It is proved that horizontally stratified media, the various materials of which do not differ in their Poisson number, can be considered as isotropic for reflections of a small dip when taking into account quasi‐longitudinal waves which have to be reckoned with in practice. Approximating the surface of the wave by an ellipsoid would, however, result in considerable errors.
Moreover, curves are presented allowing practical calculations for steeper dips to be made.
During the last two decades, the detection of coal seam discontinuities by seismic waves guided by the seam has become a special branch of exploration seismics in Europe. Waves consisting purely of SH motion (so‐called waves of Love type) are of special interest, and the rather high frequencies of the Airy phase, in thin seams, are most important because they present very high reflectivity at seam interruptions. Absorption increases with frequency in most layers, and therefore attenuates the high Airy‐phase frequencies more severely than the earlier low‐frequency part of the guided waves. Another fact additionally impairs the Airy‐phase signal: the quality factor Q is much lower in coal than in the schists and sandstones of the Carboniferous country rock. Unfortunately, most of the energy of the Airy phase is transferred by the coal, whereas the lower frequencies have their main energy conveyed by the country rock above and below the seam. In order to allow a better understanding of the influence of absorption on Love‐type seam waves, several simplified computations were carried out for the fundamental mode of a seam typical for the northwest‐German Ruhr area. The assumptions are as follows: The quality factors [Formula: see text] for coal and [Formula: see text] for the country rock do not depend upon frequency; higher powers of [Formula: see text] and [Formula: see text] can be neglected; and the distance from the source is large enough to allow the two‐dimensional plane‐wave case to be considered. The mathematics resulting from these assumptions and adequate data processing of transmission records provides the possibility to determine the quality factor [Formula: see text] of coal in‐situ, although the thickness of the seam may be much smaller than the wavelengths involved. [Formula: see text] may become of interest for practical mining problems.
It is well known that interval velocities can be determined from common‐reflection‐point moveout times. However, the mathematics becomes complicated in the general case of n homogeneous layers with curved interfaces dipping in three dimensions.
In this paper the problem is solved by mathematical induction using the second power terms only of the Taylor series which represents the moveout time as a function of the coordinate differences between shot and geophone points. Moreover, the zero‐offset reflection times of the nth interface in a certain area surrounding the point of interest have to be known. The n—I upper interfaces and interval velocities are known too on account of the mathematical induction method applied. Thus, the zero‐offset reflection raypath of the nth interface can be supposed to be known down to the intersection with the (n—1)th interface.
The method applied consists mainly in transforming the second power terms of the moveout time from one interface to the next one. This is accomplished by matrix algebra.
Some special cases are discussed as e.g. uniform strike and small curvatures.
Nonlinear sweeps have often successfully been employed in the 1960s. However, this area of sweep technology has been neglected since the introduction of digital recording techniques in the Vibroseis system. Now the advent of computerized recording instruments yields a new economical possibility of forming approximately nonlinear sweeps by combining several linear sweeps with or without time gaps to a “Combisweep”. The total duration of a Combisweep may be as long as the maximum available recording time, for example 32 s.
Beside the attenuation of correlation noise, the new method has further merits, such as the weighting of predetermined frequency ranges, in order to effect a certain kind of optimum filtering on the emitter side, or in order to compensate to some degree for frequency dependent absorption.
In all these applications the Combisweep is considered as one signal in the correlation process. But by correlating with the individual sweeps or a partial combination of them and by applying automatic switching at predetermined times within the gaps between the individual sweeps additional possibilities arise, such as obtaining in one run with a twenty‐four channel recording unit twenty‐four traces with small distances between vibrators and geophones for shallow reflections and another twenty‐four traces with larger distances for deeper reflections. Various Combisweeps and their applications are presented.
In his important paper “Computing true amplitude reflections in a laterally inhomogeneous earth” (Hubral, 1983, this issue) the author derives the fundamental equation [Formula: see text] [Hubral (25)] Here [Formula: see text] is a factor compensating the energy of a zero‐offset reflection for geometrical spreading and [Formula: see text] and [Formula: see text] are certain wavefront curvature matrices which will be explained later.
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