# Lightning Interferometry with the Long Wavelength Array
## 2 Methods
### Deconvolution
If the standard deviation of the Gaussian is equal to the angular resolution of the array (\(\sigma=\lambda/D\)), the Gaussian will be slightly larger than the central lobe of the point spread function. An estimate that is closer to equality is \(\sigma=\lambda/\sqrt{2}D\). Figure 3a-f show a graphical example of deconvolution acting on simulated sources using both of these Gaussians (details of the simulation are provided below). When the simulated sources are far enough apart (as in Figure 3d-f), deconvolution results in accurate locations for both sources. When the simulated sources are closer together than the angular resolution of the interferometer (as in Figure 3a-c), their dirty images merge and artifacts can occur. The situation shown in Figure 3a is close to the worst-case scenario, where the sources have the same amplitude, causing the maximum brightness of the dirty image to occur in between them. In this situation, even using a single point source model for the source results in significant errors, and errors of this type have been discussed in previous lightning interferometer studies [10; 11; 24]. To test the errors that might result from the deconvolution method, a simulation with two point source emitters is developed. Each point source is modeled as long-duration Gaussian random noise source, to which a randomized filter is applied to soften the spectra. The first source is randomly positioned on the sky above 45\({}^{\circ}\) elevation; the second source is positioned at a specified distance but with random bearing from the first. In addition to the 2 point sources, 50 additional low-power point noise sources are modeled in the same way but distributed randomly on the sky. These noise sources are included because only a small fraction of the noise recorded at the real antennas is uncorrelated. Simulated antennas can then be configured in the same geometry as the LWA. The delay from each source (both signal and noise) to each antenna is then computed and applied to find the received signal at each simulated antenna. Next, uncorrelated Gaussian noise is added to the signal at each antenna. Finally, the effects of sampling are applied: the signal at each antenna is shifted in time by a random amount, and the signals are converted into 8-bit integers in the frequency domain. The power ratio of signal to correlated noise to uncorrelated noise in the simulation is 1:0.5:0.01; the standard deviation of the time jitter is 2% of a sample period (approximately 0.1 ns). The distance between the simulated sources is varied, and the resulting signals are then run through the imaging and deconvolution pipeline. During imaging, a 200 sample (0.97 \(\upmu\)s) integration time is used. The aggregate results from 50 trials for each distance are shown in Figure 3g,h using \(\sigma=\lambda/\sqrt{2}D\) and \(\sigma=\lambda/D\) for the Gaussian, respectively. The blue line in each figure shows the location error in the order 1 solution. This first solution is the same solution that we would obtain if a single point source model were used; as such, the error is independent of the specific Gaussian chosen for deconvolution. The orange line shows the error for the order 2 solution; here, the size of the Gaussian used for deconvolution does affect the error. In both cases, when the separation between the sources is small enough, no order 2 solution is found; this can be seen in the figure by the absence of the orange line and the mean number of solutions (grey dashed line) falling to 1. Similarly, when the sources are separated by a distance much larger than both the angular resolution of the interferometer and the standard deviation of the Gaussian, the location of both sources is determined with good accuracy. In between these two extremes, the specific Gaussian used does have an effect. When \(\sigma=\lambda/\sqrt{2}D\), the error for closely separated sources can be quite large, but, as can be seen by the mean number of sources found, the number of these is low.