Multichannel Deconvolution of Layered Media Using MCMC methods

Multichannel Deconvolution of Layered Media Using MCMC methods Idan Ram Electrical Engineering Department Technion Israel Institute of Technology Supervisors: Prof. Israel Cohen and Prof. Shalom Raz

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

geophones source Introduction An acoustic wave is transmitted into the ground. The reflected energy resulting from impedance changes is measured. The observed trace can be modeled as a noisy convolution between a 2D reflectivity and an unknown wavelet. seismic trace Deconvolution is used to remove the effect of the wavelet

Problem Formulation.3 seismic wavelet The seismic trace Y is modeled as the convolution: Y=h*R+W. h - unknown D seismic wavelet, invariant in both horizontal and vertical directions. R - 2D reflectivity section, consist of continuous, smooth and mostly horizontal layer boundaries. W - white Gaussian noise independent 2 from R with zero mean and variance. σ w.2. -. -.2 -.3 -.4 -.5 -.6 5 5 2 25 5 reflectivity 5 2 3 4 5 6 7 8 9 The blind deconvolution problem consists in recovering the unknown seismic wavelet h and the 2D reflectivity section R from the observed seismic trace Y.

Goals Previous works: Blind Marine Seismic Deconvolution Using Statistical MCMC Methods O. Rosec, J. M. Bouceher, B. Nsiri, and T. Chonavel Multichannel seismic deconvolution Goals: J. Idier and Y. Goussard. Combine the two methods above into a blind stochastic multichannel deconvolution scheme. 2. Create a smoothing version of the proposed multichannel scheme.

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

Single Channel Deconvolution The multichannel deconvolution problem can be broken into independent D vertical deconvolution problems. Single channel blind deconvolution consists in recovering the D reflectivity sequence r and the wavelet h from a D observed trace y. qk ( ) rk ( ) D reflectivity In the vertical direction, a D reflectivity signal appears as a sparse spike train. The reflectivity sequence can be modeled as a Bernoulli-Gaussian (BG) process: 2 pqk ( ( ) = ) = λ, prk ( ( )) = λn(, σ ) + ( λδ ) ( rk ( )).5.5 D trace -.5-2 3 4 5 6 7 8 9

ML Parameter Estimation 2 2 The parameters θ= ( h, λσ,, σ w ) need to be estimated. This is carried out using ML estimation: θˆ = argmax ln p( y θ) This is an incomplete data problem: ML This maximization problem is solved using the stochastic expectation maximization (SEM) algorithm. θ θˆ = argmax ln p( rqy,, θ) ML θ

The Gibbs Sampler Suppose we wish to sample a random vector x=(x,,x n ) according to f(x) Gibbs Sample algorithm: A. For a given x t, generate y=(y,,y n ) as follows: ) Draw y from the conditional pdf f(x x t,2,, x t,n ). 2) Draw y i from the conditional pdf f(x i y,,y i-,x t,i+,, x t,n ). 3) Draw y n from the conditional pdf f(x n y,,y t,n- ). B. Let x t+ =y. Under mild conditions, the limiting dist. of the process {x t, t=,2, } is precisely f(x)

The Gibbs Sampler (cont.) Simulates observations of q and r from p(r,q y) The algorithm samples from: p( r( k), q( k) yq,, r ) ~ Bi( λ ) NmV (, ) Gibbs Sampler algorithm:. Initialization: choice of q () and r (). 2. For i I and for k=, N r Detection step: compute λ k =p(q(k)= y,q -k,r -k ) simulate q (i) (k)~ Bi(λ k ) Estimation step: k k k q () () = q () (2) = q () (3) = q () (4) = simulate r (i) (k) ~N(m,V ) if q (i) (k)=, otherwise r (i) (k)= ( ( ) ( ) q, r ) r () () = r () (2) = r () (3) = r () (4) = 2 ( ( ) ( ) q, r )

The SEM algorithm The SEM algorithm follows the steps:. Initialization: choice of r (),q (),θ () 2. For i=,,i: 3. θˆ E step: simulation of r (i),q (i) by the Gibbs sampler according to p(r,q y,θ (i-) ) M step: parameters estimation: ˆ () i () i () i argmax (,, ) θ = p r q y θ θ I = θ I I i= I + () i

The Deconvolution Process Map estimation: This maximization problem can be solved in two steps: Detection: Estimation: rˆ = ( rq ˆ, ˆ) = argmax p( rq, y) N The detection problem is hard because q has discrete configurations. A simpler criterion called maximum posterior mode (MPM) which maximizes p(r(k),q(k) y) is used instead. rq, qˆ = argmax p( q y) q argmax p( r yq, ˆ) r 2 r

The MPM algorithm The MPM algorithm follows the steps:. For i=,,i simulate (r (i), q (i) ) using the Gibbs sampler 2. For i=,,n r detection step: I () i if q ( k) >.5 qk ˆ( ) = I I i= I + otherwise estimation step: rk ˆ( ) = I () i () i q kr k i= I + ˆ = I i= I + ( ) ( ) q () i ( k), if qk ( ), otherwise

The MPM algorithm (cont.) () q (2) q (3) q (4) q (5) q () r 4-2 -2.5 4 (2) r 4.5-2 (3) r 4 (4) r 3.5 -.5 (5) r ˆq 4-2 ˆr

Reflectivity Post Processing Fuse and replace by their gravity center:. 2 successive impulses 2. 2 impulses separated by one sample ( rˆ, q ˆ ) ( rˆ, qˆ )

Advantages And Limitations Advantages:. Estimates the wavelet and BG model s missing parameters. 2. Produces good estimates for single channel traces. 3. Uses stochastic methods which are less likely to converge to local maximum points. Limitations. Does not account for the medium s stratified structure in the deconvolution process.

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

MBG-I Model Exact 2D extension of the BG representation. Comprised of:. MBRF p(t,q) models the geometric properties of the reflectivity using location and transition variables. 2. p( R TQ, ) - white Gaussian reflectivity amplitude model, defined conditionally to the MBRF. (,) Boundary representation Location variables q k, q k, t t k, / k, / t k, t k, q k, q k, + q k, + t \ k, \ t k, Transition variables q k +, q k +, + k

Markov Bernoulli Random Field Characteristics MBRF characteristics:. 2. 3. 4. 5. Amplitude field characteristics:. 2. pt (, t, t ) = pt ( ) pt ( ) pt ( ) k, k, k, k, k, k, q ~ Bi( λ ), t ~ Bi( µ ), t ~ Bi( µ ), t ~ Bi( µ ) / / \ \ k, k, k, k, λ = ( µ )( µ )( µ )( ε) p( t =, t =, t = q = ) = r k, = k, k, k, k, pq ( = t =, t =, t = ) = ε k, k, k, k+, r ~ N(, σ ) 2 k, 3. r = ar + w, w ~ N(, ( a ) σ ) 2 2 k, k+ d, k r r

Deconvolution Scheme MAP estimation: Computing the exact MAP solution is practically impossible. The following suboptimal recursive maximization procedure is used instead:. First column: 2 J 2. : ( rˆ, qˆ ) = argmax p( r, q, y ) r, q ( rˆ, qˆ, tˆ, tˆ, tˆ ) = RQT ˆ ˆ ˆ Tˆ Tˆ = RQT T T Y (,, /,, \ ) argmax p(,, /,, \ ) RQT,,, T, T arg,,,, r q t t t ax m p( r, q, t, t, t, y rˆ, qˆ ) Each partial criterion is maximized using a suboptimal SMLR type algorithm.

Deconvolution Scheme (cont.) q ˆ rˆ tˆ, tˆ, tˆ ˆq 2 2 tˆ tˆ tˆ ˆq ˆr 3 3 ˆr 2, 2, 2

Advantages And Limitations Advantages:. Produces good estimates of the 2D reflectivity Limitations:. Non blind. 2. Each partial criterion is maximized only with respect to / \ r, q, t, t, t. r 3. is determined based on observations only up to y. 4. The SMLR-type algorithm may converge to a local optimum. 5. The first reflectivity column is assumed to be known.

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

Blind Multichannel MCMC Deconvolution Uses the MBG-I reflectivity model. Uses the following suboptimal recursive maximization procedure :. First column: 2 J 2. : ( rˆ, qˆ ) = argmax p( r, q y ) ( rˆ, qˆ, tˆ, tˆ, tˆ ) = argmax p( r, q, t, t, t y, rˆ, qˆ ),,,, r q t t t The SMLR type algorithm is replaced by an extended version of the MPM algorithm which maximizes: p( t, t, t, q, r y, qˆ, rˆ ) k, k, k, k, k,

MBG I Parameters Estimation 2 2 The parameters θ= ( h, λσ, are estimated using the SEM, σ w ) algorithm. The following method is used to estimate θ = (,,,, ): MBG a µ µ µ ε. apply single channel deconvolution to each of Y s columns. 2. remove all the isolated reflectors from the obtained reflectivity section. 3. calculate: μ / = #boundary upward transitions/# samples in T / μ - = #boundary upward transitions/# samples in T - μ \ = #boundary upward transitions/# samples in T \ a = average attenuation ratio between neighboring reflectors

Multichannel Deconvolution s Gibbs Sampler Used by the extended MPM algorithm. Simulates observations of r, q, t, t, t from p( r, q, t, t, t y, rˆ, qˆ ) The algorithm samples from:. ( ) p( r ˆ ˆ k,, q,,,,,,, )~ ( ), b k, y r k, q k, r q t t t Bi λk, N mb Vb ( ) p( t t, t, t, q, r, rˆ, qˆ, y )~ Bi µ / / k, k, ( ) p( t t, t, t, q, r, rˆ, qˆ, y )~ Bi µ k, k, ( ) p( t t, t, t, q, r, rˆ, qˆ, y )~ Bi µ / \ k, k,

Multichannel Deconvolution s Gibbs Sampler s Algorithm. Initialization: choice of r, q, t, t, t () () / () () \ () 2. For i I and for k=, N r Detection step: compute simulate b λ k,, µ, µ, µ t ~ Bi( µ ), t ~ Bi( µ ), /() i / -() i k, k, \ () i \ () i b k, ~ ( µ ), k, ~ ( λk, ) t Bi q Bi transition variables t ˆ t ˆ t ˆ qˆ rˆ q ˆ ˆ,r,, Estimation step: simulation of ( ) r, ~ N m, V () i k b b () i () i if q = and r = if k, k, q = () i k,

Multichannel Deconvolution s MPM Algorithm The extended MPM algorithm follows the steps:. () i () i /() i () i \() i For i=,,i simulate r, q, t, t, t using the Gibbs sampler 2. For i=,,n r detection step: I I /() i -() i / tk, t k,, i I, ˆ I I I I k = = + tk, = i= I+ tˆ \ k, if >.5 if >.5 otherwise otherwise I \ ( i) if,.5 ˆ tk > t = I I, qˆ = i= I + otherwise k, I () i if qk, >.5 I I i= I + otherwise

Multichannel Deconvolution s MPM Algorithm (cont.) estimation step: rˆ k, = I i= I + ˆ = I i= I + q r () i () i k, k, q () i k,, if q k,, otherwise / () t / (2) t t / (3) t / (4) / (5) t ˆ/ t () r (2) (3) (4) (5) r r r r 4 4.5-2 -2.5-2 4 3.5 -.5 ˆr 4-2

Synthetic Data Results Reflectivity.3 Wavelet.2. 5 -. -.2 -.3 -.4 -.5 5 2 3 4 5 6 7 8 9 -.6 5 5 2 25 Seismic trace, SNR=5 db Seismic trace, SNR= db 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

Synthetic Results Estimated Parameters.3 Estimated wavelet (5 db).3 Estimated wavelet ( db).2.2.. -. -. -.2 -.2 -.3 -.3 -.4 -.5 real estimated -.4 -.5 real estimated -.6 5 5 2 25 -.6 5 5 2 25 λ σ / σ w a µ µ µ \ ε True.55.32.999.66.399.8.2 Estimated (5 db).533.223.94.833.8.222.96.46 Estimatd ( db).54.925.657.877.82.38.9.69

Synthetic Data Results, SNR=5 db Single channel deconvolution, SNR=5 db Multichannel deconvolution, SNR=5 db 5 5 5 2 3 4 5 6 7 8 9 5 2 3 4 5 6 7 8 9

Synthetic Data Results, SNR= db Single channel deconvolution, SNR= db Multichannel deconvolution, SNR= db 5 5 5 2 3 4 5 6 7 8 9 5 2 3 4 5 6 7 8 9

Synthetic Results Performance Quality Measures The following loss functions were used to quantify the performance of the algorithms:. 2. 3. 4. L = rˆ r + N + N L L L miss+ false miss false miss false SSQ = rˆ r + N = rˆ r + N = rˆ r 2 2 miss false miss+ false miss false 5. L2, L2, L2 give partial credit to reflectors that are close to their true positions. SNR (5dB) L miss+flase L miss L false L SSQ L 2 miss+flase L 2 miss L 2 false 5 SC 52.3 44.3 365.3 276. 293.7 235.2 86.2 MC 372.6 3.6 265.6 72.9 26. 73.6 38.6 SC 243.9 92.9 75.9 6.8 4.5 9.5 92.5 MC 96.3 49.3 45.3 75.4 23.6 92.6 89.6

Real Data Estimated Parameters Estimated wavelet.5.4.3.2. -. -.2 -.3 5 5 2 25 3 35 4 45 λ σ / σ w a µ µ µ \ ε.382 3.9488.724.8989.5.2.8.252

Real Data Results Real data Single channel deconvolution Multichannel deconvolution

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

Deconvolution By Smoothing Uses the MBG-I reflectivity prior model. Uses the same parameter estimation method as the first proposed algorithm. Uses a column recursive deconvolution scheme, similar to the one of the previous algorithm. Accounts for observation columns subsequent to y in r s estimation process.

Smoothing Scheme 2 4 6 8 2 2 4 6 8 2 4

Smoothing Windows qˆ 2 () q ˆ (2) (3) qˆ q qˆ 2 qˆ () (2) qˆ q

Smoothing Scheme (cont.) q q + r r + t t t q r t q r t q + r +

Deconvolution Scheme Uses the following estimation procedure: First column: ( rˆ, qˆ ) = argmax p( r, q, t, t, t y ) r q t t t,,,, Middle column: ( rˆ, qˆ, t ˆ, t ˆ, t ˆ ) = - - - argmax p( r, q, t, t, t y, qˆ, rˆ ),, -, -, - r q t t t - - - Last column: ( rˆ, qˆ, tˆ, tˆ, tˆ ) = J J J J J argmax p( r, q, t, t, t y, qˆ, rˆ ) J, J, J, J, J r q t t t J J J J J J J J Uses a further extended version of the MPM algorithm which maximizes: p( r, q, t, t, t y, qˆ, rˆ ) k, k, k, k, k,

Smoothing s Gibbs Sampler Used by the smoothing s MPM algorithm. Simulates observations of r, q, t, t, t from p( r, q, t, t, t y, rˆ, qˆ ). The algorithm samples from: p( r, q y, r, q, rˆ, r, qˆ, q, t, t, t, t, t, t k, k, m k, k, k, + + (, ) ~ Bi( λ ) N m V m m ) The multichannel Gibbs sampler s conditional distributions

Smoothing s Gibbs Sampler Algorithm. Initialization: choice of 2. For i I. for k=, N r r, q, t, t, t () () / () () \ () rˆ ˆ,q r ˆ,qˆ rˆ + ˆ,q + t, t, t t, t, t detection step: compute simulate estimation step: m λ k,, µ, µ, µ t ~ Bi( µ ), t ~ Bi( µ ), /() i / -() i k, k, t ~ Bi( µ ), q ~ Bi( λ ) \() i \ () i m k, k, k, () i r k, simulation of where () i if q k, = () i and r = if k, ( ) r, ~ N m, V () i k b b () i q k, = 2. for k+=2n r follow the multichannel Gibbs sampler procedure

Smoothing s MPM algorithm The smoothing s MPM algorithm follows the steps: () i () i /() i () i \() i. For i=,,i simulate r, q, t, t, t using the Gibbs sampler 2. For i=,,n r detection step: t t I I /() i -() i if tk >.5 if tk >.5 = I I i= I+, t k = I I i= I+ otherwise otherwise /,, k,, I I \ ( i) i if tk >.5 if,, q I I i= I + q I I k, = i I otherwise otherwise \, k, = = + () k >.5

Smoothing s MPM algorithm (cont.) estimation step: r I () i () i qk, rk, i= I +, if q k, = I = i q k, i= I +, otherwise k, ()

Smoothing s MPM algorithm (cont.) ( ) ( 2) q q ( 3) ( 4) ( 5) ( ) q q q r ˆq ˆq 4-2 -2.5-2 4 r ( 2) r 4.5-2 -2.5-2 4.5 ( 3) r 4 4 ( 4) r 3.5 -.5 3.5 -.5 ( 5) ˆr 4-2 4-2 ˆr

Synthetic Data Results, SNR=5 db Multichannel deconvolution, SNR=5 db Deconvolution by smoothing, SNR=5 db 5 5 5 2 3 4 5 6 7 8 9 5 2 3 4 5 6 7 8 9

Synthetic Data Results, SNR= db Multichannel deconvolution, SNR= db Deconvolution by smoothing, SNR= db 5 5 5 2 3 4 5 6 7 8 9 5 2 3 4 5 6 7 8 9

Synthetic Results Performance Quality Measures SNR (5dB) L miss+flase L miss L false L SSQ L 2 miss+flase L 2 miss L 2 false 5 SC 52.3 44.3 365.3 276. 293.7 235.2 86.2 MC 372.6 3.6 265.6 72.9 26. 73.6 38.6 Smoothing 35.4 252.4 227.4 36.4 87.7 5.2 25.2 SC 243.9 92.9 75.9 6.8 4.5 9.5 92.5 MC 96.3 49.3 45.3 75.4 23.6 92.6 89.6 Smoothing 53.8 4.8 4.8 4. 9.5 8.5 8.5

Real Data Results Single channel deconvolution Multichannel deconvolution Smoothing

OUTLINE. Introduction 2. Blind Seismic Deconvolution Using MCMC Methods 3. Multichannel Seismic Deconvolution 4. Blind Multichannel MCMC Deconvolution 5. Deconvolution By Smoothing 6. Conclusions

Conclusions We presented two stochastic blind multichannel deconvolution algorithms. The proposed parameter estimation method successfully recovers the MBG I model's parameters. Both proposed algorithms produce better deconvoltuion results than the single channel blind deconvolution method. The smoothing algorithm produces better deconvoltuion results than the proposed multichannel algorithm. The performance of both blind deconvoltuion schemes improves as the SNR increases.

Future Research Replacing the MBG I model by the MBG II model. Analyzing the smoothing algorithm s performance for larger smoothing windows Using wavelet estimation methods which can estimate both wavelet s length and maximum position.