Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
Adaptive filtering in the complex wavelet domain with unary filters: application to multiple suppression in geophysics On echoes and morphing Laurent Duval IFP Energies nouvelles, Rueil-Malmaison, France
[email protected]
January 22, 2013
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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On echoes and morphing 2 2000
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Figure 1: ... and adaptive subtraction
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Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Agenda 1. Issues in geophysical signal processing 2. Problem: multiple reflections (echoes) • adaptive filtering with approximate models
3. Complex, continuous wavelets • and how they (may) simplify adaptive filtering
4. Discretization, redundancy and unary filters (morphing) • being practical: back to the discrete world
5. Results 6. Conclusion
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Issues in geophysical signal processing
Figure 2: Seismic data acquisition and wave fields. 4/33
Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Issues in geophysical signal processing Shot number 1.8
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Figure 3: Seismic data: aspect & dimensions (time, offset) 5/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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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: • ℓ1 -promoted deconvolution (Claerbout, 1973) • wavelets (Morlet, 1975) • exabytes (106 gigabytes) of incoming data • need for fast, scalable algorithms
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models
Figure 4: Seismic data acquisition: focus on multiple reflections
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models 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(t) = p(t) + m(t)+n(t) • model-based multiple attenuation: x1 (t), x2 (t), x3 (t) • how to use approximate models?
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models
Amplitude
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Figure 5: Multiple reflections: data trace d and model x1
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Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models Multiple filtering: • multiple prediction (correlation, wave equation) has limitations • models are not accurate • m(t) ≈ ak (t)xk (t − τk (t))? • standard: identify, apply a matching filer, subtract • primaries and multiples are not (fully) uncorrelated • same (seismic) source • similarities/dissimilarities in time • similarities/dissimilarities in frequency • variations in amplitude, waveform, delay • issues in matching filter length: • short filters and windows: local details • long filters and windows: large scale effects 10/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models Amplitude
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Figure 6: Multiple reflections: data trace, model and adaptation 11/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models Shot number 2200
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Figure 7: Multiple reflections: data trace and models, 2D version 12/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Multiple reflections and models • 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)
• Main idea here: • exploit the non-stationary in the data • naturally allow both large scale & local detail matching • work in a complex domain: amplitude and phase representation • emulate an analytic signal representation (Hilbert transform)
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⇒ Complex, continuous wavelets • intermediate complexity in the transform • hyper-simplicity in the (unary) adaptive filtering
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets 0.5
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Figure 8: Complex wavelets at two different scales - 1
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Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets 0.5
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Figure 9: Complex wavelets at two different scales - 2
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Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets • Transformation group:
affine = translation (τ ) + dilation (a)
• Basis functions:
1 ψτ,a (t) = √ ψ a • • • •
t−τ a
a > 1: dilation a< √1: contraction 1/ a: energy normalization multiresolution (vs monoresolution in STFT) FT
ψτ,a (t) −→ 16/33
√ aΨ(af )e−ı2πf τ
Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets • Definition
Cs (τ, a) =
Z
∗ s(t)ψτ,a (t)dt
• Vector interpretation
Cs (τ, a) = hs(t), ψτ,a (t)i projection onto time-scale atoms (vs time-frequency) • Redundant transform: τ → τ × a “samples” • Parseval-like formula
Cs (τ, a) = hX(f ), Ψτ,a (f )i ⇒ time-scale domain operations! (cf. Fourier) 17/33
Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets Introductory example
Data
Real part
Modulus
Imaginary part
Figure 10: Noisy chirp mixture in time-scale & sampling 18/33
Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Continuous wavelets Noise spread & feature simplification 4 2 0 −2 −4
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Figure 11: Noisy chirp mixture in time-scale: scale, zoomed wiggle 19/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Continuous wavelets Amplitude
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Figure 12: Which morphing is easier: time or time-scale? 20/33
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Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Continuous wavelets • Inversion with another wavelet φ
s(t) =
ZZ
Cs (u, a)φu,a (t)
duda a2
⇒ time-scale domain processing! (back to the signal) • Scalogram |Cs (t, a)|2 • Energy conversation
E= • Parseval-like formula
hs1 , s2 i = 21/33
ZZ
ZZ
|Cs (t, a)|2
dtda a2
Cs1 (t, a)Cs∗2 (t, a)
dtda a2
Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Continuous wavelets • Wavelet existence: admissibility criterion
0 < Ah =
Z
+∞ 0
Z 0 b∗ b ∗ (ν)Ψ(C) Φ (ν)Ψ(ν) Φ dν = dν < ∞ ν ν −∞
generally normalized to 1 • Induces band-pass property:
• necessary condition: |Φ(0)| = 0, or zero-average shape • amplitude spectrum neglectable w.r.t. |v| at infinity
• examples: Morlet-Gabor (non. adm.)
ψ(t) = √ 22/33
1 2πσ 2
t2
e− 2σ2 e−ı2πf0 t
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Discretization, redundancy and unary filters Being practical again: deal with discrete signals • Can one sample in time-scale (CWT):
Cs (τ, a) =
Z
∗ s(t)ψτ,a (t)dt,
1 ψτ,a (t) = √ ψ a
t−τ a
with cj,k = Cs (kb0 aj0 , aj0 ), (j, k) ∈ Z and still be able to recover s(t)? • Result 1 (Daubechies, 1984): there exists a wavelet frame if
a0 b0 < C, (depending on ψ). A frame is generally redundant
• Result 2 (Meyer, 1985): there exist an orthonormal basis for a
specific ψ (non trivial, Meyer wavelet) and a0 = 2 b0 = 1
Now: how to choose the practical level of redundancy? 23/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Discretization, redundancy and unary filters 0.15 primary multiple noise sum
true multiple adapted multiple 0.1
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Figure 13: Redundancy selection with variable noise experiments 24/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Discretization, redundancy and unary filters • Complex Morlet wavelet:
ψ(t) = π −1/4 e−iω0 t e−t
2 /2
, ω0 : central frequency
• Discretized time r, octave j, voice v: v ψr,j [n] = √
1 2j+v/V
ψ
nT − r2j b0 2j+v/V
, b0 : sampling at scale zero
• Time-scale analysis:
X v v [n] d = dvr,j = d[n], ψr,j [n] = d[n]ψr,j n
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Discretization, redundancy and unary filters 2
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Figure 14: Morlet wavelet scalograms, data and models
Take advantage from the closest similarity/dissimilarity: • remember the wiggle: on sliding windows, at each scale, a single complex coefficient compensates amplitude and phase 26/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Discretization, redundancy and unary filters • Windowed adaptation: complex aopt compensates local
delay/amplitude mismatches: aopt
2
X
ak x k = arg min d −
{ak }(k∈K) k
• Vector Wiener equations for complex signals:
hd, xm i = • Time-scale synthesis:
ˆ = d[n]
k
ak hxk , xm i
XX r
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X
j,v
v dˆvr,j ψer,j [n]
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Results
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Figure 15: Wavelet scalograms, data and models, after unary adaptation
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Results (reminders)
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Figure 16: Wavelet scalograms, data and models
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Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Results Shot number 1.8
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Figure 17: Original data 30/33
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Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
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Results Shot number 1.8
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Figure 18: Filtered data 31/33
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Results & conclusion
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Conclusions Take-away messages: • Technical side • • • • •
Take good care of cascaded processing Non-stationary, wavelet-based, adaptive multiple filtering “Complex” wavelet transform + simple one-tap (unary) filter Redundancy selection: noise robustness and processing speed Smooth adaptation to adaptive joint multiple model filtering
• Practical side • Industrial integration • Competitive with more standard processing • Alternative results: less sensitive to random noises • Future work: better integrate incoherent noise 32/33
Context
Multiple filtering
Continuous wavelets
Discret., redundancy, unary filters
Results & conclusion
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Acknowledgements & references
Ventosa, S., S. Le Roy, I. Huard, A. Pica, H. Rabeson, P. Ricarte, and L. Duval, 2012, Adaptive multiple subtraction with wavelet-based complex unary Wiener filters: Geophysics, 77, V183–V192. Jacques, L., L. Duval, C. Chaux, and G. Peyr´e, 2011, A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity: Signal Process., 91, 2699–2730.
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