  We propose a statistical tool to compare the scaling behaviour of turbulence
in pairs of molecular cloud maps. Using artificial maps with well defined
spatial properties, we calibrate the method and test its limitations to
ultimately apply it to a set of observed maps. We develop the wavelet-based
weighted cross-correlation (WWCC) method to study the relative contribution of
structures of different sizes and their degree of correlation in two maps as a
function of spatial scale, and the mutual displacement of structures in the
molecular cloud maps. We test the WWCC for circular structures having a single
prominent scale and fractal structures showing a self-similar behavior without
prominent scales. Observational noise and a finite map size limit the scales
where the cross-correlation coefficients and displacement vectors can be
reliably measured. For fractal maps containing many structures on all scales,
the limitation from the observational noise is negligible for signal-to-noise
ratios >5. (abbrev). Application of the WWCC to the observed line maps of the
giant molecular cloud G333 allows to add specific scale information to the
results obtained earlier using the principle component analysis. It confirms
the chemical and excitation similarity of $^{13}$CO and C$^{18}$O on all
scales, but shows a deviation of HCN at scales of up to 7' (~7 pc). This can be
interpreted as a chemical transition scale. The largest structures also show a
systematic offset along the filament, probably due to a large-scale density
gradient. The WWCC can compare correlated structures in different maps of
molecular clouds identifying scales that represent structural changes such as
chemical and phase transitions and prominent or enhanced dimensions.
