## 5 Conclusions  
Coupled modeling of liquid water, water vapor, and heat transport are essential for modeling unsaturated hydrological processes, especially in semi-arid and arid regions. Precise estimation of the hydraulic and thermal properties is crucial for reliable model simulations and predictions. This study employ multi-objective algorithms AMALGAM to evaluate the worth of different observation types in simultaneous estimations of the soil hydraulic and thermal properties within coupled hydrothermal processes quantified by HYDRUS-1D model. We select several possible commonly used types of observations in soil dynamics including soil water content, temperature, matrix potential, and heat flux as the objectives for parameter estimations. Based on these four variables, the inverse simulations based on single, dual, triple, and quadruple objectives are generated to investigate the data worth of these four observations and that of different combinations of any of them.  
The simulation results show the contribution of different observations to the parameter estimates. Water movement, as the most fundamental part of the coupled hydrothermal process of water vapor and heat, plays a key role in the inversion of hydrothermal parameters, so that the optimisation of hydrothermal parameters is better when water content or matrix potential is included as an objective in either single-objective or multi-objective optimization. Thus, we see that the worst performance in dual-objective optimization is when temperature and heat flux are jointly used as objectives, when there is no water flow dynamics data involved. However, the addition of temperature to the objective is  
Figure 11: Comparison of the variation in soil heat flux, with \(\mathbf{a}\) and \(\mathbf{b}\) representing the heat flux at depths of 5 cm and 10 cm, respectively. The black circles indicate observed values, while the blue and red lines correspond to the objective variables for \(T\), \(\theta+T\), respectively.  
necessary if water vapor transport and heat transport are to be fully portrayed. Therefore, considering the cost of measurements as well as the validity of model inversion issues, we recommend constructing coupled water-vapor-heat models in practice and monitoring both soil water content and temperature data for model parameter inversion and model optimization, especially in arid and semi-arid regions. Under this objective function, we observe that the shallow water vapor flux fluctuates significantly, while the total liquid water flux increases with the amount of precipitation, and the total gaseous water flux rises with the increase in evaporation, which also better simulates the changes in heat flux. Although the ultimate goal is to optimize the hydrothermal parameters through multi-objective optimization, this study faces challenges with low computational efficiency and does not consider soil heterogeneity. In the future, multi-objective optimization should be extended to multi-layer soils to analyze the spatial heterogeneity of hydrothermal parameters. Additionally, other inversion methods, such as machine learning, can also be tried to improve computational efficiency.  
Methodology, J.Z. and N.L.; Writing--original draft, J.Z.; Writing--review and editing, N.L.; Supervision, N.L.; Funding acquisition, N.L. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the National Key R&D Program of China (Grant No. 2023 YFC3709900), the National Natural Science Foundation of China (Grant No. 42272280), and the Fundamental Research Funds for the Central Universities (Grant No. 20245 HEEAR001). We are greatly grateful to the editor, the associate editor, and anonymous reviewers for their comments.  
The data are available upon reasonable request to the authors.  
The authors declare no conflicts of interest.