ALBERT

Notes: Specific electrical resistivity of natural waters contains information on their genesis. The authors propose to conduct mass and regime observations of this parameter in river and stream beds.The electrical resistivities in streams flowing from under a glacier reveal details formed at the same time as the glacier. Observations in the beds of big rivers show a gradual increase in water salinity overlain by reductions by inflowing glacial waters.The diurnal and annual trend of changes in the electrical conductivity of water associated with the change in the balance of glacial and ground waters has been established near to glaciers.Resistivity observations help to locate discharge sites of sub-permafrost waters, for water.

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

Notes: Master curves are presented for three-layer earth-sections, the electrical resistivity in the intermediate layer of which has a linear variation with depth. A new approach is proposed to interpret the sounding data with the help of RMS difference in apparent resistivity values.

Notes: Basic Laterolog devices employ a measure-current focusing technique to provide well logs of formation resistivity that feature sharp vertical definition combined with a depth of investigation much improved over those of normals of comparable spacings.To develop the concept of measure-current focusing, a disk-electrode arrangement analogous to a guard-ring capacitor is considered first, leading finally to consideration of a real sonde in a cylindrical borehole. In the latter, auxiliary currents are adjusted to maintain substantially at zero the vertical potential gradients above and below the central measure-current electrode; this has the effect of focusing the measure-current beam.However, prediction of sonde responses on the basis of current patterns can in some cases be misleading. Indeed, for a given Laterolog array, one may use a (reciprocalmode) current pattern radically different from the one defined by the basic (direct-mode) measure-current focusing scheme and still obtain the same resistivity measurement. This feature is illustrated for Laterolog 3 by use of transfer impedances, and demonstrated for certain more general cases using the reciprocity theorem.Comparisons of Laterolog responses across typical thin beds, both non-invaded and invaded, with those of normal devices of a comparable range of spacings illustrate the improved vertical resolution obtained with Laterolog devices. The examples also show the different magnitudes of the auxiliary currents (thenceforth designated “bucking”- or “focusing” currents) used in the direct and reciprocal modes and illustrate how these currents must continuously vary in order to maintain the focused condition as the sonde moves through the bed.Finally, briefly considering possible applications of Laterolog-type arrays in surface prospecting, it is shown that a one-dimensional array laid out on the surface does not provide the same degree of measure-current focusing as is obtained in a borehole. Some improvement is obtained with a combination of two crossed linear arrays. Much more is obtained using a circular array.

Notes: Inferior reflection quality in the Gulf of Suez at the target depth interval is attributable in part to surficial multiple reflections. An excellent example of the latter is observed on a typical seismic line in the northern portion of the Gulf. An increase in prominence of the multiple reflections appears associated with decreasing depth to a dipping highvelocity layer.Inversion of a second-order polynomial time-distance function, fitted to the observed refraction onset time-distance values, gives the velocity-depth function for sediments between the water bottom and a high-velocity layer. Velocities thus determined increase non-linearly with depth from a value near water velocity at the water bottom. Depths to the high-velocity layer are obtained from the associated head-wave linear time-distance function and by ray tracing in the overlying sediments.As the high-velocity layer approaches the water bottom from sub-water depths exceeding 0.6 km to a depth of 56 m, intensity of the multiple reflections increases to the extent of completely dominating individual records to a time of at least 3 s. The estimated plane-wave normal-incident reflection coefficient at the top of the high-velocity layer increases with decreasing depth to this layer, approaching 0.5 at the shallowest depth. This strong reflection coefficient further substantiates the existence of multiple reflections between the high-velocity layer and water layer. However, existence of water-layer multiples cannot be ruled out. The estimated water-bottom reflection coefficient is approximately 0.3, a substantial value. Multiple reflections of considerably less intensity are apparent where the high-velocity layer is deepest, and it is likely that such are waterlayer multiple reflections. Unfortunately, water-layer multiple reflections and multiple reflections between the water surface and high-velocity layer cannot be separated by their coincidence with time-distance (normal moveout) curves, the configuration of each visibly matching the curves equally well.

Notes: Seismic exploration techniques which have been developed for oil prospecting contribute a valuable means for surveying coal measures. Since the object is to detect minor faults within the first 1500 m, rather than structural features at great depth, the new technique requires much higher resolution in the early part of the traditional seismic cross-section.Higher resolution means broader bandwidth, which must be obtained by extending the high frequency end of the spectrum. This is achieved (a) by scaling down the explosive charge size and using single geophones instead of groups, and (b) by reducing the sampling interval in space and time. Noise which does not scale down includes static anomalies and ground-roll. The consideration of statics, ground-roll, and the high-cut filtering effect of the near surface layers forces the use of deep shot holes and, where possible, deep detectors. This approach is confirmed by experiments and has been implemented on a regular basis in production.It is demonstrated that the present technique will clearly resolve faults with a vertical throw of about 5 m at 800 m depth.

Notes: Correlations between longitudinal velocities and rock mechanic parameters such as fracture frequencies and “Rock Quality Designation” (RQD) values have been studied, based upon velocity data from various rock types and different geographical locations.The dispersion of values at different sites studied is on average ± 0.8 cracks per meter and for the RQD values ± 3.5%. Within sites the dispersion of individual values relative to the average for the site is ± 1.0 – 2.0 cracks per meter and ± 2 – 6% for the RQD values.The deviations are rather moderate, especially when considering the variation of rock type involved in the studies: amphibolite, granite, gneiss, meta-anorthosite, pegmatite, porphyry, quartzite, and mylonite. The studies thus confirmed earlier assumptions that there is a strong correlation between longitudinal velocity and fracturing and that the velocities can be used to give rather accurate predictions of the quality of rock masses for construction purposes. The accuracy of the predictions increases if the velocity level of the more competent rock is taken into account.The correlation between velocity and fracturing is related to jointed but unweathered igneous and metamorphic rock and cannot be applied without introducing serious errors to a site where the rocks present a higher degree of alteration and weathering.Comparisons between rock permeability and longitudinal velocity proved that a more reliable general correlation is not likely to be found.By comparing the elastic moduli Edyn, μ, and k with ø, Vp/V8, and k/μ, indications have been obtained where the optimum rock conditions for a certain site are to be encountered. This has been verified by a similar comparison where the elastic moduli have been replaced by fracturing values.The value of the longitudinal velocity as a means to evaluate rock quality increases if the position of the velocity in the range of the Poisson's ratio has been established.The average relationships between longitudinal velocities and the corresponding elastic moduli proved to be:The values from each site differ from the average values with about ± 2 GPa for Edyn and about ± 1 GPa for μ and k.It was confirmed that in igneous and metamorphic rocks longitudinal velocities ≤ 4000 m/s generally indicate rock masses where heavier tunnel support will be needed. This velocity limit corresponds to an average fracture frequency of about 10 cracks per meter and a RQD value of about 65 %. The prediction of the tunnel reinforcements needed at a particular site will, however, be improved if the general velocity level of the more competent rock is considered.

Notes: The Minimum Entropy Deconvolution (MED) technique of Wiggins (1977) represents a breakthrough in deconvolution and will undoubtedly find wide application in many fields. MED does not require any phase assumptions about the disturbing function and seeks a deconvolved output which consists of the smallest number of large spikes consistent with the input data.The efficiency of MED is much improved when an exponential transformation is incorporated into the algorithm. This is particularly true when the input traces contain additive noise. In this case the noise suppression characteristics of MED are considerably enhanced by the transformation and the identification of smaller spikes is improved. This paper also presents a kurtosis criterion of simplicity rather than the varimax norm introduced by Wiggins. It appears that for a multiple trace input the kurtosis measure leads to improved results.

Notes: Long leg multiples can be suppressed by a method which provides an alternative to weighted common-depth-point stacking and multichannel stacking filtering. The suppression is achieved by coherency weighting whereby the time-dependent weighting factor decreases as the semblance of the multiple reflections increases.The algorithm of the method is described. Its efficiency is discussed in relation to the input data and results of its application to marine seismic data are presented.For practical application, the stacking velocity of the multiples has to be known. As the process is based on stacking velocities, different types of multiples can be handled, for instance water-bottom multiples or internal multiples. The parameter analysis shows that the degree of multiple suppression can easily be controlled by adapting the parameters of the procedure to the field conditions.During the suppression of multiples, the primaries are saved according to the moveout differences between the two. The non-linear behaviour of the process causes signal suppression and distortion effects, which have to be corrected by AGC normalization and low-pass filtering.Among the various applications available, only the suppression of long leg water-bottom multiples is treated here. The results show that their suppression on the basis of moveout differences is efficient even when standard length streamers are used in regions with water depth of up to 1500 m and more, if the stacking velocity of the primaries is about 10 to 20% higher than that of the multiples. Even if those parts of the primaries which are masked by the multiples are suppressed in the individual common-depth-point gathers by the procedure, the remaining primaries in the AGV stacked section are largely uncovered by the multiple suppression.

Notes: Investigation of fluctuating earth-currents has led to a simple method of eliminating the noise superimposed on the signal in geoelectrical resistivity surveys. The similarity of the potential fluctuations, simultaneously measured at different locations, is used to calculate the direction and magnitude of the potential gradient. Contour maps of equipotential lines were made, both from disjointed gradient measurements using a numerical approximation method and from a network of measured potential differences. These contour maps were compared with potential fields calculated for simple homogeneous models. A good fit was obtained with the field of a conducting plate. When this is subtracted from the measured field, the result is a map that shows anomalies that can be correlated with geological structures.

Notes: The calculation of potential field anomalies on different levels below the measuring level permits to localize the singular points related to the sources of anomalies.It is possible to achieve considerable improvement of the resolution of this method〈list xml:id="l1" style="custom"〉1) filtering of analytical downward continuation,2) the use of the magnitude of the gravitational vector,3) calculation of physe changes.The numerical examples confirm these assumptions.

Notes: The method of finite differences is applied to the computation of multi-dimensional synthetic seismograms. This paper gives a study of the mathematical and numerical formulations of the problem, the boundary conditions, the convergence conditions and how to simulate the source in both one solid or a liquid. It is shown that the numerical formulation chosen is valid both for direct and inverse problems (i.e. for modeling and migration). This formulation makes it possible to use the normal incidence reflection coefficients for P and S waves, whether they travel horizontally or vertically. The examples shown have been chosen on purpose in order to be easily interpreted. They do not give a full idea of the possibilities of the algorithm which allows to consider non-planar interfaces, except close to the vertical axis.

Notes: Finite difference migration has been developed and popularized by J. F. Claerbout of Stanford University and is now widely used in seismic processing. For most sections finite difference migration gives results comparable to those obtained by conventional Kirchhoff migration and, where events are not dipping too much, a cleaner appearance is often apparent. However, there are two practical limitations to the method, and these occur in regions of very steep dip and where there is a large variation of the velocity in the lateral direction.It is possible to develop successively more accurate equations to deal with the steep dip problem, but above third order these schemes become prohibitively expensive to implement. The finite difference method itself introduces errors and so imposes further limitations on the angle of dip. For the effective treatment of steeply dipping beds there appears to be no method available in the time domain which does not suffer from dispersion inaccuracies. However, by developing wavenumber migration, an exact one-way wave equation can be used, and this eliminates any error except that caused by finite sampling.The other difficulty with wave migration is the correct migration in regions with lateral velocity variation. A number of approaches are possible of which three are discussed here. The first uses an exact theory, the second is based on the deviation from a depth stratified model, and the third uses a transformation to a depth co-ordinate system. All methods are discussed with their advantages and limitations. Finally, some examples are shown of wave migration applied to synthetic and real data.

Notes: In seismic refraction surveys, in particular those using first arrival recording techniques, the hidden layer problem occurs where energy from a refractor of higher velocity arrives at the surface before energy from an overlying refractor. The maximum thickness of the hidden layer is referred to as the blind zone.Hypothetically, every recorded refractor has an associated blind zone which may or may not contain a hidden layer. For an assumed earth model of plane constant-velocity layers and stepwise increase of velocity with depth, the effect of a blind zone on an interpreted depth section may be evaluated by defining an intercept time for a blind zone of assumed or known velocity and by using standard time-term equations for layer thicknesses and depths.The treatment covers an arbitrary number of blind zones embedded within a multilayer sequence of horizontal or dipping refractors. Model calculations affirm the benefits of this approach compared with previous methods which, in general, have been restricted to the case of two horizontal layers with one intermediate blind zone.

Notes: For a thin highly-conducting layer with given longitudinal conductance the recurrence formulae for an n-fold horizontally stratified subsoil are established for d.c. resistivity and magnetotelluric soundings. Similarly, a thin low conductivity layer with given transverse resistance is treated in the d.c. case and a non-conducting intermediate bed in magnetotellurics. Model curves for a thin high- or low-conductivity intermediate layer in the three-layer case have been carried out, which may serve as an extension of the well-known three-layer diagrams for a Schlumberger configuration. The corresponding model curves in magnetotellurics are given.By numerical comparison of these curves with real three-layer curves some diagrams have been developed to show the allowed thicknesses of the intermediate layer in the Schlumberger case and in the case of magnetotelluric sounding.

Notes: Four radar systems at three different frequencies are described which are useful in probing into salt for finding information of interest to miners. Ranges in salt to (a) the edge of a salt dome, (b) the top of the salt dome, (c) boreholes in salt, or (d) faults or hazards ahead of mining can be determined using one or more of these radar systems. Radar wave velocities in salt are determined by radar probing through pillars of known length, and then used to determine ranges in salt to timed radar reflections. Radar probing results are shown obtained in different salt mines probing upwards and downwards. Enclosed areas in the mine are the best radar station locations to probe into salt as air reverberation of radar energy is shortened.

Notes: A simple extension of our previous work in which digital filters were developed to transform dipole resistivity measurements over layered earth to Schlumberger ones leads us to the development of filters for transforming the latter to the former. As in the previous work we use a sampling interval of 1/6 In 10 in designing the filters that are both accurate and fast in operation.

Notes: Seismic facies analysis makes use of different seismic parameters in order to get other than structural information. A review is given of possibilities and usefulness of seismic facies analysis in oil exploration.A seismic facies unit can be defined as a sedimentary unit which is different from adjacent units in its seismic characteristics. Parameters that should be taken into consideration in the seismic facies analysis are as follows: reflection amplitude, dominant reflection frequency, reflection polarity, interval velocity, reflection continuity, reflection configuration, abundance of reflections, geometry of seismic facies unit, and relationship with other units.Interpretation of seismic facies data may be either direct or indirect. The purpose of the direct interpretation is to find out geological causes responsible for the seismic signature of a seismic facies unit. So, the direct interpretation may be aimed at predicting lithology, fluid content, porosity, relative age, overpressured shales, type of stratification, geometry of the geological body corresponding to the seismic facies unit and its geological setting. The indirect interpretation is intended to reach some conclusions on depositional processes and environments, sediment transport direction, and some aspects of geological evolution (transgression, regression, subsidence, uplift, erosion).The results of the seismic facies analysis may be shown on seismic facies cross-sections and seismic facies maps. Depending on the available seismic data and geological conditions in the area under consideration, the seismic facies maps may be of different types such as general seismic facies maps showing distribution of different seismic facies units, sand-shale ratio maps, direction of cross-bedding and paleo-transport maps etc.Several kinds of seismic facies units and their geological interpretation are discussed as examples of seismic facies analysis.

Notes: The reflectivity method for computing synthetic seismograms can be applied to seismic prospecting problems with special focus on coal-mining problems. The method allows the calculation of reflected waves for point-source excitation and non-vertical incidence. It automatically includes all possible conversions of wave types and all inner multiple reflections. Synthetic horizontal- and vertical-component seismogram sections are given for a simple two-seam model and for a realistic seam sequence which is represented by 48 layers; the source-receiver distances range from 100 m to 1000 m. These seismograms show prominent PS reflections already at moderate source-receiver distances. These waves complicate the vertical-component records by producing arrivals of similar strength as the PP reflections. From this it is concluded that PS reflections in strongly layered media can cause problems in routine CDP stacking. On the horizontal-component records the PS reflections are dominant. Because of the lower velocities of S waves the time resolution of PS reflections is better than that of PP reflections. This suggests that horizontal-component recording may be useful in the investigation of subsurface regions with strong velocity contrasts, even with conventional energy sources producing mainly P waves.

Notes: Expressions for the I. P. activity coefficient for an inhomogeneous sphere with the conductivity increasing/decreasing from the core towards the periphery according to a power law have been derived.In certain geometrical situations of the electrode configurations a negative I. P. signal is obtained. The negative I. P. may be understood as due to modifications in the discharge current from the polarized medium during the discharge process.The results of the investigation may be useful for more accurate interpretation of I. P. anomalies due to isometric inhomogeneous bodies.

Notes: Magneto-telluric experiments were performed in August 1974 in the Muhos Jotnian sedimentary formation and the Koillismaa Precambrian basement complex by the Centre de Recherches Géophysiques, Garchy, and the Department of Geophysics, University of Oulu, employing the “Résistivimètre magnéto-tellurique”, an instrument which permits apparent resistivity values to be observed at eight pre-selected frequencies between 8–1700 Hz.Preliminary soundings were made on the Muhos formation and on its surroundings, after which a magneto-telluric profile was measured involving eleven sites on a line 13 km long extending across the formation. Some further soundings were made at certain sites of known stratigraphy. The resistivity values in the sedimentary formation were found to vary between 10 ωm and 200 ωm, while those in the basement were markedly higher. This resistivity contrast suggests that the method seems to be suitable for delimiting the sedimentary formation.Soundings made on the basement at Koillismaa show apparent resistivity values in the range 1000–20000 ωm. The interpretation of these soundings was in agreement with other, independently obtained geophysical results.

Notes: Multispectral recordings used for remote sensing are not without analogy with seismic records from CDP field set-ups. These seismic data may be regarded as “photographs’ of deep regions of the earth taken from various angles.The Karhunen-Loève (K.L.) transformation is commonly used for multispectral data processing, where it helps emphasize some features of remote sensing information. The same method may be applied to seismic data processing. Signal-to-noise ratio is improved on synthetic or field examples when K.L. transformation is applied instead of conventional CDP stacking. Residual statics seem to be diminished by a significant factor.

Notes: A horizontally layered non-absorptive system of homogeneous layers may be specified by giving the reflection coefficients at each interface. Provided the layers have equal vertical travel time and a perfect reflector as a free surface, the reflection coefficients are generally reconstructed from the reflected pulses by way of solving simultaneous equations of the Toeplitz matrix form with the Levinson recursion method. There exists an alternative approach to solving this problem which by simple reasoning immediately turns out the (Levinson) recursion scheme. The method is based on formulas that relate to solving the forward problem. It resembles Kunetz's (1962) original inverse solution in as much as the computation of the reflection coefficients is based on the idea of separating the contribution of a primary from the sum of all multiples.

Notes: The magneto-telluric and telluric-telluric profiling techniques are presented, and then a new method which combines these two techniques for mapping a parameter defined as the “apparent pseudo-resistivity”.The results obtained for the uranium structures in the Cluff Lake district (Saskatchewan Canada) indicate the use of these different techniques in the detection of zones of conduction for surface zones and for deeper zones, and demonstrate the advantages of these methods over the usual electrical methods in problems of structural research.

Notes: The accuracy of the two most common arrival time functions used in seismic velocity estimation is investigated. It is shown that the hyperbolic arrival time function is more accurate than the parabolic arrival time function for a horizontally layered elastic medium. An upper bound on the difference between the two arrival time functions is given.A maximum-likehood detector for estimating the arrival time of the signals is given. For the signal-in-noise model that is used the maximum-likelihood detector is equivalent to a least-squares detector which corresponds to using the signal energy as coherency measure. The semblance coefficient corresponds to a normalized least-squares detector. The semblance coefficient is very similar to a filter performance measure that is used in least-squares filter design.

Notes: The proposed system works as follows:〈list xml:id="l1" style="custom"〉1By a trial-and-error procedure using a graphic display terminal a geologically relevant layer sequence with parameters (ρj, dj) is adjusted to yield roughly the measured curve.2The resulting layer sequence is used as starting model for an iterative least squares procedure with singular value decomposition. Minimization of the sum of the squares of the logarithmic differences between measured and calculated values with respect to the logarithms of the resistivities and thicknesses as parameters linearizes the problem to a great extent, with two important implications:〈list xml:id="l2" style="custom"〉a) a considerable increase in speed (the number of iterations goes down), thus making it cheap to achieve the optimum solution;b) the confidence surfaces in parameter space are well approximated by the hyper-ellipsoids defined by the eigenvalues and eigenvectors of the normal equations.Since these are known from the singular value decomposition we do in fact know all possible solutions compatible with the measured curve and the geological concept.〈list xml:id="l3" style="custom"〉3It is possible to “freeze” any combination of parameters at predetermined values. Thus extra knowledge and/or hypotheses are easily incorporated and can be tested by rerunning step (2). The overall computing time for a practical case is of the order of 10 sec on a CDC 6400.

Notes: Ott and Meder's prediction error filter can be rederived so that it correctly handles input noise vectors which are of smaller dimension than the state vector. The poor performance obtained by Ott and Meder for their example can be explained by means of the error covariance matrix for the prediction error filter.

Notes: The geological sequence rock-coal-rock represents a seismic low-velocity channel. Channel waves generated in a coal seam and their reflections from discontinuities can be used for proving the minability of the seam. To investigate the process of reflection, two-dimensional models of the sequence rock-coal-rock have been investigated by means of the ultrasonic transducer technique. Two-dimensional models have the advantage that the wave field can be observed at an arbitrarily chosen point of the model plate. Thus, by means of these models the direct and the reflected Rayleigh channel wave can be observed along their path of propagation.From the geophysical point of view the various types of discontinuities of a coal seam can be divided into two basic types: one is restricted to the seam, the other includes in addition a fractured zone in the adjoining rock.The investigation of the symmetrical Rayleigh channel wave reflected by a discontinuity yields the following results:For dip angles γ between approximately 90° and approximately 60° the reflectivity is virtually independent of the type of discontinuity. This does not hold for the limiting case of γ= 90° (vertical dip) for which the reflectivity increases with increasing influence of the fractured zone. For dip angles γ between approximately 60° and approximately 40° the reflectivity is still independent of the fractured zone but the shape of the reflected wave deteriorates with increasing influence of the fractured zone.For dip angles γ below approximately 40° the reflected wave deteriorates such that the application of the in-seam seismic reflection method will be difficult or even impossible.The conversion of the direct wave of the symmetrical fundamental mode into a reflected wave of the antisymmetrical fundamental mode has been observed.

Notes: Analytical solutions exist for the potential of a Point Source in media with depth dependent conductivity of the form σ(z) =σ0(1 +αz)n.Of particular importance is the case of the linear gradient (n= 1). More general distributions of conductivity with depth can be approximated by equivalent sequences of homogeneous layers. Comparison between such approximations and the analytical solution for media with linear gradient of conductivity indicates that already a coarse step function yields a reasonably good approximation.

Notes: The contribution of electromagnetic phase comparison techniques to distance measuring is reviewed in an historical context. Special emphasis is accorded the new class of near infra-red electro-optic instruments suited to the measurement of distances under 2 km with an accuracy of a few millimetres. Principles of operation, sources of error, and practical applications are discussed.

Notes: A magnetotelluric digital equipment has been designed and built for operation in geophysical and applied geophysical research. The five components of the natural electromagnetic field can be digitized in the frequency range 1/1000 Hz (magnetic sensor's response limits) to some tens Hz.The magnetotelluric (MT) data are directly digitized in the field and stored in a cassette tape recorder.

Notes: An electromagnetic frequency sounding experiment with a rigid horizontal transmitter coil carrying a stabilized oscillating current was carried out in South Tunesia.The field data were interpreted in terms of the mutual impedance ratio in the horizontal coils system. Where the measurements were sufficiently accurate they generally could be interpreted to a high degree of fit. It is concluded that a vehicle-mounted electromagnetic frequency sounding system is suitable for a fast survey.

Notes: The application of approximate rules, whereby apparent resistivity space sections for two dimensional structures can be composited from spaces derived for elementary features is extended to a complex example drawn from a field survey over a fluorite mineral vein.A quantitative solution for the observed resistivity space is presented and the computational sequence involved in matching the observed space is given in detail.The interpreted results are examined in relation to the known geology, supplemented by the results of excavation, and to model tests conducted using a tank analogue.The example also illustrates how successive compositions can be employed in estimating the form of resistivity space in a relatively complex situation.

Notes: The sampling theorem in two dimensions univocally defines a surface, provided that its values are known at points disposed on a regular lattice. If the data are irregularly spaced, the usual procedure is first to interpolate the surface on a regular grid and then to contour the interpolated data: however, the resulting surface will not necessarily assume the prescribed values on the irregular grid.One way to obtain this result is to introduce a transformation of the coordinates such that all the original data points are transferred into part of the nodes of a regular grid. The surface is then interpolated in the points correspondent to the other crosspoints of the regular grid; the contour lines are determined in the transformed plane and then, using the inverse coordinate transformation, are transferred back to the original plane where they will certainly be congruent with the original data points.Nonetheless, the resulting surface is very sensitive to the interpolation method used: two algorithms for that are analyzed. The first (harmonization) corresponds to the determination of the potential of an electrical field whose contour conditions are those defined by the data points. The second method consists in two dimensional statistical estimation (krigeing); in particular, the effects of different choices for the data auto-covariance function are discussed.The solutions are compared and some practical results are shown.

Notes: Optimum stacking filters based on estimates of trace signal-to-uncorrelated noise ratios are assessed and compared in performance with conventional straight stacking. It is shown that for the trace durations and signal bandwidths normally encountered in seismic reflection data the errors in estimating signal/noise ratios largely counteract the theoretical advantages of the optimum filter. The more specific the filter (e.g. the more frequency components included in its design) the more this is true. Even for a simple weighted stack independent of frequency, the performance is likely to be better than a straight (equal weights) stack only for relatively high signal/noise ratios, when the performance is not critical anyway.

Notes: An approximate method of interpretation of resistivity sounding is presented, which may be described as a very crude manner of application of the exact direct interpretation method. The accuracy of this method is fairly low, the errors being in the order of 25%. The method is very fast in application and well suited for application to multilayer cases. The main advantage of the method is that it is in close and clear relation to the exact theory.

Notes: Electromagnetic coupling in grounded electrical prospecting systems has been studied over four decades. Recently, advanced digital electronic systems have been developed which permit both magnitude and phase measurements over four frequency decades (10−2— 102 Hz). This development has stimulated theoretical studies demonstrating behavior of EM coupling for the more commonly-used electrical prospecting arrays a wide frequency range. A comparison is made between field and theoretical results which demonstrates the validity of the fundamental assumptions involved. Additionally electromagnetic coupling is used successfully as a deep-sounding technique in a highly-conductive sedimentary basin environment.

Notes: A polygonal ray path connects the seismic source and detector positions when the intervening medium consists solely of constant velocity layers with plane interfaces which may have arbitrary orientation. The coordinates of the ray vertices satisfy a system of coupled equations resulting from the requirement that Fermat's principle be satisfied along the ray path. Solving the system of equations is equivalent to tracing the ray numerically. A notable feature of this approach is that a ray which is critically refracted over a segment of its path requires no special handling.

Notes: A satisfactory attenuation of the multiples in marine seismic may be obtained by the application of the principle of “Antiaveraging”.This principle in a first step consists in getting the model of the organized noise, which one tries to eliminate by using an averaging method, and in a second step to subtract that model from the initial information.Obviously the elimination of the model should not simultaneously cause the elimination of useful signals.The model may be obtained if the considered organized noise keeps a constant shape or if its time-space deformation is known. Besides one has to assume the time-distance curve of the organized noise can be determined. Thus noise arrivals may be detected on the records.The “antiaveraging” is very often efficient when organized noises are stronger than signals or when a signal, once identified, exploited and then considered as an organized noise, can be attenuated in order to make the detection of the other signals easier.

Notes: Dip selective migration velocity determination calculates coherency of energy migrating into an output trace or ‘MVD-axis’ as a function of the three variables time, velocity, and dip. In doing so, it provides additional information, the dip information, relative to standard MVD.

Notes: Pseudo-velocity-logs are tentative determinations of subsurface velocity variations with depth, using both information of seismic amplitude and reflection curvature.A rigorous theoretical method would consist in〈list xml:id="l1" style="custom"〉a) deconvolving the seismic traces to remove the filtering effects of the ground and of the recording equipmentb) demultiplying the deconvolved traces by a complete desynthesization with convergence criteriac) computing the velocities.While this method works with synthetic examples, it is not generally applicable to field cases, one of the reasons being the poor reliability of desynthesization in the presence of noise.The present method is a compromise between a rigorous and a practical process: the complete desynthesization is not performed; deconvolution and demultiplication are done by more classical techniques using real amplitudes; absolute velocities are determined to fit both the reflection coefficients and the rms velocities. It leads to pseudovelocity-logs, accurate enough to show lithologic variations, smoothed enough to preserve the signal/noise ratio.Examples are shown of Flexichoc profiles recorded in 2500 m (8000–9000 feet) deep areas of the Mediterranean Sea. Pseudo-velocity-logs show 1000 m (3000 feet) of a velocity-increasing-with-depth Plio-pleistocene marl formation, overlying Miocene evaporites. Intercalations of high and low-velocity layers in the evaporites seem to indicate vertical facies variations.The Pseudo-velocity-log, associated with other lithologic determination processes, should become a geological tool for deep offshore exploration.

Notes: Dynamic predictive deconvolution makes use of an entire seismic trace including all primary and multiple reflections to yield an approximation to the subsurface structure. We consider plane-wave motion at normal incidence in an horizontally layered system sandwiched between the air and the basement rock. Energy degradation effects are neglected so that the layered system represents a lossless system in which energy is lost only by net transmission downward into the basement or net reflection upward into the air; there is no internal loss of energy by absorption within the layers. The layered system is frequency selective in that the energy from a surface input is divided between that energy which is accepted over time by net transmission downward into the basement and the remaining energy that is rejected over time by net reflection upward into the air. Thus the energy from a downgoing unit spike at the surface as input is divided between the wave transmitted by the layered system into the basement and the wave reflected by the layered system into the air. This reflected wave is the observed seismic trace resulting from the unit spike input. From surface measurements we can compute both the input energy spectrum, which by assumption is unity, and the reflection energy spectrum, which is the energy spectrum of the trace. But, by the conservation of energy, the input energy spectrum is equal to the sum of the reflection energy spectrum and the transmission energy spectrum. Thus we can compute the transmission energy spectrum as the difference of the input energy spectrum and the reflection energy spectrum. Furthermore, we know that the layered system acts as a pure feedback system in producing the transmitted wave, from which it follows that the transmitted wave is minimum-delay. Hence from the computed energy spectrum of the transmitted wave we can compute the prediction-error operator that contracts the transmitted wave to a spike. We also know that the layered system acts as a system with both a feedback component and a feed-forward component in producing the reflected wave, that is, the observed seismic trace. Moreover, this feedback component is identical to the pure feedback system that produces the transmitted wave. Thus, we can deconvolve the observed seismic trace by the prediction-error operator computed above; the result of the deconvolution is the wave-form due to the feedforward component alone. Now the feedforward component represents the wanted dynamic structure of the layered system whereas the feedback component represents the unwanted reverberatory effects of the layered system. Because this deconvolution process yields the wanted dynamic structure and destroys the unwanted reverberatory effects, we call the process dynamic predictive deconvolution. The resulting feedforward waveform in itself represents an approximation to the subsurface structure; a further decomposition yields the reflection coefficients of the interfaces separating the layers. In this work we do not make the assumption as is commonly done that the surface as a perfect reflector; that is, we do not assume that the surface reflection coefficient has magnitude unity.

Notes: The theory of geoelectric sounding exploration is based on Stefanescu's kernel. In order to study the analytic properties of the kernel, necessary and sufficient conditions on it's poles and zeroes are developed. A distinction is made between “physical” and “perfect” models.It is shown that a physical model may be determined from either the set of poles or the set of zeroes of the raised kernel function at the top of the model. These roots may be of desired multiplicities.For determining a perfect model, both sets of poles and zeroes are required and they all have to be simple ones.

Notes: The Galerkin method of solving integral equations is well suited to the solution of the integral equations describing the transient response of a sphere embedded in a layered medium, which is excited by a large co-axial loop.The transient response is calculated by transforming the steady state solutions obtained in the frequency domain.The analysis shows that the scattering matrix is extremely diagonally dominant and the maximum number of modes required to obtain convergence does not rapidly increase with frequency. The number of modes required is about eight. This type of scattering matrix can be taken to be an expression of the principle of elementary superposition. This principle is reflected in the decay curves. These show that the early part of the decay curves asymptotically approach the decay curves to be expected for a layered structure without the sphere. The slope of the latter stages of the decay curve gives a decay constant that is the same as was obtained for spheres in free space excited by planar or dipolar sources.The point of departure in time of these curves from the layered ground curves is delayed either by placing the sphere at a greater depth or by placing a more conductive overburden above the sphere.

Notes: The hydrocarbon-brine contact produces a flat reflection, unconformable with the lithologic reflections from the trap boundaries, and over a limited area bounded by structural contours. When it can be reliably detected and mapped, the flat spot can provide (i) a reasonably unambiguous indication and areal extent of a reservoir and (ii) an estimate of reservoir thickness.The gas-brine contact in thick reservoirs in offshore elastic sections is the easiest target. An example of gas mapping is presented in the paper. Other reservoirs represent a continuum of increasingly elusive targets.Increasing the range of applicability of flat spot exploration will require (i) increases in signal/noise and multiple ratio, increases in the three dimensional depth point density (or grid density), increased resolution and static and dynamic correction accuracy, and (ii) processing and interpretation aimed directly at flat spot mapping.Low relief structural and stratigraphic traps should provide the most attractive targets. The approach should be at least theoretically feasible, though not necessarily cost effective, for most major reservoirs with a well-behaved hydrocarbon brine contact.

Notes: The estimation of residual static corrections is investigated as a linear system. For 1200% coverage shot one way offset weighting functions—or in other words, estimation error operators—are computed. Assuming zero time shift determination errors and zero RNMO it is seen that large residual corrections will distort the effectivity of the estimation.With an iterative procedure which includes the repeated averaging according to depth point, shot point, and geophone point the effectivity of the method becomes more favourable. This can be learnt from the evaluation of estimation error operators computed by recursive formulae.In practice, due to the time shift determination errors, the routine does not give improved results in the desired level. Errors can be reduced with a more reliable time shift determination routine.For detecting and suppressing the unfavourable effect of time shift errors still remaining in the data a self-adaptive method was developed. In this non linear routine the data are evaluated at all three averaging processes according to limit values and weighting factors computed from the data itself.Field examples processed in different ways are presented.

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

Notes: It is well known that interval velocities can be determined from common-reflection-point moveout times. However, the mathematics becomes complicated in the general case of n homogeneous layers with curved interfaces dipping in three dimensions.In this paper the problem is solved by mathematical induction using the second power terms only of the Taylor series which represents the moveout time as a function of the coordinate differences between shot and geophone points. Moreover, the zero-offset reflection times of the nth interface in a certain area surrounding the point of interest have to be known. The n—I upper interfaces and interval velocities are known too on account of the mathematical induction method applied. Thus, the zero-offset reflection raypath of the nth interface can be supposed to be known down to the intersection with the (n—1)th interface.The method applied consists mainly in transforming the second power terms of the moveout time from one interface to the next one. This is accomplished by matrix algebra.Some special cases are discussed as e.g. uniform strike and small curvatures.

Notes: The least squares estimation procedures used in different disciplines can be classified in four categories:〈list xml:id="l1" style="custom"〉a. Wiener filtering,b. b. Autoregressive estimation,c. c. Kalman filtering,d. d. Recursive least squares estimation.The recursive least squares estimator is the time average form of the Kalman filter. Likewise, the autoregressive estimator is the time average form of the Wiener filter. Both the Kalman and the Wiener filters use ensemble averages and can basically be constructed without having a particular measurement realisation available.It follows that seismic deconvolution should be based either on autoregression theory or on recursive least squares estimation theory rather than on the normally used Wiener or Kalman theory. A consequence of this change is the need to apply significance tests on the filter coefficients.The recursive least squares estimation theory is particularly suitable for solving the time variant deconvolution problem.

Notes: The hidden layer problem in seismic refraction work has been studied for three velocity configurations – the intermediate layer having (a) lower, (b) intermediate and (c) higher velocity than the underlying and overlying beds. It has been shown that conventional methods fail to locate the presence of the intermediate layer for the cases (a) and (c) and lead to errors in calculating the depth to the bedrock. For the case (b), it is possible to interpret the first arrival travel time analytically as an alternative to Green's graphical approach. It has been suggested that the hidden layer may be detected in all the three cases if converted S waves are also recorded in the seismogram.

Notes: The depth of causative bodies may be derived from the power spectrum of their magnetic anomalies. After a short review of the theoretical basis of the method, several examples of its application to synthetic cases are shown. Disturbing effects due to improper choice of the sampling interval and to anomalies only partially contained in the examined segment of the profile are studied.The spectral method is then applied to real cases; the tapering of the anomalies, the application of non-linear filters and the effects of anomaly superpositions are investigated.In conclusion, an appraisal of the method and of its possible practical impact has been given.

Notes: Some time ago, we described and implemented two methods of seismic data compression. In the first method a seismic trace is considered as being the convolution of a distribution made up of the trace peak values with a Gaussian pseudo-pulse. The second method is performed through a truncation of the sequential (Walsh, Paley or Haar) spectrum of each trace.In this paper it is shown that neither method has adverse effects on quality when traces with their information compressed undergo conventional data processing, such as stacking and deconvolution.

Notes: The effects of systematic (constant) and random errors in the observed data have been investigated analytically for rational approximation method of computing second derivative involving a summation of the products of the averages of the gravity field with the corresponding weight coefficients, both in numerator as well as in denominator. A theoretical gravity anomaly over three spheres has been analyzed to demonstrate the high accuracy in the approximation. Since the sums of the weight coefficients in numerator and denominator are zero and one respectively, the regional gravity anomaly, even though approximated by a constant value over the entire area under computation, can produce substantially large error in the calculated derivative value. This is happening because of the contribution of the regional field in the denominator. Thus, inspite of the high accuracy in rational approximation, the method has limited application to field cases where a combined gravity field consisting of regional and residual anomalies is usually used. Master curves are presented for the constant and random errors by which a rough estimate of the percentage of error in second derivative computation can be made provided one has some idea of the magnitudes of the regional field and random error.

Notes: When a rather extensive network of short distance refraction profiles was shot the following two observations were made. Firstly, the wave velocity in the weathered layer showed a quite considerable fluctuation in horizontal direction, the extreme values being of the order of 300 and 700 m/sec. A practical result of this scattering of the velocities is that an uncertainty of about 10% is introduced in the computation of the weathering corrections. The second observation is that, in the majority of cases, the travel time curve does not pass through the origin. It could be ascertained that this observation was not caused by errors in the method of observation. A probable explanation of the phenomenon is found in a theory that has been developed by Gassmann, who derived the velocity distribution for a hexagonal packing of spherical solid bodies. According to this theory extremely low velocities would occur in the first few centimeters of the crust of the earth. Qualitatively, our observations are in agreement with the theory of Gassmann.

Notes: The geophysicists who attempted to evaluate elastic anisotropy of some bedded formations most often gave but little information about recording and interpretating the measurements.May be they did not lay enough stress on the fact that any determination of the anisotropy factor is a critical operation and that, to have some certainty. it must be made under good conditions and then discussed.Due to the very favourable conditions offered: undisturbed tectonics and topography, relatively regular series the Berriane district in Northern Sahara, is rather well suited to accurate measurements and to an anisotropy study down to 1,250 meters. Since measurements have been carried out in the wildcat drilled at Berriane by S. N. Repal, it seems interesting to communicate the results obtained and to discuss them according to the influence of high velocity layers.It can be noted that in the clay and sand series which have been investigated, anisotropy coefficient amounts to 1.09 or so.To conclude, it seems desirable that determinations made by numerous operators should allow a more accurate knowledge of the anisotropy factor in the main types of rocks.

Notes: The aim of the present investigation is to examine the significance of the second derivative for the purpose of the prospective determination of the geological properties of the district of Offenburg, which are relatively wellknown as a result of both reflection seismic and bore profile findings.The calculation of the second derivative is carried out by means of three approximation formulae, which are compared with each other. In contrast to the isogam map the second derivative shows some additional characteristics, whose structural significance illuminates the comparison with the results of reflection seismics.A geological profile constructed on the basis of 6 wells completely confirms the geophysical findings.

Notes: A technique is suggested to separate the effect of conductive overburden from the recorded time-domain electromagnetic anomalies. Such a separation leads one to construct a relatively undistorted picture of the conductor underlying a conductive overburden. A parameter τ (with dimensions of time) is introduced to characterize various conductors. To illustrate the applicability of the technique, four ‘Input’ (Mark V) anomalies obtained over the Cuddapah District, A.P., India have been analyzed and different values of the time-parameter τ corresponding to the overburden and the mineralized zones are obtained.

Notes: Consider a thin, plane, finite sheet of material of thickness 2t whose electrical resistivity is σ2. It is surrounded by a medium of resistivity σ1 in which there is an electric field whose length scale is L. We consider the limit t/L→ o with 〈inlineGraphic alt="inline image" href="urn:x-wiley:00168025:GPR70:GPR_70_mu1" location="equation/GPR_70_mu1.gif"/〉, or 〈inlineGraphic alt="inline image" href="urn:x-wiley:00168025:GPR70:GPR_70_mu2" location="equation/GPR_70_mu2.gif"/〉 fixed. In the former case the sheet is an imperfect conductor and it is shown that it may be modelled by a surface on which the boundary condition is 〈inlineGraphic alt="inline image" href="urn:x-wiley:00168025:GPR70:GPR_70_mu3" location="equation/GPR_70_mu3.gif"/〉, where φ and Z are the (nondimensional) electrical potential and distance measured normal to the surface, and the subscripts + and − denote values on either side of the surface.In the latter case the sheet is an imperfect insulator and the boundary condition is 〈inlineGraphic alt="inline image" href="urn:x-wiley:00168025:GPR70:GPR_70_mu4" location="equation/GPR_70_mu4.gif"/〉

Notes: Electromagnetic depth sounding using source orientation as the sounding variable provides advantages in instrumental simplicity and operational flexibility when compared with conventional modes of electromagnetic sounding. The ease with which the technique may be simulated in a scale model permits its application to sounding over localized structures. The theoretical approach to interpretation is at present limited to structures which approximate a perfectly conducting half space. However, scale model tests suggest that the perfect conductor theory may be applicable to many localized structures.

Notes: Using a comprehensive theory of the vertical force variometer, it is shown that the mean of the two readings taken in azimuths 180 degrees apart only eliminates the error of misorientation and of inclination of the knife edge if these factors are sufficiently small. The necessary accuracy in the 180 degree rotation and in the readjustment of the levels are investigated and a practical test is developed to allow the initial adjustments to be determined. Finally, a brief description of a new variometer tripod is given.

Notes: It is known that good use can be made of calculating the second vertical derivative of gravitational and magnetic fields. The first derivative, or Vertical Gradient, has the same properties, Like the second derivative, it does not depend on the locally linear regional anomaly. Moreover, the gradient is less affected by experimental errors.The report gives a practical method of calculating the vertical gradient. Its application is identical to that in general use for calculating the second derivative, except that the numerical coefficients are different. The method also allows the establishment of formulae for downward extension of the field and its derivatives.

Notes: In the region covered by the paper, electrical methods have been applied in four different zones:〈list xml:id="l1" style="custom"〉a) Ales-Maruejols Basin (resistive lower Oligocene blanketed by conductive Upper Oligocene);b) Ledignan and Vaunage-Lunel anticlines (faulted structures in the marly and limy Cretaceous, with different resistivities;c) Camargue (thick series of conductive sediments);d) Beziers Basin (highly complicated structures in the Mesozoic, widely blanketed by conductive Miocene).It may be assumed that electrical methods can yield valuable information enabling one to define the rugged subsurface topography of resistive strata burried under a reasonably thick cover of conductive sediments especially when a part of the lower formation outcrops even if the part exposed be small. Faults may be delineated and much, more information is obtainable than can be expected from gravity meter survey at a price not very much higher.

Notes: For the treatment of the problems involved in the interpretation of gravity pictures this paper gives a formula that holds generally for any potential function. It provides an extended applicability of the relation used by Elkins for the computation of the second derivative, and also yields an analogous relation to derivatives of higher order of the gravity field.This relation has not been utilized to determine in details the course of the derivatives of higher order in a plane, but is merely applied to points of comparatively extreme curvature of the isogams where the differential values present a ratio favourable to the “noise level”. By this method the values of the derivatives of higher order will not be determined themselves but only the ratio's responsible for the depth of divergence in comparison with the course of an anomaly of masses replaced by points.It is shown by examples of intercalated masses that the divergence points are important and by law related to the form and position of the density contrasts themselves.An analysis of the total gravity picture is made possible by progressing from elements close to the surface to deeper ones. In this way the fundamental features of the earth's crust will be obtained. The application of the analysis process to a gravity anomaly actually measured in Northwest Germany is given.

Notes: When refraction time signals are transmitted over a radio link between shooter and observer there is a serious risk of interference from electrical storms. This risk can be minimised by sending a series of signals at fixed time intervals and the paper describes a simple device which will do this with the required accuracy.

Notes: Seismic reflection, telluric and gravity results are given for the lifted area of Ferrara, in Northern Italy. The agreement is shown between the available shallow seismic data and the gravity evidences. For a portion of the above area where seismic results are particularly poor the vertical gradients method gives a resolution of a gravity anomaly and confirms the telluric information. The residual gravity values, the vertical gradients and the telluric data along a seismic line crossing the zone are compared with seismic results and with evidences from adjacent wells.

Notes: Investigations into reflection seismic on the salt structure of Heide raised the following questions:〈list xml:id="l1" style="custom"〉1) How can the different directions of trend of a series of layers be established?2) How are the contour maps of reflecting horizons to be constructed if there is no general trend of the layers?To answer these questions, the case of the subsoil consisting of a series of layers with plane reflecting beds and constant layer-velocities is treated strictly. The strike lines of the reflecting planes may be at any angle whatever. However, formulas for constructing contour maps of the reflecting planes have been derived. With these formulas, the problem quoted under 1) has also been solved.The construction of contour maps is given.For special cases, formulas are obtained which have already been proved by others.

Notes: The triple integral for the attraction of a mass on a point is transformed in a double sum. This double sum serves to construct a graph for the rapid determination of the gravimetric influence of structures given by depth contours. The influence of topography can also be computed with this graph.

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

Notes: Existing techniques of deconvolution of gravity anomalies are principally based on upward and downward continuation of measured fields. It can be shown that a unique set of linear filters, depending only on geometrical parameters, relates density distribution at a given depth to gravity measured on the surface. A method to compute the filter coefficients is developed. Very accurate reconstitution of theoretical models of intricate shape, prove the validity of the linear relationship. One of these sets of linear filters is applied to a field case of underground quarries.

Notes: Model tank inline resistivity profiles with Wenner and two-electrode systems obtained over two vertical, parallel, infinitely conducting dykes submerged in water show that the two-electrode resistivity anomalies resolve better than the four-electrodes Wenner. anomalies. In contrast, the broadside resistivity anomalies obtained with a Wenner configuration resolve much better than those obtained with two-electrode configuration over an identical ground structure.

Notes: Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type:〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR876:GPR_876_mu1"/〉Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*.If the input function f(λ exp (iω)) is known to be analytic in the region o 〈 λ 〈 ∞, |ω|≤ω0 of the complex plane, we can show that the absolute error on the output function is less than (K(ω0)/r) · exp (−ρω0/Δ), Δ being the logarthmic sampling distance.Due to the explicit expansions of H* the tails of the infinite summation 〈displayedItem type="mathematics" xml:id="mu2" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR876:GPR_876_mu2"/〉 ((m−n)Δ) can be handled analytically.Since the only restriction on the order is ν 〉 − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).

Notes: For detailed determination of refractor velocities of in-line reversed profiles, two principally different systems have been employed, the ABC method and Hales's method. The two systems differ in the travel-path arrangements used. The more conventional approach to the problem, the ABC method, makes use of critically refracted rays converging on a common surface position, while Hales's method deals with the common position on the refractor surface from which critically refracted rays diverge towards the ground surface.Because of the travel-path system used, Hales's method has proved to be particularly applicable to high-relief structures and to cases where the refractor lies at considerable depth. Some of the ambiguities in more commonly used interpretation techniques can be solved by Hales's method.Some attention has also been paid to errors caused by non-critical refractions and to the diffraction problem.

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

Notes: A numerical technique is developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations by using point- as well as area-discretization of the subsurface. Potential distributions at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct explicit matrix inversion techniques. An empirical mixed boundary condition is used at the “infinitely distant” edges of the lower half-space. Accurate solutions using area-discretization method are obtained with significantly less attendant computational costs than with the relaxation, finite-element, or network solution techniques for models of comparable dimensions.

Notes: The Wiener filtering scheme is obtained for a multichannel complex system utilizing the “block-Toeplitz” property of autocorrelation matrix. A numerical example is given to elucidate the application of the filter design. The study also outlines Burg's maximum entropy method to include the multi-channel complex realm.

Notes: An algorithm for separation of signals according to their coherency is proposed. The algorithm, called Superstack, is used on common depth point data after normal moveout corrections have been applied. The algorithm can be regarded as an iterative stacking procedure. After each stack, input values are changed depending on the consistency of the output of the previous stack.The Superstack algorithm is able to provide better separation of signals showing a different degree of horizontal consistency (coherency) than the normal horizontal stack.All noise signals in a CDP-gather may be assumed to show less horizontal consistency than the primary reflections. On this basis the algorithm will give an improved primary/multiple and an improved primary/noise ratio.The Superstack is very general, should not be especially expensive in terms of computer time, and can easily be implemented in routine processing.

Notes: Computations of the time-domain electromagnetic response of a multi-layered earth have been carried out for different source-receiver coil systems. The primary excitation is a train of half-sinusoidal waveforms of alternating polarity. The conversion into the time-domain involves Fourier series summation of the matched complex mutual coupling ratios of the layered earth models computed by a digital linear filter method. As an example, the response of a perpendicular coil system on the ground surface for two source-receiver separations has been presented for a five-layer earth model. This has been compared with the responses of homogeneous, two-layer, three-layer, and four-layer models.Next, the investigations have been extended to study the problems of equivalence of three-layer models, the intermediate layer of which is either conductive or resistive. For an intermediate conductive layer (H-type), the studies show that in the early portion of the signal the interpretation of a true three-layer earth is possible to some extent, whereas the ambiguity due to equivalence persists in the late samples. On the other hand, for an intermediate resistive layer (K-type), the three-layer earth and its equivalent model cannot be distinguished from each other over the entire sampling period.On the basis of a computational approach, equivalence has been empirically established as √h/ρ=constant for H-type earth-sections, and as h2ρ=constant for K-type earth sections, where h and ρ are respectively the thickness and resistivity of the intermediate layer.

Notes: It has been found that the Wiener-Hopf least-squares method is a very successful tool for the determination of resistivity sounding filters. The values of the individual filter coefficients differ quite appreciably from those obtained by the Ghosh procedure. These differences in the filter coefficients, however, have only a negligible effect on the output of the filter. It seems that these differences in the coefficients correspond to a filter function of a rather narrow frequency band around the Nyquist frequency, which is only very weakly present in the input and output functions.