# GeoSAE: A 3D Stratographic Modeling Method Driven by Geological Constraint
## 2 Methods
### Geologically Constrained Loss Function Construction
#### 2.1.2 Stratigraphic Sequence Constraints  
In 3D geological modeling, the spatial distribution of strata is commonly represented by using the potential field method, which involves defining potential fields. The appropriate number of potential fields is determined based on the stratigraphic contact relationships, ensuring that the morphological characteristics of each stratigraphic surface are accurately captured. The stratigraphic contact relationship is determined through geological field observations, combined with the stratigraphic age relationship, and represented on the stratigraphic column. If two strata are in contact, share the same depositional environment, and exhibit integrated properties, they are represented by the same potential field due to their similar stratigraphic characteristics. Conversely, if there is an unconformity or significant depositional difference between the strata, each unconformity is assigned its own potential field to ensure that the geological features are accurately reflected in the model, as shown in Figure 1. In Figure 1, six stratigraphic surfaces are shown, including two unconformable interfaces (\(I_{2}\) and \(I_{5}\)) and four conformable interfaces (\(I_{0}\), \(I_{1}\), \(I_{3}\), and \(I_{4}\)). These are modeled by constructing four potential fields based on the stratigraphic contact relationships. The stratigraphic surfaces and units correspond to potential fields as follows: potential field \(S_{0}\) is associated with stratigraphic surfaces \(I_{0}\), \(I_{1}\), and units \(A\), \(B\), and \(C\); potential field \(S_{1}\) is associated with stratigraphic surface \(I_{2}\); potential field \(S_{2}\) is associated with stratigraphic surfaces \(I_{3}\), \(I_{4}\), and units \(D\), \(E\), and \(F\); and potential field \(S_{3}\) is associated with stratigraphic surface \(I_{5}\). Based on the vertical relationships between strata, the relative magnitude of the potential field values corresponding to sampling points at different stratigraphic surfaces within the same potential field can be inferred. The following definitions are provided in this paper:  
\"Above\" relationship: For a given potential field representing consecutive strata, the potential field value at the sampling point of the upper stratum is required to be greater than that of the lower stratum if one stratum is above another. \"Below\" relationship: In all potential fields, the potential field value at the sampling point of the lower stratum is required to be less than that of the upper stratum if one stratum is beneath another. \"Overlap\" relationship: If two stratigraphic surfaces overlap, equality of the potential field values at the sampling points along the interface of these strata is required. The \"above\" relationship is applied only within the same potential field because an unconformity is introduced by contact between strata from different potential fields. In this situation, the upper strata are influenced by the older strata beneath them, creating an erosional contact. Consequently, the potential field value of the upper strata may be lower than that of the lower strata due to the erosive effects on the older strata, as illustrated in Figure 2b. In the eroded portion of stratum \(I_{2}\) at the unconformity, the potential field value at the sampling point of stratum \(I_{2}\) is smaller than that of stratum \(I_{1}\), indicating that the \"above\" relationship does not apply in this case. However, the \"below\" relationship is unaffected. If one stratum is beneath another, the potential field values at the sampling points in the lower stratum must always be less than those in the upper stratum for the \"below\" relationship to hold.