Fighting the Curse of Dimensionality: Compressive Sensing in Exploration Seismology

Transcription

1 Fighting the Curse of Dimensionality: Compressive Sensing in Exploration Seismology Herrmann, F.J.; Friedlander, M.P.; Yilmat, O. Signal Processing Magazine, IEEE, vol.29, no.3, pp Andreas Gaich, Andrea Zabaznoska Advanced Signal Processing 1, Seminar Dec 10, 2012 Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 1/68

2 Outline Introduction Mathematical Background of CS The Wavelet-Domain Reflection Seismology Acquisition Schemes Seismic Wavefield Representation Compressive Seismic Computation and Imaging SPGL1 Solver Full Waveform Inversion and PGS Pareto Curves Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 2/68

3 Introduction On Compressive Sensing I Shortcomings of the typical workflows Current seismic techniques rely on massive data volumes as all conducted experiments produce enormous amounts of data. Most of the data consists of reflected energy within a frequency content of roughly [5-100] Hz. Moving into geologically more complex areas of the Earth which makes the correction for wavepaths along which the reflected data traveled impossible (geometry optics approximation of wave propagation). Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 3/68

4 Introduction On Compressive Sensing II Current trends Alternative sampling strategy that leverages recent insights from compressive sensing (CS) towards seismic data acquisition and processing. Introducing CS as a novel nonlinear sampling paradigm, a randomized dimensionality reduction approach effective for acquiring signals that have a sparse representation in some transform domain. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 4/68

5 Introduction On Compressive Sensing III Current trends (cont d) We need a workflow, a framework. We need to come up with sub-nyquist sampling schemes whose sampling is proportional to the sparsity of the problem and not to the dimensionality. Luckily audio, image and seismic signals admit sparse approximations, i.e. they can be well approximated by linear superposition and nonlinear recovery algorithms, such as the l1 norm. l0, l1, l2 norm Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 5/68

6 Introduction On Compressive Sensing VI Compressive sensing (CS) l1 norm Sparsity promotion Wavelets, curvelets, noiselets Exploration seismology Seismic data acquisition Seismic imaging Convex optimization Pareto curves Seismic inversion Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 6/68

7 The Math Behind CS I x R N where x 0 k y = Ψ x... Ψ is an nxn matrix n << N (#msrs.<<ambient dims.) Init. assumptions (1) x... is a high-dimensional signal Ψ... is an full-rank nxn matrix y... non-adaptive linear measurements Goal: obtain x from non-adaptive linear measurements y Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 7/68

8 The Math Behind CS II b = Ψ z where z is sparse minimize z 1 s.t. Ψ z = b Ψ... measurement matrix Sparse recovery problem (2) Goal: find x as the solution of b = Ψ z 1 st Problem: b = Ψ z has infinitely many solutions. 2 nd Problem: Sensitivity to the sparsity assumption. 3 rd Problem: Sensitivity to additive noise, thus not useful in practice. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 8/68

9 The Math Behind CS III Theorem 1 Suppose Ψ is an nxn Gaussian random matrix. If n k log( N n ) then with overwhelming probability we can recover all k-sparse x from y = Ψ x. Problem It is naive to expect signals in practice to be sparse. More realistically, the magnitude of the coefficients decays rapidly and the coefficient vector contains only few significant entries. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 9/68

10 The Math Behind CS IV Theorem 2 b = Ψ z + e, e 2 ɛ minimize z 1 subject to Ψ z b 2 ɛ x... the solution for z (3) Corollary x x 2 C 1 ɛ + C 2 k 1 2 σk (x) n = O(k log N n ) σ k (x)... the best k-term approx. error (4) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 10/68

11 The Math Behind CS V It has been empirically shown that seismic signals and CS yield small recovery errors, even in scenarios with much degraded sampling ratios. The recovered result is within the noise level and nearly as accurate as the approximation we would obtain by measuring directly the largest k entries of x. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 11/68

12 Solving the Sparse-Optimization Problem The sparsest solution of a severely underdetermined linear system can be recovered exactly by seeking the minimum one-norm (l1 ) solution. The l1 norm finds sparse solutions (whiteboard) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 12/68

13 Meanwhile in the Wavelet-Domain... I In general seismic signals admit sparse approximations in terms of curvelets: f R N f = S H x and S P XN, P N Init. assumptions b = Ψ f = Ψ S H x (5) f... a compressed data vector S H... the superscript denotes the adjoint Goal: Choose Ψ for a given S s.t. Ψ S H is a good measurement matrix. (whiteboard) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 13/68

14 Meanwhile in the Wavelet-Domain... II Problem What happens if the sparsifying dictionary is over-complete: the columns of the S H matrix are correlated? there are infinitely many x that explain the same signal f? S P xn... P > N? Solving the sparse optimization problem A universal strategy for choosing Ψ that doesn t require prior knowledge of the sparsity basis S: if we choose Ψ to be appropriate random measurement matrix, then Ψ S H is guaranteed to also be a good measurement matrix independent of S. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 14/68

15 Meanwhile in the Wavelet-Domain... III Bottomline: Many random sensing matrices are universally incoherent with any fixed basis with very high probability. The smaller the coherence between the randomly chosen matrix and the fixed basis, the fewer the samples required. This matches numerical and practical experience. Promote sparsity as a prior via one-norm regularization to overcome the singular nature of S. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 15/68

16 Reflection Seismology - Introduction Proceeding Generatation of Seismic source signal Seismic waves are reflected at the layers of the subsurface Reflected waves are measured by receivers at the ground Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 16/68

17 Reflection Seismology - Introduction - cont d Characteristic Acoustic Impedance Seismic waves travel in Earth at a speed governed by the Acoustic Impedance Z 0 = ρ c Material Property Reflection occurs at the boundary between two materials with different Characteristic Acoustic Impedances R = Z 1 Z 0 Z 1 + Z 0 Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 17/68

18 Reflection Seismology - Introduction - cont d Reflection at normal and non-normal incidence Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 18/68

19 Reflection Seismology - Introduction cont d Sources Should ideally only emit P-Waves Impulse f.e. Explotions, Airgun, Earthquakes Source hardly predictible Sweep Vibrator truck Source exactly known Correlation at the receivers necessary Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 19/68

20 Reflection Seismology - Introduction cont d Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 20/68

21 Acquisition Schemes Receiver Land receiver are so called Geophones consider only vertical movement of the earth Common Gather Types Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 21/68

22 Traditional Acquisition Encountered to the traditional Nyquist Theorem to avoid Spatial Aliasing Needs for higher resolution images leads to exponentially increasing costs Acquisition of spatio-temporal wavefield in up to five dimensions f(t, x) L 2 ((0, T ] [ L, L]) T in the order of seconds; L in the order of kilometer Sampling intervalls in the order of milliseconds and meters Dimension reduction needed Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 22/68

23 Compressive Sensing Approach Combines sampling and coding in one single step by a randomized subsampling technique Encoding is linear and does not require access to high-resolution data during encoding Based on: Randomized Sampling Sparsifying Transforms Sparsity-promotion recovery by convex optimization Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 23/68

24 Examples Periodic versus uniformly random subsampling If signals permits a sparse transform-domain representation it suffices to sample at a rate that is lower than Nyquist. Recover signals from far fewer randomly placed samples In Seismology: Use seismic arrays with fewer geophones selected uniformly random from regular sampling grids with spacings defined by Nyquist This turns coherent aliases into Gaussian white noise Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 24/68

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26 In Geophysical: subsampling-related artifacts are commonly known as spectral leakage Depend on the degree of subsampling Characteristics depend on the irregularity of the sampling Remove noisy artifacts by sparse-recovery procedures Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 26/68

27 Seismic Wavefield Representation Sparse Signal representation concerning CS Leverage structure within signals to reduce sampling Look for transform-domains that concentrate the signals energy in a few number of coefficients Consider transforms that are fast, multiscale and multidirectional Appropriate Transforms Curvelet Transform Wave Atom Transform Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 27/68

28 Wavelet Transformation Generalizes the Fourier transform by using a basis that represents both location and spatial frequency Define orthogonal basis functions as dilations and translations of Mother functions also called analyzing wavelets Φ (sl) (x) = 2 s 2 Φ(2 s x l) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 28/68

29 Wavelet Transformation - cont d Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 29/68

30 Wavelet Transformation - cont d Wavelets do not utilize geometric properties of wavefields Curved structures as superposition of Multiscale Fat Dots Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 30/68

31 Curvelet Transformation Extension of wavelets by additional Orientation in Localization ( ) ϕ j,l,k (x) = ϕ j R Θl (x x (j,l) k ) [ cos Θl sin Θ R Θl = l sin Θ l cos Θ l ], x (j,l) k = R 1 Θ l (k 1 2 j, k 2 2 j/2 ) Θ l = 2π 2 j/2 l, l = 0, 1,..., 0 Θ l < 2π Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 31/68

32 Curvelet Transformation - cont d Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 32/68

33 Curvelets Waveatoms Both obey a so called parabolic scaling Curvelets: needle-like shapes Wave Atoms: Oscillatory patterns Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 33/68

34 Performance measure for transforms Approximation Error real signals are not strictly sparse but transfer-domain coefficients often decay rapidly Orthonormal basis: Decay rate directly linked to the decay of the nonlinear approximation error σ(k) = f f k 2 This does not hold for redundant transforms Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 34/68

35 Alternative Approximation Error Based on solution of a sparsity promoting program min x 1 subject to S H x = f To account for different redundancies plot SNR as a function of the sparsity ratio ρ = k/p SNR(ρ) = 20 log f f ρ f Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 35/68

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38 Subsampling of shots Aim Breaking the periodicity of coherent sampling Oppurtunities Selections of subsets of sources Design of incoherent simultaneous-source experiments Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 38/68

39 Subsampling of shots - cont d Measurement basises Sequential-source acquisition I = I Ns I Nt Simultaneous-source acquisition Incomplete data M = G Ns I Nt R = R ns I Nt n s N s Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 39/68

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41 Subsampling of shots cont d Coherence depends on the interplay between restriction, measurement and synthesis matrices Another performance measure where δ = n/n, SNR(δ) = 20 log f f δ f f δ = S H x δ, x δ = argmin x 1 subject to A δ x = b A δ = R δ M δ S H Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 41/68

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43 Compressive Seismic Computation Overview Compressive Simulation Compressive Imaging Compressive Inversion SPGL1 - Spectral Projected-Gradient for L1 norm Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 43/68

44 Compressive Simulation Aim: Simulation of P-Waves Traditionally solved with the time-harmonic Helmholtz equation large linear system of PDEs that discretizes the underlying wave equation Use linearity in the sources to reduce the number of sequential shots into a small number of supershots complexity reduction from O(n 4 ) to O(n 3 log n) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 44/68

45 Compressive Imaging Motivation Locate mineral and oil sources Building Industry Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 45/68

46 The Seismic Image Problem Requires inversion of the linearized time-harmonic Born-scattering matrix Seismic data is decomposed through Fourier transform into monochromatic wavefields minimize b Ax 2 2 = K b i A i x 2 2 i=1 b C N f N rn s, x R M, K = N f N s Each iteration needs solutions of a big set of PDEs Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 46/68

47 Solution by batching Take Mini-Batches with K K monochromatic supershots Solve reduced system minimize b Ax 2 2 = b j A j x 2 2 K i=1 b j = K w ij b i, A j = i=1 K w ij A i i=1 Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 47/68

48 Solution by batching - cont d Sub-selection and mixing in the source-frequency space: RM(b Ax) 2 2 R = R Σ I R Ω, M = M Σ I I R Σ R n s n s, R Ω R n f n f, M R (KNr( (KNr) Solution reduced by a factor of K /K Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 48/68

49 Solution by sparsity promotion Averaging is not able to remove the source crosstalk efficiently use transform-domain sparsity promotion Additionaly: Redraw supershots at each iteration Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 49/68

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51 SPGL1 Solver large-scale sparse reconstruction solver Basis pursuit minimize x 1 subject to Ax = b Basis pursuit denoise minimize x 1 subject to Ax b 2 σ Lasso minimize Ax b 2 subject to x 1 τ Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 51/68

52 SPGL Solver - cont d Basis pursuit denoise Solve BP σ by a sequence of subproblems LS τ At each iteration k refine τ k such that τ k τ σ Derive sequence of estimates τ k by applying Newton s method to find the root of value function φ(τ) = σ Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 52/68

53 SPGL Solver - cont d LS τ solved by Spectral projected gradient P τ [b] := {argmin b x 2 subject to x 1 τ} Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 53/68

54 SPGL Solver - cont d Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 54/68

55 Full Waveform Inversion FWI I Most industry practice is based on a geometric optics approximation of the wave propagation. A smooth velocity model that describes the propagation speed of the waves in the subsurface (lack of knowledge). FWI relies on modeling the data by solving the wave equation (PDE), and adapting the model parameters in order to minimize the data misfit based on the LS-optimization. Minimization of the data misfit requires low wavenumbers. And yes, this is now becoming possible! Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 55/68

56 Full Waveform Inversion FWI II Ghost-free seismic data Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 56/68

57 Full Waveform Inversion FWI III More Wavenumbers Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 57/68

58 Full Waveform Inversion FWI IV More Wavenumbers (cont d) Broadband frequency signals propagating through the earth and the ocean. The wavenumber of any seismic signal is related in a rather complex manner to its seismic frequency, the seismic velocity and the offset between source and receiver locations, recorded along a towed streamer. Removing the receiver ghost and deep streamer towing allows recording data in a low noise environment. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 58/68

59 Full Waveform Inversion FWI V More Wavenumbers (cont d) Wavefield separation during processing allows building high resolution velocity models and high resolution imaging. LOW WAVENUMBERS: data rich in low frequencies. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 59/68

60 Full Waveform Inversion FWI VI Wavefield Separation Two interfering wavefields: up-going wavefield scattered upwards from the earth and the time-delayed version that is reflected downwards from the sea surface. Key differentiation technique of GeoStreamer. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 60/68

61 Full Waveform Inversion FWI VII Underdetermined system of linear equations: f = S H x A = RMS H Ax = b minimize x 0 s.t. Ax = b l 0... #of nonzero elements Basic problem (6) This cannot be approximated by a convex approximation problem. One of the major findings of CS is that under some conditions on A and x, the solution can be recovered by solving the convex optimization problem. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 61/68

62 Full Waveform Inversion FWI VIII Basis Pursuit (BP): minimize x 1 s.t. Ax = b (7) Recovers the correct sparse signal depending on the sparsity level of x, the #of measurements and the restricted isometry property (RIP) of A. The RIP constant measures how far the matrix A is from a unitary matrix when acting on sparse vectors (A A = I). The columns of an unitary matrix form an orthonormal basis, as well as the rows. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 62/68

63 Full Waveform Inversion FWI IX Quadratic Programming (QP): minimize 1 2 Ax b λ x 1 (8) Basis Pursuit Denoise Problem (BPDP): minimize x 1 s.t. b Ax 2 σ (9) LASSO (LS): min 1 2 b Ax 2 s.t. x 1 τ (10) For each σ, there are unique values of λ and τ so that the solutions for QP and LASSO coincide. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 63/68

64 Full Waveform Inversion FWI X LASSO (cont d) The estimate of the one norm of the solution (τ) is seldom available for geophysical problems, but is a key internal problem used by Spectral Projected Gradient Method (SPGL). The SPGL can be used to quickly solve the LS equation for very large linear systems and it has already been proven to be very successful in solving large-scale CS problems in seismic exploration. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 64/68

65 Pareto Curves I Pareto Curves: Trace the optimal trade-off between the data misfit and some prior model. Are commonly used in problems with two-norm priors and recently explored in terms of one-norm regularization. Are convex and decreasing, i.e. regular. This regularity means it is possible to obtain a good approximation to the curve with very few interpolation points. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 65/68

66 Pareto Curves II Pareto Curves (cont d) Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 66/68

67 References I F.J. Herrmann, M.P. Friedlander, O. Yilmaz, Fighting the curse of dimensionality: compressive sensing in exploration seismology, IEEE Signal Processing Magazine, vol. 29, no. 3, pp , E.J. Candes, M.B. Wakin, An Introduction to Compressive Sampling, IEEE Signal Processing Magazine, vol. 25, no. 2, pp , R. Baraniuk, Compressive Sensing, Lecture Notes in IEEE Signal Processing Magazine, vol. 24, pp A. Aravkin, X. Li, F.J. Hermann, Fast Seismic Imaging For Marine Data, The University of British Columbia Technical Report., October J. Ma, G. Plonka, The Curvelet Transform, IEEE Signal Processing Magazine, vol. 27, no. 2, pp , March J. Ma, Characterization of textural surfaces using wave atoms, Applied Physics Letters, vol. 90, no. 26, June Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 67/68

68 References II E.v.d. Berg, M.P. Friedlander, Probing the Pareto Frontier for Basis Pursuit Solutions, The University of British Columbia Technical Report., May E. Hager, Full Azimuth Seismic Acquisition with Coil Shooting, 8. Biennial International Conference and Exposition on Petroleum Geophysics Hyderabad, F.J. Herrmann, X. Li, A.Y. Aravkin, T. Leeuwen, A modified, sparsity promoting, Gauss-Newton algorithm for seismic waveform inversion, Dept. of Earth and Ocean Sciences, University of British Columbia. F.J. Herrmann, Randomized sampling and sparsity: getting more information from fewer samples, The University of British Columbia Technical Report, TR , G. Hennenfent, E.v.d. Berg, M.P. Friedlander, F.J. Herrmann, New Insights into one-norm solvers from the Pareto curve, Geophysics, vol. 73, no. 4, pp. A23 A26. Andreas Gaich, Andrea Zabaznoska Dec 10, 2012 page 68/68