# Probabilistic Analysis as a Method for Ground Freezing Depth Estimation
## 1 Introduction  
Freezing of the soil in wintertime is one of the climatic actions which must be considered when designing structures founded on soils. The actual frost depth is affected by the material type, soil thermal properties, soil water content, and climatic conditions such as temperature, wind speed, precipitation, and solar radiation. Frost depth can be estimated using numerical or analytical modeling technique ([PERSON] [1], [PERSON] [2], [PERSON] [3], [PERSON] et al. [4], ). In the past few years different numerical techniques have been used for modeling transient heat flow in pavement layers. These methods require a large number of input data [4, 5]. In most cases the input data are either not available or just too expensive to collect. In the case of missing data, higher or similar accuracy can be obtained from simpler analytical and semi-empirical models to estimate frost depth prediction--for e.g., [PERSON] model, modified Berggren model or [PERSON] and Phang empirical model [3, 5]. In those methods, the depth of frost penetration has usually been defined using the so-called air freezing index. However, it is obvious that the frost penetration into the soil is a complex random process in which many factors affect the final results, which are often divergent from the observations [6]. Those are not only air temperature but also the depth and the thermal properties of snow cover, which is changing during winter, heat contained in the soil [7, 8] and the soil properties, its structure and water content. All the climatic conditions have a random character and should be analyzed using probabilistic methods. Nowadays, the simplest approach is the probabilistic analysis of the annual (winter) maximum depth of the zero centigrade temperature. However, itshould be emphasized that it is not the same as the depth of the soil freezing. The purpose of the paper is to present the probabilistic analysis of the maximum annual depth of the zero contigrade temperature of the soils, the implemented method and some examples of the obtained results.  
Since 1955, the maps of the soil freezing depths have been given in Polish Standards. The values were calculated in centimeters according to the formula given in Soviet recommendations [9]:  
\[h_{z}=23\cdot\sqrt{\sum m+2} \tag{1}\]  
where: \(m\)--an absolute value of the winter monthly mean negative air temperature; only negative values are summarized.  
The numerical factor 23 means the type of soil. The value 23 is for loans and clays. Numbers for other types of soil were not given in the Polish Standards. According to [PERSON] [10], this formula was derived on the basis of the experiments made at six Soviet meteorological stations. A similar formula but with more detail was elaborated at the US Army Cold Regions Research and Engineering Laboratory [6].  
The results of the calculations according to the Formula (1) were implemented into the Polish Standard in 1955 (PN-55/B-03020) and its next editions with small modification in 1974. For the western half of Poland, the depth of freezing was 0.8 m, for the eastern part it was 1.0 m and for small area around Suwalki region it was 1.4 m.  
The approach, based on the air freezing index, is still recommended [1]. However, now it is possible to define the depth of the zero contigrade temperature on the basis of direct measurements of the soil temperature.