## 3 Results
### Bayesian Normal Form DA  
Particle filtering (PF), as a recursive Bayesian filtering algorithm, has gained significant attention due to its superior capabilities in handling non-Gaussian and nonlinear systems [30; 60; 170]. PF generates a set of particles through random sampling, representing the probability distribution of the system state, and approximates the posterior probability of state variables by weighting and summing these particles [171]. Its core mechanism is shown in Formulas (10)-(15):  
Hypothesis from the posterior probability distribution of the state variables \(P(x_{k}|z:k)\) to extract the N particles \(\left\{x_{K}^{i},\omega_{k}^{i}\right\}_{i=1}^{N}\), the state of the posterior probability density distribution by the following formula approximation obtained [5]:  
\[P(x_{k}|z_{1:k})\approx\omega_{k}^{i}{\sum_{i=1}^{N}}\delta\left(x_{k}-x_{k}^{ i}\right) \tag{10}\]  
where, \(\delta\) represents the Dirac function, k represents the time, \(x_{k}^{i}\) represents the particle's state value, \(x_{k}^{i}\) represents the particle's weight, \(z_{k}\) represents the observed value, and \(P(x_{k}|z_{1:k})\) represents the posterior probability distribution. Generally speaking, the posterior probability density of the state variable is unknown, so it is difficult to obtain the posterior distribution density function by direct sampling. To solve this problem, this study adopts Sequential Importance Sampling (SIS), that is, sampling from an importance distribution function \(q(x_{k}|z_{1:k})\) whose probability distribution is the same as \(P(x_{k}|z_{1:k})\), whose probability distribution is known and easily sampled. The importance weight of the particle is [172]:  
\[\omega_{k}^{i}\infty\frac{P(x_{k}|z_{1:k})}{q(x_{k}|z_{1:k})} \tag{11}\]The method mainly uses recursive ideas to calculate the weight of particle importance [173]:  
\[\omega_{k}^{i}=\omega_{k-1}^{i}\frac{P\big{(}z_{k}\big{|}x_{k}^{i}\big{)}P\big{(} x_{k}^{i}\big{|}x_{k-1}^{i}\big{)}}{q\Big{(}x_{k}^{i}\big{|}x_{k-1}^{i},z_{k} \Big{)}} \tag{12}\]  
Which \(q\Big{(}x_{k}^{i}\big{|}x_{k-1}^{i},z_{k}\Big{)}\) is the importance sampling function, \(P\big{(}z_{k}\big{|}x_{k}^{i}\big{)}\) is the likelihood function, and \(P\Big{(}x_{k}^{i}\big{|}x_{k-1}^{i}\Big{)}\) is the prior probability distribution function. In order to calculate simply, we usually choose to render \(q\Big{(}x_{k}^{i}\big{|}x_{k-1}^{i},z_{k}\Big{)}=P\Big{(}x_{k}^{i}\Big{|}x_{k- 1}^{i}\Big{)}\), there are [174]:  
\[\omega_{k}^{i}=\omega_{k-1}^{i}P\Big{(}z_{k}\big{|}x_{k}^{i}\Big{)} \tag{13}\]  
Therefore, the updated particle weights can be as follows [5]:  
\[\omega_{k}^{i}=\frac{\omega_{k}^{i}}{\sum_{i=1}^{N}\omega_{k}^{i}} \tag{14}\]  
Through this processing, the estimation of the final state variable is the weighted average of all the state values of the particles [172; 173; 174]. \[\hat{x}_{k}=\sum_{i=1}^{N}\omega_{k}^{i}x_{k}^{i} \tag{15}\]  
In agricultural research, PF has been widely applied in various CGMs (e.g., WOFOST, CERES-Wheat, APSIM, etc.) and the assimilation of key parameters such as LAI, significantly improving the accuracy of simulating crop yield, biomass, SM, and hydraulic parameters [175; 176; 177; 178; 179; 180; 181], as depicted in Figure 9, which shows a framework for assimilation based on PF. Meanwhile, research efforts have focused on the improvement and optimization of PF technology, including residual resampling, residual remapping, sensitivity analysis, PF-MCMC methods, and utilizing rigorous evaluation metrics such as RMSE and normalized RMSE [179; 182; 183; 184]. It is important to note that several assimilation methods mentioned above either rely on analytical equations, cost function minimization, or solving covari  
Figure 9: The PF assimilation method based on improved particle degradation consists of four steps: prediction, filtering, resampling, and merging [185]. depend on a series of assumptions, such as the tangent linearity assumption, initial values following normal distributions, and Gaussian error structures [186].