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We present an improved method for the precise reconstruction of cosmic ray air showers above $10^{17}$ eV with sparse radio arrays. The method is based on the comparison of predictions for radio pulse shapes by CoREAS simulations to measured pulses. We applied our method to the data of Tunka-Rex, a 1 km$^2$ radio array in Siberia operating in the frequency band of 30-80 MHz. Tunka-Rex is triggered by the air-Cherenkov detector Tunka-133 and by scintillators (Tunka-Grande). The instrument collects air-shower data since 2012. The present paper describes updated data and signal analyses of Tunka-Rex and details of a new method applied. After efficiency cuts, when Tunka-Rex reaches its full efficiency, the energy resolution of about 10% given by the new method has reached the limit of systematic uncertainties due to the calibration uncertainty and shower-to-shower fluctuations. At the same time the shower maximum reconstruction is significantly improved up to an accuracy of 35 g/cm$^2$ compared to the previous method based on the slope of the lateral distribution. We also define and now achieved conditions of the measurements, at which the shower maximum resolution of Tunka-Rex reaches a value of 25 g/cm$^2$ and becomes competitive to optical detectors. To check and validate our reconstruction and efficiency cuts we compare individual events to the reconstruction of Tunka-133. Furthermore, we compare the mean of shower maximum as a function of primary energy to the measurements of other experiments.
Tunka-Rex is a radio detector for cosmic-ray air showers in Siberia, triggered by Tunka-133, a co-located air-Cherenkov detector. The main goal of Tunka-Rex is the cross-calibration of the two detectors by measuring the air-Cherenkov light and the ra
Tunka-Rex (Tunka Radio Extension) is an antenna array for cosmic-ray detection located in Siberia. Previous studies of cosmic rays with Tunka-Rex have shown high precision in determining the energy of the primary particle and the possibility to recon
The Tunka Radio Extension (Tunka-Rex) is a digital antenna array for the detection of radio emission from cosmic-ray air showers in the frequency band of 30 to 80 MHz and for primary energies above 100 PeV. The standard analysis of Tunka-Rex includes
The recent progress in the radio detection technique for air showers paves the path to future cosmic-ray radio detectors. Digital radio arrays allow for a measurement of the air-shower energy and depth of its maximum with a resolution comparable to t
We describe a method of reconstructing air showers induced by cosmic rays using deep learning techniques. We simulate an observatory consisting of ground-based particle detectors with fixed locations on a regular grid. The detectors responses to trav