The nth Power Fourier Spectrum Analysis for the Generalized Seismic Wavelets

The generalized seismic wavelets (GSWs) are defined by fractional derivatives of the Gaussian function, whose asymmetry allows them to represent seismic signals more accurately than the commonly used symmetrical Ricker wavelet. The latter is a particular case with a second derivative of the Gaussian function. To better obtain the GSW, which could be well-matched with seismic signals, this article proposes the $\boldsymbol {n}$ th power Fourier spectrum analysis method for GSWs. First, based on the $\boldsymbol {n}$ th power Fourier spectrum of GSWs, the proposed method builds the mathematical relationship between the frequency characteristics (e.g., central frequency and bandwidth) and the statistical properties (e.g., mean frequency and deviation). Second, according to the $\boldsymbol {n}$ th power Fourier spectrum, we propose a weighting calculation method for the derivative order $\boldsymbol {u}$ of the Gaussian function. This method could be used for estimating GSWs matched the seismic first-arrival record, which is conducive to improving the accuracy of seismic imaging, inversion, and $Q$ analysis. In theory, our proposed weighting method has better robustness and noise resistance than the traditional spectrum analysis method based on the power or amplitude spectrum. The experiment of synthetic noise, including first-arrival record and vertical seismic profiling (VSP) field data, shows the effectiveness of the proposed approach.