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Section: New Results

Model Design

Sensitivity Analysis of Complex Biophysical Models

Background

Sensitivity analysis (SA) is a fundamental tool in the building, use and understanding of mathematical models [44] . Sampling-based approaches to uncertainty and sensitivity analysis are both effective and widely used [38] . For this purpose, Sobol's method is a key one [47] . Since it is based on variance decomposition, the different types of sensitivity indices that it estimates can fulfill different objectives of sensitivity analysis: factor priorization, factor fixing, variance cutting or factor mapping [37] . It is a very informative method but potentially computationally expensive [38] . Besides the first-order effects, Sobol's method also aims at determining the levels of interaction between parameters [48] . In [46] , the authors also devised a strategy for sensitivity analysis that could work for correlated input factors, based on the first-order and total-order index from variance decomposition.

Algorithm numerical implementation

Computational methods to evaluate Sobol indices sensitivity rely on Monte-Carlo sampling and re-sampling [47] , [40] . For $k$ dimensional factor of model uncertainty, the $k$ first-order effects and the 'k' total-order effects are rather expensive to estimate, needing a number of model evaluations strictly depending upon $k$ [43] . Therefore, it is crucial to not only devise efficient computing techniques, in order to make best use of model evaluations [45] , but also to have a good control of the estimation accuracy with respect to the number of samples.

With the objective of an efficient computational method for sensitivity analysis of functional-structural tree growth models, we proposed a new estimator based on Homma-Saltelli method to compute Sobol indices, which improves slightly their use of model evaluations thanks to a more balanced resampling strategy. This new estimator can be considered as an effort to improve the efficiency of SA methods for models.

We also derived a theoretical analysis of the error estimation for the sensitivity analysis for the studied class of Sobol's estimators (it can be applied to all the three Sobol's estimators mentioned in this paper) with respect to the sampling size and the number of model evaluations. An analytical test function is used to test the error estimation, and we obtained that the error estimation in this paper gives out a better 'upper bound' than the previous works related to this problem. This error estimation directly relates to the variance of the result, so it can also be used for checking the confidence interval, which is usually difficult to attain.

We derived a theoretical analysis of the error estimation for the sensitivity analysis for the studied class of Sobol's estimators (it can be applied to all the three Sobol's estimators mentioned in this paper) with respect to the sampling size and the number of model evaluations. An analytical test function is used to test the error estimation, and we obtained that the error estimation in this paper gives out a better 'upper bound' than the previous works related to this problem. This error estimation directly relates to the variance of the result, so it can also be used for checking the confidence interval, which is usually difficult to attain.

The result has been accepted to be published in J of Rel. Eng. and Sys. Safety.

Based on the published result for first order index of Sobol's, we also extended this method to the second order index computing which is very important for us to know the precise pairs parameters with interactions between them, with the objective of making best use of the input-output model evaluation matrices that involve in the main part of the sensitivity analysis computing cost. Besides the computing efficiency improvement, one side result we got from the algorithm for second order index is that it can also make the final index has smaller variance so that the result can be more reliable.

Besides Sobol's method, we also tried Morris method to complete the aim of 'screening' parameter of sensitivity analysis.

Strategies for FSPM

Global sensitivity analysis (SA) has an important role to play in functional-structural plant growth modeling by assessing the different source of uncertainty help us to gain some insights inner the models so as to explain the behavior of them. Different FSPMs have different scales of model design, which leads to all types of diverse multi-biophysical processes.

To study specifically how global SA can help for FSPMs, SA was applied on a wide variety of functional-structural plant models, typically the 3 FSPMs: firstly a simple source-sink model of maize growth, is used to specifically study the process of carbon (C) allocation among expanding organs during plant growth, with simple plant structure, multi-stage and detailed observations, secondly the GreenLab model of tree growth (applied to poplar tree) characterized by the retroaction of plant functioning on its organogenesis [41] , which describes tree structural plasticity in response to trophic competition, lastly a functional-structural model, NEMA [16] , describing C and nitrogen (N) acquisition by a wheat plant as well as C and N distributions between plant organs after flowering. This model has the specificity to integrate physiological processes governing N economy within plants: root N uptake is modeled following the transport systems high affinity transport systems (HATS) and low affinity transport systems (LATS), and N is distributed between plant organs according to the turnover of the proteins associated to the photosynthetic apparatus. C assimilation is predicted from the N content of each photosynthetic organ. Consequently, this model is more mechanistic but also more complex than the two previous ones. Another objective is to explore an effective simulation design to help the sensitivity analysis for complicate models with several logically distinct but biological functioning interacted moduls.

All these SA result shadowed a light to the models for us to diagnosis the model behavior and will bring a big step for parameter estimation and experimental simplification in our modelling next.

A model for Cecropia sciadophylla under fluctuating environmental conditions

In collaboration with Patrick Heuret (INRA, JRU Ecofog, Kourou, French Guiana), we developped a tree growth model dedicated to Cecropia sciadophylla, a neotropical species from the genus Cecropia. These trees have interesting properties from a modeller's point of view: they have a simple architecture, their number of phytomers remain limited even for old individuals, and most importantly, [39] and [49] have developped a methodology based on morphological observations to estimate tree age on C. obtusa and C. sciadophylla respectively. It is therefore possible to fully describe the tree structure and topology from morphological observations, which is very uncommon for trees: for most tree species, their high stature, complex structure, and long life span drastically increase the fieldwork required to collect data at the organ scale and hamper the development, calibration and validation of functional-structural tree growth models and their potentiel applications in the field of forest management.

We used datasets collected on 18 trees in 2007 and 2008 in french Guiana to develop and evaluate our model. Our objective was to analyse the influence of fluctuating environmental conditions on the dynamics of trophic competition within C. sciadophylla trees. We defined an integrated environmental factor that includes meteorological medium-frequency variations and a relative index representing the local site conditions for each plant. The meteorological variations were input from pluviometry data, that could be considered as the main fluctuating environmental stress under that tropical climate. The relative index was estimated based on inversion of our model using data from respectively 11 trees for model calibration (those measured in 2007) and 7 trees for model evaluation (those measured in 2008). This study provided a model that can be seen as a tool to disentangle the ontogenic variations (low-frequency trend) and the environmental variations (medium-frequency variation). One paper was accepted for publication [22] .

Using model inversion to analyze the effects of inter-tree competition on four Pine trees grown under two contrasted density conditions

In collaboration with Guo Hong and Lei Xiangdong (Chinese Academy of Forestry, Beijing, China), we analyzed the characteristics of individual tree response to competition on source-sink balance through the calibration of the GreenLab model.

Four Chinese pine trees (Pinus tabulaeformis Carr.) were destructively measured in November 2009 from the nursery garden located in the Yuanyiqi forest farm, Beijing, China. Two 13-year-old trees (T1 and T2) were from a high density plantation (3500N/ha) and two 10-year-old trees (T3 and T4) were from a low density plantation (2000N/ha). We first examined the statistical differences in the tree morphologies and topologies. Significant differences were found for internode diameter, internode biomass and needle biomass between the two densities, but not for internode length. In a second step, we studied the ability of the GreenLab model to simulate the plasticity of pine trees grown under different densities. To fulfil these objectives, it was necessary to find a way to characterize the competition conditions of each tree. Given the inherent difficulty of identifying the most relevant experimental measurements for this characterization, we proposed to represent the effects of competition on the tree growth through a single tree-specific parameter of GreenLab, called characteristic surface area, and to estimate it for each tree by model inversion, together with the more classical endogenous species parameters. This will eventually allow us to examine whether the obtained value of this characteristic surface area could be correlated to other possible indicators of competition pressure. This could pave the way to the development of an individual-based stand growth model including the effects of a competition index.

One paper was submitted to Trees - Structure and Functions.

Coffee trees and genetics

In collaboration with Sylvie Sabatier (INRA, AMAP), Philippe de Reffye (CIRAD, AMAP) and Perla Hamon (IRD Montpellier), we studied the architectural and genetic diversities in 5 Coffea species, native from Madagascar. We explore two complementary methods: the genetic diversity using molecular markers (genomic- and/or EST-microsatellites) and the variability of adaptive traits between populations with different ecological niches. We focused on 5 Coffea species endemic to Madagascar, some of which are classified as critically endangered in the World Conservation Union(IUCN) Red List. For each species, architecture and genetic comparative analyses between individuals growing in situ (natural forest) and ex situ (common garden test) are being performed. In parallel, the same populations are analysed using the GreenLab model. These results will be used to study the potential links between the parameters of GreenLab and the allelic distribution in these populations. This is the subject of the PhD of Domohina Andrianasolo (CIRAD, Montpelier and FOFIFA, Antananarivo, Madagascar). This work was presented at the XVIII International Botanical Congress (IBC) [34] .

Methods for tree crown analysis and application to young Eucalyptus

Based on the pioneer work on coffee trees of Philippe de Reffye, a stochastic model was developped to describe the topological development of trees. In the model, growth and branching processes are driven by the respective probabilities of activity, rest or death of apical and lateral buds. Because of its mathematical formulation, the model inversion can be done analytically Â which is rare Â and parameter values can be estimated from experimental data. The MATLAB softaware GLOUPS developped by Philippe de Reffye was used. We explored the feasibility of calibrating this stochastic model for eucalyptus, which presents the additional difficulty of a continuous growth with no marked endogenous cessation. Incomplete systems were also defined for the case, common with trees, of incomplete datasets. An adequate strategy was defined to sample measurements and applied to five eucalyptus trees (data collected by Pr Lei Xiangdong, Guo Hong and Diao Jun, Chinese Academy of Forestry, Beijing, China).

One paper was accepted for publication [20]