# LCNS Positioning of a Lunar Surface Rover Using a DEM-Based Altitude Constraint
## 4 Covariance Analysis
### Measurement Covariance Modelling
#### 4.5.2 DEM Covariance Model  
The uncertainty of the DEM measurement is governed by three terms:  
1. The data component, which is related to the orbital errors and interpolation;
2. A geologic component, which is related to natural terrain variations (e.g., variation of the terrain within one pixel);
3. A component that is related to the horizontal covariance of the receiver of the rover. When the covariance of the rover is larger than 5 m (i.e., the width of a pixel), the rover \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline DLL Loop bandwidth (\(B_{DLL}\)) & 0.5 Hz \\ FLL Loop bandwidth (\(B_{FLL}\)) & 10 Hz \\ Coherent integration time (\(T_{i}\)) & 20 ms \\ Early-late spacing (\(d\)) & 1 chip \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Configuration settings used for the EKF covariance analysis. can also be located in adjacent pixels, which introduces an additional uncertainty component in the height measurement. The first term (data component) is related to the geolocation uncertainty of the ground track. By iteratively adjusting the LOLA tracks to the LOLA-based DEM in a self-consistent fashion [26], the orbital geolocation errors can be reduced by a factor of 10, such that the ground track geolocation accuracy is 10-20 cm horizontally and 2-4 cm vertically. Furthermore, methods are available [26] to estimate the surface height uncertainty in the DEMs, accounting for the reduced orbital errors and interpolation errors by assuming the fractal behaviour (i.e., new emerging details when a feature is observed more closely) of the small-scale topography. This information is available for each DEM pixel. Figure 6a shows the surface height uncertainty related to the data component (accounting for the reduced orbital errors and interpolation errors). Note that the orbital tracks can be clearly observed, as for these pixels, the uncertainty is low (i.e., low interpolation error). The bright areas in Figure 6a represent areas for which the interpolation error is large. Figure 6b shows the histogram representing the altitude error due to orbital and interpolation errors. Most of the pixels have an error between 0 and 1 m.  
The second term (geological component) is related to natural terrain variations and sub-pixel sampling (i.e., variation of the terrain within one pixel that cannot be captured in the DEM). This error is typically larger than the geolocation uncertainty [26]. To capture the magnitude of this error, detailed sub-pixel information is required (i.e., to detect the presence of rocks and boulders). As this information is not available, the 3-sigma covariance is used in the variance covariance analysis for the DEM measurement (note that the 1-sigma is shown in Equation (17)), accounting implicitly for this uncertainty. The third term is related to the uncertainty of the rover receiver. In an ideal scenario, the receiver uncertainty would be zero, such that the right DEM pixel (and the associated height) could be selected and only data and geological errors remain. However, in a realistic scenario, the uncertainty of the receiver is non-zero, and as such, the selected pixel (based on the current receiver position) may not be correct. Depending on the variation of the terrain and the receiver uncertainty, this error may attain significant values (e.g., at the edge of a cliff). Figure 6: Surface Height Uncertainty (\(\sigma_{Data}\)). (**a**) shows the surface height uncertainty (data from [26]) related to the data component for each pixel of the DEM (the projection is polar stereographic, where (0,0) denotes the South Pole), while (**b**) shows the histogram for all pixels shown in (**a**). Note that the scale of the x and y axes of (**a**) is not equal. Figure 7 shows conceptually how the uncertainty of the rover is used to evaluate the associated uncertainty in the DEM measurement. It is assumed that the rover location (after the EKF propagation step) is somewhere in the pixel denoted with \(Rx\).