Thesis: SingleSine: rewrite algorithm part
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@ -122,7 +122,6 @@ This falls into the dynamic setup previously mentioned.
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The best period defects must thus be recovered from a single event.
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\\
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When doing the interferometric analysis for a sine beacon synchronised array, waveforms can only be delayed by an integer amount of periods, thereby giving discrete solutions to maximising the interferometric signal.
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\Todo{add size of shower at plane vs period defects in meters}
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\clearpage
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\section{Air Shower simulation}
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@ -212,6 +211,61 @@ The small clock defect $\tClockPhase$ is then finally calculated from the beacon
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From the above, we now have a set of reconstructed air shower waveforms with corresponding clock defects smaller than one beacon period $T$.
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Shifting the waveforms to remove these small clocks defects, we are left with resolving the correct number of periods $k$ per waveform (see Figure~\ref{fig:grid_power:repair_phases}).
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\\
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% >>>>
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\section{\textit{k}-finding} % <<<
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% unknown origin of air shower signal
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Up until now, the shower axis and thus the origin of the air shower signal have not been resolved.
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This means that the unknown propagation time delays for the air shower ($\tProp$) affect the alignment of the signals in Figure~\ref{fig:beacon_sync:period_alignment} in addition to the unknown clock period defects ($k T$).
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As such, both this origin and the clock defects have to be determined simultaneously.
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\\
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% radio interferometry
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If the antennas had been fully synchronised, radio interferometry as introduced in Chapter~\ref{sec:interferometry} can be applied to find the origin of the air shower signal, thus resolving the shower axis.
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Still, a (rough) first estimate of the shower axis might be made using this technique or by employing other detection techniques such as those using surface or fluorescence detectors.
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% (see Figure~\ref{fig:dynamic-resolve}).
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\\
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On the true shower axis, the waveforms would sum most coherently when the correct $k$'s are used.
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Therefore, around the estimated shower axis, we define a grid search to both optimise this sum and the location of the maximum power.
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In this process each waveform of the array is allowed to shift by a restricted amount of periods with respect to a reference waveform (taken to be the waveform with the highest maximum).
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\\
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Note that these grids are defined here in shower plane coordinates with $\vec{v}$ the true shower axis and $\vec{B}$ the local magnetic field.
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Searching a grid that is slightly misaligned with the true shower axis is expected to give comparable results.
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\\
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The below $k$-finding algorithm is an iterative process where the grid around the shower axis is redefined on each iteration.
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Discussion is found in the next Chapter.
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\begin{enumerate}[start=1, label={Step \arabic*.}, ref=\arabic*, topsep=6pt, widest={Step 1.}]
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\label{algo:kfinding}
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\item \label{algo:kfinding:grid}
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Define a grid around the estimated shower axis, zooming in on each iteration.
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\item \label{algo:kfinding:optimisation}
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$k$-optimisation: per grid point, optimise the $k$'s to maximise the sum of the waveforms (see Figure~\ref{fig:single:k-correlation}).
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\item \label{algo:kfinding:kfinding}
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$k$-finding: find the grid point with the maximum overall sum (see Figure~\ref{fig:findks:maxima}) and select its set of $k$'s.
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\item \label{algo:kfinding:break}
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Stop when the set of $k$'s is equal to the set of the previous iteration, otherwise continue.
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\item \label{algo:kfinding:powermapping}
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Finally, make a power mapping with the obtained $k$'s to re-estimate the shower axis (location with maximum power) (see Figure~\ref{fig:findks:reconstruction}), and return to Step~\ref{algo:kfinding:grid} for another iteration.
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\end{enumerate}
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\vspace*{2pt}
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Here, Step~\ref{algo:kfinding:optimisation} has been implemented by summing each waveform to the reference waveform (see above) with different time delays $kT$ and selecting the $k$ that maximises the amplitude of a waveform combination.\footnote{%<<<
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Note that one could use a correlation method instead of a maximum to select the best time delay.
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However, for simplicity and ease of computation, this has not been implemented.
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Other improvements are discussed in the next Section.
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} %>>>
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As shown in Figure~\ref{fig:single:k-correlation}, the maximum possible period shift has been limited to $\pm 3\,\mathrm{periods}$.
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This corresponds to the maximum expected time delay between two antennas with a clock randomisation up to $30\ns$ for the considered beacon frequency.%
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\footnote{
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Figure~\ref{fig:simu:error:periods} shows this is not completely true.
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However, overall, it still applies.
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}
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\begin{figure}%<<<
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\centering
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@ -223,51 +277,12 @@ Shifting the waveforms to remove these small clocks defects, we are left with re
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\label{fig:single:k-correlation}
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\end{figure}%>>>
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% >>>>
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\section{\textit{k}-finding} % <<<
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% unknown origin of air shower signal
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Up until now, the shower axis and thus the origin of the air shower signal have not been resolved.
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This means that the unknown propagation time delays for the air shower ($\tProp$) affect the alignment of the signals in Figure~\ref{fig:beacon_sync:period_alignment} in addition to the unknown clock period defects ($k_j T$).
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As such, both this origin and the clock defects have to be determined simultaneously.
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\\
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% radio interferometry
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If the antennas had been fully synchronised, radio interferometry as introduced in Section~\ref{sec:interferometry} can be applied to find the origin of the air shower signal, thus resolving the shower axis.
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Still, a rough first estimate of the shower axis might be made using this or other techniques.% (see Figure~\ref{fig:dynamic-resolve}).
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\\
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Starting with an initial grid around this estimated axis, a two-step process zooms in on the shower axis while optimising the interferometric signal.
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In this process each waveform of the array is allowed to shift by a restricted amount of periods with respect to a reference waveform (taken to be the waveform with the highest maximum).
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\\
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Note that these grids are defined in shower plane coordinates with $\vec{v}$ the true shower axis and $\vec{B}$ the local magnetic field.
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Searching a grid that is slightly misaligned with the true shower axis is expected to give comparable results.
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\\
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The first step consists of finding the set of period shifts $k_j$ that maximises the overall maximum amplitude on the grid.
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At each location, after removing propagation delays, each waveform and the reference waveform are summed with different time delays $kT$ to find the maximum attainable amplitude of the combined trace.%
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\footnote{%<<<
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Note that one could use a correlation method instead of a maximum to select the best time delay.
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However, for simplicity and ease of computation, this has not been implemented.
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%As shown in Figure~\ref{fig:single:annotated_full_waveform}, the air shower signal has a length in the order of a few nanoseconds.
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%Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks.
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} %>>>
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\\
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As shown in Figure~\ref{fig:single:k-correlation}, the maximum possible period shift has been limited to $\pm 3\,\mathrm{periods}$.
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This corresponds to the maximum expected time delay between two antennas with a clock randomisation up to $30\ns$ for the considered beacon frequency.
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\\
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%
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This amplitude optimisation is iterated over the grid (see Figure~\ref{fig:findks:maxima}) resulting in a grid measurement of the maximum attainable amplitude and the corresponding set of period defects $k$.
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\\
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The second step applies the obtained period defects and measures the interferometric power on the same grid (see Figure~\ref{fig:findks:reconstruction}).
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Afterwards, a new grid zooms in on the power maximum and the process is repeated (Figures~\ref{fig:findks:maxima:zoomed},~\ref{fig:findks:reconstruction:zoomed}) until the set of period defects is constant between zooming grids.
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\\
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\begin{figure}%<<< findks
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
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\caption{
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Maximum amplitudes obtainable by shifting the waveforms.
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$k$-finding: optimise the $k$'s by shifting waveforms to find the maximum amplitude obtainable at each point.
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}
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\label{fig:findks:maxima}
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\end{subfigure}
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@ -275,7 +290,7 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
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\caption{
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Power measurement with the $k$'s belonging to the overall maximum of the amplitude maxima.
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Power measurement with the $k$'s belonging to the overall maximum of the tested amplitudes.
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}
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\label{fig:findks:reconstruction}
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\end{subfigure}
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@ -283,7 +298,8 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
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\caption{
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Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
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$2^\mathrm{nd}$ $k$-finding iteration:
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Zoom in on the location in \subref{fig:findks:reconstruction} with the highest amplitude and repeat algorithm.
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}
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\label{fig:findks:maxima:zoomed}
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\end{subfigure}
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@ -299,7 +315,8 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run2.pdf}
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\caption{
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Final test grid obtaining the same $k$'s as Figure~\ref{fig:findks:maxima:zoomed}.
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$3^\mathrm{rd}$ $k$-finding iteration:
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The same set of $k$'s has been found and we stop the algorithm.
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}
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\label{fig:findks:maxima:zoomed2}
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\end{subfigure}
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@ -312,7 +329,7 @@ Afterwards, a new grid zooms in on the power maximum and the process is repeated
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\label{fig:findks:reconstruction:zoomed2}
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\end{subfigure}
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\caption{
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Iterative $k$-finding algorithm:
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Iterative $k$-finding algorithm (see page~\pageref{algo:kfinding} for explanation):
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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).
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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).
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Second \subref{fig:findks:reconstruction}, perform the interferometric reconstruction with this set of period shifts.
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@ -332,22 +349,20 @@ Phase synchronising the antennas gives a small increase in observed power, while
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The initial grid plays an important role here in finding the correct axis.
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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.
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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}
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\\
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% premature optimisation / degeneracy
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Such locations are subject to differences in propagation delays that are in the order of a few beacon periods.
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The restriction of the possible delays is therefore important to limit the number of potential axis locations.
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\\
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% fall in local extremum, maximum
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In this analysis, the initial grid is defined as $8^\circ$ wide around the true axis.
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In this analysis, the initial grid is defined as a very wide $8^\circ$ around the true axis.
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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.
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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 $\left| k \right| \leq 3$.
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Unfortunately, the above process has been observed to fall into local maxima when a too coarse and wide initial grid ($N_x < 13$ at $X=400\,\mathrm{g/cm^2}$) was used while restricting the time delays to $\left| k \right| \leq 3$.
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\\
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% Missing power / wrong k
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As visible in the right side of Figure~\ref{fig:grid_power:repair_full}, not all waveforms are in sync after the optimisation.
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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}).
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In this case, the period defects have been resolved incorrectly for two waveforms (see Figure~\ref{fig:simu:error:periods}) due to too stringent limits on the allowable $k$'s.
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Looking at Figure~\ref{fig:grid_power:repair_phases}, this was to be predicted since there are two waveforms peaking at $k=4$ from the reference waveform's peak (dashed line).
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As a result, the obtained power for the resolved clock defects is slightly less than the obtained power for the true clocks.
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\\
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@ -357,17 +372,6 @@ Figure~\ref{fig:grid_power:axis} shows the power mapping at four different atmos
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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.
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\\
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% Future: at multiple depths
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Of course, this algorithm must be evaluated at relevant atmospheric depths where the interferometric technique can resolve the air shower.
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In this case, after manual inspection, the air shower was found to have \Xmax\ at roughly $400\,\mathrm{g/cm^2}$.
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The algorithm is expected to perform as long as a region of strong coherent power is resolved.
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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.
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\\
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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.
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\\
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\begin{figure}% fig:simu:error
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\centering
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%\includegraphics[width=\textwidth]{ZH_simulation/cb_report_measured_antenna_offsets.py.time-amplitudes-missing-k.residuals.pdf}
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@ -438,6 +442,26 @@ Additionally, since the true period shifts are static per event, evaluating the
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\label{fig:grid_power_time_fixes}
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\end{figure}%>>>
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\pagebreak
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% Future: at multiple depths
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Of course, this algorithm must be evaluated at relevant atmospheric depths where the interferometric technique can resolve the air shower.
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In this case, after manual inspection, the air shower was found to have \Xmax\ at roughly $400\,\mathrm{g/cm^2}$.
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The algorithm is expected to perform as long as a region of strong coherent power is resolved.
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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.
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\\
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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.
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\\
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Further improvements to the algorithm are foreseen in both the definition of the initial grid (Step~\ref{algo:kfinding:grid}) and the optimisation of the $k$'s (Step~\ref{algo:kfinding:optimisation}).
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For example, the $k$-optimisation step currently sums the full waveform for each $k$ to find the maximum amplitude for each sum.
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Instead, the timestamp of the amplitude maxima of each waveform can be compared, directly allowing to compute $k$ from the difference.
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\\
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Finally, from the overlapping traces in Figure~\ref{fig:grid_power:repair_full}, it is easily recognisable that some period defects have been determined incorrectly.
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Inspecting Figure~\ref{fig:grid_power:repair_phases}, this was to be expected as there are two waveforms with the peak at $\left|k\right| = 4$ from the reference waveform.
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Therefore, either the $k$-optimisation should have been run with a higher limit on the allowable $k$'s, or, preferably, these waveforms must be optimised after the algorithm is finished with a higher maximum $k$.
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\begin{figure}%<<< grid_power:axis:X600
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\vspace*{-5mm}
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\centering
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