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Stable seismic data recovery Herrmann, Felix J. 2007

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Stable seismic data recovery Felix J. Herrmann* joint work with Peyman Moghaddam*, Gilles Hennenfent* & Chris Stolk (Universiteit Twente) *Seismic Laboratory for Imaging and Modeling slim.eos.ubc.ca AIP 2007, Vancouver, June 26  Combinations of parsimonious signal representations with nonlinear sparsity promoting programs hold the key to the next-generation of seismic inversion algorithms ... Since they allow for formulations that are stable w.r.t.     noise incomplete data moderate phase rotations and amplitude errors  Finding a sparse representation for seismic data & images is complicated because of     wavefronts & reflectors are multiscale & multidirectional the presence of caustics, faults and pinchouts the presence of operators (FIO’s & PsDO’s)  The seismic method  Seismic data acquisition 0  time [s]  1  2  3  4  -3000  offset [m] -2000  -1000  Exploration seismology 0 1  Depth (km)  2  • • •  create images of the subsurface  3 4 5  need for higher resolution/deeper 6 clutter and data incompleteness are problems  7 0  1  2  3 km  Exploration seismology 0 1  Depth (km)  2  • • •  create images of the subsurface  3 4 5  need for higher resolution/deeper 6 clutter and data incompleteness are problems  7 0  1  2  3 km  Forward problem F [c]u :=  • •  d  1 ∂ ∂ · − c2 (x) ∂t2 i=1 ∂x21 2  2  second order hyperbolic PDE interested in the singularities of  m = c − c¯  u(x, t) = f (x, t)  Inverse problem Minimization:  m = arg min d − F [m] m  2 2  After linearization (Born app.) forward model with noise: d(xs , xr , t) = Km (xs , xr , t) + n(xs , xr , t)  Conventional imaging:  K d (x) = T  y(x) =  K Km (x) + K n (x) T  Ψm (x) + e(x)  Ψ is prohibitively expensive to invert requires regular sampling ...  T  Sparsity promoting inversion  Formulate as inverse problem signal  y  =  A  +  n  noise  x0 curvelet representation of ideal data  ˜ = arg min x x x  1  sparsity enhancement  s.t.  Ax − y  2  ≤  data misfit  When a traveler reaches a fork in the road, the l1 -norm tells him to take either one way or the other, but the l2 -norm instructs him to head off into the bushes. John F. Claerbout and Francis Muir, 1973 New field “compressive sampling”: D. Donoho, E. Candes et. al., M. Elad etc. Preceded by others in geophysics: M. Sacchi & T. Ulrych and co-workers etc.  Sparsity promoting inversion x0 can be recovered by solving  P :  ˜ = arg minx x x ˜f = ST x ˜  1  s.t.  Ax − y  with y  = (incomplete) data  A = modeling matrix, e.g. A = RS ˜ = recovered sparsity vector x  T  = a number dependent on the noise level ST = the synthesis matrix ˜f = the recovered function f Crux lies in finding the sparse representation!  2  ≤  Curvelets & seismology  Wish list Transform that is parsimonious     detects the wavefronts localized in space and frequency (phase space) some invariance under “wave propagation”  Events correspond to curved singularities with conflicting dips    caustics faults & pinch outs  Need a transform that is     multiscale multidirectional exactly reconstructs  Representations for seismic data Transform  Underlying assumption  FK  plane waves  linear/parabolic Radon transform  linear/parabolic events  wavelet transform  point-like events (1D singularities)  curvelet transform  curve-like events (2D singularities)  Properties curvelet transform:     multiscale: tiling of the FK domain into dyadic coronae multi-directional: coronae subpartitioned into angular wedges, # of angle doubles every other scale    anisotropic: parabolic scaling principle    Rapid decay space    Strictly localized in Fourier    Frame with moderate redundancy  k2 2j/2  angular wedge  2j  fine scale data  k1  coarse scale data  2-D curvelets [Candes, Donoho, Demanet, Ying]  curvelets are of rapid decay in space  x-t  curvelets are strictly localized in frequency  f-k  Oscillatory in one direction and smooth in the others!  Wavefront detection 0  -2000  Offset (m) 0  2000  Time (s)  0.5  1.0  1.5  2.0  Significant curvelet coefficient  Curvelet coefficient~0  curvelet coefficient is determined by the dot product of the curvelet function with the data  Compression  [From Demanet ‘05]  3-D curvelets  Curvelets live in wedges in the 3 D Fourier plane...  Nonlinear approximation  Nonlinear approximation  Curvelet-based seismic data recovery joint work with Gilles Hennenfent  Sparsity-promoting inversion* Reformulation of the problem signal  y  =  H  RC  +  n  noise  x0 curvelet representation of ideal data  Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI)   P :  look for the sparsest/most compressible, physical solution KEY POINT OF THE   RECOVERY data misfit sparsity constraint sparsity constraint         = arg ˜ H H − PC minmin x 0x 0 s.t. s.t. y −yPC x 2x≤2!≤ ! x =x˜arg ˜0 )= arg minxx Wx x s.t. Ax − y 2 ≤ (P0 )(Px 1     T  ˜   ˜H f = C x  H ˜f = ˜fC= xC ˜ x˜  * inspired by Stable Signal Recovery (SSR) theory by E. Candès, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjes  Original data  85 % missing  operation by filling in the zero traces. Since seismic the missing data can be Curvelet recovered by compounding recovery  modeling operator, i.e., A := RCT . With this definit  P corresponds to seeking the sparsest curvelet vec  followed by the picking, matches the data at the n  transform (with S := C in P ) gives the interpolated  An example of curvelet based recovery is presente data volume is recovered from data with 80 % traces  traces are selected at random according to a discrete d  Observations Inverted a rectangular matrix     worked because the curvelet transform is sparse exploits the higher dimensional geometry of seismic wavefields curvelets are incoherent with the Dirac measurement basis  Data is recovered for large percentages of traces missing Is an example of an inverse problem with incomplete data Can these ideas be extended to recover migration amplitudes?    approximately invert a PsDO diagonalize zero-order PsDO’s  Stable seismic amplitude recovery “Sparsity- and continuitypromoting seismic image recovery with curvelet frames” by F.H, P. Moghaddam & C. Stolk to appear in special issue on imaging in ACHA  Migrated data  Seismic Laboratory for Imaging and Modeling  Amplitude-corrected & denoised migrated data  Existing scaling methods Methods are based on a diagonal approximation of      Ψ.  Illumination-based normalization (Rickett ‘02) Amplitude preserved migration (Plessix & Mulder ‘04) Amplitude corrections (Guitton ‘04) Amplitude scaling (Symes ‘07)  We are interested in an ‘Operator and image adaptive’ scaling method which       estimates the action of Ψ from a reference vector close to the actual image assumes a smooth symbol of Ψ in space and angle does not require the reflectors to be conormal <=> allows for conflicting dips stably inverts the diagonal  Our approach “Forward” model:  y  ≈ Ax0 + ε  with y  =  A  := C Γ  T  AA r K    = K Km + ε T  migrated data T  T  ≈  K Kr  =  the demigration operator  =  migrated noise.  diagonal approximation of the demigration-migration operator costs one demigration-migration to estimate the diagonal weighting  Solution  Solve  P: with    minx J(x) subject to  y − Ax  2    ˜ = (AH )† x ˜ m sparsity  J(x) = α x  1  +β Λ  1/2  A  H  †  continuity    need sparsity on the model invariance under the normal operator  x  p  .  ≤  Nonlinear approximation Migrated mobil data set  Nonlinear approximation Recovery from largest 3 %  Nonlinear approximation Difference  Diagonal approximation of the Hessian  Normal/Gramm operator [Stolk 2002, ten Kroode 1997, de Hoop 2000, 2003]  In high-frequency limit Ψ is a PsDO  Ψf (x) =  • •  R  −ix·ξ  e d  a(x, ξ)fˆ(ξ)dξ  pseudolocal singularities are preserved  Inversion corrects for the ‘Hessian’  Invariance under Gramm matrix  (a)  (b)  (c)  (d)  (a)  (b)  (c)  (d)  curvelets remain invariant (e) (f)  (g)  (h)  (e)  (g)  (h)  • •  (f)  approximation improves for higher frequencies  Approximation  stitutions are made for the scattering operator and the  leading behavior for their composition, the normal operator Ψ, c leading behavior their composition, the normal operator Ψ, correspond e normal operator Ψ, corresponds to that for of an  α f )∧ (ξ) = |ξ|2α · fˆ(ξ). Alter m → order-one (−∆)1/2invertible m withelliptic ((−∆) PsDO .  order-one invertible elliptic PsDO .  So let Ψ = Ψ(x, D) be a pseudodifferential operator of order 0, with homogeneous principalmade symbolzero-order a(x, ξ). this by composing thetodata side with aby 1/2-o To make PsDO amenable an approximation curv  this subPsDO amenable to an approximation by curvelets, the approximation by curvelets, To the make following  −1/2 operator and the model: stitutions are for made for the scattering −1/2 stitutions are made the scattering operator thee.g. model: → K erator and the model: K K (−∆) and the→ time coordinate, i.e., K → ∂t Kand (see 3).K After  d 2αin ˆ R . · f (ξ).  or 1/2 α f )∧ (ξ) 2α · fˆ(ξ). Alternativel 1/2 α ∧ 2α ˆ m → (−∆) m with ((−∆) = |ξ| with m → operators (−∆) mcan with ((−∆) f ) (ξ) = |ξ| · f (ξ). Alternatively, these op Alternatively, these be  operator Ψ becomes zero-order. Remark that these subsit  made zero-order by composing with a 1/2-order madeintegration zero-order by composing the datathe sidedata withside a 1/2-order fractional fra in de with a 1/2-order fractional along Lemma 1. With C some constant, the following holds  tution made in the WVD−1/2 methods, −1/2 where vaguelettes are  the time i.e., K → ∂t K → K∂ (see e.g. 3). After these substitutio thecoordinate, time coordinate, i.e., K (see e.g. 3). After these t e e.g. 3). After these substitutions, the normal  −|ν|/2 n ≤ C 2 (Ψ(x, D) − a(x , ξ ))ϕ . these subsitutions (14) 2 ν ν ν L ( R ) operator Ψ becomes zero-order. Remark that are simila mappings. Before detailing the approximate diagonalizat operator Ψ becomes zero-order. Remark that these subsitutions hat these subsitutions are similar to the substi-  tution made in the WVD methods, where vaguelettes are introduced acco tution made in the WVD methods, where vaguelettes are introd vaguelettes are introduced according the same first discuss the properties of continuous curvelets under t To approximate Ψ, we define the sequence u := (uµ )µ∈M = a(xµ , ξµ ). Let DΨ be the mappings. Before detailingdetailing the approximate diagonalization of the norm mappings. the approximate diagonalization of ate diagonalization of the normal operator, Before we diagonal matrix with entries given by u. Next we state our result on the approximation of first discuss the properties of continuous curvelets under this operator. urvelets under this operator. first discuss the properties of continuous curvelets under this ope T Ψ by C DΨ C.  APPROXIMATION OF THE NORM  Ψ by C T DΨ C.  Approximation  Theorem 1. The following estimate for the error holds (Ψ(x, D) − C T DΨ C)ϕµ  L2 (R ) n  ≤ C 2−|µ|/2 ,  where C is a constant depending on Ψ.  Allows for the decomposition  This main result proved in Appendix A shows that the approxima  Ψϕµ (x)  C DΨ Cϕµ (x) T  diagonal approximation goes to zero for increasingly finer scales. The app T  =  AA ϕµ (x)  from the property that the symbol is slowly varying over the support  with A :=  √  DΨ C and AT := C  √ T  DΨ .  approximation that becomes more accurate as the scale increases.  Approximation y(x)  =  Ψm (x) + e(x) AA m (x) + e(x) T  = Ax0 + e, Wavelet-vagulette like Amenable to nonlinear recovery  Estimation of the diagonal scaling  Diagonal estimation  • •  Define a reference vector (say conventional image).  •  Define the matrix  •  Calculate ‘data’  b = Ψr  P := C diag(v) T  with  v = Cr  Invert 1 ˜ = arg min b − Pu u u 2  2 2  + η Lu 2  2 2  Diagonal estimation Impose smoothness in phase space  extended to include q different reference vectors by making the following substitutions  L = [D1  D2  Dθ ]  b → [b1 · · · bq ]T and P → [P1 · · · Pq ]T with the “data” vector and “modeling” matrix defined by the different reference vectors r1 , . . . , rq . Calculate: b = Ψr and v = Cr. Set: η = ηmin ; while ∃ (˜ uµ )µ∈M < 0 do Solve u = arg minu  1 2  b − Pu  2 2  + η 2 Lu  Increase the Lagrange multiplier λ = η + ∆η end while  2 2  Diagonal estimation  (a)  (b)  Seismic amplitude recovery  Recovery  Final form  y = Ax0 + ε with x0 = ΓCm and Solve P: with  = Ae.    minx J(x) subject to  y − Ax  2    H †˜ ˜ m = (A ) x sparsity  J(x) = α x  1  +β Λ  1/2  A  H  †  continuity  x  p  .  ≤  Image recovery  anisotropic diffusion sotropic-diffusion penalty term (see e.g. 24) is given by [Black et. al ’98, Fehmers et. al. ’03 and Shertzer ‘03]  Jc (m) = Λ  1/2  Define  ∇m  2 2 T T D2 .  discretized gradient matrix defined as ∇ = The block 1/2 Jc (m) = Λ ∇m p s location dependent (see Fig. 10, which plots the gradients) and ro DT1  with p=2  wards the tangents of the reflecting surfaces. This rotation matrix is g          +D r   2 1  +D r −D r + υId  Λ[r] =   2 1  ∇r 22 + 2υ     −D1 r   e discretized derivative in the ith coordinate direction and υ a param  Gradient of the reference vector  500  1000  depth (m)  1500  2000  2500  3000  3500 2000  4000  6000  8000 lateral (m)  10000  12000  14000  Recovery Step 1: Update of the Jacobian of  1 2  y − Ax 22 :  x ← x + AT (y − Ax) ; Step 2: projection onto the  1  ball S = { x  1  ≤ x0 1 } by soft thresholding  x ← Tλw (x); Step 3: projection onto the anisotropic diffusion ball C = {x : J(x) ≤ J(x0 )} by x ← x − κ∇x Jc (x)  Initialize: m = 0; x0 = 0; y = KT d; Choose: M and L AT y  ∞  > λ1 > λ2 > · · ·  while y − Ax  2  >  do  m = m + 1; xm = xm−1 ; for l = 1 to L do xm = Tλm xm + AT (y − xm ) {Iterative thresholding} end for Anisotropic descent update; xm = xm − β∇xm Jc (xm ); end while x = xm ; m = AT  †  x.  Table 2: Sparsity-and continuity-enhancing recovery of seismic amplitudes.  Application to the SEG AA’ model  Example SEGAA’ data:     “broad-band” half-integrated wavelet [5-60 Hz] 324 shots, 176 receivers, shot at 48 m 5 s of data  Modeling operator     Reverse-time migration with optimal check pointing (Symes ‘07) 8000 time steps linearized modeling 64, and migration 294 minutes on 68 CPU’s  Scaling required 1 extra migration-demigration  Seismic Laboratory for Imaging and Modeling  Seismic Laboratory for Imaging and Modeling  Seismic Laboratory for Imaging and Modeling  Migrated data  Seismic Laboratory for Imaging and Modeling  Amplitude-corrected & denoised migrated data  Noise-free data  Noisy data (3 dB)  Seismic Laboratory for Imaging and Modeling  Data from migrated image  Data from amplitude-corrected & denoised migrated image  Example SEGAA’ data:     “broad-band” half-integrated wavelet [5-60 Hz] 324 shots, 176 receivers, shot at 48 m 5 s of data  Modeling operator     Reverse-time migration with optimal check pointing (Symes ‘07) 8000 time steps full modeling  Scaling required 1 extra migration-demigration  Conclusions Curvelet-domain scaling     handles conflicting dips (conormality assumption) exploits invariance under the PsDO robust w.r.t. noise  Diagonal approximation    exploits smoothness of the symbol uses “neighbor” structure of the curvelet transform  Results on the SEG AA’ show     recovery of amplitudes beneath the Salt successful recovery of clutter improvement of the continuity  Acknowledgments The authors of CurveLab (Demanet,Ying, Candes, Donoho) Dr. W. W. Symes for his reverse-time migration code This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE (334810-05) of F.J.H. This research was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron,ExxonMobil and Shell.  

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