ALBERT

Notes: A method of non-invasive NMR in the earth's field has been developed and is now used for groundwater surveys to depths of investigation of 100 m or more.A circular wire loop of diameter 100 m, laid out on the ground, is employed to excite and receive the NMR signal in the earth's field. However, in areas with high electromagnetic noise, the NMR measurements may be inaccurate.To overcome this problem, a noise-reducing figure-of-eight-shaped antenna, consisting of two touching coils each of diameter 50 m, has been utilized.Using this antenna, the NMR signal has been calculated for different depths of water-saturated layers with various inclinations of the geomagnetic field. The model calculations and experimental data have been compared and found to be mutually consistent. The two-coil antenna is shown to be suitable for studies at depths of up to 30–40 m, which is of practical importance for engineering geology.

Notes: A general inversion scheme based on a genetic algorithm is developed to invert seismic observations for anisotropic parameters. The technique is applied to the inversion of shear-wave observations from two azimuthal VSP data sets from the Conoco test site in Oklahoma. Horizontal polarizations and time-delays are inverted for hexagonal and orthorhombic symmetries. The model solutions are consistent with previous studies using trial and error matching of full waveform synthetics. The shear-wave splitting observations suggest the presence of a shear-wave line singularity and are consistent with a dipping fracture system which is known to exist at the test site. Application of the inversion scheme prior to full waveform modelling demonstrates that a considerable saving in time is possible whilst retaining the same degree of accuracy.

Notes: A numerical simulation of electromagnetic propagation through a multilayered medium is performed in order to explain and interpret the signal received from the radar sounding of a temperate glacier. During the winter of 1990, several radar profiles were obtained on the Mont-de-Lans glacier in the French Alps with a ground penetrating radar which uses a phase modulation of the transmitted pulse by coded sequences. The pulse compression is obtained by applying the matched filter to the received signal, which provides a range-resolution of about 8 m in the ice. The profiles recorded on the temperate glacier do not show a single clear reflection from the ice-bedrock interface, but they exhibit widely distributed energy decreasing with depth. This may be due to the inhomogeneous inner structure of the temperate glacier and we use a simple model of a layered medium to compute a simulation of the propagation. Thus, partial reflection at each layer and scattering from a rough basal interface may explain the observed signal. A computer-based technique is used to locate on the data the bottom of the glacier in order to estimate the ice thickness. The results from the different radar profiles are consistent and are a good fit to the thickness which has been determined by other geophysical methods.

Notes: Both approximate and exact formulations for the interaction of an incident elastic wave with a cased borehole are presented. In the approximate method, simple and explicit formulae are derived for the pressure in fluid at low frequencies. In the exact method, elastic potentials in each annulus are represented as a superposition of fundamental solutions to the Helmholtz equations. Continuity of displacements and stresses across layer boundaries are used to determine unknown coefficients. A global matrix algorithm is employed to compute simultaneously these coefficients in individual layers. Calculations show that, in cased boreholes, the borehole effects on downhole seismic measurements are more significant than in open boreholes. A strong resonance occurs in the fluid for SV-wave incidence from a soft formation. This resonance is prominent even at very high frequencies because the tube-wave velocity is raised well above the formation shear velocity by the steel pipe. At a particular angle of incidence of a plane P-wave, the pressure in the fluid is near zero at low frequencies (the cased borehole screening phenomenon). For hard formations and frequencies above 1 kHz, the cased borehole influence on a downhole geophone measurement is significant, especially at grazing incidence. For soft formations, both the pressure in the fluid and the solid displacement on the borehole wall show strong dependence on frequency and angle of incidence, even at low frequencies.

Notes: Seismograms predicted from acoustic or elastic earth models depend very non-linearly on the long wavelength components of velocity. This sensitive dependence demands the use of special variational principles in waveform-based inversion algorithms. The differential semblance variational principle is well-suited to velocity inversion by gradient methods, since its objective function is smooth and convex over a large range of velocity models. An extension of the adjoint state technique yields an accurate estimate of the differential semblance gradient. Non-linear conjugate gradient iteration is quite successful in locating the global differential semblance minimum, which is near the ordinary least-squares global minimum when coherent data noise is small. Several examples, based on the 2D primaries-only acoustic model, illustrate features of the method and its performance.

Notes: Evidence from borehole susceptibility logs and the spectral analysis of aeromagnetic data suggests that the three-dimensional distribution of magnetization within the crust can be described as fractal. This property can be exploited in magnetic interpretation methods which explicitly require statistical information on the spatial variation of magnetization. Specifically, we address the problem of magnetic source depth estimation through downward continuation and gridding aeromagnetic survey data using the method of kriging.When magnetic data are continued downwards the depth at which the power spectrum flattens out (the ‘white’ depth) can be taken to be an estimate of the top of the source distribution. This procedure assumes that individual sources are uncorrelated with each other. Taking into account the correlation of the magnetization using a fractal description leads to a reduction in this depth estimate.Gridding of randomly distributed magnetic measurements using kriging requires an estimate of the covariance of the data. Compared with the assumption of a white (uncorrelated) magnetization distribution, using fractal covariances for kriging produces gridded estimates which more closely reflect the statistics of the underlying magnetization process and produce maps with a justifiable degree of smoothness.

Notes: The superposition integral expressing the field due to a magnetic source body is relatively simple to evaluate in the case of a homogeneous magnetization. In practice this generally requires that any remnant component is uniform and the susceptibility of the body is sufficiently low to permit the assumption of a uniform induced magnetization. Under these conditions the anomalous magnetic field due to a polyhedral body can be represented in an intuitive and physically appealing manner. It is demonstrated that the components of the magnetic field H can be expressed as a simple combination of the potentials due to two elementary source distributions. These are, firstly, a uniform double layer (normally directed dipole moment density) located on the planar polygonal faces of the body and, secondly, a uniform line source located along its edges. In practice both of these potentials (and thus the required magnetic field components) are easily computed. The technique is applicable to polyhedra with arbitrarily shaped faces and the relevant expressions for the magnetic field components are suitable for numerical evaluation everywhere except along the edges of the body where they display a logarithmic singularity.

Notes: Improving the accuracy of NMO corrections and of the corresponding interval velocities entails implementing a better approximation than the formula used since the beginning of seismic processing. The exact equations are not practical as they include many unknowns. The approximate expression has only two unknowns, the reflection time and the rms velocity, but becomes inaccurate for large apertures of the recording system and heterogeneous vertical velocities. Several methods of improving the accuracy have been considered, but the gains do not compensate for the dramatic increase in computing time. Two alternative equations are proposed: the first containing two parameters, the reflection time and the focusing time, is not valid for apertures much greater than is the standard formula, but has a much faster computing time and does not stretch the far traces; the other, containing three parameters, the reflection time, like focusing time and the tuning velocity, retains high frequencies for apertures about twice those allowed by the standard equation. Its computing time can be kept within the same limits. NMO equations, old and new, are designed strictly for horizontal layering, but remain reliable as long as the rays travel through the same layers in both the down and up directions.An equation, similar to Dix's formula, is given to compute the interval velocities. The entire scheme can be automated to produce interval-velocity sections without manual picking.

Notes: The Bjøirnøya West Basin lies between latitudes 73° and 74°, longitudes 16°E and 18°E, contains at least 8 km of sediments deposited from the Late Jurassic, and is of considerable interest for hydrocarbon exploration. The Cenozoic extensional tectonics in the basin can be clearly seen from seismic data with normal faulting and from subsidence curves with rapid subsidence. The extension occurred during the Late Palaeocene with active extension lasting about 6 million years (m.y.) followed by thermal cooling. The tectonic subsidence within the study area shows a three-phase development: phase 1, synrift (58–52 Ma (million years before the present day)), is characterized by rapid subsidence; phase 2, postrift (52–5 Ma), by slow subsidence with occasional uplift; and phase 3 (5–0 Ma), by rapid subsidence. An adaptive finite-element model, with consideration of the radiogenic heat production in the lithosphere, has been used to model the subsidence and heat flow. The modelling of subsidence shows the β-factor distribution varying from 1.9 to 3.5 with an average of 2.4 for the uniform lithospheric extension. The heat-flow modelling predicts a rapid increase of heat flow during the Early Palaeocene. The maximum heat flow at about 52 Ma, which could be as much as 3.0 hfu (10−6 cal/cm2/s), was followed by a decrease in heat flow. A plate-weakening model has been proposed to explain the rapid subsidence for the last 5 m.y. by flexure of the elastic lithosphere which is weakened by a decrease in elastic thickness caused by an increase of the temperature gradient in the lithosphere. The plate-weakening model predicts a heat-flow increase at 5 Ma of up to 2.0 hfu. Our study, using quantitative modelling of the tectonic subsidence, provides a partial (if not a full) understanding of the tectonic development and thermal evolution of the Bjønøya West Basin.

Notes: We present a discrete modelling scheme which solves the elastic wave equation on a grid with vertically varying grid spacings. Spatial derivatives are computed by finite-difference operators on a staggered grid. The time integration is performed by the rapid expansion method. The use of variable grid spacings adds flexibility and improves the efficiency since different spatial sampling intervals can be used in regions with different material properties. In the case of large velocity contrasts, the use of a non-uniform grid avoids spatial oversampling in regions with high velocities. The modelling scheme allows accurate modelling up to a spatial sampling rate of approximately 2.5 gridpoints per shortest wavelength. However, due to the staggering of the material parameters, a smoothing of the material parameters has to be applied at internal interfaces aligned with the numerical grid to avoid amplitude errors and timing inaccuracies. The best results are obtained by smoothing based on slowness averaging. To reduce errors in the implementation of the free-surface boundary condition introduced by the staggering of the stress components, we reduce the grid spacing in the vertical direction in the vicinity of the free surface to approximately 10 gridpoints per shortest wavelength. Using this technique we obtain accurate results for surface waves in transversely isotropic media.

Notes: A method is described to locate secondary faults, which can be difficult to identify on the Bouguer gravity map. The method is based on cross-correlation between the theoretical anomaly due to a vertical step and the second vertical derivative of the Bouguer anomaly. Faults are located from the closed maxima and minima on the cross-correlation contour map calculated for two perpendicular directions. One-dimensional model computations show that the magnitude of the extremum of the cross-correlation is related to the depth to the top of the hanging wall and the throw of the fault. Application of the method to the Bouguer gravity map of the former mouth of the Yellow River in the Shengli Oilfield area near the Bo Hai Sea shows the effectiveness of the method.

Notes: The Fourier pseudospectral method has been widely accepted for seismic forward modelling because of its high accuracy compared to other numerical techniques. Conventionally, the modelling is performed on Cartesian grids. This means that curved interfaces are represented in a ‘staircase fashion‘causing spurious diffractions. It is the aim of this work to eliminate these non-physical diffractions by using curved grids that generally follow the interfaces.A further advantage of using curved grids is that the local grid density can be adjusted according to the velocity of the individual layers, i.e. the overall grid density is not restricted by the lowest velocity in the subsurface. This means that considerable savings in computer storage can be obtained and thus larger computational models can be handled.One of the major problems in using the curved grid approach has been the generation of a suitable grid that fits all the interfaces. However, as a new approach, we adopt techniques originally developed for computational fluid dynamics (CFD) applications. This allows us to put the curved grid technique into a general framework, enabling the grid to follow all interfaces. In principle, a separate grid is generated for each geological layer, patching the grid lines across the interfaces to obtain a globally continuous grid (the so-called multiblock strategy).The curved grid is taken to constitute a generalised curvilinear coordinate system, where each grid line corresponds to a constant value of one of the curvilinear coordinates. That means that the forward modelling equations have to be written in curvilinear coordinates, resulting in additional terms in the equations. However, the subsurface geometry is much simpler in the curvilinear space.The advantages of the curved grid technique are demonstrated for the 2D acoustic wave equation. This includes a verification of the method against an analytic reference solution for wedge diffraction and a comparison with the pseudospectral method on Cartesian grids. The results demonstrate that high accuracies are obtained with few grid points and without extra computational costs as compared with Cartesian methods.

Notes: Radio signals from very low frequency (VLF) transmitters distributed world-wide have been used for several decades to study the lateral variations of the electrical conductivity in the upper few hundred metres of the earth's crust. Traditionally, in airborne applications, the total magnetic fields from one or two transmitters are measured to form the basis for construction of maps that primarily show those conductive structures that are parallel or subparallel to the direction to the transmitters. The tensor VLF technique described in this paper makes use of all signals available in a predefined frequency band to construct transfer functions relating the vertical magnetic field and the two horizontal magnetic field components. These transfer functions are uniquely determined for a particular measuring site and contain information about the lateral conductivity variations in all directions. First experiences with real field data, acquired during a test survey in Sweden, show that maps of the so-called peaker, the spatial divergence of the transfer functions, give an image of the conducting structures. Most of the structures can be correlated to small valleys filled with conducting sediments or valleys underlain by conductive fracture zones in the crystalline rocks.

Notes: We study the geoelectrical problem of picking out the useful signal from voltage time series, monitored under conditions of a low signal-to-noise ratio and non-stationary noise. Statistical tests performed at different sites show that geoelectrical noise often belongs to the class of non-stationary phenomena with non-Gaussian probability distributions. In such cases, the application of conventional methods of geoelectrical useful signal extraction, based on the stationary white-noise assumption, gives biased estimates. For the on-line processing of geoelectrical recordings, we recommend the use of the periodogram technique combined with the Kolmogorov–Smirnov test, a suitable algorithm of which is described in detail. The suggested procedure allows data acquisition to stop as soon as the useful signal power is estimated with a relative error smaller than a predetermined value. Finally, we compare the suggested procedure with the autoregressive approach. The previously used and simpler periodogram method, applied to the solution of problems of this kind, appears to give better performances than the autoregressive analysis.

Notes: For successful prestack depth migration an accurate velocity model is needed. One method for model updating is based on image gather analysis. In an image gather all reflectors line up horizontally if the correct velocities are used for the depth migration. This is also true for dipping reflectors, as all traces of an image gather belong to the same surface coordinate. The images of the reflector in an image gather curve upwards if the velocity used for the migration is too low, or downwards if the velocity is too high. This deviation can be used for model updating. Curves which depend on depth, offset and a parameter which relates the estimated to the true model are fitted to the image. By calculating the coherence along the deviation curves, this parameter can be estimated and hence an update can be calculated.Formulae are derived for the deviation curves and the update of the velocity depth model for a multilayered model for both shot and common-offset migrated data, with and without gradients. The method is tested on synthetic data with satisfactory results.

Notes: The variation in the density of sediments with depth in a sedimentary basin can be represented by a hyperbolic function. Gravity anomaly expressions for a 2D vertical prism and an asymmetric trapezium with a hyperbolic density distribution are derived in a closed form. These are used in inverting the gravity anomaly of a sedimentary basin with variable density. Firstly, the basin is viewed as a series of prisms juxtaposed with each other. The initial thickness of each prism is obtained from the gravity anomaly at its centre, based on the gravity anomaly of an infinite slab with a hyperbolic density contrast. These thicknesses are improved, based on the differences between the observed and the calculated anomalies. For an improved rate of convergence of the solution, these thicknesses may alternatively be refined using the well-known ridge regression technique. Secondly, the basin is approximated by an asymmetric trapezium and its anomalies are inverted for the parameters of the trapezium using the ridge regression. Since this approximation serves to oversimplify the floor of the basin, it must be used only when the sediment-basement interface has minor undulations. The results of a hypothetical case and two field cases (the San Jacinto Graben, California and the Godavari Graben, southern India) are presented. In both field cases, the interpreted depths are comparable with the real ones, proving the validity of the assumption of a hyperbolic density distribution of the sediments in the two basins considered.

Notes: The polarization direction or 'sign’ of reflected converted P–S waves depends upon the angle of incidence of the incident P-wave. Sign reversal due to reversal of the angle of incidence is often encountered and is an impediment to P–S wave processing and imaging, because when P–S events or P-S migrated images with mixed signs are stacked, destructive interference occurs. We have created and demonstrated a means of correcting for this reversal. To do this, a P-wave angle of incidence is calculated for every point in the image space. This is done by calculating a P–S reflected waveform for every point, by extrapolating the reflected S-wavefield backwards from the receiver line, and then cross-correlating this waveform with the S-wave reflections observed at the receiver line. A multiplier, (sgn α) is assigned to each point in the image space, where α is the angle of incidence of the P-wave.The multiplier was applied to a set of prestack reverse time migration images derived from a cross-borehole physical elastic model data set. The improvement in the stacked image when the sign correction is applied is spectacular. The P-S image quality is comparable to, or better than, stacked migrated P-P images.The method appears to be applicable to all reflection modes and to all recording geometries.

Notes: Fluid permeability in fractured rocks is sensitive to pore-pressure changes. This dependence can have large effects on the flow of fluids through rocks. We define the permeability compliance γ= 1/k(k/δpp)pc, which is the sensitivity of the permeability k to the pore pressure pp at a constant confining pressure pc, and solve the specific problems of constant pressure at the boundary of a half-space, a cylindrical cavity and a spherical cavity. The results show that when the magnitude of permeability compliance is large relative to other compliances, diffusion is masked by a piston-like pressure profile. We expect this phenomenon to occur in highly fractured and compliant rock systems where γ may be large. The pressure profile moves rapidly when fluids are pumped into the rock and very slowly when fluids are pumped out. Consequently, fluid pressure, its history and distribution around injection and production wells may be significantly different from pressures predicted by the linear diffusion equation. The propagation speed of the pressure profile, marked by the point where δpp/δx is a maximum, decreases with time approximately as 〈inlineGraphic alt="inline image" href="urn:x-wiley:00168025:GPR693:GPR_693_mu1" location="equation/GPR_693_mu1.gif"/〉 and the amplitude of the profile also dissipates with time (or distance).The effect of permeability compliance can be important for fluid injection into and withdrawal from reservoirs. For example, excessive drawdown could cause near-wellbore flow suffocation. Also, estimates of the storage capacity of reservoirs may be greatly modified when γ is large. The large near-wellbore pressure gradients caused during withdrawal by large γ can cause sanding and wellbore collapse due to excessive production rates.

Notes: We investigate, from a theoretical point of view, the possibility of performing marine two-level magnetovariational measurements. An apparent resistivity function is denned and calculated after solving the differential equation governing the behaviour of the natural magnetic field variations inside a one-dimensional earth. In order to generalize the problem, a frequency-dependent resistivity is assumed to characterize the layers and the distortions caused by the polarization effects are carefully analysed. The computation of three-layer amplitude and phase diagrams for the apparent resistivity function shows that, in the case of an intermediate polarizable layer, sandwiched between a non-dispersive overburden and substratum, the H-type sequence results are the most affected by the dispersion phenomenon as it occurs in magnetotellurics. Finally we consider the problem of the sensitivity of the method, since, in practice, it requires top and bottom sensors separated by a vertical finite distance. It is found that in the higher-frequency range, due to the strong attenuation of the relative components of the field, the depth of the bottom sensor must be small enough to guarantee detectable signals, well above the full-scale resolution of the acquisition system. Conversely, in the lower-frequency range such a depth must be large enough to allow the difference between the top and bottom signals to be above the same recording sensitivity threshold.

Notes: Based on a Born approximation of a thin sheet integral equation, it is shown that small-scale surficial heterogeneity significantly distorts the electromagnetic field excited by electric dipoles only when either the source or the receiver are located on the heterogeneities. When a surface heterogeneity is beneath the source, the associated distortion of the electromagnetic field is manifest as a change in the effective electric dipole moment. Hence the magnetotelluric transfer functions and impedance relations remain undistorted in this case. When a surface heterogeneity is beneath the receiver, the electric field is severely distorted, but the magnetic field is only slightly distorted. The impedance tensor is therefore strongly distorted, but the tipper vector is almost unaltered. Since the controlled source tipper is a function of 1D earth conductivity, it is proposed that tipper data should be used in the first stage of 1D interpretation. For a 1D earth, the tipper vector must always point towards the source and, in the near-field limit, should have unit length. These two necessary conditions must be met by the measured tipper before it is interpreted one dimensionally.

Notes: A series of model experiments was performed in an ultrasonic laboratory to study the radiation of downhole sources in a variety of formations. Three models were used in the experiments. They were a Lucite model, a Lucite model with a free glass pipe in the centre, and a glass-cased soil model. In addition, a finite-difference modelling technique was used to simulate the wave propagation in these models and the results of the laboratory and numerical experiments are compared. In the Lucite borehole model the waveforms recorded in the experiment agree very well with the finite-difference synthetics. The snapshots of the wavefield from the finite-difference simulation show the radiation pattern of the P- and S-waves in the Lucite formation. These patterns are consistent with the theoretical calculations. In the Lucite model with the free glass pipe, the finite-difference synthetics are also in good agreement with the experimental observations, especially for the conical P-wave arrival. The angle between the wavefront of the conical P-wave and the borehole axis, observed from the snapshot, agrees with the theory. In the cased soil model, the arrival time of the finite-difference synthetics is in good agreement with the laboratory measurements. The relative amplitudes of the P-wave and the Mach wave are not correctly modelled because intrinsic attenuation is not included in the finite-difference calculation. The Mach cone angle from the snapshot agrees with the theoretical prediction. Finally, a finite-difference method was used to simulate Mach-wave propagation in a formation with two horizontal layers. In the case of two slow layers, the Mach-wave generated in the first layer is reflected back from and transmitted through the boundary and another Mach wave is generated at the second layer when the Stoneley wave travels into the second layer. In the case of a formation having one slow and one fast layer, the Mach wave generated in the slow layer is reflected back at the boundary and leaked into the fast layer. There is no Mach wave in the fast layer.

Notes: Geophysical surveys and chemical analyses on cores were carried out in three Ontario peatlands, from which we have gained a better understanding of the peat properties that control the geophysical responses. The electrical conductivity depends linearly on the concentration of total dissolved solids in the peat pore waters and the pore waters in turn bear the ionic signatures of the underlying mineral sediments. The ionic concentration, and thus the electrical conductivity, increase linearly from the surface to basement. The average bulk electrical conductivity of peatlands at Ellice Marsh, near Stratford, and at Wally Creek Area Forest Drainage Project, near Cochrane, are of the order of 25 mS/m. The Mer Bleue peatland, near Ottawa, has extremely high electrical conductivity, reaching levels of up to 380 mS/m near the base of the peat. The Mer Bleue peatland water has correspondingly high values of total dissolved solids, which originate from the underlying Champlain Sea glaciomarine clays. The dielectric permittivity in peats is largely controlled by the bulk water content. Ground penetrating radar can detect changes in water content greater than 3%, occurring within a depth interval less than 15 cm. The principal peatland interfaces detected are the near-surface aerobic to anaerobic transition and the peat to mineral basement contact. The potential for the successful detection of the basement contact using the radar can be predicted using the radar instrument specifications, estimates of the peatland depth, and either the bulk peat or the peat pore water electrical conductivities. Predicted depths of penetration of up to 10 m for Ellice Marsh and Wally Creek exceed the observed depths of 1 to 2 m. At Mer Bleue, on the other hand, we observe that, as predicted, a 100 MHz signal will penetrate to the base of a 2 m thick peat but a 200 MHz signal will not.

Notes: The investigation of water in salt-rock formations is of particular relevance to underground nuclear waste repositories. In the Asse salt-mine (Germany) a study into the relationship of in situ resistivity to water content has been made. Measurements were carried out in older rock-salt using an electrode array in boreholes, an electrode profile in a drift and small resistivity sensors in and around a drift seal. Further measurements were made on moist zones in a contact area of younger rock-salt and carnallitite and also in older rock-salt with anhydrite bands using electrode profiles in the drifts. The resistivities range from 102Ωm to 106Ωm. Corresponding probes have water contents from 0.01% to 1.3%. A definite relationship between resistivity and water content is revealed which can be described by Archie's law using a cementation factor m of 1.9. Porosities are between 0.08% and 1.4% and the saturations vary considerably. An explicit influence of saturation on resistivity cannot be discovered using the present data. The results enable us to estimate the in situ water content and the order of the in situ porosity using resistivity surveys at different scales. This increases significantly the safety of a nuclear repository site.

Notes: Minimization of seismic residuals does not guarantee uniqueness of the model, and this implies ambiguities in the inversion. Amplitude vs. offset (AVO) inversion does not lead to a unique solution of single elastic interface parameters unless converted and S-wave or critical angle reflections are available. Given the ambiguity of AVO inversion, this paper discusses the interaction between AVO and velocity estimation. The number of independent parameters necessary to describe an isolated reflection with AVO behaviour and residual velocity error is determined. Statistical analysis allows the establishment of an approximate equivalence of the effects of AVO and slight velocity variations; this equivalence cannot be solved without geological a priori information (kinematic equivalence). The data are then decomposed into compound events (i.e. sequences of N interfaces that follow each other at a fixed time lag). The decomposition is obtained by extrapolating the results of the analysis from narrowband to wideband data. Compound events decomposition demonstrates that AVO inversion is ambiguous, not only in the physical parameter space (P- and S-wave velocities, and density) but also kinematically. As an example of compound event decomposition, a medium is derived. This medium is geologically implausible but is kinematically equivalent.

Notes: Knowledge of the declination and inclination of the total and induced magnetization vectors is normally required for the interpretation and analysis of magnetic anomalies. A new method of estimating the direction of the total magnetization vector of magnetized rocks from magnetic anomalies is proposed. The unknown declination and inclination (D*T and I*T) can be found by applying a reduction-to-the-pole operator to the measured anomalies for different couples of total magnetization direction parameters (DT and IT) and by observing the variation of the anomaly minimum as a function of both DT and I*T.and D*T are estimated using the maximum of this function. Comparing our method to previous methods, one advantage is that our estimates are not zero-level dependent; furthermore, the method allows inclinations to be well estimated, with the same accuracy as declinations; finally declinations are not underestimated. Our method is applied to a real case and meaningful results are obtained; it is shown that the feasibility of the method is improved by removing the low-frequency components.

Notes: An efficient algorithm is presented to compute the Hankel transform. The algorithm yields simultaneously all the required weights for a given order of the Bessel function using the fast Fourier transform. An additional shift is introduced to the filter abscissa besides Koefoed's shift to give a better filter performance.

Notes: The stacking velocity best characterizes the normal moveout curves in a common-mid-point gather, while the migration velocity characterizes the diffraction curves in a zero-offset section as well as in a common-midpoint gather. For horizontally layered media, the two velocity types coincide due to the conformance of the normal and the image ray. In the case of dipping subsurface structures, stacking velocities depend on the dip of the reflector and relate to normal rays, but with a dip-dependent lateral smear of the reflection point. After dip-moveout correction, the stacking velocities are reduced while the reflection-point smear vanishes, focusing the rays on the common reflection points. For homogeneous media the dip-moveout correction is independent of the actual velocity and can be applied as a dip-moveout correction to multiple offset before velocity analysis.Migration to multiple offset is a prestack, time-migration technique, which presents data sets which mimic high-fold, bin-centre adjusted, common-midpoint gathers. This method is independent of velocity and can migrate any 2D or 3D data set with arbitrary acquisition geometry. The gathers generated can be analysed for normal-moveout velocities using traditional methods such as the interpretation of multivelocity-function stacks. These stacks, however, are equivalent to multi-velocity-function time migrations and the derived velocities are migration velocities.

Notes: A new definition of apparent resistivity for the presentation of magnetotelluric sounding data is proposed. The new definition is based on the frequency-normalized impedance function. Both the existing and proposed definitions of apparent resistivity are analysed theoretically and are compared using model curves computed for a 1D earth model. Apparent resistivity curves computed using the proposed definition are a better approximation to the true resistivity values of the subsurface layers. In addition, the layers are more noticeable on the apparent resistivity curves, which is an advantage, especially for the ascending and descending type of apparent resistivity curve.

Notes: A gravimetric survey, covering a site 200 m square, was carried out in order to locate karstic cavities. After eliminating the regional trend using a polynomial fit, the residual is modelled by least-squares prediction. Correlated signals for several wavelengths are detected. The inversion of these anomalies is performed by a global 3D adjustment using spherical bodies as models. The adjustment is repeated in order to obtain a stable configuration. The results show the probable presence of a system of cavities and galleries. Data collected from boreholes and the subsequent appearance of sink-holes are consistent with the results.

Notes: We have measured the velocities and attenuations of compressional and shear waves in 29 water-saturated samples of sandstones and shales at a confining pressure of 60 MPa and at frequencies of about 0.85 MHz. The measurements were made using a pulse echo method in which the samples (diameter 5 cm, length 1.5 cm to 2.5 cm) were placed between perspex buffer rods inside a high-pressure cell. The velocity of each seismic wave was determined from the traveltime difference of equivalent phase points (corrected for diffraction effects) of the signals reflected from the top and from the base of each sample. Attenuation was determined in a similar way by comparison of the diffraction corrected amplitudes of the signals. The attenuation data are presented as ‘quality factors’: Qp and Qs for compressional and shear waves respectively. The results show that Qs is strongly correlated with Vs, that Qp is weakly correlated with Vp, and that Qp is strongly correlated with Qs. Qp is strongly dependent on the volume percentage of the assemblage of intra-pore minerals, whether they are clays or carbonates. It is concluded that the attenuation mechanism is due to the local fluid flow arising from the differential dilation of the solid rock frame and the intra-pore mineral assemblage, which is a result of their very different elastic moduli.

Notes: An attempt to resolve non-uniqueness in the interpretation of transient electromagnetic (TEM) sounding data using measured data alone is made. It is shown in the various examples studied that sufficiently early time measurements can be the determining factor in reducing the ambiguity caused by model equivalence. The early delay times thus play a dual role in transient soundings: they are responsible for resolving shallow structures and they may eliminate the ambiguity in the interpretation of geoelectric parameters of deeper targets. This is illustrated by the results of a follow-up TEM survey at the Dead Sea coast of Israel where the use of supplementary early time measurements allowed non-uniqueness in the determination of the depth to fresh/saline groundwater interface to be resolved.

Notes: Flow of fluids in many hydrocarbon reservoirs and aquifers is enhanced by the presence of cracks and fractures. These cracks could be detected by their effects on propagation of compressional and shear waves through the reservoir: several theories, including Hudson's, claim to predict the seismic effects of cracks. Although Hudson's theory has already been used to calculate crack densities from seismic surveys, the predictions of the theory have not yet been tested experimentally on rocks containing a known crack distribution. This paper describes an experimental verification of the theory. The rock used, Carrara marble, was chosen for its uniformity and low porosity, so that the effect of cracks would not be obscured by other influences. Cracks were induced by loading of laboratory specimens. Velocities of compressional and shear waves were measured by ultrasound at 0.85 MHz in dry and water-saturated specimens at high and low effective pressures. The cracks were then counted in polished sections of the specimens. In ‘dry’ specimens with both dry and saturated cracks, Hudson's theory overpredicted observed crack densities by a constant amount that is attributed to the observed value being systematically underestimated. The theory made poor predictions for fully saturated specimens. Shear-wave splitting, caused by anisotropy due to both crystal and crack alignment, was observed. Cracks were seen to follow grain boundaries rather than the direction of maximum compression due to loading. The results demonstrate that Hudson's theory may be used in some cases to determine crack and fracture densities from compressional- and shear-wave velocity data.

Notes: The paper by Slob and Ziolkowski (1993) is apparently a comment on my paper (Szaraniec 1984) on odd-depth structure. In fact the basic understanding of a seismogram is in question. The fundamental equation for an odd-depth model and its subsequent deconvolution is correct with no additional geological constraints. This is the essence of my reply which is contained in the following points.〈list xml:id="l1" style="custom"〉1The discussion by Slob and Ziolkowski suffers from incoherence. On page 142 the Goupillaud (1961) paper is quoted:“… we must use a sampling rate at least double that… minimum interval…”. In the following analysis of such a postulated model Slob and Ziolkowski say that “… two constants are used in the model: Δt as sampling rate and 2Δt as two-way traveltime”. By reversing the Goupillaud postulation all the subsequent criticism becomes unreliable for the real Goupillaud postulation as well as the odd-depth model.2Slob and Ziolkowski take into consideration what they call the total impulse response. This is over and above the demands of the fundamental property of an odd-depth model.Following a similar approach I take truncated data in the form of a source function, S(z), convolved with a synthetic seismogram (earth impulse response), R̃(z), the free surface being included. The problem of data modelling is a crucial one and will be discussed in more detail below. By my reasoning, however, the function 〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR81:GPR_81_mu1"/〉 may be considered as a mathematical construction introduced purely to work out the fundamental property. In this connection there is no question of this construction having a physical meaning.It is implicit that in terms of system theory, K(z) stands for what is known as input impedance.3Our understandings of data are divergent but Slob and Ziolkowski state erroneously that:“Szaraniec (1984) gives (21) as the total impulse response…”. This point was not made.This inappropriate statement is repeated and echoed throughout the paper making the discussion by Slob and Ziolkowski, as well as the corrections proposed in their Appendix A, ineffective.Thus, my equation (2) is quoted in the form 〈displayedItem type="mathematics" xml:id="mu2" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR81:GPR_81_mu2"/〉which is in terms of the reflection response Gsc and holds true at least in mathematical terms. No wonder that “this identity is not valid for the total impulse response” (sic), which is denoted as G(z). None the less a substitution of G for Gsc is made in Appendix A, equation (A3). The equation numbers in my paper and in Appendix A are irrelevant, but (A3) is substituted for (32) (both numbers of equations from the authors’ paper). Afterwards, the mathematical incorrectness of the resulting equation is proved (which was already evident) and the final result (A16) is quite obviously different from my equation (2). However, the substitution in question is not my invention.4With regard to the problem of data modelling, I consider a bi-directional ID seismic source located just below the earth's surface. The downgoing unit impulse response is accompanied by a reflected upgoing unit impulse and the earth response is now doubled. The total impulse response for this model is thus given by 〈displayedItem type="mathematics" xml:id="mu3" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR81:GPR_81_mu3"/〉where (—r0) =— 1 stands for the surface reflection coefficient in an upward direction. Thus〈displayedItem type="mathematics" xml:id="mu4" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR81:GPR_81_mu4"/〉 that is to say, the total response to a unit excitation is identical with the input impedance as it must be in system theory.The one-directional 1D seismic source model is in question. There must be a reaction to every action. When only the downgoing unit impulse of energy is considered, what about the compensation?5In more realistic modelling, an early part of a total seismogram is unknown (absent) and the seismogram is seen in segments or through the windows. That is why in the usual approach, especially in dynamic deconvolution problems, synthetic data in the presence of the free surface are considered as an equivalent of the global reflection coefficient. It is implicit that model arises from a truncated total seismogram represented as a source function convolved with a truncated global reflection coefficient.Validation or invalidation of the truncation procedure for a numerically specified model may be attempted in the frame of the odd-depth assumption. My equations (22) and (23) have been designed for investigating the absence or presence of truncated energy.The odd-depth formalism allows the possibility of reconstructing an earlier part of a seismogram (Szaraniec 1984), that is to say, a numerical recovery of unknown moments which are unlikely designed by Slob and Ziolkowski for the data.

Notes: Gravity survey station locations are, in general, inhomogeneously distributed. This inevitably results in interpolation errors in the computation of a regular grid from the gravity data. The fractal dimension of the station distribution can be used to determine if an interpolated map is aliased at a specific wave-length and, moreover, it is often possible to determine an optimum gridding interval. Synthetic distributions of gravity station locations have been used for theoretical studies and it is found that for randomly distributed data there is a range of sizes for which the spatial data distribution has a fractal dimension of 2; that is, Euclidean. The minimum length scale at which the distribution ceases to be Euclidean is the optimum interpolation interval obeying Shannon's sampling theorem. For dimensions less than 2, the optimum interpolation interval is the shortest length at which the scaling regime is constant. In this case the gravity field cannot be interpolated without introducing some aliasing. As the fractal dimension characterizes the data distribution globally over the whole study area, the actual gridding interval, in some cases, will be smaller in order to represent short-wavelength features properly in the more densely sampled sub-areas, but this may generate spurious anomalies elsewhere. The proposed technique is applied to the station distribution of the Canadian national gravity data base and a series of sub-areas. A fractal dimension of 1.87 is maintained over a range of sizes from 15 km to over 1600 km. Although aliasing occurs, since the gravity field certainly contains much shorter wavelength anomalies, aliasing errors may be minimized by selecting the proper interpolation interval from the fractal analysis.

Notes: The VLF filtering technique of Karous and Hjelt has been applied to fixed-loop step-response transient electromagnetic data. This allows the data measured in each channel to be converted to an equivalent current-density pseudosection.For a conductive half-space, the maximum value of the equivalent current density starts near the transmitter loop and migrates outwards as a function of delay time. The rate of migration tends to increase as a function of delay time, with the increase being faster for a surficial conductive layer than it is for a half-space.Theoretical and field examples show that the currents tend to be more persistent in the relatively conductive areas, so that a pseudosection which is the average of the current densities at all delay times will highlight the more conductive zones.In resistive ground, it is not so critical to average the pseudosections as a particular delay time may give a better idea of the conductivity structure. For example, the latest possible delay time will reveal the most conductive features.

Notes: The presence of water is one of the main concerns of nuclear waste disposal in rock-salt. It can be investigated using electrical properties of the rock. Laboratory measurements of frequency-dependent resistivity and other petrophysical parameters, such as porosity, water content, and specific internal surface area, have been carried out on rock-salt from the Asse mine in Germany, in order to obtain characteristic resistivity responses for the evaluation of geoelectric field methods and to develop new methods for the estimation of the water content and saturation. The laboratory method, on a.c. half-bridge for very high impedances, allows measurements of the resistivity spectrum of rock-salt in the frequency range from 15 Hz to 10 kHz. The saturation of the samples was varied artificially and was approximately 5%, 10%, 20% and 100%.The porosity varies between 0.1% and 0.5%, the water content is approximately 0.05% or less, and the initial saturation is less than 50%. The resistivity ranges from 10 MΩm at the initial saturation down to 1 kΩm for fully saturated samples. In the low-frequency range up to 100 Hz, an Archie-type relationship may be used to estimate the water content of the rock-salt from resistivity measurements. The Archie exponent m is found to be approximately 2.The resistivity is observed to be strongly dependent on frequency. The resistivity decreases with increasing frequency, with a greater decrease for small saturations and vanishing frequency dependence at complete saturation. The relative dielectric constant was found to be 6 ± 1. Saturation dependence was not observed within this error range.The measurements imply that, by measuring resistivity in rock-salt, estimations of water content and saturation, and thus the porosity, can be made in situ. This is particularly important for the safety of nuclear waste disposal in rock-salt.

Notes: An experiment was undertaken at BP's Fulbeck Geophysical test site to compare the viability of various simultaneous vibroseis recording techniques, which are often recommended as a means of improving data acquisition production rates for 3D seismic surveys. Of particular interest were: (a) the ability to separate the signals from each source during processing, (b) the generation and suppression of harmonics and (c) the effects of any source interaction.Two vibrators were deployed with a baseplate separation of 10 m, about a borehole containing a vertical array of geophones. Our analysis concentrated on the groundforce signals measured at each vibrator and the far-field signatures measured using a vertical geo-phone at a depth of 204 m.By comparing single vibrator records with similar but separated records from a simultaneous recording sequence, signal separability, harmonic suppression and vibrator interaction could be fully studied.Separated far-field signatures from simultaneous vibroseis methods using combinations of up and downsweeps exhibited unsuppressed harmonics and substantial energy from the undesired source which leaked through the correlation process. The ‘up/down’ method was capable of separating the signal from each source by only 12.7 dB, and is therefore unsuitable as a field technique.The variphase simultaneous vibroseis methods studied afforded some harmonic suppression and gave signal separations of about 30.0 dB. Use of variphase simultaneous vibroseis methods will compromise the quality of the data recorded, when compared with single-source acquisition methods.None of the simultaneous vibroseis methods tested provided adequate signal separation and, therefore, cannot be recommended as data acquisition techniques. The ‘alternate sweeping’ method coupled with multispread recording will give the desired improvement in data acquisition rates, while preserving the necessary quality of our seismic data.

Notes: Depth migration consists of two different steps: wavefield extrapolation and imaging. The wave propagation is firmly founded on a mathematical frame-work, and is simulated by solving different types of wave equations, dependent on the physical model under investigation. In contrast, the imaging part of migration is usually based on ad hoc‘principles’, rather than on a physical model with an associated mathematical expression. The imaging is usually performed using the U/D concept of Claerbout (1971), which states that reflectors exist at points in the subsurface where the first arrival of the downgoing wave is time-coincident with the upgoing wave.Inversion can, as with migration, be divided into the two steps of wavefield extrapolation and imaging. In contrast to the imaging principle in migration, imaging in inversion follows from the mathematical formulation of the problem. The image with respect to the bulk modulus (or velocity) perturbations is proportional to the correlation between the time derivatives of a forward-propagated field and a backward-propagated residual field (Lailly 1984; Tarantola 1984).We assume a physical model in which the wave propagation is governed by the 2D acoustic wave equation. The wave equation is solved numerically using an efficient finite-difference scheme, making simulations in realistically sized models feasible. The two imaging concepts of migration and inversion are tested and compared in depth imaging from a synthetic offset vertical seismic profile section. In order to test the velocity sensitivity of the algorithms, two erroneous input velocity models are tested. We find that the algorithm founded on inverse theory is less sensitive to velocity errors than depth migration using the more ad hoc U/D imaging principle.

Notes: A vector convolutional model for multicomponent data acquired in an anisotropic earth is used as a basis for developing algebraic solutions to interpret near-offset VSP data. This interpretation of the cumulative or interval medium response (Green's tensor) for shear waves, determines a polarization azimuth for the leading shear wave and the time-delay between the fast and slow split waves. The algebraic solutions effectively implement least-squares eigenanalysis or singular value decomposition. Although the methodology for shear-wave analysis is strictly relevant to a transmission response, it can be adapted to surface data for a uniform anisotropic overburden. The techniques perform well when calibrated and tested using synthetic seismograms from various anisotropic models. Noise tests demonstrate the sensitivity of the interval measurements to local interferences, particularly if the shear waves are generated by one source. Although the algorithms are faster than numerical search routines, this is not seen as their major advantage. The solutions may have potential in near real-time interpretation of shear-wave data in well logging, where they may be coded on a microchip to provide a direct stream of separated shear waves, or polarization and birefringence information. There may also be some benefit for large prestack multicomponent surface data sets, where the solutions provide a direct transformation to the split-shear-wave components, reducing the storage space for further processing.

Notes: A new method is presented for solving the 2D problem of diffraction of a plane wave by a wedge of arbitrary angle in a purely acoustic, constant-density medium with different constant compressional wave speeds inside and outside the wedge. The diffraction problem is formulated as integral equations, and a wavenumber–frequency representation of the scattered field is obtained. With the aid of the Cagniard–de Hoop method, exact analytical expressions in the space–time domain are obtained for the different wave constituents, i.e. geometric optical scattered waves and edge diffracted waves including head waves. These expressions can be computed to any degree of accuracy within reasonable computation times on a computer, and the semi-analytical method of solution presented thus constitutes a means of constructing reference solutions for wedge configurations. Such highly accurate reference solutions are of importance for verification of results that include diffraction phenomena modelled by general numerical approximate methods, e.g. finite differences, finite elements and spectral methods. Examples of such applications of the method of solution are given.

Notes: Backus and Crampin derived analytical equations for estimating approximate phase-velocity variations in symmetry planes in weakly anisotropic media, where the coefficients of the equations are linear combinations of the elastic constants. We examine the application of similar equations to group-velocity variations in off-symmetry planes, where the coefficients of the equations are derived numerically. We estimate the accuracy of these equations over a range of anisotropic materials with transverse isotropy with both vertical and horizontal symmetry axes, and with combinations of transverse isotropy yielding orthorhombic symmetry. These modified equations are good approximations for up to 17% shear-wave anisotropy for propagations in symmetry planes for all waves in all symmetry systems examined, but are valid only for lower shear-wave anisotropy (up to 11%) in off-symmetry planes.We also obtain analytical moveout equations for the reflection of qP-, qSH-, and qSV- waves from a single interface for off-symmetry planes in anisotropic symmetry. The moveout equation consists of two terms: a hyperbolic moveout and a residual moveout, where the residual moveout is proportional to the degree of anisotropy and the spread length of the acquisition geometry. Numerical moveout curves are computed for a range of anisotropic materials to verify the analytical moveout equations.

Notes: An exact formulation for borehole coupling, which is valid for all frequencies and all azimuthally symmetric and non-symmetric components, is presented. The borehole effects on downhole seismic measurements are studied in detail as functions of frequency, angle of incidence and polarization of an incident wave as well as geophone orientation. We found that correction for the borehole effect on downhole measurements should be made for frequencies above 500 Hz in a hard formation. In a soft formation, if the angle of incidence is well away from the resonance angle for SV incidence, no borehole correction is needed for frequencies below 300 Hz, while for frequencies above 300 Hz, the borehole can cause severe problems in downhole measurements. The borehole can also significantly alter the particle motion direction which implies that horizontal component rotation from data itself is unreliable for experiments with frequencies above 1 kHz in the hard formation and around 500 Hz in the soft formation.

Notes: Seismic stratigraphy and sedimentological studies of the Gemlik Gulf in the Sea of Marmara, Turkey, have been carried out. For this purpose, 19 lines totalling 189 km of excellent quality, high-resolution seismic data were recorded.Four major acoustic units were identified in the seismic profiles. Three were sedimentary units: irregular layered, cross-layered and well-layered; and the fourth was an acoustic basement which is probably composed of crystalline volcanic rocks.Some local areas in the Neogene formation contain gas accumulations.The formation of faults in E–W and N–S directions can be explained by the existence of shear stresses in the Gulf. The bathymetric map shows good accommodation with the shore line as does the tectonic map.

Notes: Recent observations show that the differential amplitudes between the faster and slower split shear-waves in reflection surveys contain information about lateral variations of crack density in cracked reservoirs. However, the variation of amplitude with crack geometry when the crack strike changes with depth has not been reported previously. In this paper, we derive expressions for reflection and transmission coefficients of plane split shear-waves at vertical incidence at an interface separating two cracked (anisotropic) media with different crack strikes. We examine the effects on these coefficients as crack strike and crack density vary. For interfaces with large velocity-contrasts, the reflection coefficients carry little information about crack geometry, and the effects of crack strike varying with depth are negligible. In such cases, the polarization and time-delay of the split shear-waves are the only features which reliably diagnose anisotropy and contain information about the variation of crack strike and density. However, for interfaces with small velocity-contrasts, the effects of any variation of crack strike with depth cannot be neglected. In such cases, in addition to the polarization and time-delays of split shear-waves, both the differential amplitude of faster and slower shear-waves and the amplitude ratio of the two off-diagonal elements in the reflected data matrix after separation of split shear-waves, contain information about the variation of crack strike and crack density. In contrast, effects of crack strike changing with depth on transmitted waves are more sensitive regardless of the velocity-contrast and the degree of anisotropy.

Notes: In seismic migration, it is important to sample a range of dips around the local structural dip at each image point. Meaningful images are obtained only where this condition holds. For cross-hole seismic reflection surveys, the distribution of dips sampled at each image point is controlled principally by the survey geometry, including source and receiver array lengths and their element spacings. Using a real data set as an example, we show how survey geometry can limit imaging capability close to the boreholes and even in the middle of the section between the boreholes.At the processing stage, effective removal of direct waves and accurate estimation of the velocity field are essential for optimizing image quality. For migration, we propose a generalized Berryhill (GB) scheme which is based on the Kirchhoff integral and takes into account both the near-field and far-field terms. This should improve the ability to image close to source and receiver arrays, provided that the element spacing in the nearby array is small enough.

Notes: The 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.

Notes: A 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.

Notes: The 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.

Notes: Experiments 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.

Notes: The 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 104 at 3 MHz to about 102 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.

Notes: A 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.

Notes: New 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.

Notes: Two 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.

Notes: This 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.

Notes: In the case of 3D multilayered structures the 2D interval velocity analysis may be inaccurate. This fact is illustrated by synthetic examples.The method proposed solves the 3D inverse problem within the scope of the ray approach. The solution, i.e. the interval velocities and the reflection interface position, is obtained using data from conventional 2D line profiles arbitrarily located and from normal incidence time maps. Although the input information is essentially limited, the method presented reveals only minor biased velocity estimates.In order to implement the proposed 3D inversion method, we developed a processing procedure. The procedure performs the evaluation of reflection time and ray parameters along line profiles, 3D interval velocity estimation, and time-to-depth map migration. Tools to stabilize the 3D inversion are investigated.The application of the 3D inversion technique to synthetic and real data is compared with results of the 2D inversion.

Notes: Traditionally, residual static corrections are based on timeshifts estimated for individual CMP sorted traces, which are later resolved into surface-consistent statics. This is a stable and attractive procedure because the data flow is simple and the memory storage required is limited. An alternative station-oriented method maximizing the stack-power estimates surface-consistent static corrections directly. The statics evaluation in this method involves several CMP gathers, which should improve the prediction of statics on noise-contaminated data. In this paper the performance of the above methods will be compared using synthetic as well as real seismic data. Neither method is capable of estimating large statics compared to the dominating period, because local optimization might fail. Global Monte Carlo search by, for instance, simulated annealing has been used to overcome the cycle-skipping problems when proper field statics are missing. Although this procedure is computationally very heavy, it may be the only way to deal with large residual statics. In order to enlarge the operational field for local optimization, it is suggested that the stack-power in the frequency domain is maximized. This makes it easy to change the frequency band during the optimization. Making use of the frequency domain will also normally be faster than the traditional time-domain optimization even for a limited number of iterations. Moreover, the main memory storage required can be significantly reduced, since it is only necessary to keep the frequency band in the memory, where the signal-to-noise ratio is good.

Notes: The aim of seismic inversion methods is to obtain quantitative information on the subsurface properties from seismic measurements. However, the potential accuracy of such methods depends strongly on the physical correctness of the mathematical equations used to model the propagation of the seismic waves. In general, the most accurate models involve the full non-linear acoustic or elastic wave equations. Inversion algorithms based on these equations are very CPU intensive. The application of such an algorithm on a real marine CMP gather is demonstrated. The earth model is assumed to be laterally invariant and only acoustic wave phenomena are modelled. A complete acoustic earth model (P-wave velocity and reflectivity as functions of vertical traveltime) is estimated. The inversion algorithm assumes that the seismic waves propagate in 2D. Therefore, an exact method for transforming the real data from 3D to 2D is derived and applied to the data. The time function of the source is estimated from a vertical far-field signature and its applicability is demonstrated by comparing synthetic and real water-bottom reflections. The source scaling factor is chosen such that the false reflection coefficient due to the first water-bottom multiple disappears from the inversion result. In order to speed up the convergence of the algorithm, the following inversion strategy is adopted: an initial smooth velocity model (macromodel) is obtained by applying Dix's equation to the result of a classical velocity analysis, followed by a smoothing operation. The initial reflectivity model is then computed using Gardner's empirical relationship between densities and velocities. In a first inversion step, reflectivity is estimated from small-offset data, keeping the velocity model fixed. In a second step, the initial smooth velocity model, and possibly the reflectivity model, is refined by using larger-offset data. This strategy is very efficient. In the first step, only ten iterations with a quasi-Newton algorithm are necessary in order to obtain an excellent convergence. The data window was 0–2.8 s, the maximum offset was 250 m, and the residual energy after the first inversion step was only 5% of the energy of the observed data. When the earth model estimated in the first inversion step is used to model data at moderate offsets (900 m, time window 0.0–1.1 s), the data fit is very good. In the second step, only a small improvement in the data fit could be obtained, and the convergence was slow. This is probably due to the strong non-linearity of the inversion problem with respect to the velocity model. Nevertheless, the final residual energy for the moderate offsets was only 11%.The estimated model was compared to sonic and density logs obtained from a nearby well. The comparison indicated that the present algorithm can be used to estimate normal incidence reflectivity from real data with good accuracy, provided that absorption phenomena play a minor role in the depth interval considered. If details in the velocity model are required, large offsets and an elastic inversion algorithm should be used.

Notes: Linearized residual statics estimation will often fail when large static corrections are needed. Cycle skipping may easily occur and the consequence may be that the solution is trapped in a local maximum of the stack-power function. In order to find the global solution, Monte Carlo optimization in terms of simulated annealing has been applied in the stack-power maximization technique. However, a major problem when using simulated annealing is to determine a critical parameter known as the temperature.An efficient solution to this difficulty was provided by Nulton and Salamon (1988) and Andresen et al. (1988), who used statistical information about the problem, acquired during the optimization itself, to compute near optimal annealing schedules.Although theoretically solved, the problem of finding the Nulton–Salamon temperature schedule often referred to as the schedule at constant thermodynamic speed, may itself be computationally heavy. Many extra iterations are needed to establish the schedule.For an important geophysical inverse problem, the residual statics problem of reflection seismology, we suggest a strategy to avoid the many extra iterations. Based on an analysis of a few residual statics problems we compute approximations to Nulton–Salamon schedules for almost arbitrary residual statics problems. The performance of the approximated schedules is evaluated on synthetic and real data.

Notes: Knowledge of both the electrical resistivity ρ and the dielectric permittivity ɛ of the ground is important for the determination of characteristics such as granularity, porosity, moisture and salt content. Whereas the measurement of ρ is very common and can be achieved using either the d.c. resistivity method or a wide variety of EM devices, ɛ remains practically unknown in the low-frequency domain between the IP domain and the high-frequency domain. Following the principle of the quadrupole technique used in d.c. prospecting and of the quadrupolar probe for measuring the complex permittivity of the ionosphere, we propose a new approach which does not require any galvanic contact between the poles and the ground. The transfer impedance can be evaluated using the quasi-static approximation for low frequency or the full EM theory for higher frequencies. The conditions under which both calculations apply are discussed. At low frequencies and for low resistivity ground, the electrostatic quadrupole measures ρ exactly as with the d.c. technique; for higher resistivities or frequencies the simultaneous measurement of both properties becomes possible. Examples in archaeological prospecting are presented and checked against independent d.c. resistivity measurements or excavation analyses.

Notes: In seismic tomography the observed traveltimes or amplitudes of direct waves are inverted to obtain an estimate of seismic velocity or absorption of the section surveyed. There has been much recent interest in using cross-well traveltime tomography to observe the progress of fluids injected into the reservoir rocks during enhanced oil recovery (EOR) processes. If repeated surveys are carried out, then EOR processes may be monitored over a period of time.This paper describes the results of a simulated time-lapse tomography experiment to image the flood zone in an EOR process. Two physical models were made out of epoxy resins to simulate an essentially plane-layered sedimentary sequence containing a reservoir layer and simple geological structure. The models differed only in the reservoir layer, which was uniform in the ‘pre-flood’ model and contained a flood zone of known geometry in the ‘post-flood’ model. Data sets were acquired from each model using a cross-well survey geometry. Traveltime and amplitude tomographic imaging techniques have been applied to these data in an attempt to locate the extent of the flood zone.Traveltime tomography locates the flood zone quite accurately. Amplitude tomography shows the flood zone as a region of higher absorption, but does not image its boundaries as precisely. This is primarily because of multipathing and diffraction effects, which are not accounted for by the ray-based techniques for inverting seismic amplitudes. Nevertheless, absorption tomograms could complement velocity tomograms in real, heterogeneous reservoirs because absorption and velocity respond differently to changes in liquid/gas saturations for reservoir rocks.

Notes: Based on an expansion of the band-limited 3D extrapolation operator in terms of orthogonal Chebyshev polynomials, a closed form expression of the space-frequency response is presented. A key step is an evaluation of the (inverse) 2D Fourier transform of circularly symmetric functions, which is related to the (zero-order) Hankel transform. Hankel transforms of individual members of the orthogonal set of polynomials are available from tables and summation of series; hence, the real and the imaginary parts of the space-frequency response can be found in terms of cylindrical and spherical Bessel functions, respectively. The procedure permits an efficient and accurate evaluation of the space-frequency response.

Notes: Migration to zero offset (MZO) is a prestack partial migration process that transforms finite-offset seismic data into a close approximation to zero-offset data, regardless of the reflector dips that are present in the data. MZO is an important step in the standard processing sequence of seismic data, but is usually restricted to constant velocity media. Thus, most MZO algorithms are unable to correct for the reflection point dispersal caused by ray bending in inhomogeneous media.We present an analytical formulation of the MZO operator for the simple possible variation of velocity within the earth, i.e. a constant gradient in the vertical direction. The derivation of the MZO operator is carried out in two steps. We first derive the equation of the constant traveltime surface for linear V(z) velocity functions and show that the isochron can be represented by a fourth-degree polynomial in x, y and z. This surface reduces to the well-known ellipsoid in the constant-velocity case, and to the spherical wavefront obtained by Slotnick in the coincident source-receiver case.We then derive the kinematic and dynamic zero-offset corrections in parametric form by using the equation of the isochron. The weighting factors are obtained in the high-frequency limit by means of a simple geometric spreading correction. Our analytical results show that the MZO operator is a multivalued, saddle-shaped operator with marked dip moveout effects in the cross-line direction. However, the amplitude analysis and the distribution of dips along the MZO impulse response show that the most important contributions of the MZO operator are concentrated in a narrow zone along the in-line direction. In practice, MZO processing requires approximately the same trace spacing in the in-line and cross-line directions to avoid spatial aliasing effects.

Notes: Velocity estimation technique using seismic data is often based on time/distance equations which are independent of direction, and even though we now know that many rocks are quite anisotropic, useful results have been obtained over the years from these isotropic estimates. Nevertheless, if velocities are significantly direction-dependent, then the isotropic assumption may lead to serious structural interpretation errors. Additionally, information on angle-dependence may lead to a better understanding of the lithology of the rocks under measurement. VSP and cross-well data may each lack the necessary aperture to estimate more than two velocity parameters for each wave type, and if the data straddle a symmetry axis, then these may be usefully chosen to be the direct velocities (from time-and-distance measurements along the axis) and NMO velocities (from differential time-offset measurements). These sets of two parameters define ellipses, and provide intermediate models for the variation of velocity with angle which can later be assembled and translated into estimates of the elastic moduli of the rocks under scrutiny.If the aperture of the measurements is large enough e.g. we have access to both VSP and cross-well data, we divide the procedure into two independent steps, first fitting best ellipses around one symmetry axis and then fitting another set around the orthogonal axis. These sets of four elliptical parameters are then combined into a new, double elliptical approximation. This approximation keeps the useful properties of elliptical anisotropy, in particular the simple relation between group and phase velocities which simplifies the route from the traveltimes measurements to the elastic constants of the medium.The inversion proposed in this paper is a simple extension of well-known isotropic schemes and it is conceptually identical for all wave types. Examples are shown to illustrate the application of the technique to cross-well synthetic and field P-wave data. The examples demonstrate three important points: (a) When velocity anisotropy is estimated by iterative techniques such as conjugate gradients, early termination of the iterations may produce artificial anisotropy. (b) Different components of the velocity are subject to different type of artifacts because of differences in ray coverage, (c) Even though most rocks do not have elliptical dispersion relations, our elliptical schemes represent a useful intermediate step in the full characterization of the elastic properties.

Notes: Magnetic anomalies of complicated 3D sources can be calculated by using a combination of analytical and numerical integration. Two surfaces and the magnetization parameters (the amplitudes of the induced and remanent components and the direction cosines) of the source can be defined by arbitrary functions or by discrete data points in a plane. When combined with a polynomial magnetization function in the direction of the third axis, 3D magnetization distribution can also be modelled.The method gives very general equations for anomaly calculation. It can be used for direct modelling of sources interpreted by seismic or other methods and also for interactive interpretation with fast computers. It is possible to calculate anomalies of, for example, intrusives or folded sedimentary beds whose surfaces are functions of horizontal coordinates and which have polynomial magnetization variations in the vertical direction due to gravitational differentiation and arbitrarily varying magnetization in the horizontal direction due to regional metamorphosis.If the distribution of magnetization parameters in the vertical direction cannot be described satisfactorily by polynomials, models can be used whose surfaces are functions of the vertical coordinate and which can then have any arbitrary magnetization distribution in the vertical direction.

Notes: A method is presented for computing the coefficients of finite-difference operators for the elastic wave equation. As opposed to other algorithms, in this case the amplitude spectrum of the source function is taken into account. Test calculations show that this approach gives more accurate results than operators based on a Taylor expansion or on minimizing the maximum error of the group velocity in a spatial frequency interval.

Notes: Wyllie's time-average equation and subsequent refinements have been used for over 20 years to estimate the porosity of reservoir rocks from compressional (P)-wave velocity (or its reciprocal, transit time) recorded on a sonic log. This model, while simple, needs to be more convincingly explained in theory and improved in practice, particularly by making use of shear (S)-wave velocity. One of the most important, although often ignored, factors affecting elastic velocities in a rock is pore structure, which is also a controlling factor for transport properties of a rock. Now that S-wave information can be obtained from the sonic log, it may be used with P-waves to provide a better understanding of pore structure. A new acoustic velocities-to-porosity transform based on an elastic velocity model developed by Kuster and Toksöz is proposed. Employing an approximation to an equivalent pore aspect ratio spectrum, pore structure for reservoir rocks is taken into account, in addition to total pore volume. Equidimensional pores are approximated by spheres and rounded spheroids, while grain boundary pores and flat pores are approximated by low aspect ratio cracks. An equivalent pore aspect ratio spectrum is characterized by a power function which is determined by compressional-and shear-wave velocities, as well as by matrix and inclusion properties. As a result of this more sophisticated elastic model of porous rocks and a stricter theory of elastic wave propagation, the new method leads to a more satisfactory interpretation and fuller use of seismic and sonic log data. Calculations using the new transform on data for sedimentary rocks, obtained from published literature and laboratory measurements, are presented and compared at atmospheric pressure with those estimated from the time-average equation. Results demonstrate that, to compensate for additional complexity, the new method provides more detailed information on pore volume and pore structure of reservoir rocks. Examples are presented using a realistic self-consistent averaging scheme to consider interactions between pores, and the possibility of extending the method to complex lithologies and shaly rocks is discussed.

Notes: The Radon transform is applied to airborne geophysical data, which consist of parallel profiles, analogous to a seismic record. The plane-wave decomposition (PWD) thus becomes the strike-direction decomposition (SDD) since the observed spatially distributed information is represented by its strike directions in a domain achieved by the transformation. It is important that, after the SDD, we can identify anomalies and work on them according to their strikes. In particular, for gridding purposes, we may guide the second interpolation of the bi-directional gridding approach along the strike directions.In principle, the proposed Radon transform gridding method (RTGM) transforms the observed parallel profiles into a domain where information is mapped as its strike-direction ‘traces’ against its wavelengths. The number of strike directions into which the data are decomposed is equal to the number of lines to be interpolated. The Fourier spectrum of the grid is reconstructed from the strike-wavenumber domain by using the projection-slice theorem and the final square grid is obtained by performing an inverse Fourier transformation on the spectrum.The SDD is restricted to the Nyquist wavenumber bandwidth imposed by the survey line-spacing, so that there is no addition of ambiguous short wavelengths in the gridded data. A tapering window is employed to prevent any Gibb's oscillation in the final grid because of the sharp Nyquist cut-off in the reconstructed spectrum due to the survey line-spacing.The RTGM is first tested on a set of synthetic line-based data. It is also applied to aeromagnetic profile data from northern Botswana as a practical example.

Notes: The 4 × 4 T-propagator matrix of a 3D central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T-propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.

Notes: The theoretical behaviour of guided waves in cross-hole seismic surveys is examined with dispersion curves and synthetic seismograms. The modelling suggests, and field observations confirm, that the dominant energy in many cross-hole seismic surveys propagates as guided waves, as well as body waves, with the energy tied to an interface, or combination of interfaces, whenever there are interfaces with significant velocity contrasts continuous between the wells. The dispersion and waveforms of such modal solutions carry detailed information about the internal structure, particularly the stress-aligned structure of fluid-filled inclusions in the immediate neighbourhood of the waveguide. Since the information content can be controlled to some extent by the choice of frequency and source type and the level of source and receiver in the two wells, such guided waves in cross-hole surveys may be important for monitoring changes in nearly horizontal reservoirs during hydrocarbon production processes. Initially, we investigate the basic properties of propagation in isotropic strata, and then examine the behaviour of guided waves in the stress-aligned fluid-filled inclusions.

Notes: It is useful to be able to calculate synthetic primary reflection sequences from which to generate synthetic seismic sections which can be used for testing new processing algorithms. However, these synthetic reflection sequences should closely match real properties found in recent studies. Using the ARMA(1,1) model resulting from such studies to describe the correlation (or spectral) structure of the sequences, and by matching moments up to fourth order (since the sequences are non-Gaussian in practice), realistic sequences can be generated. A simple scheme is provided which also eliminates the necessity of throwing away large numbers of simulated values at start-up. The procedure is illustrated on three real sequences and is seen to reproduce all the important features.

Notes: A processing method is presented to attenuate the surface ghost using marine twin streamer data. It is an extension to the dephase and sum method which corrects for the phase of the ghost in both streamer outputs and then adds them in an attempt to fill the notches in the amplitude spectrum. The method presented corrects both the phase and the amplitude effect of the surface ghost by combining both signals as a weighted sum.This method is applicable to all types of twin streamer data, ranging from deep exploration data to very shallow high-resolution surveys. Both the synthetic and real data examples shown are of the high-resolution type, using frequencies above 2 kHz and short streamers (active sections of the order of one metre).Both the dephase and sum method and the weighted sum method are applied to synthetic high-resolution data and the results are compared. This has been done for noise-free data, data with a high noise level and data with strong geometrical spreading on the ghost reflections. From these test results it can be concluded that in general the weighted sum method gives better results. The improvement in the signal-to-noise ratio appears to be the same, due to the additive character of both methods. In the case of high-resolution twin streamer data recorded in shallow water, the delay time of the ghost reflection can be of the same order of magnitude as the traveltime of the primary. For this situation, geometrical spreading can have a considerable effect on the amplitude of the ghost reflection. If no correction can be made for the spreading function, it might be better to use the dephase and sum method.Both methods are also applied to a real data set recorded in a high-resolution survey. Because the ghost delay is of the same order of magnitude as the arrival time of the primary, the ghost reflection is strongly affected by geometrical spreading. Since the spreading function of the source is unknown, it cannot be corrected for. This causes the result of the weighted sum method to be less reliable compared to the case where no spreading is involved.

Notes: Previous studies of radiation from point sources in fluid-filled boreholes have most often been based on far-field, stationary phase analysis. In these papers, the explicit contribution of the borehole itself acting as a waveguide has not been properly considered, with a few exceptions. In general, these studies accurately describe S-wave radiation in high-velocity rocks such as granites and limestones and P-wave radiation in most rocks, and experiments have confirmed this. However, tube waves directly influence the external wavefield and in fact create a shear-wave ‘wake’ outside the borehole due to constructive interference of tube-wave emission if a velocity condition is met. This constructive interference or wake is generated when the tube-wave velocity is greater than the shear-wave velocity. When this happens, a tube-wave complex pole invalidates the mathematical assumptions for stationary phase analysis and the stationary phase predictions do not agree with experimentally derived radiation patterns. Shales at shallow depths and other soft sediments characteristically have tube-wave velocities greater than shear-wave velocities. Because the tube-wave is of relatively high amplitude compared to body waves generated directly by the source, these secondary shear waves can be the highest amplitude arrivals on receiver arrays.The shape and properties of these secondary shear waves are calculated and shown to have identical properties to Mach waves of aerodynamics and seismology. For instance, these waves are geometrically conical and the aperture of the cone and the moveout velocity can be calculated. This paper also demonstrates the important effect that casing has on the Mach waves and provides predictions about when these waves are likely to be observed. Finally, evidence of Mach waves in data sets is examined and it is shown how these waves have been confused with receiver borehole tube waves.It is possible, though rare, that the tube-wave velocity of the borehole is greater than the compressional-wave velocity of the surrounding medium. In this case secondary compressional or compressional Mach waves would be generated although this problem is not addressed here.

Notes: An approximation to plane-wave propagation through a composite material is examined using a physical model with oriented but randomly distributed penny-shaped rubber inclusions within an isotropic epoxy resin matrix. A pulse transmission method is used to determine velocities of shear and compressional waves as a function of angle of incidence and crack density. The experimental and theoretical results of Hudson were compared and limitations within the crack parameters used in this study have been determined. Results from both polarized shear waves (S1, S2) compare favourably with the theory for a composite with up to 7% crack density, but theory and experiment diverge at higher crack densities. On the other hand, compressional-wave velocities at low crack densities (1% and 3%) compare favourably with the theory. It is also shown that the velocity ratio Vp/Vs for two extreme cases, i.e. propagation normal and parallel to the cracks, as a function of crack density and porosity, has a strong directional dependence.

Notes: A quantitative AVO algorithm suitable for media with slow lateral parameter variations is developed. The method is based on a target-oriented inversion scheme for estimation of elastic parameters in a locally horizontally stratified medium. The algorithm uses band-limited PP reflection coefficients in the τ-p domain to estimate P- and S-wave velocities, densities and layer thicknesses. To obtain these reflection coefficients, a pre-processing involving the Radon transform and multiple attenuation is necessary. Furthermore, a macromodel for the velocities above the target zone must be found prior to the inversion.Various inversion tests involving synthetic data with white Gaussian noise and modelling errors that are likely to occur in conjunction with real data have been performed. In general, the inversion algorithm is fairly robust, since it is able to reproduce the main features of the reference model: main interface locations and relative contrasts in the three unknown layer parameters are recovered.From a test combining the effect of source directivity, one thin layer and 20% white Gaussian noise, it was found that neglect of the source directivity in the inversion caused the largest errors in the estimates. This indicates that it is very important either to eliminate the source directivity in a preprocessing step, or to take the directivity into account in the present algorithm. Despite these problems it was concluded that the inversion algorithm was able to reproduce the main features of the reference model.

Notes: The use of relaxation mechanisms has recently made it possible to simulate viscoelastic (Q) effects accurately in time-domain numerical computations of seismic responses. As a result, seismograms may now be synthesized for models with arbitrary spatial variations in compressional- and shear-wave quality factors (Q9, and Qs, as well as in density (ρ) and compressional- and shear-wave velocities (Vp, and Vs).Reflections produced by Q contrasts alone may have amplitudes as large as those produced by velocity contrasts. Q effects, including their interaction with Vp, Vs and p, contribute significantly to the seismic response of reservoirs. For band-limited data at typical seismic frequencies, the effects of Q on reflectivity and attenuation are more visible than those on dispersion.Synthetic examples include practical applications to reservoir exploration, evaluation and monitoring. Q effects are clearly visible in both surface and offset vertical seismic profile data. Thus, AVO analyses that neglect Q may produce erroneous conclusions.

Notes: This study investigates the propagation of borehole Stoneley waves across permeable structures. By modelling the structure as a zone intersecting the borehole, a simple 1D theory is formulated to treat the interaction of the Stoneley wave with the structure. This is possible because the Stoneley wave is a guided wave, with no geometric spreading as it propagates along the borehole. The interaction occurs because the zone and the surrounding formation possess different Stoneley wavenumbers. Given appropriate representations of the wavenum-ber, the theory can be applied to treat a variety of structures, including a fluid-filled fracture. Of special interest are the cases of permeable porous zones and fracture zones. The results show that, while Stoneley wave reflections are generated, strong Stoneley wave attenuation is produced across a very permeable zone. This result is particularly important in explaining the observed strong Stoneley wave attenuation at major fractures where it has been difficult to explain the attenuation in terms of the single planar fracture theory. In addition, by using a simple and sufficiently accurate theory to model the effects of the permeable zone, a fast and efficient method is developed to characterize the fluid transport properties of a permeable fracture zone.

Notes: A reflection response function for a 1D discretized earth model can be obtained using ray-theory and Z-transforms with the Goupillaud model. This is usually done by taking the source function as a plane wave impinging normally on the layered earth. Two important problems have been tackled with this basic idea. The first, extraction of the source wavelet, and the second, a description of the free-surface related problems.In the Goupillaud model, the one-way traveltime in each layer is taken to be the same time interval At, which is also the time unit for the Z-transform. The two-way traveltime in any layer is 2Δt, corresponding to a multiplication by Z2. The reflection impulse response therefore contains only even powers of Z. The convolution of the reflection response with the wavelet yields a seismogram whose Z-transform contains both odd and even powers of Z. However, even though the seismogram contains more coefficients than unknowns, the wavelet cannot be extracted, because the coefficients are not independent: later coefficients are functions of earlier ones, which does not make sense physically. To overcome this physical problem for the reflection seismogram, the two-way traveltime through the layer should be Δt. It is then impossible to extract the wavelet, as there are fewer coefficients in the seismogram than unknowns.Szaraniec has proposed a modification to the Goupillaud model, known as the odd-depth model, that includes the free surface and a top layer whose two-way traveltime Δt is half the two-way traveltime 2Δt of all the other layers. Using what Szaraniec calls the fundamental identity of the odd-depth model, it is possible to extract the source wavelet from the seismogram. We show that this fundamental identity holds only if reflection coefficients of deeper interfaces are functions of the reflection coefficients of shallower interfaces; that is, for extremely improbable geologies.Neither of these approaches offers a solution to the deconvolution problem. It is better to obtain the source signature from measurements in the field. Only Szaraniec's model offers the possibility of tackling the problem of the free surface but because of an inherent flaw in the model, it fails to address the problem.

Notes: P-wave velocity anisotropy determined from a cross-hole survey at the Imperial College borehole test site compares favourably with that measured in the laboratory on core from the holes. The holes penetrate a layered sequence of sandstones, shales and carbonates of the Namurian Upper Limestone Group. The Laboratory measurements of the vertical and horizontal velocities of core samples indicate that the shales exhibit P-wave anisotropies of over 20% but that the sandstones and limestones are only slightly anisotropic. These discrete measurements have been used in combination with wireline data to produce a log of P-wave anisotropy. Including the anisotropic information vastly improves the match between observed and synthetic traveltimes from the cross-hole data set. This implies that there is little frequency dependence of intrinsic P-wave anisotropy.Inversion of the cross-hole traveltimes highlights the need for good angular coverage in order to resolve the anisotropy parameters. The observed P-wave anisotropy of the field data is due to the combined effect of sedimentary layering and the intrinsic anisotropy of the rocks. The intrinsic anisotropy is found to be the dominant factor.

Notes: MT data from a detailed 84-site grid array (28 × 12 km) in the Paraná Basin (Brazil) reveal a high degree of frequency-independent and parallel amplitude distortion. The Palaeozoic sediments across the survey area are covered by 1.2 km of flood basalts. A deep well-log provides some control regarding the emplacement of thin diabase sills but information regarding vertical feeder dikes is non-existent. The degree of parallel behaviour is identified using anisotropy ratios which quantify the extent and bandwidth of the distortion characteristics.A 2D modelling study is carried out using the concept of a horizontally layered (1D) basin with superimposed, small-scale inhomogeneities. Two likely geological distortion structures are considered. The first represents at- or near-surface (0 to 100 m) inhomogeneities. The second represents thin but vertically elongate dikes. The two distortion effects are found to produce different and characteristic behaviour in the H-polarization (TM) mode model data. The modelled E-polarization (TE) mode data are undistorted and provide control. The model developed to represent vertical dikes gives rise to a characteristic frequency dependence which is distinct from that of the undistorted response. This model also generates subparallel distortion effects which give rise to a form of variance in the response estimates as a function of location. The behaviour occurs, importantly, in both the amplitude and phase of the distorted data. A comparison of the normalized amplitude data, observed and modelled, indicates a high degree of correspondence down to a low frequency limit (〈 0.1 Hz) where additional and deep contributions to the observed response become apparent. The modelling results and data characteristics indicate that the survey area is underlain by a series of closely parallel, thin intrusive dikes (a dike swarm).

Notes: A new ray-tracing method called linear traveltime interpolation (LTI) is proposed. This method computes traveltimes and raypaths in a 2D velocity structure more rapidly and accurately than other conventional methods.The LTI method is formulated for a 2D cell model, and calculations of traveltimes and raypaths are carried out only on cell boundaries. Therefore a raypath is considered to be always straight in a cell with uniform velocity. This approach is suitable to tomography analysis.The algorithm of LTI consists of two separate steps: step 1 calculates traveltimes on all cell boundaries; step 2 traces raypaths for all pairs of receivers and the shot.A traveltime at an arbitrary point on a cell boundary is assumed to be linearly interpolated between traveltimes at the adjacent discrete points at which we calculate traveltimes. Fermat's principle is used as the criterion for choosing the correct traveltimes and raypaths from several candidates routinely.The LTI method has been compared numerically with the shooting method and the finite-difference method (FDM) of the eikonal equation. The results show that the LTI method has great advantages of high speed and high accuracy in the calculation of both traveltimes and raypaths. The LTI method can be regarded as an advanced version of the conventional FDM of the eikonal equation because the formulae of FDM are independently derived from LTI. In the process of derivation, it is shown theoretically that LTI is more accurate than FDM. Moreover in the LTI method, we can avoid the numerical instability that occurs in Vidale's method where the velocity changes abruptly.

Notes: Electrical measurements are an important and integrated component of geophysical investigations connected with environmental problems. As a result of an analysis of the electrical conductivity, basic experiments on sandstones at frequencies below 10 kHz show that the complex behaviour of conductivity is caused exclusively by a complex interface conductivity. Its value is determined mainly by the internal rock interface to porosity ratio, the composition of the pore fluid and connected matrix-water interactions resulting in a specific microstructure of the interface. Therefore, it can be expected that the interface region of a soil or rock material is very sensitive to changes in composition caused by contamination. Contaminated sandstone and clay samples were investigated using a low-frequency measurement system. The investigations are directed at the influence of different contaminants and their concentration. Results show that the complex electrical conductivity (real and imaginary parts) is influenced by properties of the pore-filling contaminant. This influence results in a change of the level of both parts and the shape of their frequency dependence. The imaginary part in particular seems to provide important secondary information; in some cases this part alone allows a differentiation of the various contaminants. The different behaviour of various rock types shows that the effects observed are the result of interactions between pore fluid properties and the internal pore surface structure.

Notes: The following discussion concentrates on one of the synthetic examples used by Palmer (1991) to justify his method of seismic refraction interpretation (Fig. 2, p. 1035) and on his field data and interpretation of the first field example (a collapsed doline) which appeared on pp. 1050–1053.

Notes: A statistical technique, based on the concept of a 1D energy density spectrum of the observed gravity field, has been used to compute ensemble average depths to various horizons containing causative sources of random geometric shape, size, density, etc. The plot of the logarithm of the energy of the observed Bouguer anomaly versus the angular frequency can be approximated, over a certain frequency band, by a linear segment whose slope is related to an average ensemble depth around which a random distribution of numerous anomalous sources exists. Suitable matched filters, based on the computed values of intercepts and slopes of several linear segments approximating the spectrum, have been used to deconvolve the gravity effects associated with the causative sources, occurring around their respective mean depths. The individual deconvolved gravity effects thus separated out have been modelled using the sin x/x method by assuming a fluctuating interface between two formations.The applicability of the present method has been assessed using two observed Bouguer anomaly profiles: one from Ujjain to Mahan, and the other from Jhansi to Mandla where Deep Seismic Sounding (DSS) results are available. The proposed geological crustal models along these two profiles exhibit reasonably good agreement with those obtained from DSS results. A geologically plausible model of the crust in a virgin region has been presented along a Bouguer anomaly profile from Jaipur to Raipur.The following main conclusions have been drawn from the present analysis: (1) The depths to the Moho and Archaean basement interfaces fluctuate between 33.2 and 36.8 km and between 4.6 and 7.0 km respectively. (2) The Narmada-Son Lineament (NSL) does not coincide exactly with the Moho upwarp beneath it. However, this offset is greater in the eastern part of the NSL rather than in the western part. (3) The development of the Satpura horst structure is due to a rise in the Moho interface in a compressional regime. (4) The intrabasement feature (depth from 5 to 12 km) represents a hybrid massif possibly formed due to an admixture of sialic and simatic crust under a tensional regime in the Ujjain-Mahan section.

Notes: The paper by Li and Oldenburg (1991) gives an important insight into d.c. charge accumulation problems. Nevertheless, their derivation concerning the role of the permittivity of the medium is not as straightforward as it could be. Another question, worth discussing, is the problem of double layers, which is missing from the authors’ paper.

Notes: A back-propagation neural network is successfully applied to pick first arrivals (first breaks) in a background of noise. Network output is a decision whether each half-cycle on the trace is a first or not. 3D plots of the input attributes allow evaluation of the attributes for use in a neural network. Clustering and separation of first break from non-break data on the plots indicate that a neural network solution is possible, and therefore the attributes are suitable as network input.Application of the trained network to actual seismic data (Vibroseis and Poulter sources) demonstrates successful automated first-break selection for the following four attributes used as neural network input: (1) peak amplitude of a half-cycle; (2) amplitude difference between the peak value of the half-cycle and the previous (or following) half-cycle; (3) rms amplitude ratio for a data window (0.3 s) before and after the half-cycle; (4) rms amplitude ratio for a data window (0.06 s) on adjacent traces. The contribution of the attributes based on adjacent traces (4) was considered significant and future work will emphasize this aspect.

Notes: In-seam seismic surveys with channel waves have been widely used in the United Kingdom and elsewhere to map coal-seams and to detect anomalous features such as dirt bands, seam thinning and thickening, and particularly in-seam faulting. Although the presence of cleat-induced anisotropy has been recognized in the past, almost all previous analyses have assumed homogeneous isotropic or transversely isotropic coal-seams. Channel waves, however, exhibit properties which cannot be fully explained without introducing anisotropy into the coal-seam. In particular, Love-type channel waves are observed for recording geometries where, in a homogeneous isotropic or transversely isotropic structure, the source would not be expected to excite transverse motion. Similarly, modes of channel-wave propagation display the coupled three-component motion of generalized modes in anisotropic substrates, which would not be expected for Rayleigh and Love wave motion in isotropy or in transversely isotropic media with azimuthal isotropy.We model the observed in-seam seismic channel waves with synthetic seismograms to gain an understanding of the effects of cleat-induced anisotropy on the behaviour of channel waves. The results show a reasonable good match with the observations in traveltime, relative amplitudes, dispersion characteristics and particle motions. We demonstrate that anisotropy in the surrounding country rocks contributes significantly to the coupling of channel wave particle motion, although its effect is not as strong as the anisotropy in the coal-seam. We conclude that the effects of cleat- and stress-induced anisotropy are observed and can be modelled with synthetic seismograms, and that anisotropy must be taken into account for the detailed interpretation of channel waves.

Notes: A polarizable sphere embedded in a conducting half-space can give rise to negative voltage transients in a coincident-loop time-domain electromagnetic system. Such transients have been observed in field situations. Our results are obtained from a model in which the contributions of the host rock, the currents in the sphere, and the interaction between the sphere and the host rock are separated and superposed. This model uses approximations to the integral equation solutions rather than finite-element or finite-difference approximations, and so allows very rapid computation.The theoretical demonstration suggests that interpretation of the negative voltage transients as a polarization response is valid, but more detailed interpretation of polarization properties may not be possible, because the superposition of the polarization response on the normal response depends strongly on the position of the target.