POTSI Gaboring Advances Gary Margrave, Linping Dong, Peter Gibson, Jeff Grossman, Michael Lamoureux POTSI

This is about … The Utility of the Gabor transform Theoretical advances in Gabor theory Field testing of Gabor deconvolution

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

The key to a fast, robust Gabor transform A Partition of Unity The key to a fast, robust Gabor transform

Gabor transform via a partition of unity Given a partition Define analysis window and synthesis window so that

Gabor transform via a partition of unity Form a Gabor slice The forward Fourier transform over the set of Gabor slices gives the Gabor transform

Gabor transform via a partition of unity The inverse Gabor transform is an inverse Fourier transform, multiply by the synthesis window, sum over windows

The Utility of the Gabor Transform The Gabor transform is a natural extension of the Fourier transform into the nonstationary realm. Gabor deconvolution is enabled because the Gabor transform approximately “factorizes” the nonstationary seismic trace (Margrave et al, this report) The product of two Gabor spectra is a nonstationary convolution. A nonstationary convolution is a pseudodifferential operator. Based on a “partition of unity”, a numerical Gabor transform can easily accommodate an irregular sampling lattice (Lamoureux et al, Grossman et al, this report) Whatever you did with the Fourier transform you can “sorta” do “nonstationarily” with the Gabor transform.

What about the Wavelet Transform? Applications are largely restricted to data compression and de-noising. There is no convolution theorem for the Wavelet transform. The product of two Wavelet transforms is not anything useful.

Advancements in Gabor Theory Gabor transforms can exactly factorize certain pseudodifferential operators if “compatible” window pairs are used (Gibson et al) We have identified a small number of compatible window pairs. The use of “non-compatible” windows merely means that any factorization will be approximate.

Compatible windows The Gabor analysis and synthesis windows are called compatible if This is precisely the condition needed to solve unambiguously for the “Gabor multiplier” given the pseudodifferential operator.

Compatible windows Unity and Delta: Leads directly to K-N pseudodifferential operators

Compatible windows Gaussians: Extreme Value:

Real Seismic (200 traces) after gain for spherical spreading Receiver position

Gabor spectrum of trace 100 after gain

Attenuation and nonstationarity each reflected arrival is minimum phase source is minimum phase Attenuation depends on path length. Therefore the seismic recording is inherently nonstationary, being a linear superposition of many different minimum-phase arrivals with differing degrees of attenuation.

Gabor Deconvolution a) b) c) d)

Comparison on Synthetic

Real Data Comparison Stratigraphic line provided by Husky Energy Dynamite source Data processing provided by Sensor Geophysical (Peter Cary)

Gabor -> Stack -> Gabor (160 Hz) Real Data Comparison Standard flow: Gain->Surface Consistent Wiener -> TVSW -> Stack ->TVSW (120 Hz) Gabor flow: Gabor -> Stack -> Gabor (160 Hz)

Standard Processing 200 400 ms 600 800

Pre and post stack Gabor 200 400 ms 600 800

Standard Processing 200 400 ms 600 800

Pre and post stack Gabor 200 400 ms 600 800

Standard Processing 200 ms 400 600

Pre and post stack Gabor 200 ms 400 600

200 ms 400 600 Gabor standard Gabor

Standard Processing 200 ms 400 600

Pre and post stack Gabor 200 ms 400 600

Summary The Gabor transform is a very promising tool for exploration seismology The Gabor transform extends Fourier concepts to the nonstationary realm Gabor succeeds because it can factorize pseudodifferential operators Gabor deconvolution easily beats Wiener and edges out Wiener+TVSW

Research Goals Incorporate well information as a constraint Better phase estimation (remove more delay) Develop theory on nonstationary minimum phase filters Extend to multiple attenuation Gabor wavefield extrapolator

= 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 GEDCO, Husky Energy, Encanna =