## 3 Predicting Missing Soil Heavy-Metal Values Based on SA-FOA-BP
### SA-FOA-BP Model
Model initialization: Set the number of neurons in each layer of the BP neural network, including the number of neurons in the input layer \(m\), number of neurons in the hidden layer \(n\), and number of neurons in the output layer \(l\). Set the initial position of the fruit fly population and set the maximum number of iterations _maxgen_ and the population size. 2. SA-FOA optimization: The process is introduced in Section 2.1 and is not repeated here. In this study, the error function e in the BP neural network is taken as the objective function, represented by concentration value in the FOA; the solution vector S is used as the comprehensive vector of connection weights and threshold values in the BP network. When the SA-FOA terminates, the weight and threshold values in the solution vector S are used to calculate the weight and threshold values of the current training sample in the BP neural network. 3. BP neural network training: Assign the weight and threshold values obtained from the SA-FOA optimization to the BP neural network and train the BP neural network according to the preset parameters. Figure 2 presents a flowchart of the SA-FOA-BP neural network model. Prediction of Soil Heavy-Metal Content Using Graph Convolutional Networks Integrating Spatial Information  
Spatial data contain coordinates and geometric structural features through which the location, shape, distribution, and relationships between objects can be represented. Traditional neural network methods ignore spatial modeling, whereas graph neural networks  
Figure 2: Flowchart of SA-FOA-BP neural network model calculation. represent spatial structures in a graph structure, providing new ideas for spatial regression prediction or interpolation. When constructing a soil heavy-metal content prediction model, information from multiple sampling points can be combined to predict target points. To fully consider spatial features, a graph convolutional network model integrating spatial information was proposed to predict soil heavy-metal content at unknown points. The model is based on the graph convolutional neural network (GCN) and incorporates spatial information and points of interest [24] into the network. First, we defined a data point \(p_{i}=\{y_{i},x_{i},c_{i}\}\), where \(y_{i}\) is the dependent variable to be interpolated for prediction, \(x_{i}\) is a feature vector, and \(c_{i}\) is a coordinate vector. Thus, all points can be represented using \(P=\{p_{1},\ldots,p_{n}\}\). Using latitude and longitude information, all data points can be mapped in geometric space. Assuming that a certain data point is the center point, the distance \(d_{ij}\) between all data points and this point can be obtained using the Haversine formula. The Haversine formula is given by  
\[d_{ij}=2r\ argsin(\sqrt{sin^{2}(\frac{lat2-lat1}{2})}+cos(lat2)cos(lat1)sin^{2} (\frac{lon2-lon1}{2})) \tag{7}\]  
where \(r\) is the radius of the Earth, \(lon1\) and \(lon2\) are the longitudes of the two points, and \(lat1\) and \(lat2\) are the latitudes of the two points. The source of soil pollution directly affects heavy-metal concentrations. Thus, when predicting the heavy-metal content in soil, the features of the points of interest must be incorporated into the prediction model. In this study, the pollution source was used as the point of interest, and all pollution enterprises in the study area were regarded as a set. The interest-point matrix is defined as \(\mathbf{V}=\{v_{1},\ldots,v_{n}\}\), where \(v_{i}\) is the coordinate vector of the point of interest. Considering the dependency between the center point and other data points, which gradually weakens with an increase in the distance between the points, the correlation also becomes weaker. Thus, the k-nearest neighbor method or setting a reasonable distance threshold can be used to improve training efficiency. Using this method, the neighborhood of each data point can be obtained and an adjacency matrix A that incorporates the fusion interest-point features can be created.