Section: Scientific Foundations

Parameter Identification of the GreenLab Model from Experimental Data

The plant architecture is a target for the mathematical model, and it is the visual result of the growth process. The hidden parameters of source and sink functions must be estimated in order to fit the weights and sizes of all the organs produced by the plant development at each stage of growth. Theoretically speaking, this inverse method should be able to assess also the effect of the environment (climate and density), biomass production and biomass partitioning to each organ during the growth from the architectural data. However, the identifiability issue has not been studied yet. Only a lot of numerical tests on a wide variety of real plants have helped us define a proper experimental framework for identification, see for example [7] .

The Generalized Least Square Method was used for parameter estimation of the model. The application of this method to GreenLab was described by Zhan et al. [74] and Guo et al. [7] . Advantages of this method are that it provides rapid convergence and the standard error linked to the parameter values thus indicating the accuracy of the solution. Fitting can be done on a single architecture (single fitting), or on several stages of growth to follow the trajectory of the dynamical process (multi-fitting). This second one is more accurate, since it helps retrieving the dynamics of sink variation more easily. In both cases all the data are adjusted at the same time by the same parameter set.

This methodology for parametric identification was implemented in Digiplant software, and used on several plants: Maize [13] , Arabidopsis [53] , Pine saplings [56] , Cucumber [68] , Sugar-Beet [63] , Cotton [75] , Beech[10] , Chrysanthemum [61] , Grapevine [69] ...

We give beneath an example on the sugar beet. It corresponds to an industrial partnership with the French institute for Sugar Beet Research (ITB). Concerning the experimental data from which the model parameters are estimated, biomass measurements were carried out at seven different stages. At each date, seven individuals plants were selected and the dry mass of every individual organ (blades, petioles and root storage) was measured. The average of the seven plants was constructed. The numerical values of the parameters as well as their variances are given in [63] and are not recalled here. As an illustration, Figure 7 shows the fitting curves.

Figure 7. Fitting results: Average plants at organ level: experimental data (in red) and simulated data (lines) (a) Blade dry masses and (b) petiole dry masses according to phytomer ranks, at 7 different growth stages; (c) root dry mass as a function of the growth cycle (with experimental measures at the same 7 stages)

If the goodness of fit is excellent, the statistical analysis of the estimation results is not properly done yet. The variances of the estimates are given under the assumption that the error model chosen is the good one. But the hypothesis of a diagonal noise covariance matrix which is used in the generalized least square method, is quite gross. It corresponds to using only the deterministic part of the dynamical model, which is really restrictive since in dynamic models, errors propagate. Current studies aim at improving estimation by modelling the dynamics of the covariance matrix of the model noise.