A robust algorithm for constant-Q wavelet estimation using Gabor analysis Jeff. P. Grossman*, Gary F. Margrave, Michael P. Lamoureux, and Rita Aggarwala, University of Calgary Summary Seismic attenuation can be modeled macroscopically via an exponential amplitude decay in both time and frequency, at a rate determined by a single dimensionless quantity, Q. Current deconvolution methods, based on the convolutional model, attempt to estimate and remove the embedded causal wavelet. We propose a nonstationary seismic model, expressed in the time-frequency domain, in which (1) the embedded causal wavelet factors as the product of a stationary seismic signature and a nonstationary exponential decay; and (2) a nonstationary impulse response for the earth is tractable. Least squares fitting our model to the Gabor-transformed seismic trace yields a Q-value and an estimate of the source signature, hence an estimate of the nonstationary wavelet. These estimates lead to a smoothed version of the magnitude of the Gabor spectrum of the seismic trace, from which a least-squares nonstationary minimum-phase deconvolution filter is easily constructed. Our preliminary results on synthetic data are very promising. Introduction This paper is concerned with the application of Gabor theory (see Grochenig, 2001 for a thorough treatment of Gabor analysis) to the modelling, analysis, and deconvolution of a nonstationary, constant-Q-attenuated seismic signal. Gabor deconvolution is discussed in a broader context by Margrave, et al. (2002). Gabor theory sets the stage for nonstationary analysis of a signal, “in which time and frequency play symmetrical parts, and which contains ‘time analysis’ and ‘frequency analysis’ as special cases.” (Gabor, 1946) Our experience with sound, as in music, calls for a unified mathematical treatment of time and frequency analysis. The human ear, much like a hydrophone, combined with the brain’s processing abilities, interprets acoustic amplitude information as temporally local-ized packets of bandlimited spectral information. Conversely, in following a musical score, a musician transforms a time-frequency representation of a signal into temporally varying acoustic amplitude data. Fourier theory fails to explain what we intuitively know: that “frequency content” changes with time. “The reason is that the Fourier-integral method considers phenomena in an infinite interval, and this is very far from our everyday point of view.” (Gabor, 1946). These fundamental observations imply that we ought to be modelling the localized time and frequency characteristics of a signal simultaneously; Gabor’s milestone theory provides us with the appropriate tools. We outline the derivation of a time-variant spectral model for the Gabor transform of a constant-Q-attenuated seismic trace. Remarkably, in addition to being intuitively plausible, it generalizes the standard convolutional model. Next, we derive explicit least squares, model-based estimates for both Q and the stationary part of the wavelet, and thus an estimate of the nonstationary, Q-attenuated wavelet. The theory is then evaluated via an algorithm for the deconvolution of a Q- attenuated synthetic seismogram. Theory Let ( ) rt be a reflectivity, ( ) wt a source signature, and ( ) , tf α the time-frequency symbol of a constant-Q operator, namely ( ) ( ) / / , , f tQ iH ftQ tf e π π α − − = (1) where H is the Hilbert transform. We assume that the source signature is a stationary wavelet whose time-frequency de- composition is equivalent to its Fourier transform, ( ) ˆ , w f and that ( ) ˆ , w f is smooth; by definition, ( ) , tf α is also smooth. . A synthetic, ( ) s t , is built by nonstationary convolution of the Q- operator with the reflectivity, (Margrave, 1998) followed by stationary convolution with the source signature. This yields ( ) ( ) ( )() 2 ˆ ˆ , , or if t s f w f tf rte dt π α ∞ − −∞ = ∫ (2) () ( ) ( )() ( ) ( )() [ ] 2 2 2 ˆ , ˆ , . if u if t if t u w f u f r u e du e df w f uf rue dfdu st π π π α α ∞ − −∞ ∞ −∞ −   =     = ∫ ∫ ∫∫ (3) Now, the Gabor transform of ( ) s t is defined as ( ) () ( ) 2 , if t s f stgt e dt π τ τ ∞ − −∞ = − ∫ ) (4) where ( ) gt is the analysis window, e.g., a Gaussian (for more on Gabor analysis, see Feichtinger and Strohmer, 1998). Margrave et al. (2002) show that the Gabor transform of ( ) s t factorizes, to first order, as ( ) ( ) ( ) ( ) ˆ , , , . s w r τ ν νατν τν = ) ) (5) To first order, our Gabor-spectral model is given by (5). The problem is simplified by considering only the magnitudes, ( ) () ( ) ( ) ˆ , , , s w r τ ν ν ατν τν = ) ) . (6) The Gabor transform of the reflectivity is assumed white, with unit mean. By “white”, we mean that ( ) ˆ w ν ( ) , α τν provides the general spectral shape, while the Gabor spectrum of the reflectivity provides only details. Thus, we drop the term ( ) , r τ ν ) from (6) and seek a trace model as ( ) () / , Q S W e πντ τν ν − = , (7) where , S s = ) ˆ , W w = and / (,) . Q e πντ α τν − = The equality in (7) is taken in the least-squares sense, meaning that a residual error with minimized L 2 -norm is assumed. For a further simplification, we consider the logarithm of both sides of (7): ( ) ( ) ln , ln / S W Q τν ν πτν = −     . (8)