باز کردن عدم قطعیت خروجی یک جریان آب درختی و مدل ذخیره سازی با استفاده از روش های تجزیه و تحلیل حساسیت جهانی

The output uncertainty of two variables (stem sap flow rate and stem diameter variations) of a tree water flow and storage model is broken down into its different constituents using two global sensitivity analysis methods: extended Morris screening and an extended Fourier amplitude sensitivity test. Using these methods, quantitative and qualitative information about the parameters contributing most to the model output uncertainty was obtained. It was shown that only three parameters (out of the 11 considered) were responsible for the uncertainty on the stem sap flow rate and that three other parameters were accountable for most of the uncertainty in stem diameter variations. Furthermore, the parameters influencing the stem diameter variations were involved in significant interactions. An investigation into the effect of the magnitude of the parameter uncertainty on the results of the sensitivity analysis showed that the contribution of the parameter interactions increased with increasing parameter uncertainty. By combining several global sensitivity analysis methods the results were not only verified but more confidence was gained in the accuracy of the methods used and complementary information was obtained. This allowed a more detailed picture to be constructed of how the individual parameters interact and contribute to the output uncertainty of the model.

Most stages of the development process of a mechanistic model, in which parameters have a physical meaning, are associated with some degree of uncertainty (Beck, 1987 and Chatfield, 1995). It is well-known that models are only simplified representations of reality caused by abstract choices that are made during the development process. These choices involve (1) selecting which processes to model, and which to neglect; and (2) selecting the equations used to represent and link the processes. At this stage, choices related to spatial and temporal aggregation are also made. Many authors acknowledge that the uncertainty associated with these choices is very difficult to quantify, if at all possible (Chatfield, 1995, Jansen, 1998 and Refsgaard et al., 2006). Therefore, one could argue that the errors introduced in this way should not be regarded as uncertainties in the strictest sense of the word. Once a certain model structure has been chosen, two important sources of uncertainty contribute to the model output uncertainty: input and parameter uncertainty (Beck, 1987 and Refsgaard et al., 2006). A first source is related to the uncertainty associated with the (time-varying) driving forces of the model, the inputs. This uncertainty is not only caused by imprecise measurements, but also by the inherent variability of the inputs themselves (e.g., climatic data). A second source of model output uncertainty is associated with the model parameters. In this case, uncertainty arises when imprecise measurements are used to assign parameter values directly or through model calibration. Quantifying model output uncertainty can be very useful. Especially when judging the quality of the fit of a model to experimental data or, more importantly, when using the model for prediction and decision making. However, knowing the cause of the uncertainty can also be equally important. This knowledge allows the modeller to highlight important sources of uncertainty and, therefore, pinpoint parts of the model that require additional attention or improvement in order to reduce the overall model output uncertainty. Unravelling the model output uncertainty into its different contributions is a task that can be performed through a sensitivity analysis (Saltelli et al., 2000). In sensitivity analysis, a distinction should be made between local and global methods. A local sensitivity analysis is used to study the importance of model parameters at one point in the parameter domain. At this point, parameters are changed one at a time by a certain fraction of their value and their effect is quantified (Law et al., 2000, Dufrêne et al., 2005 and Pathak et al., 2007). The validity of the information obtained through such an analysis is, however, restricted to the selected point in the parameter domain. In order to deal with this issue, a global sensitivity analysis can be performed, which allows the entire parameter domain (or a portion of it) to be analysed. Techniques belonging to this category are regression-based methods (Levy and McKay, 2003 and Verbeeck et al., 2006), screening methods and factorial design (Morris, 1991, Campolongo and Braddock, 1999 and Pathak et al., 2007) and variance-based methods (Cukier et al., 1973, Saltelli et al., 1999, White et al., 2000, Smith and Heath, 2001 and Gottschalk et al., 2007). From the available global sensitivity analysis methods, the (extended) Morris screening method (Morris, 1991 and Campolongo and Braddock, 1999) and the (extended) FAST (Cukier et al., 1973 and Saltelli et al., 1999) stand out because of their frequent application in many scientific fields. Besides being useful for quantifying the importance of the model parameters, they also provide valuable information regarding first order, non-linear and interaction effects. It is the aim of this paper to introduce these two techniques to the field of plant science and to apply them to the RCGro tree water flow and storage model that is designed to study and simulate the water transport dynamics and related stem diameter variations in trees (Steppe et al., 2006). The contents of this paper are outlined as follows. In Section 2, we briefly describe the mathematical model. Section 3 is devoted to the methods used in this contribution, namely uncertainty analysis and the extended Morris screening and the extended FAST, both global sensitivity analysis methods. In Section 4, the results of the analysis are presented and discussed. Finally, Section 5 concludes the contribution.

In this contribution an uncertainty analysis of two of the model variables (F(stem) and D) of a tree water flow and storage model was performed. Two global sensitivity analysis methods (extended FAST and extended Morris screening) were used to determine the most influential parameters on the model output uncertainty and gain more insight into the model and the parameter interactions. Results of the uncertainty analysis showed that the propagation of the parameter uncertainties through the model did not result in large model output uncertainties for F(stem), but did result in a large uncertainty on the model output of D. Analysis of the sensitivity results for F (stem) showed that three parameters are responsible for almost all of the model output uncertainty were R x, C (crown) and L . It was also found that no interactions between the parameters existed. A similar analysis for D showed that only three parameters played an important role in the output uncertainty: Γ , β and View the MathML sourceΨx(Ψps=0). For this variable, only 80% of the model output variance could be attributed to first order effects. Therefore, the remaining 20% was due to interactions among the parameters. It could be shown that only Γ , β , View the MathML sourceΨx(Ψps=0) and φ were involved in these interactions and that all interactions were among the parameters of this group. An analysis of the effects of different parameter uncertainties on the sensitivity results showed that increasing the parameter uncertainty had no effect on the sensitivities of F(stem). However, increased uncertainty in the parameters did result in an increased contribution of the parameter interaction effects to the model output uncertainty of D. In spite of this, the general conclusions about which parameters were most influential and showed interactions with other parameters did not change. Finally, we would like to stress that combining different global sensitivity analysis methods has several advantages. Firstly, one is able to verify results and as such gain more confidence in the accuracy of the methods used and the results obtained. Secondly, one is able to obtain complementary information. Indeed the extended FAST method allows quantifying the first order, total and interaction effects of the model parameters, whereas the extended Morris screening method allows detecting the parameter combinations that are responsible for the interactions. The extended Morris screening method also provides information on whether parameters have a positive or negative impact on the model output. All this combined knowledge allows constructing a more detailed picture of how the parameters interact and contribute to the model output uncertainty and, as such, creates important insight into the model.