Three-Dimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial Basis Functions
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Three-Dimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial Basis Functions Michael J. Hillier · Ernst M. Schetselaar · Eric A. de Kemp · Gervais Perron
Received: 31 May 2013 / Accepted: 18 April 2014 / Published online: 15 July 2014 © International Association for Mathematical Geosciences 2014
Abstract A generalized interpolation framework using radial basis functions (RBF) is presented that implicitly models three-dimensional continuous geological surfaces from scattered multivariate structural data. Generalized interpolants can use multiple types of independent geological constraints by deriving for each, linearly independent functionals. This framework does not suffer from the limitations of previous RBF approaches developed for geological surface modelling that requires additional offset points to ensure uniqueness of the interpolant. A particularly useful application of generalized interpolants is that they allow augmenting on-contact constraints with gradient constraints as defined by strike-dip data with assigned polarity. This interpolation problem yields a linear system that is analogous in form to the previously developed potential field implicit interpolation method based on co-kriging of contact increments using parametric isotropic covariance functions. The general form of the mathematical framework presented herein allows us to further expand on solutions by: (1) including stratigraphic data from above and below the target surface as inequality constraints (2) modelling anisotropy by data-driven eigen analysis of gradient constraints and (3) incorporating additional constraints by adding linear functionals to the system, such as fold axis constraints. Case studies are presented that demonstrate the advantages and general performance of the surface modelling method in sparse data environments where the contacts that constrain geological surfaces are rarely exposed but structural and off-contact stratigraphic data can be plentiful.
Electronic supplementary material The online version of this article (doi:10.1007/s11004-014-9540-3) contains supplementary material, which is available to authorized users. M. J. Hillier (B) · E. M. Schetselaar · E. A. de Kemp Geological Survey of Canada, 615 Booth Street, Ottawa, ON K1A 0E9, Canada e-mail: [email protected] G. Perron Mira Geoscience Ltd. , #309–310 Victoria Avenue Westmount, Montreal, QC H3Z 2M9, Canada
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Math Geosci (2014) 46:931–953
Keywords Three-dimensional geomodelling · Structural geology · Implicit modelling · Radial basis functions · Generalized interpolation 1 Introduction Three-dimensional modelling of geological surfaces plays an important role in enhancing geological understanding of the subsurface in a wide range of earth science applications. Although geological surfaces can be built in any area where constraints on their three-dimensional position and attitude are available, three-dimensional geological surface modelling is most frequently applied in earth resource exploration, and geo
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