## Theories and Methods
### Bayesian Model Updating  
The principle of Bayesian model updating is the [PERSON] theorem. The pioneering work in the late 1990s attracted the attention of the engineering research community and gave detailed and up-to-date instructions [32; 33]. Then, the obstacle to computing in practical engineering applications was the high level of computer resources required. This problem still exists, but there has been great progress in solving it.  
The Bayesian model's updating process is based on [PERSON] theorem. In our case, likelihood is a function of the parameters in the statistical model, which represents the likelihood of model parameters.  
Figure 1: **(a)** The influence of parameters \(a\) on the soil–water characteristic curve (SWCC), and the \(n\) is 5 in this figure; **(b)** The influences of parameters \(n\) on the SWCC equation, and the \(a\) is 2 in this figure.  
The prior distribution is based on existing research assumptions. The posterior distribution can be calculated by combining the prior distribution and the likelihood function. Its general formula is given by:  
\[P(\theta|\mathbf{X}_{\text{obs}})=\frac{P(\mathbf{X}_{\text{obs}}|\theta)P( \theta)}{P(\mathbf{X}_{\text{obs}})} \tag{2}\]  
where \(P(\theta)\) is the prior distribution representing the initial knowledge about the parameters \(\theta\); \(P(\theta|\mathbf{X}_{\text{obs}})\) is the posterior distribution representing the updated knowledge based on the observation data; \(P(\mathbf{X}_{\text{obs}})\) is the normalization factor; \(P(\mathbf{X}_{\text{obs}}|\theta)\) is the likelihood function of \(\mathbf{X}_{\text{obs}}\) for an instance of the parameters \(\theta\).  
If it is possible to provide an update a prior distribution according the available knowledge, then the MAP distribution can be obtained. The Bayesian updating is a challenging problem, since it is very difficult to directly calculate the posterior probability density over the entire parameter space. However, the well-known M-H algorithm is an effective update tool [15]. The algorithm is essentially an iterative method, sampling from a series of intermediate partial differential equations that gradually converge to the MAP distribution. The \(j\)th intermediate probability density function (PDF) is expressed as a function of the likelihood probability function \(P_{L}\) as:  
\[P_{j}=P_{L}(\mathbf{X}_{\text{obs}}|\theta)^{\beta_{j}}P(\theta) \tag{3}\]  
where the index of likelihood \(\beta_{j}\) is the so-called reduction factor. Its value gradually increases from \(\beta_{o}\)= 0 in the first iteration until it reaches \(\beta_{j}\) = 1 in the last step [24]. \(\beta_{j}\) is adaptively calculated based on the sample from the previous step.  
The likelihood is a key component of the Bayesian update framework because it quantifies how relevant the model is to a given parameter instance by representing the likelihood of observation. Under the assumption of independence between observations, the probability \(P_{L}\) is theoretically defined as:  
\[P_{L}(\mathbf{X}_{\text{obs}}|\theta)=\prod_{k=1}^{N_{\text{obs}}}P(\mathbf{X }_{k}|\theta) \tag{4}\]  
where \(P(\mathbf{X}_{k}|\theta)\) is the probability density value at \(\mathbf{X}_{k}\) for a given set of parameters \(\theta\); \(\mathbf{X}_{k}\) is the \(k\)th \(\mathbf{X}_{\text{obs}}\). \(\theta\) is a set of parameters--in this analysis, the parameters \(a\) and \(n\). In this process, the PDF should be estimated separately for each instance. However, in a complete Bayesian updating process, the required computation is costly, due to the large number of evaluations that are necessary to obtain to define a statistical sample.