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 Volume 25, Issue 4, 1977
Geophysical Prospecting  Volume 25, Issue 4, 1977
Volume 25, Issue 4, 1977


VIBROSEIS SIGNALS WITH PRESCRIBED POWER SPECTRUM*
By E. RIETSCHAbstractFor 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.



ESTIMATION OF REFLECTOR PARAMETERS BY THE VIRTUAL IMAGE TECHNIQUE*
By I. KÉMÁRKYAbstractThe seismic interpretation process generally exploits three of the following independent basic assumptions:
 1 Input quantities are obtained by simplification of measured data (travel time curves).
 2 2. The geological model contains only a few parameters (for example, plane interfaces and constant interval velocities).
 3 3. Approximate transformations may be applied.
The first two are related to the simplification of the phenomena and enhance their essential features. The transformation which establishes relations between simplified data and model is required to be unique, stable, and sufficiently accurate.
Practically, the travel time curves are almost exclusively regarded as hyperbolas. We also accept this approximation.
The paper presents a simple recursive algorithm for the evaluation of the depth and dip of plane reflectors and the interval velocities.
It is a simple fact, that there exists a unique relationship between three hyperbolic parameters and a homogeneous dipping layer. Accordingly, two layers can be replaced by a single layer and the parameters of the lower boundary can be estimated when the upper one is known, initiating virtual shotpoints and geophone points (virtual surface). So, the case of multilayered media can be reduced in sequential steps to the case of a single homogeneous layer using a stripping type procedure.
Some synthetic model examples are provided to demonstrate the abilities of the algorithm.



USE OF REMOTE SENSING FROM SPACE PLATFORMS FOR REGIONAL GEOLOGIC EVALUATION AND FOR PLANNING GROUND EXPLORATION*
By R. CASSINISAbstractThe advantages and limitations of remote sensing exploration from space platforms are outlined with special reference to the geologic objectives.
The enhancement techniques of the multispectral imagery and their effects are illustrated in order to improve the interpretation of geologic linears and the discrimination of soils and rocks.
Three cases of application are shown, dealing with different geologic regions.
The ability of remote sensing from space to be employed in both regional and local problems is shown.
The planning of exploration by airborne and ground geophysical methods can be substantially helped after the results of remote sensing, and large economic gains can be reached.



SEISMIC VELOCITY ESTIMATION*
By BJØRN URSINAbstractThe 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.



A MAN/COMPUTER INTERPRETATION SYSTEM FOR RESISTIVITY SOUNDINGS OVER A HORIZONTALLY STRAFIFIED EARTH*
More LessAbstractThe 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.



A QUANTITATIVE EVALUATION OF OTT AND MEDER'S PREDICTION ERROR FILTER*
By J. M. MENDELAbstractOtt 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.



RESISTIVITY SOUNDING ON A MULTI‐LAYERED EARTH WITH TRANSITIONAL LAYERS. PART I: THEORY*
By D. PATELLAAbstractThe 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.



THE CONCEPT OF APPARENT RESISTIVITY IN LATEROLOG 7*
By A. ROYAbstractIt 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.



TIME MIGRATION—SOME RAY THEORETICAL ASPECTS*
By P. HUBRALAbstractUsing 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 NOTE ON MAGNETIZED SPHERES*
Authors B. S. R. RAO, T. K. S. PRAKASA RAO and A. S. KRISHNA MURTHYAbstractThe 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.



LIMITING DEPTH OF DETECTION IN LINE ELECTRODE SYSTEMS*
More LessAbstractAn infinitely resistive/conductive horizontal bed is assumed in an otherwise homogeneous and isotropic half space. Schlumberger, three electrode, and unipole profiles are computed at right angles to the strike of the bed. The Schwarz‐Christoffel method of conformal transformation and numerical methods of solving non‐linear differential equations are used to solve the boundary value problem. It is observed that (i) the three electrode system is the most sensitive gradient electrode configurations for electrical profiling, (ii) the apparent resistivities for Schlumberger, three electrode, and unipole methods become maximum when the depth of the bed is 0.06 L, 0.1 L, and 0.055 L for a resistive bed and minimum when depths are 0.085 L, 0.04 L‐0.02 L and indeterminate for conductive beds, respectively, (iii) the limiting depths of detection (defined in the text) by Schlumberger, three electrode, and unipole configurations are respectively 0.9 L, 6.6 L and 2.0 L for resistive beds and 0.58 L, 1.17 L and 1.5 L for conductive beds.
The electrode separation L is the distance between the two farthest active electrodes.



AN ULTRASONIC PROFILING INVESTIGATION ON SOME FRESH AND WEATHERED GRANITES OF HYDERABAD, INDIA*
Authors M. S. VIJAYA RAGHAVA, G. JAWAHAR and T. V. SHERBAKOVAAbstractThe 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.



TRANSFORMATION OF DIPOLE TO SCHLUMBERGER SOUNDING CURVES BY MEANS OF DIGITAL LINEAR FILTERS*
More LessAbstractLinear 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.

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