Likelihood Reconstruction for Radio Detectors of Neutrinos and Cosmic Rays

TL;DR

This work develops a probabilistic noise model for radio detectors that captures bin-to-bin correlations, enabling a full likelihood reconstruction for neutrino and cosmic-ray signals. By expressing the noise as a multivariate normal with a tractable covariance matrix (and using the Moore-Penrose pseudoinverse when needed), the authors obtain event-by-event parameter estimates with correct coverage and quantified uncertainties. They demonstrate the approach across in-ice neutrino signals, fast Fisher-based uncertainty estimation for detector optimization, and cosmic-ray electric-field reconstruction from dual-polarized antennas, showing reduced biases and substantial improvements in reconstruction precision over traditional chi-square or unfolding methods. The framework also supports discriminating signal from background and provides a scalable path toward differentiable, end-to-end detector design; the NuRadioReco package implements the methods for broad use in the field.

Abstract

Ultra-high-energy neutrinos and cosmic rays are excellent probes of astroparticle physics phenomena. For astroparticle physics analyses, robust and accurate reconstruction of signal parameters such as arrival direction and energy is essential. Radio detection is an established detector concept explored by many observatories; however, current reconstruction methods ignore bin-to-bin noise correlations, which limits reconstruction resolution and, so far, has prevented calculations of event-by-event uncertainties. In this work, we present a likelihood description of neutrino or cosmic-ray signals in radio detectors with correlated noise, as present in all neutrino and cosmic-ray radio detectors. We demonstrate, with simulation studies of both neutrinos and cosmic-ray radio signals, that signal parameters such as energy and direction, including event-by-event uncertainties with correct coverage, can be obtained. This method reduces reconstruction uncertainties and biases compared to previous approaches. Additionally, the Likelihood can be used for event selection and enables differentiable end-to-end detector optimization. The reconstruction code is available through the open-source software NuRadioReco.

Figures (12)

• Figure 1: Different noise spectra and resulting time domain noise realizations with an injected simulated neutrino signal. The 'realistic' band-limited spectrum is made using a Butterworth filter and results in correlated noise.
• Figure 2: Top left: The covariance matrix calculated empirically from 2000 traces of generated noise using the 'realistic' spectrum shown in Fig. \ref{['fig_noise']}. Top right: Zoom in on the first 30ns by 30ns of the covariance matrix. Bottom left: Covariance matrix calculated empirically from 2000 traces of generated noise averaged over the diagonals according to a symmetric circulant structure (zoom in on the first 30ns by 30ns). Bottom right: The values of the diagonals of the averaged covariance matrix as a function of $\Delta t = t_j - t_i$.
• Figure 3: One row of the covariance matrix (left) and its inverse (right) for simulated noise calculated empirically from 2000 traces of noise using Eq. \ref{['eq_covariance_ij']} and analytically from the spectrum using Eq. \ref{['eq_analytical_covariance']}. The difference between the two calculations is less than 4‰ of the element on the diagonal ($\Delta t = 0$), which we attribute to the statistical uncertainties of calculating the covariance matrix from a limited number of noise realizations.
• Figure 4: $-2 \Delta \ln{p}$ distributions for 10000 realizations of simulated noise using the 'realistic' spectrum in Fig. \ref{['fig_noise']}. The distributions have been calculated with the inverse covariance matrix (Eq. \ref{['eq_analytical_inverse_2']}) using the full spectrum except the first and last frequency (left) and ignoring all frequency amplitudes below 1% of the maximal amplitude of the spectrum (right). Alongside the distributions are chi-square distributions with the expected degrees of freedom for the two cases shown.
• Figure 5: Simulated neutrino signal in 16 antennas with one realization of noise for event Event 3 in Table \ref{['tab_events']}.