ToTd: Time-over-Threshold Deconvolution

• ToTd is a trigger technique that applies deconvolution to remove long waveform tails, thus reducing muon-induced background while preserving EM signals.
• It uses a digital inverse filter on FADC signals to compress the diffusely reflected Cherenkov light tail, enhancing sensitivity to neutrino-induced air showers below 10^19 eV.
• Combined with MoPS and Fisher discriminant analysis, ToTd improves overall trigger efficiency and exposure, yielding up to a 5× gain at lower energies.

Time-over-Threshold-deconvolved (ToTd) is a local trigger and signal-processing strategy in which a time-over-threshold condition is applied after deconvolution of the long exponential tail in detector waveforms. In the Surface Detector (SD) of the Pierre Auger Observatory, ToTd was introduced in mid-2013 to compress the tail associated with diffusely reflected Cherenkov light and electronics/PMT shaping before evaluating a ToT-like condition, thereby suppressing isolated-muon backgrounds while retaining sensitivity to extended electromagnetic (EM) signals from neutrino-induced extensive air showers in the downward-going low-zenith range $60^\circ < \theta < 75^\circ$, especially below $10^{19}$ eV (Sehgal, 14 Jul 2025).

1. Detector-level definition and physical motivation

The SD of the Pierre Auger Observatory uses local station-level triggers on digitized water-Cherenkov signals expressed in units of vertical equivalent muon (VEM). Historically, the dominant local triggers were Threshold (TH), optimized for narrow, high-peak muon-like signals, and Time-over-Threshold (ToT), tuned to broader, lower-peak traces characteristic of EM components in extensive air showers. In ultra-high-energy neutrino searches, particularly in the downward-going low-zenith (DGL) interval $60^\circ < \theta < 75^\circ$, the central task is to identify “young,” EM-rich showers and reject “old,” muon-dominated cosmic-ray showers (Sehgal, 14 Jul 2025).

This separation becomes difficult below $10^{19}$ eV. In that regime, neutrino-induced EM signals at ground are weaker and longer, but they can be confused with single muons that generate long exponential tails in water-Cherenkov detectors because of light reflections and electronics shaping. Standard ToT can therefore misfire on tail-dominated muon signals. ToTd was introduced specifically to reduce this failure mode: by compressing the long tail before threshold-duration counting, it reduces spurious time-over-threshold from isolated muons while preserving sensitivity to genuinely extended EM traces. This functional definition distinguishes ToTd from conventional ToT: the relevant threshold condition is not applied to the raw waveform but to a deconvolved estimate of the underlying signal.

2. Signal model and deconvolution principle

In the Auger formulation, SD signals are digitized as FADC waveforms in VEM units. The observed waveform $y[n]$ is modeled as the convolution of an underlying physical light signal $x[n]$ with an effective impulse response $h[n]$,

$y[n] = x[n] * h[n].$

The response $h[n]$ encapsulates PMT/electronics shaping together with diffusely reflected Cherenkov light. ToTd applies a deconvolution with a digital inverse filter $d[n]$, producing a compressed estimate $10^{19}$0,

$10^{19}$1

with $10^{19}$2, or equivalently $10^{19}$3, in the frequency or complex-$10^{19}$4 domain (Sehgal, 14 Jul 2025).

Conceptually, standard ToT counts consecutive FADC bins of $10^{19}$5 above a low threshold, whereas ToTd counts consecutive bins above threshold after tail compression on $10^{19}$6. The purpose is not to alter the EM-rich waveform class of interest, but to suppress the apparent broadness created by single-muon tails. The paper does not provide the exact ToTd filter coefficients, shaping constants, sampling rates, FADC bin widths, or firmware details; it limits the algorithmic description to the deconvolution step and the subsequent ToT-style threshold condition. A common misconception is therefore that ToTd in this context denotes a fully specified digital filter. In the cited neutrino-search analysis, the operational meaning is narrower: deconvolution is used to compress the exponential tail before threshold-duration logic is applied.

3. Trigger architecture and integration with event-level neutrino selection

ToTd was deployed together with Multiplicity-of-Positive Steps (MoPS), another EM-sensitive local trigger. MoPS targets long, non-smooth, low-amplitude EM signals by counting positive steps in the waveform, rather than requiring a smooth broad pulse. At the station level, ToTd and MoPS operate at a few Hz per station and are combined in a logical OR with the standard ToT, effectively lowering the array’s energy threshold for EM-rich showers (Sehgal, 14 Jul 2025).

Trigger	Signal preference	Role in DGL neutrino analysis
TH	Narrow, high-peak muon-like signals	Historical local trigger
ToT	Broader, lower-peak EM-like signals	Standard EM-sensitive trigger
ToTd	ToT-like condition after tail compression	Suppresses tail-induced muon background
MoPS	Long, non-smooth, low-amplitude EM signals	Adds sensitivity to extended step-like traces

At the event level, the DGL analysis imposes containment within the SD, a well-reconstructed zenith angle, and at least four triggered stations. A stricter EM criterion is then enforced: at least $10^{19}$7 of the stations closest to the core must have an EM-sensitive local trigger, meaning ToT, ToTd, or MoPS. This requirement is designed to impose an EM-rich footprint near the shower axis and suppress muon-dominated cosmic-ray background.

The same analysis incorporates Fisher Discriminant Analysis in five zenith sub-ranges to account for shower-age effects. The discriminant uses the sum of Area-over-Peak (AoP) from the four earliest stations near the core. AoP is the integrated signal divided by the peak; muon-like signals yield $10^{19}$8, while EM signals yield $10^{19}$9. The Fisher threshold is tuned such that the expected background is $60^\circ < \theta < 75^\circ$0 event in 20 years over the full DGL range. Within this chain, ToTd contributes by reducing the probability that narrow, tail-dominated muon signals are misclassified as broad EM traces.

4. Energy and angular dependence of the ToTd gain

Simulated neutrino events reconstructed with all triggers—ToTd, MoPS, ToT, and TH—were compared with events reconstructed using only ToT and TH. The increase in reconstructed-event yield is largest at energies below $60^\circ < \theta < 75^\circ$1 EeV, where weak, extended EM signals are most likely to benefit from the added trigger sensitivity. The gain decreases toward higher energies because ToT+TH were already efficient there (Sehgal, 14 Jul 2025).

The benefit also depends on zenith angle. At smaller zenith angles within $60^\circ < \theta < 75^\circ$2–$60^\circ < \theta < 75^\circ$3, neutrino showers retain more EM content at ground, which increases the capture probability of ToTd and MoPS. At larger zenith angles, closer to $60^\circ < \theta < 75^\circ$4, improvements arise mainly for interactions occurring close to the array, where the signal remains EM-rich at ground. More distant interactions produce increasingly muonic traces because of attenuation, and the incremental benefit of EM-sensitive triggers correspondingly decreases.

The abstract reports that adding ToTd and MoPS increased the detection capability for neutrino-induced air showers below $60^\circ < \theta < 75^\circ$5 eV by a factor of $60^\circ < \theta < 75^\circ$6–$60^\circ < \theta < 75^\circ$7. In exposure terms, the DGL analysis shows up to a $60^\circ < \theta < 75^\circ$8 increase at lower energies, especially for $60^\circ < \theta < 75^\circ$9 charged-current interactions, when all triggers are used rather than ToT+TH alone. All-trigger reconstructions also tend to have larger station multiplicity, which strengthens downstream selection, including the Fisher discriminant, and improves overall neutrino sensitivity.

5. Role in the 2014–2021 DGL search and resulting constraints

The Auger DGL search using ToTd and MoPS covered the period from 2014-01-01 to 2021-12-31. The SD array was modeled as evolving in time through the number of functional hexagons $10^{19}$0, with effective Brillouin area per hexagon $10^{19}$1. For flavor $10^{19}$2 and interaction type $10^{19}$3, the exposure is

$10^{19}$4

Over the search period, the integrated number of hexagons was $10^{19}$5, equivalent to $10^{19}$6 years of full-array operation (Sehgal, 14 Jul 2025).

The total exposure is $10^{19}$7, assuming a $10^{19}$8 flavor ratio at Earth. The electron-neutrino charged-current channel dominates at approximately $10^{19}$9 of the total, while neutral-current interactions contribute approximately $y[n]$0. With ToTd and MoPS included, exposure at lower energies increases by up to $y[n]$1 relative to ToT+TH only, especially for $y[n]$2 charged-current events; the improved Fisher discriminant also contributes at higher energies.

No neutrino candidates were found in any of the five DGL sub-ranges. For zero observed events and zero background, the diffuse single-flavor $y[n]$3 limit used the Feldman–Cousins approach extended for systematics, with $y[n]$4. The corresponding integrated limit is

$y[n]$5

and, after accounting for systematic uncertainties of $y[n]$6 on exposure, the result is

$y[n]$7

over the energy interval in which approximately $y[n]$8 of events are expected for an $y[n]$9 flux, $x[n]$0. Inclusion of ToTd and MoPS yields an approximately $x[n]$1 improvement in the DGL-only diffuse limit relative to using ToT+TH.

For point sources, the time-dependent source visibility at declination $x[n]$2 and observatory latitude $x[n]$3 is modeled as

$x[n]$4

and the point-source limit is

$x[n]$5

With ToTd, MoPS, and improved discriminants, point-source sensitivity improves by up to $x[n]$6 at the most favorable declinations compared with DGL analyses based only on ToT+TH. The DGL channel contributes directional exposure between $x[n]$7 and $x[n]$8, and portions of the sky at $x[n]$9 are visible only in DGL, complementing the DGH and ES channels.

6. Broader analytical formulations, deconvolution-like variants, and limitations

Outside the Auger implementation, the general logic of ToTd appears in analytical and DAQ studies that treat measured pulses as a convolution of an underlying signal with detector and electronics response. In plastic scintillators equipped with silicon photomultipliers, the measured voltage is modeled as

$h[n]$0

with $h[n]$1. In the frequency domain, this gives

$h[n]$2

If digitized waveforms are available, one can deconvolve by applying the inverse filter $h[n]$3, optionally with Wiener regularization to prevent noise amplification; in the time domain, this is equivalent to adding a scaled derivative term. Under that model, deconvolution largely removes the long RC recharge tail and yields a waveform closer to a single exponential, for which the threshold-crossing law becomes $h[n]$4. This suggests a route to improved amplitude mapping and reduced time-walk when waveform processing is available, although those ToTd benefits are presented as expectations under the model rather than as an implemented Auger trigger result (Karpushkin et al., 2023).

A related but distinct development appears in multi-threshold PMT DAQ. A three-threshold Multiple-Time-over-Threshold system with a 100 ps TDC was described for the HELYCON extensive-air-shower detector. That work does not explicitly use the term “ToTd,” but it states that three thresholds “compensate for the slewing effects and offer a more accurate measurement of the PMT pulses' width,” and enable energy reconstruction by estimating waveform charge. In that sense, it realizes a deconvolution-like correction of threshold and pulse-shape dependence through redundant timing information rather than explicit inverse filtering. The reported card achieved approximately $h[n]$5 ns ToT resolution, with overall output jitter of approximately $h[n]$6 ps and power consumption of approximately $h[n]$7 W (Georgakopoulou et al., 2017).

These broader formulations clarify both the scope and the limits of ToTd. In the Auger neutrino search, ToTd is a specific EM-sensitive trigger that deconvolves the waveform tail before a ToT-like condition. It is not a generic replacement for ToT, and the exact numerical thresholds, filter coefficients, and firmware implementation are not given in the cited neutrino-search paper. Its gains are most pronounced below $h[n]$8 eV and in the DGL angular band; at high energies the improvement is characterized as modest, and the overall Auger diffuse limit remains dominated by the Earth-skimming channel. Even so, the method is crucial for improving sensitivity at $h[n]$9 eV and for declinations uniquely accessible in DGL, and it motivates further refinements to reconstruction and selection algorithms (Sehgal, 14 Jul 2025).