Frequencies of the Ricker wavelet

TLDR

The Ricker wavelet is theoretically a solution of the Stokes differential equation, accounting for Newtonian viscosity, and is applicable to seismic waves in viscoelastic homogeneous media. The study defines the time‑domain breadth and frequency‑domain bandwidth of the Ricker wavelet and derives related quantities analytically using the Lambert function. The authors analytically derived the wavelet’s breadth and bandwidth with the Lambert function and related theoretical frequencies to numerical measurements. The analysis shows that the central frequency matches the mean frequency from the power spectrum, the standard deviation is not the half‑bandwidth, and theoretical and numerical frequency measures are mathematically related and expressed analytically.

Abstract

The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. In this paper, we defined the time-domain breadth and the frequency-domain bandwidth of the Ricker wavelet and developed quantities analytically in terms of the Lambert [Formula: see text] function. We determined that the central frequency, the geometric center of the frequency band, is close to the mean frequency statistically evaluated using the power spectrum, rather than the amplitude spectrum used in some of the published literature. We also proved that the standard deviation from the mean frequency is not, as suggested by the literature, the half-bandwidth of the frequency spectrum of the Ricker wavelet. Moreover, we established mathematically the relationships between the theoretical frequencies (the central frequency and the half-bandwidth) and the numerical measurements (the mean frequency and its standard deviation) and produced each of these frequency quantities analytically in terms of the peak frequency of the Ricker wavelet.