% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}

\graphicspath{
	{.}
	{../../figures/}
	{../../../figures/}
}

\begin{document}
\chapter{Air Shower Radio Interferometry}
\label{sec:interferometry}
The radio signals emitted by the air shower (see Section~\ref{sec:airshowers}) can be recorded by radio antennas.
An array of radio antennas can be used as an interferometer.
Therefore, air showers can be analysed using radio interferometry.
\\
%
Unlike, astronomical interferometry, the source of the signal is closeby.



\begin{figure}
	\centering
	\includegraphics[width=0.5\textwidth]{radio_interferometry/rit_schematic_true.pdf}%
%	\includegraphics[width=0.5\textwidth]{radio_interferometry/Schematic_RIT_extracted.png}
%	\caption{From H. Schoorlemmer}
\end{figure}

\begin{equation}\label{eq:propagation_delay}%<<<
	\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
\end{equation}%>>>


\begin{equation}\label{eq:interferometric_sum}%<<<
	S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
\end{equation}%>>>


\begin{figure}
	\centering
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
		\label{fig:trace_overlap:bad}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
		\label{fig:trace_overlap:medium}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
		\label{fig:trace_overlap:best}
	\end{subfigure}
	\caption{
		Trace overlap due to wrong positions
	}
	\label{fig:trace_overlap}
\end{figure}



\begin{figure}
	\centering
	\includegraphics[width=0.7\textwidth]{2006.10348/fig03_b.png}%
	\caption{
		From \protect \cite{Schoorlemmer:2020low}.
		$\Xmax$ resolution as a function of detector-to-detector synchronisation.
	}
	\label{fig:xmax_synchronise}
\end{figure}

\section{Time Synchronisation}
\label{sec:timesynchro}
The main method of synchronising multiple stations is by employing a \gls{GNSS}.
This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
\\

Need reference system with better accuracy to constrain current mechanism (Figure~\ref{fig:reference-clock}).

%\begin{figure}
%	\centering
%	\includegraphics[width=0.5\textwidth]{clocks/reference-clock.pdf}
%	\caption{
%		Using a reference clock to compare two other clocks.
%		\protect \todo{
%			redo figure with less margins,
%			remove spines,
%			rotate labels
%		}
%	}
%	\label{fig:reference-clock}
%\end{figure}
\end{document}