# Lightning Interferometry with the Long Wavelength Array
## 3 Results
### Multi-Source Detections  
Figure 5a shows how the fraction of windows with identifiable VHF point sources changes as a function of the number of antennas used and the solution order. Solution order here refers to the order in which the solutions were identified during deconvolution. The order 1 solution is the brightest source in the integration window, and the higher-order solutions are increasingly fainter. For this survey, the integration period was held constant at 100 samples (488 ns), while the number of antennas was changed between runs.  
The number of windows with at least one solution in them increases rapidly as the number of antennas increases. By the time 32 antennas are used to image the source, essentially all integration windows imaged have at least one source. The three-antenna case shown in Figure 5a is lower than expected based on the numerous previous studies conducted using three antenna arrays. The reason for this is that the LWA can only record 39.6 MHz of the VHF band, and this bandwidth is shifted towards the higher end of the 3-88 MHz sensing bandwidth to maximize the angular resolution. This results in the side lobes of the images being quite high (as seen in Figure 2) and dirty images that may not meet the 6\(\sigma\) deconvolution requirements specified in Section 2.3.  
The number of higher-order solutions increases more gradually as more antennas are used to image the source. When 249 antennas are used to produce the image, 51% of the windows contain more than one VHF point source. A survey was also run to investigate the dependence of this fraction on the integration time. When the integration time was reduced from 100 samples (488 ns) to 50 samples (244 ns), the results were nearly identical. Integration periods shorter than this were not tested because a single beam steering would not fully cover the entire flash. The fraction of windows containing higher-order solutions does increase as the integration period grows longer, but it increases more slowly than expected, likely due to the limited ability of the deconvolution algorithm to detect multiple point solutions.  
Some of the point source solutions in the maps are due to noise or side lobes, and a reasonable question is whether higher-order solutions are largely due to noise. Solutions due to noise and signals can be differentiated because the noise solutions are generally  
Figure 5: Effects of the number of antennas on the sensitivity of the resulting map of lightning VHF sources. (**a**) shows the fraction of integration periods (windows) for which a point solution can be found, with solution order indicating at least how many point locations were located. (**b**) shows the probability density function of received power of each source for several antenna configurations.  
scattered in space, whereas the real solutions are tightly clustered on channels. To compute the false alarm rate, differentiation was implemented by simply measuring the distance from each solution to all other solutions; if the 10 th nearest other solution was farther than 0.02 on the image plane (1.6\({}^{\circ}\) at 45\({}^{\circ}\) elevation angle), the solution was identified as noise. For the 249-antenna case, less than 1% of the order 1 point locations were due to noise or side lobes. This increased to 3-5% for higher-order solutions, depending on the integration period.