Quantitative comparison of three spectral decomposition methods

Abstract

Abstract Several time-frequency transform methods have been used for spectral decomposition of seismic data. Comparisons between methods have been largely qualitative with a focus on their relative time and frequency resolutions. Very little has been said about the fidelity of spectral decomposition results relative to the original data and the quantitative differences between methods. This paper provides some analysis of these issues as they relate to three decomposition methods, namely the S-transform (ST), Matching Pursuit (MP) and Empiric Mode Decomposition (EMD). Our quantitative comparison made use of the following methods:Total energy of the trace vs. time-frequency transform – the ideal spectral decomposition will have total energy equal to the input signal.Marginal time and frequency conditions – the ideal spectral decomposition will satisfy the marginal conditions. The integral of the spectra over frequency will equal the instantaneous energy of the trace and the integral over time will equal the power spectra of the trace. Our analysis used a combination of synthetic and acquired traces and wedge models. The single trace and wedge model results suggest that the characteristics of the spectral decomposition change with the type of time-frequency transform used in the decomposition.ST wedge model conformed to the bandlimited spike interference model with spacing dependent spectral notches. The single trace marginal conditions were satisfied best with a resolution ratio near 1.4 at the cost of a little time resolution relative to a default resolution ratio of 1.0.MP wedge model did not have spectral notches and is negatively influenced by the greediness and path dependence of the process. The single trace marginal condition details however are best satisfied by MP.EMD wedge model spectrum was very sparse with an apparent relationship between the frequency of the components and the degree of tuning. The single trace marginal conditions satisfaction is variable and dependent on the model traces having satisfied the underlying assumptions. We recommend the use of marginal condition analysis in parameter selection for spectral decomposition and to facilitate better products for quantitative analysis. Introduction Several time-frequency transform methods have been applied under the guise of spectral decomposition for the interpretation of seismic reflection data. Some examples of these methods are the short window Fourier transform, continuous wavelet transform, S-transform, MP, empiric mode decomposition, and quadratic time-frequency distributions. In this study we focused on the quantitative comparison of three methods: the S-transform, MP and empiric mode decomposition. The algorithms used in this study have been extensively tested and adjusted by the authors to minimize distortion and maximize energy conservation with the addition of appropriate scales, and application of additional signal processing techniques, beyond those discussed in the references.