Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions Adaptive filtering in wavelet frames: application to echoe (multiple) suppression in geophysics S. Ventosa, S. Le Roy, I. Huard, A. Pica, H. Rabeson, P. Ricarte, L. Duval , M.-Q. Pham, C. Chaux, J.-C. Pesquet IFPEN laurent.duval [ad] ifpen.fr Journ´ ees images & signaux 2014/03/18 1/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 2/44 In just one slide: on echoes and morphing Wavelet frame coefficients: data and model 2 2000 4 1500 Scale 8 1000 16 500 0 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Time (s) 2 2000 4 1500 Scale 8 1000 16 500 0 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Time (s) Figure 1: Morphing and adaptive subtraction required 2/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 3/44 Agenda 1. Issues in geophysical signal processing 2. Problem: multiple reflections (echoes) • adaptive filtering with approximate templates 3. Continuous, complex wavelet frames • how they (may) simplify adaptive filtering • and how they are discretized (back to the discrete world) 4. Adaptive filtering (morphing) • no constraint: unary filters • with constraints: proximal tools 5. Conclusions 3/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 4/44 Issues in geophysical signal processing Figure 2: Seismic data acquisition and wave fields 4/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 5/44 Issues in geophysical signal processing a) Receiver number 1500 1600 1700 1800 1900 1.5 2 2.5 3 Time (s) 3.5 4 4.5 5 5.5 Figure 3: Seismic data: aspect & dimensions (time, offset) 5/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 6/44 Issues in geophysical signal processing Shot number 2200 2000 1800 1600 1400 1200 1.8 2 2.2 2.4 Time (s) 2.6 2.8 3 3.2 3.4 Figure 4: Seismic data: aspect & dimensions (time, offset) 6/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 7/44 Issues in geophysical signal processing Reflection seismology: • seismic waves propagate through the subsurface medium • seismic traces: seismic wave fields recorded at the surface • primary reflections: geological interfaces • many types of distortions/disturbances • processing goal: extract relevant information for seismic data • led to important signal processing tools: • ℓ 1 -promoted deconvolution (Claerbout, 1973) • wavelets (Morlet, 1975) • exabytes ( 10 6 gigabytes) of incoming data • need for fast, scalable (and robust) algorithms 7/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 8/44 Multiple reflections and templates Figure 5: Seismic data acquisition: focus on multiple reflections 8/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 8/44 Multiple reflections and templates a) b) Receiver number Receiver number 1500 1600 1700 1800 1900 1500 1600 1700 1800 1900 1.5 1.5 2 2 2.5 2.5 3 3 Time (s) 3.5 3.5 4 4 4.5 4.5 5 5 5.5 5.5 Figure 5: Reflection data: shot gather and template 8/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 9/44 Multiple reflections and templates Multiple reflections: • seismic waves bouncing between layers • one of the most severe types of interferences • obscure deep reflection layers • high cross-correlation between primaries ( p ) and multiples ( m ) • additional incoherent noise ( n ) • d p t q “ p p t q` m p t q` n p t q • with approximate templates: r 1 p t q , r 2 p t q ,. . . r J p t q • Issue: how to adapt and subtract approximate templates? 9/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 10/44 Multiple reflections and templates −5 Data Model Amplitude 0 5 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Time (s) (a) Figure 6: Multiple reflections: data trace d and template r 1 10/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 11/44 Multiple reflections and templates Multiple filtering: • multiple prediction (correlation, wave equation) has limitations • templates are not accurate • m p t q « ř j h j ˙ r j ? • standard: identify, apply a matching filer, subtract } d ´ h ˙ r } 2 h opt “ arg min h P R l • primaries and multiples are not (fully) uncorrelated • same (seismic) source • similarities/dissimilarities in time/frequency • variations in amplitude, waveform, delay • issues in matching filter length: • short filters and windows: local details • long filters and windows: large scale effects 11/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 12/44 Multiple reflections and templates −5 Data Model Amplitude 0 5 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Time (s) (a) −2 Filtered Data (+) Filtered Model (−) Amplitude −1 0 1 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Time (s) (b) Figure 7: Multiple reflections: data trace, template and adaptation 12/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 13/44 Multiple reflections and templates Shot number Shot number 2200 2000 1800 1600 1400 1200 2200 2000 1800 1600 1400 1200 1.8 1.8 2 2 2.2 2.2 2.4 2.4 Time (s) Time (s) 2.6 2.6 2.8 2.8 3 3 3.2 3.2 3.4 3.4 Shot number Shot number 2200 2000 1800 1600 1400 1200 2200 2000 1800 1600 1400 1200 1.8 1.8 2 2 2.2 2.2 2.4 2.4 Time (s) Time (s) 2.6 2.6 2.8 2.8 3 3 3.2 3.2 3.4 3.4 Figure 8: Multiple reflections: data trace and templates, 2D version 13/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 14/44 Multiple reflections and templates • A long history of multiple filtering methods • general idea: combine adaptive filtering and transforms • data transforms: Fourier, Radon • enhance the differences between primaries, multiples and noise • reinforce the adaptive filtering capacity • intrication with adaptive filtering? • might be complicated (think about inverse transform) • First simple approach: • exploit the non-stationary in the data • naturally allow both large scale & local detail matching ñ Redundant wavelet frames • intermediate complexity in the transform • simplicity in the (unary/FIR) adaptive filtering 14/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 15/44 Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] { H t f up ω q “ ´ ı sign p ω q p f p ω q 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −4 −3 −2 −1 0 1 2 3 Figure 9: Hilbert pair 1 15/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 15/44 Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] { H t f up ω q “ ´ ı sign p ω q p f p ω q 1 0.5 0 −0.5 −4 −3 −2 −1 0 1 2 3 Figure 9: Hilbert pair 2 15/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 15/44 Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] { H t f up ω q “ ´ ı sign p ω q p f p ω q 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −4 −3 −2 −1 0 1 2 3 4 Figure 9: Hilbert pair 3 15/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 15/44 Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] { H t f up ω q “ ´ ı sign p ω q p f p ω q 3 2 1 0 −1 −2 −4 −3 −2 −1 0 1 2 3 Figure 9: Hilbert pair 4 15/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 16/44 Continuous & complex wavelets 0.5 0.5 0 0 −0.5 −0.5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Real part Imaginary part 0.5 0 −0.5 0.5 0 −0.5 2 3 0 1 −2 −1 −3 Imaginary part Real part Figure 10: Complex wavelets at two different scales — 1 16/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 17/44 Continuous & complex wavelets 0.5 0.5 0 0 −0.5 −0.5 −5 0 5 −5 0 5 Real part Imaginary part 0.5 0 −0.5 0.5 0 −0.5 6 8 2 4 −4 −2 0 −8 −6 Imaginary part Real part Figure 11: Complex wavelets at two different scales — 2 17/44
Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions 18/44 Continuous wavelets • Transformation group: affine = translation ( τ ) + dilation ( a ) • Basis functions: ˆ t ´ τ ˙ 1 ? aψ ψ τ,a p t q “ a • a ą 1 : dilation • a ă 1 : contraction • 1 {? a : energy normalization • multiresolution (vs monoresolution in STFT/Gabor) Ñ ? a Ψ p af q e ´ ı 2 πfτ ψ τ,a p t q FT Ý 18/44
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