## 3 Methods
### Ensemble-Based Four-Dimensional Variational Data Assimilation  
The POD4 DVar was proposed by [PERSON] _et al._[54] and is based on incorporating the Monte Carlo method and the proper orthogonal decomposition (POD) technique into 4 DVar. The basic idea of this method is to start with a four-dimensional ensemble obtained from the forecast ensembles at all times in an assimilation time window produced using the Monte Carlo method. The POD technique is then applied to the four-dimensional forecast ensemble so that the orthogonal base vectors can capture the spatial structure of the state and reflect its temporal evolution. After the model status is expressed by a truncated expansion of the base vectors obtained using the POD technique, the control variables in the cost function appear explicit so that the adjoint or tangent linear model is no longer required [54]. In principle, the traditional 4 DVAR analysis \(x_{a}\) is obtained through the minimization of a cost function \(J\), which measures the misfit between the model trajectory \(H(x(k))\) and the observation \(y(k)\) at a series of time points \(t_{k}\), \(k=1,\,2,\,...,\,S\),  
\[J(x_{0})=(x_{0}-x_{b})^{T}B^{-1}(x_{0}-x_{b})+\sum_{k=1}^{S}\left(y(k)-H\big{(} x(k)\big{)}\right)^{T}R(k)^{-1}(y(k)-H\big{(}x(k)\big{)}) \tag{1}\]  
with the forecast model \(M\) imposed as a strong constraint, defined by  
\[x(k)=M(x(k-1)) \tag{2}\]  
where the superscript \(T\) stands for a transpose, \(x_{b}\) is the background value, the index \(k\) denotes the observational time, \(H(*)\) is the observational model, and matrices \(B\) and \(R\) are the background and observational error covariances, respectively. The control variable is the initial condition \(x_{0}\) (at the beginning of the assimilation time window) of the model. In the CERES-Wheat model, the plant area growth function (\(PLA\)) and plant area senescence function (\(SENLA\)) are considered as state variables whose integrating functions forward with time are considered to be forecast models. The LAI function was used as an observational model in the data assimilation process. \[y(k)=LAI(k)=H\left(x^{i}(k),\theta\right),i=1,2 \tag{3}\]  
\[x^{1}(k)=PLA(k)\ =\ M\left(\ PLA(k),PLAGT(k-1)\ \right) \tag{4}\]  
\[x^{2}(k)=SENLA\ (k)\ =\ M\left(\ SENLA(k),PLAS(k-1)\ \right) \tag{5}\]  
where \(\theta\) is the initial condition of the crop model, index \(i\) denotes the number of state variables, \(PLAGT\) is the plant growth area, and \(PLAS\) is the reduced leaf area resulting from various stress factors. The POD4 DVar strategy is a non-recursive process in which all observations at multiple time nodes are used to solve all \"analyses\" in an assimilation process. In contrast, sequential assimilation uses the observations at one time node to solve only one \"analysis\" in a recursive assimilation process. To minimize the cost function, the details of POD4 DVar with the CERES-Wheat crop model are as follows (Figure 1):  
Figure 1: Flowchart of POD4 DVar with the CERES-Wheat crop model. (1) Assuming that there are S time steps in the assimilation time window (0, _T_), generate N random perturbation fields using the Monte Carlo method, add each perturbation field to the initial background field at time point \(t_{0}\), and integrate the model to produce a perturbed four-dimensional field \(x_{n}^{i}\) (\(n=1,\ \ldots,\) N; \(i\) = 1, 2) over the entire assimilation time window (S time steps) or at the observational time steps (K).