rate.tex

\chapter{Fake rate of pulse detection}
\label{ch:rate}

In \autoref{ch:snr} we measured the signal to noise ratio after filtering. The
point of using the SNR is expressing the signal height relative to the width of
the noise distribution, because the threshold required to reject noise with a
given probability is proportional to that scale. So it is convenient to express
the threshold relative to the same scale. When setting a threshold on the
filtered amplitude, as done in data acquisition to reduce the noise rate, an
SNR well above such a rescaled threshold will ensure that a good efficiency for
signal is retained.

This chapter deals with the calculation of the noise rate resulting from random
fluctuations of the noise above the threshold.

Even assuming the noise to be Gaussian, the probability that a random
fluctuation gets above the threshold is not given simply by computing the
survival function (i.e., the integral to $+\infty$) of the Gaussian
distribution at the threshold.

More precisely, the probability that any given sample is above the threshold is
given by such integral. What we need, however, is the \emph{rate} of threshold
\emph{crossings}. The noise is not white, but even if it was, after applying
the filter, which combines linearly many input samples for each output sample,
the waveform is autocorrelated at least up to the length of the filter.
Intuitively, if a smooth function crosses a threshold, it takes some time to go
down before it can cross the threshold again.

We will present the method used to compute the threshold crossing rate of the
noise, or fake rate, for a running average filter, and check that the
calculated rates are consistent with the results obtained in data.

The choice of the running average filter is due to the fact that this was the
filter implemented as part of the first prototype firmware for the FPGA of the
digitizer board. This constitutes the first processing stage in the DarkSide
DAQ chain, and the reduction of the noise is a key goal of that processing.
However, the method we present can be applied to any linear filter of choice,
and allows to extrapolate efficiently to very low rates.

\section{Model}

We expect the noise to be Gaussian. Even if it were not prior to any
processing, when filtering many samples are linearly combined, and the sum of
random variables tends to have a Gaussian distribution independently of the
initial one. So Gaussianity is a reasonable assumption in our study.

\subsection{From the continuous case}

We have a discrete sequence of samples. The continuous equivalent is a Gaussian
process. We can expect the discrete case to be equivalent to the continuous
case if the autocorrelation time is larger enough than the sampling step, which
should hold from the consideration above.

Also, even though the values are initially discrete too, after filtering the
possible non integer values between two consecutive integers are at least the
length of the filter (think about an average). So we take the formula for the
continuous case and adapt it.

The mean number of threshold upcrossings $r$ in the interval $(0,1)$ by a
zero-mean stationary and appropriately smooth Gaussian process is given by
\cite[81]{rasmussen2006}
%
\begin{align}
    r &= \sqrt{-\frac{k''(0)}{2\pi}} \operatorname{gauss}(u;0,\sigma) = \\
      &= \frac 1 {2\pi} \frac {\sqrt{-k''(0)}} \sigma
         \exp \left( -\frac12 (u/\sigma)^2 \right),
\end{align}
%
where $u$ is the threshold, $\sigma$ the noise standard deviation (the RMS),
$\operatorname{gauss}(x;\mu,\sigma)$ a Gaussian probability density on $x$ with
mean $\mu$ and standard deviation $\sigma$, and $k$ the autocovariance
function, i.e., $k(x) = \operatorname{Cov}[f(t), f(t+x)]$ for any $t$ (for
example, $k(0) = \sigma^2$), where $f(t)$ is the continuous waveform.

We have to map the second derivative of the autocovariance function to a
discrete equivalent. We first do a manipulation in the continuous realm. Since
the covariance operator is an integral, it commutes with derivation:
%
\begin{align}
    k''(x)
    &= \frac{\partial^2}{\partial x^2} \operatorname{Cov}[f(t), f(t+x)] = \\
    &= \operatorname{Cov}[f(t), f''(t+x)],
\end{align}
%
thus $k''(0) = \operatorname{Cov}[f(t), f''(t)]$. We estimate the second
derivative with a finite difference:
%
\begin{align}
    f(t \pm \Delta t)
    &= f(t) \pm f'(t) \Delta t + \frac12 f''(t) \Delta t^2 + O(\Delta t^3)
    \rightarrow \\
    \rightarrow f''(t) \Delta t^2 &=
    f(t + \Delta t) + f(t - \Delta t) - 2 f(t) + O(\Delta t^3).
\end{align}
%
Choosing $\Delta t = 1/f_s$, where $f_s$ is the sampling frequency, and calling
$y_i = f(t_0 + i\Delta t)$ the samples, we have:
\begin{align}
    k''(0) &\mapsto f_s^2 k_2, \\
    k_2 &\equiv \operatorname{Cov}[y_i, y_{i+1}+y_{i-1}-2y_i], \label{eq:k2} \\
    r &= f_s \frac 1 {2\pi} \frac {\sqrt{-k_2}} \sigma
         \exp \left( -\frac12 (u/\sigma)^2 \right).
    \label{eq:rcont}
\end{align}
%
Discretizing directly $k''(0)$ yields the same result.

The covariance in \autoref{eq:k2} can be estimated with the sample
covariance on a filtered waveform array~$\mathbf y$.

\subsection{Direct discrete derivation}

Since the formula we derived is approximate, as a cross check we derive another
approximate one following a different path.

A threshold crossing happens when a sample is below the threshold and the next
one is above: $y_i \leq u$, $y_{i+1} > u$. Fix $i=0$ and let $p(y_0,y_1)$ be the
joint distribution of the two samples. The probability of crossing at any given
point then is
%
\begin{equation}
    P =
    \int_{-\infty}^u \mathrm d y_0\,
    \int_u^\infty \mathrm d y_1\,
    p(y_0, y_1).
    \label{eq:crossingprob}
\end{equation}

In general we can not obtain the crossing rate just by multiplying $P$ by the
sampling frequency because of correlations. However in practice we are
interested in low crossing rates, less than~\SI{10}{cps}, to be compared to the
filter length $\SI{2}{\micro s} = 1/(\SI{500}{kHz})$. If the typical time
between crossings is much longer than the autocorrelation time, then we can
ignore correlations. Thus the crossing rate is $r = f_s P$.

The integrand in \autoref{eq:crossingprob} is a bivariate Gaussian
distribution, which explicitly is
%
\begin{equation}
    p(y_0,y_1) =
    \frac 1 {2\pi \sqrt{\sigma^4 - c^2}}
    \exp \left(
    \frac 1 2
    \begin{pmatrix}
        y_0 & y_1
    \end{pmatrix}
    \begin{pmatrix}
        \sigma^2 & c \\
        c & \sigma^2
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        y_0 \\ y_1
    \end{pmatrix}
    \right),
\end{equation}
%
where $c = \operatorname{Cov}[y_0, y_1]$.

We do not know how to the integral analytically, so we break down the joint
distribution as $p(y_0,y_1) = p(y_1|y_0) p(y_0)$ and discretize the integral
over~$y_0$:
%
\begin{align}
    p(y_0) &= \operatorname{gauss}(y_0; 0, \sigma), \\
    p(y_1|y_0) &= \frac {p(y_0, y_1)} {p(y_0)}
    = \operatorname{gauss} \left(
        y_1; \frac c {\sigma^2} y_0, \sqrt{\sigma^2 - \frac {c^2} {\sigma^2}}
    \right), \\
    P &\approx
    \sum_{k=0}^{N-1} \Delta u\, p(y_0(k,u,\Delta u))
    \int_u^\infty \mathrm d y_1\, p(y_1|y_0(k,u,\Delta u)), \label{eq:rdisc} \\
    y_0(k,u,\Delta u) &\equiv u - k \Delta u.
\end{align}

The integral on $y_1$ can be computed using the error function. $\Delta u$
should be chosen small compared to $\sigma$, while $N$ large relative to
$\sigma / \Delta u$.

In \autoref{fig:crossingprob} we compare formula~\eqref{eq:rcont} (with $f_s
= 1$) with $P$ and with the Gaussian survival function. To make the comparison
we have to use a $k_2$ that matches $c$:
%
\begin{align}
    k_2 &= \operatorname{Cov}[y_i, y_{i+1}+y_{i-1}-2y_i] = \notag \\
    &= \operatorname{Cov}[y_i,y_{i+1}]
    + \operatorname{Cov}[y_i,y_{i-1}]
    - 2 \operatorname{Cov}[y_i,y_i] = \notag \\
    &= 2 (c - \sigma^2). \label{eq:c2k2}
\end{align}
%
We use $\sigma=1$, $c = \SI{99}\%$, $\Delta u = 1/100$, $N=500$ (we decreased
$\Delta u$ until convergence). We see that our derivation and the formula
for Gaussian processes agree very well, while differing visibly from the
survival function. We will henceforth use the continuous formula for its
simplicity.

\begin{figure}
    
    \widecenter{\includempl{figcrossingprob}}

    \figcaption{crossingprob}{The threshold upcrossing rate expressed as
    per-sample crossing probability (i.e., the rate if the sampling frequency
    is~1) for an autocorrelated Gaussian waveform, estimated using three
    formulae: 1) the probability for a single sample to be higher than the
    threshold, 2) a formula for continuous processes (\autoref{eq:rcont}),
    3) an approximation of the discrete case (\autoref{eq:rdisc}).}
    
\end{figure}

\subsection{Dead time}

One part of the data acquisition system (DAQ) for which the study of threshold
crossings is particularly relevant are the digitizers. Due to limited logic
resources in the installed FPGAs, a digitizer can not do complicated
processing. Instead, a simple filter is applied, and the filtered waveform is
compared to the threshold. Whenever the threshold is crossed, a fixed slice of
waveform is sent to the front end processing (FEP) for further analysis
(identify multiple signals, locate them precisely, use a better filter, etc.).
A threshold crossing that happens too close in time to a previous crossing will
be ignored.

This means that we have a dead time $T$. We model it as a non-restartable dead
time, i.e., a crossing that happens within $T$ of a previous one is ignored
only if the latter has not been ignored itself.

Assuming that the crossings are a Poisson process, the formula to correct a
rate $R$ for the effect of the dead time is \cite[120]{knoll2010}:
%
\begin{equation}
    R \mapsto \frac R {1 + RT}. \label{eq:deadrate}
\end{equation}

We note that the crossings of a Gaussian process are not in general a Poisson
process. We just need one counterexample to show this. Consider the process
with autocovariance function $k(x)=\cos(x)$. This is positive definite because
it is a linear combination of external products: $\cos(x-y) = \cos x \cos y +
\sin x \sin y$. Since $\cos$ is orthogonal to $\sin$, they are the
eigenfunctions, so a realization of the process is a random linear combination
of harmonic functions, which means that it is a shifted cosine, so the
crossings are exactly periodic.

However, in practice we expect that there will just be a ``repulsion'' or
``attraction'' of crossings within the scale of the autocorrelation time, so
for low crossing rate the formula should work.

\section{Application to real electrical noise data}

We want to test formula~\eqref{eq:rcont} on actual electrical noise.

\subsection{Data}

We will use the Proto0 run 886, collected when operating the SiPMs below their
breakdown voltage, see \autoref{sec:dataproto0}. Tiles 53, 57 and 59 (used in
Proto0) will be also studied with LNGS data, while for Tile~15 just LNGS data
are available. The LNGS data files are:
%
\begin{verbatim}
FBK/NUV/MB2-LF-3x/NUV-LF_3x_53/nuvhd_lf_3x_tile53_77K_64V_6VoV_1.wav
FBK/NUV/MB2-LF-3x/NUV-LF_3x_53/nuvhd_lf_3x_tile53_77K_66V_7VoV_1.wav
FBK/NUV/MB2-LF-3x/NUV-LF_3x_57/nuvhd_lf_3x_tile57_77K_64V_6VoV_1.wav
FBK/NUV/MB2-LF-3x/NUV-LF_3x_59/nuvhd_lf_3x_tile59_77K_64V_6VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_55V_0VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_59V_2VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_63V_4VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_67V_6VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_71V_8VoV_1.wav
LFOUNDRY/pre-production-test/TILE_15/LF_TILE15_77K_73V_9VoV_1.wav
\end{verbatim}

The Proto0 data consists purely of noise, so no preprocessing is required. For
the LNGS data we take the pre-trigger part of the events, and ignore events
where any pre-trigger sample is less than~750 (860 for Tile~15).

We plot the time-value histogram of the data for Tile 53 in Proto0 and LNGS
(\autoref{fig:hist2dtile53}), for Tile 15 at maximum overvoltage, and for
Tiles 57 and 59 (\autoref{fig:hist2dtile155759}).

\subsection{Filter}

We filter using a \SI1{\micro s} moving average with \SI1{\micro s} of baseline
and \SI1{\micro s} of dead time, without delay between the baseline and the
signal averages. In \autoref{fig:sqfilt} we show an example filtered
waveform and the filter shape.

\begin{figure}

    \widecenter{\includempl{figsqfilt}}

    \figcaption{sqfilt}{A filtered waveform. The computation is split into a
    moving average $m$ (dotted line) and its finite difference $y$ (solid black
    line). The right plot shows the overall filter shape. (We use an event with
    a signal just to see how the filter behaves, the analysis is done on noise
    only.)}

\end{figure}

Although we know from \autoref{ch:snr} and~\ref{ch:timeres} that this is
not the optimal filter by SNR neither by temporal resolution, this is a simple
filter that would be suitable for implementation in the digitizer's FPGA.

Since on LNGS data the events are short compared to the filter length
(\SI9{\micro s} vs.\ \SI2{\micro s}), we need to take into account boundary
effects. The output length of the filtered waveform is $(\text{initial length})
- \SI2{\micro s} + \SI1{sample}$; to compute the rate we have to divide by this
quantity instead of the initial waveform length.

The dead time does not play nicely with borders, because a hypothetical unseen
crossing within \SI1{\micro s} before the event start could kill a crossing in
the event. Moreover, at low thresholds, the first crossing will happen
almost immediately, and again immediately after the dead time ends, thus the
number of crossings per event is quantized.

A complete solution to these problems would be to avoid counting the crossings
which happen within \SI1{\micro s} after the event start (although they are
detected and do project a dead time on the following ones), and in the
remaining region further select a subregion which is \SI1{\micro s} shorter but
has a uniformly random starting position.

However we are mostly interested in the low-rate regime, in which the dead time
boundary effects are negligible, so we keep the whole filtered region.

\subsection{Algorithm}

The simplest way to count the threshold crossings as a function of the
threshold is to repeat the calculation varying the threshold. The computational
complexity is $O(nN)$ where $n$ is the number of thresholds and $N$ the number
of samples.

To produce a smooth curve (large $n$) we use instead a reverse histogram. We
choose an evenly spaced range of thresholds. For each pair of consecutive
samples, if the second sample is higher than the first, we determine the
subrange of thresholds that falls between the samples, and increment their
counts. To apply the dead time, we keep a per-threshold last occurence time.

Since the range of thresholds is evenly spaced, the subrange can be found
arithmetically, so the complexity is just $O(N)$, i.e., it does not depend on
the number of thresholds.

\subsection{Results}
\label{sec:rateconcl}

In \autoref{fig:fakerate1} we compare the measured threshold crossings with
the continuous-derived formula~\eqref{eq:rcont} for a single Tile over the
entire threshold range. The coefficients $\sigma$ and $k_2$ for the formula are
computed on each filtered event and then averaged. Finally, the dead time is
accounted for with~\eqref{eq:deadrate}. For the data, the conversion from count
to rate is done dividing by the length of the filtered waveform (so \SI2{\micro
s} less than the initial length per event).

A discrete agreement can be observed at low and high rates, less so in the
intermediate region. The agreement at high rate is probably improved by
saturation due to dead time, since the maximum rate allowed by dead time is
$1/(\SI1{\micro s}) = \SI1{Mcps}$. Although it cannot be fully appreciated from
\autoref{fig:fakerate1} due to the logarithmic scale, even where the theory and
data lines are closer they still differ by a factor~1.3, while the Poisson
error is approx.~\SI3\% since the count is~1000.

\begin{figure}
    
    \widecenter{\includempl{figfakerate1}}

    \figcaption{fakerate1}{Measured and predicted (with \autoref{eq:rcont})
    fake rate for Tile~31 in Proto0 with a noise-only acquisition. The right
    scale shows the actual count of threshold crossings for the data, with a
    gray band marking one crossing.}
    
\end{figure}

In \autoref{fig:fakerate} we show the same comparison together for all
datasets, divided in three groups (Tile~15 LNGS data, LNGS other Tiles, Proto0
data). The parameters for the formula, and the rates at threshold~$4\,\sigma$,
are listed in \autoref{tab:fakerate}.

For comparison, in \autoref{fig:fakerate} we also show the rate predicted by
\cite[98]{savarese2018}, which provides the following formula:
%
\begin{align}
    r &= \frac{R_0}2 \exp\left(-\frac12 (u/\sigma)^2\right), \\
    R_0 &= \frac 2{\sqrt 3} f_u, \quad f_u = \SI{40}{MHz}.
\end{align}
%
We see that it overestimates the true rate about by a factor of~5-10. Note,
however, that the formula was derived without taking into account filtering, so
the overestimation is expected.

In the second group above some threshold the measured rate stops decreasing and
remains constant at approximately 10 counts. This is probably due to real
pulses which our very simple preprocessing can not filter away, most likely
\SI1{PE} dark noise or random pulses from light leaks.

In \autoref{fig:fakerate} we highlight the measured rates for Tile~53 because
they are evident outliers. The rate remains higher than the theory predicts and
than the other Tiles as the threshold increases. This is visible in Proto0, and
in LNGS at \SI6{VoV}, but not at \SI7{VoV}. Analogously, from
\autoref{tab:fakerate} we see that the quantity $f_s \sqrt{-k_2}/(2\pi\sigma)$
that multiplies the exponential in~\eqref{eq:rcont} is different from the
others for Tile~53, in the same cases as the data, but with the opposite trend.
This variation seems to depend only on an increase in $\sigma$ and not on $k_2$.

Comparing the 2D histograms for Tile~53 (\autoref{fig:hist2dtile53}) to the
others (\autoref{fig:hist2dtile155759}) there is no apparent difference. Three
possible explanations come to mind: 1)~a violation of Gaussianity, 2)~stray
pulses, 3)~low frequency electrical noise.

\marginpar{Looking at the spectrum it is evident that there's a very low
frequency component ($<\SI{10}{kHZ} = 1/(\SI{100}{\micro s})$) absent in other
Tiles. Maybe they are sudden changes of baseline? I should look at events where
crossings with high threshold happen. Try to do the spectrum with windowing to
see if it's a boundary problem.}

Regarding Gaussianity, we checked the distribution of the samples before
filtering and it agrees very well with a Gaussian. If they are stray pulses,
they do not have the same height, otherwise they would show up as a flat rate
curve. It could be oscillating noise with a frequency of
approx.~$1/(\SI2{\micro s}) = \SI{500}{kHz}$, since our filter would be very
good at picking that up, or sudden variations of the baseline.

We note that this behavior arises for the same Tile in different setups,
showing up both as an increased noise RMS and an higher threshold crossing
rate. We also note that the Proto0 data were collected after the LNGS data. As
for the absence of this behavior in the LNGS data collected at \SI{7}{VoV}, we
cannot formulate hypotheses as we do not know in what order the LNGS data at
\SI{6}{VoV} and \SI{7}{VoV} were collected, i.e., whether the tile was damaged
during the testing procedure.

We did not investigate further the discrepancy for Tile~53, since for all
the other Tiles the agreement between our model and data is consistent and
satisfactory.

We now want to estimate the minimum amount of data required for the procedure.
From \autoref{tab:fakerate} we see that the relative error on $k_2$ times the
square root of the time, $C \equiv
\operatorname{Std}[\bar{k_2}]\sqrt{T}/|k_2|$, is approximately \SI8{ns^{1/2}}
in all cases. The error should be proportional to $T^{-1/2}$, so to have an
\SI1\% error we need $T_{\SI1\%} = (100 C)^2 = \SI{0.7}{ms}$.

We conclude this chapter by summarizing all the steps we took to compute the
fake rate:
%
\begin{enumerate}
    
    \item Acquire at least \SI1{ms} of noise data.
    
    \item Filter the data (including baseline subtraction) producing a
    filtered waveform $\mathbf y$.
    
    \item Compute the standard deviation $\sigma$ and $k_2 =
    \operatorname{Cov}[y_i, y_{i+1} + y_{i-1} - 2y_i]$.
    
    \item Compute the fake rate for threshold $u$ using $r =
    f_s\sqrt{-k_2/(2\pi)}\operatorname{gauss}(u;0,\sigma)$ where $f_s$ is the
    sampling frequency.
    
\end{enumerate}
%
Alternatively, if one has a noise spectrum available but not the noise
waveform, it is possible to obtain the autocovariance by computing the discrete
Fourier transform of the power spectrum \cite[84]{ferrante2015} and then
normalizing it to be $\sigma^2$ in 0. Then $k_2$ is obtained by the covariance
at lag 1 $c$ with \eqref{eq:c2k2}. If the spectrum was obtained from the
discrete Fourier transform of a noise waveform, $c$ is the first coefficient
after the central one in the autocovariance.

The model was satisfactory in 24 of the 25 Tiles we considered. The uncertainty
in the low rate regime is \SI{\pm50}\% and the result is an overestimate with
probability \SI{90}\%. These statements are educated guesses based on
\autoref{tab:fakerate}.

\begin{table}
    
    \widecenter{%
    \begin{tabular}{
        c
        S[table-format=2]
        S[table-format=>1]
        S[table-format=3]
        S[table-format=2.1]
        S[table-format=+1.4]
        S[table-format=2.3]
        S[table-format=1.1]
        *2S[table-format=1.2]
    }
        \toprule
        \multicolumn3c{Data}
        &
        &
        &
        &
        &
        & \multicolumn2c{Rate @ $4\,\sigma$} \\
        \cmidrule(r){1-3} \cmidrule(l){9-10}
        
        Setup
        & {Tile}
        & {Overvoltage}
        & {$T$}
        & {$\sigma$}
        & {$k_2$}
        & {$\operatorname{Std}[\bar{k_2}]\sqrt T$}
        & {$f_s \sqrt{-k_2}/(2\pi\sigma)$}
        & {Theory}
        & {Data} \\
        
        &
        & {[\si{V}]}
        & {[\si{ms}]}
        & {[\si{u}]}
        & {[\si{u^2}]}
        & {[\si{u^2 ns^{1/2}}]}
        & {[\si{Mcps}]}
        & {[\si{kcps}]}
        & {[\si{kcps}]} \\
        \midrule
        
  LNGS & 15 &  0 & 138 &  1.6 & -0.0017 & 0.012 & 4.2 &  1.4 &  1.1 \\
  LNGS & 15 &  2 & 138 &  1.5 & -0.0017 & 0.012 & 4.3 &  1.4 &  1.2 \\
  LNGS & 15 &  4 & 138 &  1.5 & -0.0017 & 0.012 & 4.3 &  1.4 &  1.3 \\
  LNGS & 15 &  6 & 138 &  1.5 & -0.0017 & 0.012 & 4.4 &  1.5 &  1.1 \\
  LNGS & 15 &  8 & 138 &  1.5 & -0.0017 & 0.012 & 4.5 &  1.5 &  1.4 \\
  LNGS & 15 &  9 & 138 &  1.4 & -0.0017 & 0.012 & 4.6 &  1.5 &  1.3 \\  \midrule
  LNGS & 53 &  6 &  69 &  3.8 & -0.0044 & 0.030 & 2.7 & 0.92 &  2.6 \\
  LNGS & 53 &  7 &  69 &  2.1 & -0.0043 & 0.029 & 5.0 &  1.7 &  1.0 \\
  LNGS & 57 &  6 &  68 &  2.2 & -0.0043 & 0.031 & 4.8 &  1.6 &  1.2 \\
  LNGS & 59 &  6 &  69 &  2.2 & -0.0040 & 0.028 & 4.6 &  1.5 &  1.0 \\  \midrule
Proto0 & 29 & <0 & 499 &  8.3 &   -0.86 &   10. & 2.2 & 0.74 & 0.77 \\
Proto0 & 30 & <0 & 499 &  6.9 &   -0.75 &   6.0 & 2.5 & 0.84 & 0.67 \\
Proto0 & 31 & <0 & 499 &  7.7 &   -0.82 &   7.0 & 2.3 & 0.78 & 0.60 \\
Proto0 & 32 & <0 & 499 &  7.0 &   -0.76 &   6.6 & 2.5 & 0.83 & 0.67 \\
Proto0 & 34 & <0 & 499 &  6.5 &   -0.78 &   6.2 & 2.7 & 0.90 & 0.62 \\
Proto0 & 36 & <0 & 499 &  7.7 &   -0.84 &   7.1 & 2.4 & 0.79 & 0.63 \\
Proto0 & 37 & <0 & 499 &  6.5 &   -0.69 &   5.4 & 2.5 & 0.85 & 0.67 \\
Proto0 & 38 & <0 & 499 &  7.7 &   -0.84 &   7.1 & 2.4 & 0.80 & 0.59 \\
Proto0 & 39 & <0 & 499 &  7.6 &   -0.91 &   7.5 & 2.5 & 0.84 & 0.68 \\
Proto0 & 41 & <0 & 499 &  7.5 &   -0.87 &   7.1 & 2.5 & 0.83 & 0.71 \\
Proto0 & 42 & <0 & 499 &  7.0 &   -0.82 &   6.8 & 2.6 & 0.87 & 0.63 \\
Proto0 & 52 & <0 & 499 &  7.8 &   -0.82 &   6.5 & 2.3 & 0.77 & 0.61 \\
Proto0 & 53 & <0 & 499 & 10.1 &   -0.90 &   8.0 & 1.9 & 0.63 &  2.3 \\
Proto0 & 54 & <0 & 499 &  8.0 &   -0.89 &   7.4 & 2.4 & 0.79 & 0.59 \\
Proto0 & 55 & <0 & 499 &  8.0 &   -0.85 &   7.1 & 2.3 & 0.77 & 0.63 \\
Proto0 & 57 & <0 & 499 &  7.8 &   -0.88 &   7.4 & 2.4 & 0.80 & 0.69 \\
Proto0 & 58 & <0 & 499 &  7.6 &   -0.82 &   6.4 & 2.4 & 0.79 & 0.65 \\
Proto0 & 59 & <0 & 499 &  7.6 &   -0.79 &   6.6 & 2.3 & 0.77 & 0.55 \\
Proto0 & 60 & <0 & 499 &  7.5 &   -0.77 &   6.3 & 2.3 & 0.78 & 0.61 \\
Proto0 & 61 & <0 & 499 &  8.0 &   -0.89 &   7.4 & 2.4 & 0.79 & 0.60 \\
Proto0 & 62 & <0 & 499 &  7.6 &   -0.80 &   6.6 & 2.3 & 0.78 & 0.62 \\
Proto0 & 63 & <0 & 499 &  8.1 &   -0.86 &   7.1 & 2.3 & 0.76 & 0.61 \\
Proto0 & 64 & <0 & 499 &  7.9 &   -0.81 &   6.9 & 2.3 & 0.76 & 0.66 \\
Proto0 & 65 & <0 & 499 &  7.5 &   -0.80 &   6.4 & 2.4 & 0.80 & 0.62 \\
Proto0 & 66 & <0 & 499 &  8.2 &   -0.91 &   7.5 & 2.3 & 0.77 & 0.61 \\
        
        \bottomrule
    \end{tabular}}
    
    \tabcaption{fakerate}{The coefficients measured on the filtered waveforms
    needed to evaluate the formula for the threshold upcrossing rate. $T$ is
    the total duration, $\sigma$ the standard deviation, $k_2$ the covariance
    of the waveform with its second derivative (\autoref{eq:k2}),
    $\operatorname{Std}[\bar{k_2}]$ the uncertainty on the value of $k_2$
    determined as the standard deviation of the sample mean of $k_2$ values
    across events. The unit ``\si{u}'' is the ADC digit.}
    
\end{table}

\begin{figure}
    
    \widecenter{\includempl{figfakerate}}
    
    \figcaption{fakerate}{The threshold upcrossing rate both counting directly
    the crossings on data and computing it with formula~\eqref{eq:rcont}. The
    gray band marks the rate corresponding to a single crossing counted in the
    data. The thick gray curve is the rate predicted by
    \cite[98]{savarese2018}.}
    
\end{figure}

\begin{figure}
    
    \widecenter{\includempl{fighist2dtile53-0}}

    \widecenter{\includempl{fighist2dtile53-1}}

    \widecenter{\includempl{fighist2dtile53-2}}
    
    \figcaption{hist2dtile53}{Time-value histograms of Tile 53 noise in Proto0
    with a baseline acquisition, and in LNGS pre-trigger at overvoltage \SI6V
    and \SI7V.}
    
\end{figure}

\begin{figure}
    
    \widecenter{\includempl{fighist2dtile155759-0}}

    \widecenter{\includempl{fighist2dtile155759-1}}

    \widecenter{\includempl{fighist2dtile155759-2}}

    \figcaption{hist2dtile155759}{Time-value histograms of Tiles 15, 57 and 59
    noise in LNGS data.}
    
\end{figure}