ALBERT

Notes: According to a study of seismic velocities in the Alpine Foreland of Eastern Switzerland, the initial velocity is rather high in comparison with other Tertiary basins and shows an exceptionally high increase rate. When analysing the average and the interval velocities, especially of Tertiary strata, and when comparing them with velocities of strata of the same age and a very similar facies of the Northern Rheintalgraben, it has been found that the increase of velocity is closely related to the distance to the Subalpine Molasse. The conclusion is that the velocity of the Tertiary strata is strongly influenced by the folding pressure of the Subalpine Molasse or of the Alps. The same method has been applied to a relatively large number of wells in the area of the “German Molasse”. Not only the results in Eastern Switzerland have been confirmed, but also it has been proved that the diagenesis of the Tertiary strata and, hence, their velocities are influenced only partially by the specific depth of the basin. Velocities increase towards the Folded Molasse or the Alps. Consequently they depend on lateral folding pressure, which decreases from west to east with the increasing width of the basin. The tertiary strata of the basin have been affected by lateral folding pressurefrom south to north. However, structures with lateral compression have not been discovered yet in the German Alpine Foreland.

Notes: In a paper by Koefoed and Kegge (1968), which was based on previous work of Wesley (1958), the electrical current pattern has been derived that is induced by an oscillating magnetic dipole in a semi-infinite thin plate of infinitesimal resistivity.In the present paper, the range of validity of the assumptions, on which the work of Wesley is based, is subjected to a theoretical analysis. It is found that the decisive factor for the validity of Wesley's derivation is the quotient of the square of the penetration depth of the electrical current over the product of the thickness of the plate and a distance that is indicative of the size of the current loops in the plate. Wesley's derivation is shown to be valid only when this factor is negligible. It is also shown that in this condition the imaginary component of the anomaly must be negligible.Model experiments are described in which the electrical current pattern is studied also in the range in which the derivation of Wesley is not valid. The procedure used in these model experiments was to measure the tangential component of the magnetic field strength very close to the metal plate that simulated the conductive dyke.In order to express the results of the measurements in terms of the imaginary to real ratio, these results are compared with an interpretation graph for field measurements that was published by Hedström and Parasnis (1958). It is found that the current pattern in the plate is essentially the same as that which follows from Wesley's derivation, provided that the imaginary to real ratio is less than one third. The measurements do not permit to draw conclusions regarding the current pattern in the plate in conditions that correspond to larger values of the imaginary to real ratio.

Notes: Electrical resistivity anomalies of a symmetrical four-electrode co-linear system across a single vertical discontinuity are treated in relation to: a) the ratio of potential electrode separation to the current electrode separation that are employed in the system and b) the angle which the electrode alignment makes with the discontinuity. Several conclusions are extracted from this treatment and methods for obtaining an optimum sensitivity of the system, with respect to these parameters, are shown. Disadvantages of special arrangements, such as the Wenner configuration, are indicated. Methods are outlined to utilise variations in the apparent resistivity plot for determining the angle between the electrode alignment and the discontinuity, quantitatively or qualitatively. These variations include certain deviations from the standard curves obtained in longitudinal traverses made at right angles to the discontinuity. Also, a comparison is made between longitudinal and cross traverses, in relation to the discontinuity.

Notes: Studies made on head waves in a large number of two-dimensional horizontal and inclined layer seismic models are described. The assumption that the amplitude of head wave decreases exponentially with distance is found to be valid in horizontal layers. For the correct estimation of the head wave decay coefficient, the decrease of amplitude due to the divergence of wave energy should always be considered.The variations of head wave amplitude at large distances from the source appear to follow the theory of Heelan (1953). It is, however, concluded that the theory can only be verified from the measurements in models which have large velocity contrast and thick layer at the top.A prominent interference phenomenon between the direct and head wave has been observed near the critical distance. This is more marked for up-dip profile in the case of inclined layer models. The rate of head wave amplitude decrease with distance beyond the zone of interference is found to depend on dip angle, direction of the dip, and velocity contrast.

Notes: Velocity Spectra which were originally developed for the optimum stacking of seismic data have been found to give considerable information concerning lithologic and stratigraphic changes in the geologic section. In the Gulf of Mexico shale sections and sand bodies have been recognized on the Velocity Spectra display, and in the Caribbean last year a first attempt was made to utilize Velocity Spectra information for the determination and mapping of lithology. Since that time, a new program has been developed which takes dip into account when computing the interval velocity. This program has been applied to a seismic section in the North Sea which has resulted in a geologic model derived from interval velocities which were found to be quite consistent. Such a model can be of great value in geological interpretation.

Notes: The transient fields from a finite horizontal loop excited by a half sine wave current pulse have been computed numerically for a particular source receiver configuration at a height of 100 meters above a layered ground. The amplitude of the vertical component of the magnetic field has been chosen for the interpretation. Curves of apparent conductivity vs. time, plotted during the off-time of the signal, show that layering is easily resolved, that resonance effects are present and that polarization effects are detectable for certain types of polarization.

Notes: Seismic refraction measurements were made in two areas of the Baltic Sea in June 1967. The refraction data were obtained in the course of the transmission measurements program of Operation MILOC BALTIC 67.Three profile pairs were obtained, two in the area south of Öland Island, Sweden, the third to the east, north of the peninsula of Hel, Poland. The water depths vary from 60 to 90 meters between the areas.The receiving positions for the two profiles south of Oland Island are only 46 km apart but the structures differ markedly. The northern section, only 500 m thick, shows a rather thin sedimentary cover above 350 m of 3.7 km/sec material that in turn overlies 5.6 - 5.9 km/sec velocity material. The southern section, almost 2 km thick, has an equivalent amount of low velocity material, and a layer about 1 km thick having velocity 4.8 km/sec that overlies 6.0 km/sec velocity material. The eastern profile shows high velocity material, 5.6 km/sec, at 2.5 km depth.Correlation of the layers determined by seismic refraction with nearby geology suggests that the structural change south of Öland Island may represent the boundary of the Sarmatian Shield in this region.

Notes: In designing as well as in the construction of hydro-electric power plants and water-supply projects in karst, the knowledge of the location of groundwater flows is invaluable. Attempts made so far in this direction have not produced the desired results and all the methods known so far have shown to be more or less ineffective, especially with respect to deep-seated underground flows.An interesting suggestion on the possible determination of the groundwater flow location by using a specific geophysical prospecting procedure is based on the studies and knowledge of groundwater flows in the karstland of Yugoslavia. The procedure is as follows: a time-bomb thrown down a swallow-hole travels down to the groundwater flow and explodes at a predetermined moment. Elastic waves generated by the explosion propagate reaching the surface, where they are detected by properly arranged geophones that are connected with a standard seismic apparatus. By a specific graphical or arithmetical method the coordinates of the epicentre and hypocentre of the underground explosion can be worked out, and thus the location of the groundwater flow determined. By successive repetition of the experiment, i.e. by different time activation of underground explosion of the time-bomb, the location of the groundwater flow can be defined over longer paths.The success of the experiment depends upon a number of factors all of which must be met. If only one of them is overlooked, the experiment will not succeed. All this calls for great care both in preliminary work and in the course of the experiment. Substantial funds invested in and considerable time required in executing the experiment are multiply recompensed by the success of the experiment. The information obtained about the location of groundwater flows will directly affect the maximum operating pool adopted for hydro-electric power projects and the most favourable point in cutting off groundwater flows and bringing them to the surface for water-supply purposes.

Notes: This paper deals with electromagnetic fields in the so called “Transient Induced Polarization Technique”. Field equations are integrated in the case of a polarizable sphere inbedded in a sterile overburden. The existence of a remanent polarization vector P which slowly decays once the charging current has been cut off is taken as the initial condition. When the surface between the air and overburden is (as a first step) disregarded, the Laplace transform of the EM fields is given. The integral of the electric field (which is fairly often the result of prospection work) is independent of the various time constants related to electrochemical processes, but it stands as a good measurement of the total electrostatic dipole created by the charging current.We investigate the geometrical circumstances that can bring negative values of I.P. signal. Such negative values can be found in two cases: 1. The discharge currents are. distorted by the ground surface. 2. The I.P. signal is picked up by electrodes inside the polarizable material.The last part is dedicated to an account of experimental work performed on models and of an actual case history.The mathematical derivations are included in an appendix.

Notes: Laboratory measurements, and a few in situ observations, show that saturated marine sediments have interdependent mechanical and acoustical properties, Acoustically, of particular importance are the acoustic impedance, velocity of sound and the sound attenuation coefficient of the sediment. The first two properties can be measured relatively easily from a surface ship; the measurement of attenuation however, is more problematical. It is suggested that this can be achieved by a quantitative treatment of the acoustic data collected during routine sub-bottom profiling over a variable thickness of superficial sediments. In the assessment of four different sediment locations in the Irish Sea it was found that quantitative treatment of the acoustic signals yielded both a value of the attenuation coefficient as well as a measure of the frequency dependence of the attenuation. In addition a statistical analysis of the signal intensities seems to provide an indication of the relative roughness of the bottom and sub-bottom interfaces. From the wide range of information provided the mechanical properties of the sea-floor sediment may be estimated.

Notes: Book reviewed in this article:Courant-Hilbert, Methoden der mathematischen Physik I, Heidelberger Taschen-bucher Vol. 30

Notes: The tools used in digital processsing have received little application, to date, to refraction seismic.It seems however that some problems encountered in the analysis of refraction records could be solved through techniques which are of common use in processing of reflection seismic data (correlation, stacking, f and k filtering).The experiments made with some of these techniques revealed the possibility of a more complete refraction record analysis, allowing the study of later arrivals.The interpretation of these records could also, to a fairly large extent, be automated by means of digital processing.While recording, some care ought to be taken to allow identification and interpretation by digital processing of as many events as possible. However, the results shown here have been obtained from fairly old analog records, for which no particular care had been taken, which makes it possible to complete and revise the interpretation of previous studies using the same processing techniques.Long line refraction shooting is justified for marine surveys in some areas, particularly where reflection is disturbed by singing; improvements in analysis and interpretation would therefore prove very useful for a faster and more thorough interpretation.

Notes: Using plane wave theory and assuming a given input wavelet the shape of the reflected (or transmitted) wavelet from a layered boundary is derived. Several types of boundaries are considered, among them the weathered layer and a wedge shaped intermediate layer. Different angles of incidence and all internal multiples are taken into account. The examples shown in the figures can be used for a direct comparison between theoretical and observed shapes of reflected (or transmitted) wavelets from special boundaries.

Notes: During the years from 1965 until 1967, a German/Afghan team carried out a number of geoelectrical resistivity measurements according to the “Four-Point-Method” within the framework of Technical Assistance to Afghanistan.On two selected objects of investigation, i.e. Kabul (69°10 E, 34°30′N) and Zarkašan (67°40′E, 32° 54.5′N), the possibilities and limits of the resistivity method in the exploration of useful groundwater resources, of the geological structure, and in the treatment of problems pertaining to mineral deposits in Afghanistan are demonstrated. In connection with geological mapping these geophysical investigations provided certain conceptions on the structure and the stratigraphy of the intramontane basins in the region south of the Hindukush mountains. Field interpretations are checked by boreholes and by computed model graphs.

Notes: A high resolution Boomer system was used to investigate the bedrock configuration of part of Port Jackson, New South Wales. The continuous reflection profiling technique was successful in delineating a channel incised in bedrock under a thickness of up to 60 m. of unconsolidated sediment covered by up to 20 m. of water.

Notes: Total gamma airborne radiometric survey of El Alamein area in the Western Desert has been carried out by the Atomic Energy Establishment. The flight lines are directed north-south with 500 m spacing, while the ground clearance is about 50 m. Superimposition of the radioactivity profiles over El Alamein Oilfield shows a general decrease of gamma radiation in the central part of the structure framed by zones of relatively higher values. The aerial radioactivity of El Alamein Oilfield is described and it is correlated with some reported radiometric characteristics of other oil bearing structures. The distribution of aeroradiometric measurements over El Alamein Oilfield and its environment has been found to be lognormal.

Notes: This paper gives an interpretation of a specific type of refracted-diffracted waves. The solution is supported by the good accord of data in seismic and magnetic interpretation. Beyond solving the problem of the phenomenon, the interpretation of such events shows that in the exploration of dykes and intruded formations the refracted-diffracted waves can—under favourable conditions—also render some useful information.

Notes: Resistivity surveys including surface measurements and electric well logging are at present carried out as routine techniques by a Groundwater Project of the United Nations Special Fund in arid valleys of the Argentine eastern Andes.The aims of the Project are the evaluation of ground water supplies in 5 pilot areas and the training of Argentine professionals in modern techniques of groundwater investigation.In Zonda and Tulüm Valleys, one of the five pilot areas, the groundwater body is essentially represented by alluvium of San Juan River and upper Tertiary gravels lying upon impervious clayey Tertiary sediments.Profiles of electrical soundings—spread in the valleys normal to the structural trends—were completed as a broad reconnaissance type investigation. In the outwash plain, a thick clayey overburden sometimes covered by saline material produces a masking effect on the deep sands and gravels and makes the evaluation of true resistivities difficult. Under certain conditions, appropriate use of electric well logging results helps solving the ambiguities found in the interpretation of the electrical soundings. Two examples are discussed.Also a typical development of surface resistivity interpretation as a whole is presented. Emphasis is placed upon the significance of the transverse resistance values to better interpret the hydrogeological picture. The last stage in data processing, presented on a map of the resistive fill and the conductive basement, provides evidence of the strong structural control of the valleys and shows the main variations in the ground water body constitution.

Notes: Book reviewed in this article:Hot Brines and Recent Heavy Metal Deposits in the Red Sea, edited by Egon T. Degens &- David A. RossH. Israel, Einführung in die Geophysik, Springer- Verlag, Berlin-Heidelberg-New YorkVerständliche Wissenschaft, Band 98: H. W. Franke

Notes: A mathematical analysis is given of the phenomenon of equivalence in resistivity sounding, which is based upon the properties of the raised kernel function. Analysis of this function instead of the apparent resistivity function is justified because, as has been shown in a previous publication, variations in the apparent resistivity function lead to variations in the raised kernel function with relative values of the same order of magnitude The expression for the raised kernel function is expanded into a Mac Laurin series. Equivalence can occur only if the second order term of this series is negligible. The coefficient of the first order term depends on the resistivity and the thickness of the layer under consideration. There is an infinite set of combinations of values for these two quantities, for which the coefficient of the first order term has the same value. All these combinations represent equivalent layer distributions.

Notes: One of the main objectives of seismic digital processing is the improvement of the signal-to-noise ratio in the recorded data. Wiener filters have been successfully applied in this capacity, but alternate filtering devices also merit our attention. Two such systems are the matched filter and the output energy filter. The former is better known to geophysicists as the crosscorrelation filter, and has seen widespread use for the processing of vibratory source data, while the latter is. much less familiar in seismic work.The matched filter is designed such that ideally the presence of a given signal is indicated by a single large deflection in the output. The output energy filter ideally reveals the presence of such a signal by producing a longer burst of energy in the time interval where the signal occurs. The received seismic trace is assumed to be an additive mixture of signal and noise. The shape of the signal must be known in order to design the matched filter, but only the autocorrelation function of this signal need be known to obtain the output energy filter. The derivation of these filters differs according to whether the noise is white or colored. In the former case the noise autocorrelation function consists of only a single spike at lag zero, while in the latter the shape of this noise autocorrelation function is arbitrary.We propose a novel version of the matched filter. Its memory function is given by the minimum-delay wavelet whose autocorrelation function is computed from selected gates of an actual seismic trace. For this reason explicit knowledge of the signal shape is not required for its design; nevertheless, its performance level is not much below that achievable with ordinary matched filters. We call this new filter the “mini-matched” filter.With digital computation in mind, the design criteria are formulated and optimized with time as a discrete variable. We illustrate the techniques with simple numerical examples, and discuss many of the interesting properties that these filters exhibit.

Notes: In the case of simple time series it is shown that prediction operators can be considered as deconvolution operators which are easily obtained.These operators possess the special feature of having a minimum phase, and their spectrum modulus represents, with a good dynamic range, the reciprocal of the square root of the modulus of the original autocorrelation spectrum.A generalization of the simple time series theory to the functions of two independent variables is possible in as much as, on a given section, the application of the multiple time series method enables the function of time and distance variables to have well-defined statistical properties; it is necessary, in particular, that the processes involved are stationary with respect to the two independent variables.In the case of multiple time series the application of the Prediction Theory permits greater uniformity of the traces because it enhances the events which show a good correlation between traces and, on the other hand, tends to minimize the random events which cannot be correlated between traces. The signal-to-noise ratio is thus increased to a great extent.

Notes: Considerable water leakages from reservoirs make it difficult to attain the planned storage capacity. In some cases water leakages give rise to suffusion followed by catastrophes. Until recently methods for locating water leakages were extremely imperfect. Geophysical methods offer good prospects in this direction. For solving these problems, it is effective to use streaming potential measurements, water flow rate observations and thermometry.Laboratory experiments were carried out in connection with the fact that water leakages from reservoirs are characterized by negative anomalies of natural currents; the more filtration discharge, the higher these anomalies are. As a result, the relationship governing the intensity of streaming potential and sand granulometric composition, electrolyte concentrations and other factors were revealed. To determine the velocity of water flowing to leakage sites, a special device, based on the relationship between the temperature of a heated body and its resistance, is applied. This device simultaneously makes it possible to measure the water medium temperature.Observations at reservoir sites were effected by moving along the reservoir non-polarizable potential electrodes and water velocity devices. Recordings were carried out automatically by the recording device of the logging apparatus. Under the conditions of ice cover on water surfaces, measurements were made through separate points by digging holes in the ice cover.Practical field observations were carried out at reservoir sites located in regions where fissured massive rocks as well as loose sediments predominate. In the first case field experiments were carried out in alpine reservoirs, in Armenia. The major water leakages were found to be concentrated on the right bank of the reservoir. In this connection it was not only possible to locate water leakage sites, but also to evaluate their relative intensity. These data were used for planning antifiltration measures. In the second case water leakages from a reservoir located in Uzbekistan in the submontane part of the Pamirs were studied. Streaming potential anomalies and high benthonic flow rates made it possible to discover high filtrations in the base and walls of the dam.Further perfection of these methods should not only permit the determination of water leakage sites and their relative intensity, but also filtration discharges in absolute units.

Notes: Recordings were made with three types of detector of the primary compressional (P) and shear (S) wave pulses generated by explosions in boreholes. Charge weights varied from 0.08 kg to 9.5 kg and detector distances varied from about 3 m to about 80 m. Scaling by the simple factor W1/3 where W is the charge weight, enabled observations from different sized charges to be fitted to a single expression.Experiments were carried out in the Bunter sandstone and the London clay and both fluid and solid tamping were used. This variation in tamping had no significant effect on the P-waves but it may have affected the generation of SV-waves. In both media the P-wave energy carried at 30 m from the shot by frequencies less than 100 Hz decreased rapidly with depth and was usually 1–2 % of the available chemical energy for a shot depth of 15 m. The S-wave energy was much less than this, but was highly directional.The P-wave pulse had the appearance of a damped sinusoid in very good agreement with the predictions of the ‘equivalent radiator’ hypothesis. However, the surface of this radiator should be identified not with the blown cavity but with the surface at which the tensile stresses associated with the stress wave become less than the tensile strength of the rock.The predominant frequency for a 1 kg charge at a depth of 15 m was 24 Hz in the clay and 52 Hz in the sandstone. In these and similar media, therefore, an effort should be made to keep individual charges less than 1 kg in reflection shooting and less than 10 kg in refraction shooting.The value of Q was about 50 in clay and about 25 in the sandstone. These estimates are rather uncertain because of the small distances over which the pulses were observed.The Z-transforms of the sampled pulses indicated that they were all of minimum phase, or very near to it.

Notes: The paper presents a discussion of some of the noise features of the VIBROSEIS SYSTEM*** and their bearing on the determination of the optimally weighted stack.

Notes: De nombreuses études ont été effectuées jusqu'à ce jour sur la Transformée de Fourier et ses applications en géophysique. En particulier, depuis ľavènement des ordinateurs qui ont entraîe la digitalisation de la principale méthode, sismique réflexion, les différentes techniques de traitement numérique ont fait ľobjet ?innombrables communications. La gravimétrie a toujours été une méthode numérique. Mais il est évident que pour elle également, de nouvellés possibilités se sont ouvertes. Aussi les gravimétriciens s'intéressent de plus en plus à la théorie et aux applications fondées sur ľétude des spectres en fréquences que permet la Transformée de Fourier. Depuis les travaux de C. A. Schwartz, W. Sokoloff et W. C. Dean, ?autres idées intéressantes, concernant le traitement de ľinformation gravimétrique, sont apparues: calcul du gradient vertical (V. Baronov), interprétation des anomalies à partir de leur spectre (M. E. Odegard et J. Berg Jr.), calcul des isopaques gravimétriques (J. L. Bible) etc….Toutes ces idées, toutes ces techniques ont le même but: faciliter ľinterprétation. Mais tout en ayant le même but et ayant affaire au même genre de difficultiés, il semble que le gravimétricien et le sismicien se soient elabores, chacun de leur côté, des outils sans se soucier du fait que leurs méthodes, et plus spécialement leur méthode de traitement de mesures brutes, sont, sinon identiques, du moins très semblables et que par conséquent les perfectionnements atteints dans une méthode peuvent servir ľautre.Ľobjet de la présente communication sera de repenser la philosophie des deux méthodes, sismique et gravimétrie, depuis ľexécution des mesures en passant par les principales méthodes de traitement, pour aboutir àľinterprétation, en se plaçant à deux points de vue complémentaires ?ailleurs:– Bien que les traitements numériques se fassent le plus souvent dans le domaine fonctionnel par les convolutions, il est bien plus facile de comprendre et de concevoir ces traitements si ľon raisonne alternativement grâce à la Transformée de Fourier dans le domaine fonctionnel et dans le domaine spectral.– En raisonnant en fréquences, il n'y a pas de différence de principe entre la sismique et la gravimétrie. Une courbe, graduée en milligals fonction de distance, et une trace sismique, représentant les variations de tension ?un galvanomètre en fonction du temps, sont, du point de vue de la Transformée de Fourier, identiques.Dans cet esprit on abordera les sujets suivants:– Signal sismique et signal gravimétrique– Problème ?échantillonnage en gravimétrie (espacement de mesures variable)– Filtrages.•Présentation et commentaires de spectres de quelques exemple syntétiques et pratiques•Filtrage en nombre ?onde•Filtrage en fréquences.•Problème de la fréquence o (dérivées horizontale et verticale)– Autres applications de la Transformée de Fourier en gravimétrie.

Notes: An efficient method of computing spectrum and cross-spectrum of large scale aero-magnetic field (or of any other two-dimensional field) has been developed and programmed for a digital computer. The method uses fast Fourier transform techniques. Briefly, the method is as follows: a digitized aeromagnetic map is divided into a number of rectangular blocks. Fourier transforms of these blocks are computed using a two-dimensional fast Fourier transform method. Finally, the amplitude of the Fourier transforms is averaged to give the desired spectrum. Computation of cross-spectrum follows the same lines. In fact, the same programme may be used to a compute the spectrum as well as cross-spectrum. The method has a number of computational advantages, in particular it reduces greatly computational time and storage requirements. The programme has been tested on synthetic data as well as on real aeromagnetic data. It took less than 30 seconds on an IBM 360/50 computer to compute the spectrum of an aeromagnetic map covering an area of approximately 4500 square miles.

Notes: The widespread use of common depth point techniques has emphasized the need for accurate static corrections. Manual interpretation methods can give excellent results, but a computer technique is desirable because of the great volumn of data recorded in common depth point shooting. The redundancy inherent in common depth point data may be used to compute a statistical estimate of the static corrections. The corrections are assumed to be time-invarient, surface-consistent, and independent of frequency. Surface consistency implies that all traces from a particular shot will receive the same shot static correction and all traces from a particular receiver position will receive the same receiver correction.Time shifts are computed for all input traces using crosscorrelation functions between common depth point traces. The time shift for each trace is composed of a shot static, a receiver static, residual normal moveout if present, and noise. Estimates of the shot and receiver static corrections are obtained by averaging different sets of the measured time shifts. Time shifts which are greatly in error are detected and removed from the computations.The method is useful for data which has a moderate to good signal to noise ratio. Residual normal moveout should be corrected before estimating the statics. The program estimates the statics for correctly stacking common depth point traces but it is not sensitive to constant or very slowly changing static errors.

Notes: It was found in Part I of this paper that approximating the sharp cut-off frequency characteristic best in a mean square sense by an impulse response of finite length M produced a characteristic whose slope on a linear frequency scale was proportional to the length of impulse response, but whose maximum overshoot of ±9% was independent of this length (Gibbs' phenomenon). Weighting functions, based on frequency tapering or arbitrarily chosen, were used in Part II to modify the truncated impulse response of the sharp cut-off frequency characteristic, and thereby obtain a trade-off between the value of maximum overshoot and the sharpness of the resulting characteristic. These weighting functions, known as apodising functions, were dependent on the time-bandwidth product Mξ, where 2ξ, corresponded to the tapering range of frequencies.Part III now deals with digital filters where the number 2N–1 of coefficients is directly related to the finite length M of the continuous impulse response. The values of the filter coefficients are taken from the continuous impulse response at the sampling instants, and the resulting characteristic is approximately the same as that derived in Part II for the continuous finite length impulse response. Corresponding to known types of frequency tapering, we now specify a filter characteristic which is undefined in the tapering range, and determine the filter coefficients according to a mean square criterion over the rest of the frequency spectrum. The resulting characteristic is dependent on the time bandwidth product Mξ= (N–1/2)ξ up to a maximum value of 2, beyond which undesirable effects occur. This optimum partially specified characteristic is an improvement on the previous digital filters in terms of the trade-off ratio for values of maximum overshoot less than 1%. Similar to the previous optimum characteristic is the optimum partially specified weighted digital filter, where greater “emphasis is placed on reducing the value of maximum overshoot than of maximum undershoot”. Such characteristics are capable of providing better trade-off ratios than the other filters for maximum overshoots greater than 1/2%. However these filters have critical maximum numbers 2.NC–1 of coefficients, beyond which the resulting characteristics have unsuitable shapes. This type of characteristic differs from the others in not being a biassed odd function about its cut-off frequency.

Notes: In 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.