# Displacement Patterns and Predictive Modeling of Slopes in the Bayan Obo Open-Pit Iron Mine
## Appendix A  
GRU contains two key gating structures:  
Update Gate: The update gate determines how much of the previous hidden state is retained and how much new information is introduced, as shown in Equation (A1).  
\[z_{t}=\sigma(W_{z}\cdot[h_{t-1},x_{t}]+b_{z})\] (A1)  
where \(x_{t}\) is the current input, \(h_{t-1}\) is the previous hidden state, \(W_{z}\) and \(b_{z}\) are the weight matrix and bias term of the update gate, and \(\sigma\) is the Sigmoid activation function with outputs in [0, 1].  
Reset Gate: The reset gate determines how much of the previous hidden state is \"forgotten\" to generate the candidate hidden state, as shown in Equation (A2).  
\[r_{t}=\sigma(W_{r}\cdot[h_{t-1},x_{t}]+b_{r})\] (A2)  
where \(x_{t}\) is the current input, \(h_{t-1}\) is the previous hidden state, \(W_{r}\) and \(b_{r}\) are the weight matrix and bias term of the reset gate, and \(\sigma\) is the Sigmoid activation function.  
The core of GRU lies in the collaboration between the update gate and reset gate, as well as the progressive update of the hidden state. The candidate hidden state integrates information from the reset gate to generate a temporary hidden state at the current time step, as shown in Equation (A3).  
\[\widetilde{h}_{t}^{\sim}=\tanh(W_{h}\cdot[r_{t}\odot h_{t-1},x_{t}]+b_{h})\] (A3)  
where \(\odot\) denotes element-wise multiplication (Hadamard product) and \(W_{h}\) and \(b_{h}\) are the weight matrix and bias term.  
The final hidden state combines the output of the update gate, fusing previous information with the current candidate state, as shown in Equation (A4).  
\[h_{t}=(1-z_{t})\odot h_{t-1}+z_{t}\odot\widetilde{h}_{t}\] (A4)  
where \(z_{t}\) controls the weights of old and new information. If \(z_{t}\) approaches 0, more historical information is retained.  
As shown in Figure A1, GRU first combines the current input \(x_{t}\) and the previous hidden state \(h_{t-1}\) to generate two gating signals. Specifically, the update gate \(z_{t}\) and reset gate \(r_{t}\) output values between 0 and 1 via the Sigmoid function, corresponding to \"the degree of retaining old states\" and \"the degree of forgetting historical information,\" respectively.  
Next, the reset gate value \(\mathrm{r_{t}}\) acts on the previous hidden state \(\mathrm{h_{t-1}}\) through element-wise multiplication (\(\mathrm{r_{t}}\odot\mathrm{h_{t-1}}\)), selectively filtering out irrelevant historical information. This operation resembles \"resetting\" certain long-term memories while retaining content useful for the current time step. Subsequently, the filtered historical state and current input \(\mathrm{x_{t}}\) are combined to generate a candidate hidden state \(\mathrm{\overset{\sim}{h_{t}}}\), which uses the \(\tanh\) activation function to capture potential new information at the current time step, achieving preliminary fusion of current input and historical information.  
Finally, the update gate \(\mathrm{z_{t}}\) performs a weighted average of the old state \(\mathrm{h_{t-1}}\) and candidate state \(\mathrm{\overset{\sim}{h_{t}}}\). If the value of the update gate is close to 1, the candidate state dominates the current output, that is, it tends to learn new information. If it is close to 0, more historical states will be retained, that is, long-term dependence will be maintained.