## 3 Results
### Model Parameter Optimization DA  
In contrast, model parameter optimization methods (also known as continuous assimilation methods) are more suitable for complex and refined agricultural scenarios. At the heart of this approach lies the iterative adjustment of model initial parameters within a given reanalysis time window T, where all available observation data and model state values are used as a foundation to strive for the best possible fit between the model trajectory and observed data [109; 122; 123; 124]. Particularly noteworthy among these methods is the Four-Dimensional Variational (4D-Var) algorithm, which builds upon the three-dimensional version by incorporating temporally evolving background information. It transforms the assimilation problem into a search for the optimal solution of the dynamic model, effectively integrating all observational results over a specified time period [125; 126; 127]. The core mechanism of 4D-Var is shown in Formulas (1)-(3) [128; 129]:  
\[J(X_{0})=\frac{1}{2}\Big{[}X_{0}-X_{0}^{b}\Big{]}^{T}B^{-1}\Big{[}X_{0}-X_{0}^{ b}\Big{]}+\frac{1}{2}{\sum_{i=1}^{n}}\left[H(X_{i})-Y_{i}\right]^{T}R_{i}^{-1} [H(X_{i})-Y_{i}] \tag{1}\]  
where \(J(X)\) is the cost function of 4D-Var, \(X\) and \(Y\) represent the model state variable and observation, respectively, \(X_{0}\) represents the initial value of the state variable, \(X_{0}^{b}\) is the background field of the state variable, \(X_{0}^{b}\) is the prior estimate of \(X_{0}\), \(B\) and \(R\) represent the covariance matrix of the background field and the observation field, respectively, and \(H\) is the observation operator. \(X_{i}\) is the model prediction value at time \(i\) and \(Y_{i}\) is the observation vector at time \(i\). \(X_{i}\) is obtained from the integral prediction equation \(M\):  
\[X_{i}=M_{i-1}(X_{i-1}) \tag{2}\]  
In 4D-Var, the minimum objective function \(J\) is solved to obtain the optimal \(X_{0}\). First, the gradient information of the objective function is required:  
\[\
abla J(X_{0})=B^{-1}\Big{(}X_{0}-X_{0}^{b}\Big{)}+{\sum_{i=0}^{n}}M_{0\to i }^{T}H^{T}O^{-1}[Y_{i}-(H_{i})] \tag{3}\]  
where the adjoint of the second term on the right solves the gradient through the inverse integral of the adjoint equation, and the following assumptions are made in the calculation process:  
* \(M_{0\to i}^{T}\) represents the mode prediction operator from 0 to \(i\), and has \(X_{i}=M_{0\to i}^{T}X_{0}\), and the forward prediction mode can be expressed as the product of the intermediate prediction mode, that is, \(X_{i}=M_{i-1\to i}\cdots M_{0\to 1}X_{0}\). * Under the tangent linearity hypothesis, the pattern operator and the observation operator are linearized, that is, \(X_{i}-X_{0}\approx M_{0\to i}^{\prime}\Big{(}X_{i}^{b}-X_{0}^{b}\Big{)} \approx H^{\prime}\Big{(}X_{i}^{b}-X_{0}^{b}\Big{)}\), where \(M_{0\to i}^{\prime}\) and \(H^{\prime}\) are the tangent operators of the pattern operator and the observation operator, respectively, and the minimum value of \(J\) is solved by gradient descent or conjugate gradient method. The obtained \(X_{0}\) is the optimal solution of the initial state. The current 4D-Var research either focuses on the selection of assimilation parameter variables [44; 62; 130] or on the optimization of cost function [108; 131]. As shown in Figure 7, the assimilation framework based on the 4D-Var model is shown in detail. Based on this model, many scholars have achieved satisfactory results. However, the latest research shows that there are challenges in the application of precision agriculture [132; 133; 134; 135]. First, based on the statistical normal distribution hypothesis and tangent linear hypothesis, 4D-Var simplifies the complexity of the model, but when dealing with strong nonlinear or significant model errors, linearization approximation may lead to gradient information distortion and increase the uncertainty in the process of assimilation [135].