# Lightning Interferometry with the Long Wavelength Array
## 2 Methods
### Deconvolution  
The image computed above is the convolution of the antenna array response (or point spread function) with the true image of the sky, referred to as a dirty image. To recover the source image without the contribution from the antenna response, the dirty image is deconvolved. Unlike computing the dirty image, deconvolution does not have a unique solution, particularly for collections of closely spaced sources or extended sources. Indeed, in some cases, it is better to work with the dirty images directly. Even in cases  
Figure 2: Images of lightning VHF sources using 3, 32, and 249 antennas in the same projection format used in Figure 1a. For each image, the color map is normalized with red showing directions that are positively correlated and blue showing directions that are negatively correlated. **(a–c)** are of a single source near the beginning of the flash. **(d–f)** are also of a single source near the beginning of the flash, but a very weak one. **(g–i)** are of a period with 2 point VHF sources. **(j,k)** are of a period with many VHF sources. Divisions in all panels are 10\({}^{\circ}\). Peak image power for panels **(c,f,i,l)** is 28.8, \(-\)10.4, 29.7, and 3.1 dBGB, respectively. Image time for panels **(c,f,i,l)** is 87.084, 87.792, 277.660, and 255.370 ms after 00:15:15 UT. where deconvolution is desired, there are multiple methods to choose from, each of which performs best in different situations. In previous interferometric studies of lightning, 'deconvolution' was achieved by modeling each integration period as a single radio point source located where the dirty image was brightest. This location was possible to calculate without explicitly computing the image of the source. It also had the benefit of reducing the sequence of images of the lightning sources into a sequence of point locations, or a map, of the flash. Maps in aggregate can appear similar to an image and make the analysis of lightning flashes much easier. The LWA has enough antennas that it is easily capable of imaging multiple VHF sources in a single integration period. Therefore, instead of modeling each integration period as a single VHF point source, it is modeled as multiple VHF point sources. Finding the locations of these sources proceeds iteratively as follows. First, the brightest pixel of the dirty image is found. A sub-pixel location for the VHF point source is then calculated by fitting a paraboloid to the brightest pixel. Then, a Gaussian distribution is centered at the sub-pixel location and subtracted from the dirty image, leaving a residual image. To determine whether the sub-pixel location is likely to be from lightning, the brightest pixel amplitude is compared to the standard deviation of the residual image. If the brightness is more than a factor of 6 greater than the standard deviation of the residual, the sub-pixel location is appended to the map of the lightning flash; this is the order 1 solution. The factor of 6 threshold used here is a noise filter; lower threshold values will have more solutions and also more noise. The above procedure is repeated for increasingly higher-order solutions until no more sources can be identified. The procedure is very similar to the well-known CLEAN algorithm described in [PERSON] [35], except that instead of subtracting off a point spread function, we subtract off a Gaussian. The size of the Gaussian used in this process is related to the angular resolution of the interferometer, \(\theta\sim\lambda/D\), where, for a planar interferometer, \(\theta\) is a measure on the image plane. A close enough estimate of the angular resolution for deconvolution is obtained if \(\lambda\) is taken to be the shortest wavelength and \(D\) is the diameter of the array. For the LWA, the diameter is larger in the north-south direction (111 m) than the east-west direction (89 m), so we use an elliptical Gaussian.