## 3 Adaptation of the Model Function for GUM-S1  
According to GUM, there are two approaches to obtain the estimated value of the measurand _Y_: method 1 and method 2 [24]. Method 1 refers to obtaining \(y\) from  
\[y=f\big{(}\overline{X}_{1},\overline{X}_{2},\ldots,\overline{X}_{N}\big{)} \tag{5}\]  
where \(\overline{X}_{1},\overline{X}_{2},\ldots,\overline{X}_{N}\) are the arithmetic means of each input quantity and \(f\) is the model function. Method 2 refers to obtaining the estimate \(y\) as  
\[\overline{Y}=\big{(}\sum_{k=1}^{n}Y_{k}\big{)}/n \tag{6}\]  
In general, if the model is not linear both methods are not compatible. If the repeatability range of the input quantities is low, both can be compatible. Figure 1: Calibration set-up scheme. Figure 2: Photograph of the tank and a flow meter under calibration in the facility. The GUM-S1 method cannot implement method 2. If both methods are not compatible, only method 1 would be meaningful in the MCM (Monte Carlo method) framework as it requires a model function to obtain the measurand. In case of lack of repeatability, both approaches are incompatible, the model cannot be linearized, and GUM cannot be used: GUM-S1 should be used with method 1. In this work, it is shown how the application of GUM-S1 affects the measurand representation, considering the model function of the meter factor (_MF_) of a calibrated meter against a proving tank. When applying the GUM method, uncertainty calculation and measurand estimation can be independent; that is, method 1 can be used to estimate uncertainty and method 2 can be used to estimate the measurand. In fact, this is the situation in this case. However, if MCM is used, method 1 should be applied both for measurand and uncertainty estimation. The _MF_ calculation is performed in Equation (2) by using method 2. So, in order to apply GUM-S1, Equation (2) should be adapted [9]. The proposal of this work is substituting the type A evaluation of uncertainty of the measurand by a global additive contribution of a rectangular shape. Therefore, the adapted model function of the calibration against a proving tank is as follows:  
\[MF=\frac{V_{\text{TANK}}\{1+k_{v}[(t+\delta t_{e})-t_{\text{REF}}]\}}{V_{ \text{METER}}}+\delta M_{\text{REP}} \tag{7}\]  
where the quantities, \(t\), \(V_{\text{METER}}\) and \(V_{\text{TANK}}\) are detailed in the following equations:  
\[t=\overline{t}+\delta t_{cal}+\delta t_{res}+\delta t_{drift}+\delta t_{corr} \tag{8}\]  
\[V_{\text{TANK}}=\overline{V}_{\text{TANK}}+\delta V_{\text{TANK,all}}+\delta V _{\text{TANK,res}}+\delta V_{\text{TANK,drift}}+\delta V_{\text{TANK,range}} \tag{9}\]  
\[V_{\text{METER}}=\overline{V}_{\text{METER}}+\delta V_{\text{METER,res}} \tag{10}\]  
Also, \(\delta M_{\text{REP}}\) is a zero-mean random variable whose range is twice the repeatability tolerance and whose probability distribution is uniform (rectangular). This global correction is due to random effects. When applying the Monte Carlo method, the _PDF_s of the input quantities must be assigned as prior quantities based on the metrologist's judgement [25]. The explanation of every input quantity as a random variable and its related probability density function _PDF_ is detailed in Table 1.