# Extreme Ground Snow Loads in Europe from 1951 to 2100
## 2 Material and Methods
### The Factor of Change Approach  
Trends will be assessed in Section 3.2 according to common practice for engineering applications by means of the FC approach [20; 21; 22; 23]. FCs are synthetic measures of changes in the statistics of climate variables and can be derived also for characteristic values of climatic actions investigating the \(n-\)th time window and the initial time window. The main assumption in the FC approach is that errors in measurements and predictions of climate variables are nearly constant over time. Of course, the higher the quality of the data the more reliable are the results obtained with the proposed method.  
To be consistent with the actual definition of climatic hazard maps for structural design [6; 7; 8; 58; 59], a 40-year period is considered in the present study for the definition of the time windows and a time shift of 10 years between each other is adopted.  
The fundamental steps of the procedure are the following:  
1. Extraction of annual maxima of the investigated climate variable for the investigated time windows (e.g., 1951-1990, 1961-2000, 1971-2010,..., 2050-2090);
2. Elaboration of extreme values for each time period, via the block maxima method [60]. Of course, several extreme value distributions can be assumed for the considered variable. Since ground snow load extremes in Europe are usually described by the extreme value type I (EVI) distribution [31; 33; 36], an EVI distribution was assumed in the present study too. The cumulative distribution function (CDF) of the EVI distribution, also known as Gumbel distribution, is \[F(x<X)=\exp\biggl{\{}-\exp\biggl{[}-\biggl{(}\frac{x-\mu}{\sigma}\biggr{)} \biggr{]}\biggr{\}}\ \ \ \text{with}\ \mu\ \in\ \mathbb{R};\ \sigma>0,\] (2) being \(\mu\) the location, and \(\sigma\) the scale of the distribution.
3. Characteristic values of ground snow load, \(q_{k}(n)\), are thus evaluated for the \(n\)-th time window, from \[q_{k}(n)=\mu+\sigma\cdot\{-\ln[-\ln(1-p)]\}.\] (3) In the equation, \(p\) is the given probability of exceedance, i.e., \(p=0.02\) in one year according to EN1990 [10].
4. FC are derived in terms of ratios between the characteristic value in the \(n\)-th time window and that obtained in the first one \[\text{FC}(n)=\frac{q_{k}(n)}{q_{k}(n=1)}\.\] (4)  
FCs can provide advice for the update of climatic maps given in codes and standards, relying on the analysis of past observations. In this context, it is important to underline once more that engineering works shall cope with climatic hazards during their physical life, which can obviously be longer than the design service life [8]. Therefore, to guarantee adequate reliability levels over time for the designed structures, an adaptation strategy for structural design was proposed in [8], which consists of adopting an envelope of characteristic values in the future time windows for the updated climatic load maps [8; 59].  
According to the proposed approach, the updated values of ground snow load \(q_{k}^{\prime}\), including influence of climate change, are obtained multiplying the maximum FC in the future time windows by the current value of \(q_{k}\) obtained from the analysis of observations (\(q_{k,obs}\))  
\[q^{\prime}_{k}=\max(\text{FC}(t))\cdot q_{k,obs} \tag{5}\]