For discrete-interpolation schemes of implicit modeling with a special mesh, the GoCAD (https://app.pdgm.com/product-category/skua-gocad/, last access: 30 June 2022) software was developed based on the discrete smooth interpolation (DSI) method to meet the needs of geological, geophysical, and petroleum reservoir engineering modeling (Mallet, 2004; Frank et al., 2007). Caumon et al. (2013) proposed a discretizing finite-element method (FEM) to generate 3D models of horizons on a tetrahedral mesh, using stratigraphic interface traces of unknown attribute values and strike and dip measurements from 2D geological maps, remote sensing images, and digital elevation models. Hillier et al. (2013) presented a structural field interpolation (SFI) algorithm using an anisotropic inverse distance-weighted (IDW) interpolation scheme derived from eigen analysis of strike and dip measurements. Renaudeau et al. (2019) proposed an implicit structural modeling method using locally defined moving-least-squares shape functions and solved a sparse sampling problem without relying on a complex mesh. Irakarama et al. (2020) introduced a new method for implicit structural modeling by regularization operators on the Cartesian grid using finite differences. Grose et al. (2021b) presented LoopStructural, a new open-source 3D geological modeling Python package, in which discrete interpolators and polynomial trend interpolators can be mixed and matched within a geological model.

Since the continuous-interpolation scheme does not depend on a mesh for its definition, the stratigraphic interfaces can be extracted at any desired resolution in the specific volume of interest. There is already a dual kriging or cokriging formulation for continuous potential-field modeling of multiple stratigraphic interfaces. Lajaunie et al. (1997) proposed an implicit potential-field modeling method using the dual formulation of kriging interpolation that considers known points on a geological interface and plane strike and dip data such as stratification or foliation planes. Calcagno et al. (2008) cokriged the location of geological interfaces and strike and dip data from a structural field to interpolate a continuous 3D potential-field scalar function describing the geometry of geological bodies. Geomodeller 3D (http://www.geomodeller.com, last access: 30 June 2022), an implicit geological modeling application, utilizes the implicit potential-field method by cokriging or the dual formulation of kriging (Lindsay et al., 2012; Hassen et al., 2016). Gonçalves et al. (2017) proposed a vector potential-field solution from a machine learning perspective, recasting the problem as multivariate classification in a compositional data framework, which alleviates some of the assumptions of the cokriging method. De la Varga et al. (2019) presented GemPy (https://github.com/cgre-aachen/gempy, last access: 30 June 2022), an open-source implementation, to generate 3D geological models based on an implicit potential-field cokriging interpolation approach and to enable stochastic geological modeling and inversions of gravity and topology in machine learning and Bayesian inference frameworks. To reduce the impact of regularly occurring modeling artifacts that result from data configuration and uncertainty, Von Harten et al. (2021) proposed an approach that is a combination of an implicit interpolation algorithm with a local smoothing method based on the concepts of nugget effect and filtered kriging known from conventional geostatistics.

For continuous radial basis function (RBF) or HRBF interpolation schemes of implicit modeling without a mesh, Cowan et al. (2003) constructed an implicit model of the orebody or stratigraphic interface using a volumetric RBF interpolation function with an equipotential surface that includes the interface points and conventionally assigned an attribute value of zero and a “±” sign to indicate the inside and outside of the interface. Hillier et al. (2014, 2016) presented a generalized interpolation framework using RBF in Surfe, an open-source library, to implicitly model 3D continuous geological interfaces from on-surface points with gradient constraints as defined by strike–dip data with assigned polarity. Leapfrog Geo (http://www.leapfrog3d.com, last access: 30 June 2022) is an implicit geological modeling software package that models scattered data for interfaces using fast RBF interpolation methods (Vollgger et al., 2015; Basson et al., 2016, 2017; Creus et al., 2018; Stoch et al., 2020). Martin and Boisvert (2017) developed a RBF-based implicit modeling framework using domain decomposition to locally vary orientations and magnitudes of anisotropy for geological boundary models. Zhong et al. (2019, 2021) introduced combination constraints for modeling ore bodies based on multiple implicit field interpolation through RBF methods, in which a multiply labeled implicit function was defined that combines different implicit sub-fields by the combination operations to construct constraints honoring geological relationships more flexibly. Guo et al. (2016, 2018, 2020, 2021) proposed an explicitly and implicitly integrated 3D geological modeling approach for the geometric fusion of different types of complex geological structure models; therein, the HRBF-based implicit method was used to model general strata, faults, and folds, and the skinning method and the free-form surface were used to model local detailed structures. Wang et al. (2018) proposed an implicit modeling approach to automatically build a 3D model for orebodies by means of spatial HRBF interpolation directly based on geological borehole data.

However, the above RBF and HRBF interpolants, which use only the on-contact point datasets for each geological interface or assign an approximate gradient vector for each on-contact point according to its nearest strike and dip measurements, cannot be accurately consistent with actual strike and dip survey data. To maintain topological consistency between geological bodies and represent their internal burial depth and structural orientations, our AdaHRBF interpolation scheme yields an HRBF linear system that is analogous in form to the previously developed implicit potential-field interpolation method based on cokriging of contact increments using parametric isotropic covariance functions.

The geological boundaries and structural orientations on planar geological maps and cross-sections are the most common data used for 3D geological modeling. Besides the geological boundaries extracted from boreholes, cross-sections, and geological maps, structural orientation (including strike direction, dip direction, and dip angle) data from geological maps play important roles in characterizing the shape and distribution of geological bodies, as shown in Fig. 1. The SPF modeling method can jointly reconstruct a 3D geological model using these data extracted from geological maps and cross-sections.

A field in a spatial domain Rn defines the function f=f(p) at a point p∈Rn in domain Rn, and f(p) is also called field function. The SPF defines the 3D space as a scalar function f(p) at any point p; meanwhile, the stratigraphic interfaces are simulated and expressed as specific equipotential surfaces satisfying f(p)=fk(i=1, …, K) in the SPF. In practice, this specific function value fk may correspond to the age of the stratigraphic interface or a relative distance from a reference interface (Mallet, 2004). Therefore, a stratum occupies the space between its bottom surface fk and top surface fk+1, while there are countless disjoint equipotential surfaces in each stratum (Mallet, 2004). A well-known problem is how to interpolate unknown points by a function f(⋅) using known points of the space Rn. The key problem of SPF modeling is to obtain surfaces that are consistent with known on-contact points on the stratigraphic interfaces and the off-contact strike and dip directions of the strata. The stratigraphic interface points define the distribution of reference equipotential surfaces, while the strike and dip points define the gradient vectors of the scalar field.

The SPF modeling by the HRBF interpolant satisfies both the on-contact attribute constraint and off-contact strike and dip constraint. To fit an implicitly defined SPF from known attribute values pi,fii=1N∈Rn×R and gradients (pj,gj)}j=1M∈Rn×Rn derived from strike and dip data, we can search for a function f:Rn→R which satisfies both the on-contact constraints fpi=fi for each i=1,…,N and the off-contact gradient constraints ∇fpj=gj for each j=1, …, M. In particular, pi=pixpiypiz, and gj=gjxgjygjz in space R3.

Generally, the basic RBF reconstructs an implicit function with constraint fpi=fi; however, the HRBF reconstructs an implicit function which interpolates scattered multivariate Hermite–Birkhoff data (i.e., unstructured points and orientations) (Macedo et al., 2011). With the joint constraints of fpi=fi and ∇fpj=gj, the optional solution is to obtain equipotential surfaces that are as smooth as possible, that is, to ensure the energy function, which represents the degree of equipotential surface smoothness and unevenness of SPF, is as small as possible. Therefore, the energy function E of the SPF is defined by further combining the thin-plate regularizer of f (Wahba, 1990; Walder et al., 2006) as 1E=∑i=1Nf(pi)-fi2+∑j=1M∂f(pj)∂x-gjx2+∂f(pj)∂y-gjy2+∂f(pj)∂z-gjz2+∫R3∂2f(p)∂2x+∂2f(p)∂2y+∂2f(p)∂2z+2∂2f(p)∂x∂y+2∂2f(p)∂y∂z+2∂2f(p)∂z∂xdxdydz, where ∂f(pj)∂x, ∂f(pj)∂y, and ∂f(pj)∂z are the first-order partial derivatives of implicit function f(p) at point pj, and ∂2f(p)∂2x, ∂2f(p)∂2y, ∂2f(p)∂2z, ∂2f(p)∂x∂y, ∂2f(p)∂y∂z, and ∂2f(p)∂z∂x are the second-order partial derivatives of implicit function f(p). The first and second components of energy function represent the misfit between the estimated values and observed contact and orientation points, respectively, and the third component of a second-order derivative of implicit function guarantees the smoothness of SPF implicit function.

When using the HRBF interpolation method, we usually add a first-order polynomial C(p) to ensure the smoothness and continuity of equipotential surfaces. In particular, Cp=c1+c2px+c3py+c4pz. The HRBF interpolation function has a concrete estimation form f∗p: 2f∗p=∑i=1Nαiφp-pi+∑j=1M〈βj,∇φ(p-p)〉+Cp3∇f∗p=∑i=1Nαi∇φp-pi+∑j=1M∇2φp-pjβj+∇Cp, where p-pi denotes the Euclidean distance between locations p and pi; φr is the radial basis function, herein, for which the cubic function φr=r3 is used in this study; ∇ is the Hamiltonian operator; ∇2 is the Hessian operator, in particular, ∇=∂∂x∂∂y∂∂zT and ∇2=∂2∂2x∂2∂x∂y∂2∂x∂z∂2∂y∂x∂2∂2y∂2∂y∂z∂2∂z∂x∂2∂z∂y∂2∂2z; and 〈a,b〉 is the inner product of vectors a and b. The scalar weight coefficients αi∈R, vector weight coefficients βj∈Rn, and c∈Rn+1 (in particular, βj=βjxβjyβjzT and c=c1c2c3c4T) are unknown and uniquely determined by the joint constraints f∗pi=fi for each i=1, …, N and ∇f∗pj=gj for each j=1, …, M.

The HRBF interpolant defines the implicit function as a sum of chosen basic functions with their linear weights. Furthermore, the type of basic functions (e.g., Gaussian, multi-quadric, and thin-plate spline) affects the result of spatial interpolation (Wendland, 2005; Rasmussen and Williams, 2006), which is split into two categories, i.e., strictly positive definite (SPD) and conditionally positive definite (CPD) functions (Hillier et al., 2014). We adopt the cubic function as the basis function in this study, i.e., φR=r3, since it minimizes the curvature in three dimensions (Eq. 1).

According to the joint constraints, the weight coefficients α, β, and c of the interpolant are determined by the following linear system: 4Φ∇ΦC(∇Φ)T∇2Φ∇CCT(∇C)T0αβc=fg0, where Φ=φ11φ12⋯φ1Nφ21φ22⋯φ2N⋮⋮⋱⋮φN1φN2⋯φNNN×N, whose element φij=φpi-pj represents the RBF value between a pair of contact points; ∇Φ=∇φ11∇φ12⋯∇φ1M∇φ21∇φ22⋯∇φ2M⋮⋮⋱⋮∇φN1∇φN2⋯∇φNMN×nM, whose element ∇φij=∇φpi-pj represents the differential RBF value between a contact point and an orientation point; ∇2Φ=∇2φ11∇2φ12⋯∇2φ1M∇2φ21∇2φ22⋯∇2φ2M⋮⋮⋱⋮∇2φM1∇2φM2⋯∇2φMMnM×nM, whose element ∇2φij=∇2φpi-pj represents the second-order differential RBF value between a pair of orientation points, C=C(p), in particular, C=1p1xp1yp1z1p2xp2yp2z⋮⋮⋮⋮1pNxpNypNzN×(n+1)and∇C=0∇p1x∇p1y∇p1z0∇p2x∇p2y∇p2z⋮⋮⋮⋮0∇pMx∇pMy∇pMznM×(n+1), whose elements are ∇pix=100T, ∇piy=010T, and ∇piz=001T, respectively. α=α1α2⋯αNT;β=β1β2⋯βMT;f=f1f2⋯fNT; and g=g1g2…gMT. Once we have the weight coefficients (αi, βj) and the polynomial coefficients (c1, c2, c3, c4) by solving the above HRBF linear system, we can substitute the weight coefficients and polynomial coefficients into the HRBF equations; then the interpolant function f(p) and its gradient function ∇fp can be easily obtained.

The study area is located in the Lingnian–Ningping manganese ore zone, in Debao County, southwestern Guangxi Zhuang Autonomous Region, China (Fig. 6). The study area mainly consists of strata from the late Paleozoic to the late Triassic–Pliocene (T3–N2). The middle Permian (P2) strata are in para-unconformity contact with early Triassic (T1) strata; the middle Triassic (T2) strata are in angular unconformity contact with the Quaternary. There is a left strike–slip inverse fault, the Nacha Fault, in the middle of the study area. It dips to the southeast, with a northeastern strike direction of 45∘; a dip angle of about 70∘; and a total length of about 12 km, extending outside the study area. The footwall slid to the west relative to the hanging wall, and the slip distance is about 600 m. There are two synclines (I and III) and an anticline (II) in the study area. Syncline III is located in the middle of the study area with a high symmetry. The axis of syncline III strikes nearly northeast, and its southern limb is cut by the Nacha Fault. Anticline II is located in the northwest of the study area with a good symmetry, the fold axis striking about 30∘ northeast.

Faults, unconformable strata, and intrusive rocks all cause discontinuities in a SPF (Calcagno et al., 2008). We used the fault surface samplings to interpolate the potential field and extract the surface model of the Nacha Fault (Fig. 7).

According to the comprehensive stratigraphic column, the burial depth of each stratigraphic interface relative to the top surface of the Quaternary was used as the attribute value of the SPF (Fig. 8) for implicit-function interpolation. The SPF defines the 3D space as a scalar function f(p) at any point p, where f is defined as the relative burial depth in this study. Each point inside of a stratum has its own burial depth relative to the top surface of the Quaternary; therefore, the depth values in the field decrease gradually from bottom to top in strata. When the relative burial depth is used as the attribute value of the SPF, we can set the initial gradient magnitude g≅1 if the strata underwent heterogenous deformation. However, if we use geological age as the attribute value of the SPF, g can no longer be initially assumed to be 1 because the stratigraphic age and distance along the gradient direction are from different measured variables.

Based on the geological map and DEM of the study area, we produced a series of cross-sections (Fig. 9). However, the cross-sections were presented in 2D form. According to the necessary geographic projection parameters and scale, we derived the mapping relationship between 2D and 3D. Finally, we extracted the geological boundary points with 3D coordinates from 2D cross-sections.

The attribute points and strike and dip points of each stratigraphic interface and fault plane extracted from the geological map and cross-sections were used as the original dataset for 3D SPF modeling. The 3D points of stratigraphic interfaces extracted from the geological map and cross-sections were regarded as samplings of the SPF. The gradient vectors which are transformed from the off-contact stratigraphic strike and dip points were regarded as the samplings of the gradient of SPF.

There are 1410 known on-contact attribute points and 34 off-contact strike and dip points scattered throughout the study area (Fig. 10a). The known strike and dip sampling points are scattered on the southern limb of fold I, the northern and southern limbs of fold II, and the northern and southern limbs of fold III. There are 17 strike and dip sampling points on the northern side of the Nacha Fault and 17 strike and dip sampling points on the southern side. The distribution of the dip directions and dip angles is shown in Fig. 10b.

First, we set the initial gradient magnitude to 1.0 and calculated the x-, y-, and z-axis components of the gradient vector field according to the dip direction and angle of the strike and dip points. We constructed HRBF solution matrices on the northern and southern side of the Nacha Fault, respectively. Then, we iterated to converge toward the optimized gradient magnitudes by adding an optimization term to the HRBF linear system. The termination conditions were met after 200 iterations in the northern sub-domain and 300 iterations in the southern sub-domain. The gradient magnitudes became stable, and finally the optimized magnitudes of the gradient were obtained. The changes in gradient magnitude are shown in Fig. 11.

On a specific grid resolution, we modeled the scalar field of gradient magnitude before and after optimization for each strike and dip point (Fig. 12). Furthermore, we cut four cross-sections of the gradient magnitude scalar field, as shown in Fig. 13.

After the optimized gradient magnitude for each strike and dip point was obtained, all scattered attribute points and strike and dip points were finally substituted into the HRBF linear system to respectively solve the HRBF coefficients (αi, βj) and the polynomial coefficients (c1c2c3c4) for each side of the Nacha Fault. On a specific grid resolution, we generated the regular discrete grids as interpolated points in 3D space. Then the points above the digital elevation model (DEM) were removed from the interpolated points. Finally, we reconstruct the SPFs in 3D space before and after optimization of the gradient magnitude according to the respective HRBF interpolant of each sub-domain. In this study, the SPF represents the relative burial depth in 3D space. The larger field value represents earlier-deposited strata with larger relative burial depth, and vice versa. The same stratigraphic interfaces in different sub-domains share the same field value. The field values change abruptly at the Nacha Fault because the conformable strata were cut by the fault plane.

The SPFs are both constrained so that the interpolated SPF values at the attribute points are equal to the initial relative burial depths, but the SPF values may abruptly change or produce outliers at some locations. Obviously, the SPF values change nonuniformly with gradient magnitude before optimization (Fig. 14a), which caused the SPF values that originally belonged to the Carboniferous strata to be interpolated as those of other strata and sequentially resulted in incorrect extraction of the stratigraphic interfaces. This nonuniform gradient change in stratigraphic potential field causes separated, discontinuous, and dispersed stratigraphic interfaces to be extracted through equipotential surface tracking. However, reconstructing the SPF through optimization of gradient magnitude for each strike and dip point (Fig. 14b) avoids the generation of either abnormal field values or of the wrong equipotential surfaces. This geologically plausible SPF can be appropriately constrained by the known gradient direction and the optimized gradient magnitude at the strike and dip sampling points.

We cut the SPF along four section lines, and the SPF value also changes more uniformly from older to younger strata after gradient magnitude optimization than using a fixed gradient magnitude of 1, as shown in Fig. 15.

Once the field was interpolated in 3D space, the specific equipotential surfaces were extracted from the implicit volumetric function as stratigraphic interfaces within each main-structure-bounded sub-domain. We used the marching cube method to extract the equipotential surfaces with a specific relative burial depth from the stratigraphic interfaces by connecting all the points with the same field value in the stratigraphic potential field (Fig. 16). The interface model on both sides of the Nacha Fault restores the location of the fault in the southern limb of syncline III.

Sequentially, according to the range of relative burial depth of stratigraphic top and bottom, two solid stratigraphic models were reconstructed from these equipotential surfaces before and after optimization of gradient magnitude for each strike and dip point, respectively, combined with sub-domain boundaries and the DEM (Fig. 17). The HRBF interpolation with the initial fixed gradient magnitude of 1 roughly reflects stratigraphic on-contact information and captures the structure of syncline I in the north. However, several details are different from the stratigraphic structure on the geological map. Where the Nacha Fault passes through syncline III, the strata on the southern side of the fault plane should correspond to the same strata on the northern side. However, the Devonian strata corresponded to the Permian strata in area B, as shown in Fig. 17a and b, which is inconsistent with the geological structure. The geological model extracted using the optimized gradient magnitude for each strike and dip point is shown in Fig. 17c. Overall, the obtained geometries follow the shape of the folds and stratigraphic on-contact lines more closely. From north to south in the study area, anticline II and syncline III were successfully modeled with the Nacha Fault correctly represented as an inverse fault that cuts syncline III. On both sides of the Nacha Fault, the sequence of the strata is the same, and the model exhibits traces of the fault plane passing through the stratigraphic surfaces.

Four cross-sections through the solid models (see the geological map for cross-section lines) were cut, and the cross-sections of the solid model are more consistent with the original structural relationships on the geological map after gradient magnitude optimization than using HRBF with a fixed gradient magnitude of 1 and RBF without gradient constraint, as shown in Fig. 18.

The highest stratum and section coincidence percentages on cross-sections are 74.50 % (T2) and 78.03 % (Section 16) before optimization, respectively, as shown in Table 1. However, the highest stratum and section coincidence percentages on cross-sections are 98.99 % (D1) and 98.01 % (Sections 13 and 15) using the optimized gradient magnitude for each strike and dip point, respectively, as shown in Table 2. The total coincidence percentage on cross-sections increases from 67.03 % to 98.27 % after optimizing the gradient magnitude.