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We consider the high-resolution seismic imaging method called full-waveform inversion (FWI). FWI is a data fitting method aimed at inverting for subsurface mechanical parameters. Despite the large adoption of FWI by the academic and industrial communities, and many successful results, FWI still suffers from severe limitations. From a mathematical standpoint, FWI is a large scale PDE-constrained optimization problem. The misfit function that is used, which measures the discrepancy between observed seismic data and data calculated through the solution of a wave propagation problem, is non-convex. After discretization, the size of the FWI problem requires the use of local optimization solvers, which are prone to converge towards local minima. Thus the success of FWI strongly depends on the choice of the initial model to ensure the convergence towards the global minimum of the misfit function. This limitation has been the motivation for a large variety of strategies. Among the different methods that have been investigated, the use of optimal transport (OT) distances-based misfit functions has been recently promoted. The leading idea is to benefit from the inherent convexity of OT distances with respect to dilation and translation to render the FWI problem more convex. However, the application of OT distances in the framework of FWI is not straightforward, as seismic data is signed, while OT has been developed for the comparison of probability measures. The purpose of this study is to review two methods that were developed to overcome this difficulty. Both have been successfully applied to field data in an industrial framework. Both make it possible to better exploit the seismic data, alleviating the sensitivity to the initial model and to various conventional workflow steps, and reducing the uncertainty attached to the subsurface mechanical parameters inversion., Comment: 18 figures

Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting $v$-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).

In the last ten years, full-waveform inversion has emerged as a robust and efficient high-resolution velocity model-building tool for seismic imaging, with the unique ability to recover complex subsurface structures. Originally based on a data fitting process using a least-squares cost function, it suffered from high sensitivity to cycle-skipping and was therefore of poor efficiency in handling large time shifts between observed and modelled seismic events. To tackle this problem, a common practice is to start the inversion using the low temporal frequencies of the data and selecting diving wave events. Complementary to this, the use of other cost functions has been investigated. Among these, cost functions based on optimal transport appeared appealing to possibly handle large time shifts between seismic events. Several strategies inspired by optimal transport have been proposed, taking into account the specificities of seismic data. Among them, the approach based on the Kantorovich-Rubinstein norm offers the possibility of the direct use of seismic data and an efficient numerical implementation allowing for a multidimensional (data coordinate space) application. We present here an analysis of the Kantorovich-Rubinstein norm, discussing its theoretical and practical aspects. A key component of our analysis is the back-propagated adjoint-source. We highlight its piecewise linearity, analyze its frequency content and amplitude balancing. We also emphasize the benefit of having a multidimensional implementation. Furthermore, we give practical rules for setting the tuning parameters of the numerical implementation. Our set of synthetic and field data examples demonstrate the improvements brought by the use of the Kantorovich-Rubinstein norm over least-squares full-waveform inversion, and highlight the improvements brought by the multidimensional approach over the one-dimensional one.

To train a deep neural network to mimic the outcomes of processing sequences, a version of Conditional Generalized Adversarial Network (CGAN) can be used. It has been observed by others that CGAN can help to improve the results even for deterministic sequences, where only one output is associated with the processing of a given input. Surprisingly, our CGAN-based tests on deterministic geophysical processing sequences did not produce a real improvement compared to the use of an $L_p$ loss; we here propose a first theoretical explanation why. Our analysis goes from the non-deterministic case to the deterministic one. It led us to develop an adversarial way to train a content loss that gave better results on our data., Comment: NeurIPS 2020, ICBINB Workshop

Evaluating structural uncertainties associated with seismic imaging and target horizons can be of critical importance for decision-making related to oil and gas exploration and production. An important breakthrough for industrial applications has been made with the development of industrial approaches to velocity model building. We propose an extension of these approaches, sampling an equi-probable contour of the tomography posterior probability density function (pdf) rather than the full pdf, and using non-linear slope tomography (rather than standard tomographic migration velocity analysis as in previous publications). Our approach allows to assess the quality of uncertainty-related assumptions (linearity and Gaussian hypothesis within the Bayesian theory) and estimate volumetric migration positioning uncertainties (a generalization of horizon uncertainties), in addition to the advantages in terms of efficiency. We derive the theoretical concepts underlying this approach and unify our derivations with those of previous publications. As the method works in the full model space rather than in a preconditioned model space, we split the analysis into the resolved and unresolved tomography spaces. We argue that the resolved space uncertainties are to be used in further steps leading to decision-making and can be related to the output of methods that work in a preconditioned model space. The unresolved space uncertainties represent a qualitative byproduct specific to our method, strongly highlighting the most uncertain gross areas, thus useful for QCs. These concepts are demonstrated on a synthetic data. Complementarily, the industrial viability of the method is illustrated on two different 3D field datasets.

We remind and emphasize the connections and differences between Kirchhoff and Born modelling. We discuss how they lead to a general expression for the conversion of a velocity model perturbation into a reflectivity through the "generalized reflectivity" concept. We seize the opportunity to clarify some aspects related to possibly non-smooth propagating media and the linearity approximation on reflectors. The generalized reflectivity offers opportunities: on FWI approaches that include a reflectivity or least squares migration approaches that can be based on Kirchhoff or Born modelling; in the framework of traditional Kirchhoff modelling scheme, to model first-order effects that go beyond first-order reflections (like first-order diffractions); in the framework of traditional Kirchhoff inversion or true amplitude migration, i.e. for the interpretation of seismic-migrated images., Comment: 16 pages, 2 figures, 26 references

The recently developed "internal" Density Functional Theory provides an existence theorem for a local potential that contains the center-of-mass correlations effects. The knowledge of the corresponding energy functional would provide a much cheaper way than projection techniques to treat these correlations. The aim of this article is to construct such a functional. We propose a well-founded method, suitable for Fermions as well as for Bosons, which does not require any free parameters., Comment: 17 pages, 4 figures, 4 tables, 36 references

We generalize the recently developped "internal" Density Functional Theory (DFT) and Kohn-Sham scheme to multicomponent systems. We obtain a general formalism, applicable for the description of multicomponent self-bound systems (as molecules where the nuclei are treated explicitely, atomic nuclei and mix of 3He and 4He droplets), where the fundamental translational symmetry has been treated correctly. The main difference with traditional DFT is the explicit inclusion of center-of-mass correlations in the functional. A large part of the paper is dedicated to the application to molecules, which permits among other to clarify the approximations that underly traditional DFT., Comment: 20 pages, 46 references