Generalized seismic wavelets

Abstract

The Ricker wavelet, which is often employed in seismic analysis, has a symmetrical form. Seismic wavelets observed from field data, however, are commonly asymmetric with respect to the time variation. In order to better represent seismic signals, asymmetrical wavelets are defined systematically as fractional derivatives of a Gaussian function in which the Ricker wavelet becomes just a special case with the integer derivative of order 2. The fractional value and a reference frequency are two key parameters in the generalization. Frequency characteristics, such as the central frequency, the bandwidth, the mean frequency and the deviation, may be expressed analytically in closed forms. In practice, once the statistical properties (the mean frequency and deviation) are numerically evaluated from the discrete Fourier spectra of seismic data, these analytical expressions can be used to uniquely determine the fractional value and the reference frequency, and subsequently to derive various frequency quantities needed for the wavelet analysis. It is demonstrated that field seismic signals, recorded at various depths in a vertical borehole, can be closely approximated by generalized wavelets, defined in terms of fractional values and reference frequencies.