The Pierre Auger Collaboration recently introduced a deep-learning-based method to reconstruct the depth of maximum air shower profiles, X_{max}, using the Surface Detector (SD) of the Pierre Auger Observatory. This depth, at which the number of particles in an air shower induced by an ultra-high-energy cosmic ray (UHECR), reaches the maximum is one of the most powerful observables for inferring the nature of primary particles. Therefore X_{max} measurements play a key role in the determination of the mass composition of UHECRs.

In our publication, the performance of the deep-learning-based X_{max} reconstruction was extensively studied on simulated air showers and measured data. It was shown that the method is precise and can be used to obtain detailed information about the mass composition of UHECRs. This makes the developed method a promising candidate to provide new insights into the nature of cosmic rays with energies up to 100 EeV.

So far, measurements of X_{max} are mainly based on observations with the Fluorescence Detector, which operates only 15% of the time compared to nearly 100% duty cycle of SD. This work introduces a new algorithm based on deep learning, a form of artificial intelligence developed on the basis of neural networks. The algorithm was designed to exploit the signal footprint measured by the SD (see Figure 1) to reconstruct X_{max}, enabling an increase in statistics by a factor of 10 to 15 compared to the FD.

In detail, computer vision techniques are used to analyze the footprint and the arrival time of the shower front at the detector stations, and methods from speech recognition are used to analyze their detected signal traces. The approach is based on the fact that various particle components of the air shower contain different information about X_{max} and leave characteristic signals in the detector. Those are analyzed using the deep-learning-based algorithm.

**Design and training of the algorithm**

The algorithm is based on deep neural networks (DNNs), which have gained popularity in recent years. As the algorithm's first part, the signal traces are exploited using a sub-network. It consists of recurrent Long-Short Term Memory (LSTM) layers, which are particularly suitable for analyzing time series. In the next part, the extracted information is combined with the arrival times and information about the status of the detector station. Finally, after a network part very similar to ResNet (a powerful architecture designed by Microsoft Research) but modified to use hexagonal convolutions to exploit the hexagonal structure of the SD grid, X_{max} is predicted.

Almost half a million air-shower events simulated using the hadronic interaction model EPOS-LHC are used for training the machine learning algorithm. The simulated air showers feature energies between 1 and 160 EeV and are induced by either protons, helium, oxygen, or iron nuclei. Training the network for 150 epochs using an Nvidia Geforce 1080 GTX takes around 60 hours.

**Evaluation of the algorithm on simulations and data**

The network's performance was extensively investigated using showers simulated by various hadronic interaction models. A high correlation between simulated (true) and reconstructed X_{max} was found. In Figure 2, the correlation between the reconstruction of the DNN and the Monte Carlo simulation around 15 EeV is shown. Here, the correlation amounts to around 0.82 for protons and 0.52 for iron nuclei. This is a significant improvement compared to the more traditional analysis of the SD time traces with the Delta method.

The resolution is energy and composition-dependent and ranges from 30-45 g/cm² (at 3 EeV) to 15-30 g/cm² (above 100 EeV). This behavior is very similar for all investigated hadronic interaction models (EPOS-LHC, Sibyll 2.3, and QGSJetII-04). However, a shift is found when applying the DNN trained using EPOS-LHC showers to events simulated using Sibyll 2.3 and QGSJetII-04, which amounts to -15 g/cm². This finding indicates that a calibration of the algorithm is needed to ensure an unbiased estimate of X_{max} using the SD.

One core feature of the Pierre Auger Observatory is its hybrid design (fluorescence telescopes overlook the large SD array) that provides two independent reconstructions, one from the SD and one from the FD, for one and the same event. This so-called *Auger hybrid data set*, consisting of measurements using two complementary detection mechanisms, provides the opportunity for accurate tests, calibrations, and crosschecks. For example, this set is used to calibrate the well-established energy reconstruction of the SD. A similar procedure can be performed using these data to calibrate the DNN X_{max} reconstruction and ensure that the reconstruction is independent of shortcomings in the simulation (to which the DNN may be sensitive).

The calibration using the hybrid data is depicted in Figure 3. On the left side, the energy-dependent difference between the DNN reconstruction and the FD observation is shown as a function of energy. It was found that a constant offset of roughly -30 g/cm² can describe this bias precisely and can be used to calibrate the algorithm. The finding of this offset, moderately above the expectations from simulation studies (-15 g/cm²), indicates that the current generation of hadronic interaction models and/or the detector simulation can not describe air shower measurements in full detail.

The estimation of the resolution is shown in Figure 3 (b). After correcting the Auger hybrid measurements for the FD resolution (dashed grey line), the resolution of the DNN on data is extracted (dashed red line). The resolution improves from 40 g/cm² at low energies to 25 g/cm² above 20 EeV, well in line with the observation on simulations.

This precision will enable new insights into the UHECR mass composition at the highest energies and opens up new possibilities for statistically significant analysis strategies that use mass information at the event-by-event level.

**Related paper:**

**Deep-Learning based Reconstruction of the Shower Maximum X _{max} using the Water-Cherenkov Detectors of the Pierre Auger Observatory**

The Pierre Auger Collaboration, JINST 16 P07019 (2021)

[arxiv.org/abs/2101.02946] [doi: 10.1088/1748-0221/16/07/P07019]