## 2 Background  
### Frost Penetration and Hawve  
There are several methods to calculate frost penetration depth; almost all these methods are related to the freezing index (FI), which is a measure of the severity of freezing condition during the winter season. FI is calculated as the summation of daily average air temperatures over the freezing period [3]. Figure 2 shows an example of the calculation of the FI over the winter of 1996/1997 in Beatrice, ON. According to the Transport Canada Civil Aviation (TCCA) manual of pavement structural design ASG-19, the frost protection depth should be calculated using a minimum 10-year average freezing index [3]. Site air freezing index measures should be available from airfield data records, based on meteorological observations. If not available, Transport Canada allows an estimation to be made from a map of freezing indexes provided at the ASG-19 manual. Limiting the depth of frost action into the subgrade soils can limit the potential for frost heave andthaw weakening. This is achieved by defining a minimum pavement thickness and excavating the soil susceptible to frost penetration, preferably to the full depth. However, when the latter approach is not possible, ASG-19 advises that at least 1 m should be excavated [3]. To identify whether a soil is frost susceptible or not, TCCA relies on soil gradation, where the subgrade soil is non-frost-susceptible if it has less than 10 percent passing the 0.075 mm sieve. TCCA uses an empirical equation to calculate frost penetration depth. The method is based on the results published by [PERSON] and [PERSON] [4], who evaluated 39 sites and 93 observations to predict frost depth in asphalt-surfaced pavements. The calculation of frost penetration depth in asphalt-surfaced pavements, according to the Pavement Structural Design Manual ATR-021 proposed by TCCA, is presented in Equation (1) [5]. \[T_{min}=-61+6.8\sqrt{F} \tag{1}\]  
where \(T_{min}\) is the minimum thickness of the pavement structure (cm) and \(F\) is the average annual air freezing index (\"C-days). Another available method to classify the soil frost susceptibility is the Chamberlain method, used by the Ministry of Transportation of Ontario (MTO) [6]. This method defines  
Figure 2: Determination of the freezing index for Beatrice, Ontario, over the winter of 1996/1997. the soil, according to the material proportions of sand, silt, and clay, as acceptable, borderline, or unacceptable material. To calculate the frost penetration depth, the MTQ uses the empirical curve presented in Equation (2). \[P=-32.8+5.78\sqrt{F} \tag{2}\]  
where \(P\) is the frost penetration depth (cm) and \(F\) is the freezing index (\({}^{\circ}\)C-days). On the other hand, Ministere des Transports du Quebec (MTQ) proposes a mechanistic approach to predict frost depth and frost heave as a function of climate, subgrade soil properties, and pavement thickness. This analysis method requires more detailed site investigation than other methods, but it has been shown, through validation procedures, to yield good results in predicting frost penetration depth [7]. The MTQ approach is based on the SSR model developed in Finland [8], in which the calculation of the frost depth is based on the heat balance at the frost front. This method accounts for the segregation potential, which is a measure of the susceptibility of a soil to frost action, that can be measured in the laboratory or estimated through empirical equations. Material properties variables, such as dry density, particle mineralogy, water content, and specific surface area, are some of the inputs, providing the basis for calculation of thermal conductivities, latent heat, and other important material-dependent parameters. This model is utilized in i3C-me, a mechanistic software for pavement design and performance prediction, where a specific module is available to predict frost penetration and frost heave as a function of climate, subgrade soil properties, and pavement design [7; 9].