There are three related problems with our approach to signature deconvolution. First, there is a confusion among geophysicists about the basis of the convolutional model itself, which leads to doubts about the value of measurements of the source signature. Secondly, it is not generally recognized that statistical methods of wavelet estimation are unreliable. Thirdly, many explorationists are unaware that it is practical in many cases to make meaningful measurements of the source signature. The convolutional model of the reflection seismogram applies only for a point source, and is the convolution of the source signature with the impulse response of the earth, of Green’s function, which contains all possible arrivals, including reflections, refractions, multiples and diffractions. Stabilized deconvolution of the data with a known band‐limited signature is straightforward. The signature can be obtained by independent measurements, as described in the literature. The recovery of the elastic layer parameters from the band‐limited impulse response of the earth, after removal of the source signature by deconvolution, is the problem of inversion, and is not discussed in this paper. The theory of wave propagation does not support the commonly held view that a reflection seismogram can be regarded as a convolution of a wavelet with the series of normal‐incidence primary reflection coefficients. This is true of both prestack and poststack data. Poststack seismic inversion schemes, based on this model, that use well logs to extract the wavelet for predicting lateral variations in lithology away from the wells, rely on the wavelet to be laterally invariant. Even if there is perfect shot‐to‐shot repeatability, this model must yield a different wavelet at every well, and therefore the extracted wavelet does vary laterally. These schemes are therefore self‐contradictory and, in the worst cases, their results are likely to be worthless. Published methods for determining the source signature from measurements for the land vibrator, marine seismic source arrays, and dynamite on land are summarized. None of these methods appears to be in use. A Vibroseis example is included to show that the signal transmitted into the ground by the vibrators does not closely resemble the predetermined sweep, as is normally assumed. The transmitted signal could be determined in processing from measurements of the vibrator behaviour that are made in production for vibrator control, if only these measurements were recorded. Normally they are not. Instead of using measurements to determine the signature, the exploration industry relies on wavelet estimation methods that depend on both a model and statistical assumptions that have no theoretical justification.
Fully probabilistic seismic source inversion – Part 1: Efficient parameterisation
