Showing papers in "Geophysical Prospecting in 1977"

Time migration—some ray theoretical aspects*

Abstract: Using an elementary theory of migration one can consider a reflecting horizon as a continuum of scattering centres for seismic waves. Reflections arising at interfaces can thus be looked upon as the sum of energy scattered by interface points. The energy from one point is distributed among signals upon its reflection time surface. This surface is usually well approximated by a hyperboloid in the vicinity of its apex. Migration aims at focusing the scattered energy of each depth point into an image point upon the reflection time surface. To ensure a complete migration the image must be vertical above the depth point. This is difficult to achieve for subsurface interfaces which fall below laterally in-homogeneous velocity media. Migration is hence frequently performed for these interfaces as well by the Kirchhoff summation method which systematically sums signals into the apex of the approximation hyperboloid even though the Kirchhoff integral is in this case not strictly valid. For a multilayered subsurface isovelocity layer model with interfaces of a generally curved nature this can only provide a complete migration for the uppermost interface. Still there are various advantages gained by having a process which sums signals consistently into the minimum of the reflection time surface. The position of the time surface minimum is the place where a ray from the depth point emerges vertically to the surface. The Kirchhoff migration, if applied to media with laterally inhomogeneous velocity, must necessarily be followed by a further time-to-depth migration if the true depth structure is to be recovered. Primary normal reflections and their respective migrated reflections have a complementary relationship to each other. Normal reflections relate to rays normal to the reflector and migrated reflections relate to rays normal to the free surface. Ray modeling is performed to indicate a new approach for simulating seismic reflections. Commonly occuring situations are investigated from which lessons can be learned which are of immediate value for those concerned with interpreting time migrated reflections. The concept of the ‘image ray’ is introduced.

A man/computer interpretation system for resistivity soundings over a horizontally strafified earth*

Abstract: The proposed system works as follows: 1 By 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. 2 The 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: 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. 3 It 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.

Interpretation of potential field data a generalized inverse approach

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

Inversion of seismograms and pseudo velocity logs

Abstract: Pseudo velocity logs can be obtained by seismogram inversion, using true amplitude processing and detailed investigation of move-out velocities. The precision of the results depends on the quality of the seismic data and on the possibility of deconvolving without increasing the noise. An investigation is made of the deformation of pseudo logs due to seismic signal variations and to imperfections of deconvolution. Both marine and land examples are shown, in some cases with adjustment on well logs. When the dips are large, time sections must be migrated and pseudo velocity logs must be computed from migrated sections. Comparison of sonic logs with pseudo velocity logs obtained in the same area is usually good enough to obtain information on lithological parameter variations by adjustment of pseudo velocity logs on sonic logs. Even when no well is available, pseudo velocity logs can give some indications on the nature of sediments between seismic horizons.

The Old and the New in Seismic Deconvolution and Wavelet ESTIMATION

Abstract: Various seismic deconvolution operators can be determined by estimating a seismic wavelet and subsequently designing an appropriate inverse filter which converts the wavelet to a spike. Seismic wavelets and deconvolution operators must be estimated in a time adaptive sense due to the nonstationarity of the seismic trace. The wavelet estimation methods considered either use the assumption of a minimum phase wavelet and a random impulse response, or the assumption that the wavelet cepstrum is readily separable from the cepstrum of the seismic trace. The former assumption is required in using the Hilbert transform and Wiener-Levinson wavelet estimations, while the latter assumption is used in homomorphic deconvolution. These wavelet estimates can be used in the design of multichannel Wiener and Kalman deconvolution operators. Multichannel usage of homomorphic deconvolution can also be implemented through various types of cepstral stacking. The discussion of deconvolution filter design focuses on the problems of filter length degree of prewhitening and nonstationarity. In designing time adaptive deconvolution filters, the autocorrelation function can be used to monitor the nonstationarity of the seismic trace. The autocorrelation function, which is used in the computation of least squares inverse filters, can be estimated in an optimum fashion by using the maximummore » entropy method. Differences between minimum phase Wiener deconvolution and maximum entropy deconvolution become more pronounced for shorter data gates. As a result the maximum entropy approach is preferred for time adaptive deconvolution. The performance of various wavelet estimators and inverse filters is discussed using real and synthetic seismic data. Discussions of homomorphic deconvolution and maximum entropy prediction error filtering are merged with descriptions of conventional approaches to deconvolution.« less

Resolving anisotropy in layered media by joint inversion

Abstract: A finite number of layers must normally be used to represent the electrical properties of a horizontally-layered region. It is well known that these representative layers must include anisotropy. Applying the concept of joint inversion to DC resistivity and magnetotelluric (electromagnetic) data from a structure consisting of many fine layers, we show how the resistivities in the horizontal and vertical directions can be separately determined in each layer. The results are related to the contrasts existing in the interbeds which comprise the composite layers. The method has applications in both engineering and exploration.

On the Migration of Reflection Time Contour MAPS

Abstract: After the sampling of a reflection time contour map, i.e. after times and time gradients at the grid points of a square sampling grid have been determined, its conversion into true depth contours can be performed by normal incidence ray tracing. At each grid point the spatial orientation of the ray is uniquely defined by a corresponding time gradient vector, whereas its continuation into the subsurface is controlled by Snell's law. For arbitrarily orientated velocity interfaces the 3 – D ray tracing problem can systematically be solved with the aid of vector algebra, by expressing Snell's law as an equation of vector cross products. This allows to set up a computer algorithm for migration of contour maps. Reliable sampling of reflection time contour maps in the presence of faults is essential for the realization of a practical map migration system. A possible solution of the relevant sampling problem requires a special map editing and digitization procedure. Lateral migration shifts cause a translation and distortion of the original sampling grid. On the transformed grid the true positions of faults can be related to their apparent ones on the reflection time contour map. Errors in the time domain correlations or an incorrect velocity distribution or a combination of both these effects may cause migration failures due to total reflection and time deficiencies, or give rise to an anomalous distortion of grid cells, the latter signifying a violation of the maximum convexity condition. Emphasis is placed upon the significance of map migration as an interpretive tool for solving time to depth conversion problems in the presence of severely faulted or salt intruded overburdens.

The spectral function of a layered system and the determination of the waveforms at depth

Abstract: Consider the mathematical model of a horizontally layered system subject to an initial downgoing source pulse in the upper layer and to the condition that no upgoing waveforms enter the layered system from below the deepest interface. The downgoing waveform (as measured from its first arrival) in each layer is necessarily minimum-phase. The net downgoing energy in any layer, defined as the difference of the energy spectrum of the downgoing wave minus the energy spectrum of the upgoing wave, is itself in the form of an energy spectrum, that is, it is non-negative for all frequencies. The z-transform of the autocorrelation function corresponding to the net downgoing energy spectrum is called the net downgoing spectral function for the layer in question. The net downgoing spectral functions of any two layers A and B are related as follows: the product of the net downgoing spectral function of layer A times the overall transmission coefficient from A to B equals the product of the net downgoing spectral function of layer B times the overall transmission coefficient from B to A. The net downgoing spectral function for the upper layer is called simply the spectral function of the system. In the case of a marine seismogram, the autocorrelation function corresponding to the spectral function can be used to recursively generate prediction error operators of successively increasing lengths, and at the same time the reflection coefficients at successively increasing depths. This recursive method is mathematically equivalent to that used in solving the normal equations in the case of Toeplitz forms. The upgoing wave-form in any given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding prediction error operator with the surface seismogram. The downgoing waveform in this given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding hindsight error operator (i.e., the time reverse of the prediction error operator) with the surface seismogram.

On tracing seismic rays with specified end points in layers of constant velocity and plane interfaces

Abstract: 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.

A new approach to vibroseis deconvolution

Abstract: A method is proposed to obviate the shortcomings of conventional deconvolution approaches applied to vibroseis data. The vibroseis wavelet reduces the time domain resolution of the earth's impulse response by restricting its passband. The spectrum of the wavelet is assumed to be a “low quefrency” phenomenon, and hence it can be estimated by low cut cepstral filtering. The wavelet's amplitude spectrum can then be removed by spectral division. By using an approach which is consistent with the principle of maximum entropy, the undetermined portions of the seismogram's Fourier transform can be filled in by autoregressive prediction. The process of initially deconvolving in a restricted passband reduces the enhancement of noise contaminated parts of the spectrum, and the spectral extension scheme increases the time domain resolution of the process.

a Note on Magnetized SPHERES

Abstract: The interpretation of total field anomalies becomes somewhat complicated, especially when an arbitrarily magnetized spherical ore mass happens to be the causative body. Even though some attempts have been made to analyze total field anomaly maps, they are often too complicated and their underlying assumptions in respect of permanent and induced components of magnetism are far from realistic. In this note, an attempt has been made to show that vertical magnetic anomalies are capable of yielding interpretation with ease and precision as far as magnetized spheres are concerned. An empirical method has been outlined for computing the magnetization inclination in the plane of the profile using the measured distances between principal maximum, principal minimum, and zero anomaly positions on a magnetic anomaly profile.

Vibroseis signals with prescribed power spectrum

Abstract: For a swept frequency signal with constant envelope the power spectral density is approximately inversely proportional to the signal\\\'s rate of frequency change. Hence, by means of this relation, the phase function which describes the sweep\\\'s frequency variation may be derived from a predefined power spectrum. This is in contrast to the present use where the power spectrum is the result of a rather arbitrarily chosen phase function. A numerical algorithm based on the relation between power spectrum and phase function was found to lead to sweeps whose power spectra matched the prescribed ones rather closely.

Computerized analysis of vibroseis signal similarity

Abstract: Computerized evaluation of Vibroseis similarity test data is the logical consequence of the increasing quality requirements for signal reproducibility and for synchronization of vibrators. The differences in start time and phase as well as an indication of local phasing problems between any two Vibroseis signals can be obtained from an analysis of the difference of their respective phase spectra. This method appears to be accurate and stable with respect to harmonics which usually plague the signals from transducers monitoring the motion of the vibrators'base plates.

Resistivity sounding on a multi‐layered earth with transitional layers. part i: theory*

Abstract: The electrical potential generated by a point source of current on the ground surface is studied for a multi-layered earth formed by layers alternatively characterized by a constant conductivity value and by conductivity varying linearly with depth. The problem is accounted for by solving a Laplace's differential equation for the uniform layers and a Poisson's differential equation for the transitional layers. Then, by a simple algorithm and by the introduction of a suitable kernel function, the general expression of the apparent resistivity for a Schlumberger array placed on the surface is obtained. Moreover some details are given for the solution of particular cases as 1) the presence of a infinitely resistive basement, 2) the absence of any one or more uniform layers, and 3) the absence of any one or more transitional layers. The new theory proves to be rather general, as it includes that for uniform layers with sharp boundaries as a particular case. Some mathematical properties of the kernel function are studied in view of the application of a direct system of quantitative interpretation. Two steps are considered for the solution of the direct problem: (i) The determination of the kernel function from the field measurements of the apparent resistivity. Owing to the identical mathematical formalism of the old with this new resistivity theory, the procedures there developed for the execution of the first step are here as well applicable without any change. Thus, some graphical and numerical procedures, already published, are recalled. (ii) The determination of the layer distribution from the kernel function. A recurrent procedure is proposed and studied in detail. This recurrent procedure follows the principle of the reduction to a lower boundary plane, as originally suggested by Koefoed for the old geoelectrical theory. Here the method differs mainly for the presence of reduction coefficients, which must be calculated each time when passing to a reduced earth section.

Estimation of depth to conductors by the use of electromagnetic transients

Abstract: The transient response of a layered structure to plane wave excitation can be considered to be composed of a series of waves and a ground wave. For the case of a half-space of conductivity σ and permeability μ the maximum in the electric field is found at a depth z and time t when t=z2σμ/2. This formula can be used to estimate the depth to a buried horizontal conductor with an accuracy that depends upon the resistive contrast at the conductor's surface. The above ray type of solution can be converted to a solution composed of a number of modes by the use of a Poisson transform and the transformed solutions yield decay constants that are consistent with the previously reported results. In the case of a finite source, the maximum in the electric field is strongly directed. The direction depends upon the geometry of the source and the air-earth interface. Although the maximum varies with direction it can be shown that in some directions similar laws to that above are valid. The depth to a conductor can be estimated from the early part of the transients when the ground wave is removed. The removal of the ground wave from the transient is facilitated by the use of an apparent conductivity formula. Although these results were obtained under restrictive conditions they do provide some insight into the electrical transients that are encountered by studying more complex models.

The performance of optimum stacking filters in suppressing uncorrelated noise

Abstract: 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.

Analysis of the gravity effect of two‐dimensional trapezoidal prisms using fourier transform*

Abstract: The gravity effect of an infinite horizontal trapezoidal prism is derived and its Fourier spectrum is analyzed so as to yield information about four parameters of the causative structure, namely the depths to the upper and lower surfaces, width of the upper surface, and the inclination of the sides. In order to test the applicability of the method, synthetic data are constructed by digitizing the theoretical gravity effect. Subsequently, the corresponding Discrete Fourier Transform (DFT) is obtained. The parameters evaluated from the DFT are observed to be sufficiently close to the chosen values.

Transformation of dipole to schlumberger sounding curves by means of digital linear filters

Abstract: Linear relationship between dipole and Schlumberger sounding resistivities leads to the use of digital filters to transform the former to the latter. This transformation is of importance from the viewpoint that Schlumberger interpretational techniques and know-how could then be applied to the pseudo-Schlumberger field curve. Filters for this transformation are presented for the radial, perpendicular, and parallel (30°) dipole method. The characteristics of these filters are similar to the ones for transforming dipole data to the kernel and are favourable in that they do not amplify noise. A sampling interval of (In 10) /6 has been used in determining the filter yielding good accuracy. Like previous filters the present one is handy and fast in operation.

Adaptive predictive deconvolution of seismic data

Abstract: The Wiener prediction filter has been an effective tool for accomplishing dereverberation when the input data are stationary. For non-stationary data, however, the performance of the Wiener filter is often unsatisfactory. This is not surprising since it is derived under the stationarity assumption. Dereverberation of nonstationary seismic data is here accomplished with a difference equation model having time-varying coefficients. These time-varying coefficients are in turn expanded in terms of orthogonal functions. The kernels of these orthogonal functions are then determined according to the adaptive algorithm of Nagumo and Noda. It is demonstrated that the present adaptive predictive deconvolution method, which combines the time-varying difference equation model with the adaptive method of Nagumo and Noda, is a powerful tool for removing both the long- and short-period reverberations. Several examples using both synthetic and field data illustrate the application of adaptive predictive deconvolution. The results of applying the Wiener prediction filter and the adaptive predictive deconvolution on nonstationary data indicate that the adaptive method is much more effective in removing multiples. Furthermore, the criteria for selecting various input parameters are discussed. It has been found that the output trace from the adaptive predictive deconvolution is rather sensitive to some input parameters, andmore » that the prediction distance is by far the most influential parameter.« less

An Ultrasonic Profiling Investigation on Some Fresh and Weathered Granites of Hyderabad, INDIA*

Abstract: The ultrasonic profiling method of measuring the compressional and shear wave velocities in cylindrical rock samples is extended to measurements in some weathered and fresh granite blocks collected from the Hyderabad (India) region. This possibility of the method provides a means of investigating the elastic properties of the less compact rocks, of which the near-surface formations are particularly important. In this article the important parts of the ultrasonic profiling instrument developed are described and the relevant aspects of the seismic wave fields and identification of the individual waves in the wavetrain responses to longitudinal excitation are considered. Compressional, shear and surface (Rayleigh) wave velocities in some fresh and weathered granites are detailed. The compressional velocities range from 4.8 km/s to 5.5 km/s in fresh granites and lie between 1.1 km/s and 2.5 km/s in weathered granites. Young's modulus and Poisson's ratios computed from the measured velocities are also presented. An empirical relation of the form log E= 4.27 + 2.11 log Vp between Young's modulus E and compressional velocities Vp in the fresh granites studied is deduced. The versatility of the approach is thus demonstrated.

Seismic velocity estimation

Abstract: 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.

The Concept of Apparent Resistivity in Laterolog 7

Abstract: It appears that Doll (1951) and N. N. (1958, 1969, 1972) on the one hand, and Roy (1975) and Roy and Apparao (1976) on the other, used different formulas for computing the apparent resistivity for the Laterolog 7 sonde. This would partly explain the contra-dictory nature of LL7 results from these two groups of workers. The first group use a formula that relates the measured potential to a system of currents that are largely fictitious and non-existent in the ground at the time of measurement. The second group, on the other hand, employ a formula that combines the observed signal with the currents that actually exist in the ground and produce that signal. We believe that the second procedure is the right one.

Electromagnetic depth sounding experiment

Abstract: 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.

Magnetometric and Electrometric Methods for the Investigation of the Dynamics of Landslide PROCESSES

Abstract: Geophysical methods for the investigation of the dynamics of landslide processes are most effective in long-term observations. Magnetometric and electrometric methods for observing the displacement of man-made and natural markers are most sensitive and most highly developed at the present time. A detailed magnetic survey carried out near boreholes where magnets have been emplaced makes it possible to establish the direction and velocities of the displacement of landslide masses at different depths. Observations of the dislocation of the stable anomalies of the natural electric field allow us to elucidate the character of landslide deformations in the zone of capillary uplift of ground waters.

Weighted vertical stacking in crustal seismic reflection studies on the canadian shield

Abstract: Seismic reflection methods are being developed at the University of Manitoba to aid in determining fine crustal structures in the Precambrian of Manitoba and northwestern Ontario. Present-day environmental concern as well as mineshaft conditions necessitate the detonation of several smaller charges repeated, say, I times and followed by ''vertical'' stacking. To obtain the familiar ..sqrt..I improvement in signal-to-noise (S : N) amplitude ratio applying the straight-sum (SS) method, one assumes, among other things, that both S : N ratio and signal variance are the same on all traces. Dropping these assumptions, as we must for our data, it becomes necessary to apply weighting coefficients to optimize the S : N ratio of the stacked trace. We still assume the signal shapes to be the same for repeated shots, so for the jth trace on the record of the ith shot we model the time series as: t/sub ij/ = a/sub i/(s/sub j/ + n/sub ij/); where a/sub i/ is a scaling factor. The proper weights w/sub i/ are then shown to be proportional to sigma/sub si//sigma/sup 2//sub ni/ where sigma/sup 2/ is variance, or to ..gamma../sub i//a/sub i/ where ..gamma../sub i/ is S : N power ratio. Applying the weighted-stackmore » (WS) method gives S : N amplitude ratios which are, on average, 55% of the optimal ratios expected from WS theory compared with only 24% for the SS method. The 45% shortfall in WS performance is ascribed mainly to trace-alignment (or time-delay) errors. Varying noise levels on individual traces, slight dissimilarity of signal shape, and correlated noise may also contribute to a lesser extent (in decreasing order of significance). This WS method appears to strike a good practical balance between S : N improvement and processing efficiency.« less

Characteristics of audio and sub-audio telluric signals*

Abstract: Telluric current measurements in the audio and sub-audio frequency range, made in various parts of Canada and South America over the past four years, indicate that the signal amplitude is relatively uniform over 6 to 8 midday hours (LMT) except in Chile and that the signal anisotropy is reasonably constant in azimuth.

Induced polarization response of a horizontally multilayered earth with no resistivity contrast

Abstract: The induced polarization (IP) response of a horizontally multilayered earth with no resistivity contrast can rapidly be calculated on a desk calculator or minicomputer for any electrode array. The formulation is a simple series summation of the products of weighting coefficients and the true induced polarization responses for each of the layers. The coefficients are directly derivable from the corresponding resistivity model. This series approach to IP formation was originally described by Seigel but has not been treated extensively in the present-day geophysical literature. This method can be applied to either time or frequency domain induced polarization measurements. Once the coefficients are known, apparent induced polarization response can readily be obtained by judicious substitution of known, suspected, or assumed values of the true induced polarization of each layer. Basic formulation is presented for the IP potential coefficients (pole--pole or two array) with no resistivity contrast between the layers. From these coefficients, response of any number of layers for any electrode array can be obtained by suitable differentiation. Some examples of Wenner array for a three-layered earth and dipole-dipole array for a four-layered earth are used to illustrate the application. The results of this technique are valid for many natural situations more » of modest resistivity contrast. However, they definitely cannot be used if there are highly contrasting resistivity layers present. Such an approach is conceptually simple and is useful for survey planning, checking or setting the depth-of-penetration of a given array. For field induced polarization data that fits reasonably well to the no-resistivity-contrast model, this simple approach facilitates quantitative interpretation. « less

A quantitative evaluation of ott and meder's prediction error filter*

Abstract: 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.