Dynamic crop models such as EPIC [1], SALUS [2], and STICS [3] are non-linear models that describe the growth and development of a crop interacting with environmental factors (soil and climate) and agricultural practices (crop species, tillage type, fertilizer amount…). They are developed to predict crop yield and quality or to optimize the farming practices in order to satisfy agricultural objectives, as the reduction of nitrogen lixiviation. More recently, crop models are used to simulate the effects of climate changes on the agricultural production. Nevertheless, the prediction errors of these models may be important due to uncertainties in the estimates of initial values of the states, in input data, in the parameters, and in the equations. The measurements needed to run the model are sometimes not numerous, whereas the field spatial variability and the climatic temporal fluctuations over the field may be high. The degree of accuracy is therefore difficult to estimate, apart from numerous repetitions of measurements. For these reasons, the problem of state/parameter estimation represents a key issue in such nonlinear and non-Gaussian crop models including a large number of parameters, while measurement noise exists in the data.
For example, it is useful to predict the evolution of variables, such as the biomass and the grain protein content during the crop lifecycle. State estimation techniques can be of a great value to solve that problem since they have the potential to estimate simultaneously the variables and several parameters. As an example, involved parameters are the radiation use efficiency, the maximal value of the ratio of intercepted to incident radiation, the coefficient of extinction of radiation, the maximal value of LAI.
Several estimation techniques, such as the extended Kalman filter [4], unscented Kalman filter [5] and more recently the Particle filtering [6] method have been developed and utilized in many applications. The EKF and UKF algorithms, which are presented in details in [7], do not always provide a satisfactory performance, especially for highly nonlinear processes, because linearizing the process model does not necessarily provide good estimates of the mean of the state vector and the covariance matrix of the estimation error which are used in state estimation.
These issues are addressed by the particle filter (PF). PF methods approximate the posterior probability distribution by a set of weighted samples, called particles. Since real world problems usually involve high dimensional random variables with complex uncertainty, the nonparametric and sample-based estimation of uncertainty has thus become quite popular to capture and represent the complex a posterior distribution in nonlinear and non-Gaussian models [8]. PF methods offer a number of significant advantages over other conventional methods. However, since they use the prior distribution as the importance distribution [6], the latest data observation is not considered and not taken into account when evaluating the weights of the particles. While the importance sampling distribution has computational advantages, it can cause filtering divergence. In cases where the likelihood distribution is too narrow compared to the prior distribution, few particles will have significant weights. Hence, a better proposal distribution that takes the latest observation data into account is needed. In other words, new adaptive methods that incorporate better feedback and smoothing in the selection or deletion of particles and their weights need to be investigated.
The objectives of this paper are threefold. The first objective is to develop a new Particle filtering (IPF) for improving nonlinear and non-Gaussian crop model predictions. In case of standard PF, the latest observation is not considered for the evaluation of the weights of the particles as the importance function is taken to be equal to the prior density function. This choice of importance sampling function simplifies the computation but can cause filtering divergence. In cases where the likelihood function is too narrow compared to the prior distribution, very few particles will have significant weights. Hence, a better proposal distribution that takes the latest observation into account is needed. The main novelty of this task is to develop new Bayesian algorithm for nonlinear and non-Gaussian state/parameter estimation with better proposal distribution based on minimizing Kullback-Leibler divergence.
The second objective is to investigate the effects of practical challenges on the performances of state estimation algorithms PF and IPF. Such practical challenges include (i) the effect of measurement noise on the estimation performances and (ii) the number of states and parameters to be estimated.
The third objective is to apply the proposed state estimation techniques PF and IPF for predicting biomass and grain protein content. In a first step, we present an application of the new IPF to a simple dynamic crop model with the aim to predict a single state variable, namely winter wheat biomass. In a second step, we apply the new IPF for updating predictions of complex nonlinear crop models in order to predict protein grain content.
