The estimation of spectra from time-frequency transforms for use in attenuation studies

We introduce the signal dependent time–frequency distribution, which is a time–frequency distribution that allows the user to optimize the tradeoff between joint time–frequency resolution and suppression of transform artefacts. The signal-dependent time–frequency distribution, as well as the short-time Fourier transform, Stockwell transform, and the Fourier transform are analysed for their ability to estimate the spectrum of a known wavelet used in a tuning wedge model. Next, the signal-dependent time–frequency distribution, and fixed- and variable-window transforms are used to estimate spectra from a zero-offset synthetic seismogram. Attenuation is estimated from the associated spectral ratio curves, and the accuracy of the results is compared. The synthetic consisted of six pairs of strong reflections, based on real well-log data, with a modeled intrinsic attenuation value of 1000/Q = 20. The signal-dependent time–frequency distribution was the only time–frequency transform found to produce spectra that estimated consistent attenuation values, with an average of 1000/Q = 26±2; results from the fixed- and variable-window transforms were 24±17 and 39±10, respectively. Finally, all three time–frequency transforms were used in a pre-stack attenuation estimation method (the pre-stack Q inversion algorithm) applied to a gather from a North Sea seismic dataset, to estimate attenuation between nine different strong reflections. In this case, the signal-dependent time-frequency distribution produced spectra more consistent with the constant-Q model of attenuation assumed in the pre-stack attenuation estimation algorithm: the average L1 residuals of the spectral ratio surfaces from the theoretical constant-Q expectation for the signal-dependent time-frequency distribution, short-time Fourier transform, and Stockwell transform were 0.12, 0.21, and 0.33, respectively. Based on the results shown, the signal-dependent time-frequency distribution is a time–frequency distribution that can provide more accurate and precise estimations of the amplitude spectrum of a reflection, due to a higher attainable time–frequency resolution.