An efficient frequency-domain full waveform inversion. method using simultaneous encoded sources

Transcription

1 An efficient frequency-domain full waveform inversion method using simultaneous encoded sources Hafedh Ben-Hadj-Ali, Stéphane Operto and Jean Virieux Géoazur, Université de Nice Sophia-Antipolis, CNRS, IRD, OCA, Valbonne, France. Géoazur, Université de Nice Sophia-Antipolis, CNRS, IRD, OCA, Villefranche-sur-mer, France. Laboratoire de Géophysique Interne et Tectonophysique, Université Joseph Fourier, CNRS, Grenoble, France. now at Total (August, ) Running head: Efficient frequency-domain FWI with source encoding ABSTRACT Three-dimensional (D) full waveform inversion (FWI) still suffers from its prohibitive computational costs that arise because of the seismic modeling for multiple sources that is performed at each nonlinear iteration of FWI. Building super-shots by assembling several sources allows mitigation of the number of simulations per FWI iteration, although this adds cross-talk artifacts because of interference between the individual sources of the supershots. These artifacts can themselves be mitigated by encoding each individual source with a random phase shift during source assembling. The source encoding method is applied to an efficient frequency-domain FWI, where a limited number of discrete frequencies or coarsely-sampled frequency groups are successively inverted following a multiscale approach.

2 Random codes can be regenerated at each FWI iteration or for each frequency of a group during each FWI iteration to favor the destructive summation of cross-talk artifacts over FWI iterations. Either a limited number of sources (partial assembling) or the total number of sources (full assembling) can be assembled into super-shots. Wide-aperture acquisition geometries such as land or marine node acquisitions are considered to allows for stacking a large number of shots in the full computational domain and to test different partial assembling strategies involving close or distant sources. Two-dimensional case studies show that partial source assembling of distant shots has a limited sensitivity to noise, for a computational saving roughly proportional to the number of shots assembled into the super-shots. In contrast, full assembling is less robust to noise, and it requires successive inversions of finely sampled frequency groups with a large number of FWI iterations. In contrast, refining the shot interval to improve the fold degrades the models when full assembling is applied to noise data. Preliminary D case studies suggest comparable behaviors than for D ones.

3 INTRODUCTION As three-dimensional (D) prestack depth-imaging methods are computationally expensive, such as prestack depth migration and full waveform inversion (FWI), the simultaneous-shot technique can provide an interesting trade-off between computational efficiency and imaging accuracy (Capdeville et al., 5). The simultaneous-shot technique takes advantage of the linear relationship between a seismic wavefield and its source, by summing the individual sources to mitigate the number of wave simulations during imaging. This is equivalent to the simultaneous operating of several spatially distributed shots in the field (Krebs et al., 9). The computational cost of one migration or one FWI iteration is reduced in proportion to the number of sources gathered into each shot assembly, or super-shot. However, imaging is altered by cross-talk noise that results from the interference between the individual sources of a super-shot. The imaging kernel of FWI is basically the same that of prestack depth migration based on the wavefield-continuation imaging principle (Claerbout, 985; Lailly, 98), and is computed by cross-correlating the incident wavefield with the back-propagated residual wavefield. When the simultaneous-shot technique is used, the cross-talk noise arises from the correlation between the incident wavefield emitted by a given source and the wavefields back-propagated from the receivers associated with other sources of the super-shot. Additional inversion iterations can be required here to mitigate the footprint of the cross-talk noise, and to achieve convergence towards an acceptable model. This footprint of the cross-talk noise can be mitigated by applying judicious source encoding during assembling of the super-shot. The codes are chosen such that the correlation between one incident wavefield and the corresponding back-propagated wavefields is left unchanged, while the cross-talk terms are mitigated during the summation by an incoherent sum. This technique, called phase encoding (Morton and Ober, 998; Jing et al., ; Romero et al.,

4 ), was originally developed for prestack migration. Romero et al. () discussed several frequency-dependent and frequency-independent phase encodings, and proposed two main approaches for prestack shot-record migration. The first consists of the migration of several super-shots, where a limited number of shots are summed into each super-shot. In the second approach, all of the shots are summed into one super-shot, multiple migrations are performed, and all of the migrated images are stacked to reduce the cross-talk noise. In both approaches, phase encoding significantly reduces the computational costs and leads to acceptable images. Liu et al. () proposed a phase-encoding method for prestack depth migration, which was recast as the migration of a limited number of plane waves. Their approach was validated with the D Marmousi velocity model. A -fold computational speed-up was obtained with the plane-wave phase-encoding method. Since the imaging kernels of prestack migration and FWI are basically the same, the simultaneous-shot technique with source encoding can be applied to FWI algorithms without any particular modifications. Krebs et al. (9) studied the combination of simultaneousshot and phase-encoding (SSP E) techniques in time-domain FWI. They demonstrated the efficiency of SSP E implemented with random codes with D acoustic case studies. They addressed the sensitivity of SSP E to the length of the encoding function, the starting model, and the noise. The robustness of SSP E was shown when all of the individual shots were assembled into one super-shot for a signal-to-noise (S/N) ratio corresponding to an ocean-bottom cable survey in the Gulf of Mexico. The length of the codes has a minor effect on imaging quality. Therefore, short codes of length one can be used to derive accurate FWI models without extra computational costs. Codes of length one are equivalent to randomly multiplication of seismograms by or -. Krebs et al. (9) showed that a key issue is the regeneration of the random encoding of the individual sources at each nonlinear iteration of

5 the FWI. They obtained a 5-fold computational speed-up, which corresponds to the total number of shots of the D acquisition used for the D Marmousi case study. Herrmann et al. (9a) recasts the simultaneous-shot technique with source-encoding in the theoretical framework of compressive sensing and discusses the implications of these techniques for FWI. Neelamani et al. (8); Herrmann et al. (9b) propose to speedup the forward problem by using simultaneous-shot technique with phase encoding followed by the nonlinear recovery of the source-separated data volumes exploiting the transform-domain sparsity property of the wavefields. The use of the nonlinear recovery of the source-separated data volumes for FWI remains however to be shown. An alternative strategy to reduce the volume of the data processed by FWI is to apply a plane wave decomposition of the data and to invert a limited number of plane waves. With such plane-wave decomposition, Vigh and Starr (8) obtained a speedup ranging between and compared to conventional prestack FWI. In the present study, we assess the SSP E strategy in the framework of efficient frequencydomain FWI applied to wide-aperture seismic data, where a limited number of increasing frequencies are inverted hierarchically following a multiscale approach. The multiscale approach is designed to mitigate the nonlinearity of the inverse problem by injecting progressively higher wavenumbers into the subsurface model as the inversion proceeds towards higher frequencies (Pratt and Worthington, 99; Pratt et al., 996; Pratt, 999). A computationally efficient inversion is designed by limiting the number of inverted frequencies that take advantage of the redundant control of frequency and aperture angle on the wavenumber coverage when wide-aperture geometries are considered (Sirgue and Pratt, ; Brenders and Pratt, 7). The most efficient strategy consists of the successive inversion of coarsely sampled single frequencies. However, it has been shown in several contexts that improved 5

6 imaging with a higher S/N ratio can be obtained by successive inversions of slightly overlapping frequency groups of increasing high-frequency content (Brossier et al., 9), where a frequency group denotes several frequencies that are simultaneously inverted. Multiscale approaches have been also proposed in the time domain where subdatasets of increasing frequency bandwith are successively inverted (Bunks et al., 995). Two main differences exist, however, between the time-domain and the frequency-domain multiscale approaches: first, the frequency bandwidth of the subdatasets that are simultaneously inverted in the time-domain approach increases over the successive steps. At the last step, the full bandwidth is involved in the inversion; second, a higher number of frequencies are simultaneously inverted in the time-domain approach during each step, through the time-domain sampling. One objective of the present study is to assess the impact on the SSP E technique of the data decimation or fold-reduction underlying efficient frequency-domain FWI, and in particular in the presence of noisy data. In the frequency domain, phase encoding simply consists of applying a phase shift to the complex-valued source. Several strategies can be viewed in frequency-domain FWI and will be extensively discussed here. One of these issues are the type of the source assembling (full versus partial). In full source assembling, all of the shots are gathered into one super-shot, while in partial source assembling, only a limited number of shots with different possible spatial distributions are assembled into super-shots. We shall use random encoding which was shown to provide the best results compared to deterministic encoding (Ben Hadj Ali et al., 9a,b). When random encoding is used, the codes can be regenerated at different locations of the multiscale frequency-domain FWI algorithm, which is expected to have significant impact on frequency-domain FWI, as seen for the time domain (Krebs et al., 9). We focus here on wide-aperture/wide-azimuth acquisition geometries such as offshore node surveys or land surveys. These geometries imply 6

7 that each shot is recorded by all the receivers, and are, therefore, suitable for simultaneousshot techniques since all the shots can be stacked in the full computational domain. The computational savings provided by SSP E in frequency-domain FWI depends on the numerical approach used for the seismic modeling. Seismic-wave modeling for frequencydomain FWI can be performed with different numerical approaches that are implemented in the frequency or the time domains (Virieux et al. (9) for a review). In the frequency domain, seismic modeling reduces to the resolution of a large and sparse linear system that results from the discretization of the wave equation (Marfurt, 98). The right-hand side of the system is the source, and the solution is the monochromatic wavefield. The linear system can be solved with a direct solver (e.g., Operto et al., 7), an iterative solver (e.g., Plessix, 7), or a hybrid solver (Sourbier et al., 8) that combines the use of the direct and iterative solvers in the framework of a domain-decomposition method (Smith et al., 996; Saad, ). Alternatively, seismic modeling can be performed in the time domain and monochromatic wavefields are extracted by the discrete Fourier transform integrated within the time loop (Sirgue et al., 8), or by phase-sensitive detection once the steady-state regime has been reached (Nihei and Li, 7). All of these approaches can be classified according to the relative costs of the source-independent tasks and the source-dependent tasks during seismic modeling. In the direct-solver approach, the source-independent task consists of the LU decomposition of the impedance matrix, which is computationally intensive, while the source-dependent task consists of computing the solution by forward/backward substitutions, which is computationally efficient. In contrast, the time-domain and the frequency-domain iterative approaches do not embed any source-independent tasks, which implies that the computational cost of these approaches increases linearly with the number of sources. Therefore, the time-domain and the iterative frequency-domain approaches are 7

8 expected to take full advantage of the SSP E method, unlike the direct-solver approach. In the present study, we use the hybrid-solver approach that represents a compromise between the direct and iterative approaches. In the first part of this study, we briefly review the hybrid-solver seismic modeling approach and the main steps of efficient multiscale frequency-domain FWI. Secondly, we review the principles of the SSP E technique and the implementation of SSP E in frequencydomain FWI. Third, we assess the method with a D synthetic case study performed with a dip section of the SEG/EAGE overthrust model. We show that the random encoding must be regenerated during each FWI iteration. We then compare the efficiency of the full source assembling and the partial source assembling for noise-free and noisy data. The results show a strong sensitivity of the full source assembling to noise, in contrast to partial shot assembling. The simultaneous inversion of multiple frequencies is efficient to mitigate the sensitivity of the full source assembling to noise, at the expense of the computational efficiency. In the last section, a preliminary application to D efficient frequency-domain FWI is presented, to highlight some differences with the D applications. A table of mathematical symbols is presented in Table. METHOD Frequency-domain seismic wave modeling In the frequency domain, acoustic wave modeling can be recast in matrix form as Ap = s, () 8

9 where A is the large and sparse complex-valued impedance matrix, p is the pressure field and s is the source term (Marfurt, 98). For small-scale problems (D or D at low frequencies), this sparse system can be solved efficiently for multiple sources using direct solvers (Marfurt, 98). Direct solvers first perform a LU factorization of the impedance matrix before computing the solution by forward and backward substitutions (e.g., Duff et al., 986). This approach is efficient for modeling with multiple sources because LU decomposition, which is the most expensive part of the process, is independent of the source, and therefore is performed only once. However, the time and memory cost of the LU factorization together with its limited scalability on large-scale distributed memory platforms prevents the application of the direct solver to large-scale D problems (Operto et al., 7; Brossier et al., ). In the present study, we use a highly scalable hybrid direct-iterative solver that is based on a substructuring domain-decomposition method and the algebraic Schur complement approach (Smith et al., 996; Saad, ) to solve equation. Although a detailed description of the method is beyond the scope of this study, a brief review is given here, and for additional details, the reader is referred to Haidar (8), Sourbier et al. (8), and Virieux et al. (9). The wave equation is solved with the finite-difference scheme of Operto et al. (7). The computational domain is subdivided into subdomains, and one subdomain is assigned to one processor in the framework of parallel computing. The unknowns (i.e., the values of the monochromatic wavefield at each node of the finite-difference grid) are sorted into interior unknowns and interface unknowns. Interior unknowns belong to the interior of the subdomains, while interface unknowns belong to the edges of the subdomains and are shared by two adjacent subdomains. This sorting of the unknowns in equation according to interior and interface nodes allows us to build the reduced Schur complement system for 9

10 the interface unknowns. This reduced system is solved with a Krylov-subspace iterative solver, such as the generalized minimal residual method (GMRES) (Saad, 986; Frayssé et al., 997). Of note, the right-hand side of the Schur complement system depends on the source, and therefore a Schur complement system must be solved for each shot. Once the interface unknowns have been computed, the interior unknowns are efficiently computed by matrix-vector products and local substitutions for each shot. In this study, the aim of the SSP E method will be to mitigate the computational cost associated with the resolution of the Schur complement system for multiple right-hand sides during iterative frequencydomain FWI. In the hybrid approach, source-independent tasks consist of sequential LU factorization of small local impedance matrices assembled on each subdomain/processor and the building of the preconditioner of the Schur complement system. The SSP E method will not provide any computational savings for these tasks. The relative computational cost of the source-dependent and source-independent tasks in the hybrid approach depends on the number (i.e., on the size) of the subdomains. More the number of the subdomains is increased, more the elapsed time for the source-independent tasks is made negligible because the complexity of the sequential LU factorizations on each subdomain decreases in O(N 6 ), where N denotes the size of a D cubic N subdomain (Ben Hadj Ali, 9). In contrast, the elapsed time for the source-dependent tasks remains roughly constant when the number of subdomains is increased because the fact that each processor performs a numerical problem of smaller size during the iterative solve of the Schur complement system is balanced by the increase of the number of GMRES iterations resulting from the degradation of the Schur complement preconditioner. Therefore, the computational saving provided by the SSP E technique is expected to be more significant, when the hybrid approach is used on a large number of processors.

11 Frequency-domain FWI In this study, FWI consists of the minimization of the weighted least-squares misfit function (Tarantola (98); Pratt et al. (998) and Virieux and Operto (9) for a review) given by, C(m) = d W d d + γ (m m prior) W m (m m prior ), () where m is the model, d is the data residual vector, the difference between the recorded data d o and the modeled data d c (m), W d and W m are data-space and model-space weighting operators, the inverse of the covariance matrices in the framework of maximum likelihood methods, denotes the complex conjugate transpose, γ is a damping factor. The minimization of the misfit function is performed by iterative local-optimization. Minimization of the misfit function in the vicinity of a starting model m gives for the model perturbation vector: m [H ] C(m ) = [ ) ] ] R (J W dj + γw m R [J W d d, () where J is the sensitivity or Fréchet derivative matrix computed for a starting model m, R denotes the real part of a complex number and we use m = m prior. In the right-hand side term of equation, H and C(m ) are the approximate Hessian and the gradient of the misfit function, respectively. Of note, the second-order term of the Hessian which accounts for double-scattering effects in non linear problems was dropped off in equation (Pratt et al., 998). Since the approximate Hessian is computationally intensive to compute, we use here a preconditioned steepest-descent method (Tarantola (987), page ) instead of a Gauss- Newton method, equation : m = αmwm C(m ) m, ()

12 where α is the so-called step length (Ben Hadj Ali et al., 8a). For the gradient preconditioner M, we use the inverse of the diagonal terms of the approximate Hessian to compensate for geometrical spreading effects (Ravaut et al., ). M = { } diag J J, (5) + γi where the damping factor γ is defined by trial-and-error (Ravaut et al., ) and I is the identity matrix. The computation of M requires the explicit computation of the sensitivity matrix, that is, the modeling for each source and receiver positions. To mitigate the computational burden resulting from the modeling for each source and receiver position, M is computed only at the first iteration of one frequency-group inversion and is kept constant over iterations (Ravaut et al., ; Operto et al., 6). The operator W m is generally a roughness operator in the framework of Tikhonov regularization (e.g., Hansen, 998). In this study, we use a smoothing Gaussian function G m for W m to filter out high-frequency artifacts in the gradient. The horizontal and vertical correlation lengths of the Gaussian function are defined as fractions of a mean propagated wavelength for one given frequency (Ravaut et al., ). The step length is computed by parabolic fitting (e.g., Tarantola, 987). In this study, we also shall use the identity matrix for the data weighting operator W d. The gradient can be efficiently computed by the adjoint-state method (Tarantola, 987; Plessix, 6; Chavent, 9). In the frequency domain, the contribution of one shot and one frequency to the gradient is given by the product of the incident monochromatic wavefield emitted by the shot with the back-propagated residual wavefield and with a sparse radiationpattern matrix D, which is obtained by differentiating the impedance matrix with respect to the model parameter (Pratt et al., 998). For inversion of one frequency i and of multiple

13 sources, the expression of the gradient for the parameter m l is given by: ( ) C(m ) m l i = = N s j= N s j= p T ijd T il A i P d ij p T ijd T il r ij, (6) where p ij = A i s j is the incident monochromatic wavefield associated with the source j and frequency i, r ij is the associated adjoint back-propagated wavefield, and T denote the complex conjugate and the transpose, respectively. P is an operator with augments with zeroes the vector d to match the dimension of the matrix A (Pratt et al., 998). For simultaneous inversion of multiple frequencies, the regularized gradient is the sum of the smoothed gradient contribution of each frequency, W C(m ) m = N f i= ( ) C(m ) G i. (7) m i Simultaneous-shot technique The computational burden in FWI that results from multisource simulations can be mitigated by summing monochromatic sources into super-shots s: N s s = s j, (8) j= where s denotes the monochromatic super-shot. The computational time saving during one FWI iteration is, ideally, proportional to the number of sources summed into the super-shot (equation 6). In virtue of the linear relationship between the seismic wavefield and the source, the seismic wavefield, p, associated

14 with a super-shot is the sum of the individual wavefields associated with each source: p = A s N s = p j. (9) The same reasoning applies to the corresponding adjoint wavefield where the adjoint residual source is the sum of the adjoint residual sources associated to each individual shot. j= Plugging the expression of the super-shot wavefield, equation 9, and of the corresponding adjoint-wavefield in the expression of the gradient, equation 6, gives: ( ) C(m ) m l i = N s j= p T ijd T li r ij + N s k= k j p T ijd T li r ik. () The first term in the right-hand side of equation corresponds to the standard gradient formed by stacking the contribution of each individual shot, while the second term corresponds to cross-talk interferences between sources j and k. This extra term alters the imaging result. The minimization of the related artifacts is achieved by the so-called phase-encoding technique. Note that this reasoning would apply in a same manner to the (diagonal) Hessian computation. It is worth noting here that the number of correct terms in the gradient increases linearly with the number of sources, while the number of crosstalk terms increases as a quadratic with the number of sources. Phase encoding Encoding sources with arbitrary phase shifts (referred to as codes in the following) during their assembly into a super-shot can reduce cross-talk artifacts. The monochromatic supershot with phase encoding is given by: N s s = a j s j, () j=

15 where a j = exp(ιφ j ) = and ι =. The corresponding super-shot incident and adjoint wavefields are respectively given by: p = r = N s j= N s j= a j p j, () a jr j. () The conjugate of a j is taken in the expression of r, equation, because the source of the adjoint wavefield is the conjugate of the data residuals. Plugging the expression of the encoded super-shot and adjoint wavefields, equation, in the expression of the gradient, equation 6, gives: ( ) C(m ) m l i = N s j= p T ijd T li r ij + N s k= k j a ij a ik pt ijd T li r ik. () Of note, the frequency index i was added to the codes in equation because one may consider to regenerate the phase encoding for each frequency during simultaneous inversion of multiple frequencies, equation 7, as we shall show later. The codes impact on only the crosstalk terms. Therefore, judicious choice of the code phases, φ j, can minimize the crosstalk noise while leaving the correct gradient unchanged. Several phase-encoding strategies have been proposed, such as deterministic phase encoding (DP E: Deterministic Phase Encoding) (Jing et al., ) and random phase encoding (RP E: Random Phase Encoding) (Morton and Ober, 998; Romero et al., ). RP E generates random phases in the interval [, π]. DP E relies on the assumption that sources in a super shot are closelyspaced, and it defines the phase φ j of the j th shot as a function of the previous phases φ n ; n =,,..., j, as: j n= tan(φ j ) = cos(φ n) j n= sin(φ n). (5) 5

16 An alternative phase encoding (P P E: Periodic Phase Encoding) can be viewed, which consists of taking equidistant phases in the interval [, π], i.e. φ j = (j )π/n, where N is the number of sources in a super-shot. The DP E, RP E and P P E were assessed in the framework of frequency-domain FWI in Ben Hadj Ali et al. (9a,b) who concluded that the RP E was providing the best results. Therefore, we shall use RP E in this study. Phase encoding and frequency-domain FWI In the frequency domain, the encoding is performed simply through multiplication by a complex term a j = exp(ιφ j ). Therefore, neither the generation nor the application of phase encodings significantly impact on the computational cost of SSP E. In multiscale frequency-domain FWI, where only a few discrete frequencies or a few groups of several frequencies are inverted successively, phase codes can be regenerated at different stages of the inversion. A conventional frequency-domain FWI algorithm embeds three main loops (Algorithm ): the outer loop is over frequency groups, the second loop is over FWI nonlinear iterations and the inner loop is over the frequencies of the frequency group. More detailed descriptions of such algorithms are provided in Sourbier et al. (9a), Ben Hadj Ali et al. (8a), and Brossier et al. (9). Phase codes can be regenerated within the outer loop of the algorithm when FWI starts processing a new group of frequencies. Alternatively, phase codes can be regenerated within the middle loop over FWI iterations. The last option is to regenerate the codes in the inner loop over the frequencies of the group. The shot assembling can be performed by summing all of the shots in one super shot. In the following, this strategy will be referred to as full source assembling. In this case, many 6

17 Algorithm Multiscale frequency-domain FWI algorithm. N group : number of frequency groups. N f : number of frequencies in one frequency group. These frequency components are simultaneously inverted. k: iteration number. k max : maximum number of iterations. : for i group = to N group do : while (NOT convergence OR k < k max ) do : Set starting model : for i = to N f (i group ) do 5: Build encoded super-shots 6: Propagate wavefields from super-shots 7: Compute residuals d and misfit function C m 8: Back-propagate residual sources 9: Compute gradient-update associated with frequency i : Smooth gradient-update associated with frequency i : Update the total regularized gradient Wm C(m ) m : end for : if iter = then : Compute diagonal Hessian 5: end if 6: Scale gradient vector by diagonal Hessian, δm = MWm C(m ) m 7: Compute step length α by parabola fitting ( estimations of the misfit function) 8: Update model m k+ = m k + α m k 9: end while : end for 7

18 FWI iterations would be required before convergence towards an acceptable model. In the opposite strategy, only a limited number of shots with different possible spatial distributions can be assembled into super-shots. In the following, this opposite strategy will be referred to as partial source assembling. Two main kinds of spatial distribution of sources can be used during partial source assembling: closely-spaced sources, and distant sources, which are referred to as the cluster and the coarse approaches, respectively. In the coarse approach, the spacing between two next shots of a super shot is the shot spacing of the survey multiplied by the number of shots assembled in the super shots. This criterion allows us to use a uniform shot spacing within each super shot. In the case of partial source assembling, fewer FWI iterations should be required to reach convergence towards an acceptable model, but more super-shots must be modeled during each FWI iteration. Compared to closely-spaced sources, distant-source assembling allows to confine the interferences between individual shots at longer offsets and to preserve the directivity of each individual sources but does not provide the same flexibility to limit the computational aperture used in the inversion. Efficient frequency-domain FWI processes a reduced data volume by inverting a limited number of coarsely sampled frequencies at a time. The impact of this data reduction in efficient frequency-domain FWI should be assessed when the SSP E is used. In the frequency domain and for a given acquisition spread, the data redundancy is controlled by the number of sources and the number of inverted frequencies. All the above-mentioned factors that should impact on the efficiency of frequencydomain FWI with SSP E lead to the following definition of the computational speed-up S provided by the SSP E approach: S = T ref T SSP E = ref T pre + Ns T pre + N SSP E s N ref it N SSP E it N ref f T sol N SSP E f T sol, (6) 8

19 where T ref, Ns ref, N ref it, N ref f and T SSP E, N SSP E s, Nit SSP E, Nf SSP E denote the total FWI elapsed time, the number of sources, the number of FWI iterations and the number of inverted frequencies when FWI is applied without and with SSP E, respectively. T pre and T sol denote the elapsed time for the source-independent task and for one source-dependant task in the forward problem. For example, T pre and T sol are the elapsed times for the LU decomposition and for the substitution phase, respectively, in frequency-domain direct methods. In the following, the speed-up will be the criterion to define the efficiency of the SSP E method in the framework of frequency-domain FWI for noise-free and noisy data. The quality of the FWI models will be assessed by the square root of the normalized least-squares norm of the differences between the true model and the FWI model given by: ( ) mf W I m true / err m =, (7) m true where m F W I and m true denote the FWI and the true models, respectively. The number of iterations in conventional FWI was fixed before the inversion and was the same for each frequency group. It was defined such that the final FWI model was considered of sufficient quality from a qualitative viewpoint. To define the number of iterations, frequencies and super-shots in SSP E, we shall use the final FWI model obtained by conventional FWI as a reference such that the quality of the final FWI models obtained by SSP E and conventional FWIs are of comparable quality. The number of iterations in SSP E FWI was also kept constant over frequency groups (see equation 7) although the crosstalk noise may not have the same impact at different frequencies. Although this tuning may not be optimal, it is difficult to predict accurately the minimum number of FWI iterations for each frequency group that will lead to a given quality of the final FWI model. Therefore, the speedups and the model qualities provided in the following sections should be viewed as guidelines 9

20 to bring out the main features of the SSP E method rather than accurate estimations of computational costs. NUMERICAL EXPERIMENTS In all the synthetic numerical examples now presented, the same frequency-domain seismic modeling code that is based on the hybrid solver is used to compute the recorded data in the true model and the modeled data during the FWI iterations. However, we use different stopping criteria for the iterations of the GMRES Krylov-subspace iterative solver for computing the recorded and the modeled data. In GMRES, the stopping criterion of iterations is given by ɛ = Ap(n) s s, where n denotes the GMRES iteration. We use a small value of ɛ for the computation of the recorded data to guarantee accurate recorded wavefields (ɛ = ), while we use a higher value of ɛ (ɛ = ) for the modeled wavefields during FWI iterations, to save on computational time. Ben Hadj Ali et al. (8b) showed that the wavefield accuracy for ɛ = was sufficient for FWI applications, which will be further confirmed by the numerical examples of the present study. D overthrust model case study: noise-free data In this section, we first assess the method for noise-free data. Two-dimensional experiments can be designed considering.5d velocity models (laterally invariant in the y direction) and an infinite-line shot in the y direction, to speed up the computation, although we consider D numerical simulations (Ben Hadj Ali et al., 8a). We apply D FWI to a dip section of the SEG/EAGE overthrust velocity model (Figure a), discretized on a 8 87 grid with a grid spacing h = 5m. For D application, the dip section of the overthrust

21 model is duplicated three times in the y direction, which leads to a D 8 87 finite-difference grid. Perfectly matched layer (PML) absorbing boundary conditions are set on the four edges of the D model, and periodic conditions are implemented in the y direction to mimic an infinite medium and an infinite line of sources. The computational domain was subdivided into and subdomains in the horizontal and vertical directions, respectively, for seismic modeling with the hybrid solver. For this domain decomposition, a sequential LU factorization takes around.5 s per subdomain and an iterative solve of the Schur complement system takes around.6 s for one right-hand side. The D acquisition geometry consists of a line of 99 sources and receivers, equally spaced on the surface with an interval of meters. The starting model for inversion is built by smoothing the true velocity model with a Gaussian function of horizontal and vertical correlation lengths of 5 m (Figure b). The true model is set in the first meters of the starting model to avoid instabilities in the weathered near surface which result from the vicinity of the sources and receivers to the PML layers (Ravaut et al., ). Such instabilities can also be removed by artificially augmenting the model on the topside to move away the sources and the receivers from the PMLs (Sourbier et al., 9b). We invert sequentially seven frequencies, ranging from.5 to Hz (.5,.76, 7., 9.6,., 6.97,.6 Hz). For each frequency, we compute 5 iterations. Smoothing regularization did not improve the results of conventional FWI, and therefore, was not applied during either conventional or SSP E FWI for a fair comparison between the two inversions. The final FWI model obtained without shot assembling is shown in Figure c. This model will be used as a reference FWI model to quantify the impact of the SSP E technique on the FWI model qualities.

22 Partial source assembling We first consider partial source assembling. Twenty five super-shots of eight sources each are inverted. We build super-shots by gathering eight sources as either close (cluster supershot) or distant (coarse super-shot), and we perform 5 iterations per frequency. The FWI model obtained with shot assembling using the cluster approach but without phase encoding is shown in Figure a. The footprint of the source assembling is clearly visible, especially in the shallower part of the model. Summing closely spaced sources without phase encoding has a similar footprint as increasing the shot spacing, because the resulting super-shot can be viewed as a single source that is spatially distributed in the horizontal direction. Indeed, this footprint affects the shallow part of the model more, where a horizontal periodic pattern can be clearly seen with a period corresponding to the spatial support of eight sources (i.e., 8 m). The FWI models obtained with the RP E, are shown in Figure (b-c), and they do succeed in reducing the cross-talk noise. The random codes were regenerated within the outer loop over frequencies and within the middle loop over FWI iterations in Figures b and c, respectively. Comparisons between these two models clearly shows that the best strategy is to regenerate the random codes at each iteration of the FWI iterations, which is consistent with the conclusions of Krebs et al. (9) that were inferred from time-domain FWI. The final FWI velocity models obtained with the coarse approach are shown in Figure. The shot spacing in a super-shot is 8 x = 8 m. The FWI model obtained with shot assembling but without phase encoding is shown in Figure a. The footprint of the crosstalk artifacts is less obvious than for the cluster approach (Figure a). As the assembled sources are distant, this allows for the preservation of the full directivity of the explosive

23 sources, and therefore of the aperture illumination of the subsurface. Furthermore, the significant distance between the assembled sources helps to make the sum of the crosstalk incoherent. The FWI models obtained with RP E is shown in Figure b. The RP E approach provides an accurate FWI model, which is slightly better than that obtained with the coarse approach, especially in the shallow part of the model (Figure c and Table ). FWI with partial source assembling is nearly 8-time faster than FWI without shot assembling, which is consistent with the ratio between the number of seismic modelings performed during FWI without and with SSP E (Table ). In partial source assembling, the source-independent task in the hybrid modeling approach has a minor impact on the speedup, equation 5, because the overall cost of the modeling remains dominated by the source-dependent tasks. Full source assembling We now consider full source assembling. All of the 99 sources are summed into one supershot. The same experimental set-up is used as in the previous experiment. The final FWI model obtained without phase encoding is shown in Figure a. The number of iterations per frequency inversion was increased to. As expected, the inversion did not converge towards an acceptable velocity model because a super-shot formed by closely spaced sources is equivalent to a horizontal plane-wave source. Together with the coarse sampling of the frequencies in efficient frequency-domain FWI, the limited apertureangle illumination provided by a single plane wave leads to a narrow bandwidth and coarse sampling of the wavenumber spectrum of the model (see for example, Sirgue and Pratt () for a resolution analysis of FWI). The coarse sampling of the model wavenumbers

24 leads to the wrap-around of the reflectors in the spatial domain (Mulder and Plessix, ). To further verify this statement, we simultaneously inverted 9 frequencies within the [.5 Hz -.6 Hz] frequency band. Compared with the case of the sequential inversion of seven frequencies (Figure a), the final FWI model is greatly improved (Figure b), although a significant deficit in the small wavenumbers is observed because of the poor illumination of the wide-aperture angles by the horizontal plane-wave source. The FWI models inferred from full assembling and RP E are shown in Figure 5, when the seven above-mentioned frequencies are inverted sequentially. Two hundred iterations per frequency were computed. FWI models are shown after inversions of frequencies.5 Hz (starting frequency), 7. Hz and.6 Hz (final frequency). Some artifacts remain in the final FWI model in the low-velocity layer at a -km depth on the right-hand side of the model (Figure 5c). Of note, a significant amount of cross-talk noise is still present in the FWI models after inversion of the starting (.5 Hz) and intermediate (7. Hz) frequencies, whereas the footprint of the cross-talk noise is minor in the final FWI model. This suggests that the summation of the multiscale FWI models over frequencies also contributes to the mitigation of the footprint of the cross-talk noise generated by previous frequencies. In an attempt to improve the FWI results with the full source assembling, we applied FWI using the same frequencies as above, but with four frequency groups now successively inverted, instead of the single frequencies. The frequency groups without overlap are [.5,.76], [7.,9.6], [.,6.97],.6 Hz. The random phase encoding was regenerated at each iteration, which implies that the same encoding was applied to each frequency of one group. Final FWI models are shown in Figure 6(a-b) when 5 and FWI iterations per frequency group are computed, respectively. The final FWI model obtained when iterations per frequency group are performed is greatly improved upon and closely matches

25 the reference FWI model (Figure 6b and Table ). Comparison of the FWI models obtained with 5 (Figure 6a) and (Figure 6b) FWI iterations per frequency group confirms that, when full source assembling is used, a high number of FWI iterations is required to achieve an acceptable S/N ratio and spatial resolution. We obtained a speed up S of when FWI iterations per frequency group are performed. Ideally, we should have reached a speed up of 5 according to the number of sources and the number of FWI iterations, without and with SSP E (5 iterations and 99 sources, against iterations and super-shot) (Table ). However, unlike in the partial source assembling case described before, the cost of the source independent tasks has a significant impact on the speedup of the full assembling approach because the number of super-shots was decreased from 5 to. As mentioned in the section Frequency-domain seismic wave modeling, one strategy to limit the cost of the source-independent tasks would have been to increase the number of subdomains in the domain decomposition. D overthrust model case study: noisy data We now assess the sensitivity of efficient frequency-domain FWI to noise when the SSP E technique is applied to the D overthrust case study. In the following, we define the signalto-noise (S/N) ratio as S/N = P S P N, (8) where P S and P N denote the signal power and the noise power, respectively. We consider colored random noise with a uniform spatial distribution in the source-receiver plane. By colored noise is meant that the same S/N for each frequency component is used: the power of 5

26 the noise is adapted to the power of the frequency component. The noise will be white only if the source wavelet is a Dirac function. Use of colored noise may be not representative of real data, where frequencies below 5 Hz can have a poor S/N. However, within the relatively-narrow frequency range [ Hz] investigated in this study, the assumption of the colored white noise is reasonable. Figure 7 shows a receiver gather of a D OBC (Ocean Bottom Cable) data set from the Valhall field. The S/N measured on this receiver gather suggests that the S/N is of the same order of magnitude over frequencies within the [.5-5 ] Hz frequency band. One motivation behind the use of the colored noise was to assess the sensitivity of the inversion to crosstalk noise as a function of frequency during the multiscale reconstruction. Monochromatic data with or without noise are shown in Figure 8 for three S/N ratios:., 5. and.. The reference FWI models obtained without source assembling are shown in Figure 9, when the four groups of frequencies defined in the previous section are successively inverted using 5 FWI iterations per frequency group ([.5,.76], [7.,9.6], [.,6.97],.6 Hz). The quality of these models is given in Table. Full source assembling FWI with full-source assembling and RP E is now applied to the noisy data considering the four frequency groups just described. Two hundred iterations are computed per frequency group. In contrast to the previous examples, the FWI results are shown when a smoothing regularization implemented with a Gaussian smoother is applied to the gradient at each FWI iteration, as in equation 6. The final FWI models obtained for the three S/N ratios are shown in Figure. Comparisons between FWI models obtained with full assembling from noise-free (Figure 6b) and noisy data (Figure ) shows the strong sensitivity of efficient frequency-domain FWI to noise when full source assembling is used. 6

27 To further mitigate the cross-talk noise in full source assembling, we investigate the impact of the number of sources in the acquisition, and of the number of frequencies in the frequency groups. Sensitivity of full source assembling SSP E to the number of sources. Refining the number of shots in the acquisition modifies, on the one hand, the redundancy of the summation over shots performed during gradient building, as in equation 6, and, on the other hand, the ratio between the number of correct terms and the number of crosstalk terms in the gradient, as in equation. These two factors can have antagonistic effects in the quality of the FWI model. We consider a S/N ratio of 5 and compare the FWI results obtained with 99, 99 and 99 sources in Figure. The best model is obtained for 99 sources which seems to provide the best trade-off between the need to stack sufficiently-redundant data and the need to mitigate the crosstalk noise. The worse model is obtained for 99 shots, that shows that increasing the number of shots can dramatically increase the crosstalk noise because of the quadratic increase of the number of crosstalk terms with the number of shots, equation. Sensitivity of full source assembling SSP E to the number of inverted frequencies. We now increase the number of frequencies within each frequency group. For each experiment, the inverted frequencies are evenly sampled within the [ Hz] frequency band and non-overlapping frequency groups are defined such that the smallest frequency of each group is roughly.5 Hz, 5 Hz, Hz and 7 Hz. A S/N ratio of 5. is still considered, and FWI iterations are computed per frequency group. One hundred ninety-nine shots are used in the acquisition. The final FWI models obtained for the inversion of a total of 7, 6 and 9 frequencies are shown in Figure (a-c). The FWI models are seen to be greatly improved when more frequencies are injected in the inversion (Table ). For 9 inverted 7

28 frequencies, the speedup drops to. while the theoretical speedup is.6. Of note, the quality of the model inferred from the inversion of 9 frequencies is higher than that of the reference FWI model. This suggests that the velocity amplitudes were better recovered by SSP E FWI than by conventional FWI because of the large number of iterations used in SSP E. The better amplitude recoverage in the SSP E model compensates for the crosstalk noise in the assessment of the model quality. A final improvement can be obtained by regenerating the random codes within the innermost loop over frequencies during each iteration of a frequency-group inversion rather than at the beginning of each iteration. If the frequencies within a frequency group are sufficiently close, the crosstalk noise generated by each frequency should have a similar frequency content. Regeneration of the random encodings for each frequency should favor the incoherent summation of the crosstalk terms during the summation of the partial gradients associated with each frequency of one group during a FWI iteration (see equation 7). The final FWI model obtained using this encoding strategy when 9 frequencies are inverted is shown in Figure d, and is slightly improved when compared with that obtained when the encoding is regenerated at the beginning of each FWI iteration (Figure c and Table ). In particular, the imaging of a channel at a distance of km and a depth of.7 km has been clearly improved in Figure d. Partial source assembling Since the full source assembling shows a strong sensitivity to noise, the partial source assembling is now considered for noisy data. The three S/N ratios previously considered are used (., 5. and.). We apply FWI to the frequency groups: [.5,.76], [7.,9.6], 8

29 [.,6.97] and.6 Hz, and we perform 5 iterations per frequency group. The final FWI models show limited sensitivity of the partial source assembling to noise (Figure ), as suggested by the comparison with the reference FWI models obtained without source assembling (Figure 9). The speed-up obtained with the partial source assembling is 8, while a speed-up of. was obtained with the full source assembling. The quality of the FWI model obtained by partial source assembling is, however, less than obtained with full assembling and 9 frequencies (Table ), and additional FWI iterations would have been required to reach an equivalent quality. We conclude, however, that partial source assembling outperforms the full source assembling when efficient frequency-domain FWI with SSP E is applied to noisy data. A D overthrust model case study We now consider the application of D FWI to the SEG/EAGE overthrust velocity model. The finite-difference grid is with a grid spacing 75 m. An additional six grid points are added all around the models for the PML absorbing boundary conditions. The computational domain was subdivided into subdomains for the domain decomposition used by the hybrid solver for seismic modeling. The acquisition is composed of (5) sources and 6 6 (676) receivers. The 5 sources are summed to build one super-shot in the framework of full source assembling. Vertical and horizontal slices of the true and initial velocity models are shown in Figure (a-b). The starting model was built by smoothing the true model with correlation lengths of 5 m without including the true velocity structure in the near surface. Three single frequencies were inverted:.5, 5. and 7. Hz. The same grid interval was used for each of the three frequencies. 9

30 In the first application, the data are noise free. Two hundred fifty and 5 FWI iterations were performed for the first frequency and for the last two frequencies, respectively. A vertical section and a horizontal slice of the FWI velocity models after the inversion of the first and the third frequencies are shown in Figure (c-d). Crosstalk noise still impacts the FWI models at these low frequencies, which is consistent with the D results shown in Figure 5. Qualitative comparisons between the vertical sections obtained by D FWI (Figure b) and D FWI (Figure 5b) at the frequency of 7 Hz suggests that the amplitudes of the cross-talk noise in the D-FWI model are similar to those in the D one. However, the model inferred from the D FWI is better resolved and more accurate than the model inferred by the D FWI in the shallow part, whereas the deep part is better resolved in the model inferred by D FWI. One reason might be that the increase fold provided by D acquisition allows for improved reconstruction of the shallow part of the model during the early stages of the inversion at low frequencies. In contrast, more FWI iterations might be required by D FWI to build the deeper part of the model. Indeed, the deep parts of the model are generally more difficult to reconstruct because they are mainly constrained by long-offset data which have weaker amplitudes than short-offset data and, which leads to poorly conditioned data. Three-dimensional FWI might be more sensitive to these poorly conditioned data than D FWI because of the increased number of model parameters and data involved in the D FWI. In the second application, we added random noise. S/N ratio is.. At each frequency, FWI iterations were performed. The FWI velocity models after the inversions of frequencies.5 Hz and 7. Hz are shown in Figure (e-f). As in D, the quality of the FWI velocity models is degraded when noisy data are inverted. Of note, no smoothing regularization was applied for this numerical example because the smoothing tends to spread an

31 instability triggered in the weathered layer. Velocity profiles extracted from the true velocity model, the starting model and the 7-Hz FWI velocity model are compared in Figure 5, and these show that the long-wavelengths of the true model have been reasonably well reconstructed. The quality of the starting models and of the FWI models inferred from the noise-free and noisy data are outlined in Table. Although the FWI velocity models are affected by crosstalk noise, their quality is higher that of the starting model thanks to the improved resolution of the velocity estimation. Figure 6 shows the misfit functions for the two FWI applications plotted as a function of iteration number. The misfit function does not decrease monotonically, because the phase encoding modifies the misfit function at each FWI iteration as already noticed by Krebs et al. (9). The curves also show that the convergence was not fully reached and that an increased number of FWI iterations would have been necessary to obtain an acceptable S/N ratio. As in D, inverting frequency groups instead of single frequencies should help to improve the accuracy of the SSP E method. A more detailed parametric analysis of D FWI using source assembling and phase encoding is still required. The two D applications were performed on the IBM Blue Gene/P of the IDRIS computer center. The simulation needed 6 cores with GB of memory each. The average time for one inversion iteration was about 5 s (nearly days per frequency). DISCUSSION In this study, we have assessed a simultaneous-shot and phase-encoding strategy in the framework of efficient frequency-domain FWI where a limited number of frequencies are inverted independently. We have tested two main strategies for shot assembling: partial

32 source assembling where a limited number of shots are summed in several super-shots, and full source assembling, where all of the shots are summed into one single super-shot. We first showed that the random codes must be regenerated within the innermost loop of the algorithm, i.e., in the loop over simultaneously-inverted frequencies. When partial source assembling is used with eight shots per super-shot, the same number of frequencies and the same number of iterations as for conventional FWI without source assembling provide models of acceptable quality, even when the data are noisy. In this case, the speed-up provided by SSP E is of the order of the number of shots in one super-shot: eight for the case study performed in the present study. The quality of the model can be further improved by increasing the number of iterations, at the expense of the computational efficiency. When full-source assembling is used, the SSP E is more sensitive to crosstalk noise. When no noise is added to the data, successive inversions of the frequency groups, rather than successive inversions of the single frequencies, improves the quality of the FWI velocity model without increasing the number of inverted frequencies. Even if the number of FWI iterations required by full source assembling is significantly higher than that for partial source assembling ( versus 5, respectively), a higher speed-up is obtained with full source assembling for noise-free data. However, the full source assembling shows strong sensitivity to noise in the data. In the case of noisy data, increasing the number of frequencies per frequency group is the best remedy to improve the quality of the FWI models, at the expense of the computational costs. In contrast, increasing the number of shots in the acquisition beyond a given limit can dramatically degrade the results. Increasing the number of inverted frequencies in full source assembling makes this approach less computationally efficient than partial source

33 assembling in the case of noisy data with a S/N ratio of 5. The results obtained with full source assembling in this study might appear contradictory to those of Krebs et al. (9), who applied time-domain FWI with full source assembling and RP E. The results of Krebs et al. (9) did not show any significant footprint of noise on time-domain FWI with SSP E when a realistic amount of noise was added to the data. This is, however, consistent with our conclusion that refining the frequency interval in frequency-domain FWI significantly improves the FWI results in the presence of noise. Indeed, frequency-domain FWI with a fine frequency sampling can be viewed as being equivalent to time-domain FWI. It is worth noting than Krebs et al. (9) simultaneously inverted the full frequency content of the data in one go without multiscale strategy. With such approach, only one final FWI model is produced and the assessement of the footprint of the crosstalk noise over FWI iterations is easy. In multiscale frequency-domain FWI, several inversions applied to difference subdatasets are hierarchically performed, where the final model of one iterative inversion is used as the starting model for the next inversion. This makes more difficult the appraisal of the SSP E method in terms of speedup and model quality, in particular, because we cannot predict the optimal number of iterations of each intermediate inversions that will give a final model of desired quality. The need of a significant number of frequencies in the SSP E method when full assembling is used requires to develop efficient frequency-domain iterative modeling methods for multifrequency modeling where the solution at a given frequency is exploited to speedup the modeling of the next frequency. This is required to make frequency-domain modeling competitive with time-domain one. Another conclusion of this study is that FWI is less sensitive to cross-talk noise at high

34 frequencies than at low frequencies. This statement is tentatively illustrated in Figure 7 which shows the difference between the model error obtained by conventional FWI and the model error obtained by SSPE FWI, ( err = err mref - err mssp E ), as a function of the inverted frequency groups. These curves are shown for partial and full source assembling without noise (corresponding FWI results are shown in Figures b and 6b, respectively). The curves are shown at the first and at the last iteration of each frequency-group inversion. If the error difference is the same at the first and last iterations of one frequency-group inversion, the crosstalk noise injected during the frequency-group inversion was fully removed. At low frequencies, the error difference increases between the first and the last iterations suggesting that the crosstalk noise was not fully removed. As the inversion proceeds towards higher frequencies, the gap between the error differences at the first and last iterations decreases progressively and can vanish. In the case of partial source assembling, this gap is removed after the third frequency inversion, whereas it vanishes at the last frequency-group inversion in the case of full source assembling. This illustrates on the one hand that partial source assembling is less sensitive to crosstalk noise as already shown and that crosstalk noise is more efficiently removed at high frequencies. Therefore, the best strategy might be to invert the low frequencies without SSP E with an efficient multishot approach, such as the direct-solver approach, and to make uses of SSP E only for the inversion of higher frequencies. A preliminary application of D FWI with SSP E is also presented. Comparisons between the D and the D results suggest that the crosstalk noise has a similar footprint in D and D FWI. However, a higher number of iterations can be required in D FWI compared with D FWI because of the higher dimensionality of both the model space and the data space in D applications.

35 CONCLUSION In this study, we have assessed the combination of efficient frequency-domain FWI where a limited number of discrete frequencies are inverted with a data-reduction technique based on shot assembling and source encoding. When all of the shots are assembled into one super-shot, and when noisy data are inverted, a significant number of finely-sampled frequencies must be involved in the inversion to converge towards velocity models with an acceptable S/N ratio. This conclusion is consistent with previous studies on time-domain FWI with source encoding, which have shown stable results when full source assembling is used with noisy data. Time-domain seismic modeling might be the most natural strategy to efficiently compute the frequencies required by robust D frequency-domain FWI with full source encoding, but iterative frequency-domain modeling engines can efficiently perform modeling for multiple frequencies. Alternatively, only a limited number of distant shots can be assembled into several super-shots. In this case, efficient frequency-domain FWI is robust with respect to noise, even without increasing the number of frequencies involved during the inversion. Data reduction techniques, such as source encoding, have been shown to be powerful tools to mitigate the computational burden of FWI associated with modeling for multiple sources. Although acoustic D FWI is feasible today at low frequencies, source encoding techniques should provide a valuable tool to perform exhaustive parametric analyses of D acoustic FWI and to begin investigations into D elastic FWI. ACKNOWLEDGMENTS This study was funded by the SEISCOPE consortium sponsored by BP, CGG-VERITAS, EXXON-MOBIL, SHELL and TOTAL, and by Agence Nationale de 5

36 la Recherche (ANR) under project ANR-5-NT5--7. Access to the high-performance computing facilities of the SIGAMM (Observatoire de la Côte d Azur) and IDRIS computer center (project 88) provided the required computer resources, and we gratefully acknowledge both of these facilities and the support of their staff. We would like to thank R. Brossier (Geoazur, now at LGIT) for very fruitful discussions. We would like the Associate Editor, Bill Harlan, J. Krebbs(ExxonMobil Upstream Research), and anonymous reviewers for their detailed and constructive comments. 6

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43 TABLES

44 Table : Symbols used in this study. Symbol Type Description RP E, DP E, P P E - Random, deterministic, periodic phase encodings i Integer Frequency index j Integer Shot index N f Integer Number of simultaneously-inverted frequencies N s Integer Number of shots A Complex Square impedance matrix associated with one frequency p Complex Monochromatic pressure wavefield vector s Complex Monochromatic source vector d Complex Data residual vector r = AP d Complex Adjoint wavefield vector P - Prolongation operator from the receiver domain to the computational domain C Real Misfit function C Real Gradient of the misfit function m Real Velocity model vector m Real Velocity perturbation model vector m Real Starting model for FWI m prior Real A priori model for FWI

45 Table : Following of Table W d, W m Real Data-space and model-space weighting operators k Integer FWI iteration number γ Real Prewhitening or damping factor α Real Step length J Complex Sensitivity matrix H = J W dj Complex Approximate Hessian M Real Gradient preconditioner G m Real Gaussian smoothing operator D li = A i m l Complex Radiation pattern matrix s = N s j= s j, Complex Monochromatic super-shot s p = N s j= p j, Complex Monochromatic wavefield associated with super shot a j Real random code S Real Speedup provided by the SSP E method compared to conventional FWI err m Real Square-root of the normalized least-squares error in the FWI model ɛ Real Stopping criterion in GMRES for hybrid-solver modeling S/N Real Signal to Noise ratio 5

46 Table : Results of the D overthrust case study. S.A: source assembling strategy. N S : number of sources. N S S : total number of super shots. S/N: signal-to-noise ratio. err m (%): model error. N its : number of FWI iterations. S: Observed speed-up. The theoretical one is between brackets. Figure S.A N S N S S S/N err m (%) N its S (b) Starting (c) reference (.) (c) cluster (8.) (b) coarse (8.) (5c) full (5.) (6b) full (5.) (9a) reference (.) (9b) reference (.) (9c) reference (.) (b) full (5.) (c) full (.6) (d) full (.6) (b) cluster (8.) 6

47 Table : Results of the D overthrust case study. S.A: source assembling strategy. N S : total number of sources. N S S : number of super shots. S/N: signal-to-noise ratio. err m (%): model error. N its : number of FWI iterations. S: Observed speed-up. The theoretical one is between brackets. Figure S.A N S N S S S/N err m (%) N its (b) Starting (c) full (e) full

48 FIGURE CAPTIONS 8

49 Figure : Two-dimensional overthrust case study. (a) True model. (b) Starting model for FWI. (c) Final velocity model inferred from conventional FWI. Figure : Two-dimensional overthrust case study. (a-c) Final FWI models obtained with partial source assembling and cluster approach. (a) Without P E. (b-c) With RP E. The random codes were regenerated within the outer loop over frequency groups (b) and within the middle loop over FWI iterations (b). (d) For comparison, reference model obtained by conventional FWI (Figure c). 9

50 Figure : Two-dimensional overthrust case study. (a-d) Final FWI models obtained with partial source assembling and the coarse approach. (a) Without P E. (b) with RP E. (c) For comparison, reference model obtained by conventional FWI (Figure c). Figure : Two-dimensional overthrust case study. Final FWI model obtained with full source assembling without phase encoding. (a) Seven frequencies were successively inverted. (b) Twenty-nine frequencies were simultaneously inverted. Figure 5: Two-dimensional overthrust case study. FWI models obtained with full source assembling and RP E. Seven frequencies were inverted sequentially. (a) FWI model after inversion of the starting frequency of.5 Hz. (b) As (a) for an intermediate frequency of 7. Hz. (c) As (a) for the final frequency of.6 Hz. Of note, the crosstalk noise is not fully cancelled at the starting and intermediate frequencies. 5

51 Figure 6: Two-dimensional overthrust case study. (a-b) Final FWI model obtained with full source assembling and RP E. Four frequency groups were successively inverted.(a) 5 and (b) iterations per frequency group. (c) For comparison, reference model obtained by conventional FWI (Figure c). Figure 7: (a) Raw receiver gather from the Valhall field. (b) Noise window extracted from (a). (c) Amplitude spectrum of the signal plus noise (gray) shown in (a) and of the noise (black) shown in (b). The amplitude spectra was low-pass filtered to highlight the main trend of the spectra. (d) Ratio between the power of the signal plus noise and the power of the noise. Figure 8: Two-dimensional overthrust case study. Monochromatic data in the sourcereceiver domain. (a) Without noise. (b-d) With noise. S/N ratios: (b).; (c) 5.; (d).. 5

52 Figure 9: Two-dimensional overthrust case study. Final reference models obtained by conventional FWI. S/N ratios: (a).; (b) 5.; (c).. Figure : Two-dimensional overthrust case study. Final FWI models obtained with full source assembling and RP E. S/N ratios: (a).; (b) 5.; (c).. Seven frequencies distributed over four frequency groups were inverted. Two hundred iterations per frequency group were computed. 5

53 Figure : Two-dimensional overthrust case study. Final FWI models obtained with full source assembling and RP E. S/N ratio is 5.. (a) Acquisition with 99 sources and a spacing of 5 m. (b) Acquisition with 99 sources and a spacing of m (Figure b). (c) Acquisition with 99 sources and a spacing of m. Seven frequencies distributed over four frequency groups were inverted. Two hundred iterations per frequency group were computed. Figure : Two-dimensional overthrust case study. Final FWI models obtained with full source assembling and RP E. S/N ratio is 5. (a) Seven frequencies distributed over four frequency groups were inverted (Figure b). (b) Sixteen frequencies distributed over four frequency groups were inverted. (c) Twenty-nine frequencies distributed over four frequency groups were inverted. (d) As (c), the random encoding is regenerated in the loop over frequencies at each FWI iteration. (e) For comparison, final reference FWI model obtained by conventional FWI for a S/N ratio of 5 (Figure 9b). 5

54 Figure : Two-dimensional overthrust case study. Final FWI models obtained with partial source assembling and RP E. Seven frequencies distributed over four frequency groups were inverted. Fifteen FWI iterations per frequency groups were computed. S/N ratios:(a).; (b) 5.; (c).. Figure : Three-dimensional overthrust case study. (a) Vertical (top) and horizontal (bottom) slices of the true overthrust model. The horizontal slice is at.5-km depth. The vertical slice is at y =. km. (b) Same as (a) for the starting model. (c) Same as (a) for the FWI model after inversion of the 7-Hz frequency in the case of noise-free data. (d) Same as (c) after inversion of the.5- Hz frequency. (e-f) Same as (c-d) in the case of noisy data. (b) As (a) for the FWI starting model. Figure 5: Three-dimensional overthrust case study. Vertical profiles extracted from velocity models at (a) x =.5 km and y = 7.5 km, and (b) x = 7.5 km and y = 7.5 km. Solid line, true model; coarse dashed line, starting model; thin dashed line, FWI model inferred from noise-free data; dotted line, final FWI model inferred from noisy data. 5

55 Figure 6: Three-dimensional overthrust case study. Misfit function versus iteration numbers for the three inverted frequencies indicated. (a) Noise-free data. (b) Noisy data. Figure 7: Difference between the model error of the reference FWI model (Figure c) and the model error of the SSP E FWI model at the first (filled diamond) and the last (open diamond) iteration as a function of the frequency group. The SSP E FWI model was obtained with partial source assembling (Figure ) (a) and with full source assembling (Figure 6b) (b). The aim of this figure is to assess the ability of the inversion to remove the crosstalk noise over frequency group inversions. 55

56 FIGURES FIGURES 56

57 a) b) c) Figure : 57

58 a) b) c) d) Figure : 58

59 a) b) c) Figure : 59

60 a) b) Figure : 6

61 a) b) c) Figure 5: 6

62 a) b) c) Figure 6: 6

63 a) Time (s) b) Time (s) Offset (km) 5 Offset (km) Modulus (S+N)/N c) d) Frequency (Hz) 5 5 Frequency (Hz) Figure 7: 6

64 Shots 5 5 Shots Receivers Receivers 5 5 Shots 5 5 Shots Receivers Receivers 5 5 Figure 8: 6

65 Figure 9: 65

66 Figure : 66

67 a) b) c) Figure : 67

68 a) b) c) d) e) Figure :

69 Figure : 69

70 a) k 6 5 k c) e) Crossline (km) 8. Crossline (km) 8 Crossline (km) b) d) f) Crossline (km) 8 Crossline (km) 8 Crossline (km) Figure : 7

71 a) 5 6 b) 5 6 Figure 5: 7

72 a)..5-hz 5.-Hz 7.-Hz Objective function. b) Objective function Number of iterations.5-hz 5.-Hz 7.-Hz 5 5 Number of iterations Figure 6: 7

73 a)..5 Error (%)..5 b) Frequency number. Error (%)... Frequency-group number Figure 7: 7