## 1 Introduction
The resulting simulated crop growth minimizes the error between simulated leaf areas and values of leaf areas obtained from remote sensing. An advantage of this procedure is the capability to use infrequent observations to calibrate the model. The GRAMI model was further developed and was applied to simulate cotton growth and lint yield under limited irrigation conditions [16; 17], as well as geospatial projections of rice productivity [18]. The crop modelling technique formulated in GRAMI was applied to assess and monitor crop conditions and yields at regional scales, using imagery from operational satellites [19; 20; 21; 22; 23]. This within-season calibration methodology was also used to estimate evaporation and biomass production [24; 25]. The remote sensing data either from airborne or satellite platforms is spatiotemporal data consisting of many pixels but a relatively small number of time points. For example, the reflectance can be obtained from the RapidEye satellite with a ground resolution of 5 m at \(800\times 800\) pixels five times during a crop-growing season in a monsoonal climate region. A hybrid approach utilizing both remote sensing and crop modelling can fill the temporal gap. Parameters that characterize a crop model are not necessarily constant. However, pixel-by-pixel estimation is not a good choice for studying the spatial variation in the parameters for the following two reasons. First, the model can be unidentifiable when the sample size is smaller than the number of unknown parameters. Second, as illustrated in [PERSON] and [PERSON] [26] and [PERSON] and [PERSON] [27; 28], the pixel-by-pixel estimate is not optimal with respect to the mean-squared loss. An empirical Bayesian method is introduced to overcome the above-mentioned difficulties. Bayesian methods are characterized by the following concepts and procedures. Random variables are used to model all sources of uncertainty in statistical models, e.g., uncertainty due to lack of information. Determination of the prior probability distribution is required to take into account the prior information. Use of [PERSON] formula is sequentially performed, calculating the posterior distribution when more data become available. In Bayesian statistics, the probability can range from 0 to 1. In this paper, a random-effect model is used to describe the location-dependent unknown parameters, and the empirical Bayesian approach is adopted. A random effect is a factor whose levels are considered a random sample from some population, technically considered to have a normal distribution. Moreover, the relationship between reflectance and LAI is incorporated into the statistical model. These models allow us to monitor the crop growth and estimate the crop model using the remote sensing data from various platforms and optical sensors with different ground resolutions. We report a statistical hybrid approach of crop modelling and remote sensing as well as a method to project spatiotemporal crop growth information using paddy rice (_Oryza sativa_). It is one of the main food crops for more than half of the world's population, including about 557 million people in Asia.