Attenuation imaging by wavefield reconstruction inversion with bound constraints and total variation regularization Hossein S. Aghamiry 12, Ali Gholami 1 and St´ephaneOperto 2 INTRODUCTION ABSTRACT Wavefield reconstruction inversion (WRI) extends the Full waveform inversion (FWI) is a high-resolution non- search space of Full Waveform Inversion (FWI) by al- linear imaging technology which can provide accurate sub- lowing for wave equation errors during wavefield re- surface model by matching observed and calculated wave- construction to match the data from the first iteration. forms (Tarantola, 1984; Pratt et al., 1998; Virieux and Then, the wavespeeds are updated from the wavefields Operto, 2009). However, it is well acknowledged that it by minimizing the source residuals. Performing these suffers from two main pathologies. The first one is the two tasks in alternating mode breaks down the non- nonlinearity associated with cycle skipping: When the linear FWI as a sequence of two linear subproblems, distance between the observed and calculated data is the relaying on the bilinearity of the wave equation. We least-squares norm of their differences, FWI remains stuck solve this biconvex optimization with the alternating- into spurious local minima when the initial velocity model direction method of multipliers (ADMM) to cancel out does not allow to match traveltimes with an error lower efficiently the data and source residuals in iterations than half a period. To mitigate cycle skipping, many vari- and stabilize the parameter estimation with appropri- ants of FWI have been proposed with more convex dis- ate regularizations. Here, we extend WRI to viscoa- tances such as those based on matching filters (Warner coustic media for attenuation imaging. Attenuation and Guasch, 2016; Guasch et al., 2019) or optimal trans- reconstruction is challenging because of the small im- port (M´etivieret al., 2018) among others. The second print of attenuation in the data and the cross-talks with pathology is ill-posedness resulting from uneven subsur- velocities. To address these issues, we recast the multi- face illumination provided by limited-aperture surface ac- variate viscoacoustic WRI as a triconvex optimization quisitions (e.g., Tang, 2009) and parameter cross-talks and update wavefields, squared slowness, and attenua- during multiparameter reconstruction (see Operto et al., tion factor in alternating mode at each WRI iteration. 2013, for a tutorial). Mitigating this ill-posedness requires This requires to linearize the attenuation-estimation to account for the Hessian in local optimization methods (e.g. M´etivieret al., 2017) and regularize the inversion arXiv:1909.05170v2 [math.OC] 6 Jan 2020 subproblem via an approximated trilinear viscoacoustic wave equation. The iterative defect correction embed- with prior information such as physical bound constraints ded in ADMM corrects the errors generated by this (e.g. Asnaashari et al., 2013; Duan and Sava, 2016). linearization, while the operator splitting allows us to Among the methods proposed to mitigate cycle skipping, tailor `1 regularization to each parameter class. A toy wavefield reconstruction inversion (WRI) (van Leeuwen numerical example shows that these strategies mitigate and Herrmann, 2013, 2016) extends the parameter search cross-talk artifacts and noise from the attenuation re- space of frequency-domain FWI by processing the wave construction. A more realistic synthetic example rep- equation as a soft constraint with a penalty method. The resentative of the North Sea validates the method. resulting wave equation relaxation allows for data fitting with inaccurate velocity models through the reconstruc- tion of data-assimilated wavefields, namely wavefields sat- isfying the observation equation relating the wavefields to 0 1 University of Tehran, Institute of Geophysics, Tehran, Iran, the observations (Aghamiry et al., 2019a). The algorithm email: [email protected], [email protected] 2University Cote d'Azur - CNRS - IRD - OCA, Geoazur, Valbonne, then updates the model parameters by least-squares min- France, email: [email protected], [email protected] imization of the wave equation errors (or source residu- 1 2 A PREPRINT Aghamiry et al. als) so that the assimilated wavefields explain both the above and beneath. On the other hand, Hak and Mul- wave equation and the data as well as possible. Perform- der (2011) show that wavespeed and attenuation can be ing wavefield reconstruction and parameter estimation in decoupled during nonlinear waveform inversion of multi- an alternating mode (van Leeuwen and Herrmann, 2013) offset/multi-frequency data provided that the causality rather than by variable projection (van Leeuwen and Her- term is properly implemented in the attenuation model. rmann, 2016) recasts WRI as a sequence of two linear This conclusion has been further supported by several re- subproblems as a result of the bilinearity of the wave alistic synthetic experiments and real data case studies in equation in wavefield and squared slowness. The reader is marine and land environments, which manage to recon- also referred to Aghamiry et al. (2019b) for a more gen- struct trustworthy attenuation models (Hicks and Pratt, eral discussion on the bilinearity of the elastic anisotropic 2001; Askan et al., 2007; Malinowski et al., 2011; Tak- wave equation. Aghamiry et al. (2019e) solved this bi- ougang and Calvert, 2012; Kamei and Pratt, 2008; Prieux convex problem with the alternating direction method of et al., 2013; Stopin et al., 2016; Operto and Miniussi, multipliers (ADMM) (Boyd et al., 2010). ADMM is an 2018; Lacasse et al., 2019). This decoupling between ve- augmented Lagrangian method which makes use of oper- locity and attenuation can be further argued on the ba- ator splitting and alternating directions to solve convex sis of physical considerations. In the transmission regime separable multi-variate constrained problems. The aug- of wave propagation, wavespeeds control the kinematic mented Lagrangian function combines a penalty function of wave propagation. This implies that FWI is domi- and a Lagrangian function (Nocedal and Wright, 2006a, nantly driven toward wavespeed updating to match the Chapter 17). The penalty function relaxes the constraints traveltimes of the wide-aperture data (diving waves, post- during early iterations as in WRI, while the Lagrangian critical reflections) and update the long wavelengths of the function progressively corrects the constraint violations subsurface accordingly, while attenuation has a secondary via the action of the Lagrange multipliers. The lever- role to match amplitude and dispersion effects (e.g., see age provided by the Lagrange multipliers guarantees to Operto and Miniussi, 2018, for an illustration). This weak satisfy the constraints at the convergence point with con- imprint of the attenuation in the seismic response was il- stant penalty parameters (Aghamiry et al., 2019e). Ac- lustrated by the sensitivity analysis carried out by Kurz- cordingly, Aghamiry et al. (2019e) called their approach mann et al. (2013) who concluded that a crude homo- iteratively-refined WRI (IR-WRI). Alternatives to satisfy geneous background attenuation model may be enough to the constraints at the convergence point with penalty meth- perform reliable FWI, while da Siva et al. (2019) proposed ods rely on multiplicative (da Silva and Yao, 2017) or to reconstruct an under-parametrized attenuation model discrepancy-based (Fu and Symes, 2017) approaches. Aghamiryby semi global FWI. When a high-resolution attenuation et al. (2019d) implemented bounding constraints and total model is sought, the ill-posedness of the attenuation recon- variation (TV) regularization (Rudin et al., 1992) in IR- struction may be managed with different recipes includ- WRI with the split Bregman method (Goldstein and Os- ing data-driven and model-driven inversions (joint versus her, 2009) to improve the imaging of large-contrast media, sequential updates of the velocity and attenuation of se- with however undesirable staircase imprints in smooth re- lected subdatasets), parameter scaling, bound constraints gions. To overcome this issue and capture both the blocky and regularizations (e.g. Prieux et al., 2013; Operto et al., and smooth components of the subsurface, Aghamiry et al. 2013). (2019c) combine in IR-WRI Tikhonov and TV regulariza- In this context, the contribution of this study is two fold: tions by infimal convolution. first, we show how to implement velocity and attenuation The objective of this paper is to extend frequency-domain reconstruction in frequency-domain viscoacoustic IR-WRI IR-WRI to viscoacoustic media for attenuation imaging. when equipped with bound constraints and nonsmooth Attenuation reconstruction by FWI raises two potential regularizations. Second, we discuss with numerical exam- issues. The first is related to the cross-talks between ples whether the alternating-direction algorithm driven by wavespeed and attenuation. The ambiguity between ve- the need to expand the search space is suitable to manage locity and attenuation perturbation in least-squares mi- ill-conditioned multi-parameter reconstruction. It is well gration has been emphasized by Mulder and Hak (2009). acknowledged that viscous effects are easily included in Many combination of velocity and attenuation perturba- the time-harmonic wave equation with frequency-dependent tions can fit equally well reflection amplitudes since they complex-valued velocities as function of phase velocity