# System Dynamics Modeling for Estimating the Locations of Road Icing Using GIS
## 2 Materials and Methods
### Modeling Road Icing
#### 2.4.2 Road Moisture  
Road moisture is an important factor in the formation of road icing along with road temperature [47]. When moisture forms on a road, a decrease in road temperature below zero forms road icing. There are two cases in which moisture is formed on a road as follows. Based on these, two input factors were derived. The input factors that affect the formation of road moisture are precipitation (mass) and condensation.  
* Case (1) when water is collected on a road by precipitation (Precipitation)
* Case (2) when water vapor condenses on a road as the road temperature drops below the dew point (Condensation)  
The formula to obtain road moisture using precipitation (mass) and condensation is as follows. C is the condensation, MoP is the mass of precipitation, and Fr is the amount of freezing water. The road moisture formed by precipitation and condensation freezes and decreases.  
\[\text{M}_{\text{r}}=\sum_{\text{t=0}}^{\text{t=13}}(\text{MoP}+\text{C}\ -\ \text {Fr})_{\text{t}} \tag{9}\]  
The mass of precipitation in Equation (9) is the actual mass of precipitation per hour on an area of 1 m\({}^{2}\). This is expressed in Equation (10). P is the precipitation (mm) and MoP is the mass of precipitation (g/m\({}^{2}\)).  
\[\text{MoP}\Big{(}\text{g/m}^{2}\Big{)}=\frac{\pi\,\text{r}^{2}\text{P}}{10} \tag{10}\]  
The condensation in Equation (9) is calculated using the equation between the current amount of water vapor in the atmosphere and the amount of saturated water vapor on the road. The temperature of the water vapor contained in the atmosphere drops below the dew point near the road surface, and condensation occurs. The content on condensation is shown in Equation (11). C is the condensation (g/m\({}^{2}\)), E is the evaporation (g/m\({}^{3}\)), V is the amount of vapor (g/m\({}^{3}\)), and Vs is the amount of saturated water vapor (g/m\({}^{3}\)).  
\[\text{C}=\sum_{\text{t=0}}^{\text{t=13}}(\text{E}+\text{V}\ -\ \text{V}_{\text{s}}) \tag{11}\]  
For the evaporation in Equation (11), [PERSON]'s law formula based on [PERSON]'s aerodynamic was used. It is a theory that the movement of water molecules from a free surface is proportional to the vapor pressure gradient in the vertical direction [48]. This is expressed in Equation (12). E is the amount of evaporation from the reservoir, e\({}_{\text{s}}\) is the saturated vapor pressure at the air temperature (mmHg), P\({}_{\text{v}}\) is the actual vapor pressure at the air temperature (mmHg), and Sw is the wind speed (m/s) at a height of 2 m from the water surface (m/s) [48].  
\[\text{E}=0.345*(\text{e}_{\text{s}}-\text{P}_{\text{v}})*(0.5+0.54\text{S}_{ \text{w}}) \tag{12}\]  
The amount of vapor (V) in Equation (11) is obtained using Equation (13). P\({}_{\text{v}}\) is the vapor pressure and T\({}_{\text{a}}\) is the air temperature [49].  
\[\text{V}=217*\frac{\text{P}_{\text{v}}}{\text{T}_{\text{a}}+273.15} \tag{13}\]  
P\({}_{\text{v}}\) in Equation (12) is the vapor pressure, and it is entered into the model as an integral value to which the differential values over time are added. This is expressed in Equation (14). The integrated value P\({}_{\text{v}}\) was obtained by adding the amount of change per time to the initial value (P\({}_{\text{v}}\))\({}_{\text{t=0}}\).  
\[\text{P}_{\text{v}}=\int{(\text{P}_{\text{v}^{{}^{\prime}}})_{\text{t}}\text{ dt}}+(\text{P}_{\text{v}})_{\text{t=0}} \tag{14}\]\(S_{w}\) in Equation (12) is the wind speed. It is entered into the model as an integral value to which the differential values over time are added. This is expressed in Equation (15). The integrated value \(S_{w}\) was obtained by adding the amount of change per time to the initial value (\(S_{w}\))\({}_{t=0}\).  
\[S_{w}=\int{(S_{w}}^{\prime})_{t}dt+(S_{w})_{t=0} \tag{15}\]