The nth Power Fourier Spectrum Analysis for the Generalized Seismic Wavelets

Abstract

The generalized seismic wavelets (GSW) 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 paper proposes the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum analysis method for GSW. Firstly, based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum of GSW, the proposed method builds the mathematical relationship between frequency characteristics (e.g., central frequency and bandwidth) and the statistical properties (e.g., mean frequency and deviation). Secondly, according to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum, we propose a weighting calculation method for the derivative order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i> of the Gaussian function. This method could be used for estimating GSW 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.