Edge preserved denoising and singularity extraction from angles gathers

Transcription

1 Edge preserved denoising and singularity extraction from angles gathers Felix Herrmann, EOS-UBC Martijn de Hoop, CSM

2 Joint work Geophysical inversion theory using fractional spline wavelets: ffl Jonathan Kane (ERL-MIT) Geophysical inversion theory by Radon transforms ffl Daniel Trad (EOSC-UBC)

3 Objectives Application of new data analysis and estimation techniques: detect and characterize singularities. ffl solve inverse problems preserving the ffl singularities/edges. Optimal basis functions and non-linear estimation

4 Leading edge This year s Leading edge: Edge-preserving smoothing and applications by Yi Luo et. al. and Fast structural interpretation with structure-oriented filtering by Cristian Höcker et. al.

5 Denoising example

6 Denoising example

7 Inversion One is much better set to solve an inversion problem when one is able to sparsely represent data and model! Sparse representation and non-linear thresholding improve app. error by an order of magnitude ffl Approximation () Estimation ffl increase SNR error by an order of magnitude

8 ^f = (I + flc 1 ) 1 h Estimation Denoising problem: h = f + n ffl linear estimation by generalized least-squares: Problem: edges are NOT preserved! Can we come up with an edge-preserving method?

9 ^f = X T (hh; ψ i)ψ Approaches ffl non-linear estimation by iterative methods a ffl non-linear estimation by thresholding b : ffl T ( ) threshold operator. ffl only works when ψ s are local and optimal. a Claerbout, Scales, Ulrych, Sacchi, Trad. b Donoho [1995], Candès and Donoho [2001] and Jonathan Kane, this conference.

10 Denoising example

11 Denoising example

12 Denoising Example Generalized Least Squares Denoising noisy measurement denoised measurement with γ=1 denoised measurement with γ=10 denoised measurement with γ=50 original

13 Denoising Example Non smooth wavelet thresholding with δ= noisy measurement denoised measurement original

14 Wavelet s miracle a : Wavelets represent piece-wise continuous functions at virtually no additional cost! no prior info. on location discontinuities! ffl ffl non-adaptive and optimal basis functions local in time and frequency a ffl David Donoho.

15 How does it work? Optimal basis functions For piece-wise continuous functions: wavelets are orthogonal w.r.t. the smooth ffl polynomial part (vanishing mom. property). ffl smooth part ends up in a few scaling coefficients. non-smooth jumps and noise end up in a few ffl wavelet coefficients. Noise and data separate!

16 Denoising Example 1 Thresholding with Θ=

17 Problem When zooming out wavelet coefficients of (blue) reflection events decay faster then those of the (white) noise. They do not stand a chance! Exploit ffl reflectivity is relative smooth along reflectors. ffl distinct waveforms in the singular direction.

18 Our approach Design a redundant dictionary of parametric seismic waveforms/geologic patterns: ffl very sparsely and adaptively represent reflectivity ffl bear geologic relevance Data-adaptive Matching pursuit ffl to find coherent structures in noise a to characterize reflector s abruptness a ffl [Mallat, 1997]

19 Imaged Seismic Reflectivity Common Image Gathers: ux(z; e) = K Λ (e)d(x; z) ffl ideal domain to denoise. ffl singularities are preserved a. ffl singularities are aligned after migration. data is piece-wise smooth along the angular ffl direction. a Brandsberg-Dahl and de Hoop [1998], de Hoop and Brandsberg- Dahl [2000]

20 Seismic Imaging We separate noise from imaged reflectivity non-data adaptive thresholding in angular direction. ffl impose smoothness. ffl preserve discontinuities for critical reflectivity. ffl data-adaptive atomic decomposition in the spatial ffl (vertical) direction ffl locate reflectors (stratigraphy) ffl characterize reflectors (geology)

21 Simple Stack very noisy data

22 Simple Stack estimated original Simple Stack

23 Non-linear denoised waveform estimated original Wavelet Packet estimated wavelet

24 ff 8 <, χ 0 x» 0 x ff (x) + : Parametric transition model a ffl Fractional Splines (ff+1) x > 0 Fractional Spline Wavelets ffl Multifractals ffl Precise control over scaling! a [Gel fand and Shilov, 1964, Herrmann, 1997, Unser and Blu, 2000, Herrmann, 2001, Herrmann et al., 2001]

25 Causal B-splines α = 0 : 0.2 :

26 f (x), X n2n Generalized subsurface model We assume superposition of n + χff n + (x x n) c fractional spline ffl representation. have to locate the (x knots n ffl ) have to estimate the (ff orders n ffl ) have to estimate the (c magnitudes ) n ffl

27 Parametric representation of geologic transitions transition with α [0,1] Reflectivity with Ricker

28 Causal Orthogonal Fractional Spline 1 Wavelets 0.5 alpha = 0 : 0.2 :

29 Valhall

30 Common Image Gather depth angle

31 Common Image Gather after denoising depth angle

32 8 original denoised

33

34 8 denoised atom. denoised

35 original stack

36 denoised stack

37 atom stack with the first 100 atoms

38 Amplitude Attribute

39 Order Attribute

40 Observations ffl Singularities are detected. ffl Some noise is removed. ffl Singularities are characterized. ffl Stratigraphy partially resolved.

41 Common Image Gathers: Observations ffl are too smooth in the radial direction. ffl smoothing due to stacking and filtering. ffl damage has already been done. ffl is noise really a problem? Only find out if we conduct noisy experiments!

42 Bottom line Remaining challenge is to detect singularities on curves! Design basis functions align themselves with reflectors. ffl do not require prior info on location and dip. ffl are directional selective. ffl explore relative smoothness along reflectors. ffl

43 Ridglets a Orthogonal basis functions on lines radial position (intercept time) and angle ffl (ray-parameter) ffl radial and angular scale ffl directional selective To detect events in SNR = 1 noise! ffl optimal for slant events a Donoho [1998], Candès and Donoho [2001]

44 Ridglets data 1 noisy data wavelet denoised data (500 coeff) ridgelet denoised data (20 coeff)

45 Ridgelets Single Slant Event

46 Ridgelets Single Slant Event with std

47 Ridgelets Non linear Radon + thresholding with 10 coef. yielding std

48 Ridgelets Wavelet thresholding with 1000 coef. yielding std

49 Ridgelets Ridgelet thresholding with 1000 coef. yielding std

50 References S. Brandsberg-Dahl and M. de Hoop. Focusing in dip and ava compensation on scattering-angle/azimuth common image gathers. Geophysics, E. J. Candès and D. L. Donoho. Recovering Edges in Ill-posed Problems: Optimality of Curvelet Frames. Technical report, Department of Statistics, Stanford University, M. de Hoop and S. Brandsberg-Dahl. Maslov asymptotic extension of generalized radon transform inversion in anisotropic elastic media: a least-squares approach. Inverse problems, 16(3): , 2000.

51 D. Donoho. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. App. and Comp. Harmonic Analysis, 2, D. Donoho. Orthonormal ridgelets and linear singularities. Technical report, Department of Statistics, Stanford University, I. M. Gel fand and G. E. Shilov. Generalized functions, volume 1. Academic press, F. Herrmann. A scaling medium representation, a discussion on well-logs, fractals and waves. PhD thesis, Delft University of Technology, Delft, the Netherlands, URL

52 F. J. Herrmann. Singularity characterization by monoscale analysis. Appl. Comput. Harmon. Anal., 11(4):64 88, July F. J. Herrmann, W. Lyons, and C. Stark. Seismic facies characterization by monoscale analysis. Geoph. Res. Lett., 28(19): , Oct URL felix/preprint/wellseis.ps.gz. S. G. Mallat. A wavelet tour of signal processing. Academic Press, M. Unser and T. Blu. Fractional splines and wavelets. SIAM Review, 42 (1):43 67, 2000.