A derivation is presented of the response of an embedded thin layer to a vertically incident seismic pulse. The reflected pulse has the shape of the time derivative of the incident-wavelet, and its amplitude is proportional to the two-way traveltime in the thin layer and to a factor depending upon the ratio of acoustic impedances. The influence of spectral bandwidth on interface resolution and thin-layer response is investigated by means of zero-phase signals, and a filtering philosophy is proposed which enables thickness estimation either from amplitude or from peak-totrough time
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.
Approximate deconvolution by means of Wiener filters has become standard practice in seismic data‐processing. It is well‐known that addition of a certain percentage of noise energy to the autocorrelation of the signal wavelet leads to a filter that does not increase, or even reduces, the noise level on the seismogram. This noise addition will, in general, cause a minimum phase signal to become mixed phase. A technique is presented for the calculation of the optimum‐lag shaping filter for a contaminated signal wavelet. The advantages of this method over the more conventional approach are that it needs less arithmetic operations and that it automatically gives the filter with the optimum combination of shaping performance and noise reduction.
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