Quantifying energy fluence and its uncertainty for radio emission from particle cascades in the presence of noise

Measurements of radio signals induced by an astroparticle generating a cascade present a challenge because they are always superposed with an irreducible noise contribution. Quantifying these signals constitutes a non-trivial task, especially at low signal-to-noise ratios (SNR). Because of the randomness of the noise phase, the measurements can be either a constructive or a destructive superposition of signal and noise. To recover the electromagnetic energy of the cascade from the radio measurements, the energy fluence, i.e. the time integral of the Poynting vector, has to be estimated. Conventionally, noise subtraction in the time domain has been employed for energy fluence reconstruction, yielding significant biases at low signal-to-noise ratios. In several analyses, this bias is mitigated by imposing an SNR threshold cut, though this option is not ideal as it excludes valuable data. Additionally, the uncertainties derived from the conventional method are underestimated, even for large SNR values. To address this known issue, the uncertainties have so far typically been approximated and corrected by using ad-hoc terms. This work tackles these challenges by detailing a method to correctly estimate the uncertainties and lower the reconstruction bias in quantifying radio signals, thereby, ideally, eliminating the need for an SNR cut. The development of the method is based on a robust theoretical and statistical background, and the estimation of the fluence is performed in the frequency domain, allowing for the improvement of further analyses by providing access to frequency-dependent fluence estimation.