## 3 Results
### Bayesian Normal Form DA
The top-level parameter model is mainly used to describe the parameter uncertainty (the error term of the data model and/or the coefficient of the dynamic model). It usually satisfies a certain prior distribution, based on our preliminary understanding or empirical estimation of them. For example, the coefficient \(\theta\) of the dynamic model satisfies [203; 204]:  
\[\omega_{k}^{i}\infty\frac{P(x_{k}|z_{1:k})}{q(x_{k}|z_{1:k})} \tag{11}\]2  
Note: \(p(\theta)\) is not only Poisson distribution, but may be uniform distribution, normal distribution, or Gamma distribution, which depends on prior knowledge. For the observation error covariance matrix \(\mathrm{R}\) in the data model, if there is a correlation between its elements, the inverse Wishart distribution may be used as its a priori. \[\omega_{k}^{i}\infty\frac{P(x_{k}|z_{1:k})}{q(x_{k}|z_{1:k})} \tag{11}\]3  
where \(\
u\) represents the degree of freedom parameter and \(S\) represents the status matrix. In practical DA procedures, initial prior distributions of parameters, combined with observational data, are updated through a Bayesian theorem to yield posterior distributions, thereby obtaining optimal parameter estimates or their probability distributions [205]. Thanks to its flexibility, efficiency, strong explanatory power, and robust handling of uncertainties, HBM has found wide application in fields including environmental science, oceanography, meteorology, and others [206; 207; 208; 209]. However, its adoption in precision agriculture remains nascent, with numerous challenges still to overcome. For instance, dealing with multi-level uncertainties and computing high-dimensional posterior distributions within HBMs often employs MCMC sampling techniques. [PERSON] et al., within an HBM framework for assimilating IMTA models, successfully estimated the posterior distributions of state variables and parameters along with their uncertainties using Particle MCMC [211]. Nonetheless, when models have numerous parameters and dense observation data, this can significantly escalate computational costs during assimilation [212; 213]. Addressing this, [PERSON] et al. employed Hamiltonian Monte Carlo (HMC) sampling to effectively mitigate such high computational expenses [214]. Moreover, gradient-based inference methods like variational inference, approximate Bayesian computation, and rapid assimilation algorithms such as PF are also worth exploring and applying [215; 216]. In practice, while simplifying assumptions are often made based on linear problems, the presence of nonlinearity, non-stationarity, and multi-scale effects can, in reality, lead to intricate prior distributions and covariance structures in parameter models, complicating model construction and solution [217; 218]. Techniques such as mode averaging and Principal Component Analysis (PCA) for dimensionality reduction are commonly used to decrease the number of parameters and alleviate computational loads [219]. With advancements in computational power and deeper understanding of physical processes,emerging methods like Bayesian analysis based on DL and Approaches combining Visual Salience with Hierarchical Bayesian hold promise to further advance HBM assimilation techniques in precision agriculture, ushering in new application opportunities [220; 221].