A Bayesian Method for Air-Shower Reconstruction using Information Field Theory

Abstract

The radio detection of extensive air showers provides a powerful method for studying the origin of high-energy cosmic rays. The Low-Frequency Array (LOFAR) offers unprecedentedly detailed measurements of the radio emission footprint. However, fully exploiting this information requires advanced reconstruction techniques. In this paper, we introduce a novel framework for air shower reconstruction based on Bayesian inference and Information Field Theory (IFT). Our method is built on a fully differentiable forward model of the radio signal, which incorporates a physical emission parameterization and a precise wavefront model. Additionally, we augment this physical model with Gaussian processes to account for systematic uncertainties in both the signal fluence and arrival timing. By leveraging gradient information, our approach enables efficient (three orders of magnitude acceleration w.r.t.\ the legacy method) and robust inference of the underlying physical shower parameters, such as primary energy and the depth of shower maximum, $X_\text{max}$. This work provides not only point estimates but also a rigorous quantification of uncertainties. We achieve a resolution in $X_\text{max}$ of $25\,\mathrm{g/cm^2}$ and a radiation energy resolution of $12\%$ on simulations for LOFAR.

Figures (11)

• Figure 1: The radio pattern of the two emission mechanisms in the 30-80MHz band, the Geomagnetic effect (left) and the Charge Excess effect (middle) along with the direction of their E-Field indicated by arrows. The total emission (right) shows a characteristic asymmetric pattern due to the interference of the two effects. Patterns produced using parameterization from Glaser:2019.
• Figure 2: Dependence of the optimal timing parameter $C$ on the slant distance to the shower maximum, $D_{X_{\mathrm{max}}}$. Each point represents an individual CoREAS simulation, where the signal times were determined via the Hilbert envelope maximum of the total electric field trace, colored by the root-mean-square (RMS) of the timing residuals for the best fit. The solid black line shows the fit using a 3rd-degree polynomial (Tab. \ref{['tab:poly_fit']}), with the shaded region indicating the root-mean-square (RMS) deviations from the fit. The dashed gray line indicates the fixed value used by LOPES, $C = 25200~\mathrm{g/cm}^{2}$).
• Figure 3: Reconstruction summary for a simulated air shower event, an iron primary with an energy of 2.44e+17eV. The first two columns display the ground distributions of fluence and arrival timing (with the plane-wave component subtracted to better emphasize the hyperbolic shower front), showing (top to bottom) input data at LOFAR positions, CoREAS truth, reconstructed Posterior means, relative uncertainties, and residuals. The third column maps the inferred spatially Correlated Fields (CF). The fourth column presents the marginal posterior distributions for the depth of shower maximum $X_{\text{max }}$, radiation energy ($E_{\text{rad}}$), core position ($X_0, Y_0$), and arrival direction ($\theta, \phi$), with ground truth values indicated. The last row compares obtained differences with the corresponding pull.
• Figure 4: Performance of the radiation energy $E_\text{rad}$ reconstruction on CoREAS simulations. The left side displays both the reconstructed values compared to the CoREAS truth for all 390 successful reconstructions, as well as the pull from the reconstruction. The right side of the plot shows a histogram of the reconstruction errors at the top and how the reconstruction uncertainties correlate with reconstruction error.
• Figure 5: Performance of the $X_\text{max}$ reconstruction on CoREAS simulations. (Left) Reconstructed vs. true $X_\text{max}$ and pull distribution. (Right) Error distribution and uncertainty correlation.