  The present article proposes a data analysis method for determining surface
structures from experimental results, which consists of a non-linear
optimization procedure and a sensitivity analysis. This method solves
numerically the partial differential equation in the fully-dynamical quantum
diffraction theory with many trial surface structures. In the sensitivity
analysis, we focused on the experimental uncertainties and the variation over
individual fitting parameters, which was analyzed by solving the eigenvalue
problem of the variance-covariance matrix. A modern massively parallel
supercomputer was used to complete the analysis within a moderate computational
time. The sensitivity analysis provides a basis for the choice of variables in
the data analysis for practical reliability. The significance of the present
analysis method was demonstrated in the structure determination of a
Si$_4$O$_5$N$_3$ / 6H-SiC(0001)-($\sqrt{3} \times \sqrt{3}$) R30$^\circ$
surface in a total-reflection high-energy positron diffraction (TRHEPD)
experiment.
