## 4 Results and Validation
### Quality Assessment and Discussion
Of course, the results will improve when more data points are used for the fitting process. However, then the validation is no longer done by independent input data. When comparing the cross-validation results with the internal errors derived in the AWAX approach (provided in Table 5) a very good agreement can be seen. For Lake Ray Roberts, the area error derived from the AWAX approach is 2.18 km\({}^{2}\) (1.59%) where the cross-validation yields a RMS of 3.04 km\({}^{2}\) (2.21%) for the linear fit, respectively 2.85 km\({}^{2}\) (2.07%) for the polynomial fit. Another example is Poco da Cruz, Lake where the linear and polynomial fit behaves differently. Here, the cross-validation shows an RMS of 3.53 km\({}^{2}\) (6.07%) for the linear fit, respectively 1.10 km\({}^{2}\) (1.90%) for the polynomial fit. The surface area error derived from the AWAX approach is 3.23 km\({}^{2}\) (5.56%) which is within both results of the cross-validation. The external validation thus confirms the order of magnitude of the internal AWAX errors and proves their high reliability. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Target Name, Country (ID)** & \multicolumn{6}{c}{**Hypository (Linear)**} & \multicolumn{2}{c}{**Hypository (2 nd Deg.)**} \\ \cline{2-7}  & **Model Points [\#]** & **Validation Points [\#]** & **RMS [km\({}^{2}\)]** & **wrt. Area [\%]** & **RMS [km\({}^{2}\)]** & **wrt. Area [\%]** \\ \hline Aragon, Spain (10297) & 14 & 111 & 1.14 & 5.16 & 1.14 & 5.16 \\ Bankim, Cameroon (3560) & 9 & 71 & 21.30 & 6.33 & 19.26 & 5.72 \\ Boston, China (226) & 13 & 102 & 18.70 & 1.52 & 18.20 & 1.48 \\ Chalborne, USA (10472) & 23 & 189 & 0.65 & 2.53 & 1.07 & 4.20 \\ Clear, USA (10496) & 16 & 135 & 3.92 & 7.85 & 3.74 & 7.48 \\ Cooper, USA (10505) & 19 & 153 & 2.03 & 2.48 & 1.94 & 2.35 \\ Doyge, China (10425) & 7 & 60 & 3.54 & 1.08 & 3.68 & 1.13 \\ Enriquilho, Dom. Rep. (11521) & 11 & 84 & 13.97 & 3.94 & 11.59 & 3.27 \\ Euclumpe, Australia (64) & 11 & 84 & 2.58 & 1.83 & 2.67 & 1.90 \\ Fairfield, USA (6672) & 11 & 85 & 0.28 & 2.85 & 0.28 & 2.88 \\ Forgeon, Germany (10341) & 19 & 160 & 0.83 & 4.94 & 0.84 & 5.02 \\ Jaceari, Brazil (10345) & 8 & 57 & 1.56 & 3.17 & 1.75 & 3.55 \\ Jennipenapore, Brazil (5881) & 10 & 72 & 0.26 & 2.33 & 0.26 & 2.33 \\ Lagdo, Cameroon (1472) & 11 & 90 & 39.19 & 5.16 & 40.07 & 5.28 \\ Lewisville, USA (11327) & 28 & 237 & 2.85 & 2.19 & 2.42 & 1.86 \\ Massigni, Mozambique (606) & 8 & 56 & 6.30 & 3.33 & 4.03 & 2.13 \\ Mead, USA (204) & 16 & 129 & 5.21 & 0.87 & 3.97 & 0.66 \\ Murray, USA (8852) & 31 & 266 & 2.64 & 1.34 & 2.34 & 1.19 \\ Nova Pune, Brazil (10351) & 20 & 169 & 6.07 & 1.52 & 5.47 & 1.37 \\ Orellana, Spain (11415) & 7 & 50 & 1.26 & 2.39 & 1.26 & 2.38 \\ [PERSON], Brazil (8692) & 7 & 52 & 0.80 & 6.89 & 0.51 & 4.40 \\ [PERSON], Brazil (8671) & 7 & 60 & 5.45 & 6.41 & 6.80 & 7.99 \\ Poco da Cruz, Brazil (8702) & 5 & 39 & 3.53 & 6.07 & 1.10 & 1.90 \\ [PERSON], USA (10146) & 30 & 255 & 2.88 & 2.18 & 2.74 & 2.07 \\ [PERSON], USA (8814) & 23 & 198 & 2.34 & 1.28 & 2.19 & 1.20 \\ Salton Sea, USA (71) & 34 & 292 & 4.58 & 0.46 & 4.52 & 0.46 \\ [PERSON], USA (10246) & 26 & 220 & 10.07 & 2.25 & 5.96 & 1.33 \\ Tavakori, USA (8813) & 22 & 185 & 2.74 & 1.70 & 2.36 & 1.47 \\ Tharthar, Iraq (122) & 27 & 232 & 20.11 & 1.59 & 16.71 & 1.21 \\ Toledo Bend, USA (10247) & 15 & 122 & 10.48 & 0.81 & 7.96 & 0.67 \\ Tundes Grandes, Argentina (4475) & 12 & 93 & 8.43 & 2.55 & 8.11 & 2.45 \\ Zujar, Spain (10301) & 10 & 77 & 5.75 & 3.74 & 5.75 & 3.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the cross-validation of 32 selected study areas. For each study area, the used number of points for the calculation of the hypsometry and the number of points for the validation are shown. Furthermore, the resulting RMS and relative RMS with respect to the maximum surface area using a linear and polynom of degree 2 are presented.