## 3 Results
### Bayesian Normal Form DA
To avoid particle degradation or falling into local optimization, a priori knowledge, historical data, or model simulation results can be used to intelligently select or generate the initial particle distribution. In addition, PF involves multiple parameter adjustments, and the development of adaptive parameter adjustment algorithms, such as joint probabilistic data association-particle filter, can help to better meet these challenges [171; 198]. Although these optimization methods do improve the performance of PF, they also increase the complexity and computational requirements of the algorithm. However, the core advantage of PF in dealing with non-Gaussian processes is always the key to gain a foothold in practical applications [199]. Therefore, in practical applications, we should comprehensively weigh various factors according to the specific situation and select the most appropriate algorithm and optimization strategy. Hierarchical Bayesian Methods (HBMs), pioneered by [PERSON] et al. and [PERSON] et al. [200], are particularly adept at handling complex agricultural datasets with hierarchical or nested structures. Their essence lies in decomposing problems into distinct levels and employing conditional probability distributions to precisely describe the inherent relationships between these levels. This approach transforms the daunting task of solving a complex joint probability distribution into a series of more manageable posterior probability calculations, significantly streamlining the process [201]. HBMs typically consist of three components: a lower-level data model, a mid-level dynamic model, and a top-level parameter model [202]. Its core mechanism is shown in Formulas (16)-(21): In order to ensure the observation error, \(\epsilon\) is introduced to represent the error term [203]. \[y=\mathrm{H}x+\epsilon \tag{16}\]  
In the formula, \(y\) represents the observation vector, which contains a series of observations for the real state \(x\), which represents the actual state of the system at a certain time or space (temperature, wind speed), and H is the observation operator (or observation matrix). On the questions of how the observed data are generated from the real state, it can be a case of simple scalar multiplication (for example, a single observation point directly corresponds to a state variable), or it can be a more complex function (for example, RS observation is associated with the real state through the sensor response function). \(\epsilon\) is the observation error vector, which represents the deviation between the observed data and the real state. In general, we assume that \(\epsilon\) obeys multivariate normal distribution. \[\epsilon\,\sim\mathrm{N}(0,\mathrm{R}) \tag{17}\]  
Among them, R is the observation error covariance matrix, which reflects the correlation between observation errors and their respective variances. Then we give two expressions of the middle dynamic layer, including as many different application scenarios as possible. The first one represents the dynamic model as a set of partial differential equations [53]. That is:  
\[\omega_{k}^{i}\infty\frac{P(x_{k}|z_{1:k})}{q(x_{k}|z_{1:k})} \tag{11}\]0  
In the formula, \(T(x,y,t)\) is expressed as a temperature field, \(\mathrm{D}\) is a heat conduction coefficient, and \(f(x,y,t)\) represents an external heat source or boundary condition. This kind of equation is usually solved by finite difference and finite element method discretization. In the second way, we regard dynamic behavior as a random walk process (considering one-dimensional random walk model), and consider the trend and fluctuation of state variables, whose state evolution follows [204]:  
\[\omega_{k}^{i}\infty\frac{P(x_{k}|z_{1:k})}{q(x_{k}|z_{1:k})} \tag{11}\]1  
where \(x_{t}\) is the state variable of \(t\) at the time point, \(\mu\) is the drift rate (long-term trend), \(\sigma\) is the diffusivity (fluctuation amplitude), and \(W_{t}\) is the standard Brownian motion (Wiener process), indicating random disturbance.