## 3 Methods
### Model Design  
Artificial neural networks (ANNs), as statistical learning models, are a research focus in image processing and the remote sensing retrieval of parameters on the Earth's surface based on remote sensing images. The water depths of remote sensing models based on ANN have found a wide range of applications around the world [22; 24]. [PERSON] and [PERSON] [21] noted three advantages of ANNs in the remote sensing inversion model. ANNs are more accurate and efficient than other forms of linear statistics, especially for very complex feature spaces or multi-source data with different statistical distributions; ANNs allow prior knowledge (e.g., a geomorphology value) to be used as an input to build the model; and ANNs can couple different types of data, facilitating collaborative research. Therefore, theoretically, an ANN is an effective method to simulate the elevation of tidal flats based on remote sensing images.  
A multi-layer feed-forward back propagation network was proposed by [PERSON] [33] and is widely used for data prediction. The mechanism is that the weights and thresholds of the network were modified to minimize the error between the real and simulated values. The basic BP algorithm includes two processes: forward propagation of the signal and backward propagation of the error. That is, the error output is calculated in the direction from input to output, and the adjustment weights and thresholds are performed in the direction from output to input. During forward propagation, the input signal acts on the output node through the hidden layer and undergoes a nonlinear transformation to produce the output signal. If the actual output does not match the expected output, the error is transferred back to the reverse propagation process. After repeated learning and training, the network parameters (weights and thresholds) corresponding to the minimum error are determined.  
Designing the structure of a BP neural network model, however, is still a problem. BP neural networks with at least one hidden layer are necessary for arbitrary nonlinear function approximation. In practice, one or two hidden layers (that is, three layers or four layers including the input and output layers) are commonly used. In this study, the DEM (Figure 2d) generated in the previous section is considered as the expected output vector. Using the HJ-1 spectral bands (B1, B2, B3, B4), the geographical coordinates (X, Y), and the distance to the waterline (D), a total of seven vectors were prepared as the input layer. Taking the HJ-1 image of 21 March 2014 as the input, single-hidden, and double-hidden layers with nodes (from 3 to 22) for each layer network were developed. The root mean square error (RMSE in Equation (1)) and coefficient of determination (R\({}^{2}\) in Equation (2)) were used as evaluation indices to select the optimal model structure. The RMSE shows the dispersion of the sample and reflects the degree of deviation from the true value of the simulation data. R\({}^{2}\) reflects the degree of correlation between the two sets of data. Therefore, when the RMSE is smaller, the R\({}^{2}\) is larger, and the structure is better. The results in Figure 3 show that the double-hidden layer network is obviously better than the single-hidden layer network. A model with double-hidden layers with 15 nodes per layer model was chosen. The topology of the BP neural network with double-hidden layers is shown in Figure 4.  
\[RMSE=\sqrt{\frac{\sum\left(Z_{i}-\hat{Z}_{i}\right)^{2}}{n-1}}\quad\mathrm{i }=1,\;2,\;3,\;\ldots,\;\mathrm{n} \tag{1}\]  
\[R^{2}=\left(\frac{\sum\left(Z_{i}-\overline{Z}\right)\times\sum\left(\hat{Z}_ {i}-\dot{Z}\right)}{\sqrt{\sum\left(Z_{i}-\overline{Z}\right)^{2}\times\sum \left(\hat{Z}_{i}-\dot{Z}\right)}^{2}}\right)^{2}\quad\mathrm{i}=1,\;2,\;3,\; \ldots,\;\mathrm{n} \tag{2}\]