# Evaluation of the Urban Weather Generator on the City of Toulouse (France)
## 2 Materials and Methods
### UWG Evaluation
#### 2.3.2 UWG Sensitivity Analysis Methodology  
To analyze the sensitivity of UWG with respect to its input parameters, the Morris sensitivity analysis method has been exploited [55]. Also known as \"the elementary effects\" (EE) method, the Morris technique is particularly adapted to models with quantitative inputs/outputs and identifies, in our situation, the input parameters that mainly explain the air temperature outputs. More precisely, it distinguishes parameters with (i) negligible effect on the output; (ii) linear and/or additive effect on the output, and (iii) non-linear effects and/or including interaction with other ones. In general, a sensitivity analysis method consists of varying all the input parameters in a given range (defined manually based on prior physical knowledge of extreme values of each parameter) and analyzing the sensitivity of the output. As the number of input parameters is large with UWG, computing simulations with all combinations of parameters is in practice impossible from a computational point of view (the problem complexity grows exponentially with the number of parameters). The [PERSON] method proposes an alternative strategy to limit the number of simulations by designing a subset of the entire set of all possible parameter combinations on which to perform the sensitivity analysis. It relies on screening methods [56] where a set of \(R\) trajectories of parameters is computed. A single trajectory \(\Theta^{(k)}=[\Theta^{(k)}(0),\Theta^{(k)}(1),\ldots\Theta^{(k)}(N)],k=\{1, \ldots,R\}\) is a set of \(N+1\) parameter variations carefully chosen. Here, \(N\) refers to the total number of parameters to analyze. In practice, we focus on atmospheric and urban surface parameters, i.e., \(N=N_{a}+N_{u}\). As mentioned previously, the influence of the rural station is indeed analyzed in a specific section and the associated parameters \(\Theta_{r}\) are not included in the sensitivity analysis. The idea consists of \((N+1)\times R\) simulations of \(UWG\) along all \(R\) trajectories and then performs a sensitivity analysis on this subset of parameters instead of on the entire range of possible variations. This leads to a linear complexity with respect to the number of input parameters instead of an exponential one. It has been shown in [57] that this process based on trajectories ensures a reliable representation of input parameter variations and is therefore reliable for a sensitivity analysis. In practice, the set of trajectories is designed as follows:  
1. **Discretization of the parameter space**. Each parameter \(\theta_{j}\in\{\Theta_{u},\Theta_{u}\}\) is uniformly split into \(p\) levels with a step \(\Delta_{j}\) (this latter depends on each parameter's range of magnitude). We then obtain a grid \(X\) of size \(N\times p\) containing all possible parameter values;
2. Computation of \(R\) trajectories. For \(i=\{1,\ldots,R\}\), do the following steps: 1. **Definition of an initial set of parameters \(\Theta^{(i)}(0)\)**. For the \(i\)-th trajectory, a starting set of parameters \(\Theta^{(i)}(0)\) is selected by choosing, for all lines in \(X\), a random value associated with parameter \(\theta_{j},j=\{1,\ldots,N\}\) among the discretized \(p\)-levels; 2. **Generation of a trajectory \(\Theta^{(i)}=[\Theta^{(i)}(0),\Theta^{(i)}(1),\ldots,\Theta^{(i)}(N)]\)**. From the initial set of parameters \(\Theta_{(i)}(0)\), a sequence of \(N+1\) parameter set is designed in a sequential way for \(j=\{1,\ldots,N\}\) as follows: * The parameter set \(\Theta^{(i)}(j)\) is derived from \(\Theta^{(i)}(j-1)\) by adding to the value of parameter \(\theta_{j}\) in \(\Theta^{(i)}(j-1)\) a random step \(\pm\Delta_{j}\) in a positive or negative direction. Other values of parameters \(\theta_{k},k=\{1,\dots,N\},k\) remain unchanged.