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We reconstructed the energy and the position of the shower maximum of airshowers with energies $E \gtrsim 100 $PeV applying a method using radiomeasurements performed with Tunka-Rex. An event-to-event comparison toair-Cherenkov measurements of the same air showers with the Tunka-133photomultiplier array confirms that the radio reconstruction works reliably.The Tunka-Rex reconstruction methods and absolute scales have been tuned onCoREAS simulations and yield energy and $X_{\mathrm{max}}$ values consistentwith the Tunka-133 measurements. The results of two independent measurementseasons agree within statistical uncertainties, which gives additionalconfidence in the radio reconstruction. The energy precision of Tunka-Rex iscomparable to the Tunka-133 precision of $15 %$, and exhibits a $20 %$uncertainty on the absolute scale dominated by the amplitude calibration of theantennas. For $X_{\mathrm{max}}$, this is the first direct experimentalcorrelation of radio measurements with a different, established method. At themoment, the $X_{\mathrm{max}}$ resolution of Tunka-Rex is approximately $40$g/cm$^2$. This resolution can probably be improved by deploying additionalantennas and by further development of the reconstruction methods, since thepresent analysis does not yet reveal any principle limitations.

Radio measurements of the energy and the depth of the shower maximum of cosmic-ray air showers by Tunka-Rex