## Gabor deconvolution

### Gary F. Margrave, Michael P. Lamoureux

The Gabor transform decomposes a 1-D temporal signal onto a time-frequency plane. Temporal localization is accomplished by windowing the signal with a Gaussian analysis window translated to any particular time. The Fourier transform of the windowed signal provides frequency information for the time of the window centre. The inverse Gabor transform is an integration over the time-frequency plane using an optional synthesis window. The discrete Gabor transform has an approximate realization based on the fact that a shifted suite of Gaussians can sum to unity. Gabor filters are nonstationary filters that can be implemented by multiplying the Gabor transform of a signal by a time-frequency filter specification. Extreme cases of Gabor filters are shown to correspond to normal or adjoint pseudodifferential operators. Gabor deconvolution is developed for exploration seismology based on a nonstationary convolutional model of a seismic trace. This model predicts that the Gabor transform of a seismic signal decomposes as the Fourier spectrum of the source signature times the symbol of a forward Q-filter times the Gabor transform of the reflectivity. Gabor deconvolution requires the spectral factorization of the Gabor transform of the seismic signal into a reflectivity part times a propagating wavelet part. This factorization can be done on magnitude spectra with phase being determined from a minimum-phase condition. Testing on synthetic data shows the nonstationary Gabor deconvolution is an effective tool that combines the processes of stationary deconvolution, inverse Q-filtering, and gain adjustment.