancing the computational effort expended on minor and major features of the model.For processing problems, the operator representation shows the relation of processing stages to the seismic wave field and thereby indicates effective sequences of operations. For migration it is possible to specify an ideal pre-stack migration procedure in terms of the inverse of the propagation operators and to examine the problems which need to be overcome by practical algorithms.

Notes: A numerical method is presented for direct interpretation of resistivity sounding measurements. The early part of the resistivity transform curve derived from field observations by standard methods is approximated by a two-layer curve. The resistivity of the first layer is determined from the arithmetic mean of the successive computations which are carried on each of three successive discrete values of the resistivity transform curve. Using this mean value of the resistivity, the thickness of the first layer is computed from the sample values in pairs of the resistivity transform curve. After these determinations, the top layer is removed by Pekeris's reduction equation. The parameters of the second layer are obtained from the discrete values of the reduced transform curve (which corresponds to the second part of the resistivity transform curve) by the same procedure as described for the first layer.The same computational scheme is repeated until the parameters of all intermediate layers are obtained. The resistivity of the substratum is determined from the reduction equation.

Notes: In odd-depth structure the two-way traveltime to each boundary is constrained to be an odd integer. The odd-depth property of a model is exposed to possible refutation under a seismogram test. Test function is a simple transformation of a synthetic seismogram. For an odd-depth model the test function has identically the value 1.The testability of a synthetic seismogram over an odd-depth structure provides a method of deterministic deconvolution. There is no need of specialized assumptions, like the minimum-phase property, about the source wavelet. The deconvolution may be performed in the absence of the early segment of a seismogram.

Notes: Zero-offset-source VSP surveys provide information about the subsurface only within the Fresnel zone centered at the well. Offsetting the source location moves the reflection zones away from the well thus providing lateral cover.Conventional processing of this type of data gives rise to a distorted image of the subsurface. Using a simple ray-tracing scheme, this image may be reconstructed into the more familiar coordinate system of the surface seismic section. This simple data-independent mapping is based on the assumption of horizontal layering and requires a vertical velocity profile.The technique of placing the source away from the borehole was first applied to the single-offset-source VSP survey. However, data from any survey geometry (such as deviated well with rig source, walkaway VSP, etc.) can be mapped to the coordinate system defined by the appropriate seismic section.To obtain the best results from this type of survey the target area must be defined and simple modeling techniques used to optimize the source location(s). These pre-survey modeling methods may also be used to anticipate—and hence avoid a number of problem areas which experience has highlighted.The data from any VSP survey is the result of a realizable experiment and as such obeys the wave equation. This implies that the wave equation may be used to migrate the data to its true subsurface location. Theoretically, such a process is more secure than ray-tracing techniques, although its practice presents many difficulties.

Notes: A mathematical expression for potential of a direct current point source in an inhomoge-neous anisotropic earth is derived. The coefficient of anisotropy is given by f= (σr/σz), where σr and σz are the conductivities parallel and perpendicular to the bedding plane. It is assumed that σz varies with depth, whereas σr varies transversely. This potential may be useful in interpretation of geoelectrical data in specified geological situations. Master curves for Wenner and Schlumberger configurations are presented

Notes: Total magnetic intensity contour maps for the study region (between 2°E to 10°E and 56°N to 60°N) were digitized and converted to a regular grid of 285 × 285 points. The study area measures approximately 444 km × 444 km and the grid spacing is thus 1. 56 km. The International Geomagnetic Reference Field for 1975 was gridded for the above-used net, and from the two data sets a further grid of the ▵T field was generated. A large number of profiles were constructed which were suitable for depth determinations. The regular grid ▵T data is also convenient for the computation of the second vertical derivative. Using the method of vertical prisms of Vacquier et al. (1963), a large suite of curvature-depth indices was measured to complement the depths obtained from the intensity slopes and from boreholes which reach the crystalline basement. The depth to the magnetic basement has been contoured, and the resulting map is shown to be in good agreement with what is known about the deeper geology of the study area.The work reported here is part of a research project supported by Amoco Norway, BP Petroleum Development Ltd, Elf Aquitaine, Esso Exploration and Production, Norwegian Gulf, Norsk Hydro, Mobil Exploration Norway, Norwegian Petroleum Directorate, Royal Norwegian Council for Scientific and Industrial Research (NTNF), Norske Shell, and Statoil.

Notes: Geophysical inversion involves the estimation of the parameters of a postulated earth model from a set of observations. Since the associated model responses can be nonlinear functions of the model parameters, nonlinear least-squares techniques prove to be useful for performing the inversion. A common type of inversion applies iterative damped linear least squares through use of the Marquardt-Levenberg method. Traditionally, this method has been implemented by solving the associated normal equations in conventional ways. However, Singular Value Decomposition (SVD) produces significant improvements in computational precision when applied to the same system of normal equations. Iterative least-squares modeling finds application in a wide variety of geophysical problems. Two examples illustrate the approach: (1) seismic wavelet deconvolution, and (2) the location of a buried wedge from surface gravity data. More generally, nonlinear least-squares inversion can be used to estimate earth models for any set of geophysical observations for which an appropriate mathematical description is available.

Notes: In this study we derive expressions for particle displacement or particle velocity anywhere inside a stratified earth and at its surface due to horizontal torque source located in the top layer. Equivalently, invoking Green's function reciprocity theorem, the solution applies also to the case of a surface or subsurface source when the resulting displacement or velocity is measured within the top layer.In order to evaluate the closed-form analytical solution economically and accurately it is advisable to introduce inelastic attenuation. Causal inelastic attenuation also lends the necessary realism to the computed seismic trace. To provide proof that the analytical solution is indeed correct and applicable to the multilayer case, a thick uniform overburden was assumed to consist of many thin layers. The correctness of the computed particle velocity response can be very simply verified by inspection. The computed response can also serve as a check on other less accurate methods of producing synthetic seismograms, such as the techniques of finite differences, finite elements, and various sophisticated ray-tracing techniques.It is not difficult to construct horizontal surface torque source. It appears that such source is well suited for seismic exploration in areas with a high-velocity surface layer. A realistic source function is analyzed in detail and normalized displacement response evaluated at different incidence angles in the near and the far fields.In an effort to distinguish the features of an SH torque seismogram from a pressure seismogram two models with identical layerings and layer parameters have been set up. As expected the torque seismogram is very different from the compressional seismogram. One desirable feature of a torque seismogram is the fast decay of multiples.Exact synthetic seismograms have many uses; some of them, such as the study of complex interference phenomena, phase change at wide angle reflection, channeling effects, dispersion (geometrical and material), absolute gain, and inelastic attenuation, can be carried out accurately and effortlessly. They can also be used to improve basic processing techniques such as deconvolution and velocity analysis.The numerical evaluation of the analytical solution of the wave equation as described in this paper has a long history. Most of the work leading to this paper was carried out by one of us (M. J. K.) in the years 1957 to 1968 at the Geophysical Research Corporation. However, the full testing of the various computer codes was carried out only very recently at the Phillips Petroleum Company.

Notes: In South and Central Goa iron ore occurs in two parallel belts with the general NW-SE Dharawar trend. The ore occurrence, however, is not continuous. There are barren zones as well as zones of very high concentration in some of which there are mining activities.Landsat MSS data have been interpreted over a zone covering both mineral belts in order to delineate the ore occurrences. As a guide line a known ore-bearing area has been considered along with the unknown zones.On the basis of two-dimensional plotting of gray level values it has been found that the MSS bands 4 and 7 are most suitable for the studies over iron-rich areas in Goa. Two techniques are described here for the processing of the MSS data; the separation of residual from the regional and MSS band-ratioing. It is observed that (i) the gray level residual maps of MSS bands 4 and 7 are of use in demarcating the iron-ore-bearing zones, and (ii) an existing mine, an abandoned mine, and a proved iron ore zone could be delineated by MSS band-ratioing. On the basis of the latter technique, a few areas with ore occurrence potential have been indicated.

Notes: The most common source of seismic energy is an explosion at some depth in a borehole. The radiated waves are reflected not only at the subsurface layers but also at the free surface. The earth's surface acts as a generator of both P- and S-waves.If the source depth is much less than the dominant wavelength the reflected waves resemble closely the waves generated by a single force. Theoretical seismograms were computed with different methods to look for the relevance of the surface-reflected waves. The numerical experiments show reflected shear waves even for small shotpoint—receiver distances. Due to their polarization these waves can be detected most easily on in-line horizontal geophones. The existence of these waves was examined during a conventional survey in Northern Germany. Conventional data analysis shows a large variability in the νp/νs ratio. The method used here produced a shear-wave section with a rather good signal-to-noise ratio down to 4 s S-wave reflection time.

Notes: We describe the concept of physico-geological models (PGMs) in geophysical exploration. They represent a “general model”, a spatial combination of a set of particular models (disturbing bodies). The modeling is called complete, incomplete or approximate, depending on the degree of characterization of the PGM by parameters such as dimension, shape and petrophysical property. Each of the three modeling types can be realized as a conceptual, and analytical, or a material PGM. Both deterministic and stochastic PGMs exist; deterministic models are mainly used to investigate the possibilities of a geophysical method, while stochastic models serve to substantiate complex geophysical interpretations.Depending on the geological problem, PGMs are subdivided into multi-alternative models (geological mapping, prediction, general prospecting) and double alternative models (specialized prospecting).An exploration-oriented classification of the PGMs of mineral deposits is discussed. According to this classification the variety of known genetic deposit types is reduced to a limited number of generalized PGM types. The development of typical PGMs is illustrated with examples of magnetitic deposits of Siberia.

Notes: A new mode of operation for the Turam electromagnetic exploration system is proposed in which the transmitter loop is placed across the expected trend of a conductor and the receiver is operated along lines parallel to one side of the transmitter. The concept appears to offer several benefits which include greatly extended traverse length, the use of large coil spacing, rejection of the effects of conductive environments, and consistency in the indication of target dip.

Notes: Classical methods of interpretation of reflection seismic data are such that interpretation and processing usually occur in the “collected” frame of reference. However, in recent times other data planes have gained increasing acceptance in seismology as a viable alternative. Through linear transformations applied to a record section, both the t—p- and p—x-planes can be produced. The r—p-domain may be obtained from the t—x-plane by a transformation known as slant stacking. Normal practice has been to do most of the data processing in the t—x-plane and then transforming to the r—p-plane. However, many of the procedures used in the t—x-domain can be modified for use in the t—p-plane to increase the coherence.Velocity inversion may be carried out either in the r—p-domain or further transformed to the p—x-plane where the modified Herglotz-Wiechert inversion may be applied. To perform the inversion, the t—p-wavefield is converted to a p—x-representation by the use of a new linear transformation technique, the cross-stack. By a simple sampling process along a particular p—x-trajectory, the Herglotz-Wiechert method can be used to reconstruct an acceptable velocity model of the subsurface. A comparison of derived velocity structures is made between that produced by the Herglotz-Wiechert technique and that of the Dix method.

Notes: 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.

Notes: Forward filters to transform the apparent resistivity function over a layered half-space into the resistivity transform have been derived for a number of sample intervals. The filters have no apparent Gibbs' oscillations and hence require no phase shift. In addition, the end points of the filter were modified to compensate for truncation. The filters were tested on simulated ascending and descending two-layer cases. As expected, “dense” filters with sample spacing of In (10)/6 or smaller performed very well. However, even “sparse” filters with spacing of In (10)/2 and a total of nine coefficients have peak errors of less than 5% for p1:p2 ratios of 10–6 to 106. If a peak error of 5.5% is acceptable, then an even sparser filter with only seven coefficients at a spacing of 3 In (10)/5 may be used.

Notes: The concept of minimum phase is clarified for geophysicists by collecting in one place the properties of minimum phase functions. The “earliest energy arrival” property in the time domain, the “minimum phase-slope property” in the frequency domain, and of the role of causal all-pass filters are demonstrated. The emphasis is placed on keeping the mathematics within the realm familiar to geophysicists and on making clear the somewhat imperfect match between physical continuous time functions and their associated discrete time representations.

Notes: Volcanic maar structures sometimes contain important groundwater reserves. The volcanic formations of the Beaunit maar have a much higher magnetization than their granite substrata. The structure of the maar and its volcanoclastic infill can therefore be defined by magnetic surveying and resistivity soundings. Resistivities of the geological formations present in the Beaunit maar are comparable to those recorded in other volcanic systems. The geophysical data, confirmed by drilling tests, contain useful information on the hydrogeological regime of the volcanic system under investigation.

Notes: In migration procedures, the velocity profile of the subsurface is the most important input information. Since, in general, this information is only approximately known, errors in the migration output due to errors in the velocity input occur in all practical applications. In migration, velocity errors and depth errors can be interchanged. This interchange property is perfect in the paraxial approximation. From this result it follows that migration with incorrect velocities may still yield correctly migrated data if the imaging principle is modified. This attractive property can be used in the stripping version of migration (recursive migration).

Notes: The presence of the water layer in marine seismic prospecting provides an effective waveguide for acoustic energy trapped between the sea-bed and the sea-surface. This energy persists to large ranges and can be the dominant early feature on far-offset traces. On airgun records, there is commonly a lower frequency set of arrivals following the water-trapped waves. These arrivals are not as obvious with higher frequency watergun sources. By using a combination of intercept-time/slowness (τ—p) mapping on observational data and theoretical modelling, we are able to identify the origin of the events. If a very rapid increase in a seismic wavespeed occurs beneath the sea-bed sediments, a new waveguide is formed bounded by the sea surface and this transition zone. The low frequency waves are principally guided within this thicker waveguide. Numerical filtering in the τ—p domain followed by trace reconstruction is very effective in removing the low frequency noise.

Notes: The efficacy of the magnetotelluric and audiomagnetotelluric (MT/AMT) methods for detailing the structure of a hypothetical geological section is investigated by using the singular value decomposition (SVD) technique. The section is representative of southeastern Turkey, which is mostly covered by basalt and is a prime area for oil exploration. One of the geological units, the Germav shale at a depth of 600 m, is a problem layer for electromagnetic surveys because of its very low resistivity (on average 3 Ωm) and highly variable thickness across the area (200–900 m). In the MT frequency range (0.0004–40 Hz) its total conductance—or, since its resistivity is known from resistivity log information, its thickness—is the best resolved model parameter. The total depth to the Germav shale and the resistivity of the Cambrian/Precambrian basement are the marginally resolved parameters. In the AMT frequency range (4–10000 Hz) the resistivity of the surface basalt layer strongly affects the resolution of the other, less important, model parameters which are the total depth to the Germav shale and the total conductance of the Germav shale. The errors in the measurements determine the number of model parameters resolvable, and are also important for interpretation of the geological model parameters to within a desired accuracy.It is shown that statistical evaluation of the MT and/or AMT interpretations by using an SVD factorization of the sensitivity matrix can be helpful to define the importance of some particular stage of the interpretation, and also provides a priori knowledge to plan a proposed survey. Arrangements of MT and AMT observations, together with some Schlumberger resistivity soundings, on a large grid will certainly provide three-dimensional detailed information of the deep geoelectric structure of the area.

Notes: A method for interval velocity analysis is formulated on the basis of wavefield extrapolation, i.e., on the basis of wave-equation migration. When this scheme is applied to multioffset seismic sections or to an ensemble of CMP gathers, it allows for the proper treatment of dipping events. The underlying assumptions are that local velocities should be derived from data associated with events within the interval under consideration. To minimize the effect of the region above the layer of interest, the data are first extrapolated to the top of the analysis interval. Subsequent analysis of these data then pertains to the events within this interval. Velocity estimation consists of repeated wavefield extrapolations through the analysis interval using a set of trial velocities. The optimal velocity is chosen on the basis of coherency measures designed to express the collective phase agreement among a set of offset Fourier modes. The reliability of this approach to interval velocity estimation is demonstrated on synthetic multi-offset data.

Notes: 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.

Notes: A principal use of the vertical seismic profile (VSP) is to determine the variation of seismic velocity with depth. Presented here is a discussion of the errors involved with the time picks of a VSP survey and several methods currently used to calculate a velocity section from these time picks. Another technique is proposed, based on the least-squares inversion of the traveltimes, to arrive at a better estimate and statistical description of the velocity section. This technique uses the Levenberg-Marquardt damped least-squares formulation and ray tracing through a horizontally-layered medium to iteratively refine the velocity section. The accuracy and robustness of the procedure are investigated by inverting noisy traveltime curves and comparing these results to the original model velocity section. Agreement is found to be good. One interesting feature of the inverse procedure is that, for certain geometries, it can resolve a few velocity layers, even though there are no measurements made inside those layers. Three actual VSP surveys are analyzed and compared to their corresponding sonic logs. In two of the surveys, it appears that there is some velocity dispersion. Velocity changes associated with gas saturation are evident on one of the surveys.

Notes: It is well recognized that in order to realize the full potential of the Vibroseis technique, one needs to ensure accurate phase locking and a meaningful cross-correlation. To achieve these two important objectives we require an accurate estimate of the compressional stress wave radiated by the vibrator into the ground.In this paper a simple method (subject of a patent application) is developed for predicting the compressional stress waves radiated by a vertical vibrator. The main feature of the proposed method is that it involves the field measurement of the acceleration of the reaction mass and the baseplate, respectively.The method is illustrated by computing the compressional stress waves generated by a typical vertical vibrator radiating into ice, chalk, sand, and mud. It is shown that for a seismic vibrator radiating into hard ground the pressure of the downgoing P-wave is 180° out of phase with the baseplate velocity. It is also shown that when the driving force of the seismic vibrator has a flat amplitude spectrum, the amplitude spectrum of the downgoing P-wave falls off by 6 dB/octave towards low frequencies.

Notes: Bedding on a scale small in comparison to the wavelength results in transverse isotropy. On the other hand, anisotropy has been observed in the field, particularly with horizontally polarized shear waves. In this article velocity data from 23 wells are used to estimate the resulting transverse isotropy and to compare these data with anisotropy observations reported in the literature. Since the well data were compressional velocity (or rather transit time) only, the corresponding shear velocity data were estimated on the basis of a reasonable—though arbitrary—assumption of Poisson's ratio. Actual densities were used wherever they were available; for the remainder of the data, density was assumed to be constant throughout the averaging interval.The anisotropy data estimated on this basis were generally considerably smaller than the observational data that have been reported in the literature. Although both the above assumptions—functional dependence of Poisson's ratio on transit time and constant density—tend to underestimate anisotropy, and although 23 wells is but a small subset of all available data, it appears unlikely that bedding alone could cause anisotropy of the magnitude reported.

Notes: In land seismic surveys spectrum equalization can increase the quality of seismic data in a selected frequency band. The power of lower frequencies in the spectrum of input traces is generally greater than that of higher frequencies, particularly in land seismic surveys because of ground roll. In order to improve the quality of seismic data it is necessary to raise the energy of higher frequencies to the same level as that of lower frequencies, without alteration of the phases.The first step of the method is to compute the amplitude spectrum of each input trace to determine a weighting function which is then applied to the amplitude spectrum in order to balance it. The function is the inverse of the short wavelength variation of the amplitude spectrum. The short wavelength variation can be obtained by interpolation between average values of the modulus of the amplitude spectrum computed in narrow bands within a selected band of frequencies. Another way of obtaining the short wavelength variation is to apply a low-pass filter to the amplitude spectrum. The calculations are readily performed in the frequency domain by the Fourier transform.Spectrum equalization is automatically adjusted to each trace and does not modify the average amplitude in the time domain. However, as the frequency band and energy of the ground roll both vary according to the distance from the shot, spectrum equalization tends to make the spectrum of output traces independent of the offset distance.The use of spectrum equalization before any two-dimensional filtering improves ground roll elimination. Continuity and resolution of horizons are also increased by spectrum equalization before CDP stack.Several examples of applications of spectrum equalization to seismic land and marine surveys are shown.

Notes: A thin superparamagnetic layer on the earth's surface greatly affects the transient electromagnetic response of a conducting ground. The effect of the layer is most evident for singleloop transient electromagnetic data where transient voltages decay as 1/t. Even when a separate transmitter and receiver are used, the effect of the superparamagnetic layer is still pronounced. In this case the effect of the 1/t term in the equation is much less. More dominant now is a 1/t2 term. The effect of the superparamagnetism can readily be seen in the analytical expressions for the apparent resistivities. If the presence of the superparamagnetic layer is not recognized, then the apparent resistivities decrease with time rather than approach the true value of the host rock.

Notes: The use of very long streamers in marine seismic surveys makes the response offset dependent. Line equalization networks are used to balance this effect.In this paper the response of the transformerless streamer system driving a charge amplifier is analyzed. Examples are given for a seismic cable 2500 m long, showing that the delay and the reduction in amplitude of the streamer impulse response in the absence of leakage are of the order of 1 ms and 0.6 dB, respectively. This means that the line equalization network currently used with the transformerless streamer system is not needed. We have also demonstrated that the presence of leakage resistance across the hydrophone group render the sensitivity of the transformerless streamer system offset dependent.

Notes: 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.

Notes: Field results of shallow seismic reflections obtained with a propane-oxygen detonator (POD) are presented. The survey site was in a tin-mining area of the Kinta Valley in Malaysia. The shallow and irregular limestone bedrock is overlain by alluvial “tailing” and virgin sediments. Sizes of such mining areas can range from about 320 ± 320 m2 to 900 ± 900 m2. The survey was intended to delineate the topography of the bedrock, which is of vital importance in tin ore exploration and exploitation. The equipment included single- and 12-channel signal enhancement seismographs, the POD, a hammer and thumper. The inexpensive and portable POD generates directional waves of reproducible form, variable energy of high frequency, and only a few surface waves at short offsets. Reflections at around 200 Hz were obtained from the shallow bedrock at about 25 m as well as from very shallow lithological interfaces. The interpretation of seismograms is supported by drill-hole lithological sections and synthetic seismograms. The data illustrate the successful use of shallow reflections for mapping irregular bedrock. Reflection seismics can provide better horizontal and vertical details than the refraction method. Further improvements based on the data-processing flexibility of new signal enhancement seismographs and synthetic seismograms are suggested.

Notes: Dynamic elastic moduli like E, μ, K and μ of the foundation rock of a dam have been determined by finding Vp- and Vs-velocities by seismic refraction with a hammer as source. Some parameters such as “fracture frequency” and “rock quality designation” (RQD) of the foundation rock have been derived using “average regression curves” and Vp-velocities. By comparing K/μ with Vp/Vs, a few locations showing weathered conditions have been demarcated. This compares well with RQD values of those locations.

Notes: Using homogeneous full-space and half-space models, we show that induced-polarization properties of the medium influence the inductive coupling between two circuits. It is suggested that existing methods to interpret electromagnetic sounding data should be viewed with caution if the electro-chemical dispersion is not taken into account.

Notes: A synthetic seismogram that closely resembles a seismic trace recorded at a well may not be at all reliable for, say, stratigraphic interpretation around the well. The most accurate synthetic seismogram is, in general, not the one that displays the smallest errors of fit to the trace but the one that best estimates the noise on the trace. If the match is confined to a short interval of interest or if the seismic reflection wavelet is allowed to be unduly long, there is considerable danger of forcing a spurious fit that treats the noise on the trace as part of the seismic reflection signal instead of making a genuine match with the signal itself. This paper outlines tests that allow an objective and quantitative evaluation of the accuracy of any match and illustrates their application with practical examples.The accuracy of estimation is summarized by the normalized mean square error (NMSE) in the estimated reflection signal, which is shown to be(/n)(PN/PS)where PS/PN is the signal-to-noise power ratio and n is the spectral smoothing factor. That is, the accuracy varies directly with the ratio of the power in the signal (taken to be the synthetic) to that in the noise on the seismic trace, and the smoothing acts to improve the accuracy of the predicted signal. The construction of confidence intervals for the NMSE is discussed. Guidelines for the choice of the spectral smoothing factor n are given.The variation of wavelet shape due to different realizations of the noise component is illustrated, and the use of confidence intervals on wavelet phase is recommended.Tests are described for examining the normality and stationarity of the errors of fit and their independence of the estimated reflection signal.

Notes: There are numerous modeling techniques commonly employed for the computer simulation of seismic wave propagation. The capabilities of these techniques vary according to the theoretical foundations and subsequent approximations upon which the algorithms are based. This paper constitutes a comparative review of seven modeling methods: geometric ray theory, asymptotic ray theory, generalized ray theory, Kirchhoff wave theory, Fourier synthesis, finite differences, and finite elements.These methods can be categorized as ray or wave, acoustic or elastic, and can be contrasted according to their relative abilities to simulate such behavior as wave interference effects, diffractions, and mode conversions. As is implied by their names, geometric ray theory and asymptotic ray theory are both ray methods. The other five methods provide wave theory simulations. Geometric ray theory and Kirchhoff wave theory are normally implemented in acoustic form, while the other methods are readily adapted for computing elastic theory solutions. Generalized ray theory and Fourier synthesis are more limited in the complexity of geological model they can accommodate than are the other techniques. The methods which typically demand the greatest computer resources are the finite-difference and the finite-element techniques. All methods can incorporate at least some multiple events. Diffractions, however, are only inherent in the solutions computed by Kirchhoff wave theory, finite differences and finite elements. Attenuation is readily incorporated in both the Fourier synthesis and the finite-element methods.As an example of the application of seismic modeling, a geological model representative of a typical petroleum exploration target is used to compare vertical seismic profiles calculated by different modeling methods.

Notes: The cross-hole variant of the magnetometric resistivity (MMR) method requires two bore holes in the vicinity of a conductive target. In the first, two fixed current electrodes are located, one above the other. They are linked to a low frequency current source by cables, the whole system forming a vertical current bipole. In the second, a sensitive coil measures the axial magnetic field as a function of depth. For a uniform earth, if both holes are vertical, the measured component vanishes by symmetry. However, the presence of a local conductor channels the current and causes an anomalous magnetic component which is interpreted to indicate the position, shape and relative conductance of the target.Mineral deposits are often lamellar in form. The conductive disc is the simplest bounded lamella for which MMR responses may be computed. It is excited by a single current source on its axis. The second source and the surface of the earth are assumed to be far away, a valid assumption for down-hole measurements.The numerical method introduces a new integral equation describing the interaction of current dipoles located in the plane of the disc. The equation is solved analytically for a disc of infinite radius, a layer, and the result is compared with a corresponding known boundary value solution.The computed radial current in the disc and the magnetic field generated by it are described in terms of a current channelling number. The magnitude of the computed field is of the order of one nanoTesla for a typical mining problem.

Notes: Time horizons can be depth-migrated when interval velocities are known; on the other hand, the velocity distribution can be found when traveltimes and NMO velocities at zero offset are known (wavefront curvatures; Shah 1973).Using these concepts, exact recursive inversion formulae for the calculation of interval velocities are given. The assumption of rectilinear raypath propagation within each layer is made; interval velocities and curvatures of the interfaces between layers can be found if traveltimes together with their gradients and curvatures and very precise VNMO velocities at zero offset are known.However, the available stacking velocity is a numerical quantity which has no direct physical significance; its deviation from zero offset NMO velocity is examined in terms of horizon curvatures, cable length and lateral velocity inhomogeneities.A method has been derived to estimate the geological depth model by searching, iteratively, for the best solution that minimizes the difference between stacking velocities from the real data and from the structural model.Results show the limits and capabilities of the approach; perhaps, owing to the low resolution of conventional velocity analyses, a simplified version of the given formulae would be more robust.

Notes: From a great variety of in situ shear wave experiments, i.e., reflection, refraction and borehole surveys in the shallow sediments of the north German plains, several specific properties have been derived. Shear waves (S) differ from compressional waves (P) in that:〈list xml:id="l1" style="custom"〉1they are not affected by the degree of water saturation. Thus, they provide a better correlation between the velocity Vs and (solid) lithology;2they generally have lower frequencies, but shorter wavelength and, hence, a better resolution of thin layers;3they have lower absorption Qs−1 and hence a better penetration in partially saturated and gas-containing sediments than P-waves.Correlations have been established between Vs and the confining pressure and between reduced Vs values and several lithological parameters like the grain size of sandy material. More lithological and hydrological information is obtained by using S- and P-wave surveys along the same profile. The best information on a sedimentological structure is obtained by the simultaneous observation of Vs, Vp, Qs and Qp.

Notes: In the design of linear airgun arrays the interaction between the airguns is usually neglected. We review the different formulae which have been proposed for the minimum separation between airguns at which the interaction is negligible. These formulae can all be approximated by a linear function of a single variable.We have analyzed a large number of measurements in order to establish the amount of interaction between two airguns of various volumes at different pressures and depths. The resulting far-field signature has been measured and compared with the sum of the signatures from the two airguns measured in the same experimental situation. The changes in primary pulse amplitude, bubble period and primary/bubble peak-to-peak amplitude ratio were computed from the measurement data as a function of airgun separation, chamber volume, chamber pressure and airgun depth. The influence of a waveshape kit was investigated, and the effects of interaction and the effects of using a waveshape kit were compared.

Notes: Almost all ray-tracing methods ignore the analysis of the amplitudes of seismic arrivals and therefore utilize only half of the available information. We propose a method which is a combination of ray-tracing imaging and transformation of the amplitudes of wide-aperture data.Seismic data in the conventional X-T domain are first transformed to the domain of intercept time τ and ray parameter p to recover the plane wave response. The next step is the derivation of a series of plane wave reflection coefficients, which are mapped as a function of τ and p. The reflection coefficients R(τ, p) for two arbitrarily chosen traces can then be used in our inversion method to derive a slowness-depth and a density-depth profile. It is shown that the inclusion of amplitudes of seismic arrivals (in this method, we consider the acoustic case) makes the inverse method highly stable and accurate. In a horizontally stratified medium one can recover separate profiles of velocity and density. Since this method utilizes large-offset data, it can be used for separate recovery of velocity and density to a greater depth.

Notes: Random amplitude and phase errors in seismic input data introduce a coherent distribution of migration half-circles or “smiles”, the occurrence of which may cause a significant decrease of signal-to-noise ratio. In addition, the effect of quantization errors is discussed for different wordlengths, used both during acquisition and during data processing. Results of sign-bit recordings are shown.

Notes: A hybrid method for wave field computation in two-dimensional heterogeneous media is proposed. The proposed method is a combination of analytical and numerical techniques. The method is based upon the separation of wave propagation and scattering and upon the description of each process by the most suitable technique. The SH wave scattering problem is used to elucidate the proposed method.Examples of numerical computations using the hybrid method are considered for a number of simple models. The analysis of the results shows that the hybrid method gives both a detailed and a reasonably accurate description of the total wave field.

Notes: After a brief discussion of MT modeling methods, we expose a particular application of finite elements that may be assimilated to a finite-difference method. The resulting linear equations obtained are similar to the transmission network equations for two-dimensional media. The introduction of resistivity discontinuities is more rigorous than in the usual finite-difference development. The consequence is an improved accuracy.We also present some conclusions about various problems encountered in modeling, such as the choice of network boundary conditions, linear system resolution and the final derivation of apparent resistivity. Application of successive over-relaxation is discussed and we detail rules for mesh design that control result accuracy and iterative convergence.

Notes: The renewed interest in the self-potential method of exploration for mineral deposits gives an understanding of the self-potential mechanism new importance. The cause of SP anomalies in general lies in the interference between simultaneously occurring nonequilibrium phenomena. However, theories of the mechanism of mineral SP anomalies generally relate the SP anomaly to the equilibrium potential of the chemical reaction supposed to occur on the ore body surface. In this paper, I reformulate these equilibrium mechanisms in terms of nonequilibrium thermodynamics. The result is that the SP anomaly depends not on the equilibrium potential alone, but also on the potential resulting from current transferred across the ore body—electrolyte interface. It is not possible to calculate the overpotential theoretically because of the number of complicating factors, and experimental data are not available. This does not imply that SP data are uninterpretable quantitatively. SP data may be interpreted similarly to other potential field data.

Notes: The detectability of an intermediate layer in a three-layer earth model in the time domain has been investigated. The calculations were made for the perpendicular loop (designated system II) and vertical-coplanar (designated system III) electromagnetic (EM) sounding systems. The primary excitation employed is a train of half-sinusoidal and square waveforms of alternating polarity. The time-domain response has been determined by Fourier transformation of the matched complex mutual coupling ratios into the time domain and by linear digital filtering. Top and bottom layers have equal resistivity. EM responses have been computed for conductive and resistive intermediate layer with a wide range of thickness and for two values (500 m and 1000 m) of loop-separation. For the detectability analyses, the root mean square (rms) difference between three-layer and homogeneous-earth responses is adapted. The threshold value for detectability is defined as an rms difference of 10% and the measurement error is arbitrarily assumed to be of the order of 3%. It is observed that the perpendicular-loop system is better than the vertical-coplanar system in detecting thin intermediate layers (either conductive or resistive). For a loop separation of 1000 m and half-sinusoidal pulse excitation, the detectable thickness ratio (h2/h1) is 0.10 by system II for the conducting middle layers; for square pulse excitation the corresponding thickness ratios are 0.06 for system II and 0.12 for system III. For a loop separation of 1000 m and half-sinusoidal pulse excitation in detecting the resistive intermediate layers, the corresponding thickness ratios are 0.9 for system II and 2.25 for system III; while for square pulse excitation the thickness ratios are 0.55 for system II and 1.55 for system III.Results in the frequency domain and time domain (for half-sinusoidal and square pulsed field) have also been presented for systems II and III for detecting conducting layers by considering an earth model where p1≠ p3 and p3 〉 p1 (p is the resistivity). The loop separa- tion used is 1000 m. Direct comparisons between the frequency domain and time-domain results clearly demonstrate the superiority of frequency-domain systems for detecting con- ducting intermediate layers.

Notes: In this paper we develop a recursive algorithm to obtain the layer parameters of an elastic medium (density, P-wave velocity, S-wave velocity) from reflection coefficient matrices in terms of energy flux ratios for a non-normal incidence case. We define a layer impedance matrix, analogous to the impedance of an acoustic medium. Next we derive a matrix relationship between the layer impedance matrix of the n+ 1st layer and the reflection coefficient and parameter matrices of the nth layer. This relationship leads to recursively computing the parameters of the subsurface. We show that the elastic case—unlike the acoustic case—allows one to recover the layer parameters from the impedance matrix for non-normal incidence. The results of this work play a key role in the solution of the inverse problem with non-normal-incidence plane-wave seismic data when using a downward continuation technique.

Notes: Frequency-dependent attenuation of compressional waves within the earth has been estimated in the vicinity of wells from〈list xml:id="l1" style="custom"〉1spectral power ratios of the coherent events in separate time gates on the seismic section2matching a broadband synthetic trace with seismic data at the well, and3determining the operator that transforms one down(up) going pulse recorded in the well into another recorded at a deeper (shallower) level.The accuracy of estimation of all three methods was insufficient to estimate attenuation over small depth intervals, and it was not possible to distinguish between the contribution due to internal multiples and that of genuine absorption with much confidence. Spectral ratios from (1) showed a smoother variation with frequency—and one more consistent with other estimates—when they were compensated for the spectra of the reflectivities over the time gates employed, but they did not provide more than a broad indication of attenuation over a substantial depth interval. Approach (2) was hampered by the restricted durations over which synthetic trace and seismic data can be reliably matched; approach (3) gave the best results. Here matching is a much more powerful tool than the spectral-ratio techniques that are commonly applied since it can yield the form of the attenuation operator, i.e., both its amplitude and phase response, together with properly defined measures of its accuracy, while at the same time it minimizes the influence of noise and local interference effects at each recording level.For seismic target depths where internal multiple activity was low the logarithms of the amplitude responses of the estimated attenuation operators decreased approximately linearly with frequency and the phase responses showed no significant dispersion. Application of approach (3) to downgoing and upgoing waves estimated from a vertical seismic profile revealed the importance of changes in frequency-dependent geophone coupling and their effect on values of Q determined from downgoing pulses only.

Notes: From the wealth of information which can be deduced from VSP, the information most directly comparable to well logs is considered: P-wave and S-wave interval velocity, acoustic impedance, and the velocity ratio γ=Vs/Vp. This information not only allows better interpretation of surface seismic sections but also improves processing.For these results to be usable a number of precautions must be taken during acquisition and processing; the sampling in depth should be chosen in such a way that aliasing phenomena do not unnecessarily limit the spectra during the separation of upwards and downwards travelling waves. True amplitudes should be respected and checked by recording of signatures, and the interference of upwards and downwards travelling waves should be taken into account for the picking of first arrivals.The different steps in processing and the combination of results in the interpretation of surface seismic results are described with actual records.

Notes: The arrival-time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x - t-domain.If the subsurface is one-dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly true, though the arrival-time curves are still hyperbola-like. In the case of transverse isotropy, however, classical interpretation of these curves fails. Interval velocities calculated from t2 - x2-curves do not always approximate vertical velocities and therefore cannot be used to calculate depths of reflectors.To study the relationship between velocities calculated from t2 - x2-curves and the true velocities of a transversely isotropic layer, we approximate t2 - x2-curves over a vertically inhomogeneous transversely isotropic medium by a three-term Taylor series and calculate expressions for these terms as a function of the elastic parameters. It is shown that both inhomogeneity and transverse isotropy affect slope and curvature of t2 - x2-curves. For P-waves the effect of transverse isotropy is that the t2 - x2-curves are convex upwards; for SV-waves the curves are convex downwards. For SH-waves transverse isotropy has no effect on curvature.

Notes: A combination of ray theory and diffraction theory has been used to simulate reflections of seismic pulses from a full plane and reflections and diffractions from a half plane buried horizontally in earth models with linear increase of velocity with depth. Wave forms of signals reflected from a full plane in this case are modifications of the source pulse, even if the medium is assumed to be perfectly elastic. The extent of pulse modification increases with increase of the velocity gradient. The duration of the reflected pulse is always longer than that of the source pulse. The pulses diffracted from the edge of the half plane in such cases do not lie along hyperbolic curves in the time-offset plane. Diffracted signals can be observed only up to a limited distance from the edge of the half plane equalling half the horizontal range of the ray which is tangential to the half plane and has its end points at the level of the source and the receiver.

Notes: A new method with general applications for seismic data is presented for spectral extrapolation. The method gives a restored image that is optimum in the sense of minimum norm and can be adapted to incorporate any constraints on the reconstruction. As an illustration of the technique it is first applied to a synthesized (noise-free) image. Then it is shown that enhancement of a density log (run in a hole drilled entirely in Coal Measures rocks) is possible using constraints taken from the geologist's log. A synthetic trace with significantly improved event arrival times and general character is obtained.

Notes: The dynamic finite-element method allows frequency-dependent reflection and transmission coefficients to be computed for Love waves scattered by two-dimensional inhomogeneities in coal seams.Clean faults of zero hade angle show a throw-dependent cross-over frequency in reflection spectra, and throw-dependent conversion from fundamental to first higher mode energy in transmission spectra. Oblique faults show mode conversion in reflection spectra which is relatable to the fault hade angle by simple Huygens-theory models. Thin fracture zones or dykes normal to the seam show a reflection maximum when thickness of the zone is of order one quarter of the seam wave wavelength.Published field data from two known faults and a dyke are compared with the modeling results and support the belief that broad-band seam-wave data are capable of characterizing a seam discontinuity (throw, hade angle, dyke thickness) as well as locating it. Development of such procedures will require extensions to existing field practice and processing.This research was funded in part by the National Energy Research Development and Demonstration Council of Australia. The authors thank The Broken Hill Proprietary Company Limited for permission to quote from company case histories.

Notes: The response of a rectangular plate to the Crone PEM system has been investigated for different values of depth, dip and conductance parameters. The conductance aperture diagram is presented to estimate the conductance and depth to the top of a vertical sheet. Information regarding dip and depth may be obtained using the appropriate nomogram.

Notes: In areas where steep dips are encountered conventional practice in resistivity work has involved orienting arrays favourably in relation to the geological strike. In concealed conditions, however, the geological strike may not be known; moreover, strike may change with depth. Considerable advantage is to be gained, in such circumstances, by the use of crossed square arrays in that these yield orientationally insensitive resistivity measurements and also allow strike determinations and measurements of the effective vertical anisotropy.Two traverses of crossed square array observations are presented, together with one deeper sounding. The results show that, in favourable circumstances, reliable data on concealed strike directions can be obtained, and that the anisotropy findings greatly assist the subsequent interpretation. Model results pertinent to the field material are presented and discussed.

Notes: Absorption of seismic energy in the earth reduces amplitudes and changes phases of the propagating seismic waves. Amplitudes are usually recovered according to an estimated exponential decay curve, while phase distortions are generally disregarded. Therefore, accurate processing of seismic data requires a careful investigation of the relationship between absorption and phases.In this paper a procedure is suggested to achieve this goal, and some related topics are worked out. A method is outlined for computing synthetic seismograms and vertical seismic profiles with phase distortion due to absorption.The algorithm works in the frequency domain, and it provides for absorption according to the usual model of exponential decay of amplitude with distance. The absorption coefficient is a linear function of frequency and is related to the quality factor Q of the rocks. Complex seismic velocities are introduced and minimum-phase delay due to absorption is assumed for all cases considered.Methods for estimating Q profiles from seismic well surveys and seismic data are described. Comparison between field and synthetic data shows the effectiveness and benefits of the procedure. Some applications of the method to phase distortion recovery and wavelet processing are presented.

Notes: 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.

Notes: A numerical method is given for calculating resistivity and induced polarization anomalies produced by a surface polarization model. Surface polarization is generated when a purely electronic conductor is located in an electrolyte environment. The system that develops on the boundary between the conductor and the electrolyte is described macroscopically by a net surface charge distribution and an electric double layer. An integral equation is derived for the potential by assuming that the electronic conductor forms an equipotential system and that the polarization impedance across the boundary is linear. The integral equation is solved by means of the method of subsections. As an application some numerical modeling results are presented. The surface impedance values used in calculations are based on laboratory measurements that are briefly described. Implications of the results for scale modeling are discussed.

Notes: Offset continuation is a technique that was recently proposed for the dip moveout correction. This correction can be carried out in the time-wavenumber domain using a proper partial differential equation that links sections with different offset.It has been shown that a single spike in a common-offset section—corresponds to a semi-elliptically shaped reflector with foci located at the source and receiver in the section migrated after dip moveout correction.The sections that result after offset continuation, stack, and migration are thus a superposition not only of semicircles, but also of semi-ellipses with different lengths of axes. This effect smears the migration alias-noise which, without offset continuation, would appear as migration circles not close enough together to interfere destructively.Offset continuation can improve the quality of seismic sections in several ways:—the velocity analyses are more readable, less dispersed and dip independent; diffraction tails arrive with the same normal moveout velocity as the apex and thus diffraction-noise can be “stacked out”;—noise produced by aliasing in the migrated section is reduced.In this paper we give a practical and conceptual interpretation of the offset continuation method, with a generalization to three-dimensional volumes of data. A critical examination of several synthetic and field data examples shows the actual possibilities and advantages of offset continuation.

Notes: There is a significant increase in terrestrial heat flow with depth in the Hinton-Edson area of the deep part of the western Canadian sedimentary basin in Alberta. This is especially true near the Rocky Mountain foothills which is an area of high relief, high hydraulic head and regional water recharge. Gravity-imposed downward movement of meteoric water through the thick sedimentary strata with velocities as low as 10–10 m/s to 0.5 × 10–9 m/s may cause an increase of heat flow with depth. Such disturbance of heat flow with depth on a regional scale in the sedimentary strata means that it is not possible to determine the background conductive steady-state heat flow associated with crustal or upper mantle heat sources in such an area from measurement of conductive heat flow in the part of the sedimentary column where water movement occurs. This is because the convective portion cannot be determined, particularly when measurements are made in only part of the regional hydrodynamic system of the basin.

Notes: The heat flow equation in cylindrical coordinates is solved numerically for any general distribution of thermal diffusivity. The temperature stabilization of a borehole is considered, and solutions for the case where thermal diffusivity is a function of radial distance from the borehole are obtained and compared to solutions for uniform diffusivity. The results are discussed in terms of thermal diffusivities that are different for the well contents and for the surrounding material. It is found that the approach to formation temperature is affected by differences between well contents and the surrounding region.

Notes: 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.

Notes: This paper is concerned with an interpretation of the Bouguer anomaly on the Western flank of the Brabant Massif (Belgium). The position, shape, and density contrast of elementary bodies in the upper part of the earth's crust are determined in a purely numerical manner. A batholith-like body and a basin-type structure both adequately account for the observed surface field, at least from a one-sided geophysical and mathematical point of view. Since seismological control is lacking the two models are representative of the ambiguity and indeterminacy of the definitions of the geologic cause of the anomalous features, if the data are restricted to gravity information alone.

Notes: This paper presents a digital linear filter which maps composite resistivity transforms to apparent resistivities for any four—electrode array over a horizontally layered earth. A filter is provided for each of three sampling rates; the choice of filter will depend on resistivity contrasts and computational facilities.Two methods of filter design are compared. The Wiener-Hopf least-squares method is preferable for low sampling rate filters. The Fourier transform method is more successful in producing a filter with a high sampling rate which can handle resistivity contrasts of 100 000: 1.