The discrete prolate spheroidal sequences and a series expansion for seismic wavelets

Abstract

The discrete prolate spheroidal sequences (DPSSs) are a set of time-limited, band-limited, and mutually orthogonal sequences. I exploit their favorable properties to introduce a series expansion for seismic wavelets. By modifying the definition of the DPSSs, I generate new sequences that are tailored to the character of the seismic wavelet but retain the DPSSs’ favorable properties. I show that a series expansion composed of these new sequences can be thought of as a logical extension of a familiar practice, assuming the wavelet is a linear combination of a Ricker wavelet and its Hilbert transform ([Formula: see text] rotation). Two examples, one synthetic and one real, show how the series expansion can be used to tie well data to seismic data. This leads to a robust method to determine a wavelet. The examples also illustrate the properties of the series expansion, some of which can be related to familiar geophysical concepts such as thin-bed tuning. I conclude that this series expansion is a natural and useful way to describe the seismic wavelet.