Three-dimensional Stratigraphical Potential Field Modeling By Generalized Radial Basis Functions(GRBFs) Interpolant

There are only two solutions for modeling the potential field of multiple continuous interfaces based on dual Kriging and co-kriging interpolation.The existing radial basis functions(RBFs)interpolation methods can only extract zero level iso-surface of the potential field to get the stratigraphic surface model.This paper proposes a novel modeling method of three-dimensional stratigraphical potential field modeling by generalized radial basis functions(GRBFs)interpolant under adaptive gradient constraint,including RBFs interpolation for fault plane modeling,Hermite-Birkhoff radial basis functions(HRBFs)with added gradient data constraints and GRBFs with added tangent data constraints for modeling multiple continuous stratigraphic interfaces.The method proposed in this paper fully guarantees the geological plausibility and topological consistency of the reconstructed 3D geological model.The main research results obtained in this paper are as follows:(1)This paper summarizes a set of modeling data extraction methods based on geological map,cross-sections and digital elevation model.Combined with the comprehensive stratigraphic histogram,the value of the relative burial depth of the stratigraphic bottom surface relative to the top surface of the Quaternary is selected to get attributes discrete points sampling data.According to the dip and angle of dip of the attitude sampling points,the gradient vectors are obtained by deriving the geometric relationship in 3D space.According to the principle of calculus,the tangent data of the stratigraphic boundary lines are obtained.(2)This paper proposes a modeling method of stratigraphic potential field based on HRBFs and GRBFs interpolation.This method introduces the idea of potential field modeling,and regards stratum with uniform properties as scalar fields.It can model multiple continuous stratigraphic interfaces and fill the gap that it is difficult to construct a single implicit function to model multiple stratigraphic interfaces based on RBFs.(3)Both HRBFs and GRBFs interpolation for stratigraphic potential field modeling need to use gradient data to constrain the variation of the stratigraphic potential field.The gradient data as a 3D vector has the attributes of vector direction and magnitude.The direction of the gradient vector is from the stratum with a small potential field value to the stratum with a large potential field value.The gradient magnitude is unknown in the obtained gradient data but the gradient magnitude controls the variation’s degree of the potential field value.This paper set the initial gradient magnitude first,and then the gradient magnitude is optimized in the HRBFs interpolation linear system by adding an optimizing term.The gradient magnitude is converged to the true value via iteration.In this study,theoretical model experiments are designed to verify the proposed algorithm and we get the ideal result.(4)Based on different original modeling data,the gradient magnitude scalar field model,gradient vector field model and stratigraphic potential field model of the study area are obtained through the interpolation based on HRBFs and GRBFs under the adaptive gradient constraint proposed in this paper.By introducing the concept of regularized stratigraphic thickness index,we obtained the stratigraphic thickness index scalar field of the study area.It is used to jointly analyze the relationship between the gradient magnitude and the ratio of the difference between the potential field values of the stratigraphic top and bottom surfaces and the stratigraphic thickness.(5)By tracking specific isosurfaces in the 3D stratigraphic potential field to obtain the stratigraphic interfaces model.We obtain the geological volume model by connecting the isosurfaces.The obtained modeling results are basically consistent with the geological map and cross-sections,and restore the complex geological fold structure in the study area.Meanwhile,we compare and analyze the model results based on HRBFs and GRBFs interpolation using different data constraints.(6)The gradient vector field is established by the method proposed in this paper,and the strike vector lines and the dip vector lines are extracted according to the properties of the vector field.The strike and dip of the stratum can be clearly and intuitively analyzed on the whole or in detail according to the vector lines,which are of great significance for analyzing the shape of the stratum.The RBFs,HRBFs and GRBFs interpolation methods are implemented,analyzed and compared in the study area.By gradually adding constraints on the original modeling data and optimizing the gradient magnitude,an accurate stratigraphic potential field model can be effectively established.It can fully express the attribute and attitude information inside the stratum,reconstruct a high-precision 3D model that conforms to the constraints of geological observations and geological rules,and provide a basis for subsequent underground space planning,mineral resource evaluation,and oil and gas storage simulation.