# Detection of Cracks in Arch Dams Using Monitoring Data and Numerical Models
## 4 Discussion and Conclusions
### Estimation of Crack Length and Depth  
It was assumed that the crack was detected: \begin{table}
\begin{tabular}{l c c c c} \hline  & \multicolumn{2}{c}{**Level 35 m**} & \multicolumn{2}{c}{**Level 45 m**} \\ \cline{2-5}  & **Error with 0.1** & **Error with 0.02** & **Error with 0.1** & **Error with 0.02** \\ \hline No crack length & \(\infty\) & 0\% & 0\% & 0\% \\ No crack depth & \(\infty\) & 0\% & 0\% & 0\% \\ Crack A length & \(-\)37\% & +65\% & +3\% & \(-\)31\% \\ Crack A depth & \(-\)65\% & \(-\)25\% & \(-\)20\% & +40\% \\ Crack B length & \(-\)100\% & +10\% & +71\% & \(-\)50\% \\ Crack B depth & \(-\)100\% & +20\% & \(-\)40\% & +200\% \\ Crack C length & +1\% & +53\% & +56\% & +54\% \\ Crack C depth & \(-\)18\% & \(-\)40\% & \(-\)31\% & \(-\)33\% \\ Crack D length & \(-\)28\% & +12\% & +30\% & +20\% \\ Crack D depth & \(-\)20\% & \(-\)36\% & \(-\)18\% & \(-\)24\% \\ \hline \end{tabular}
\end{table} 
Table 7: Errors of estimation by the proposed methodology for useful devices with 0.1 or 0.02 accuracy. Any error in the case of length 0 is indicated by an infinite percentage error. \begin{table}
\begin{tabular}{l c c c c} \hline  & \multicolumn{2}{c}{**Level 35 m**} & \multicolumn{2}{c}{**Level 45 m**} \\ \cline{2-5}  & **Error with 0.1** & **Error with 0.02** & **Error with 0.1** & **Error with 0.02** \\ \hline No crack length & \(\infty\) & 0\% & \(\infty\) & 0\% \\ No crack depth & \(\infty\) & 0\% & \(\infty\) & 0\% \\ Crack A length & \(-\)10\% & +63\% & \(-\)38\% & \(-\)31\% \\ Crack A depth & \(-\)55\% & \(-\)25\% & +20\% & +40\% \\ Crack B length & \(-\)100\% & +11\% & +5\% & +48\% \\ Crack B depth & \(-\)100\% & +25\% & +40\% & +200\% \\ Crack C length & \(-\)2\% & +55\% & +54\% & +56\% \\ Crack C depth & \(-\)20\% & \(-\)36\% & \(-\)31\% & \(-\)33\% \\ Crack D length & \(-\)26\% & +10\% & +71\% & +20\% \\ Crack D depth & \(-\)22\% & \(-\)33\% & \(-\)40\% & \(-\)24\% \\ \hline \end{tabular}
\end{table} 
Table 8: Errors of estimation by the proposed methodology for minimum devices with 0.1 or 0.02 accuracy. Any error in the case of length 0 is indicated by an infinite percentage error. * The proposed methodology makes an average error of 41% length and 25% depth using all monitoring points with 0.1 mm precision devices. Eliminating monitoring points with non-relevant information reduces the error length to 33% but increases the depth error to 30%. Although it may seem a big error, the estimation nevertheless gives an order of magnitude of the crack importance, which is useful for the dam safety assessment. * Using 0.02 mm precision devices does not improve the results; the estimation average error becomes 35-37% depth and 52% depth with any number of measuring points. The bigger depth error is caused by the case of crack B with reservoir level 45 m and can be attributed to the fact that, in this case, the random noise applied greatly distorts the most relevant measurements. It can be seen that there is no real improvement in employing the more precise devices because, in approximately half of the cases, it was the lower precision devices that gave an estimate closer to reality. * Applying the alternative methodology only produces a significant improvement for length estimation with 0.02 devices. It can also be seen that when cracks are present, the methodology overestimated the length two out of three times and underestimated the depth three out of four times. So, the tendency is to overestimate the length and underestimate the depth. A clear relationship between estimated values for crack size and actual dimensions can be seen in the results obtained, considering all the monitoring points. A higher real dimension of the crack corresponds to a higher value estimated by the methodology. This relationship has an exception in the case of crack B, with reservoir level 45 m and 0.02 mm accuracy devices. As mentioned before, this exception can be attributed to random noise greatly distorting the most relevant measurements for this case.