# Analysis of the Relationship between Scintillation Parameters, Multipath and ROTI
## 2 Materials and Methods
### Variograms  
Geostatistics provides a set of tools and methods to analyze spatially referenced data. Of particular interest is a tool known as the sample variogram, which is defined as:  
\[\hat{\gamma}(h)=\frac{1}{n(h)}\ \sum_{k=1}^{n(h)}(y(u_{k})-y(u_{k}+h))^{2} \tag{21}\]  
where \(\hat{\gamma}(h)\) is the semi-variance for two points separated by distance \(h\), \(y(u)\) is the attribute value at location \(u\) and \(n(h)\) is the number of observations separated by distance \(h\). The variogram provides information about the spatial dependence and spatial structure in the data. When \(\hat{\gamma}(h)\) is small, the difference between two observations of \(y\), separated by \(h\) is expected to be small. When \(\hat{\gamma}(h)\) is large, the difference is expected to be large. An example variogram for field observations of the heavy metal, Cadmium, is shown in Figure 2.  
The solid line in Figure 2 is a model that has been fitted to the sample variogram. This allows us to estimate \(\gamma(h)\) for all values of \(h\). The model is parameterized by the sill, nugget and range. The variogram still represents the maximum variability in the data. The nugget represents a combination of non-spatial variability and micro-scale variability (variability at lags less than \(\min(h)\)). The range is the limit of spatial correlation. We expect two observations separated by distances larger than the range to be uncorrelated. A flat variogram (commonly referred to as \"pure nugget\") indicates that there is no spatial structure in the data.  
Figure 2: Illustration of the sample variogram (black points) and modelled variogram (black line). This is for an example dataset of heavy metals from the River Meuse in the Netherlands ([PERSON], 2004).  
The concept behind the variogram can be extended to consider the spatial cross-correlation between two variables. This allows us to investigate how correlated \(y_{1}(u)\) is with \(y_{2}(u+h)\). The sample cross-variogram is given as:  
\[\gamma_{yz}(h)=\frac{1}{n(h)}\ \sum_{i=1}^{n(h)}(y_{1}(u_{k})-y_{1}(u_{k}+h))(y_{2 }(u_{k})-y_{2}(u_{k}+h)) \tag{22}\]  
The sample variogram and modelled variogram can be used as exploratory tools to investigate the spatial structure in a dataset. They can also be used in predictive mapping to interpolate datasets. The concepts are further explained in detail by [PERSON] et al. (2013); [PERSON] and [PERSON] (2014) and [PERSON] et al. (2015). [PERSON] (2016) gives a more detailed theoretical treatment. Examples in remote sensing are given by [PERSON] and [PERSON] (2017); [PERSON] (2018) and [PERSON] et al. (2019).  
Here we use the variogram and cross-variogram to investigate the spatial structure in the four variables (MP, ROTI, S4, \(\sigma_{\varphi}\)) and to investigate the cross-correlation between the four variables. In all calculations we used degrees as the unit of distance. We acknowledge that it is preferred to work with a projected coordinate system. However, we defend this choice because we work with reasonably small distances and use the variogram for data exploration rather than for prediction.