  The Tunka Radio Extension (Tunka-Rex) is an array of 63 antennas located in
the Tunka Valley, Siberia. It detects radio pulses in the 30-80 MHz band
produced during the air-shower development. As shown by Tunka-Rex, a sparse
radio array with about 200 m spacing is able to reconstruct the energy and the
depth of the shower maximum with satisfactory precision using simple methods
based on parameters of the lateral distribution of amplitudes. The LOFAR
experiment has shown that a sophisticated treatment of all individually
measured amplitudes of a dense antenna array can make the precision comparable
with the resolution of existing optical techniques. We develop these ideas
further and present a method based on the treatment of time series of measured
signals, i.e. each antenna station provides several points (trace) instead of a
single one (amplitude or power). We use the measured shower axis and energy as
input for CoREAS simulations: for each measured event we simulate a set of
air-showers with proton, helium, nitrogen and iron as primary particle (each
primary is simulated about ten times to cover fluctuations in the shower
maximum due to the first interaction). Simulated radio pulses are processed
with the Tunka-Rex detector response and convoluted with the measured signals.
A likelihood fit determines how well the simulated event fits to the measured
one. The positions of the shower maxima are defined from the distribution of
chi-square values of these fits. When using this improved method instead of the
standard one, firstly, the shower maximum of more events can be reconstructed,
secondly, the resolution is increased. The performance of the method is
demonstrated on the data acquired by the Tunka-Rex detector in 2012-2014.
