# Lightning Interferometry with the Long Wavelength Array
## 2 Methods
### Projection Imaging  
Once the time series is generated, the data can then be used to image the VHF lightning source using the imaging techniques applied in aperture synthesis (although the actual synthesis part will be less effective since the Earth rotates a negligible amount during a lightning flash). Imaging techniques have been used in previous lightning interferometric studies [32; 33; 34], but the methods used are not well documented and differ somewhat from standard imaging techniques [18]. Specifically, the image is formed by projecting the cross-correlation of antenna pairs onto the image plane, instead of aggregating the complex visibility observations in the diffraction plane and transforming back to the image. The study herein uses the same imaging technique as the previous lightning interferometry works, here referred to as 'projection imaging'. The connection between projection imaging and more traditional interferometry methods has not been well established, so below we provide a brief formalization showing that projection imaging is mathematically equivalent to aperture synthesis. The image of a source in two dimensions is given by inverting Equation (1-6) of [PERSON] et al. [18]:  
\[\frac{I(l,m)}{\sqrt{1-l^{2}-m^{2}}}=\int\int V(u,v)e^{2\pi i(ul+vm)}dudv \tag{1}\]  
Here, \(I\) is the image of the incident radiation power at the \(l\), \(m\) coordinates of the image plane (which are simply the direction cosines \(\alpha\) and \(\beta\)), and \(I_{m}=I/\sqrt{1-l^{2}-m^{2}}\) is the modified intensity. In this study, as in previous studies, we will ignore the \(\sqrt{1-l^{2}-m^{2}}\) term and instead solve for the modified intensity. \(V\) is the complex visibility at the \(u\), \(v\) coordinates of the diffraction plane. In terms of the baseline, \(b\), and the observed radio wavelength, \(\lambda\), the diffraction plane coordinates are \(u=b_{x}/\lambda\) and \(v=b_{y}/\lambda\). For practical radio interferometer arrays, the aperture is not completely sampled because there is a finite number of antenna elements, leading to the discrete form of the imaging equation:  
\[I_{m}(l,m)=\frac{1}{N}\sum_{u}\sum_{v}V(u,v)e^{2\pi i(ul+vm)}\;. \tag{2}\]  
where \(N\) is the number of \(u,v\) coordinates that the sum is computed over (this is very similar to Equation (7-3) of [PERSON] et al. [18]). From these equations, it is evident that the image of the source is a two-dimensional Fourier transform of the complex visibility \(V\). The complex visibility is measured by correlating the signals arriving at pairs of antennas. For each pair of antennas and each wavelength, the complex visibility is sampled at a single \(u,v\) coordinate. For digital interferometers such as the LWA, \(V\) can be found for all frequencies using a Fourier transform of the input signal:  
\[V_{ij}=\mathcal{F}(E_{i})\mathcal{F}(E_{j})^{*} \tag{3}\]  
where \(E_{i}\) and \(E_{j}\) are the electric field signals incident on the \(i\)th and \(j\)th antennas, respectively. The image can then be obtained by computing the sums in Equation (2) for every \(l,m\) coordinate of interest by looping over the measured complex visibilities. These sums are commonly computed using a two-dimensional discrete inverse Fourier transform. Because the diffraction plane is sampled sparsely, the resulting image is the true image of the source convolved with the point-spread function determined by the antenna configuration. Alternatively, and equivalently, we can re-order the sums to express the measured complex visibility in terms of frequency:  
\[I_{m}(l,m)=\frac{1}{N_{b}}\sum_{ij}\frac{1}{N_{f}}\sum_{f}V_{ij}(f)e^{2\pi i( u_{f}l+v_{f}m)}\;, \tag{4}\]  
where \(ij\) indicates each antenna pair, and \(f\) is the frequency; similarly, \(N_{b}\) is the number of baselines, and \(N_{f}\) is the number of frequency components used in the Fourier transform.