A Nonlinear Sparsity Promoting Formulation and Algorithm for Full Waveform Inversion

Transcription

1 A Nonlinear Sparsity Prooting Forulation and Algorith for Full Wavefor Inversion Aleksandr Aravkin, Tristan van Leeuwen, Jaes V. Burke 2 and Felix Herrann Dept. of Earth and Ocean sciences University of British Colubia Vancouver, BC, Canada 2 Dept. of Matheatics University of Washington Seattle, WA January 4, 2 Abstract Full Wavefor Inversion (FWI) is a coputational procedure to extract ediu paraeters fro seisic data. FWI is typically forulated as a nonlinear least squares optiization proble, and various regularization techniques are used to guide the optiization because the proble is illposed. In this paper, we propose a novel sparse regularization which exploits the ability of curvelets to efficiently represent geophysical iages. We then forulate a corresponding sparsity prooting constrained optiization proble, which we call Nonlinear Basis Pursuit Denoise (NBPDN) and present an algorith to solve this proble to recover ediu paraeters. The utility of the NBPDN forulation and efficacy of the algorith are deonstrated on a stylized cross-well experient, where a sparse velocity perturbation is recovered with higher quality than the standard FWI forulation (solved with LBFGS). The NBPDN forulation and algorith can recover the sparse perturbation even when the data volue is copressed to 5% of the original size using rando superposition. Introduction Full Wavefor Inversion (FWI) is a data-fitting procedure based on full wavefield odeling designed to extract ediu paraeters (velocity and density) fro seisogras. Coputational ethods for wavefor inversion go back ore than 2 years (see, e.g, Tarantola (984)) and the proble has been consistently forulated as a nonlinear least squares or siilar type of optiization proble (Virieux and Operto (29)). It is useful at this point to provide an explicit fraework for a typical FWI approach: φ() := D PH[] Q 2 F () where 2 F is the Frobenius nor, is a vector of velocity paraeters in a 2D or 3D grid, D R k l contains results of l source experients (as k-diensional coluns), H[] is a discretization of the Helholtz operator with boundary conditions, Q R p l specifies l source experients, H []Q describes the solution of the Helholtz equation for the sources Q, and P is a restriction of this solution to the surface where the data was observed. FWI is widely known to be an ill-posed proble, and so regularization strategies are applied in practice (see Virieux and Operto (29) and sources within). A coon strategy is least squares regularization, where given reasonable guess of prior paraeters, one solves the proble φ()+( ) T W( ) (2) where W is soe weighting atrix that encodes the confidence in the prior guess as well as relationship (correlations) between the paraeters. Alternative odels using total variation (TV) regularization have also been proposed (see e.g. Vogel and Oan (996)). The ai of the present paper is to forulate an alternative regularization approach based on sparsity

2 5% of coeff. % of coeff. Figure Partial Maroussi odel in curvelets: 5% of the largest curvelet coefficients capture ost of the features of the full representation. prootion, to develop an algorith for the solution of the resulting optiization proble, and to show the results of the new algorith on a toy exaple. Sparsity prootion for seisic data using curvelets The curvelet frae was presented as an effective nonadaptive representation for objects with edges in the seal paper Candes and Donoho (2). The key result in that paper is that the curvelet frae provides a stable, efficient, and near-optial representation of otherwise sooth objects having discontinuities along sooth curves. While there ay be liitations to this characterization of geophysical iages, it is iportant to note that such iages are layered due to geological sedientation, and this feature allows for efficient representations using curvelets. Motivated by this observation, researchers have used curvelet representations in igration, diensionality reduction, siulation, and sparse sapling applications (see Hennenfent et al. (2); Herrann et al. (29, 28, 27)). See also Figure for a siple deonstration. The notion that velocity paraeters should be sparse (or at least copressible) in the curvelet representation leads to a sparse regularization of FWI (copare with eq. (2)): x φ(c x)+λ x, (3) where C denotes the curvelet basis and x is the vector of curvelet coefficients corresponding to the velocity paraeters, i.e. = C x, the ter x serves to proote sparsity in this representation, and λ is a paraeter that balances sparsity in curvelets vs. odel fit. While this is a reasonable forulation, λ ust be known ahead of tie, and it is not clear how to choose it. Rather than working with eq. (3), we go to a closely related constrained reforulation x x s.t. g(c x) σ, (4) where as before the objective x serves to proote sparsity in this representation, and the paraeter σ is a regularization paraeter that deteres the acceptable value of the residual D RH[C x] Q F (i.e. noise level in the data). Unlike λ in eq. (3), the paraeter σ in eq. (4) is likely to be known to scientists working with inverse probles in geophysics. Note that the forulation (4) is a natural nonlinear extension to the Basis Pursuit Denoise (BPDN) forulation used in copressive sensing literature to for sparse signal recovery fro under-sapled noisy data (see van den Berg and Friedlander (28)). The optiization forulation (4) is harder to solve then (), and requires a custo algorith. The ain contribution of this paper is to describe such an algorith and deonstrate its perforance on a siplified proble of the for s.t. D PH[ + ] Q F σ, (5) 73 rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2 Vienna, Austria, May 2

3 for a situation where we are trying to recover a velocity perturbation relative to a constant background velocity, and so the perturbation is sparse in the physical doain. The algorith for (5) can also be used to solve (4), but requires special care to aintain the feasibility of transfored velocities C x. Nonlinear Basis Pursuit Denoise (NBPDN) algorith To solve (5), we ipleent an iterated algorith of the for ν+ = ν + τ ν s ν, (6) where s ν is the solution to a particular subproble at step ν, and τ ν is a step size chosen by a line search strategy. In developing the algorith, especially in the line search for τ ν, we follow ideas presented in Burke (989) and Burke (992). To obtain the subproble, at each step ν, we linearize the functions and D RH[ + ] Q, and solve the resulting optiization proble, using a ν-dependent paraeter σ ν : ν + s s s.t. D F( ν ) F( ν (7) )(s) F σ ν, where F() = PH[ + ] Q and F( ν ) denotes the linearized Born scattering operator. The solution to this proble is the direction s ν that appears in eq. (6). To solve this proble, we use the substitution y= ν + s to obtain y y (D F( ) s.t. ν )+ F(ν)ν F( ν )(y) σ ν. F (8) For a fixed ν, this proble is now equivalent to the basis pursuit denoise (BPDN) proble detailed in van den Berg and Friedlander (28). The algorith in that paper, called SPGl, allows us to solve (8) quickly, and oreover allows a functional representation of F to be provided (specifying its action on vectors y) rather than requiring an explicit atrix representation. The paraeters σ ν are chosen to start large and decrease untill it reaches the σ paraeter specified by the user. To obtain the step paraeter τ ν, we first define an auxiliary penalty function P α ()= + α( D F() 2 σ) +, which includes both the sparsity prooting objective and a easure of the distance fro optiality. The paraeter α ν is then selected to ensure that s ν, the solution to (5), is a descent direction for P αν (). In other words, the choice of α ν ensures P αν ( ν + s ν ) P αν ( ν ) <. We then use the backtracking Arijo line search (see e.g. Nocedal and Wright (999)) using the erit function P αν (). The resulting step τ ν is used to update the odel as described in (6). Results To illustrate the new algorith and the power of sparsity regularization, we considered a stylized crosswell proble. The true velocity consists of three sall features ebedded in a constant background of 2k/s and is depicted in figure 2. The features are sparse in the pixel-basis so we can directly enforce sparsity on the recovered perturbation. We use a 9-point discretization of the Helholtz operator with absorbing boundaries on a grid with spacing. The data are generated for equispaced sources and receivers located in vertical wells 8 apart for (randoly chosen) frequencies [5., 6.,.5, 4., 5.5, 7.5, 23.5] Hz. We consider two different scenarios: inversion with all the sources and inversion using only 5 randoly synthesized supershots. These are generated by weighting all the sources with rando Gaussian weights and stacking. Such techniques have recently been proposed to draatically reduce the costs of FWI (Krebs et al., 29; Moghadda and Herrann, 2; Haber et al., 2) (see also other contributions of the authors to these proceedings). We copare the use of unregularized L-BFGS on () and the newly proposed algorith on (8). The results are depicted in figure 2. We see that when using all the sources the unregularized approach produces a reasonable 73 rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2 Vienna, Austria, May 2

4 iage. The resolution is not very high, as expected, and the vertical sides of the circle are not well recovered. The NBPDN algorith, however, produces a nearly exact recovery. The circle is now recovered copletely but the horizontal bar is soewhat distorted. When using only 5 supershots (a reduction of a factor 2 in the data volue) the L-BFGS approach produces an unusable iage. The artifacts introduced by the crosstalk between the shots copletely obscures the recovered velocity perturbations. The NBPDN forulation, rearkably, gives us alost the sae result as before. Conclusion We forulated FWI as a non-linear, sparsity prooting optiization proble. The underlying assuption is that the ediu paraeters that we are trying to recover have a sparse representation in soe basis. In particular, we envision that typical velocity structures are sparse in curvelets. Instead of adding a penalty ter to the isfit ter with a regularization paraeter, as is coonly done in for exaple TV regularization, we propose to iize the penalty subject to the isfit being saller than soe preset error level. The advantage of this forulation copared to other sparsity prooting strategies (e.g. LASSO) is that this error level ay be easier to detere than the regularization paraeter. We deonstrate the algorith on a toy cross-well exaple, where the unknown velocity perturbation is sparse in the pixel-basis. Copared to an unregularized least-squares inversion, our approach gives a superior result with uch higher resolution. We also consider using randoly synthesized data to reduce the coputational cost of the inversion. Such a reduction coes at the cost of introducing crosstalk between the shots. In the unregularized inversion, this crosstalk overshadows the reconstructed velocity perturbations. With the regularized inversion, however, we obtain a result nearly identical to the earlier case at roughly 5% of the coputational cost. The latter result ay be tentatively explained by invoking results fro copressive sensing; a sparse signal ay be reconstructed fro severely undersapled data by solving a linear sparsity prooting progra as long as the sapling satisfies soe additional criteria. Most notably, the sapling ust be rando. The current forulation is a direct generalization of the sparsity prooting linear forulation used in copressive sensing. Future research will be aied at further exploiting the connection to copressive sensing. 73 rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2 Vienna, Austria, May 2

5 (a) (b) (c).5.5 (d).5 (e).5.5 Figure 2 (a) True odel for cross-well experient; asterisks are sources and triangles are receivers. (b) LBFGS recovery using full data ( shots). (c) NBPDN recovery using full data ( shots). (d) LBFGS recovery using five supershots (2 x speedup). (e) NBPDN recovery using five supershots (2 x speedup). References Burke, J.V. [989] A sequential quadratic prograg ethod for potentially infeasible atheatical progras. Journal of Matheatical Analysis and Applications, 39(2), Burke, J.V. [992] A robust trust region ethod for constrained nonlinear prograg probles. SIAM Journal on Optiization, 2(2), , doi:.37/826. Candes, E.J. and Donoho, D.L. [2] Curves and Surfaces, Vanderbilt University Press, Nashville, TN., chap. Curvelets - a surprisingly effective nonadaptive representation for objects with edges. Haber, E., Chung, M. and Herrann, F.J. [2] An effective ethod for paraeter estiation with pde constraints with ultiple right hand sides. Tech. Rep. TR-2-4, UBC-Earth and Ocean Sciences Departent. Hennenfent, G., Fenelon, L. and Herrann, F.J. [2] Nonequispaced curvelet transfor for seisic data reconstruction: a sparsity-prooting approach. Tech. Rep. TR-2-2, UBC-Earth and Ocean Sciences Departent. Herrann, F.J., Brown, C., Erlangga, Y. and Moghadda, P. [28] Curvelet-based igration preconditioning and scaling. Tech. Rep. TR-28-, The University of British Colubia, subitted for publication. Herrann, F.J., Erlangga, Y.A. and Lin, T. [29] Copressive siultaneous full-wavefor siulation. Geophysics, 74, A35. Herrann, F.J., Moghadda, P.P. and Stolk, C.C. [27] Sparsity- and continuity-prooting seisic iage recovery with curvelet fraes. Accepted for publication in the Journal of Applied and Coputational Haronic Analysis. Krebs, J.R. et al. [29] Fast full-wavefield seisic inversion using encoded sources. Geophysics, 74(6), WCC77 WCC88, doi:.9/ Moghadda, P.P. and Herrann, F.J. [2] Randoized full-wavefor inversion: a dienstionality-reduction approach. SEG Technical Progra Expanded Abstracts, 29(), , doi:.9/ Nocedal, J. and Wright, S.J. [999] Nuerical Optiization. Springer Series in Operations Research, Springer. Tarantola, A. [984] Inversion of seisic reflection data in the acoustic approxiation. Geophysics, 49(8), , doi:.9/ van den Berg, E. and Friedlander, M.P. [28] Probing the pareto frontier for basis pursuit solutions. SIAM Journal on Scientific Coputing, 3(2), 89 92, doi:.37/ Virieux, J. and Operto, S. [29] An overview of full-wavefor inversion in exploration geophysics. Geophysics, 74, 27 +, doi:.9/ Vogel, C.R. and Oan, M.E. [996] Iterative ethods for total variation denoising. SIAM J. Sci. Coput., 7, , ISSN , doi:.37/ rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2 Vienna, Austria, May 2