## 3 Research Methods
### Long Short-Term Memory Network  
The LSTM network as a type of RNN addressed the issue of gradient vanishing and exploding in conventional recurrent neural networks (RNNs) while handling lengthy sequential data [23]. LSTM incorporates gate control mechanisms to effectively retain and modify the internal state of the network, enabling it to capture long-term dependencies (Figure 6).  
Figure 5: Flowchart of SBAS-InSAR.  
The LSTM module is composed of a memory unit and three gates, namely the input gate, the forget gate, and the output gate [24]. The input gate regulates the transmission of data from the input layer to the memory unit, while the forget gate determines the information that should be preserved or eliminated. Additionally, the output gate governs the transfer of data from the memory unit to the output layer. These gate structures enable LSTM to selectively remember or forget information while learning in sequences, effectively capturing and maintaining long-term dependency.  
In each time step, LSTM takes in the present input along with the hidden state and memory unit contents from the previous time step. It then produces updates for the current time step's hidden state and memory unit by utilizing these gates, which finally produce the output. These operations are implemented through nonlinear activation functions with learnable parameters. As a result, the LSTM model can be properly trained to effectively maintain information flow, and it performs well in applications when dealing with long series of data.  
In temporal InSAR (TS-InSAR) data analysis, the utilization of the LSTM model enables the accurate prediction of land subsidence trends and patterns. By learning and training historical surface subsidence data, the LSTM model can extract hidden patterns and rules in time series information, which predicts future surface subsidence. Compared to conventional statistical techniques, the LSTM model exhibits enhanced capability in capturing nonlinear associations and long-term dependencies within time series data, thereby elevating prediction accuracy.  
However, the traditional LSTM may have the following limitations in the surface settlement prediction [25]. (1) Long-term dependency modeling problems are shown as follows. An unoptimized LSTM may have difficulty dealing with long-term dependencies. Long-term dependence refers to the existence of a large time interval among a series of events. The memory unit of the LSTM model may not be able to effectively capture information, causing a decline in the prediction accuracy of such long-term context information. (2) Lack of data: Surface subsidence prediction studies usually require a large amount of observational data to train models, but real-world geological monitoring data may be missing or incomplete. Unoptimized LSTM may be sensitive to missing data, which can affect the training and prediction ability of the model in inaccurate prediction results. (3) Overfitting problem: Unoptimized LSTM may have the risk of overfitting when fitting training data. The LSTM model has a strong memory ability. It is possible to excessively fit the training dataset, which leads to a lack of generalization when encountering new and unfamiliar data. This results in poor performance of the predictive model in practical applications. (4) Parameter selection problem:  
Figure 6: Schematic diagram of LSTM principle.  
Unoptimized LSTM requires careful parameter selection, including network structure, learning rate, and number of iterations. The inadequate parameter selection can result in subpar model performance and impact the accuracy and stability of prediction outcomes.