# Assessment of Frost Depth and Frost Heave in Romania
## 3 Frost Depth Prediction
### USA Standards/Regulations  
There are several frost depth prediction models that have been developed and implemented by the US authorities. Some of these models are empirical in nature, while others are semi-empirical. The reference methodology developed by the NOAA (National Oceanic and Atmospheric Administration) is based on the AFI calculation at each weather station. Using the NOAA's National Climatic Data Center (NCDC), [PERSON] et al. 2015 [6] used the complete  
Figure 2: Number of days with daily minimum temperatures less than 0 \({}^{\circ}\)C for the Arad station for the 1950–2020 period. dataset of daily maximum and minimum temperatures that were utilized to calculate the NOAA's 1981-2010 freezing degree days (FDDs) and the corresponding AFI values. The air-freezing index (AFI) is a common metric for determining the freezing severity of the winter season and estimating frost depth. AFI values represent the seasonal magnitude and duration of below-freezing air temperature. Departures from the daily mean temperature above or below 0 \({}^{\circ}\)C are accumulated over each August-July cold season are accumulated and can be plotted on a seasonal time curve. The difference between the highest and lowest extrema points on this seasonal curve is defined as the seasonal AFI value. For example, in Figure 3, the most extreme AFI value for the Arad station over the 1900-2020 period occurred during the 1953/54 season, with an AFI value of 521 FDDs. AFI is converted to frost depth using the [PERSON] (1964) [5] Formula (1). \[\text{d}_{\text{frost}}=0.0174(\text{AFI}_{100})^{0.67} \tag{1}\]  
where \(\text{d}_{\text{frost}}\) is the depth of frost for a bare ground surface (m) and \(\text{AFI}_{100}\) is the 100-year return AFI (\({}^{\circ}\)C). The \(\text{AFI}_{100}\) is computed using generalized extreme value distribution analysis. In this study, we calculate all the parameters, with \(x_{p}\) corresponding to the return periods determined via the Gumbel distribution for maxima [7]. The Gumbel distribution for maxima is defined by its cumulative distribution function, CDF:  
\[F(x)=e^{-e^{-a(x-u)}} \tag{2}\]  
where \(u\) and \(a\) are the parameters of the distribution. \[\text{u}=\mu_{\text{x}}-\frac{\gamma}{\alpha};\gamma=0.5772\text{--Euler constant};\ \gamma=\lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\right);\alpha= \frac{\pi}{\sqrt{6}}\cdot\frac{1}{\sigma_{\text{x}}}\]  
and \(\mu_{\text{x}}\) and \(\sigma\)x are the mean and the standard deviation of the distribution. The fractile \(x_{p}\) that is defined as the value of the random variable X with p non-exceedance probability (\(P(X\leq x_{p})=p\)) is computed as follows:  
\[ \begin{array}{l}F(x_{p})=P(X\leq x_{p})=p=e^{-e^{-a(x_{p}-u)}}\\ x_{p}=u-\frac{1}{\alpha}\cdot\ln(-\ln p)=\mu_{x}-0.45\sigma_{x}-\frac{\sigma_{ x}}{1.282}\cdot\ln(-\ln p)=\mu_{x}+k_{p}^{G}\cdot\sigma_{x}\\ k_{p}^{G}=-0.45-0.78\cdot\ln(-\ln p)\end{array}  \tag{3}\]  
Figure 3: FDDs for the Arad station in the 1953–1954 season (seasonal time curve). The values of \(k_{p}{}^{G}\) for different non-exceedance probabilities are given in Table 2.  
Table 3 presents the statistical parameters (mean, standard deviation, and maximum recorded values) for the measured AFI and the corresponding return periods values for the Arad station. The depth of frost (in cm) is also presented for the four periods of time investigated. The 100-year return AFI values were calculated for all 111 meteorological stations in Romania for the 1950-2020 period. Based on the simple kriging geostatistical method, Figure 4 presents the AFI\({}_{100}\) map of Romania. The statistical model takes into consideration the yearly average temperatures and the location and average temperature of the coldest quarter for all administrative units of Romania. The reference methodology to calculate frost depth (cm) developed by the U.S. Corps of Engineers ([PERSON], 1975) [8] requires only the freezing index (FI) or the cumulative freezing degree day (CFDD), Equation (4).