ANÁLISIS DE LA INTERVENCIÓN

Before adjusting the parameters of 3-PG2, the application of a sensitivity analysis (SA)

will identify those parameters which are most important and therefore require the most precise estimation. Model sensitivity is defined as the standard deviation of output X

( Xp) with respect to parameter(p) as:

Xp =

X

X /

p

p (6.1)

i.e. the ratio of fractional change in output to fractional change in the parameter ( Xp)

is calculated by running the model for a range of p for each p (Saltelli et al., 2004). The use of positive and negative values for p are used to capture possible non-linearity in the model outputs. The process can be repeated for sites with radically di↵erent environmental (climactic and physical) conditions to capture any variability introduced through interaction of these variables with p.

Xp can be divided into main e↵ects and joint e↵ects. Main, or primary, e↵ects are

those direct interactions between p and X. Joint e↵ects are caused by interactions between two or more parameters which can cause a greater X than through varying each parameter individually.

A sensitivity analysis is capable of providing information on a model’s sensitivity on several layers. Structural sensitivity is defined by how sensitive the performance of the model is to structural assumptions, or processes, it includes. Parameter sensitivity is the response shown in the model’s performance to the values of parameters that characterise relationships included in the model. Finally, the sensitivity of the model’s performance to variations in the data required to drive the model, such as climatic and

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environmental variables, is known as the input sensitivity.

Sensitivity analyses are vital for the validation model structure. A SA that indicates that the majority of parameter sensitivity lies within the primary interactions demonstrates a good model structure (Sobol, 2001). Performing a SA is a key step in refining parameter selection (Landsberg and Sands, 2010). In the case of PROMOD, a process based forest growth model, the results of multiple sensitivity analyses have been invaluable in strengthening the acceptance of the model’s structure an enhancing understanding of it’s behaviour. Sensitivity analyses have been used to facilitate adaptation of the model to novel situations or species e.g. PROMOD for Eucalyptus nitens and P. radiata (Sands et al., 2000).

There are two methods of SA. Local SA test the model output sensitivity by varying one parameter at a time whilst the others remain at their central values (‘one factor-at- a-time’, OAT). Several local SA have been carried out on 3-PG (Battaglia and Sands, 1997; Sands and Landsberg, 2002; Almeida et al., 2004b; Esprey et al., 2004; Almeida et al., 2007a). The combination of these studies does not cover the full parameter space of 3-PG as each individual study only tested the e↵ect of varying 9–27 of the model’s 60 parameters. These parameters were selected for their contribution to specific aspects of 3-PG, namely those characterising canopy structure and canopy quantum efficiency; allometric relationships and biomass partitioning; branch and bark fractions; basic wood density; litterfall and root turnover rates and various environmental modifiers.

These a↵ect 3-PG2 outputs in various ways. Some parameters only influence a subset of

outputs, whilst many a↵ect most outputs and, in some cases, combinations of parameters interact in their e↵ects. These OAT methods are only informative at the central point where the calculation is executed and does not cover the whole input parameter space (Song et al., 2012). Therefore, they are unsuitable for analysing complex biophysical process models comprised of several dimensions and non-linear responses (Saltelli and Annoni, 2010) as they cannot quantify the e↵ects of parameter interaction.

The elementary e↵ects method or ‘Morris Method’ (Morris, 1991) determines which input factors may be considered to have e↵ects which are (a) negligible, (b) linear and additive, or (c) non-linear or involved in interactions with other factors (feedbacks). It is used to reduce the number of parameters to be analysed in a more detailed SA

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of a computationally costly model, or a model such as 3-PG2 with a large number of

inputs.

A preferable SA technique is a model-independent global SA. In a global SA the e↵ect of each parameter within the full parameter space is considered. This is achieved by changing the value of all input parameters in each simulation. This captures the variation in outputs with respect to the multiple input variation. These methods also allow examination of interactions between parameters when varied simultaneously. Common variance based global SA techniques include: Fourier amplitude sensitivity analysis (FAST) (Cukier et al., 1975; Saltelli et al., 1999), the Sobol method (Sobol, 2001; Saltelli, 2002; Saltelli and Annoni, 2010), Saltelli’s method (Saltelli et al., 1999), the Derivative-based Global Sensitivity Measure (DGSM) (Sobol and Kucherenko, 2010) and the Bayesian Analysis of Computer Code Outputs (BACCO) sensitivity analysis, which creates a model emulation (Oakley and O’Hagan, 2004). These variance based techniques work within a probabilistic framework by decomposing the variance of the model output into fractions that can be attributed to individual inputs or groups of inputs. The percentages by which each input a↵ects the output, can be used as a measure of sensitivity. In these methods importance of a model input is based on the expected reduction in the model output variance provoked by knowing the value of the model input with certainty.

In a Sobol SA these measures of sensitivity are expressed as (Sobol) sensitivity indices (SIs). This method utilises Monte Carlo (or quasi-Monte Carlo) to sample the entire parameter space requiring thousands of model simulations. Despite these high compu- tational demands, this powerful SA technique has recently become more popular in forest growth modelling (e.g. Wu et al., 2012; Wang et al., 2013a), because of its ability to incorporate parameter interactions and its relatively straightforward interpretation (Nossent et al., 2011).

Although computationally di↵erent, the FAST method uses the same measure of sensitivity as Sobol’. The main advantages of FAST is its robustness, especially at low sample size, and its capability in ascribing the output variance to individual inputs in the model. FAST and Sobol are both independent of the model structure. However, FAST is computationally complex for a model with a large number of inputs (Christopher

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Frey and Patil, 2002) such as 3-PG2.

The DGSM method relies on averaging local derivatives, the e↵ects of one variable whilst holding others constant, using Monte Carlo or preferably quasi-Monte Carlo sampling methods. This technique is more accurate than the Morris method in computing the elementary e↵ects as each derivative is evaluated as strict local derivatives with small increments in comparison to the variable uncertainty ranges used by the Morris method (Kucherenko et al., 2009). The computational time required for numerical evaluation of DGSM is many orders of magnitude lower than that for estimation of the Sobol sensitivity indices, as the method benefits from much higher convergence rate if quasi-Monte Carlo sampling methods are used. It becomes especially efficient if automatic calculation of derivatives is used (Sobol and Kucherenko, 2010).

These methods require Monte Carlo simulations to sample the entire range of the param- eter space. This high computational cost has been a barrier to using these techniques in the past (Verbeeck et al., 2006), however the availability of high-performance computing (HPC) resources have lowered this barrier (Dietze et al., 2013). In the absence of a HPC, a screening technique, e.g. the Morris method, can be applied to reduce the number of parameters under consideration.

The BACCO method uses a Bayesian analysis approach. It builds a statistically based representation of the model from a set of training data points derived from runs of the model of interest. A space-filling algorithm is used to fill in the gaps not sampled in the input parameter space. This model emulation is used to compute the sensitivity indices of interest. This method is designed to be faster, easier and more efficient to run across the entire parameter space than running the model under investigation thousands of times. However, this method requires information on the probability distribution function for each model input to provide a representational emulator designed from only a few hundred runs (Petropoulos et al., 2009).

The sensitivity of ten of the 3-PG2’s outputs, namely: mean DBH (avDBH), basal

area (BA), LAI, StandVol, evapotranspiration, fraction of available soil water (fASW) , transpiration and biomass pools, were assessed by Song et al. (2012). Using a variance- based global sensitivity analysis to calculate the elementary e↵ects, this study assessed the impact of key parameters linked to the allocation of biomass in di↵erent parts of a

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tree; stand volume; leaf area; photosynthesis and water availability; biomass removal and turnover rates in above ground components and root growth. Whilst the majority of SA carried out on 3-PG involve simple regression analysis (Esprey et al., 2004; Zhao et al., 2009; Rodr´ıguez-Su´arez et al., 2010) which follow the same procedure as local SA in their ‘OAT’ sampling and share it’s limitations.