P016 Toward Gauss-Newton and Exact Newton Optiization for Full Wavefor Inversion L. Métivier* ISTerre, R. Brossier ISTerre, J. Virieux ISTerre & S. Operto Géoazur SUMMARY Full Wavefor Inversion FWI applications classically rely on efficient first-order optiization schees, as the steepest descent or the nonlinear conjugate gradient optiization. However, second-order inforation provided by the Hessian atrix is proven to give a useful help in the scaling of the FWI proble and in the speed-up of the optiization. In this study, we propose an efficient atrix-free Hessian-vector foralis, that should allow to tackle Gauss-Newton GN and Exact-Newton EN optiization for large and realistic FWI targets. Our ethod relies on general second order adjoint forulas, based on a Lagrangian foralis. These forulas yield the possibility of coputing Hessianvector products at the cost of 2 forward siulations per shot. In this context, the coputational cost per shot of one GN or one EN nonlinear iteration aounts to the resolution of 2 forward siulations for the coputation of the gradient plus 2 forward siulations per inner linear conjugate gradient iteration. A nuerical test is provided, ephasizing the possible iproveent of the resolution when accounting for the exact Hessian in the inversion algorith.
Introduction Full Wavefor Inversion FWI is becoing an efficient tool to derive high resolution quantitative odels of the subsurface paraeters. The ethod relies on the iniization, through an iterative procedure, of the residual between recorded data and synthetic data coputed by solving the two-way wave equation in a subsurface odel. The growth of available coputational resources and recent developents of the ethod akes now possible applications to 2D and 3D data in the acoustic approxiation see for exaple Prieux et al., 2011; Plessix et al., 2010 and even in the elastic approxiation Brossier et al., 2009. Most of the FWI applications rely on fast optiization schees as preconditioned steepest descent or preconditioned conjugate-gradient ethods PCG. Second-order inforation provided by the Hessian is often neglected in FWI, due to the high coputational cost to build this atrix and solve the noral equation syste. However, a significant iproveent of the results can be obtained using this inforation: Pratt et al. 1998 have shown the iproved resolution of Gauss-Newton ethod copared to the steepest descent one in a canonical application. Hu et al. 2011 have shown results iproveent provided by a non-diagonal truncated Hessian in PCG. Brossier et al. 2009 have shown the estiated Hessian s ipact of a quasi-newton l-bfgs Nocedal, 1980, on iage resolution and convergence speed copared to PCG. Epanoeritakis et al. 2008; Fichtner and Trapert 2011 have also discussed the interest of Hessian for inversion and uncertainty estiation, and also the prohibitive cost of Hessian coputation and storage. In this study, we develop the atheatical fraework to propose an efficient atrix-free Hessian-vector product algorith for FWI, and give an illustration of the interest of accounting for the exact Hessian in the inversion process. The final ai is to tackle Gauss-Newton and Full-Newton ethod for large scale applications. Our developent relies on a general second-order adjoint-state forula, valid either in the tie or in the frequency doain. Proble stateent We consider the forward proble equation Spu = ϕ, 1 where p denotes the subsurface paraeters odel space, Sp denotes the linear forward proble operator corresponding to the two-way wave equation discretization 1, ϕ is the source vector, and u is the wavefield vector. In the following, we consider that p and u have N coponents 2. These notations are general and can be applied either in the tie doain or in the frequency doain. The FWI proble is expressed as the least-square proble in f p = 1 p 2 N s R s u s p d s 2, 2 where d s and u s p are respectively the recorded dataset and the solution of the forward proble associated with the source ϕ s, N s is the total nuber of sources in the dataset, and R s is a restriction operator that aps the wavefield u s to the receivers locations. The isfit gradient is f p = R J s R s R s u s p d s 3 1 Note that the operator Sp depends non-linearly on p 2 This siplification is satisfied whenever u is discretized on the sae grid than p. However, the forulas can be straightforwardly extended to the situation where p and u do not have the sae nuber of coponents.
where J s p denotes the Jacobian atrix p u s p. The coplex conjugate transpose operator is denoted by the sybol and the real part application by R. The Hessian operator is Hp = R J s R s R s J s + [R s R s u s p d s ] j H s j, H s j = 2 ppu s j p, 4 and its Gauss-Newton approxiation is Bp = R J s R s R s J s 5 In our study, the proble 2 is solved using a Newton or a Gauss-Newton algorith, a local iterative optiization ethod that coputes a sequence p k fro an initial guess p 0 using the update forula where d k is the solution of p k+1 = p k + α k d k, 6 Hp k d k = f p k, or Bp k d k = f p k in the Gauss-Newton approxiation, 7 and α k is coputed through a globalization procedure linesearch or trust region. Explicit coputation and storage of atrices Jp, Hp and Bp is prohibitive for large scale probles and realistic FWI applications. Hence, the equation 7 should be solved using atrix-free linear iterative solvers, as proposed in Epanoeritakis et al. 2008; Fichtner and Trapert 2011, such as the conjugate gradient CG ethod. This requires to copute efficiently both the gradient f p and Hessian-vector products Hpv or Bpv for arbitrary vectors v. While the coputation of f p can be perfored efficiently through the classical adjoint-state forula Plessix, 2006 where λp the adjoint state is the solution of f p i = R pi Spup,λp, i = 1,..., 8 Sp λ = R d Ru, 9 we propose general second-order adjoint-state forula for the coputation of Hpv and Bpv related to the Lagrangian foralis and the nonlinear constrained optiization theory. Coputing Hessian vector products For the sake of clarity, we consider in the following that N s = 1 and we drop index s. Forulas for N s > 1 are directly obtained by suation. First, we define the function g v p such that g v p = f p,v = R J R Rup d,v, 10 where up is the solution of 1. We have g v p = Hpv. We define the Lagrangian function associated with the functional g v p L v p,u,α,λ,µ = R R Ru d,α + R Spu ϕ, µ + R Spα + v j p j Spu,λ 11 The Lagrangian L v is coposed of three ters: the first one accounts for the function g v, the second one accounts for the constraints on the wavefield u, solution of the forward proble, the third one accounts for the constraints on the first-order derivatives of the wavefield u with respect to p. For ũ and α such that Spũ = ϕ, Sp α = v j p j Spũ, 12
we have g v p = p L v p,ũ, α,λ,µ + u L v p,ũ, α,λ,µ p ũp + α L v p,ũ, α,λ,µ p αp. 13 We define λ and µ such that We have and Hpv i = R u L v p,ũ, α, λ, µ Sp µ = R R α pi Spũ, µ + = 0, α L v p,ũ, α, λ, µ = 0. 14 v j p j Sp λ, Sp λ = R Rũ d 15 p pi Sp α, λ + v j j pi Sp ũ, λ,i = 1,...,. 16 Note that λ corresponds to the adjoint state defined for the coputation of f. In addition, it can be proved that the coputation of Bpv aounts to setting λ to 0 in equations 15 and 16. Bpv i = R pi Spũ, µ,i = 1,..., with Sp µ = R R α, 17 The coputation of one atrix vector product Hpv or Bpv thus requires to solve one additional forward proble for α and one additional adjoint proble for µ, as reported in Epanoeritakis et al. 2008; Fichtner and Trapert 2011. For N s > 1, the overall coputation cost is ultiplied by N s. Nuerical results We consider the estiation of the pressure wave velocity v p using a 2D acoustic FWI algorith in the frequency doain. The exact odel v p is coposed of a hoogeneous background v 0 p = 1500.s 1 defined on a 2000 length square, and two inclusions where v p = 4500.s 1. PML are all around. The two inclusions are distant only fro 40 see fig.1. The high velocity contrast generates high aplitude ulti-scattered waves. We use a full acquisition with four lines of 29 sources placed each 50 on each side of the doain. Each source is associated with four lines of 29 receivers placed each 50 all around the doain. The odel is discretized over a 101 101 grid with a spatial step of 20. The initial guess is the background v 0 p. We use one dataset corresponding to the frequency of 5 Hz. The associated average wavelength is λ 300. At this frequency, the distance between the two inclusions is saller than the expected resolution of classical FWI algoriths based on l-bfgs or Gauss-Newton approxiation. We copare the results obtained perforing 50 iterations of the l-bfgs algorith, 20 iterations of Gauss-Newton inversion and 20 iterations of Exact-Newton inversion 3. The corresponding estiated odels are plot in Figure 1. Figure 1 Acoustic FWI for pressure wave velocity. Fro left to right, exact odel, l-bfgs result, Gauss-Newton result, Exact-Newton result. 3 This corresponds approxiately to the sae coputation cost for 2.5 conjugate gradient inner iterations per Newton nonlinear iteration
These results ephasize the role of the second-order part of the Hessian, which is neglected in the Gauss-Newton approxiation, and hardly estiated in the l-bfgs approxiation. As entioned by different authors, this part of the Hessian allows to account for double scattering during the inversion Pratt et al., 1998; Virieux and Operto, 2009. An enhanceent of the resolution is obtained: only when using the exact Hessian, the two inclusions are identified. In this case, the noralized isfit is 7 10 4. In the case of the Gauss-Newton inversion, the final noralized isfit is around 0.25: neglecting the double scattered waves prevent the optiizer to converge. In the case of the l-bfgs inversion, the final noralized isfit is around 10 3 : the l-bfgs Hessian approxiation allows to account for a part of the scattered wavefield. Conclusions and perspectives The preliinary results we have obtained illustrate how accounting for the Hessian effect can iprove the FWI results. The second-order adjoint forula is an efficient tool to ipleent Gaus-Newton or Exact-Newton algoriths in a atrix-free fashion. The next step consists now in developing globalization ethods either based on linesearch or trust region that render the overall ethod copetitive with the l-bfgs ethod. Two ain probles have to be tackled: the first one is the definition of a proper criterion to stop the inner conjugate gradient iterations, in order to iniize the coputation cost. The second one is the developent of efficient preconditioners to speed-up the convergence of the inner conjugate gradient loop. Aong several possibilities, we are interested in the Newton Steihaug algorith Steihaug, 1983, based on the trust-region ethod, that provides naturally a stopping criterion for the inner iterations. In addition, for each resolution of the inner linear systes, a l-bfgs preconditioner can be coputed siultaneously, which is applied to the inner linear syste raised by the next outer nonlinear iteration. Acknowledgeents This research was funded by the SEISCOPE consortiu sponsored by BP, CGG-VERITAS, ENI, EXXON MOBIL, PETROBRAS, SAUDI ARAMCO, SHELL, STATOIL and TOTAL. The linear systes were solved with the MUMPS package. This work was perfored by accessing to the high-perforance coputing facilities of CIMENT Université de Grenoble, France and to the HPC resources of GENCI- CINES under Grant 2011-046091. References Brossier, R., Operto, S. and Virieux, J. [2009] Seisic iaging of coplex onshore structures by 2D elastic frequency-doain full-wavefor inversion. Geophysics, 746, WCC63 WCC76, doi:10.1190/1.3215771. Epanoeritakis, I., Akçelik, V., Ghattas, O. and Bielak, J. [2008] A Newton-CG ethod for large-scale threediensional elastic full wavefor seisic inversion. Inverse Probles, 24, 1 26. Fichtner, A. and Trapert, J. [2011] Hessian kernels of seisic data functionals based upon adjoint techniques. Geophysical Journal International, 1852, 775 798, ISSN 1365-246X, doi:10.1111/j.1365-246x.2011.04966.x. Hu, W., Abubakar, A., Habashy, T.M. and Liu, J. [2011] Preconditioned non-linear conjugate gradient ethod for frequency doain full-wavefor seisic inversion. Geophysical Prospecting, 593, 477 491, ISSN 1365-2478, doi:10.1111/j.1365-2478.2010.00938.x. Nocedal, J. [1980] Updating Quasi-Newton Matrices With Liited Storage. Matheatics of Coputation, 35151, 773 782. Plessix, R.E. [2006] A review of the adjoint-state ethod for coputing the gradient of a functional with geophysical applications. Geophysical Journal International, 1672, 495 503. Plessix, R.E., Baeten, G., de Maag, J.W., Klaassen, M., Rujie, Z. and Zhifei, T. [2010] Application of acoustic full wavefor inversion to a low-frequency large-offset land data set. SEG Technical Progra Expanded Abstracts, 291, 930 934, doi:10.1190/1.3513930. Pratt, R.G., Shin, C. and Hicks, G.J. [1998] Gauss-Newton and full Newton ethods in frequency-space seisic wavefor inversion. Geophysical Journal International, 133, 341 362. Prieux, V. et al. [2011] On the footprint of anisotropy on isotropic full wavefor inversion: the Valhall case study. Geophysical Journal International, 187, 1495 1515, doi:doi: 10.1111/j.1365-246X.2011.05209.x, 2011. Steihaug, T. [1983] The conjugate gradient ethod and trust regions in large scale optiization. SIAM Journal on Nuerical Analysis, 20, 626 637. Virieux, J. and Operto, S. [2009] An overview of full wavefor inversion in exploration geophysics. Geophysics, 746, WCC127 WCC152.