We are very pleased to announce two PhD defenses and a small workshop on geophysics and structural modeling on April 24 and 25, 2019.

- April 24, 13h (ENSG, Amphi G): PhD Defense of Julien Renaudeau: Continuous formulation of implicit structural modeling discretized with mesh reduction methods.

- April 25, 9h (ENSG, Amphi H): Geophysics and structural modeling workshop

- 9h Sophie Viseur (CEREGE, Aix-Marseille Universite): Extra-terrestrial modeling
- 9h30 Florian Wellmann (RWTH Aachen): Quantification of uncertainties in geological models
- 10h Laetitia Le Pourhiet (Université Pierre et Marie Curie): 3D thermo-mechanical forward modelling of long term tectonic processes
- 10h30 Break
- 10h45 Alexandrine Gesret (Mines Paris Tech): Uncertainty estimation by stochastic seismic tomography
- 11h15 Pierre Thore (Total): On time-laspe seismic and history matching.
- 11h45 Thomas Bohlen (KIT): Applications of Full Waveform Inversion at different spatial scales

- April 25, 14h30 (ENSG, Amphi H): PhD Defense of Modeste Irakarama: Towards Reducing Structural Interpretation Uncertainties Using Seismic Data.

### Continuous formulation of implicit structural modeling discretized with mesh reduction methods

by Julien Renaudeau

PhD Committee: Julie Digne (Univ. Lyon), Florian Wellmann (RWTH Aachen), Dominique Bechmann (Univ. Strabourg), Laetitia Le Pourhiet (Uviv. Pierre et Marie Curie), Sophie Viseur (Univ. Aix-Marseille), Frantz Maerten (YouWol), Bruno Lévy (INRIA Nancy Grand Est), Guillaume Caume (Univ. Lorraine)

**Abstract**: Implicit structural modeling consists in approximating geological structures into a numerical model for visualization, estimations, and predictions. It uses numerical data interpreted from the field to construct a volumetric function on the domain of study that represents the geology. The function must fit the observations, interpolate in between, and extrapolate where data are missing while honoring the geological concepts. Current methods support this interpolation either with the data themselves or using a mesh. Then, the modeling problem is posed depending on these discretizations: performing a dual kriging between data points or defining a roughness criterion on the mesh elements. In this thesis, we propose a continuous formulation of implicit structural modeling as a minimization of a sum of generic functionals. The data constraints are enforced by discrete functionals, and the interpolation is controlled by continuous functionals. This approach enables to (i) develop links between the existing methods, (ii) suggest new discretizations of the same modeling problem, and (iii) modify the minimization problem to fit specific geological issues without any dependency on the discretization. Another focus of this thesis is the efficient handling of discontinuities, such as faults and unconformities. Existing methods require either to define volumetric zones with complex geometries, or to mesh volumes with conformal elements to the discontinuity surfaces. We show, by investigating local meshless functions and mesh reduction concepts, that it is possible to reduce the constraints related to the discontinuities while performing the interpolation. Two discretizations of the minimization problem are then suggested: one using the moving least squares functions with optic criteria to handle discontinuities, and the other using the finite element method functions with the concept of ghost nodes for the discontinuities. A sensitivity analysis and a comparison study of both methods are performed in 2D, with some examples in 3D. The developed methods in this thesis prove to have a great impact on computational efficiency and on handling complex geological settings. For instance, it is shown that the minimization problem provides the means to manage under-sampled fold structures and thickness variations in the layers. Other applications are also presented such as salt envelope surface modeling and mechanical restoration.

### Towards Reducing Structural Interpretation Uncertainties Using Seismic Data

by Modeste Irakarama

PhD Committee: Alison Malcolm (Memorial Univ. Newfouldland), Thomas Bohlen (Karlruhe Institute of Technology), Alexandrine Gesret (Mines Paris Tech), Isabelle Lecomte (Univ. Bergen), Pierre Thore (Total), Paul Sava (Colorado School of Mines), Paul Cupillard (Univ. Lorraine), Guillaume Caumon (Univ. Lorraine).

**Abstract**: Subsurface structural models are routinely used for resource estimation, numerical simulations, and risk management; it is therefore important that subsurface models represent the geometry of geological objects accurately. The first step of building a subsurface model is usually interpreting structural features, such as faults and horizons, from a seismic image; the identified structural features are then used to build a subsurface model using interpolation methods. Subsurface models built this way therefore inherit interpretation uncertainties since a single seismic image often supports multiple structural interpretations. In this manuscript, I study the problem of reducing interpretation uncertainties using seismic data. In particular, I study the problem of using seismic data to determine which structural models are more likely than others among an ensemble of geologically plausible structural models. I refer to this problem as \emph{appraising structural models using seismic data}.

The first Part of the manuscript is devoted to seismic imaging. I first propose to use reverse-time migration (RTM) as a preconditioner for waveform inversion. Numerical experiments show that the proposed preconditioner accelerates both linearized waveform inversion (least squares reverse time migration) and nonlinear waveform inversion (full waveform inversion) by at least an order of magnitude. I justify the positive numerical performance of the proposed preconditioner by showing algebraically that a low pass filter of the RTM image can approximate the diagonal elements of the Hessian matrix of the objective function under appropriate assumptions. However, I am still unable to propose a physical meaning of the low pass filtering and how it relates the RTM image to the elements of the diagonal of the Hessian matrix. Then, I propose a generalized extended Kirchhoff imaging operator for velocity modeling; the operator is generalized in the sense that it describes multiple data-domain extensions (e.g. shot, offset, and angle extensions) and image-domain extensions (e.g. time-lag and space-lag extensions) simultaneously. The advantages of the proposed generalized extended operator are twofold: firstly, it allows a unified implementation for multiple extensions (i.e. a single implementation that is valid for multiple extensions); secondly, the operator leads to a unified gradient-based migration velocity analysis (MVA) scheme. I confirm the ability of the proposed generalized extended operator to capture image distortion due to inaccurate velocity by applying it to a ray-based MVA experiment.

The second Part of the manuscript is devoted to structural modeling, particularly implicit structural interpolation. I introduce Finite Difference Structural Implicit Modeling (FDSIM). The advantages of FDSIM are twofold: firstly, it is relatively easy to implement and to optimize since it is based on finite differences on regular grids; secondly, because it handles discontinuities by rasterization, FDSIM has shown to easily handle very complex fault-networks. The main disadvantage of the method is that it may require a very fine resolution depending on the complexity of the fault network, sometimes leading to memory limits. I also propose new regularization operators; the particularity of these operators is that they do not need to be implemented on boundary nodes, a property which is very appealing in implicit modeling where boundary conditions are usually unknown. I then introduce Finite Element Structural Implicit Modeling (FESIM). FESIM is based on a finite element implementation of the newly proposed regularization operators. I show that the conventional finite element familiar for solving boundary value problems has to be slightly modified for implicit modeling where boundary conditions are usually unknown.

The third Part of the manuscript is devoted to appraising structural models/interpretations using seismic data. I introduce and formalize the problem of appraising structural interpretations using seismic data. I propose to solve the problem by generating synthetic data for each structural interpretation and then compute misfit values for each interpretation; this allows to rank the different structural interpretations. The main challenge of appraising structural models using seismic data is to propose appropriate data misfit functions. I derive a set of conditions that have to be satisfied by the data misfit function for a successful appraisal of structural models. I argue that since it is not possible to satisfy these conditions using vertical seismic profile (VSP) data, it is not possible to appraise structural interpretation using VSP data in the most general case. The conditions imposed on the data misfit function can in principle be satisfied for surface seismic data. In practice, however, it remains challenging to propose and compute data misfit functions that satisfy those conditions. Numerical experiments illustrating practical issues of appraising structural interpretations using surface seismic data. I propose a general data misfit formula that is made of two main components: (1) a residual operator that computes data residuals, and (2) a projection operator that projects the data residuals to the image-domain. This misfit function is therefore localized in space as it outputs data misfit values in the image-domain. However, I am still unable to propose a practical implementation of this misfit that satisfies the conditions derived for a successful appraisal of structural interpretations; this is a subject of ongoing investigations.