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20
documents/thesis/chapters/beacon_discipline.tex
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@ -513,9 +513,8 @@ Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase
It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below). It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below).
The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}. The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}.
\\ \\
Note that these distributions have non-zero means. Note that these distributions have non-zero means,
This might be a systematic offset. this systematic offset has not been investigated further in this work.
However, this has not been investigated.
\\ \\
% Signal to Noise definition % Signal to Noise definition
@ -559,16 +558,20 @@ For gaussian noise, the measurement of the beacon phase $\pTrue$ can be shown to
\end{equation} \end{equation}
where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function. where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function.
\cite{goodman1985:2.9} names this equation ``Constant Phasor plus a Random Phasor Sum''. \cite{goodman1985:2.9} names this equation ``Constant Phasor plus a Random Phasor Sum''.
For sake of brevity, it will be referred to as ``Random Phasor Sum''.
\Todo{use Phasor Sum instead}
\\ \\
This Random Phasor Sum distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude. This distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}. This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}.
There, the phase residuals of the simulated waveforms closely follow the distribution. There, the phase residuals of the simulated waveforms closely follow the distribution.
\\ \\
From Figure~\ref{fig:sine:snr_time_resolution} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$. From Figure~\ref{fig:sine:snr_time_resolution} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$.
Since the time accuracy is derived from the phase accuracy, slightly lower frequencies could be used, but they would require a stronger signal to resolve to the same degree. Since the time accuracy is derived from the phase accuracy with
\begin{equation}\label{eq:phase_accuracy_to_time_accuracy}
\phantom{,}
\sigma_t = \frac{\sigma_\pTrue}{2\pi \fbeacon}
,
\end{equation}
slightly lower frequencies could be used instead, but they would require a comparatively stronger signal to resolve to the same degree.
Likewise, higher frequencies are an available method of linearly improving the time accuracy. Likewise, higher frequencies are an available method of linearly improving the time accuracy.
\\ \\
@ -578,8 +581,9 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
\begin{figure} \begin{figure}
\includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf} \includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
\caption{ \caption{
Timing accuracy for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise. Phase accuracy (right y-axis) for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise.
It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$. It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$.
The time accuracy is converted from the phase accuracy using \eqref{eq:phase_accuracy_to_time_accuracy}.
The green dashed line indicates the $1\ns$ level. The green dashed line indicates the $1\ns$ level.
Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$. Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$.
} }

6
documents/thesis/chapters/introduction.tex
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@ -12,8 +12,6 @@
\label{sec:introduction} \label{sec:introduction}
%\section{Cosmic Particles}%<<<<<< %\section{Cosmic Particles}%<<<<<<
%<<< %<<<
\phantomsection
\label{sec:crs}
% Energy and flux % Energy and flux
The Earth is bombarded with a variety of extra-terrestrial particles, with the energy of these particles extending over many orders of magnitude as depicted in Figure~\ref{fig:cr_flux}. The Earth is bombarded with a variety of extra-terrestrial particles, with the energy of these particles extending over many orders of magnitude as depicted in Figure~\ref{fig:cr_flux}.
The flux of these particles decreases exponentially with increasing energy. The flux of these particles decreases exponentially with increasing energy.
@ -57,8 +55,6 @@ Unfortunately, aside from both being much less frequent, photons can be absorbed
%>>> %>>>
%\subsection{Air Showers}%<<< %\subsection{Air Showers}%<<<
\phantomsection
\label{sec:airshowers}
When a cosmic ray with an energy above $10^{3}\GeV$ comes into contact with the atmosphere, secondary particles are generated, forming an air shower. When a cosmic ray with an energy above $10^{3}\GeV$ comes into contact with the atmosphere, secondary particles are generated, forming an air shower.
This air shower consists of a cascade of interactions producing more particles that subsequently undergo further interactions. This air shower consists of a cascade of interactions producing more particles that subsequently undergo further interactions.
Thus, the number of particles rapidly increases further down the air shower. Thus, the number of particles rapidly increases further down the air shower.
@ -144,8 +140,6 @@ It is therefore important for radio detection to obtain measurements in this reg
%>>>>>> %>>>>>>
%\subsection{Experiments}%<<< %\subsection{Experiments}%<<<
\phantomsection
\label{sec:detectors}
As mentioned, the flux at the very highest energy is in the order of one particle per square kilometer per century (see Figure~\ref{fig:cr_flux}). As mentioned, the flux at the very highest energy is in the order of one particle per square kilometer per century (see Figure~\ref{fig:cr_flux}).
Observatories therefore have to span huge areas to gather decent statistics at these highest energies on a practical timescale. Observatories therefore have to span huge areas to gather decent statistics at these highest energies on a practical timescale.
In recent and upcoming experiments, such as the~\gls{Auger}\cite{Deligny:2023yms} and the~\gls{GRAND}\cite{GRAND:2018iaj}, the approach is typically to instrument a large area with a (sparse) grid of detectors to detect the generated air shower. In recent and upcoming experiments, such as the~\gls{Auger}\cite{Deligny:2023yms} and the~\gls{GRAND}\cite{GRAND:2018iaj}, the approach is typically to instrument a large area with a (sparse) grid of detectors to detect the generated air shower.

15
documents/thesis/chapters/radio_interferometry.tex
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@ -15,26 +15,25 @@
\begin{document} \begin{document}
\chapter{Air Shower Radio Interferometry} \chapter{Air Shower Radio Interferometry}
\label{sec:interferometry} \label{sec:interferometry}
The radio signals emitted by an \gls{EAS} (see Section~\ref{sec:airshowers}) can be recorded by radio antennas. The radio signals emitted by an \gls{EAS} (see Chapter~\ref{sec:introduction}) can be recorded by radio antennas.
For suitable frequencies, an array of radio antennas can be used as an interferometer. For suitable frequencies, an array of radio antennas can be used as an interferometer.
Therefore, air showers can be analysed using radio interferometry. Therefore, air showers can be analysed using radio interferometry.
Note that since the radio waves are mainly caused by processes involving electrons (see Section~\ref{sec:airshowers}), any derived properties are tied to the electromagnetic component of the air shower. Note that since the radio waves are mainly caused by processes involving electrons, any derived properties are tied to the electromagnetic component of the air shower.
\\ \\
In Reference~\cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using radio interferometry.% In Reference~\cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using radio interferometry.%
\footnote{ \footnote{
Available as a python package at \url{https://gitlab.com/harmscho/asira}. Available as a python package at \url{https://gitlab.com/harmscho/asira}.
} }
Figure~\ref{fig:radio_air_shower} shows a power mapping of a simulated air shower. A power mapping of a simulated air shower is shown in Figure~\ref{fig:radio_air_shower}.
It reveals the air shower in one vertical and three horizontal slices. It reveals the air shower in one vertical and three horizontal slices.
Analysing this mapping, the shower axis and particle densities can be computed. Analysing the power mapping, we can then infer properties of the air shower such as the shower axis and $\Xmax$.
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. 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. 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.
For detector synchronisations under $2\ns$, the atmospheric depth resolution is competitive with techniques from fluorescence detectors ($\sigma(\Xmax) ~ 25\,\mathrm{g/cm^2}$ at \gls{Auger} \cite{Deligny:2023yms}). For detector synchronisations under $2\ns$, the atmospheric depth resolution is competitive with techniques from fluorescence detectors ($\sigma(\Xmax) ~ 25\,\mathrm{g/cm^2}$ at \gls{Auger} \cite{Deligny:2023yms}).
With a difference in $\langle \Xmax \rangle$ of $\sim 100\,\mathrm{g/cm^2}$ between iron and proton initiated air showers, this depth of shower maximum resolution allows to study the mass composition of cosmic rays. With a difference in $\langle \Xmax \rangle$ of $\sim 100\,\mathrm{g/cm^2}$ between iron and proton initiated air showers, this depth of shower maximum resolution allows to study the mass composition of cosmic rays.
However, for worse synchronisations, the $\Xmax$ resolution for radio antennas degrades linearly. However, for worse synchronisations, the $\Xmax$ resolution for interferometry degrades linearly.
\\ \\
An advantage of radio antennas with respect to fluorescence detectors is the increased duty-cycle. An advantage of radio antennas with respect to fluorescence detectors is the increased duty-cycle.
Fluorescence detectors require clear, moonless nights, resulting in a duty-cycle of about $10\%$ whereas radio detectors have a near permanent duty-cycle. Fluorescence detectors require clear, moonless nights, resulting in a duty-cycle of about $10\%$ whereas radio detectors have a near permanent duty-cycle.
@ -93,8 +92,8 @@ The time delay due to propagation can be written as
where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal. where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
\\ \\
% time delays: particular per antenna % 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}). Note that unlike in astronomical interferometry, the source cannot be assumed at infinity, instead it is close-by (see Figure~\ref{fig:rit_schematic}).
This requires us to compute the time delays for each test location $\vec{x}$ separately. Therefore the time delays for each test location $\vec{x}$ have to be computed separately.
\\ \\
% Features in S % Features in S

29
documents/thesis/chapters/radio_measurement.tex
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@ -31,12 +31,8 @@ The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] +
% Filtering before ADC % Filtering before ADC
The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate. The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate.
For frequencies at or above this Nyquist frequency, the \gls{ADC} records aliases that appear as lower frequencies in the waveform. In addition, various frequency-dependent backgrounds can be reduced by applying a bandpass filter before digitisation.
To prevent such aliases, these frequencies must be removed by a filter before sampling. For example, in \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.
\\
For air shower radio detection, very low frequencies are also not of interest.
Therefore, this filter is generally a bandpass filter.
For example, in the \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.
\\ \\
In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies. In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies.
For example, in \gls{GRAND} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$. For example, in \gls{GRAND} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$.
@ -50,9 +46,9 @@ Thus to reconstruct properties of the electric field signal from the waveform, b
\\ \\
% Analysis, properties, frequencies, pulse detection, shape matching, % Analysis, properties, frequencies, pulse detection, shape matching,
Different methods are available for the analysis of the waveform and the antenna and filter responses. Different methods are available for the analysis of the waveform, and the antenna and filter responses.
A key aspect is determining the frequency-dependent amplitudes (and phases) in the measurements to characterise the responses and, more importantly, select signals from background. A key aspect is determining the frequency-dependent amplitudes (and phases) in the measurements to characterise the responses and, more importantly, select signals from background.
With \acrlong{FT}s these frequency spectra can be produced. With \glspl{FT}, these frequency spectra can be produced.
This technique is especially important for the sinewave beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement. This technique is especially important for the sinewave beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement.
\\ \\
The detection and identification of more complex time-domain signals can be achieved using the cross correlation, The detection and identification of more complex time-domain signals can be achieved using the cross correlation,
@ -61,7 +57,7 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse
\section{Fourier Transforms}% <<<< \section{Fourier Transforms}% <<<<
\label{sec:fourier} \label{sec:fourier}
The \gls{FT} allows for a frequency-domain representation of a time-domain signal. \glspl{FT} allow for a frequency-domain representation of a time-domain signal.
In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency. In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency.
By evaluating the \gls{FT} at appropriate frequencies, the frequency spectrum of a waveform is calculated. By evaluating the \gls{FT} at appropriate frequencies, the frequency spectrum of a waveform is calculated.
This method then allows to modify a signal by operating on its frequency components, i.e.~removing a narrow frequency band contamination within the signal. This method then allows to modify a signal by operating on its frequency components, i.e.~removing a narrow frequency band contamination within the signal.
@ -197,7 +193,6 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
= \arctantwo\left( X_I(f), X_R(f) \right) = \arctantwo\left( X_I(f), X_R(f) \right)
. .
\end{equation} \end{equation}
\\
% Recover A\cos(2\pi t[n] f + \phi) using above definitions % Recover A\cos(2\pi t[n] f + \phi) using above definitions
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains
@ -207,7 +202,7 @@ When the minus sign in the exponent of \eqref{eq:fourier} is not taken into acco
Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}. Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}.
It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies. It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies.
By missing the correct frequency bin for the sine wave, it estimates a too low amplitude for the sine wave. By missing the correct frequency bin for the sine wave, it estimates both a too low amplitude and the wrong phase for the input function.
\\ \\
@ -217,7 +212,7 @@ Therefore, at the cost of an increased memory allocation, these terms can be pre
% .. relevance to hardware if static frequency % .. relevance to hardware if static frequency
Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors, Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors,
opening the way to efficiently measuring the phases in realtime. opening the way to efficiently measuring the amplitude and phase in realtime.
% >>>> % >>>>
@ -225,9 +220,7 @@ opening the way to efficiently measuring the phases in realtime.
\section{Cross-Correlation}% <<<< \section{Cross-Correlation}% <<<<
\label{sec:correlation} \label{sec:correlation}
The cross-correlation is a measure of how similar two waveforms $u(t)$ and $v(t)$ are. The cross-correlation is a measure of how similar two waveforms $u(t)$ and $v(t)$ are.
By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay. By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay,
It is defined as
\begin{equation} \begin{equation}
\label{eq:correlation_cont} \label{eq:correlation_cont}
\phantom{,} \phantom{,}
@ -236,7 +229,6 @@ It is defined as
\end{equation} \end{equation}
where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$. where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
Still, $\tau$ remains a continuous variable. Still, $\tau$ remains a continuous variable.
\\
% Figure example of correlation and argmax % Figure example of correlation and argmax
Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation. Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation.
\\ \\
@ -247,7 +239,6 @@ When the sampling rates are equal, the time delay variable is effectively shifti
\\ \\
% Upsampling? No % Upsampling? No
Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree. Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree.
However, for the purposes in this document, these methods are not used.
\\ \\
% Approaching analog \tau; or zero-stuffing % Approaching analog \tau; or zero-stuffing
@ -257,7 +248,7 @@ This allows to approximate an analog time delay between two waveforms when one w
\begin{figure} \begin{figure}
\centering \centering
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf} \includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf}
%\caption{ %\caption{
% Two waveforms. % Two waveforms.
@ -265,7 +256,7 @@ This allows to approximate an analog time delay between two waveforms when one w
\label{subfig:correlation:waveforms} \label{subfig:correlation:waveforms}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{methods/correlation/correlation.pdf} \includegraphics[width=\textwidth]{methods/correlation/correlation.pdf}
%\caption{ %\caption{
% The correlation of two Waveforms as a function of time. % The correlation of two Waveforms as a function of time.

8
documents/thesis/chapters/single_sine_interferometry.tex
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@ -20,7 +20,7 @@ This poses an opportunity to use a ``free'' beacon to synchronise the radio ante
Due to the periodicity of sine beacons, the ability to synchronise an array is limited up to the beacon period $T$. Due to the periodicity of sine beacons, the ability to synchronise an array is limited up to the beacon period $T$.
As previously mentioned, the correct periods can be ascertained by choosing a beacon period much longer than the estimated accuracy of another timing mechanism.\footnote{For reference, \gls{GNSS} timing is expected to be below $30\ns$} As previously mentioned, the correct periods can be ascertained by choosing a beacon period much longer than the estimated accuracy of another timing mechanism.\footnote{For reference, \gls{GNSS} timing is expected to be below $30\ns$}
Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\ns$.. Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\us$.
\\ \\
In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon. In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
@ -348,7 +348,7 @@ Unfortunately, the above process has been observed to fall into local maxima whe
% Missing power / wrong k % 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. 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}). 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. As a result, the obtained power for the resolved clock defects is slightly less than the obtained power for the true clocks.
\\ \\
% directional reconstruction % directional reconstruction
@ -447,7 +447,7 @@ Additionally, since the true period shifts are static per event, evaluating the
\hfill \hfill
\includegraphics[width=0.46\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.X400.trace_overlap.zoomed.repair_full.pdf} \includegraphics[width=0.46\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.X400.trace_overlap.zoomed.repair_full.pdf}
\caption{ \caption{
Fully resolved clocks Resolved clocks
} }
\label{fig:grid_power:repair_full} \label{fig:grid_power:repair_full}
\end{subfigure} \end{subfigure}
@ -502,7 +502,7 @@ Additionally, since the true period shifts are static per event, evaluating the
\label{fig:grid_power:axis:X800} \label{fig:grid_power:axis:X800}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Interferometric power for the fully resolved clocks at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}). Interferometric power for the resolved clocks (from Figure~\ref{fig:grid_power:repair_full}) at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}).
The simulation axis is indicated by the red plus, the maximum power is indicated by the blue cross. The simulation axis is indicated by the red plus, the maximum power is indicated by the blue cross.
Except for \subref{fig:grid_power:axis:X800}, the shower axis is resolved within $0.1^\circ$ of the true shower axis. Except for \subref{fig:grid_power:axis:X800}, the shower axis is resolved within $0.1^\circ$ of the true shower axis.
} }