## 3 Results
### Dual-Objective Optimisation Parameters  
Then, in this section, we compare and analyze the results of parameter optimization under dual-objective functions, with paired combinations of \(\theta\), \(T\), \(h\), and \(q\) as the objective variables. When there is more than one objective variable, parameter inversion produces a set of Pareto optimal solutions that represent trade-offs between different objectives. These solutions are characterized by movement along the trade-off surface, where improving one objective results in the deterioration of others. Plotting a Pareto chart provides an intuitive way to analyze the trade-offs between objective variables. Due to the magnitude differences among these variables, we normalize the model solutions and plot the Pareto chart, as shown in Figure 5. We select 30,000 sets of samples, where the gray circles represent the final model solutions closest to the optimal solutions, and the Pareto front is represented by solid black circles. \(F_{1}\) and \(F_{2}\) represent the dual objective functions, with the subscripts \(\theta\), \(T\), \(h\), and \(q\) indicating which variable is used as the objective variable. In most cases, the shape of the Pareto front can be categorized as either steep or smooth. However, the Pareto front in Figure 5 is asymmetrical. This may be because numerical simulation errors are almost negligible, and the model achieves a high degree of fit. In practice, simulations often involve significant uncertainties, including model structure, measurement data errors, and equifinality for different parameters. In detail, when the objective variables are \(\theta+h\) (Figure 5b), the Pareto front is a curve in dual-dimensional space, indicating that there are considerable trade-off when fitting two data types simultaneously. It can be seen that as the error of one objective decreases, the error of the other often increases, i.e., it is difficult to find a solution that satisfies both variables. The Pareto front extends to its extremes (blue and green), minimizing each objective function and taking the compromise solution (red circle) as its geometric center. However, when the objective variables are \(\theta+T\), \(\theta+q\), \(T+h\), and \(h+q\) (Figure 5a,c,d,f), the Pareto front is asymmetric, with the optimal solutions almost aligned along a straight line, indicating that one dimension of the objective variables has already reached its optimal value, and further optimization in this dimension will no longer improve the overall efficiency. Based on the obtained Pareto solutions, we show the normalized parameter ranges of the Pareto solution samples obtained by the AMALGAM algorithm (Figure 6). Due to some  
Figure 5: The Pareto optimal solutions in the dual-dimensional Pareto trade-off space. The gray, black, green, blue, and red circles represent model solutions, Pareto optimal solutions, extreme solutions of objective functions \(F_{1}\) and \(F_{2}\), and compromise solution, respectively. The objective variables of the graphs are: (**a**) \(\theta+T\), (**b**) \(\theta+h\), (**c**) \(\theta+q\), (**d**) \(T+h\), (**e**) \(T+q\), and (**f**) \(h+q\). parameters in the Pareto solutions obtained from the dual-objective optimization have the same values, only the results of the objective variables \(\theta+T\) (Figure 6a), \(\theta+q\) (Figure 6b), \(T+h\) (Figure 6c), and \(h+q\) (Figure 6d) are shown. \(F_{1}\) and \(F_{2}\) represent the dual objective functions, with the subscripts \(\theta\), \(T\), \(h\), and \(q\) indicating which variable is used as the objective function. The hydraulic parameter \(\theta_{s}\) and \(l\) show a clear bifurcation. The identifiability of thermal parameters is poor, with \(b_{2}\) and \(b_{3}\) showing significant divergence. More specifically, with \(\theta+T\) (Figure 6a) as the objective variables, the empirical parameter \(b_{2}\) is much smaller than \(b_{3}\). However, when the objective variables are \(\theta+q\) (Figure 6b), \(b_{2}\) becomes much larger than \(b_{3}\).