I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION A constrained-based optimization approach for seismic data recovery problems ICASSP 2014 — SS5: Seismic signal processing M-Q. PHAM, L. DUVAL* IFP Energies nouvelles 1 et 4 av. de Bois-Pr´ eau, 92852 Rueil-Malmaison - France C. CHAUX Univ. Aix-Marseille, I2M UMR CNRS 7373 39 rue F. Joliot-Curie, 13453 Marseille - France J-C. PESQUET Univ. Paris-Est, LIGM UMR CNRS 8049 5 bd Descartes, 77454 Marne-la-Vall´ ee - France 4-9 May 2014 1 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION The fast way ◮ Here: signal = primary; disturbance = multiples ◮ Use of approximate disturbance models (given) ◮ similar to acoustic echo cancellation, speech dereverberation, pattern matching. . . ◮ imperfect models: adaptation (and regularization) required ◮ Two aims: ◮ to recover highly under-determined signals ◮ to alleviate and fasten hyper-parameter determination ◮ An entwined approach: ◮ pragmatic: with geophysically-sound functions/models ◮ heuristic: explore other suitable penalties/constraints ◮ Focus on the “sparsity” term (signal wavelet coefficients) 2 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Outline I NTRODUCTION C ONTEXT Seismic data Problem formulation G ENERAL SCHEME MAP estimation Hyperplane constraints R ESULTS Synthetic data Real data C ONCLUSION 3 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Seismic data propagation & acquisition Hydrophone Towed streamer • • • • • • • Primaries + multiple reflection disturbances 4 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Multiple problem formulation z ( n ) s ( n ) y ( n ) + b ( n ) ¯ + ¯ = ���� ���� ���� ���� observed signal multiple noise primary where ◮ n : time index (somehow related to underground depth) ◮ z = ( z ( n ) ) 0 ≤ n < N : observed data, combining: 1. primary ¯ y = (¯ y ( n ) ) 0 ≤ n < N (signal of interest, unknown) 2. multiples (¯ s ( n ) ) 0 ≤ n < N (sum of undesired bouncing signals) ◮ b = ( b ( n ) ) 0 ≤ n < N : additive noise realization. 5 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Availability of different approximate models ◮ Several ( J ) models (or templates) r ( n ) are given j ◮ Imperfect in time, amplitude and frequency s ( n ) and corrected ◮ Assumption: models linked to ¯ throughout time-varying (FIR) adaptive filters p ′ + P j − 1 J − 1 � � s ( n ) = h ( n ) ( p ) r ( n − p ) ¯ ¯ j j p = p ′ j = 0 where h ( n ) ◮ ¯ : unknown impulse response of the filter j corresponding to model j and time n ( P j tap coefficients) 6 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Model examples s ( n ) ¯ r ( n ) 1 r ( n ) 2 100 200 300 400 500 600 700 800 900 1000 Simulated Multiple, 1st model, 2nd model 7 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Model examples Receiver number Receiver number a) b) 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 Real data Model 7 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Multiple problem reformulation ¯ ¯ z = R h + y + b ���� ���� ���� ���� observed signal filter noise primary where s = � J − 1 � ¯ s ( N − 1 ) � ⊤ j = 0 R j ¯ h j = R ¯ ◮ ¯ s ( 0 ) h = · · · ¯ � � ⊤ � � and ¯ ¯ 0 · · · ¯ h ⊤ h ⊤ ◮ R = R 0 · · · R J − 1 h = J − 1 � ◮ ¯ h ( 0 ) ¯ ( p ′ ) · · · ¯ h ( 0 ) ( p ′ + P j − 1 ) · · · h j = j j � ⊤ · · · ¯ h ( N − 1 ) ( p ′ ) · · · ¯ h ( N − 1 ) ( p ′ + P j − 1 ) j j ◮ R j is a block diagonal matrix. 8 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Maximum-a-Posteriori (MAP) estimation ◮ Assumption : ¯ y is a realization of a random vector Y , whose probability density is given by: ( ∀ y ∈ R N ) f Y ( y ) ∝ exp ( − ϕ ( y )) ◮ Assumption : ¯ h is a realization of a random vector H , whose probability density can be expressed as: ( ∀ h ∈ R NP ) f H ( h ) ∝ exp ( − ρ ( h )) ◮ Assumption : b is a realization of a random vector B , of probability density: ( ∀ b ∈ R N ) f B ( b ) ∝ exp ( − ψ ( b )) B is assumed to be independent from Y and H 9 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Maximum-a-Posteriori (MAP) estimation MAP estimation of ( y , h ) � � minimize ψ z − Rh − y + ϕ ( y ) + ρ ( h ) . y ∈ R N , h ∈ R NP ���� � �� � ���� a priori on the signal a priori on the filters fidelity: linked to noise 9 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Constraints Assumption : Gaussian noise, B ∼ N ( 0 , σ 2 ) � w ∈ R N | � w � 2 ≤ N σ 2 � ψ = ι S where S = Assumption : slow variations of adaptive FIR filters along time � � | h ( n + 1 ) ( p ) − h ( n ) ∀ ( j , n , p ) ( p ) | ≤ ε j , p ( hyperslab C ) j j Assumption : limits on (functions of) primary coefficients Fy ∈ D , where F ∈ R K × N : analysis frame operator 10 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Constraints Assumption : Gaussian noise, B ∼ N ( 0 , σ 2 ) � w ∈ R N | � w � 2 ≤ N σ 2 � ψ = ι S where S = Assumption : slow variations of adaptive FIR filters along time � � | h ( n + 1 ) ( p ) − h ( n ) ∀ ( j , n , p ) ( p ) | ≤ ε j , p ( hyperslab C ) j j Assumption : limits on (functions of) primary coefficients Fy ∈ D , where F ∈ R K × N : analysis frame operator MAP estimation of ( y , h ) � � y ∈ R N , h ∈ R NP ϕ ( y ) + ρ ( h ) + ι S z − Rh − y + ι C ( h ) + ι D ( Fy ) . minimize 10 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Hard constraints on primaries y : convex set D F : analysis frame operator, with L subbands ( x k = ( Fy ) k ) D = D 1 × · · · × D L ∀ ℓ ∈ { 1 , . . . , L} 1. D ℓ = { ( x k ) k ∈ K ℓ | � k ∈ K ℓ φ ℓ ( x k ) ≤ β ℓ } , where, β ℓ ∈ ] 0 , + ∞ [ , φ ℓ : R card ( K ℓ ) → [ 0 , + ∞ [ is a proper lsc convex function. 2. D ℓ = { ( x k ) k ∈ K ℓ | � k ∈ K ℓ φ ℓ (( Lz ) k ) x k = β ℓ } , where φ ℓ : R → R , L ∈ R N × N : linear operator, e.g. a Wiener-like filter � ( 1 + λ 1 + λ 2 � R ( 0 ) � 2 ) − 1 , . . . , L = λ 1 Diag ( 1 + λ 1 + λ 2 � R ( N − 1 ) � 2 ) − 1 � ( λ 1 , λ 2 ) ∈ ] 0 , + ∞ [ 2 , R ( n ) : n -th row of matrix R . 11 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Qualitative results 1.5 1 0.5 0 −0.5 −1 −1.5 0 100 200 300 400 500 600 700 Observed signal z (red; σ = 0 . 01), original y (blue). D is the intersection of two hyperplanes defined from the identity (Id) and the sign functions. 12 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Qualitative results 1.5 1 0.5 0 −0.5 −1 −1.5 0 100 200 300 400 500 600 700 Estimated signal ˆ y (magenta), original signal y (blue). D is the intersection of two hyperplanes defined from the identity (Id) and the sign functions. 12 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Qualitative results 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0 100 200 300 400 500 600 700 Estimated multiples ˆ s (magenta), original multiples s (blue). D is the intersection of two hyperplanes defined from the identity (Id) and the sign functions. 12 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Quantitative results σ 0 . 01 0 . 04 φ α SNR y SNR s α SNR y SNR s 0 0.4 23.98 15.79 0.9 15.03 9.60 ℓ ∞ 0.6 24.48 15.81 0.8 16.24 8.69 ℓ 2 0.6 25.59 16.02 0.8 17.84 9.20 ℓ 1 0.4 25.98 16.16 0.9 18.19 6.61 sign 0.3 24.43 14.73 0.1 14.75 4.58 Id 0.4 26.19 15.81 0.2 19.74 8.84 Id + sign 0.3 26.40 15.56 0.1 18.43 5.94 Upper table part: “classical constraints” and lower table part: hyperplane contraints. ( ϕ = ( 1 − α ) ℓ 1 , ρ = αℓ 2 ) 13 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Real data Portion of a common receiver gather: (left) recorded seismic data with a partially appearing primary (right) multiple wavefield template. 14 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Real data Subtraction results, low field-noise case: primaries (separated from multiples) with (left) [Ventosa et al. 2012] (right) with the proposed method. 14 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Real data Subtraction results, high field-noise case: primaries (separated from multiples) with (left) [Ventosa et al. 2012] (right) with the proposed method. 14 / 17
I NTRODUCTION C ONTEXT G ENERAL SCHEME R ESULTS C ONCLUSION Conclusion: in symbols minimize ( 1 − α ) ϕ ( y ) + αρ ( h ) y ∈ R N , h ∈ R Q ψ ( z − y − Rh ) ≤ 1 subject to h ∈ C Fy ∈ D with easier bounds inferred from data & coarser processing 15 / 17
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