## 1 Introduction  
Regional crop yield information is critical for food security and sustainable agriculture [1; 2; 3]. However, obtaining such data in a timely and accurate manner is challenging if global climate change is considered. Recently, remote sensing data assimilations based on crop growth simulations [4; 5] have furnished a potential approach to improving regional crop yield monitoring and forecasting. By making better use of crop-growth models such as the World Food Studies (WOFOST) [6], Erosion Productivity Impact Calculator (EPIC) [7], and Decision Support System for Agro-technology Transfer (DSSAT) [8], crop growth processes can be effectively simulated under different environmental and management conditions while incorporating various limiting factors (e.g., soil, weather, water, and nitrogen) in a dynamic manner [9]. However, the use of spatial observations from remotely sensed data is an ideal option for reducing regional simulation uncertainties [10]. Accordingly, several remote sensing data assimilation strategies based on crop-growth models have been proposed to accurately estimate regional crop yields. Most data assimilation strategies focus on reducing the discrepancy between observations and simulations by adjusting the uncertain model parameters, initial conditions [11; 12], or model state variables [13; 14; 15]. Early data assimilation strategies based on optimum estimation algorithms failed to consider errors in observations and the model itself. In addition, the iterative process of minimization between the modeled and observed values may require excessive computing time. To address these shortcomings, several researchers have proposed advanced strategies based on sequential Bayesian filters, _i.e._, the ensemble Kalman filter (EnKF) [16], particle filter (PF) [17], or variational algorithm [18]. More accurate performance of these strategies was also obtained for operational numerical weather predictions [19; 20; 21; 22], ocean prediction systems [23; 24; 25; 26; 27; 28], and land surface processing simulations [29; 30; 31; 32; 33; 34; 35]. Although the abovementioned strategies have been successfully applied at regional scales, many well-known drawbacks limit the efficiency and feasibility of data assimilation with crop-growth models. Sequential data assimilation becomes prohibitively expensive in terms of computational cost for high-dimensional state spaces because of recursive calculations of the posterior distributions of simulated variables [17; 36]. The EnKF, in particular, relies on the assumption that the posterior density at every time step is a Gaussian distribution parameterized by a mean and a covariance, which results in an obvious deficiency when addressing complicated estimation issues in a realistic, nonlinear, and non-Gaussian dynamic system [37]. In data assimilation with crop-growth models, the assumptions described above are unlikely to hold true. The PF-based strategy appears to be a better choice for crop-growth models in nonlinear and non-Gaussian systems. However, a particle filter is a suboptimal algorithm whose accuracy and stability depend on the choice of importance density or resampling method [17]. For traditional four-dimensional variational (4 DVar) data assimilation, the most attractive feature is the reduced computing cost that can be attributed to the ability to simultaneously assimilate the observational data at multiple times in an assimilation window. However, 4 DVar still facesnumerous challenges in coding, maintaining, and updating the adjoint model of the forecast model, and it requires linearization of the forecast model [38]. Accordingly, it has not been used frequently in crop-growth models. To address these problems and maximally exploit the strengths of both sequential and variational data assimilation while simultaneously offsetting their respective weaknesses, a 4 DVar method is proposed on the basis of the proper orthogonal decomposition (POD) and ensemble forecasting techniques [39].