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 materialThe online version of this article (

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Volcanic rocks in, and around, the Ansil mining camp host a large number of massive sulfide mineral deposits. A high‐resolution seismic reflection profile was shot across the camp with the objective of mapping the contacts between the different volcanic units at which most of the ore bodies have been found. Numerous exploration boreholes define the geology to a depth of 1600 m and allow a precise comparison with the recorded reflections. Geophysical logs obtained in one deep borehole suggest that reflection coefficients between the andesite‐rhyolite units of the volcanic stratigraphy are around 0.05, but few corresponding reflections can be identified in the seismic data. Those reflections in the surface seismic profile that can be correlated with the subsurface geology originate from diorite sills, at which reflection coefficients are between 0.05 and 0.11. We suggest that reflections are observed from the diorite sills because the sills were intruded as sheets, some along fault planes, resulting in interfaces that extend over an area much greater than the first Fresnel zone. The contacts between the rhyolite‐andesite volcanic units may be highly variable spatially, preventing any strong reflection response, in contrast to the results of 1-D synthetic seismograms calculated from the borehole logs. Thus, the strength of a reflection from a lithological contact in igneous rock is likely to be related as much to the way the contact was created as to the magnitude of any local change in seismic impedance across it. Although it did not prove possible to map the volcanic stratigraphy of the Ansil mining camp, reflections interpreted to be from the disused, 1300-m-deep mine galleries were recorded. The seismic reflection data also indicate that the tonalitic Flavrian pluton, which underlies the volcanics, is an imbricated, tabular body, crosscut by diorite sills. Seismic reflections in the pluton arise from the sills and, possibly, primary magmatic layering.

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