374 A.G. Woolard et al. Velocity (inches/sec) 0 500 1000 1500 2000 2500 3000 RMSE (inches) 0 100 200 300 400 500 600 700 Cross correlation, Config. 1 Cross correlation, Config. 2 Peak-difference, Config. 1 Peak-difference, Config. 2 Fig. 36.7 RMSE over a 3000 in./s velocity range for two TDOA methods and two sensor configurations The RMSE results are shown in Fig. 36.7; however, they are not a good measure of the accuracy of the TDOAs due to indiscriminate outliers. For example: the RMSEs for cross-correlation in the second configuration are orders of magnitude above the first configuration because of one particular impact case. Since the reliance of any localization method is of considerable importance, all outliers are retained. Instead of assessing accuracy based on RMSE, the coefficient of variation (COV) is considered for the perceived propagation velocity, along with the number of SOA errors. The COV represents the consistency of the perceived propagation velocity, i.e.: the ability to scale the time-differences by a single wave speed and get accurate results, while the number of SOA errors represents the ability to accurately discern a wave source as originating from one side or another for any sensor pair. Considering the RMSE over the entire velocity range, there are several drops in the error as the velocity increases. These drops are either associated with wave speeds that dominate that particular response, or sporadic events that occur due to the nonlinear nature of hyperbolic localization. An example of this is shown in Fig. 36.8 for a particularly good localization case (COV 0.081), where we see RMSE values below 2 in. for a wave speed of 807 in./s. The low COV accurately suggests the existence of a particular wave speed that provides accurate results. The drops in RMSE above the ideal wave speed are inconsistent, and can be recognized by the gradient leading up to a particular drop off. Several other cases are included for comparison, showing inconsistency in which wave speed produces the minimum RMSE. Examining the two impacts at (9.26, 5.85) and (12.2, 8.78), the error of the calculated TDOAs are so large that no wave speed solution is better than assuming the middle of the plate as the source. The percent error for SOAs represents the percent of SOAs that are reported incorrectly out of 126 (6 sensor pairs, 21 impacts), and is a good measure of the ability to isolate the source to an area by constructing a grid of sensor pairs. For the cross-correlation method in the first configuration, there is a COV of 0.79, with 7.1 % error of SOAs. In the second configuration, the COV increases to 2.81 (highly variable wave speed), and the SOAs error also increases to 8.7 %. This suggests that the increase in reflections from the boundaries in the second configuration increases the variance in the perceived wave speed and reduces the ability to isolate the source to an area from the SOAs. For the peak-difference method in the first configuration, there is a COV of 0.74, and SOAs error of 4.0 %, while the second configuration has a COV of 4.68 with 9.5 % SOAs error. This shows a similar trend as the cross-correlation method between the first and second configurations: that the increase in reflections increases the COV and SOAs error. A noticeable difference in the peak-difference method, however, is the magnitude that the COV and SOAs increase between the first and second configurations, suggesting that the accuracy of the peak-difference method may be more negatively affected when reflections are significantly large. For both cases, moving the sensors closer to the boundaries decreases the overall accuracy. When the sensors are very near to the boundaries and reflections are large, it appears that the cross-correlation method may perform better than peakdifference. This could be the result of the symmetry of the structure since all sensors are symmetric at the boundaries, the
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