# Kalman Filter and Wavelet Cross-Correlation for VHF Broadband Interferometer Lightning Mapping
## 3 Results and discussion
### Simulation Results  
The performance of the lightning mapping estimation was measured based on the number of sliding windows over the time course of the analyzed ITF lightning signals, where each pair of \(Az-El\) angles are correctly estimated relative to the ground-truth simulated lightning map. The Euclidean distance between the estimated and the actual-simulated lightning map was used to measure the estimation error,  
\[dist=\frac{1}{W_{n}}\sum_{i=1}^{W_{n}}\sqrt{\left(Az_{i}-\hat{Az}_{i}\right)^{2 }+\left(El_{i}-\hat{El}_{i}\right)^{2}} \tag{9}\]  
where \(W_{n}\) is the total number of sliding windows in the processed lightning data. To summarize the performance of the simulated dataset overall, we compute the average of Euclidean distances.  
Table 2 summarizes the performance comparisons of the suggested lightning mapping estimation methods. In Table 2, The Euclidean distances were calculated for each simulated ITF datum and then averaged over the entire available augmented lightning dataset to show the overall benchmarking performance of the suggested approaches. We compare three main filtering approaches listed in the first column of Table 2. It can be seen that, when using the CCTD method, the KF and BPF show the lowest Euclidean distances of 3.55\({}^{\circ}\) and 3.63\({}^{\circ}\) respectively. The linear-interpolation factor used to achieve these results was 8. The WT denoising also shows relatively low distances of 3.94\({}^{\circ}\) when using the Fejer-Korovkin function with cubic-interpolation of factor 8.  
For the CCFD method, the best performance was achieved using the Kalman filter without interpolation with a distance of 3.59\({}^{\circ}\), while the BPF and wavelet denoising performed comparably to that of the CCTD method showing minimum distances of 3.76\({}^{\circ}\) and 3.94\({}^{\circ}\), respectively. On the other hand, using CCWD, the WT denoising shows the best performance with a distance of 3.46\({}^{\circ}\) using the Symlet function without any interpolations. BPF and KF also provide comparable results to that of the WT with 3.76\({}^{\circ}\) and 3.47\({}^{\circ}\).  
Generally, from Table 2, the filtering methods do not show significant enhancements in the performance when increasing the cross-correlation interpolation factor due to the relatively small window size adapted from [PERSON] et al. (2018). The introduced wavelet filtering shows better performance than that of the BPF. However, the variation of wavelet functions and the thresholding methods show almost similar performances. On the other hand, the KF outperforms both the BPF and the WT in general. This is because KF can recursively estimate the noise regardless of the frequency contents.  
In comparing the cross-correlation techniques, the CCWD method outperforms the other methods in all cases. This is due to the inherent capability of wavelet transforms to enhance the signals further and reduce the noise effects, providing a better time-domain resolution and reliable estimation of lighting maps. Moreover, comparing several wavelet functions in WT denoising helps to identify the best combination of WT parameters that fits the analyzed signals. A noticeable enhancement of cross-correlation methods was found introducing cross-correlation in the wavelet domain for estimating the lightning maps.