ALBERT

Notes: This second part deals with a subject analogous to the one developed in the first part, but related to surface irregularities with a non-cylindrical structure. In this part we have taken into consideration both the effects due to potential electrodes and the effects due to current electrodes.The study of these cases has been made with the help of mathematical theory, but it has been necessary to extend this theory from the case of cylindrical bodies to the case of non-cylindrical ones.Finally, as conclusion to both parts of the paper, some pages have been dedicated to the details of the calculation procedure, in order to demonstrate its practical htness for the study of this problem.

Notes: The usual computation methods in refraction seismic in the Sahara often involve considerable shot distances. Therefore, in addition to a high consumption of explosives, difficulties of operation and interpretation arise.In order to cut down the shot distances, it was necessary to study with great care the refracted waves having the shortest possible traveling path along the marker. Such paths correspond to the zone of great relative energy resulting, on one hand, from the sudden increase of reflected energy beyond the critical angle (total reflection) and, on the other hand, from the sudden appearance of refracted energy. The proposed method is founded on correlations of traces located in this “critical zone” on the different records shot on a profile; in order to get around certain difficulties due to insufficient knowledge of the marker velocity, the interpreter is led to select a constant correlation distance, at least for sections of the profile.The paper describes in detail the setting up and utilization of the proposed method, analyses a few examples and gives an estimate of the advantages and the restrictions of this method.

Notes: It was felt that Faust's first law was the best guide in classifying data concerning velocities and their correlation with lithology, even if, on a local basis, other laws fit the observed values better.Data have been re-examined which were provided chiefly by borehole measurements, both conventional and continuous, but also by some particularly reliable velocity profiles. “Faust's coefficient” has been computed for a series of geological formations, i.e. the ratio of true velocity to the value resulting from Faust's law, according to the age and present depth of the formation.Some graphs are shown, in the first place concerning the Sahara, and especially the Palaeozoic. It can be seen that the only formation showing values notably less than I disappears.On the other hand, values greater than I can be noticed for the quartzitic Ordovician of limited thickness.This comparison has been continued on a broader basis, using, in particular, data published by Faust and Wyrobek. The data concerning the very homogeneous Westphalian result in an average value slightly less than I.The Aquitanian basin has been studied in greater detail, owing to the closer spacing of the measurements.Generally speaking, the Middle and Upper Jurassic, the Cretaceous and the Eocene-Oligocene show values notably higher than the Palaeozoic in the Sahara. This seems to be connected with the higher proportion of limestones and dolomites. Values as high as 1.7 are observed in compact dolomites.Four maps are given, concerning the Oligocene, the Eocene-Paleocene (one of them excluding the Danian) and the Upper Cretaceous (exclusive of Danian), and a cross-section showing the link with lithology.Some discrepancies with regard to data from borehole samples are discussed.The question may be raised as to whether such studies can be carried out at the very beginning of exploration of a basin, on the basis of only a few logged wells. In the Aquitaine basin, data given by velocity profiles are in close agreement with those from wells as regards the Tertiary; but as regards the Mesozoic, information is meagre. In basins of the Sahara type, synthetic records have shown that interference of true and multiple reflections with each other is a major drawback. But on the other hand, the possibilities of a combined use of reflection and refraction in order to define lateral changes in velocities may not have been fully appreciated. Some computed examples are shown, since the results of a survey at present being carried out on this basis cannot yet be published.Finally, mention is made of the part played by the determination of anisotropy, and an appeal is made for more coöperation between geophysicists in this sphere.

Notes: The vertical component of the attraction exerted by a horizontal homogeneous polygon is obtained as the sum of terms proportional to the difference, for each of the vertices, between the external angle and its spherical perspective. A set of computation charts is presented, each for a fixed value of the angle of the polygon (22° 30′, 30°, 45°, 67° 30’ and 90°). These charts give the difference between the external angle and its spherical perspective as a function of the quotient of the distance from the sides to the vertical through the station over the depth. The chart for the right angle is an alignment chart, permitting a more accurate reading; the others are graphs in rectangular coördinates.For a remote station the result would be obtained as a small difference between relatively large quantities, and consequently with a not quite satisfactory precision. For these stations it is preferable to assume the entire mass to be concentrated in the centre of gravity.

Notes: In the laboratories of the agip mineraria a new apparatus has been developed which, utilizing reflected energy collected by several groups of seismometers, provides on a single trace an indication of the total energy level. This indication is controled in time according to an exponential characteristic, thus compensating for the attenuation which is effected in the subsurface. The apparatus enables indications of successive energy levels to be obtained for different shots, thus furnishing a time section even during the routine field work.

Notes: The factors affecting the response of a velocity-sensitive well geophone are discussed and it is shown that the geophone movement in response to a seismic pulse is primarily due to the pressure difference between its ends. Several practical examples are calculated to show that the first arrival waveform recorded by a geophone depends on the distance from an interface and that these changes in recorded waveform may give errors of up to 3 msec in well-velocity surveys. These errors may be minimised by using pressure geophones or by avoiding measurements at geological interfaces.

Notes: Surface charges in seismic refraction work in the sand dunes of the Sahara have been necessitated by the many problems involved in shot hole driving. These include lost circulation, unconsolidated surface sands and gravels, caving, poor shot efficiency etc.Tests were conducted in September 1958 in an attempt to evaluate some of the problems involved in surface shooting. This method is complicated by the large number of variables, 8 major ones, influencing the shot pattern.It is difficult to evaluate each of such a large number of variables by an exhaustive and systematic test procedure due to the prohibitive number of shots that this would involve. However, two series of tests contributed greatly to an understanding of the problem, although they left some points to be cleared up.Firstly shots were fired using small charges (less than 25 kg). These gave interesting results, however, they are not applicable to the heavy charges currently in use (500-1500 kg). These results refer primarily to the use of suspended charges which gave superior results to those obtained from surface charges layed on the ground. They also tend to indicate the influence of the weight of the charge. This is manifested by a saturation in seismic energy, as the size of the charge increases.The results obtained from the shooting (200 kg) of several individual charges are not as pronounced. The following conclusions are drawn:Nitrate explosives, in current use in the Sahara for several years, give good results in refraction.– The use of suspended shots does not result in sufficient gain in energy over the ground shots to justify their more complicated utilization: preparation, partial destruction of supports, safety etc. Hence further shooting of suspended charges has been abandoned.– The detonation of individual charges by use of primacord layed parallel to the line, in preference to simultaneous electrical detonation, gave superior results from the energy point of view (especially for high frequency arrivals) and simplified lay-out procedures. Hence this method was adopted.– As for the small charges the relation between seismic energy and weight of the individual charges shows a saturation effect which is less pronounced.–The distance between the individual charges appears to be the most important variable. The relation, seismic energy-distance between charges, indicates an increase in a linear manner for distances of 5 to 20 meters. It would appear as though the beneficial effect for greater distances diminishes beyond this point.Finally, it would appear as though it would be preferable for a given overall charge to reduce the weight of the individual charges and increase the distance between the individual charges. Obviously this rule has certain practical and economic limits. If the cost of the unit of seismic energy is evaluated it is preferable to utilize large distances between charges (20 m) but also relatively large individual charges (about 25 kg).The testing program was followed by nine months of surveying in different types of terrain. The parameters were etablished from the results of the testing program.Economically the following conclusions may be drawn: The average explosives consumption increased from 1000 kg for the shot–hole shots to 1500 kg for the surface shots. This increase, costwise, represents almost exactly the cost of shot hole drilling at the present price of explosives. Efforts are being made to have the price of explosives reduced and should this realize the net result will be a definite cost advantage in favor of surface shooting.In addition, the introduction of magnetic recording in refraction work to the Sahara in 1959 appears to indicate a substantial reduction in the size of the charges needed, hence, a further advantage to surface shooting.

Notes: The purpose of the automatic gain control in seismic amplifiers is to balance the slow decrease of the signal in such a manner that its energy does not exceed the capacity of the recording system. At the same time the interesting seismic content of the signal has to be preserved as undistortedly as possible. At first, the technical data of AGC circuits are calculated. The characteristic curves of circuits with one, two, and three time constants are computed. In circuits with two and more time constants an oscillating of the AGC is possible. In order to guarantee aperiodic operation, the ratio of the time constants must not fall below a certain value. As long as these technical conditions are complied with, the free choice of the largest time constant remains open and can be accommodated to the geophysical conditions.From the endeavour to balance the slow decrease of the signal as well as possible, an upper limit for the choice of the time constants can be derived. A lower limit is obtained from the condition that the seismic content of the signal has to be preserved. The determinant time constant should lie between the two limits and differ from them as much as possible.Finally, the problem of the amplifier “squeezed” by energy pulses is touched.

Notes: Book Review in This Article:P. F. Shokin, Gravimetry (Apparatus and Methods for Measuring Gravity)

Notes: The paper deals with quantitative seismic measurements near underwater explosions. The blastings were carried out in an artificial small pond, the records were taken in adits, situated vertically below the charge. The range of charges extended from one detonator to 200 g of Ammon-Gelit explosive.Conventional seismic scaling by comparing amplitude and size of charge does not yield a simple power law, the exponent being larger for smaller charges and vice versa. This result is due to the different frequency-spectra of the seismic pulses, as shown by harmonic analysis. In contrast to this, the seismic energy as calculated from the records is directly proportional to the size of charge over the whole range.During the Lago Lagorai experiments in 1962 some quantitative seismic measurements near the shot point were also carried out. The relevant results are compared with the preceding ones and some differences are discussed.Some conclusions can be drawn:〈list xml:id="l1" style="custom"〉1The normal methodes, investigating the amplitude only, gives in many cases no clear description; this is especially true for nearby measurements with widely varying frequencies. Therefore it is suggested to extend conventional seismic scaling by including information on the frequency-spectrum and the kinetic energy of the seismic signal.2In order to get clear experimental data for theoretical investigations, it is necessary to extend the recording over a half-space, i.e. in adits or boreholes.3Influences of shape and size of the lake, in which the shots are fired, should be taken into account, too.

Notes: The magnitude of the streaming potential due to the passage of a liquid through a porous medium is examined, under conditions similar to those found in an aquifer under the action of a pumping well. Three models simulate flow in channels, conduits, and unconfined aquifers for several grain sizes of sands and for several resistivities of fluids. The ratio of the streaming potential to the pressure difference is constant for a selected sand and fluid, decreases as grain size increases, is essentially independent of the geometry of the flow, increases slightly as the permeability decreases, and decreases as the salt concentration increases. Extremely low streaming potentials can be expected near a pumping well.

Notes: Book received in this Articles:P. E. Potter and F. J. Pettijohn, Paleoccurrents and Basin Analysis

Notes: A method is described which allows to measure the elastic wave motion in the “interior” of two-dimensional solid models. It is based on the measurements of the lateral deformations of a plate that are related to the state of compression in the plane of the plate. A capacitive detector measuring the motion has been developed.Wave front studies have been carried out in a half space, in two half spaces in contact and in a single-layered half space. The experimental results are compared with theory. Special attention has been paid to the head wave.

Notes: All sorts of organic matter preserved under anaerobic conditions and transformed in reducing environments may form components of natural crude oils; such oils are linked to sapropel deposits by their contents in porphyrins and trace metals.

Notes: Offshore reflection records, even if they are of medium quality, can be used to determine the thickness of unconsolidated sediments. The method is described and its limits are specified. An example of application referring to the Gulf of Lyons is given.

Notes: A transistorized combination of multivibrators is described, by which the gain is set according to a time programme. This equipment allows quantitative recording of seismic events with extremely large amplitude variations. Some possible applications of this recording system to seismic research are discussed.

Notes: The theoretical aspects of a general configuration of the tripotential prospecting method are described while particular cases of the method were described by Carpenter and Haberjam (1956) and Sarma (1961).

Notes: The accuracy of an airborne magnetic survey is limited by the following factors:〈list xml:id="l1" style="custom"〉1Slow (diurnal) variations of the earth's field.2Fast variations of the earth's field (earth field “noise”).3Drift in measuring equipment.4Magnetometer “noise” (electronics, orientors …).5External disturbances, miscompensation.6Irregularities in flight altitude.7Location errors.The relative importance of these different causes of errors is analyzed from two viewpoints:〈list xml:id="l2" style="custom"〉a)Direct interpretation of anomalies on actual records,b)Compilation of isogram maps and their quantitative interpretation (second derivative, reduction to the pole, …).In conclusion, these errors do not affect the direct interpretation of profiles of standard quality. For instance, if the average altitude of the plane above the magnetic bodies is 1000 m, a variation of its flight level of about 100 m does not introduce serious errors in the interpretation.On the other hand, compilation of contoured isogam maps is often difficult, and compensation of the previous errors is not always possible except by approximate, rule-of-thumb methods. Furthermore, when the interval between lines is greater than the depth of magnetic masses below the plane of flight, the contours are interpolated and do not represent the magnetic field. Sometimes the contouring can offer different solutions, and the map is therefore not objective; this could then result in errors larger than those from the other causes.To evaluate the quality of the map and its interpretation we must know the location of flight lines and the compilation process employed. It is also wise to avoid quantitative interpretation of small anomalies and limit oneself to a qualitative interpretation.

Notes: Books reviewed in this article: Geomagnetismus und Aeronomie, Vol. II, Edited by Prof. Dr. G. Fanselau“Geomagnetic instruments and measuring methods” (Geomagnetische Instrumente und Meßmethoden) by Dr. H. Wiese, Dr. H. Schmidt, Dr. O. Lucre, Dr. Dr. F. Frölich Lehrbuch der Angewandten Geologie (Textbook for applied geology). Edited by Prof. Dr. A. Bentz Band I: Allgemeine Methoden, Kartierung, Petrographie, Paläontologie, Geophysik, Bodenkunde, written by well known authorities Continental Drift, edited by S. K. Runcorn

Notes: For a given refraction time-distance curve, the range in thickness for a hidden intermediate layer is given together with the effect the layer has on the total thickness of the two layers above the recorded refractor.

Notes: The method of Far Terrain Corrections is described which permits contouring of terrain effects for distant topography; near terrain effects must be computed separately.The method is only slightly less accurate than standard graphical techniques and saves computing time. It is most useful in medium size surveys involving 100 to 2,000 observation points where the application of electronic computers is uneconomic.

Notes: Books reviewed in this article:Emmons, Allison, Stauffer, Thiel, Geology, Principles and Process, 5th EditionB. N. Ivakin, The Microstructure and Macrostructure of Elastic Waves

Notes: Book Review in This Article:Robert L. Miller and James Steven KahnYe. P. Fedorov, Nutation and Forced Motion of the Earth's Pole. Translated from the Russian by Bertha Swirles Jeffreys, Foreword by Sir Harold Jeffreys.

Notes: Case history of the use of fan filtering for improving reflection continuity without destroying their character. The field technique is described: wide-band, large dynamical range recording with a programmed gain control and a very slow A.G.C., small distance between trace centers, high cutoff wave-number.Although the total amount of mixing due to fan filtering is rather high, a broad spectrum was obtained and inverse convolution filtering was used to improve trace resolution.

Notes: Graticules are known for the calculation of the magnetic field of homogeneously magnetized “two-dimensional” structures. For bodies whose lateral dimensions are comparable with the dimensions of the cross section and/or the distance between the point of observation and the structure, these graticules allow an approximative computation of the field in the plane of symmetry only. The degree of approximation can be improved by means of end corrections, but the amount of work involved increases rapidly. Moreover, even with end corrections the applicability of the graticules is still restricted to cases of rather simple geometry.In this paper, a set of graticules is proposed which already incorporates all end corrections. The set consists of three graticules, and it can be shown that at any point of observation lying outside the body each graticule needs be applied once only in order to calculate any component of the magnetic field produced by a homogeneously magnetized cylinder of arbitrary cross section, finite lateral extension and arbitrary spatial orientation.Curves and tables have been prepared, which allow the construction of the graticules under consideration. The application is illustrated by a hypothetical example. Possibilities for further improvements of the method are discussed.

Notes: By this method an approximation of gravity-anomaly of an inhomogeneous mass with any form is calculated. The mass is inbedded in a homogeneous medium. The density of the medium differs from the density-distribution of the mass.

Notes: Book Review in this Article:H. Israel and A. Krebs, Nuclear Radiation in GeophysicsB. A. Tkhostov, Initial Rock Pressures in Oil and Gas DepositsJohn S. Steinhart and Robert P. Meyer, Explosion Studies of Continental Structure

Notes: Pour caractériser les anomalies de la carte initiate et de la carte transformeé, on définit des indices de netteté désingés respectivement par I0 et I et on étudie le rapport I/I0 pour diverses formules appliquées à trois anomalies typiques (sphère, anticlinal, faille). Les courbes obtenues permettent d'introduire, par analogie avec les circuits électriques, des notions telles que profondeur de coupure et pente.

Notes: Condition nécessaire et suffisante pour qu'un sismogramme obtenu à la surface ?un milieu stratifé puisse être considéré comme résultant de l'ensemble des réflexions, directes et multiples, subies par une onde plane de compression se propageant normalement aux strates, supposés parfaitement élastiques. Modification minimale à faire subir à un sismogramme donné pour qu'il satisfasse à cette condition.Possibilité, quand cette condition est satisfaite, de déduire du sismogramme la loi de variation de la vitesse des ondes élastiques dans le milieu considéré.Exposé détaillé du processus pratique sur un exemple et quelques autres résultats de son application.

Notes: Based on simple scaling laws, a method is developed for the routine interpretation of magnetic anomalies arising from uniformly magnetised dykes. The method can be applied to any observed component of the field, i.e. the vertical, horizontal or the total field anomaly and it is applicable for any resultant direction of magnetisation of the dyke itself and for any strike direction. The necessary data for the interpretation are given in the form of curves covering a range from o up to 15 for the ratio of the thickness of the dyke to the depth to its top. It is also shown that, by integrating in a suitable manner the observed anomaly due to inclined bodies having a horizontal upper surface and enclosed by two pairs of parallel and inclined planes, a curve is obtained which corresponds to a dyke anomaly and which can be interpreted by the same technique.

Notes: In a recent paper, H. A. Winkler developed a suitable method for determining longdistance terrain corrections in gravity by interpolation. Such a technique implies that it is necessary to take into account the changes of the correction g not only with the location (X, Y) but also with the elevation Z of a station. Study of the function g(Z) shows that its curve, with amply sufficient accuracy, may be represented as a parabola. Moreover, the parabolas corresponding to different stations are all equal. From these properties a method has been derived which saves computing time and yields long-distance terrain corrections with convenient accuracy.

Notes: Books reviewed in this article:V. Fritsch, Elektrische Messungen an räumlich ausgedehnten Leitem (besonders in der Angewandten Geophysik), (Electrical measurements on 3-dimensional extended conductors especially in applied geophysics)D. S. Parasnis, Principles of Applied GeophysicsN. A. Plotnikov, Ressources en eaux souterraines: Classification et Methodes ?Evaluation

Notes: In the case where a medium of velocity α1 is underlain by a medium of higher velocity α2, no propagation of compressional elastic wave energy into the lower medium takes place at angles of incidence exceeding a certain critical value. This so-called critical angle –which is a measure of the actual velocity-contrast –is furthermore marked by a sharp increase in the amount of reflected compressional energy. An investigation has been made to find out whether this critical-angle effect might be usable as a velocity-contrast indicator in seismic exploration.Model experiments confirmed the expectation that this effect should be manifest at the surface under ideal conditions.Two small-scale field set-ups showed that the effect is actually measurable in the field (in one case after applying corrections for planting variations).Finally a seismic line along the shore of the North Sea provided the data for contrast analysis over a considerable distance along the base of the Tertiary. Comparison of the measured changes in contrast and subcrop velocities obtained at those locations where subsurface data from independent sources were available, showed good agreement.It is concluded that critical-angle shooting may be used to indicate subcrop changes along a reflector, or even to obtain the same sub-stratum velocity as is sought in conventional refraction work, at considerably shorter operating distances.

Notes: Nach einer kurzen Einführung über das Prinzip der Vibratortechnik werden Ergebnisse vibratorseismischer Arbeiten auf Platten (Eis, Betonplatten, Straßendecken) mitgeteilt. Der Auswertung wurden Biegewellen zugrunde gelegt und die ermittelten experimentellen Ergebnisse mit theoretischen Dispersionskurven verglichen.

Notes: Using optimum filter theory as a starting point, we describe a method for the design of practical multi-trace seismic data processing systems. We assume the inputs to be the superposition of signal, coherent noise, and incoherent noise. The signal and coherent noise moveouts are described statistically by their probability densities. Our approach is to split the system into two stages. The first stage achieves optimum noise suppression but distorts the signal. The signal distortion is reduced in the second stage by an optimum finite memory inverse filter.The system that is obtained using our method of design depends upon the form of the probability density functions. We show two examples, ghost suppression and velocity filtering. In ghost suppression we choose a model with moveouts known exactly, which corresponds to delta functions for the probability densities. In velocity filtering the signal and coherent noise moveouts are equally probable within non-overlapping ranges.The resulting system in each case is both simple and effective. In ghost suppression a simple shift and subtract cancels the coherent noise. The signal distortion is reduced by an inverse filter. The velocity filter system consists of differentiated moving averages applied to each trace, followed by a 90° phase shift and a low pass filter.

Notes: A graphical method of interpreting refraction data is presented based upon wave front construction in which different overburden velocity distributions can be used. The wave fronts are constructed with the aid of an Isochron Chart. By using two or more Isochron Charts, the effect of velocity variations above the refractor of interest can be allowed for.

Notes: This study is an attempt to find a phenomenological model which will behave in a manner similar to that of common earth materials in regard to the dependence of attenuation and phase velocity on frequency.An analysis of two commonly employed rheological models, Maxwell's and Kelvin-Voigt's and their variants, is presented indicating that these models are too simple to conform phenomenologically to the behavior of rocks. It is shown that a linear combination of the Maxwell and Kelvin-Voigt models, known as the standard linear solid, comes closer to giving an adequate description of the behavior of some earth materials, than either one of the two models.A method of determining the viscoelastic constants of a solid is presented provided that its attenuation and dispersion curves are known. The viscoelastic constants (relaxation times and/or viscosity coefficients) of the solid are determined by superimposing the experimental curves on master curves, and reading the constants off the theoretical curves.An analysis of published experimental data is presented, indicating that the present model is capable of accounting for at least some attenuation and dispersion phenomena.

Notes: For a group of n geophones at a single station, a signal-to-noise improvement of √n can only be expected when the background noise, between the individual detectors in the group, is completely uncorrected. This paper gives results, from four different sites, for the variation of correlation coefficient with distance between the detectors. It is found that at these locations the geophone spacings required to reduce the correlation coefficient to zero varied from 30 to 300 feet.These results are in agreement with the predictions of the theoretical model of a two-dimensional isotropic noise field described by Hills and Faran (1952). This model provides estimates of the spacing required between geophones to reduce the correlation coefficient to zero, provided that the velocity of the noise is known and that its spectrum is of a simple form.Often it is impractical to reduce the correlation coefficient to zero because of the large spacing required between the geophones in the group. For this case a simple method is described in which the signal-to-noise improvement can be calculated given that the correlation coefficients between the geophones in the group are known. This procedure can easily be used in the field because the correlation coefficients can rapidly be computed from paper records by using Tomoda's method (Tomoda, 1956) which gives values of sufficient accuracy to be used for on-the-spot calculation.

Notes: Modern marine gravimeters enable us to measure gravity at sea with a mean error of 2 to 5 mgals, depending on sea conditions and navigational accuracy.Good results could be obtained, even in rough seas, by means of Graf's ASKANIA Sea Gravimeter mounted on an ANSCHÜTZ gyrostabilized platform.When a DECCA Navigator System is used, the gravity measured at the points of intersection agrees within 5 mgals at wind-force Beaufort 4 and vertical accelerations of about 60.000 mgals.The southern North Sea has been surveyed with VFS “GAUSS”, the research vessel of the German Hydrographic Institute, Hamburg. Since 1959 about 4.000 miles of continuous gravity measurements combined with simultaneous magnetic measurements have been carried out in this area. Here nearly all remote-control gravimeter stations of the Netherlands Geodetic Commission were passed once or several times. Hence they could be used as basic points. However, there are a few cases in which the Dutch data could not be confirmed.The results are given as a free air isogam map covering the area between the meridian of Greenwich and 8° east, and between 54° and 56° latitude, with decreasing density of the measurements from east to west.

Notes: The theoretical radiometric anomalies caused either by a pinpoint source or by a larger one, which could be detected 75 metres above the source by a mobile detector with a very low velocity and time constant, were treated by an analog simulator which gave the curves transformed by the effect of velocity and time constant. The characteristics of these curves are discussed.An actual airborne survey above a radioactive anomaly of small surface yielded results which are analyzed and compared with the previous theoretical data. The degree of agreement between theoretical and experimental results seems satisfactory.This study shows that the deformation of curves by velocity and time constant in most actual surveys is small enough when the product of velocity and time constant is less than 75 m for a flying altitude of 75 m.

Notes: The paper deals with the calculation of the potential distribution over various polarized bodies; the potential profiles are plotted on a double-logarithmic net. For a quantitative evaluation the field graphs, plotted likewise in a logarithmic scale, are compared with model graphs. The method is explained by means of several examples, in which potential graphs of a single, as well as of several disturbing bodies lying close to each other, are interpreted. The practical examples have been derived from selfpotential measurements which were carried out above graphite deposits in the southern Bavarian woods.This method is generally valid for the interpretation of potential graphs of arbitrary dipoles. Therefore it can also be applied – in a slightly modified form – for the interpretation of magnetic measurements. An appropriate method of interpretation for this purpose is being prepared.

Notes: Certain travel time readings in shallow engineering seismology can be attributed to a seismic wave which has propagated through the air, but in which the energy has been transmitted into the ground before reaching the detector. Waveform evidence for this conclusion is presented, based upon the effects of altering the source conditions to delay the direct wave and of embedding the geophone in sound-absorbent material. Erroneously large depths may be computed if these air wave readings are interpreted as readings produced by direct waves through the ground. Potential difficulty exists only if velocity in the near-surface material is less than 330 meters/second (1100 feet/second).

Notes: The geometric description of wave fronts and ray paths has previously always started from the velocity distribution which has been chosen as simple as possible in order to be able to solve the integrals occurring in the basic equations. As the velocity is derived from the measured values of path and time, it is preferable to start from the path-time function. There are exponential, parabolic and hyperbolic path-time functions; the first and the second case correspond to the known linear and parabolic velocity functions. The hyperbolic case, on the other hand, has not yet been covered in literature. In the first two cases, two parameters–a and vo–can be chosen arbitrarily. In the hyperbolic case, we have 3 parameters (a, b and v)o at our disposal; this makes for a better approximation to actual conditions. A special advantage is seen also in the fact that the velocity does not become infinite with increasing depth but approaches a finite limit.

Notes: It has been discovered that certain flowing oilwells in the Eakring area (Nottinghamshire, England), which have not otherwise been disturbed, show distinct annual variations in the flow of total fluid. There appears to be a definite connection between these variations and the seasonal movements of the earth's crustal layers as evidenced by observations of the tilts of the earth's surface. The investigation has been extended further to show that a similar relation exists between the earth tilts and the production of a whole oilfield which has not been disturbed unduly by outside influences.A correlation has also been established between the earth tilt observations and the periodicity of shallow earthquakes.

Notes: A method has been developed for the measurement, in the laboratory, of the variation of the elastic properties of rocks in the frequency range 40–120 cycles/sec. In addition, the observations yield information concerning the internal dissipation of energy over this range. A number of rock specimens have been examined and the results are compared with those of other investigators.

Notes: In the mathematical treatment of the propagation of seismic waves the process in the medium surrounding the point of explosion is introduced as initial condition into the differential equation. It is the purpose of this paper to contribute to the knowledge of this process.The treatment of the process of explosion is a difficult one because the elastic limit is exceeded in the surroundings of the point of explosion. To gain a better view on the acting forces the space concerned is replaced by a simplifying model. It is shown that a permanent deformation is caused by the explosion. This deformation will be especially intensive if the rock does not offer any resistance to tensile forces.To determine the extent of such a permanent deformation a sudden release of pressure — the opposite of an explosion — was recorded by seismographs. It has been proved that any seismic waves of measurable magnitude are not produced. This leads to the conclusion that permanent deformation plays an essential part in the generation of seismic waves.

Notes: A description is given of the development, during the previous fifteen years, of gravity surveying methods in marshlands and marine areas. The methods described include the use of marshbuggies, weasels, helicopters, tripods, diving bells and remote control instruments. Special emphasis is placed on remote control instruments and a brief description of their construction is given. Finally, the corrections involved in marine gravity surveying are discussed, and the costs of the several methods are compared.

Notes: Uphole shooting during a reflection or refraction survey provides additional information about the velocities within the weathered layer if recorded with a complete spread of geophones on the surface. Wave-front diagrams can be constructed which give a genuine picture of the wave paths in the subsurface. Their construction is based on the assumption that the travel time from a shot in the hole to a geophone on the surface is the same as that from a fictitious shot on the surface to a fictitious geophone in the subsurface.

Notes: A filtered seismic trace often appears as an almost sinusoidal curve. The reflected energy arrivals are superimposed and interfere with the continuous oscillations of the trace, and are therefore often difficult to distinguish. This is the chief difficulty in picking reflections. The situation is similar to that met with in gravimetry when a strong regional anomaly conceals small local anomalies. However, a regional anomaly is regular and broad, and owing to these two characteristics it can be removed. In reflection seismology batches of energy may also be concealed by oscillations of a continuous character. The main difference lies in the fact that the background of continuous seismic vibrations is not static, as is the case in gravimetry, but variable with time. However, a fairly constant physical quantity corresponds to these vibrations, which are composed partly of noise, partly of undesired secondary reflections. This quantity is the energy of motion of the surface layer of the ground. It is this troublesome energy that we intend to remove, in order to keep only the useful, actually reflected, energy.These remarks clearly demonstrate the importance of an investigation of the energy contained in the surface layer of the ground. After showing that, in the simplest case, the density of this energy can be expressed almost exactly by the formula u ′2−uu′′in which u′ is the velocity of a ground particle, we show how the non-linear filtering defined by this formula can be realized, and we provide a few examples.

Notes: The method of induced polarization is being more and more used for sulphide ore exploration. The purpose of the present paper is to show that the decay-time of the polarization curve represents a characteristic property of polarized media and consequently of various ores. The decay-times for various minerals, determined by laboratory measurements, are given in table form. Although the intervals of the decay-times partially overlap, the decay-time does to a certain degree supplement the data of induced polarization. Further, a method of interpretation in cases in which there is a superposition of the polarization effects of bodies with different decay-times, is discussed. Laboratory results as well as examples of field and bore hole measurements, in cases in which standard geophysical methods did not yield satisfactory results, are given.

Notes: Laboratory investigation has shown that the resistivity of massive sulphide ores at specimen dimensions is of the order of 0–10 ohmcm. Several writers, on the basis of field results obtained with various electromagnetic methods, deny the possibility that natural orebodies could have correspondingly low resistivities. The question is of interest, since such conductors, because of their small imaginary response, would be undetectable by electromagnetic methods measuring only imaginary components or phase shifts.In the present paper anomalies obtained with the Loop-Frame electromagnetic method on massive sulphide bodies in different parts of Canada are discussed. It is shown, by means of model experiments, that the conductors all have very low resistivities which, where the effective thickness can be determined, appear to correspond to the order of magnitude of specimen resistivity. It is demonstrated that confusion of the total width of a conductor with its effective thickness may lead to erronous resistivity determinations.

Notes: The shape of the seismic signal generated by an explosive charge is assumed to be governed by shock-wave phenomena in a non-linear region around the shot hole.The dimensions of this shock-wave region are shown to depend on the weight of the explosive charge and the properties of the medium in which the shot is fired.The presented theory enables us to elucidate the power law relationship between the amplitude of the seismic pulse and the weight of the charge fired, in the literature often denoted by A = c.Qn, in which A is the pulse amplitude, Q the charge weight and c a constant.Experiments have been made to check the theory and the results are promising.

Notes: Let Oxyz be a system of rectangular axes with origin at the earth's surface and with the z axis pointing vertically downwards. If a body B lies wholly] between the planes z=h, z=I then for all x, y and for n= 1, 2, 3 it is proved that 〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR607:GPR_607_mu1"/〉 where a=h/d, β=l/d and K is the gravitational constant. Dn are very easily computed from the Bouger anomaly and Jn are tabulated in this paper.

Notes: Distinguishing between diffractions and true reflections is often difficult and may lead to misinterpretations. In the Molasse zone of Upper Austria, numerous faults were established by seismic surveying. Diffractions were observed at several antithetic faults but not at synthetic faults. As an example, a seismic record section of the Steindlberg structure is shown. The reflections from the base of Tertiary arid from the Cretaceous-Jurassic contact run parallel over long distances, and so do the less important reflections lying above and between. If, contrary to the general trend, the reflection from the base of the Tertiary approaches the underlying reflection from the Cretaceous-Jurassic contact, or if the latter diverges from the former, this is considered a criterion for a diffraction.

Notes: In some mines of the Siegerland, besides dilatational and shear waves, an additional direct wave has sometimes been found the velocity of which was somewhat lower than that of the shear wave. Investigations were carried out with the object to establish the directions of oscillations and the velocities of all three kinds of waves. It was the purpose to draw conclusions as to the character and the causes of the appearence of the third wave.By these investigations the wave having the lowest velocity can be identified as a second shear wave. This points to the fact that double refraction of the shear wave is taking place, the wave of higher velocity being a pure transverse wave and the wave of lower velocity being a quasi-transverse wave.As a result of these measurements the examined medium is shown to have a transverse isotropy due to stratification. It is, however, not possible to decide whether there exists also an anisotropy due to schistosity.Some further calculations show that the wave of lowest velocity can not be explained as a surface wave propagated on the walls of the gallery.

Notes: An example is given from the Gulf of Aden of a total intensity magnetic anomaly which cannot be explained by a body assumed to be magnetized in the direction of the Earth's present magnetic field. It is enferred that the natural remanent magnetization is much greater than the induced magnetization and computations have been made which suggest the presence of igneous rocks with a reverse magnetization. The example illustrates that it is important to know the ratio of remanent to induced magnetization and the direction of remanent magnetization for the correct interpretation of magnetic anomalies.

Notes: By means of demonstration material, the author reports on the successful application of a shooting technique for seismic reflection operations, which has proved to work satisfactorily in those parts of the North-West German prospective oil area which are unfavourable to reflections. The shooting technique is characterized by uniformly linear shotpoint arrangements which are parallel with the geophone spread and reach from one end of the spread to the opposite. All charges are detonated simultaneously.Technical details of the method are discussed, particularly the problem of the most favourable geophone spreads in continuous profiling as well as the position of the uphole geophone. The difference in quality of reflections obtained by the normal pattern shooting and the linear shotpoint arrangement is demonstrated by seismograms. The special problems involved in the interpretation of the seismic records are referred to.

Notes: This paper is intended as a basic introduction to synthetic seismogram techniques. Their principles, uses and limitations are discussed and the relative advantages of the different methods are compared.

Notes: The method of conformal transformation is applied to determine the distribution of telluric current density at the surface for several subsurface situations. To this end one starts of from a configuration of boundaries between media of different resistivities and of two equal and opposite current poles, for which the potential distribution can be easily determined. Then a suitable conformal transformation is applied, by which the two current poles are displaced to infinity, and the boundary between the media are made to correspond to typical layer configurations in the subsurface. Several examples of the method are discussed in detail, leading to typical subsurface structures as well as to typical topographic conditions.

Notes: The amplitudes of reflection coefficients as computed from well velocity logs are distributed according to approximately normal laws. The distribution in time of the reflection coefficients, the absolute value of which is greater than a certain threshold, is Poissonian. The amplitudes of real seismogram traces obey a normal law.The autocorrelation of a seismic trace is usually quite different from the autocorrelation of the corresponding impulse.

Notes: Comparison of synthetic seismograms with field records has shown, that, under favorable circumstances, a large part of what is present on the latter may be ascribed to direct or multiple reflections due to velocity contrasts as they appear on a nearby velocity log.Therefore it does not seem unreasonable to submit good field records to a series of transformations, inverse to those which lead from a velocity log to the synthetic seismogram, with the purpose of getting somewhat more detailed and more accurate information on the variation of velocities with depth.The main difficulty in this kind of problem is generally the lack of stability of the results, i.e. the great influence on the final outcome of even small and unavoidable inaccuracies in the data or in the assumptions.For this reason a theoretical case has first been examined, where both the data and the physical hypotheses, as well as the final result were perfectly well known,—as this allows estimation and if necessary improvement in the stability of the method employed.Thus the successive steps of the inverse procedure, leading from a filtered synthetic record with multiples back to the initial velocity log are briefly discussed and the results obtained are shown.Two main stages are distinguished:—suppression of the effect of filtering, called “deconvolution”—discrimination between direct and multiple reflection, called “analysis”.For the former stage, the largely unknown filtering effect of the earth has also to be taken into account. First the way of deducing the total filtering effect from the filtered record itself is examined; second (a certain shape of the “signal”, the impulsional response of this filter, being assumed) an inverse filter is calculated ensuring a compromise between a good recovery of the original impulse and a minimum amplification of noise.A complete example of the results of the inverse procedure is given, in the case of a filtered synthetic record without multiples. The only data, supposedly known, were the sampled ordinates of this record, drawn on the usual scale of a field record. Good correlation was nevertheless found between the output of the inverse filtering and the original impulsional record. Integration yielded a pseudo-velocity as a function of time, showing again good agreement with the true velocities, except for their absolute values. If, in addition, check-shot data are supposedly available, these pseudo-velocities can be tied in, and a second integration yields the values of depth in function of time.The analysis itself, discrimination of direct and multiple reflection, starts by a step by step reversal of the recurrent procedure used for introducing multiple reflections (p. 4 ref. 3). The rapidly increasing accumulation of errors, due to the noise on the record and to the approximate nature of the physical assumptions, is partially accounted for by a continous readjusting of the results obtained. Nevertheless, if this method allows, as shown by a last example, a very satisfactory analysis of a rather noisy synthetic record, it is still too unstable to be applied to field records. An alternative method of successive approximations is finally outlined.As a conclusion, the necessarily approximate nature of the solutions of these problems is stressed. If some of the methods presented are still too theoretical, others have already been applied with success to field records.

Notes: For the propagation of acoustic waves in formations, laboratory experiments yield an expression of the form:〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR370:GPR_370_mu1"/〉in which A is the amplitude (strain or stress) A 0 some constant coefficient x the coordinate according to which propagation takes placeα the attenuation coefficient C velocityω the angular frequency of sinusoidal oscillation.No analogy between this law and that of electromagnetic waves can be made, since:〈list xml:id="l1" style="custom"〉1) coefficient α is not independent of frequency: its expression is α=Kω (K constant);2) however, with present recording technique, no noticeable dispersion can be found.The present paper shows how results are compatible if losses of energy are accounted for by a hysteresis phenomenon that is analyzed (see Figure). Stress T is in abscissae, and strain δu/δx, multiplied by a coefficient E′ characteristic of the elastic properties of the solid, is in ordinates. The horizontal part of the curve is θ. It is supposed that the absorption properties of the ground are given by some dimensionless coefficientλ=θ/T−Then one getsα=ωλ/4CIf λ≪ 1, we get the propagation law〈displayedItem type="mathematics" xml:id="mu2" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR370:GPR_370_mu2"/〉An attenuation takes place without phase shift, and consequently without dispersion.The author reverts to the inapplicability of the superposition principle, and foresees theoretically, for instance, that a strain T′ cos ω′t, where T′ 〉 T and ω′ 〉 ω, can completely cancel coefficient α.

Notes: The synthetic seismogram is a geophysical instrument. Examples taken from several areas are presented in order to define this tool and to show how it may be utilised in all stages of the prospection.The synthetic seismogram translates the stratigraphic information obtained from exploration wells into seismic terms; it therefore deserves to be taken into consideration when choosing the methods to be employed for solving a given problem.The synthetic seismogram enables one to know the limitations imposed by the reflecting power of the different subsurface layers. The knowledge of these limitations, combined with an understanding of the difficulties inherent in the surface conditions, helps to determine the frame in which the survey and the interpretation must be set.During the actual survey the synthetic seismogram illuminates somehow the work of the seismologist. It is a reference base which defines the correlation criteria which permit him to analize the real traces and, at the same time, to increase their resolving power.The methods of utilisation are examined both with respect to the field work (attempts to determine the technique to be used in any area by reference to the synthetic traces; determination of the volume control; choice of filters) and in the field of interpretation (identification and picking of reflections; analysis of the signals in order to obtain stratigraphic information from them; study of the character of reflections on a regional scale and density of the synthetic seismograms to be made in any region; re-evaluation of the information contained in old seismic records).

Notes: Abnormal phenomenons such as refractions and diffractions are more readily seen on cross-sections produced in magnetic processing offices than on single records. The result is that the understanding of these phenomenons intervenes more and more in the interpretation of the data.Our purpose is to proceed here to a restatement of the geometric properties related to these phenomenons and to make an attempt to distinguish between what is useful for the interpretation and what is not.To begin with we shall call our attention on diffractions and refractions over faults permitting to recognize the faults or to precise them, then on diffractions caused by irregularities of the surface forming the common boundary of heterogeneous media (more or less deep erosion surface, basement rock, …). The following phenomenons are classified as parasites:〈list xml:id="l1" style="custom"〉1) refractions following very shallow horizons, mostly related to topographic accidents,2) diffractions on the upward path caused by shallow faults,3) diffractions in the horizontal plane, found either in marine seismic or in case of heterogeneities “wrapt up” inside of an homogeneous overthrust.

Notes: It has been observed in the course of a detailed investigation that Indian granites exhibit high ultrasonic velocities in contrast to the specimens obtained from other countries. An attempt has been made in this paper to offer an explanation of this peculiar behaviour. Firstly, the initial internal pressure and initial velocity at the time of formation of rocks are calculated by using the experimental results for the changes in velocity with pressure in Indian, American, Japanese and Russian rocks. Besides this, standard curves have been computed using the above named parameters. Also, their possible influence on the high velocity values of Indian rocks, even under atmospheric conditions is studied. The effect of time of the formation of rocks at specific depths is also calculated in detail. As one of the results of this investigation, it comes out that under high pressure (deeper depths) it requires longer time to complete the processes of formation of rocks than under low pressure (shallow depths) and under high pressure a lot of stress energy can be stored i.e., the initial velocity becomes high when the rock is formed at great depths while at shallow depths a less stress energy can be stored and consequently less velocity values. Thus the combined effect of time and physical conditions at the time of formation of rock on the velocity in the rock at a given time is investigated. It can be seen here that the elastic behaviour of rocks with pressure will provide a clue to the possible petrogenetic history and also the approximate age of the rock.

Notes: A method is developed for determining the propagation of a wave of arbitrary shape in an inhomogeneous medium in which both the density and the velocity change with depth according to an arbitrary continuous function. The solution is obtained by means of successive approximations. The successive terms in this approximation correspond to the direct wave, single reflections and multiple reflections of successive order. The general formulae are then applied to three particular types of transition layer.

Notes: It has been found that ore prospecting by suitable electromagnetic and electrical methods may under favourable conditions—such as occur in parts of Norway—be carried out to greater depths than have so far been realized. By 500 c.p.s. electromagnetic measurements at the surface there was located a deep ore body, occurring in the continuation of, but separate from, an old ore body, which had been worked down to its end at a depth of some 500 m. The new ore body, starting at a depth of some 700 m and continuing downward with a slightly more dipping axis, has been traced over a considerable distance by further electromagnetic measurements at the surface, with the energizing current led directly to the ore in drill holes. Possibilities, problems and further improvements in deep E.M. techniques of measurement and interpretation are discussed. By D.C. equipotential mapping at the surface, with the current led directly to the ore, the location and direction of the deep ore body was established in agreement with the electromagnetic findings. Problems and developments in deep Charged Potential prospecting and interpretation are discussed. By 500 c.p.s. electromagnetic and D.C. potential measurements in drill holes essential information has been achieved both in locating ore formations beside a drill hole and in deciding whether ore formations in different drill holes are connected or not.

Notes: Book Reviewed in this article:Louis de Vries, Technical and Engineering Dictionary.Mrs S. Duclaux, Seismétrie théorique.

Notes: A graphical method is presented for construction of trajectories and wave-front diagrams for any velocity distribution.The analytical solution is given for the case when the time-depth curve shows a flexure, that is, when the the velocity increases down to a certain depth and, after reaching a maximum, begins to decrease.A numerical example is considered for the computation of a trajectory.

Notes: In continuation of a publication by Pierau and Mueller (1960), theoretical and experimental comparisons are made about the energy content of seismic waves for central and linear shotpoint arrangements. The following results are obtained:Using shotpoint arrangements as chosen for the investigation, the energy content of deep reflections is greater in linear than in central pattern shooting, provided that the total quantity of charge is the same in both cases. Furthermore the seismograms obtained with linear shotpoint arrays are less disturbed by surface waves than those obtained in central pattern shooting. Both effects are superimposed and improve considerably the signal-to-noise ratio, thus explaining the improvement in the quality of deep reflections by uniformly linear shotpoint arrangements.〈section xml:id="abs1-2"〉〈title type="main"〉ZUSAMMENFASSUNGIn Fortführung einer Publikation von Pierau und Müller (1960) werden theoretische und experimentelle Vergleiche über den Energiegehalt seismischer Wellen bei Zentralund Reihenschüssen durchgeführt, die folgendes Ergebnís liefern:Bei den gewählten Anordnungen der Schussbohrungen und unter der Voraussetzung gleicher Gesamtladungen sind die tiefen Reflexionen beim Reihenschuss energiereicher als beim Zentralschuss. Ausserdem sind die Seismogramme von Reihenschüssen weniger durch Oberflächenwellen gestöit als diejenigen von Zentralschüssen. Beide Effekte zusammen verbessern wesentlich das Verhältnis von Nutz- zu Stör-Energie und erklären die Verbesserung der Qualität tiefer Reflexionen durch gleichmässig lineare Anordnung der Schussbohrungen.

Notes: The four problems considered are the definition and simulation of the filtering effects which occur in the earth between the explosive and the case of the geophone, the problem of errors in the time scale of the synthetic record, the simulation of multiple reflection effects, and the difficulty of finding a representative comparison trace on the field record. A partial solution to the first of these is offered by Siran, examples of whose operation are presented. A method of assessing allowable errors in the time scale is discussed. Multiple reflections may be introduced by the preparation of multiple-corrected reflection-coefficient logs on a digital computer such as the IBM 709. The particular problems of multiple reflections from the base of the weathered layer (including “ghost” reflections) are discussed. The preparation of a “composite” trace is shown to be a partial solution to the problem of the comparison with the field record.

Notes: Book reviews in this articleP. Lasfargues, Prospection Electrique par courants continus,Milton B. Dobrin, Introduction to Geophysical Prospecting,M. M. Slotnick, Lessons in seismic Computing,

Notes: Reflected refractions are considered as the limiting case for ordinary steeply dipping reflections. Particularly, reflected refractions may be constructed (migrated) according to the same principles as ordinary reflections. In view of the inherent inaccuracy of the velocity data, there is no discrimination between reflected refractions and other steep events by methods of construction. Velocity determinations from reflection data also show a continuous transition from steep reflections to reflected refractions. I.e., it is impossible to discriminate by seismic ray methods between reflected refractions and other steep events.If disturbing, steep events may be attenuated to a great extent by special shot and geophone set-ups. Also in this connection it is unnecessary to distinguish between reflected refractions proper and steep reflections.It seems to be more rational to consider steep events as ordinary steep reflections in the beginning of a survey, because otherwise real structural indications might possibly be thrown away.

Notes: Some tests carried out on a Worden “Educator” gravimeter (n. E 302) showed that pressure variations had a significant effect on the readings.The changes of pressure cause changes in readings of the same sign. Corrections are given which may be applied to the instrument readings to compensate for this effect.

Notes: Normally only operational checks are made on geophones in the field because of the lack of suitable test equipment. The test method described here was developed to enable accurate (better than 1 %) measurements to be made, in the field, of all geophones in common use which have natural frequencies of i c/s or greater.The portable field instrument is simple to operate and can be used to test refraction and reflection geophones, either singly or in series strings.

Notes: An analytical expression for the time-distance curve of seismic waves travelling in a medium consisting of intrinsically anisotropic layers with arbitrarily dipping plane interfaces can be given in terms of the “co-ordinates” of the interfaces (length h of the perpendicular from the shotpoint to the interface, strike ν and dip α of the interface) if for each layer the velocity is given as a function of the orientation of the wave normal. The interpretation of the time-distance curve is understood as the inverse process, namely finding an expression for the co-ordinates in terms of some characteristics of the time-distance curve, e.g. intercept times and apparent velocities. In addition, it is useful to know where the “limiting ray”, which is the ray connecting shotpoint and last geophone, enters and leaves a specific layer, for it is only on the medium between these two points that information can be obtained by interpretation. As ray and wave normal do not generally coincide in anisotropic media, the location of these points cannot be calculated from the co-ordinates and the direction of the wave normal without recourse to the functional dependence between the directions of ray and wave normal.An analytical solution of this problem would involve the solution of a number of equations, implicitly containing several parameters. Successive approximation would be rather cumbersome. Instead, a graphical method is proposed which yields all pertinent information without calculation.