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 Volume 16, Issue 2, 1968
Geophysical Prospecting  Volume 16, Issue 2, 1968
Volume 16, Issue 2, 1968


ESSAI DE FILTRAGE NON‐LINEAIRE APPLIQUE AUX PROFILS AEROMAGNETIQUES*
More LessABSTRACTDigital aeromagnetic profiles recorded with a high accuracy can be filtered with an electronic computer. Linear filtering is not very efficient, because the frequency spectrum of a given anomaly is very wide. But other methods are possible which, through a step by step analysis of the profile, leave some categories of anomalies unaltered while they remove the others completely.
Our method uses as a criterium the width of the anomalies; it distinguishes and processes differently “bell‐shaped” and “multi‐legged” anomalies. Some examples of the use of the method are shown.



TRANSFORMATION OF MAGNETOMETRIC DATA INTO TECTONIC MAPS BY DIGITAL TEMPLATE ANALYSIS*
More LessABSTRACTResults of gravimetric surveys can be interpreted by comparing the Bouguer field values with master curves based on simplified geological models. It has been shown in a previous paper how this procedure can be transformed into routine processes which can be computerized. The application of this method has yielded useful results in detailed gravity surveys.
The present paper discusses the application of the same interpretation principles to magnetic data. After some modifications, the method elaborated for the gravimetric data can be used for the interpretation of magnetometric survey results. Magnetometric‐tectonic maps are obtained which show the structural picture by common geological symbols. In the case of faults, the dimensions of depth of burial and throw are indicated on the maps.
The method is illustrated by an example where these procedures have been applied to gravimetric and magnetometric data of the same area. Two different maps are obtained: One shows the tectonics according to density contrasts and the other map depicts the tectonic situation on the basis of magnetic susceptibility contrasts.



REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*
More LessAbstractSharp cut‐off frequency filtering is carried out in the discrete time domain on digital computers. A convolution of the digital filter impulse response with the sampled input yields the output. For practical reasons, the length of the filter inpulse response, corresponding to the number of filter coefficients, is limited, and consequently the resulting frequency characteristic will no longer be identical to that originally specified. This is analogous to synthesising some specified frequency characteristic with a finite number of resistive, capacitative and inductive components.
In Part I of this paper, we examine the effect of approximating the sharp cut‐off frequency characteristic best in a mean square sense by an impulse response of finite length. The resulting frequency characteristic corresponds to the truncated impulse response of the specified frequency characteristic. It has a cut‐off slope proportional to, and a mean square error inversely proportional to, the length of the impulse response, and is a biassed odd function about the cut‐off frequency point. Because of the Gibbs phenomenon for discontinuous functions, the resulting frequency characteristic will always have a maximum overshoot with respect to the specified characteristic of ± 9%, regardless of the length of the corresponding impulse response. Equal length truncated impulse responses of specified filters with different cut‐off frequencies yield frequency characteristics which are almost identical about their respective cut‐off points. Now on a log frequency scale (as against a linear frequency scale implied previously) such characteristics may be made almost identical about the respective cut‐off points by having the truncated impulse responses composed of an equal number of zero crossings. Results for the low‐pass filter are applicable to the high‐pass and band‐pass characteristics.
In the latter case, the mean square error is double that for a single slope characteristic (low‐pass or high‐pass) and the slopes at both edges of the passband are approximately equal in magnitude to the length of the impulse response (linear frequency scale).
Part II of this paper is concerned with reducing the ± 9% overshoot that results from the discontinuous nature of the sharp cut‐off frequency characteristic and which is not dependent on the length of the truncated impulse response. The reduction is achieved, at the expense of the steepness of cut‐off for the resulting frequency characteristic, by the use of functions which weight the truncated impulse response of the specified frequency characteristic. These functions are called apodising functions. Among other variables, the length of the truncated weighted impulse response will determine the amount of maximum overshoot since the effective frequency characteristic being approximated is no longer a discontinuous function. The digital realization of the finite length impulse responses of Parts I and II is discussed in Part III, together with the optimum partially specified digital filter approximation to the desired frequency characteristic.



REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*
More LessAbstractIn Part I of this paper, we examined the properties of the best mean square approximation to the sharp cut‐off frequency characteristic by an impulse response of finite length. It was found that the sharpness of cut‐off for the resulting frequency characteristic depended on the length of the impulse response–but because of the discontinuous nature of the specified frequency characteristic, this best mean square approximation always had a maximum overshoot of ± 9%, independent of the length of the impulse response (Gibbs phenomenon).
In Part II, we investigate ways of reducing this ± 9% overshoot at the expense of a reduced sharpness of cut‐off. The discontinuous frequency characteristic is first approximated by a continuous characteristic with linear or cosine frequency tapering. The impulse response for such tapered characteristics consists of the impulse response of the discontinuous frequency characteristic weighted by a certain function corresponding to the type of tapering employed. The best mean square approximation to the tapered characteristic by an impulse response of finite length M will produce a frequency characteristic whose properties are now dependent on the time‐band width product Mζ, where 2ζ is the tapering range.
A trade‐off exists between the maximum overshoot and the sharpness of cut‐off for the resulting characteristic for both forms of frequency tapering. Instead of considering other forms of tapering in the frequency domain, we now investigate arbitrarily chosen weighting functions in the time domain to determine the minimum length of impulse response for a minimum value of maximum overshoot and a maximum value of sharpness of cut‐off.
Part III will discuss the digital realization of the above finite length impulse responses together with the optimum partially specified digital filter approximation to the desired frequency characteristic.



VERFAHREN ZUR AUSWERTUNG REFLEXIONSSEISMISCHER UNTERTAGEMESSUNGEN NACH DER SPIEGELPUNKTMETHODE*
More LessAbstractIn underground reflection seismic measurements use is made of geophone spreads and shot points in the interior of a mine. The possibilities to do so are restricted due to the limitations of working place and due to the measuring method. Nevertheless statements about a reflector's position in three dimensional space are oftenly required. It must be possible, therefore, to put a great strain on construction procedures.
With a view on the measuring accuracy it is above all necessary to distinguish between favourable and unfavourable arrangements of spreads and shot points. This question is discussed briefly in the introduction.
The description of all‐round constructions is the primary objective of this paper in its present version. It is shown, how plane reflecting elements located arbitrarily in space can be constructed according to the image point method by means of Descriptive Geometry. This, depending on the correlation, can be done in two different ways which are described separately in sections I and 2.
In the first section, the case is discussed of two geophone spreads connected like the branches of an angle. They may be shot from a shot point in any position (see the left part of figure I). This is illustrated in some more detail by means of an example taken from the potash mine of Salzdetfurth near Hildesheim, Western Germany. In the cases of known strike or known vertical inclination of the reflector, already one geophone spread leads to an end and the construction is much simplified.
The second section gives the case of two not necessarily connected spreads which need not lie within a common plane. By applying the technique of countershooting it is even then possible to gain a reflecting plane element (see the right part of figure 1). Of course, in this case the position of the two shot points is not arbitrary.



NOUVELLE METHODE DE RESTITUTION D'UN HORIZON REFRAGTEUR PAR SISMIQUE REFRACTION*
Authors FRANCESCO A. PANCIROLI and ET MAURICE L. ARNAUDABSTRACTThe purpose of this report is to show a method of determining the top of a refractor departing from the times and slopes of the direct and inverse dromocrones. The method does not need topographical correction and can be applied without knowledge of the distance between the geophone and the shot point.
These results having been obtained, the commonly accepted point of view is upset: instead of looking for two points on the surface corresponding to one point of the refractor, we try to etablish, starting with only one point from the surface, the two corresponding points from the top of the refractor.
This method can be applied to isolated points and does not demand interpretative hypotheses of any kind, excluding the velocity evaluation of the overburden and of the refractor.
The necessary calculations can be easily executed by means of a digital computer to which the dromocrone times and the distances between the geophones must be given. These calculations can also be executed by a person having no knowledge of refraction seismology.
This report also examines the validity of the approximations involved in the method proposed.



COMPUTATIONS OF NETWORK MODELS OF POROUS MEDIA*
Authors MANFRED RINK and JÜRGEN R. SCHOPPERABSTRACTIn a paper presented at last year's Amsterdam meeting (viz. Geophysical Prospecting 14 (1966), 3, 301–341), J. R. Schopper derived formulas relating formation factor, permeability and porosity, by means of a statistical‐network approach and by treating the electric and hydraulic resistance analogously.
The model of the porous medium consists of a network of branch resistors, their values being statistically distributed about a mean Ro with a relative standard deviation (variation coefficient) s. A properly defined total resistance R of the network can be expressed by the relationship:
(1′)Here α is a geometrical factor dependent only on the shape of the network (i.e. the number of meshes in the longitudinal and transversal direction), ε is a characteristical constant dependent only on the individual mesh shape (i.e. the number of nodes and branches within a mesh).
This network constant ε enters the equations relating formation factor, permeability and porosity, ε had been found to be in the range zero to one by calculating algebraically two special limiting network cases. However, for a better understanding of which value exactly this constant will have in actual porous media, networks with various mesh shapes have to be treated generally.
Because of the basically statistical approach, the networks have to be large so that a general algebraic treatment is precluded. Hence numerical methods using digital computers must be applied.
The determination of the total resistance R of any resistance network leads to the problem of solving a system of linear, inhomogeneous equations; i.e. Ohm's law written in matrix form:
(2′) (R) is the matrix of the coefficients, composed of the individual branch resistances.
 (I) is the column vector, its components being fictitious circular mesh currents.
 (U) is the inhomogeneity column, its components being source voltages within the individual meshes.
The matrix (R) has characteristic properties that depend on the mesh shape on the one hand and on the number and arrangement of the meshes on the other hand. With the regular arrangement of identical meshes investigated here, the matrix always has a banded structure and is symmetrical with respect to the main diagonal, positive definite, and non‐singular.
For the numerical determination of the wanted constant ε the coefficients matrix is provided with values having a known distribution. Here, in particular, a computer‐generated pseudorandom homogeneous distribution is used. The system, of equations is solved for R by a modified Cholesky method. Equation (1′) can then be solved fore. The main features of an ALGOL program written for this purpose and optimized with respect to storage space requirement and computer time are discussed.
Networks of triangular, square and hexagonal meshes have been investigated. The results are discussed.



BOOK REVIEWS
Book reviewed in this article:
Le Filtrage en Sismique, Tome I
J. AUBOUIN “Geosynclines” (Developments in Geotectonics 1). Elsevier Publishing Company
Earth and Planetary Science Letters Vol. 1. Nr. 2
H. Ramberg Gravity, Deformation and the Earth's Crust Academic Press
“Potassium Argon Dating” Compiled by O. A. Schaeffer and J. Zähringer

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