In-plane, fractional-order diffraction-data sets from thin Au layers on Si(111) with (square-root 3 X square-root 3)R 30-degrees and 6 X 6 structures were measured at the wiggler beamline W1 at the Hamburg synchrotron radiation laboratory. For the square-root 3 X square-root 3 structure the trimer model is confirmed with an Au-Au distance of 2.8 angstrom. In the square-root 3 X square-root 3 unit cell, two additional sites beside the Au trimer were found which can be identified with distorted substrate layers or additional partially occupied Au sites. The 6 X 6 structure is a sixfold twinned structure. The observed Patterson function clearly indicates the main features of the structural units. Each consists of three trimer clusters of Au atoms, forming a nearly equilateral triangle. The local structure of each trimer is either the original square-root 3 X square-root 3 structure or a twin structure where the Au trimers are rotated by 60-degrees. Three of these structural units form the 6 X 6 unit cell. Model calculations with incoherent superposition of twin domains lead to Patterson maps very similar to the one observed.