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International audience; We introduce a new method for implicit structural modeling. The main developments in this paper are the new regularization operators we propose by extending inherent properties of the classic one-dimensional discrete second derivative operator to higher dimensions. The proposed regularization operators dis-cretize naturally on the Cartesian grid using finite differences, owing to the highly symmetric nature of the Cartesian grid. Furthermore, the proposed regulariza-tion operators do not require any special treatment on boundary nodes, and their generalization to higher dimensions is straightforward. As a result, the proposed method has the advantage of being simple to implement. Numerical examples show that the proposed method is robust and numerically efficient.

International audience; In this paper, an implicit structural modeling method using locally defined moving least squares shape functions is proposed. The continuous bending energy is minimized to interpolate between data points and approximate geological structures. This method solves a sparse problem without relying on a complex mesh. Discontinuities such as faults and unconformities are handled with minor modifications of the method using meshless optic principles. The method is illustrated on a two-dimensional model with folds, faults and an unconformity. This model is then modified to show the ability of the method to handle sparsity, noise and different reliabilities in the data. Key parameters of the shape functions and the pertinence of the bending energy for structural modeling applications are discussed. The predefined values deduced from these studies for each parameter of the method can also be used to construct other models.

International audience; In geology, implicit structural modelling constructs the geometry of geological structures (e.g. layers) by interpolating between sparse field data. A model is represented by a volumetric scalar field which is discontinuous on structural discontinuities such as faults or stratigraphic unconformities. The management of such discontinuities may involve boolean operations on several scalar fields or the creation of conformal meshes. Instead, we propose a ghost cell technique for the cartesian grid together with a set of relations between the ghost points, the regular nodes, and the discontinuities. Consequently, poor quality meshes are avoided and only the resolution of the grid has an impact on the solution. The modelling problem is posed as a least square’s minimization of a bending energy penalization on data mismatch functions and approximated by finite difference. As all relationships in the grid are implicitly defined, except close to the discontinuities, this algorithm is computationally efficient. We provide some benchmarks of the method on two-dimensional examples with folds, faults, and erosions.

Summary We introduce a new method for implicit structural modeling. The method belongs to the Discrete Smooth Interpolation class of methods. We exploit the highly symmetric nature of Cartesian grids to propose new regularization operators that discretize very naturally on the Cartesian grid using finite differences: these operators do not have to be defined on boundary nodes, and their generalization to higher dimensions is straightforward. Numerical examples show that the proposed method is both robust and numerically efficient.