Sensitivity analysis methods for the evaluation of the models

which these data was acquired are usually not well recorded, adding uncertainty right from the first step of model building.

In the case of Lavrio, the majority of data used were collected with own efforts and under conditions known to the modeller. Data from other sources has also been available, with those data being pre-processed before being used in the models. For that reason, the information that is used as an input to the Lavrio groundwater flow model is considered to be of very good quality. In general, it is desirable that the modeller has an active role to the methodologies that are used when taking information that are going to be used for modelling. This can optimize the routine of data collection and improve the model structure and the use of the acquired data.

karstic aquifer. Therefore, the measurements are considered to have very good quality and the cumulative error would not, in any case, exceed the value of 10 cm.

The statistics used in the sensitivity analysis (and later in the parameter estimation) is the standard deviation (σ), meaning that the measurement cumulative error could not be more than 10 cm. The weight is calculated using Eq. 2.

ωii= ¹

σ²

where ωii is the assigned weight and σ the standard deviation of the measurement.

3.5.2 Sensitivity analysis basic concepts

The sensitivity analysis of the models was performed using the UCODE 2014 code (Poeter et al., 2014). The code can be used for the parameter estimation also and its usability is not limited in groundwater flow models but can be expanded in other types of models. The code uses a number of statistical indicators to quantify the sensitivity of each model parameter by comparing changes in the hydraulic heads and the volumetric flows that are simulated with different parameters set. Each time, the hydraulic heads and volumetric flows are compared to the relevant observed values that are used in the model. The outputs are used in order to make the selection of the parameters that are most important and are going to be used in the parameter estimation process. The use of such a powerful mathematical tool should be done with care in order to avoid misinterpretation of the results and, eventually, unrealistic modelling approach to the physical system. Further information on the code, its structure and useful guidelines on model building, can be found in the literature (Poeter et al., 2005; 2014; Hill and Tiedman, 2007).

One of the strong advantages of UCODE 2014 is the fact that the user can perform sensitivity analysis to as many parameters desired. In comparison, when using the more traditional trial and error method, the modeller has to select in prior the parameters that are expected to have high sensitivity and perform the sensitivity analysis. In that way, the sensitivity analysis may not be able to include many or all of the available parameters due to lack of flexibility or inability to explore the parameter value acceptable limits. This characteristic can be of vital importance in the case of very complex models with a large number of parameters. In addition, parameters that may initially be considered

Eq. 2

having low importance can be overseen by the user, so the code results can highlight the parameters that are most important for the modeller. Finally, the way the sensitivities are produced makes the choice for the parameters that are going to be used in the parameter estimation more defendable due to the statistical methodology that is followed.

The results of the sensitivity analysis are presented in the following section. Before the sensitivity analysis run, a manual calibration step was necessary in order to achieve smaller residuals and have more credibility in the results of the sensitivity. This step can also significantly reduce the time and computational effort needed in the parameter estimation process.

3.5.3 Scaled sensitivities

The chosen method for the sensitivity analysis is the use of the fit independent statistics. The method has the advantage of not requiring the initial model results to be extremely accurate while the results can be directly evaluated and used in the parameter estimation. However, having initial results that are not completely unrealistic is still helpful for the simulation. More information about the fit independent statistics can also be found in Hill and Tiedman (2007).

3.5.3.1 Dimensionless and composite scaled sensitivities

One of the main indicators used in the sensitivity analysis is the dimensionless scaled sensitivity (DSS), which is a measure that reflects how effective a change in a specific parameter is to the model.

The comparison is with the initial parameter value and the perturbed value that is used in the specific iteration. The DSS is calculated using the Eq. 3:

where y’i is a simulated value, bj the jth parameter, the ^∂y′i

∂b_j derivative is the “sensitivity” of the ith observation to the jth parameter, b is a vector containing that parameter values and ωii is the weight that is assigned to the ith observation (Hill and Tiedman, 2007). When summing all the DSS for a single parameter the scaled sensitivity (CSS) is calculated. This indicator is mainly used for the selection of the parameters that are the model is mostly sensitive to.

Eq. 3

2 /

' 1

ii i b j

i

ij b

b

dss y  



 







 

3.5.3.2 Parameter correlation coefficient

The correlation between different parameters is one of the most important aspects of the sensitivity analysis since highly correlated parameters cannot be estimated independently. In that case, when used in the parameter estimation, one of them should be assigned with a fixed value when the other is estimated and vies versa. The choice of which parameter to fix can be taken in combination with the CSS i.e. if two parameters have high correlation coefficient, the parameter with low sensitivity could be fixed when the parameter with high sensitivity is estimated. The parameter correlation coefficient (PCC) is calculated using Eq. 4:

Cov 𝑏 _jk/ Var 𝑏 _jj Var 𝑏 _kk

where Cov b _jk is the covariance between two parameters and Var b _jj,Var b _kk the variances of each parameter (Hill and Tiedman, 2007).

For the UCODE 2014 to be able to estimate a pair of values independently, their PCC is suggested to be below 0.95 (Hill and Tiedman, 2007). When parameters with higher PCC are used, a justification on the reasons why this option is chosen should be presented.