## 2 Methodology
### Ordinary Kriging (OK)  
Kriging has been shown to be the most accurate interpolation method for road weather variables [5; 18]. The unique property of this method is that it considers the spatial covari  
Figure 4: Training data with 500-m separation distance. The map shown is powered by ESRI. Sources: City of Edmonton, Esri Canada, Esri, HERE, Garmin, SafeGraph, Geotechnologies, Inc, METI/NASA, USGS, EPA, USDA, NRCan, Parks Canada.  
ance structure of the measurements and uses it to estimate unmeasured values. Equation (1) below depicts the general kriging formula.  
\[\hat{Z}(x)=m(x)+\sum_{i=1}^{m}\lambda_{i}[Z(x_{i})-m(x_{i})] \tag{1}\]  
where \(\hat{Z}(x)\) is the interpolated value at an unknown \(x\) location, \(m\) is the number of observation points, \(m(x)\) is the expected value at unknown location \(x\), \(\lambda_{i}\) is the kriging weight for observation \(i\), \(Z(x_{i})\) is the observed value at location \(i\), and \(m(x_{i})\) is the expected value at location \(i\).  
OK [19] is derived from Equation (1) above by forcing the weights to sum to one, i.e., \(\sum_{i=1}^{m}\lambda_{i}=1\). This change allows us to rewrite the kriging formula as follows:  
\[\hat{Z}(x)=\sum_{i=1}^{m}\lambda_{i}(x)Z(x_{i}) \tag{2}\]  
\[\begin{bmatrix}\lambda_{1}\\ \vdots\\ \lambda_{2}\\ u\end{bmatrix}=\begin{bmatrix}C(x_{1},x_{1})&\cdots&C(x_{1},x_{m})&1\\ \vdots&\ddots&\vdots&\vdots\\ C(x_{m},x_{1})&\cdots&C(x_{m},x_{m})&1\\ 1&\ldots&1&0\end{bmatrix}^{-1}\begin{bmatrix}C(x_{1},x)\\ \vdots\\ C(x_{m},x)\\ 1\end{bmatrix} \tag{3}\]  
\[\sum_{i=1}^{m}\lambda_{i}(x)=1 \tag{4}\]  
where \(C(x_{n},x_{m})\) is the covariance between the observation at location \(n\) and \(m\).  
In order to determine the covariance structure, a semivariogram [20] must first be constructed to model the degree of dissimilarity between two measurements based on their separation distance. The dataset used for this purpose must be trend free to meet the modeling assumption of mean stationarity [21]. Upon removing the trend, an empirical semivariogram can be assembled using the following formula:  
\[\hat{\gamma}(h)=\frac{1}{2n}\sum_{i=1}^{n}[Z(x_{i}+h)-Z(x_{i})]^{2} \tag{5}\]  
where \(\hat{\gamma}(h)\) is the semivariance at lag distance \(h\), \(n\) is the number of samples, and \(Z(x_{i}+h)\) is the observed value at h lag distance away from observation at \(x_{i}\).  
Based on the developed empirical semivariogram structure, a theoretical variogram with a similar structure must be fitted, from which the covariance values used in kriging are obtained. This is necessary as using the empirical model may result in a non-invertible covariance matrix that makes kriging impossible [22]. The spherical model was selected for this study (Equation (6)) as it had the best fit.  
\[C\left(\frac{3h}{3a}-\frac{1}{2}\left(\frac{h}{a}\right)^{3}\right) \tag{6}\]  
where \(C\) is the sill, \(h\) is the distance, and \(a\) is the range.  
Regardless of the model chosen, all semivariogram models contain three parameters: the nugget, sill, and range. The nugget is the amount of semivariance at a lag distance of 0, typically due to measurement error. In comparison, the sill is the point at which semivariance plateaus, beyond which observations are no longer considered spatially correlated. Lastly, the range is the lag distance where the sill is reached. In other words, it is the maximum distance at which spatial autocorrelation is present. A stereotypical semivariogram is illustrated in Figure 5.