Gabor Deconvolution. Gary Margrave and Michael Lamoureux

Transcription

1 Gabor Deconvolution Gary Margrave and Michael Lamoureux =

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

3 Gabor Papers in the Research Report 1. Gabor Deconvolution by Margrave and Lamoureux 2. Constant-Q wavelet estimation via a nonstationary Gabor spectral model by Grossman, Margrave, Lamoureux, and Aggarwalla 3. Gabor deconvolution applied to a Blackfoot dataset by Iliescu and Margrave 4. A ProMAX implementation of nonstationary deconvolution by Henley and Margrave =

4 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. =

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

6 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.

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

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

9 The Continuous Gabor Transform ( τ, ) ( ) ( τ) 2 The signal Gaussian analysis window π if t Vs f = st gt e dt g The Gabor transform Forward transform Fourier exponential and integration over t The Gabor transform is a 2D time-frequency decomposition of a 1D signal.

10 The Continuous Gabor Transform () = (, ) ( ) 2 π if t s t V s τ f γ t τ e dfdτ The signal Inverse transform g The Gabor transform Gaussian synthesis window Fourier exponential and integration over f and τ The inverse Gabor transform is a 2D integration over the timefrequency plane.

11 The Continuous Gabor Transform The analysis and synthesis windows need not be Gaussians and are subject only to the condition: g() t γ () t dt = 1 The continuous Gabor transform generalizes the Fourier transform to a nonstationary setting.

12 Gabor Filtering Gabor filtering is implemented by multiplying the Gabor spectrum by a mask or filter function. We show that Gabor filters are nonstationary filters that generalize the concept of pseudodifferential operators.

13 Gabor Filtering The analysis window filter () ( ) ( ) 2 = π if t s t α τ, f V s τ, f e dfdτ α1 g Nonstationary filter transfer function We show that the analysis window filer is a generalized adjoint-form pseudodifferential operator. As such, it is a form of nonstationary convolution.

14 The synthesis window filter Nonstationary filter transfer function () = ( ) ( ) ( ) 2 πif t s t α τ, f sˆ f γ t τ e dfdτ α 2 Gabor Filtering Ordinary Fourier transform We show that the synthesis window filer is a generalized standard-form pseudodifferential operator. As such, it is a form of nonstationary combination.

15 Gabor Filtering These Gabor filters are prototypical end-members of a possible continuum of filter styles. We choose to analyze them for their simplicity.

16 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.

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

18 A discrete Gabor transform The Gaussian summation property. We show the the sum of a discretely spaced set on Gaussians Summation curve on an infinite interval is given by k! ( k τ ) 1 g t = + Gaussian half-width.1 s window increment.05 ( τ) 2cos 2 / [ T / ] 2 πt e π τ +" First-order error term T is the Gaussian half-width τ is the Gaussian spacing

19 A discrete Gabor transform The Gaussian summation property. T is the Gaussian half-width τ Summation curve is the Gaussian spacing T / τ First-order error term Gaussian half-width.1 s.5 window -21 increment decibels decibels decibels -340 decibels The inter-gaussian spacing should well less than the Gaussian half-width.

20 Gaussian summation test Gaussian half-width.1 seconds. T / τ

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

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

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

24 Gabor transform example Reflectivity Q decay curve Wavelet No clipping

25 Gabor transform example Q decay curve Clipping

26 Variations of Gabor Gaussian half-width sec Gaussian spacing 0.05 sec 0.01 sec

27 Stationary Convolution Model () ( ) ( ) s t = w t τ r τ dτ

28 Stationary Convolution Model in the Fourier domain sˆ( f ) = wˆ( f ) rˆ( f )

29 Nonstationary Convolution Model ( ) ( ) ( ) ( ) 2 sˆ f = wˆ f α τ, f r τ e dτ Q (, ) = f / α τ f e π τ Q nonstationary Q πif τ

30 Nonstationary Convolution Model We have proven that ( τ, ) ˆ ( ) α ( τ, ) ( τ, ) Vs f w f f Vr f g Q g Gabor transform of seismic signal is approximately given by Fourier transform of wavelet Q-filter Gabor transform of reflectivity

31 Proof Vs w u r u g u e [ ] du ( τν (, 2 τν,) ) ˆ ( ˆ ν( ) ν) α α( ( ττν, ), ν) (( τ) ( ) ( τ) ) 2πντ i u g πν i u Vs = g = w ru gu e du+ " + " [ ft ] 1 [ ft ] [ T / ] 2 it / 2 it / F δ f e π δ f e π e π τ e π τ + + = + e π τ τ [ τ][ ] [ ] ˆ ( ) πν i t Vs st gt e dt 2 πit' + τ ν f 2πiτ ν f I g t e dt e g f = () ( ) 2πif u 2πif t s t = wˆ ( f) α ( u, f) r( u) e du e df ( ) =, ( ) ( ) ν 2 g τν = τ w = + 2 πν i u Vs g ( τν, ) = wˆ ( ν) α( u, ν) ru ( ) gu ( τ) e du+ " π if t ( ) ˆ τν, ˆ ( ν) ( υ) ˆ( ) ( ) = ˆ ft/ Q ih( ft/ Q t, f e π ( ) α ( = π ν, ) ( ) 2 Vs w f g s f w f t f r t e dt α therefore g ( τν, ) ˆ ( ν) α( τν, ) ( τν, ) Vs = w Vr +" g

32 Nonstationary Convolution Model example Gabor transform of seismic signal

33 Nonstationary Convolution Model example Gabor transform of reflectivity

34 Nonstationary Convolution Model example Q filter transfer function

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

36 Nonstationary Convolution Model example wavelet surface

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

38 Nonstationary Convolution Model example Gabor transform of seismic signal

39 Gabor Deconvolution Gabor transform of the seismic trace Gabor transform of the reflectivity Given this Estimate this Recall the nonstationary trace model: ( τ, ) ˆ ( ) α ( τ, ) ( τ, ) Vs f w f f Vr f g Q g

40 Gabor Deconvolution Vr g ( τ, f) Vs g ( τ, f) ( ) α ( τ f ) wˆ f, Q

41 Gabor Deconvolution So, given this: Vs g ( τ, f) We must estimate this: ( ) α ( τ f ) wˆ f, Q

42 Gabor Deconvolution Estimation of ( ) α ( τ f ) wˆ f, Q the propagating wavelet Can be done by: Smoothing Vs g ( τ, f) Time-variant Burg + smoothing Least squares (, ) modeling of Vs τ f g Lotsa other ways

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

44 Gabor Deconvolution Phase calculation ( ) Let Vsτ, f g sep symbolize a suitably input Gabor transform, then a minimum phase estimate is obtained by the Hilbert transform (, f ) ϕτ = ln Vs, g ( τ f ) f f sep df

45 Gabor deconvolution example nonstationary nonstationary seismic seismic signal signal reflectivity

46 Initial Gabor spectrum

47 Initial Gabor/Burg spectrum

48 Smoothed Gabor spectrum

49 Smoothed Gabor/Burg spectrum

50 Deconvolved Gabor spectrum Fourier algorithm

51 Deconvolved Gabor spectrum Burg algorithm

52 Deconvolved Gabor spectrum Actual reflectivity

53 Gabor deconvolution result Fourier algorithm Gabor deconvolution result reflectivity

54 Gabor deconvolution result Burg algorithm Gabor deconvolution result reflectivity

55 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.

56 Least squares Q and wavelet estimates The nonstationary convolution model predicts g ( τ, ) ˆ ( ) α( τ, ) ( τ, ) Vs f w f f Vr f g This, together with the nearly random nature of suggests that we impose the model Vr g ( τ, f) g (, ) = ˆ ( ) α( τ, ) Vsτ f w f f where the equality is taken to hold in the least-squares sense.

57 Least squares Q and wavelet estimates f τ Ω 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: e π f τ / Q

58 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.

59 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.

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

61 Blackfoot testing Gabor

62 Blackfoot testing Wiener

63 Amplitude Amplitude Blackfoot testing Typical trace spectra 0 db 0 db -10 db -10 db -20 db -20 db -30 db -30 db -40 db -50 db -40 db 0 Hz 20 Hz 40 Hz 60 Hz 80 Hz 100 Hz 120 Hz After Wiener 0 Hz 20 Hz 40 Hz 60 Hz 80 Hz 100 Hz 120 Hz After Gabor

64 Blackfoot testing Gabor/Burg stack

65 Blackfoot testing Wiener stack

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

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

68 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.

69 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.

70 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

71 Q and wavelet estimation Q = π Ω Ω 2 ( ) τ f τ τ f dτdf [ ln ]( ) ln (, ) τ f S f S τ f dτdf. $ π f w( f ) = exp [ ln S] ( f ) + τ ( f ) Ω. Ω Q (, ) (, ) S τ f V s τ f g