We present an improved method for the precise reconstruction of cosmic-ray air showers above $10^{17}\mathrm{eV}$ with sparse radio arrays. The method is based on the comparison of measured pulses to predictions for radio pulse shapes by CoREAS simulations. We applied our method to the data of Tunka-Rex, a $1{\mathrm{km}}^2$ 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 an updated data analysis of Tunka-Rex and details of the new method applied. After quality 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 has improved compared to the previous method based on the slope of the lateral distribution and reaches a precision of better than $35\mathrm{g}/{\mathrm{cm}}^2$. We also define conditions of the measurements at which the shower maximum resolution of Tunka-Rex reaches a value of $25\mathrm{g}/{\mathrm{cm}}^2$ 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 the shower maximum as a function of primary energy to the measurements of other experiments.

• Received 5 April 2018

DOI:https://doi.org/10.1103/PhysRevD.97.122004