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Thesis: Single: more discussion

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documents/thesis/chapters/single_sine_interferometry.tex
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@ -133,25 +133,25 @@ When doing the interferometric analysis for a sine beacon synchronised array, wa
To test the idea of combining a single sine beacon with an air shower, we simulated a set of recordings of a single air shower that also contains a beacon signal.
\footnote{\url{https://gitlab.science.ru.nl/mthesis-edeboone/m-thesis-introduction/-/tree/main/airshower_beacon_simulation}}
\\
The air shower signal (here a $10^{16}\eV$ proton) was simulated by \acrlong{ZHAires} on a grid of 10x10 antennas with a spacing of $50$\,meters.\Todo{cite ZHAires?}
Each antenna recorded a waveform of $500$ samples with a samplerate of $1\GHz$ for each of the X,Y and Z polarisations.\Todo{polarisation names}
The air shower signal was simulated by \acrlong{ZHAires}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
Each antenna recorded a waveform of $500$ samples with a samplerate of $1\GHz$ for each of the X,Y and Z polarisations.
The air shower itself was generated by a $10^{16}\eV$ proton coming in under an angle of $20^\circ$ from zenith.
%Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.
\Todo{verify numbers in paragraph}
\\
Figure~\ref{fig:single:proton_grid} shows the maximum electric field measured at each of the antennas.
The ring of antennas with maximum electric fields in the order of $30\mathrm{\;\mu V/m}$ at the center of the array is the Cherenkov ring.
The Cherenkov ring forms due to the forward beaming of the radio emissions of the airshower.\Todo{expand/refer}
The ring of antennas with maximum electric fields in the order of $25\uVperm$ at the center of the array is the Cherenkov ring.
The Cherenkov ring forms due to the forward beaming of the radio emissions of the airshower.
Outside this ring, the maximum electric field quickly falls with increasing distance to the array core.
\Todo{geometry, vertical shower}
As expected for a vertical shower, the projection of the Cherenkov ring on the ground is roughly circular.
\\
%% add beacon
A sine beacon ($\fbeacon = 51.53\MHz$) was introduced at a distance of approximately $75\mathrm{\,km}$ northwest of the array, primarily received in the X polarisation.
A sine beacon ($\fbeacon = 51.53\MHz$) was introduced at a distance of approximately $75\,\mathrm{km}$ northwest of the array, primarily received in the X polarisation.
The distance between the antenna and the transmitter results in a phase offset with which the beacon is received at each antenna.%
\footnote{%<<<
The beacon's amplitude is also dependent on the distance. Although simulated, this effect has not been incorporated in the analysis as it is negligible for the considered grid and distance to the transmitter.
} %>>>
The beacon signal was recorded over a longer time ($10240$\;samples), to be able to distinguish the beacon and air shower later in the analysis.
The beacon signal was recorded over a longer time ($10240\,\mathrm{samples}$), to be able to distinguish the beacon and air shower later in the analysis.
\\
The final waveform of an antenna (see Figure~\ref{fig:single:annotated_full_waveform}) was then constructed by adding its beacon and air shower waveforms and band-passing with relevant frequencies (here $30$ and $80\MHz$ are taken by default).
Of course, a gaussian white noise component is introduced to the waveform as a simple noise model (see Figure~\ref{fig:sine:time_accuracy} for a treatise on the timing accuracy of a sine beacon).
@ -161,9 +161,9 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
\centering
\includegraphics[width=0.5\textwidth]{ZH_simulation/array_geometry_shower_amplitude.pdf}
\caption{
\protect\Todo{text}
The 10x10 antenna grid used for recording the air shower.
Colours indicate the maximum electric field recorded at the antenna.
The Cherenkov-ring is clearly visible as a circle of radius $100\metre$ centered at $(0,0)$.
}
\label{fig:single:proton_grid}
\end{figure}% >>>
@ -181,11 +181,14 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
\textit{Left:}
%\textit{Right:}
%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
Excerpt of a fully simulated waveform ($N=10240\mathrm{\;samples}$) (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (green, $\fbeacon = 51.53\MHz$) and noise.
Excerpt of a fully simulated waveform ($N=10240\,\mathrm{samples}$) (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (green, $\fbeacon = 51.53\MHz$) and noise.
The part of the waveform between the vertical dashed lines is considered airshower signal and masked before measuring the beacon parameters.
\textit{Right:}
Fourier spectra of the waveforms.
The green dashed lines indicate the measured beacon parameters.
The amplitude spectrum clearly shows a strong component at roughly $50\MHz$.
The phase spectrum of the original waveform shows the typical behaviour for a short pulse.
\protect\Todo{ft amplitude airshower x1000}
}
\label{fig:single:proton}
\end{figure}% >>>
@ -256,7 +259,7 @@ At each location, after removing propagation delays, each waveform and the refer
%Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks.
} %>>>
\\
As shown in Figure~\ref{fig:single:k-correlation}, the maximum possible period shift has been limited to $\pm 3\mathrm{\; periods}$.
As shown in Figure~\ref{fig:single:k-correlation}, the maximum possible period shift has been limited to $\pm 3\,\mathrm{periods}$.
This corresponds to the maximum expected time delay between two antennas with a clock randomisation up to $30\ns$ for the considered beacon frequency.
\\
%
@ -318,7 +321,7 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
\caption{
Iterative $k$-finding algorithm:
First \subref{fig:findks:maxima}, find the set of period shifts $k$ per point on a grid that returns the highest maximum amplitude (blue cross).
The grid starts as a $8^\circ$ wide shower plane slice at $X=400\mathrm{\,g/cm}$, centred at the true shower axis (red cross).
The grid starts as a $8^\circ$ wide shower plane slice at $X=400\,\mathrm{g/cm^2}$, centred at the true shower axis (red cross).
Second \subref{fig:findks:reconstruction}, perform the interferometric reconstruction with this set of period shifts.
Zooming on the maximum power \subref{fig:findks:maxima:zoomed},\subref{fig:findks:reconstruction:zoomed} repeat the steps until the $k$'s are equal between the zoomed grids \subref{fig:findks:maxima:zoomed2},\subref{fig:findks:reconstruction:zoomed2}.
}
@ -331,11 +334,12 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
\subsection{Strategy / Result} %<<<
Figure~\ref{fig:grid_power_time_fixes} shows the effect of the various synchronisation stages on both the alignment of the air shower waveforms, and the interferometric power measurement near the true shower axis.
Phase synchronising the antennas gives a small increase in observed power, while further aligning the periods after the optimisation process significantly enhances this power.
\\
The initial grid plays an important role here in finding the correct axis.
Due to selecting the highest maximum amplitude, and the process above zooming in aggressively, wrong candidate axes are selected when there is no grid-location sufficiently close to the true axis.
Figure~\ref{fig:findks:reconstruction} shows such a potential point near $(-1, 0.5)$ with a maximum that is comparable to the selected maximum near the true axis.
Figure~\ref{fig:findks:reconstruction} shows such a potential point near $(-1, 0.5)$ with a maximum that is comparable to the selected maximum near the true axis.\Todo{no longer shown}
\\
% premature optimisation / degeneracy
Such locations are subject to differences in propagation delays that are in the order of a few beacon periods.
@ -345,22 +349,38 @@ The restriction of the possible delays is therefore important to limit the numbe
% fall in local extremum, maximum
In this analysis, the initial grid is defined as $8^\circ$ wide around the true axis.\Todo{why?}
As the number of computations scales linearly with the number of grid points ($N = N_x N_y$), it is favourable to minimise the number of grid locations.
Unfortunately, the above process has been observed to fall into local maxima when a too coarse initial grid ($N_x < 13$ at $X=400\mathrm{\,g/cm}$) was used while restricting the time delays to $\left| k \right| \leq 3$.
Unfortunately, the above process has been observed to fall into local maxima when a too coarse initial grid ($N_x < 13$ at $X=400\,\mathrm{g/cm^2}$) was used while restricting the time delays to $\left| k \right| \leq 3$.
\\
Of course, this algorithm must be evaluated at proper\Todo{word} atmospheric depths where the interferometric technique can resolve the shower.
Additionally, since the true period shifts are static per event, the evaluation at multiple atmospheric depths allows to compare the obtained sets thereof.
% Missing power / wrong k
As visible in the right side of Figure~\ref{fig:grid_power:repair_full}, not all waveforms are in sync after the optimisation.
In this case, the period defects have been resolved incorrectly for two waveforms, lagging 1 and 3 periods respectively (see Figure~\ref{fig:simu:error:periods}).
As a result, the obtained power for the fully resolved clock defects is slightly less than the obtained power for the true clocks.
\\
% directional reconstruction
This does not impede resolving the shower axis.
Figure~\ref{fig:grid_power:axis} shows the power mapping at four different atmospheric depths for the resolved clock defects.
Except for the low power case at $X=800\,\mathrm{g/cm^2}$, the shower axis is found to be $<0.1^\circ$ of the true shower axis.
\\
% Future: at multiple depths
Of course, this algorithm must be evaluated at relevant atmospheric depths where the interferometric technique can resolve the air shower.
In this case, after manual inspection, the air shower was found to have \Xmax\ at roughly $400\,\mathrm{g/cm^2}$.
The algorithm is expected to perform as long as a region of strong coherent power is resolved.
This means that with the power in both Figure~\ref{fig:grid_power:axis:X200} and Figure~\ref{fig:grid_power:axis:X600}, the clock defects and air shower should be identified to the same degree.
\\
Additionally, since the true period shifts are static per event, evaluating the $k$-finding algorithm at multiple atmospheric depths allows to compare the obtained sets thereof to further minimise any incorrectly resolved period defect.
\\
\Todo{directional reconstruction}
\Todo{missing power}
\Todo{res k Figure}
\begin{figure}% fig:simu:error
\centering
\includegraphics[width=\textwidth]{ZH_simulation/cb_report_measured_antenna_time_offsets.py.time-periods.residuals.pdf}
%\includegraphics[width=\textwidth]{ZH_simulation/cb_report_measured_antenna_offsets.py.time-amplitudes-missing-k.residuals.pdf}
\includegraphics[width=0.5\textwidth]{ZH_simulation/cb_report_measured_antenna_time_offsets.py.time-periods.residuals.pdf}
\caption{
Errors in the resolved clock defects with respect to the true clock defects.
Errors in the resolved period defects with respect to the true period defects.
}
\label{fig:simu:error:periods}
\end{figure}