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Thesis:Radio Interferometry: incorporated feedback

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documents/thesis/chapters/radio_interferometry.tex
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@ -1,6 +1,11 @@
% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}
%%%%%
%%%%%
%%%%%
\graphicspath{
{.}
{../../figures/}
@ -17,39 +22,46 @@ Note that since the radio waves are mainly caused by processes involving electro
\\
In \cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using interferometry.%
\footnote{
Available as a python package at \url{gitlab}.
Available as a python package at \url{gitlab}\Todo{url}.
}
As shown in Figure~\ref{fig:radio_air_shower}, the shower axis and particle densities along that axis can be observed.
Figure~\ref{fig:radio_air_shower} shows a power mapping of a simulated air shower.
It reveals the air shower in one vertical and three horizontal slices.
Analysing this mapping, the shower axis and particle densities can be computed.
From these, the energy, composition and direction of the cosmic particle can be derived.
\\
The accuracy of the technique is primarily dependent on the timing accuracy of the detectors.
In Figure~\ref{fig:xmax_synchronise}, the estimated atmospheric depth resolution as a function of detector synchronisation is shown as simulated for different inclinations of the air shower.
According to Figure~\ref{fig:xmax_synchronise}, to be able to distinguish the iron and proton showers from Figure~\ref{fig:airshower_depth} ($\Delta\Xmax \sim 40\;\mathrm{g/cm^2}$), we need a synchronisation better than $2\ns$.
For detector synchronisations above $1\ns$, the atmospheric depth resolution is degrading rapidly.
\\
Note that the values in Figure~\ref{fig:xmax_synchronise} are particular to the simulation setup of \cite{Schoorlemmer:2020low}.
Generally, this will depend on the antenna density of the array.
\\
\begin{figure}
\centering
\begin{subfigure}[t]{0.47\textwidth}
\begin{minipage}{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{2006.10348/fig01.no_title}%
\caption{
\captionof{figure}{
From \protect \cite{Schoorlemmer:2020low}.
Radio interferometric power analysis of an \gls{EAS}.
\protect \Todo{describe and expand caption, remove title}
Radio interferometric power analysis of a simulated air shower.
\textit{a)} shows the normalised power of $S(\vec{x})$ mapped onto a vertical plane.
while \textit{b)}, \textit{c)} and \textit{d)} show the horizontal slices on different heights.
On \textit{b)}, \textit{c)} and \textit{d)}, the orange and blue dot indicate the true shower axis and the maximum power respectively.
}
\label{fig:radio_air_shower}
\end{subfigure}
\end{minipage}
\hfill
\begin{subfigure}[t]{0.47\textwidth}
\begin{minipage}{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{2006.10348/fig03_b}%
\caption{
\captionof{figure}{
From \protect \cite{Schoorlemmer:2020low}.
$\Xmax$ resolution as a function of detector-to-detector synchronisation.
A typical noise (gaussian) background is simulated.
\protect \Todo{describe and expand}
}
\label{fig:xmax_synchronise}
\end{subfigure}
\end{minipage}
\end{figure}
\section{Radio Interferometry}
@ -57,7 +69,7 @@ According to Figure~\ref{fig:xmax_synchronise}, to be able to distinguish the ir
Radio interferometry exploits the coherence of wave phenomena.
\\
In a radio array, each radio antenna records its ambient electric field.
A simple interferometer can be achieved by summing the recorded waveforms $S_i$ with appropriate time delays $\Delta_i(\vec{x})$ to compute a coherent\Todo{word} waveform for a location $\vec{x}$,
A simple interferometer can be achieved by summing the recorded waveforms $S_i$ with appropriate time delays $\Delta_i(\vec{x})$ to compute the coherency of a waveform at $\vec{x}$,
\begin{equation}\label{eq:interferometric_sum}%<<<
\phantom{.}
S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
@ -69,8 +81,7 @@ The time delays $\Delta_i(\vec{x})$ are dependent on the finite speed of the rad
Being an electromagnetic wave, the instantaneous velocity $v$ depends solely on the refractive~index~$n$ of the medium as $v = \frac{c}{n}$.
In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through, and the frequencies of the signal.
\\
In many cases, the refractive index can be taken constant over the trajectory to simplify models.
As such, the time delay due to propagation can be written as
The time delay due to propagation can be written as
\begin{equation}\label{eq:propagation_delay}%<<<
\phantom{,}
\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_\mathrm{eff}
@ -80,30 +91,32 @@ where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of
\\
% time delays: particular per antenna
Note that unlike in astronomical interferometry, the source of the signal is not in the far-field (see Figure~\ref{fig:rit_schematic}).
Thus, instead of introducing a geometric phase, this requires us to compute the time delays for each antenna location separately.
This requires us to compute the time delays for each test location $\vec{x}$ separately.
\\
% Features in S
Features in the combined waveform $S(\vec{x})$ are enhanced according to the coherence of that feature in the recorded waveforms with respect to the time delays.
Features in the summed waveform $S(\vec{x})$ are enhanced according\Todo{word} to the coherence of that feature in the recorded waveforms with respect to the time delays.
\\
Figures~\ref{fig:trace_overlap:best} and~\ref{fig:trace_overlap:bad} show examples of this effect for the same recorded waveforms.
At the true source location, the recorded waveforms are aligned.
The combined waveform therefore shows the
Meanwhile, at a far away location, the waveforms add up incoherently resulting in a low amplitude combined waveform.
At the true source location, the recorded waveforms align and sum coherently to result in a summed waveform with enhanced features and amplitudes.
Meanwhile, at a far away location, the waveforms sum incoherently resulting in a summed waveform with low amplitudes and without clear features.
\\
% Noise suppression
An additional effect of the summing is the suppression of noise particular to individual antennas as this is adds up incoherently.
\Todo{rephrase}
An additional effect of interferometry is the suppression of noise particular to individual antennas as this adds up incoherently.
The signal in the summed waveform grows linearly with the number of detectors, while the incoherent noise in that same waveform scales with the square root of the number of detectors.
\\
\begin{figure}% fig:trace_overlap %<<<
\centering
\begin{subfigure}[b]{0.47\textwidth}
\includegraphics[height=8cm, width=\textwidth]{radio_interferometry/rit_schematic_far.pdf}%
\caption{}
\label{fig:rit_schematic}
\end{subfigure}
\begin{minipage}[c][9cm][t]{0.47\textwidth}
\begin{subfigure}{\textwidth}
\includegraphics[height=8cm, width=\textwidth]{radio_interferometry/rit_schematic_far.pdf}%
\caption{}
\label{fig:rit_schematic}
\end{subfigure}
\end{minipage}
\hfill
\begin{minipage}[b][7cm][s]{.47\textwidth}
\begin{minipage}[c][9cm][t]{.47\textwidth}
\begin{subfigure}{\textwidth}
\includegraphics[height=2.5cm, width=\textwidth]{radio_interferometry/trace_overlap_best.png}
\caption{}
@ -117,13 +130,9 @@ An additional effect of the summing is the suppression of noise particular to in
\end{subfigure}
\end{minipage}
\caption{
\textit{Left:}
Schematic of radio interferometry.
The antennas the time delays for a location $\vec{x}$ not trained on the source $S_0$.
\protect \Todo{describe}
\textit{Right:}
Overlap between the recorded waveforms for the source location~\subref{fig:trace_overlap:best} and a far away location~\subref{fig:trace_overlap:bad}.
\protect\Todo{include sum}
Schematic of radio interferometry \subref{fig:rit_schematic}
and the overlap between the recorded waveforms at the source location~$S_0$~\subref{fig:trace_overlap:best} and a far away location~\subref{fig:trace_overlap:bad}.
$\Delta_i$ corresponds to the time delay per antenna from \eqref{eq:propagation_delay}.
}
%\hfill
%\begin{subfigure}[t]{0.3\textwidth}
@ -140,11 +149,12 @@ An additional effect of the summing is the suppression of noise particular to in
% Spatial mapping of power
In the technique from \cite{Schoorlemmer:2020low}, the air shower is identified using the power in the combined waveform.
In the technique from \cite{Schoorlemmer:2020low}, the summed waveform $S(\vec{x})$ is computed for multiple locations.
For each location, the power in $S(\vec{x})$ is determined to create a power distribution.
%\\
An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
\\
Here,
Computing the combined waveform $S$ for multiple locations, and analysing the power in it, a source region can be identified as a maximum
At locations with high power, the recorded waveforms interfere constructively while for low power locations, the interference is destructive.
The region of high power identifies strong coherent signals related to the air shower.
By mapping this region, the shower axis and shower core can be resolved.
Later, with the shower axis identified, the power along the axis is used to compute \Xmax.
\end{document}