Finite Difference Implicit Structural Modeling of Geological Structures
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Finite Difference Implicit Structural Modeling of Geological Structures Modeste Irakarama1 Guillaume Caumon1
· Gautier Laurent2 · Julien Renaudeau1 ·
Received: 15 July 2019 / Accepted: 10 August 2020 © International Association for Mathematical Geosciences 2020
Abstract 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 discretize naturally on the Cartesian grid using finite differences, owing to the highly symmetric nature of the Cartesian grid. Furthermore, the proposed regularization 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. Keywords Implicit modeling · Structural modeling · Regularization operators · Finite-differences · Interpolation
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Modeste Irakarama [email protected] Gautier Laurent [email protected] Julien Renaudeau [email protected] Guillaume Caumon [email protected]
1
GeoRessources Laboratory, Université de Lorraine, CNRS, 2 Rue du Doyen Marcel Roubault, 54518 Vandœuvre-lès-Nancy, France
2
Université d’Orléans, CNRS, BRGM, ISTO, UMR 7327, 45071 Orléans, France
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Math Geosci
1 Introduction Structural implicit modeling, in this paper, is defined as the interpolation of randomly distributed, and possibly sparse, structural data. For simplicity, structural data, also referred to as structural constraints, will be limited to the following. value constraints : φ(x j ) = a j ,
(1)
orientation constraints : ∇φ(x j ) = ||∇φ(x j )||u j ;
(2)
where φ(x) is the unknown function to be interpolated, x is a point in space, a is some given scalar, and u is some given unit vector. We refer to Eq. (2) as the normal form of an orientation constraint, which informs both about the direction and the norm of φ, as opposed to the tangential form, which informs only about the orientation of φ. Let l l=N u1 be a given unit vector, then there are N − 1 unit vectors {ul }l=N l=2 such that {u }l=1 forms an orthogonal basis in N ≥ 2 dimensions. In that case, the tangential form of the normal orientation constraint ∇φ(x) = ||∇φ(x)||u1 is ul · ∇φ(x) = 0, for l = 2, . . . , N .
(3)
In implicit structural modeling, the object being modeled is obtained by extracting a hypersurface along an iso-value of the interpolated function φ(x) (Newman and Yi 2006). This principle can be distinguished from explicit structural modeling, where geological interfaces defined by a two-dimensional planar graph are embedded in three-dimensional space (Caumon et al. 2009). Implicit structural modeling has extensively been used in geosciences for contour mapping (Briggs 1974;
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