# GeoSAE: A 3D Stratographic Modeling Method Driven by Geological Constraint
## 2 Methods
### Geologically Constrained Loss Function Construction
#### 2.1.2 Stratigraphic Sequence Constraints
Therefore, based on the definitions of the \"above\", \"below\", and \"overlap\" relationships, an example of the layer ordering relationships based on potential field values between the various stratigraphic layers in Figure 1 is illustrated in Figure 3.  
The relationships of \"above\", \"below\", and \"overlap\" between layers at each location are determined based on potential field values, and corresponding loss functions are constructed as outlined below. For each stratigraphic surface, the difference between the potential field value \(S_{x}^{I_{X}}\) of any point \(x\) in the sampling set \(X\) of stratigraphic surface \(I_{K}\) and the average potential field value \(\overline{S_{K}^{I_{K}}}\) of stratigraphic surface \(I_{K}\) is used to indicate the following: if the difference \(D_{I_{x},I_{K}}\), \(I_{K}\) is greater than 0, point \(x\) is considered to be above stratigraphic surface \(I_{K}\); if the difference \(D_{I_{x},I_{K}}\) is less than 0, point \(x\) is considered to be below stratigraphic surface \(I_{K}\)  
Figure 1: Potential field representation based on stratigraphic contact relationships, showing six stratigraphic surfaces, including two unconformable and four conformable interfaces. and if the difference \(D_{I_{x},I_{K}}\) is equal to 0, point \(x\) is considered to be on stratigraphic surface \(I_{K}\), meaning the two overlap. The relevant formula is as follows. The hierarchical relationship between the surfaces is represented by the constraint inequality provided in Equation (3):  
\[ \begin{array}{l}D_{I_{x},I_{K}}>0\to Above\\ D_{I_{x},I_{K}}<0\to Below\\ D_{I_{x},I_{K}}=0\to Overlap\end{array}  \tag{3}\]  
where the difference in potential field between stratigraphic surfaces, denoted as \(D_{I_{x},I_{K}}\), is given by Equation (4):  
\[D_{I_{x},I_{K}}=S_{x}^{I_{X}}-S_{K}^{I_{K}} \tag{4}\]  
where the potential field value of point \(x\) within the potential field at stratigraphic surface \(I_{X}\) is represented by \(S_{x}^{I_{X}}\), and the average potential field value at stratigraphic surface \(I_{X}\) is denoted by \(\overline{S_{K}^{I_{K}}}\), as shown in Equation (5):  
\[\overline{S_{K}^{I_{K}}}=\frac{1}{|K|}\sum S^{I_{K}}(K)=\frac{1}{|K|}\sum_{k \in K}^{|K|}S_{k}^{I_{K}} \tag{5}\]  
The number of points in the sampling point set \(K\) at the stratigraphic surface \(I_{K}\) is represented by \(|K|\), with \(k\) being one of the sampled points in the set. If the difference \(\frac{D_{I_{x},I_{K}}}{S_{K}^{I_{K}}}\) between the potential field value \(S_{x}^{I_{X}}\) at point \(x\) and the average potential field value \(\overline{S_{K}^{I_{K}}}\) at stratigraphic surface \(I_{K}\) is non-zero, the loss of the average potential field value \(\overline{S_{K}^{I_{K}}}\) at point  
Figure 3: Layer ordering relationships between stratigraphic surfaces based on potential field value sequence, illustrating the “above”, “below”, and “overlap” relationships in the stratigraphic model. Figure 2: Schematic illustration of the potential field for the erosion section: (**a**) original stratigraphy; (**b**) stratigraphic surfaces \(I_{1}\) and \(I_{2}\) corresponding to the extracted potential fields and erosion section. \(x\) relative to the stratigraphic surface \(I_{K}\) is represented by the approximate signed distance \(\delta_{I_{x},I_{K}}\). The relevant formulae are presented as follows. The formula for the approximate signed distance \(\delta_{I_{x},I_{K}}\) is given in Equation (6):  
\[\delta_{I_{x},I_{K}}=\frac{D_{I_{x},I_{K}}}{\left\|\
abla S_{x}^{I_{X}}\right\| }=\frac{S_{x}^{I_{X}}-\overline{S_{K}^{I_{K}}}}{\left\|\
abla S_{x}^{I_{X}}\right\|} \tag{6}\]  
The scalar gradient of point \(x\) in the potential field corresponding to the stratigraphic surface \(I_{X}\) is denoted by \(\left\|\
abla S_{x}^{I_{X}}\right\|\). The change in thickness between the units of interest is represented; a smaller scalar gradient is indicated for thicker units, while a larger gradient is related to thinner units.