Random Coding for Fast Forward Modeling

Transcription

1 Random Coding for Fast Forward Modeling Justin Romberg with William Mantzel, Salman Asif, Karim Sabra, Ramesh Neelamani Georgia Tech, School of ECE Workshop on Sparsity and Computation June 11, 2010 Bonn, Germany J. Romberg (GaTech) Random Fast Forward Bonn 10 1 / 23

2 Forward simulations as acquisition problems Fast forward modeling for seismic exploration Given a candidate model of the earth, simulate a field acquisition to infer the responses between each source/receiver pair Matched field processing (matched filtering) for acoustic source localization Quickly test different source locations in a complicated environment J. Romberg (GaTech) Random Fast Forward Bonn 10 2 / 23

3 Seismic imaging,+.)/0#&'1)2*34'(4#2)*"4-'!"#$%#&"' ()"*"+,")-' y k p 1 p 2 p 3 5/)1%' h 1,k h 2,k h 3,k J. Romberg (GaTech) Random Fast Forward Bonn 10 3 / 23

4 Forward modeling/simulation Given a candidate model of the earth, we want to estimate the channel between each source/receiver pair p 1 p 4 p 2 p 3 ""!""""""""""%""""""""""".""""""""""/" #$#" h :,1 h :,2 h :,3 h :,4 1456(643*")76*8" '()*"+,-" %$&" 0*1*(2*0",*3",()9843*6"793:93" J. Romberg (GaTech) Random Fast Forward Bonn 10 4 / 23

5 Simultaneous activation Run a single simulation with all of the sources activated simultaneously with random waveforms The channel responses interfere with one another, but the randomness codes them in such a way that they can be separated later #"!%" ""!""""""""""%""""""""""".""""""""""/" p 1 p 2 p 3 p 4! '()*"+,-" #$#" %$&" h :,1 h :,2 h :,3 h :,4 0*1*(2*0",*3" #" 4*5167(589"!%" y 1:57(7:3*")67*;" 0*1*(2*0",*3" J. Romberg (GaTech) Random Fast Forward Bonn 10 5 / 23

6 Multiple channel linear algebra m y k = G 1 G 2 h G 1,k p h 2,k.!"#$$%&'()*(( &%$+,"((n!)$-)&./)$(01,"(2.&'%( p j h c,k How long does each pulse need to be to recover all of the channels? (the system is m nc, m = pulse length, c =# channels) Of course we can do it for m nc J. Romberg (GaTech) Random Fast Forward Bonn 10 6 / 23

7 Restricted isometries for multichannel systems m y k = G 1 G 2 h G 1,k p h 2,k.!"#$$%&'()*(( &%$+,"((n!)$-)&./)$(01,"(2.&'%( p j y k = Φh k h c,k Theorem: With each of the pulses as iid Gaussian sequences, Φ obeys when (1 δ) h 2 Φh 2 2 (1 + δ) h 2 2 s-sparse h R nc m C δ s log 5 (nc) + n Consequence: we can separate the channels using short random pulses (using l 1 min or other sparse recovery algorithms) J. Romberg (GaTech) Random Fast Forward Bonn 10 7 / 23

8 Multichannel theory F = DFT matrix, G i = diagonal matrices of iid Gaussians We can write and 2 F Φ Φ = 6 4 I Φ Φ = c k=1 F m Φ = F [ G 1 F G 2 F G c F ] G 1 G 1 G 1 G 2 G 1 Gc 3 2 F G 2 G 1 G 2 G 2 G 2 Gc F G c G 1 G c G 2 G c Gc ω=1(1 g k (ω) 2 )f k,ω f k,ω + j k F... F m g k (ω)g j (ω)f k,ω fj,ω ω=1 This is a sum of rank-1 matrices; can control action on s-sparse signals using tools closely related to Rudelson and Vershynin s uniform operator law of large numbers J. Romberg (GaTech) Random Fast Forward Bonn 10 8 / 23

9 Seismic imaging simulation (a) Simple case. (b) More complex case. Array of (8192) sources activated simultaneously (1 receiver) Sparsity enforced in the curvelet domain urce and receiver geometry. We use 8192 (128 64) sources and 1 receiver. Figure 2: Desired band-limited Greens s functions obtained by sequential-source modeli (a) Simple case and (b) More complex case. J. Romberg (GaTech) Random Fast Forward Bonn 10 9 / 23

10 Seismic imaging simulation (a) Estimated (16x faster, SNR=9.6 db). (b) Estimation error (Figure 2b minus 5(a)) (c) Cross-correlation estimate. Result produced with 16 compression in the computations Figure 5: Simulation results for the more complex Green s function and the random impulsive-source Can evenapproach take this example down to 32 J. Romberg (GaTech) Random Fast Forward Bonn / 23

11 MIMO channel estimation!"#$%&'()"%* -.-/*+0#$$)1* ")+)',)"%* j k Estimate all channel responses h j,k = between source j and receiver k Activating with diverse source signatures allows us to separate the cross-talk Reduces the total amount of time we spend probing the channels Other applications: underwater/wireless MIMO channel eq. MIMO radar imaging, etc. J. Romberg (GaTech) Random Fast Forward Bonn / 23

12 Application: Increased field-of-view with coded apertures Architecture proposed by Marcia et al 08 J. Romberg (GaTech) Random Fast Forward Bonn / 23

13 Acoustic source localization We measure the response through a known, complicated channel Source located a γ, the response is g γ Matched Field Processing: we test each location λ by correlating the measurements against a simulated response max g τ, g γ = G T g γ τ Given g τ, time reversal can be used to calculate G T g τ J. Romberg (GaTech) Random Fast Forward Bonn / 23

14 Complicated channel response Complicated channel proximate locations might have totally uncorrelated responses The slice G T G γ of G T G has a main lobe at γ, then is random-looking away from γ m m J. Romberg (GaTech) Random Fast Forward Bonn / 23

15 Localization model Example (1D, synthetic) slice f 0 of G T G Model this as Gaussian + side lobes Find the max using a matched filter * = With F = {shifts of a Gaussian}, we are solving arg min f F f f J. Romberg (GaTech) Random Fast Forward Bonn / 23

16 Multiple realizations Framework: we receive a series of measurement vectors g γ1, g γ2,..., g γl for the same environment γ i = possible different source locations Calculating Gx or G T y can be expensive: it requires the solution of a large PDE A naïve approach given observations g γi : test by calculating G1 τ = g τ for all τ on a grid; re-use calculations at each time step Idea: we can use ideas from compressive sampling to significantly reduce the amount of computation required (Think: every G1 τ is an expensive measurement...) J. Romberg (GaTech) Random Fast Forward Bonn / 23

17 Coded simulations Pre-compute the responses to a dense set of randomly and simultaneously activated sources b 1 = Gφ 1, b 2 = Gφ 2,..., b K = Gφ K Given observations g γ, correlate with the b i and find the row of Φ which is closest to this set of correlations Correlate b 1, g γ b 2, g γ y =. = ΦGT g γ = Φf 0 b K, g γ J. Romberg (GaTech) Random Fast Forward Bonn / 23

18 General mathematical framework Given a signal f 0 and a class of signals F, we want to find the closest point in F to f 0 ˆf1 = arg min f F f 0 f 2 2 But...we are only given y = Φf 0. We solve instead ˆf 2 = arg min f F y Φf 2 2 = arg min f F Φ(f 0 f) 2 2 If F=shifts of the same function, than this is the smashed filter (Davenport et. al 09) Q: When are the solutions ˆf 1 and ˆf 2 close to one another? A: When Φ preserves the distances between g γ and all points in F. J. Romberg (GaTech) Random Fast Forward Bonn / 23

19 Preserving distances Set F 0 = F g γ Fact (BDDW 08) Suppose the for any fixed f F 0 we have P { Φf 2 2 f 2 2 > δ f 2 2} γ(δ). Then { P sup f F 0 Φf 2 2 f 2 2 } > δ f 2 2 2N δ/4 (F 0 ) γ(δ/2) γ( ) = tail bound (concentration function) N δ (F 0 ) = covering number for F 0 For subgaussian Φ, we have γ(δ) e C mδ2 Need to estimate the covering numbers... J. Romberg (GaTech) Random Fast Forward Bonn / 23

20 Net of Gaussians Distance between two Gaussians, width σ, shifts τ 1, τ 2 when f(t τ 1 ) f(t τ 2 ) 2 τ 1 τ 2 2σɛ ɛ This gives us an easy estimate for the size of the best ɛ-net: N ɛ (F 0 ) T 8σɛ T = length of interval you are searching over J. Romberg (GaTech) Random Fast Forward Bonn / 23

21 How many tests to guarantee accuracy? Theorem: The functions f g γ 2 and Φ(f g γ ) 2 are within δ of each other (w/ probability p) uniformly when ( ) ( ) ) T 1 K Const δ (log 2 + log + C. σ p!" #$!%&'" ()*"+#)',-./0" # " For complicated channels we just need δ Const J. Romberg (GaTech) Random Fast Forward Bonn / 23

22 Demo Grid resolution n 3000, number of coded experiments k ambiguity function 20 coded localization (independent) (G T g γ )(λ) (Φ T ΦG T g γ )(λ) J. Romberg (GaTech) Random Fast Forward Bonn / 23

23 Summary Two problems: forward modeling for seismic imaging source localization in complicated channels Both have similar computational bottlenecks (simulations require solving large PDEs) Randomly coding the inputs allows us to push through these computational bottlenecks more efficiently Analogous to expensive acquisitions made easier with compressive sensing J. Romberg (GaTech) Random Fast Forward Bonn / 23