## 1 Introduction
However, in contrast with EnKF, PF does not rely on the Gaussian assumption of distributions, can perfectly accommodate the propagation of non-Gaussian distributions through nonlinear models [39; 44; 45; 46; 47; 48] and is thus adapted to the potentially highly nonlinear crop models [49; 50]. Accordingly, the PF algorithm appears to be a better choice for complicated crop growth models. On the other hand, the advantages of the traditional four-dimensional variational (4 DVar) methods are that they can provide the temporal smoothness constraint and have the ability to simultaneously assimilate the observational data at multiple times in an assimilation window so that the computational cost is lower than EnKF. However, traditional 4 DVar methods also have some challenges in maintaining and updating the adjoint models of the forecast models, and they require linearization of the forecast models [9; 41]. To maximally use the advantages of the EnKF and variational data assimilations, while simultaneously offsetting their respective weaknesses, an ensemble 4 DVar method is proposed on the basis of the proper orthogonal decomposition (POD) [50]. Although a Gaussian distribution assumption of the posterior density of the simulated variable is also given, this method outperforms both the 4 DVar and EnKF methods under various nonlinear and non-Gaussian model scenarios with lower computational costs than EnKF and does not need the adjoint or tangent linear model, and thus, the higher accuracy and computational efficiency of the crop yield estimations would be obtained in crop model data assimilation [9; 51]. Accordingly, the assimilation of remotely-sensed information into crop growth models with the POD-based ensemble 4 DVar (POD4 DVar) has also been presented as a better approach to improve the estimation of regional crop yields. Thus, both the PF- and POD4 DVar-based strategies were chosen to assimilate the crop model for estimating the regional crop yields. When a crop model is used to estimate crop yields early in the growing season, climate and model uncertainties are the two main sources of uncertainties [52; 53]. Climate-related uncertainties are greatest early in the growing season, but can be reduced when meteorological data become available [54]. Model-related uncertainties are subject to the model structure, the model assumed and data errors; hence, they are imperfect in simulating the true growth process. Without altering the crop models' structure, data assimilation can be used to improve the model performance by periodically updating state variables within the growing season with remote sensing observations. Remote sensing of vegetation (e.g., leaf area index (LAI)), as the major indicator for crop models, is potentially useful for data assimilation because of its obvious influence on crop growth and, hence, on crop yields [18; 53]. To obtain better estimates of regional crop information, a series of remotely-sensed LAIs during the crop growing season must be accurately estimated [6; 22; 54]. However, current remote sensing LAI products [55; 56; 57] have some issues in the practical applications for crop model data assimilation: (1) the observations during the crop growing season are usually limited by cloud coverage, rain and long revisit periods; (2) the high frequencies of observations would need a high computational cost, while the low frequencies at critical crop growth stages may also provide relatively accurate results; (3) the commonly-used LAI data were from single sensors, while data combined from multiple sensors' LAI results were rarely used for assimilation; and (4) the spatial resolutions of the remotely-sensed LAIs are commonly fairly coarse with mixed pixels [6], which may result in various uncertainties in the spatial distributions of the crop parameters, soil properties, management parameters and, finally, result in poor performance of the assimilation model.