## 2 Methods  
### Umap  
Uniform Manifold Approximation and Projection (UMAP) is a recently developed dimensionality reduction technique [28; 29]. As a manifold learning method, UMAP is designed to preserve both the global and local structures of high-dimensional data during projection to a lower-dimensional space. This includes maintaining the topological relationships and inter-point distances inherent in the original data. UMAP has been successfully applied across a diverse range of data types, including gene expression profiles, image datasets, and textual information, often yielding visually compelling and interpretable low-dimensional representations. UMAP is founded on three key assumptions: (1) the data are uniformly distributed on a Riemannian manifold, (2) the Riemannian metric is locally constant, and (3) the manifold is locally connected. Based on these assumptions, given a high-dimensional seismic datasets with \(N\) input vectors \(\mathbf{X}=\{x_{1},x_{2},\cdots x_{n}\}\) and \(x_{i}\in R^{d}\), we can achieve low-dimensional output vectors \(\mathbf{Y}=\{y_{1},y_{2},\cdots y_{m}\}\) through the UMAP algorithm, where \(y_{i}\in R^{m}\), \(i\in[1,N]\), and \(m<d\). The specific realization of UMAP can be decomposed into the following steps:  
(1) Construct probability distributions in high-dimensional spaces. UMAP finds the embedded manifold by identifying the fuzzy topological structure of a low-dimensional projection. To construct this structure, UMAP represents data points as a weighted high-dimensional graph, where edge weights indicate the likelihood of connections between points. UMAP uses an exponential probability distribution function to compute the similarity between high-dimensional data points:  
\[p_{i|j}=exp\left(-\frac{d\left(x_{i},x_{j}\right)-\rho_{i}}{\sigma_{i}}\right) \tag{1}\]  
where \(d\left(x_{i},x_{j}\right)\) is the distance between the data points \(x_{i}\) and \(x_{j}\), \(\rho_{i}\) is the distance between \(x_{i}\) and its first nearest neighbor, \(\sigma_{i}\) is a scaling factor that depends on the density of the data. When the weight between the data points \(x_{i}\) and \(x_{j}\) in the graph differs from the weight between \(x_{j}\) and \(x_{i}\), UMAP applies a symmetrization process to the high-dimensional probabilities. After UMAP aggregates points with locally varying metrics, the weight from node A to B may not be equal to the weight from B to A. Therefore, joint probability is used to ensure symmetry. \[p_{ij}=p_{i|j}+p_{j|i}-p_{i|j}p_{j|i} \tag{2}\]  
In addition, UMAP uses the number of nearest neighbors \(k\) instead of perplexity. The number of nearest neighbors is defined as the \(k\) nearest data points around a given data point, which determines how many adjacent nodes each data point has in the KNN graph:  
\[k=2^{\sum\limits_{i}p_{ij}} \tag{3}\]  
(2) Construct probability distributions in low-dimensional spaces. Design a probability function to compute the relationship between low-dimensional data points. Note that \(y_{i}\) is the embedding corresponding to \(x_{i}\). In the embedding space, the probability that a point \(y_{i}\) has the point \(y_{j}\) as its neighbor can be computed by the similarity of these points:  
\[q_{ij}=\left(1+a\left(y_{i}-y_{j}\right)^{2b}\right)^{-1} \tag{4}\]  
where \(a\approx 1.93\) and \(b\approx 0.79\) for default UMAP. (3) Construct a cost function to make the relationship between high-dimensional sample points as similar as possible to the relationship between low-dimensional sample points. Binary cross entropy (CE) is used as the cost function in the UMAP with the goal of minimizing the distance between high-dimensional and low-dimensional space [30].