Iterative Thickness Regularization of Stratigraphic Layers in Discrete Implicit Modeling

Abstract

International audienceDiscrete implicit modeling consists in representing structural surfaces as isovalues of three-dimensional piecewise linear scalar fields, which are interpolated from available data points. Data are expressed as local constraints that can enforce the value of the scalar fields as well as their gradients. This paper illustrates some limitations of published discrete implicit methods, related to the difficulty of controlling the norm of scalar field gradient and its evolution over the interpolated domain. It is shown that important artifacts may arise due to the intrinsic dependence between variations in the norm and the direction of the scalar field gradient, from one element to its neighbors. Evidence that these artifacts are related to mesh facet direction with respect to gradient direction are given. The artifacts lead to rapid and uncontrolled variations of thickness that may induce erroneous interpolations. This paper proposes two original approaches to overcome these problems. The first one consists in iteratively adjustingthe norm of scalar field gradients in the direction obtained after previous iterations. The second solution consists in optimizing the mesh used by the interpolation. This requires finding appropriate mesh facet orientationwith respect to scalar field gradient. These methods demonstrate that the results of discrete implicit surface interpolation can be improved and call for further development of available interpolation schemes

3-D Structural geological models: Concepts, methods, and uncertainties

“…In mesh-based implicit methods, the kriging formalism is generally replaced by a discrete formulation of thin-plate spline energy or some other roughness criterion, see Sec. 3.2.3 and (Moyen et al, 2004;Frank et al, 2007;Souche et al, 2013;Caumon et al, 2013a;Laurent, 2016). Independently of how exactly a mesh-based implicit model is computed, a simple and direct way to perturb the model away from the observations is to add a spatially correlated random field (x) to the reference implicit scalar field s(x) (Caumon et al, 2007;Caumon, 2010;Mallet and Tertois, 2010;Cherpeau et al, 2010;Cherpeau and Caumon, 2015;Aydin and Caers, 2017).…”

“…• Mesh-based methods have also been proposed to compute the scalar field s(x) (Moyen et al, 2004;Frank et al, 2007;Souche et al, 2013;Caumon et al, 2013a;Laurent, 2016). In this case, the domain is covered by a linear tetrahedral mesh conformable to discontinuities.…”

“…A continuum of conformable stratigraphic surfaces can be generated, as in extrusion approaches, but without limitations in the type of fault network. Conversely, the management of thickness variations is complex to handle in implicit methods (Laurent, 2016), as the various regularization terms used in interpolation tend to penalize simultaneously the variation of surface curvature and the variation of layer thickness. Even if recent work on anisotropy may provide interesting solutions for this (Martin and Boisvert, 2017), current implicit methods can only represent a limited number of unconformities owing to computational and storage limitations.…”

Implicit modeling of folds and overprinting deformation

“…The mesh could then be adapted to the position in the axial surface field of the current fold. Unfortunately, discrete linear approaches also suffer from limitations related to the underlying mesh as illustrated in Laurent [2016].…”

“…(11) and similar fold regularization Eq.(13). Two additional value data points are introduced in the northern and southern borders of the model to help the interpolated values to stretch in the whole model and limit problems of gradient norm diffusion due to the limited number of value constraints [Laurent, 2016]. In addition, a similar fold gradient norm constraint Eq.…”