## 1 Introduction
And compared with a number of multiscale numerical methods to calculate the effective constitutive relations, which are formed and solved numerically, the proposed upscaling approaches is based on analytical formulations, which are preferred in practical applications because they are efficient in computation and convenient to use. Simultaneously, within the Bayesian inversion, the using of upscaling method avoids solving the model at fine scales, thus improving the computational efficiency of the forward models. In addition, the local one-dimension (LOD) [32] is used to solve the multidimensional Richards equation to further improve the efficiency of the inversion process. Within the \(\text{DREAM}_{\text{(ZS)}}\) framework, the incorporation of the upscaling method can efficiently integrate the hydraulic parameters and state variables at different scales, but it still has to face the problem of model uncertainty due to the large number of parameters and thus the consequent problems of convergence and efficiency of inversions. This study employs KL expansion to reduce the parameter dimension and thus the uncertainty of the inversion simulation. Karhunen-Loeve (KL) expansion is a technique used in data analysis to represent a data set in a more efficient way by decomposing the data into a set of simpler, uncorrelated functions capturing the most significant variations within the data. In Bayesian inference, KL expansion has been an effective approach to represent the random field and is successfully applied in various fields (e.g., [33; 34; 35; 36; 37; 38]). Through KL expansion, thousands of spatially distributed hydraulic parameters can be reduced to dozens of quantities, which greatly reduces the number of predicted parameters and thus the uncertainty of the inverse simulation. Therefore, in the framework of Bayesian inference \(\text{DREAM}_{\text{(ZS)}}\), combined with the upscaling algorithm and KL expansion, we propose an efficient and effective inversion method to downscale the observations on coarse scales to obtain higher-resolution saturated hydraulic conductivity. The applicability and effectiveness of the proposed method are investigated through a series of numerical examples. Different flow boundary conditions, including constant head boundaries, variable flux boundaries, and mixed boundaries, are considered for two-dimensional soil profiles. We consider soil heterogeneity, characterized by statistical spatial variability include variance, correlation length, degree of anisotropy. We set different levels of mesh coarsening, i.e., inversion of saturated hydraulic conductivity field on fine scales based on different coarse-scale data, to explore the effects of different levels of mesh coarsening on the inverse results. And we investigate the effect of a smaller number of data on the inverse simulation by reducing the number of observation points in the data. Overall, the proposed methodology is validated in a comprehensive manner through a large number of numerical experiments. The paper is structured as follows. Section 2 describes our numerical inverse model, revisits the upscaling method and KL expansion, and introduces the \(\text{DREAM}_{(\text{ZS})}\) to solve the inverse problem. Section 3 describes the case setup for numerical simulation, including the selection of boundary conditions for water flow, characterization of soil heterogeneity, scaling between coarse and fine scales, and the setup of observation point density. Section 4 provides a thorough assessment of our method's performance under different model settings. Section 5 summarizes the key findings of this paper.