## Section: New Results

### Inverse Problems

#### Complex-frequency domain Full Waveform Inversion

Participants : Florian Faucher, Maarten V. de Hoop, Henri Calandra.

We study the seismic inverse problem for the (complex) frequency-domain elastic isotropic wave equation; in particular the recovery of the Lamé parameters and density. We employ a Full Waveform Inversion where the iterative minimization is based on a gradient descent. The elastic inverse problem shows a Lipschitz-type stability where the Fréchet derivative has a strictly positive `lower bound'. This bound is connected to the stability constant and can be approximated using the Gauss-Newton Hessian. The successive stability estimates provide a control of the convergence and decide the parameters of inversion. We develop a multi-level approach based on a structured domain partitioning of the sub-surface. The coefficients (Lamé parameters and density) are assumed to be piecewise constant functions following the domain partitioning, which is naturally defined with the successive stability estimates to maintain the radius of convergence, while refinement provides resolution. It allows us to start with minimal prior information for the coefficients and the algorithm is perfectly suitable for complex frequency. We have carried out numerical experiments in two and three dimensions; those results have been presented during the following conferences in 2014: [48] , [49] .

#### Imaging of complex media with elastic wave equations

Participants : Jérôme Luquel, Hélène Barucq, Henri Calandra, Julien Diaz.

Even if RTM has enjoyed the tremendous progresses of scientific computing, its performances can still be improved, in particular when applied to strong heterogeneous media. In this case, images have been mainly obtained by using direct arrivals of acoustic waves and the transition to elastic waves including multiples is not obvious essentially because elastic waves equations are still more computationally consuming. The accuracy of numerical wave fields is obviously of great importance. We have thus chosen to consider high-order Discontinuous Galerkin Methods which are known to be well-adapted to provide accurate solutions based upon parallel computing. Now one of the main drawback of RTM is the need of storing a huge quantity of information which is redhibitory when using elastic waves. For that purpose, we apply the Griewank algorithm following Symes' ideas for the acoustic RTM. The idea is to find a compromise between the number of wave equations to solve and the number of numerical waves that we have to store. This is the so-called Optimal Checkpointing. By reducing the occupancy of the memory, RTM should be efficient even when using elastic waves. By this way, one may wonder if considering elastic waves including multiples in order to improve images of heterogeneous media is a valid option. It must involve a careful numerical analysis including the evaluation of the impact of the imaging condition. It is thus necessary to derive accurate imaging conditions, which could take advantage of all the information contained in the wavefield. For acoustic media, Claerbout proposed an imaging condition which is widely used and turns out to be sufficient to accurately reproduce interfaces. But Claerbout conditions do not take wave conversions into account and it is not clear wether conversions do or do not contain interesting information to get accurate images of heterogeneous media.

Since P-wave and S-wave interact with each other, it might be relevant to use an imaging condition including these interactions. In fact, this has been done successfully by J.Tromp and C. Morency for seismology applications based upon the inversion of the global Earth. Their approach is based upon the state adjoint and it involves sensitivity kernels which are defined from the propagated and the back-propagated fields. Now it has been shown that full wave form inversions using these sensitivity kernels may be polluted by numerical artefacts. One solution is to use a linear combination of the sensitivity kernels to delete artefacts. In this work, we propose then a new imaging condition which construction is inspired from with some approximations required to keep admissible computational costs. We illustrate the properties of the new imaging condition on industrial benchmarks like the Marmousi model. In particular, we compare the new imaging condition with other imaging conditions by using as criteria the quality of the image and the computational costs required by the RTM. This work was presented at the the WCCM XI - ECCM V - ECFD VI - Barcelona 2014 Conference and SIAM Conference on IMAGING SCIENCE (SIAM-IS14) Hong Kong Baptist University[67] .

#### Helioseismology

Participants : Juliette Chabassier, Marc Duruflé, Thorsten Hohage.

We have begun to write a software interface that allows to solve an inverse problem using adjoint and regularization methods (iTReg software) while using Montjoie software for the direct problem that must be solved at each iteration of the inversion process.