# Lightning Interferometry with the Long Wavelength Array
## 2 Methods
### Deconvolution
For a small range of separations, when the sources are of the same brightness, deconvolution will result in 3 solutions even though only 2 sources were simulated. The order 1 source will be found between the 2 simulated sources, with an order 2 and order 3 source on either side of it in a line. An example of this is shown in Figure 3b. As the source separation increases, the error in the order 2 source very quickly matches that of the order 1 source. When \(\sigma=\lambda/D\), the sources have to be separated by a larger distance before an order 2 source can be found. This also results in no situation in the simulation where more than 2 locations are produced by the deconvolution algorithm. However, while the error of the order 2 source still approaches that of the order 1 source, it does so more gradually. From Figure 3, it may appear that using \(\sigma=\lambda/\sqrt{2}D\) in the deconvolution will produce more accurate results than \(\sigma=\lambda/D\). When the signals are imaged with 249 antennas, this can be true; however, the situation degrades when the number of antennas used is reduced. In these cases, the side lobes of the dirty image are higher, and the deconvolution algorithm is more likely to misidentify a side lobe as a true source when a smaller Gaussian is used. An example of this issue can be seen in Figure 2b,h, where the side lobes of the image resemble point sources. As a result, this study uses \(\sigma=\lambda/D\) for the deconvolution Gaussian. The deconvolution method described above is applied to the example flash used in this study, and the resulting map is shown in Figure 1. In this example, the deconvolution method is applied to images produced using 1000 sample (4882 ns) integration periods. Figure 1a shows the map in the LWA's native image plane colored by time. This is a two-dimensional, angular depiction of the flash, where the zenith is at the origin and the horizon falls on a unit circle. The axes of the image plane are the direction cosines, which means that the distances measured on the image plane are unitless. In this projection, the angular resolution of the LWA is constant, but the distances between sources do not correspond to a constant angular distance; the azimuthal angle is compressed near the zenith, and the elevation angle is compressed near the horizon. Moreover, 10\({}^{\circ}\) increments in the azimuthal and elevation angles are indicated by the division lines. Figure 1b shows an angular elevation-time projection of the same map using the same coloring. The map depicts a fairly typical normal (positive) polarity intra-cloud flash. The positive and negative channels of the flash overlap in areas but are fairly easy to distinguish, with the negative leaders being fuzzier and generally at higher elevation angles [8; 36]. The lightning flash was also located by the ENTLN approximately 7.6 km to the south east. The blue dots in Figure 1a show the directions to the ENTLN-located pulses, and Figure 1d shows when they happened during the flash. The ENTLN detected, but did not locate, the initial breakdown pulses. Overall, the locations and distances of the located ENTLN pulses are very consistent with the LWA map and indicate that the lightning channels are approximately 6200 m above the ground (7600 m above sea level). Figure 3: Depiction of the effects of deconvolution determined through simulation. **(a)** is a dirty image of two simulated sources separated by 0.042 on the image plane (3.4\({}^{\circ}\) at 45\({}^{\circ}\) elevation). The black \(+\) marks the true location of the simulated sources. **(b)** is the residual image after deconvolving with a Gaussian whose standard deviation \(\sigma=\lambda/\sqrt{2}D\). The red \(\times\) show the locations determined by the deconvolution algorithm. **(c)** is the residual image after deconvolving the simulated dirty image with a Gaussian of standard deviation \(\sigma=\lambda/D\). The red \(\times\) show the locations determined by the deconvolution algorithm.