Three-dimensional (3D) stratigraphic modeling is capable of modeling the shape, topology, and other properties of strata in a digitalized manner. The implicit modeling approach is becoming the mainstream approach for 3D stratigraphic modeling, which incorporates both the off-contact strike and dip directions and the on-contact occurrence information of stratigraphic interface to estimate the stratigraphic potential field (SPF) to represent the 3D architectures of strata. However, the magnitudes of the SPF gradient controlling the variation trend of SPF values cannot be directly derived from the known stratigraphic attribute or strike and dip data. In this paper, we propose a Hermite–Birkhoff radial basis function (HRBF) formulation, AdaHRBF, with an adaptive gradient magnitude for continuous 3D SPF modeling of multiple stratigraphic interfaces. In the linear system of HRBF interpolants constrained by the scattered on-contact attribute points and off-contact strike and dip points of a set of strata in 3D space, we add a novel optimizing term to iteratively obtain the optimized gradient magnitude. The case study shows that the HRBF interpolants can consistently and accurately establish multiple stratigraphic interfaces and fully express the internal stratigraphic attribute and orientation. To ensure harmony of the variation in stratigraphic thickness, we adopt the relative burial depth of the stratigraphic interface to the Quaternary as the SPF attribute value. In addition, the proposed stratigraphic-potential-field modeling by HRBF interpolants can provide a suitable basic model for subsequent geosciences' numerical simulation.

The three-dimensional (3D) stratigraphic modeling and visualization technology is of great importance for the intelligent management of subsurface space (e.g., mineral resource assessment, reservoir characterization, groundwater management, and urban subsurface space planning) (Houlding, 1994; Mallet, 2002). The two main ways of representing 3D stratigraphic surface are so-called explicit and implicit modeling (Lajaunie et al., 1997). Traditional explicit modeling can be described as a way of representing 3D geological boundaries that relies heavily on a complicated and time-consuming process of human–computer interaction for connecting the geological boundary lines to form a 3D model of geological surfaces, and it is difficult to update the model. Implicit modeling defines a continuous 3D stratigraphic potential field (SPF) that describes the stratigraphic distribution and represents geological boundaries using an implicit mathematical function. The increasing importance of an implicit method in stratigraphic modeling stems from not only the advantages of efficiency, reproducibility, and topological consistency over the traditional explicit modeling method but also the full representation of stratigraphic structure through SPF. Although implicit modeling often requires a large solution system of linear equations to consume more computational time than explicit modeling, e.g., the Delaunay triangulation (Mallet, 2002; De Berg et al., 2008), we can overcome this difficulty with the help of increasing computational ability of computers. Three-dimensional stratigraphic-potential-field modeling is to implicitly represent the nature, shape, topology, and internal property of a given set of strata. The stratigraphic interface is expressed by a specific equipotential surface of the SPF. Therefore, using SPF to express a set of conformable strata and their attribute distribution in 3D space is convenient for spatial analysis, statistics, and simulation.

The strike and dip information can be incorporated into implicit modeling by setting up the gradients of implicit function. To control the orientation of the modeled strata, the dip and strike directions are encoded as the gradient directions. The difference between Hermite–Birkhoff radial basis function (HRBF) and standard radial basis function (RBF) is the presence of gradients; however, the existing HRBF method constructs implicit field functions separately for each geological interface and extracts the zero-value equipotential surfaces to locate the geological interface. Therefore, it is difficult to maintain topological and semantic consistency between geological bodies. For modeling multiple strata in an integrated and unique framework, however, setting up the gradient magnitudes to be adaptive to the orientation and thickness variations in strata is rather challenging. Assigning the adaptive gradient magnitudes to HRBF interpolant function is a “chicken-and-egg” problem: while the implicit function results from the gradients, the suitable gradient magnitudes are estimated from the reasonable implicit function.

In this study, we propose a gradient-adaptive HRBF framework for SPF modeling, AdaHRBF, which interpolates multiple interfaces among a set of conformable strata by a unified one-step process. In this linear system of HRBF interpolants, we iteratively obtain the optimized gradient magnitudes. The particular case where the SPF was reconstructed from geological maps and cross-sections demonstrates the advantages and general performance of stratigraphic-potential-field modeling using the AdaHRBF method, comparing with HRBF interpolants using constant unit normal gradients and RBF interpolants only using contact locations without orientations. The SPF attribute value is set to the relative burial depth of strata, i.e., mean distance from a given stratigraphic surface to the top surface of the Quaternary. The distributions of burial depth, thickness, and strike and dip of strata in 3D space can be fully expressed by the SPF and its gradient vector field.

The key of implicit modeling methods is to interpolate a 3D scalar field function whose equipotential surfaces indicate the boundaries of geological bodies. These surfaces can represent ore-grade boundaries or stratigraphic interfaces. This scalar field is interpolated from stratigraphic interface points and strike and dip data with either discrete-interpolation schemes or continuous-interpolation schemes.

For discrete-interpolation schemes of implicit modeling with a special mesh,
the GoCAD (

Data commonly used in

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 (

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 “

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

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

Generally, the basic RBF reconstructs an implicit function with constraint

When using the HRBF interpolation method, we usually add a first-order
polynomial

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.,

According to the joint constraints, the weight coefficients

The gradient of the SPF is an important feature of stratum shape because it
indicates the strike and dip of a stratum. For construction of a scalar
field

The gradient vector

The gradient

The exact definition of gradient magnitude (

Simply optimizing Eq. (7) would lead to a linear system as

Pseudo-code of iterative algorithm for optimizing gradient magnitude.

On the other hand, to optimize

Experimental field 1:

In this study, we calculate the increment of

We use two stopping criteria to finish the iterations. Firstly, for all
observed strike and dip points, if the sum of differences in gradient
magnitudes between two consecutive iterations is less than or equal to a
small enough threshold

Experimental field 2:

Two experimental fields in 2D space, with the gradient changing in direction or
magnitude, were designed to verify the AdaHRBF method. The experimental
results show that the different gradient magnitude settings apparently
affect the modeled fields; moreover, the AdaHRBF method is effective to
iteratively obtain the optimized gradient magnitude of the fields. We
modeled an analytic field of

Geological map of the study area.

We also modeled a potential field of

Model of Nacha Fault:

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 (T

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

Comprehensive stratigraphic column of the study area. In this context, the SPF is fitted by a scalar function of the relative burial depth. Burial depth decreases as geological time progresses; therefore, earlier-deposited strata are assigned a relatively larger burial depth, while later-deposited strata are assigned a relatively smaller burial depth.

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.

Geological cross-sections, mapped according to the planar geological map and DEM of study area. The cross-sections were mapped by vertical extension according to the boundaries and strike and dip points of strata along the layout lines of 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.

Scattered attribute points and strike and dip points of strata:

First, we set the initial gradient magnitude to 1.0 and calculated the

Changes in optimization coefficient

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.

Scalar field of

Cross-sections of the gradient magnitude field:

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 (

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.

Stratigraphic potential field

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.

Cross-sections of the stratigraphic potential field

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.

Three-dimensional model of the bottom surfaces of strata. The 3D surface model extracted from the potential field shows that the geometrical shape of each equipotential (iso-depth) surface is smooth, and the topology is consistent.

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.

Three-dimensional stratigraphic models using

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.

Cross-sections of the solid models.

The highest stratum and section coincidence percentages on cross-sections
are 74.50 % (T

Coincidence percentages on cross-sections using RBF without gradient constraint.

Coincidence percentages on cross-sections using HRBF with an initial fixed gradient magnitude of 1 for each strike and dip point.

Coincidence percentages on cross-sections using AdaHRBF with optimized gradient magnitude.

The AdaHRBF proposed in this study improves the use of strike and dip data in SPF modeling by optimization of gradient magnitudes. In addition to use of strike and dip information as the gradient directions of SPFs, we use the gradient magnitude as a new constraint to control the rate of change in SPF values. The gradient of a SPF is a vector with certain direction and magnitude, in which the gradient magnitude provides constraints on the thickness of deformed strata. Therefore, it is extremely important to construct HRBF linear systems with accurate gradient magnitudes in 3D SPF modeling. As a “chicken-and-egg” problem, it is difficult to determine the exact gradient magnitude through the geological measurements or prior structural knowledge. We proposed an iterative optimization method which alternates between estimation of SPF and gradient magnitudes so that the gradient magnitudes progressively converge towards the values and are adaptive to the stratigraphic architecture. The optimized gradient magnitudes more accurately simulate the variations in the SPF between the top and bottom surfaces. Besides constraints of scattered multivariate Hermite–Birkhoff data, the generalized RBF (Hillier et al., 2014) reconstructs an implicit function with more constraints of lithologic markers (inequality) and lineations (tangent). How to integrate these constraints in our solution to utilize more kinds of modeling data shall be studied in future work.

Jessell et al. (2014) highlighted two limitations of current implicit modeling schemes: (1) they are incapable of interpolating or extrapolating a fold series within a continuous structural style; (2) the shape of fold hinges they produce is not controlled and may yield inconsistent geometries. To overcome these two limitations, we adopted two strategies: (1) a 3D stratigraphic-potential-field modeling method based on HRBF interpolants was used to interpolate a fold series within a structurally continuous domain; (2) a number of structural strike and dip points were sampled on both limbs of the folds to control the geometries of fold hinges. A novel method for modeling folds uses a fold coordinate system based on fold axis direction, fold axial surface, and extension direction and incorporates a parametric description of fold geometry (e.g., fold wavelength, amplitude, tightness, and rotation angle) into the interpolation algorithm (Laurent et al., 2016; Grose et al., 2017, 2019), which would be our future research direction of fine fold modeling based on AdaHRBF.

There are several choices for the value of the potential field, e.g., the sorted serial number, burial depth, or depositional time for each stratigraphic interface (Mallet, 2004). However, the thickness of the stratum is not necessarily proportional to the sorted serial number and deposition time. Compared with using the sorted serial number or depositional age of stratigraphic interfaces as the potential-field value, choosing the burial depth is more in line with 3D SPF modeling. We derived the gradient direction from the strike and dip points; moreover, we used the gradient magnitude as a constraint to control the rate of change in the SPF.

The purpose of this study is to establish a framework for 3D SPF modeling by using the HRBF interpolant with adaptive gradient optimization constrained by on-contact attribute points and off-contact structural strike and dip points. We applied this method to a study site in the Lingnian–Ningping area, and a geological map, four cross-sections, and a DEM were used as original data to model a SPF whose field value was taken from the relative burial depth of the stratigraphic interfaces. The results show that the implicit modeling of the SPF by HRBF interpolants and optimization of gradient magnitude can be effectively adapted to 3D geological modeling using the sampling points from a geological map and cross-sections. A SPF can express the parameters of a stratum such as property, shape, and topology in 3D space.

However, the modeling process is complicated because the sub-domains are required to be divided manually. In actual geological surveys, the geological structure may be more complex and include a large number of faults, unconformable strata, and intrusive rocks. Therefore, it is necessary to separately identify the boundary of the sub-domains according to the fault interfaces, unconformable strata, and intrusive rocks before the 3D geological modeling work. A goal for future work is to introduce a way to integrate faults (Grose et al., 2021a) into the implicit model to accommodate discontinuity of fault planes. In addition, the 3D orientations are usually surveyed on the outcrop strata; however, it would introduce uncertainty if we were to assume that the orientations of a totally subsurface terrain are consistent with its conformable outcrop strata. Therefore, this uncertainty in the model should be considered in the modeling process, and additional geophysical exploration data and geological interpretation should be incorporated into the modeling constraints.

The source code for the AdaHRBF is available in MATLAB at GitHub
(

The data for the
AdaHRBF are available
at GitHub (

BZ and HD initiated the conception of the study and advised the research on it. LD and YT programmed the AdaHRBF code and carried out the data analyses for real-world case studies. UK contributed significantly to analysis and manuscript preparation. YT performed both verification and real-world experiments, created all plots, carried out the initial analysis, and wrote the manuscript. HD and LW helped perform the analysis with constructive discussions. All authors provided critical feedback and helped to shape the whole study.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study was supported by grants from the National Natural Science
Foundation of China (grant nos. 42072326 and 41972309), the China Geological
Survey Project (grant no. DD20190156), and the National Key Research and
Development Program of China (grant no. 2019YFC1805905). The authors express
their sincere gratitude to Italo Goncalves, Lachlan Grose, and
Michal Michalak (referees) for their constructive reviews, which benefited
this paper. The authors thank the MapGIS Laboratory co-constructed by
the National Engineering Research Center of Geographic Information System
of China and Central South University for providing
MapGIS^{®} software (Wuhan Zondy Cyber-Tech Co.
Ltd., Wuhan, China). We also thank Shang-guo Zhou
(Institute of Mineral Resources Research, China Metallurgical Geology
Bureau) and Xian-cheng Mao (Central South University) for their
kind assistance with data collection and Jeffrey Dick (Central
South University) for revising scientific English writing of this
paper.

This research has been supported by the National Natural Science Foundation of China (grant nos. 42072326 and 41972309), the China Geological Survey (grant no. DD20190156), and the National Key Research and Development Program of China (grant no. 2019YFC1805905).

This paper was edited by Thomas Poulet and reviewed by Lachlan Grose, Michal Michalak, and Italo Goncalves.