  The analysis of high-energy air shower data allows one to study the
proton-air cross section at energies beyond the reach of fixed target and
collider experiments. The mean depth of the first interaction point and its
fluctuations are a measure of the proton-air particle production cross section.
Since the first interaction point in air cannot be measured directly, various
methods have been developed in the past to estimate the depth of the first
interaction from air shower observables in combination with simulations. As the
simulations depend on assumptions made for hadronic particle production at
energies and phase space regions not accessible in accelerator experiments, the
derived cross sections are subject to significant systematic uncertainties. The
focus of this work is the development of an improved analysis technique that
allows a significant reduction of the model dependence of the derived cross
section at very high energy. Performing a detailed Monte Carlo study of the
potential and the limitations of different measurement methods, we quantify the
dependence of the measured cross section on the used hadronic interaction
model. Based on these results, a general improvement to the analysis methods is
proposed by introducing the actually derived cross section already in the
simulation of reference showers. The reduction of the model dependence is
demonstrated for one of the measurement methods.
