Gary Margrave and Michael Lamoureux Gabor Deconvolution Gary Margrave and Michael Lamoureux =

= Outline The Gabor idea The continuous Gabor transform The discrete Gabor transform A nonstationary trace model Gabor deconvolution Examples =

Gabor Papers in the Research Report Gabor Deconvolution by Margrave and Lamoureux Constant-Q wavelet estimation via a nonstationary Gabor spectral model by Grossman, Margrave, Lamoureux, and Aggarwalla Gabor deconvolution applied to a Blackfoot dataset by Iliescu and Margrave A ProMAX implementation of nonstationary deconvolution by Henley and Margrave =

The Gabor Idea Dennis Gabor proposed (1946) proposed the expansion of a wave in terms of “Gaussian wave packets”. The mathematics for the continuous Gabor transform emerged quickly. The discrete Gabor transform emerged in the 1980s due to Bastiaans. =

The Gabor Idea A seismic signal Multiply A shifted Gaussian A Gaussian slice or wave packet.

The Gabor Idea A seismic signal A suite of Gaussian slices Remarkably, the suite of Gaussian slices can be designed such that they sum to recreate the original signal with high fidelity.

The Gabor Idea The Gabor transform time Window center time Suite of Gabor slices Window center time frequency Gabor spectrum or Gabor transform Fourier transform

Inverse Fourier transform Inverse Fourier transform The Gabor Idea The inverse Gabor transform done two ways Window center time frequency Inverse Fourier transform Sum time Window center time Inverse Fourier transform Sum

The Continuous Gabor Transform Forward transform Gaussian “analysis” window The Gabor transform The signal Fourier exponential and integration over t The Gabor transform is a 2D time-frequency decomposition of a 1D signal.

The Continuous Gabor Transform Inverse transform Gaussian “synthesis” window Fourier exponential and integration over f and t The signal The Gabor transform The inverse Gabor transform is a 2D integration over the time-frequency plane.

A discrete Gabor transform In the discrete case, Gabor transforms have proven difficult to implement without loss. The issue was finally “solved” in the 1980’s using “frame theory”. We have implemented a simpler, approximate transform that is based on a convenient summation property of Gaussians.

A discrete Gabor transform The Gaussian summation property. Summation curve Gaussian half-width .1 s window increment .05

A discrete Gabor transform The Gaussian summation property. We show the the sum of a discretely spaced set on Gaussians on an infinite interval is given by Summation curve Gaussian half-width .1 s window increment .05 First-order error term is the Gaussian half-width is the Gaussian spacing

A discrete Gabor transform The Gaussian summation property. is the Gaussian half-width is the Gaussian spacing Summation curve Gaussian half-width .1 s window increment .05 First-order error term .5 1.0 1.5 2.0 -21 decibels -85 decibels -150 decibels -340 decibels The inter-Gaussian spacing should well less than the Gaussian half-width.

Gaussian summation test Gaussian half-width .1 seconds. 0.1 0.25 0.5 1.5 2 1

Gabor transform test Unnormalized transform Difference Seismic signal After forward and inverse Gabor transform Difference

Gabor transform test Normalized transform Difference Seismic signal After forward and inverse Gabor transform Difference Normalization is a simple process that reduces end effects.

Gabor transform example Minimum phase wavelet Reflectivity Nonstationary synthetic Q = 25

Gabor transform example Wavelet Q decay curve Reflectivity No clipping

Gabor transform example Q decay curve Clipping

Stationary Convolution Model

Stationary Convolution Model in the Fourier domain

Nonstationary Convolution Model

Nonstationary Convolution Model We have proven that Gabor transform of seismic signal Fourier transform of wavelet Q-filter Gabor transform of reflectivity is approximately given by

Proof therefore

Nonstationary Convolution Model example Gabor transform of seismic signal

Nonstationary Convolution Model example Gabor transform of reflectivity

Nonstationary Convolution Model example Q filter transfer function

Nonstationary Convolution Model example Q filter times Gabor transform of reflectivity

Nonstationary Convolution Model example wavelet surface

Nonstationary Convolution Model example wavelet times Q filter times Gabor transform of reflectivity

Nonstationary Convolution Model example Gabor transform of seismic signal

Gabor Deconvolution Gabor transform of the seismic trace Gabor transform of the reflectivity Given this Estimate this Recall the nonstationary trace model:

Gabor Deconvolution

Gabor Deconvolution So, given this: We must estimate this:

Gabor Deconvolution Estimation of “the propagating wavelet” Can be done by: Smoothing Time-variant Burg + smoothing Least squares modeling of Lotsa other ways …

Gabor Deconvolution Phase calculation The phase can be locally minimum, implemented by a Hilbert transform over frequency, or locally zero.

Gabor deconvolution example reflectivity nonstationary seismic signal nonstationary seismic signal reflectivity

Initial Gabor spectrum

Initial Gabor/Burg spectrum

Smoothed Gabor spectrum

Smoothed Gabor/Burg spectrum

Deconvolved Gabor spectrum Fourier algorithm

Deconvolved Gabor spectrum Burg algorithm

Deconvolved Gabor spectrum Actual reflectivity

Gabor deconvolution result Fourier algorithm reflectivity Gabor deconvolution result

Gabor deconvolution result Burg algorithm Gabor deconvolution result reflectivity

Least squares Q and wavelet estimates Constant-Q wavelet estimation via a nonstationary Gabor spectral model by Grossman et al. We fit a constant-Q wavelet model to the Gabor transform of the seismic signal to obtain expressions for a Q estimate and the wavelet amplitude spectrum.

Least squares Q and wavelet estimates The nonstationary convolution model predicts This, together with the nearly random nature of suggests that we impose the model where the equality is taken to hold in the least-squares sense.

Least squares Q and wavelet estimates The Q and wavelet estimates emerge as integrations of the Gabor transform of the signal over a domain in the time-frequency plane. The integration domain is bounded by hyperbolae that are level-lines of the constant Q operator:

Least squares Q and wavelet estimates This method is still in development but we are hopeful that is will prove superior to simple smoothing as the method of spectral separation in Gabor deconvolution.

ProMAX Implementation A ProMAX implementation of nonstationary deconvolution by Henley and Margrave This module is in the current CREWES software release. The FORTRAN computation module is self-contained and can be easily ported to other systems. Both Fourier and Burg methods are included with spectral separation by smoothing. The Q estimator is not there yet. A word of caution: unwise choice of parameters can mean very long run times.

Blackfoot testing Gabor deconvolution applied to a Blackfoot dataset by Iliescu and Margrave Gabor/Burg deconvolution Wiener spiking deconvolution

Blackfoot testing Gabor

Blackfoot testing Wiener

Blackfoot testing Typical trace spectra After Wiener After Gabor

Blackfoot testing Gabor/Burg stack

Blackfoot testing Wiener stack

Gabor/Burg (pre and post stack) and final migration Blackfoot testing Gabor/Burg (pre and post stack) and final migration

Wiener (pre stack), TVSW and final migration Blackfoot testing Wiener (pre stack), TVSW and final migration

Conclusions The discrete Gabor transform can be developed by simple Gaussian slicing. The nonstationary extension of the convolution model predicts that the Gabor transform of a seismic signal decomposes into the product of the Fourier transform of the wavelet, the symbol of the Q filter, and the Gabor transform of the reflectivity.

Conclusions Gabor deconvolution is implemented by a spectral factorization applied to the Gabor transform of a trace. Gabor deconvolution generalizes Wiener deconvolution to the nonstationary setting. Within the Wiener design gate Gabor and Wiener are similar; elsewhere, Gabor seems better. A new Gabor deconvolution code is released and shows promising results.

Acknowledgements All of the following provided support CREWES: Consortium for Research in Elastic Wave Exploration seismology NSERC: Natural Sciences and Engineering Research Council of Canada MITACS: Mathematics of Information Technology and Complex Systems PIMS: Pacific Institute of the Mathematical Sciences