# Lightning Interferometry with the Long Wavelength Array
## 2 Methods
### Projection Imaging
The sum over \(f\) is simply the cross-correlation of the signals arriving at the \(i\) and \(j\) antennas, rotated in the direction indicated by \(u_{f}\), \(v_{f}\), so we can write Equation (4) as  
\[I_{m}(l,m)=\frac{1}{N_{b}}\sum_{ij}X_{ij}(\tau_{lm})\, \tag{5}\]  
where \(X_{ij}\) is the cross-correlation between the \(i\) and \(j\) antennas, and \(\tau_{lm}\) is the time delay for signals arriving from the \(l,m\) direction. Conceptually, the way that this imaging method works is that the cross-correlation from each baseline is projected onto the image plane, and they are summed together. This works for any sky geometry in which there is a unique \(\tau\) for each baseline for every direction in the sky; the cosine projection used in this paper is especially convenient because the transform is linear. From Equation (5), it is evident that the amplitude of the image is the VHF energy arriving at the array (or the power if we normalize by the integration period). In general,  
\[I_{max}\leq\frac{1}{N_{a}}\sum_{i}E_{i}^{2} \tag{6}\]  
where \(N_{a}\) is the number of antennas. Equality will happen when all of the VHF signal is due to a single point source in the center of the beam and in the absence of any noise. Figure 2 shows the results of imaging various different lightning signals with 3, 32, and 249 antennas. How the antennas were selected, and why some antennas were omitted, is discussed in Section 3. All images were created using a 100 sample (488 ns) integration window with the beam steered at the center of the image. Panels a-c show the results of imaging a single point source that emerged close to the beginning of the flash. At this point in time, the flash had not grown enough for multiple sources to be likely. The location of the source is quite evident for each of the 3, 32, and 249 antenna images, although the side lobes decrease as the number of antennas increases. When 249 antennas are used to image the source, more short baselines are included in the array, causing the size of the primary lobe to be somewhat larger than the image produced with only 32 antennas. At the same time, including these short baselines also serves to reduce the side lobes of the image. Many of the integration periods contain only a single source, but, even for single sources, the sensitivity is improved when more antennas are used to compute the image. This is shown in Figure 2d-f, where the image is still a single point source near the beginning of the flash but is less bright than the galactic background. In these cases, it is impossible to even identify that a source is present when imaging with only 3 antennas. The image is clearly improved but still noisy when computed with 32 antennas, and it is quite good when imaged with 249 antennas. Figure 2g-i shows a case with two point sources of similar brightness. In the 3-antenna image, we can see that a source exists, but it is difficult to determine exactly where. If one of the sources were substantially brighter than the other, the 3-antenna case would start to resemble what is seen in panel a. With 32 antennas, both point sources can be clearly identified, but the side lobes are increased in comparison to panel b. Figure 2j-l shows a case with many VHF sources. Again, it is impossible to determine whether there is a source at all in the 3-antenna image, although the signal amplitude suggests that there may be. The situation is improved in the 32-antenna image, but not significantly. The confusion added by the side lobes of the image makes it very difficult to identify where the sources are. Even in the 249-antenna image, the side lobes are quite high and not all sources are clearly distinct from one another.