The computation method described in this paper is based on the existence of a linear relationship between the mutual coupling ratio and the kernel function in the integral expression for it. Accordingly, the mutual coupling ratio can be determined by first computing sample values of the kernel function and then subjecting these to a digital linear filter. In the present paper the appropriate sampling distance is determined and the values of the digital filter coefficients are computed, both for electromagnetic sounding with horizontal coils and for electromagnetic sounding with perpendicular coils.
In a previous publication (Koefoed 1968) a function called the “raised kernel function” has been introduced as an intermediate function in the interpretation of resistivity sounding data, and methods have been described both for the determination of the raised kernel function from the apparent resistivity function, and for the determination of the layer distribution from the raised kernel function.
In the present paper a procedure is described by which the second step in this interpretation method–i.e. the determination of the layer distribution from the raised kernel function–is considerably accelerated. This gain in interpretation speed is attained by the use of a standard graph for a function which defines the reduction of the raised kernel function to a lower boundary plane.
Calculations are presented of reflection coefficients of plane longitudinal waves incident at oblique angles on boundary planes between elastic media. It is shown that the manner in which these reflection coefficients vary with the angle of incidence is strongly affected by the values of the Poisson's ratios of the two media. It appears that, contrary to a conclusion arrived at by Muskat and Meres, the reflection coefficient may vary appreciably with the angle of incidence in the range between 00 and 300. Possibilities of practical application of this phenomenon are discussed.
It has been found that the Wiener‐Hopf least‐squares method is a very successful tool for the determination of resistivity sounding filters. The values of the individual filter coefficients differ quite appreciably from those obtained by the Ghosh procedure. These differences in the filter coefficients, however, have only a negligible effect on the output of the filter. It seems that these differences in the coefficients correspond to a filter function of a rather narrow frequency band around the Nyquist frequency, which is only very weakly present in the input and output functions.
Thin layers are considered from the point of view of the quasi-linear relation that exists between their thickness and their reflection response to a seismic signal. The range within which this quasi-linearity exists is investigated; for a continuous sine wave, this is done by means of the equation for the response given in Rayleigh (1945), and for a seismic wavelet by means of a synthetic seismogram program. For a wavelet, the limiting value of the dominant frequency is found to be smaller than that for a continuous sine wave, the difference being in the order of magnitude of 15 percent.Within the linearity range, a thin layer may be replaced by an equivalent layer which gives the same reflection response but differs in thickness and in acoustic impedance. In the construction of synthetic seismograms over coal seams, this equivalent replacement may be utilized to replace the seams by layers, for which the two-way traveltime is equal to an integer number of sampling intervals; by this procedure the usual rounding-off errors are avoided. The method of equivalent replacement is also applicable when the host rock above and below the seam have different velocities.
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