  The multifractal geometry remains an under-exploited approach to describe and
quantify the large-scale structure of interstellar clouds. In this paper, the
typical tools of multifractal analysis are applied to Herschel far-infrared
(70-500 $\mu$m) dust continuum maps, which represent an ideal case of study.
This dust component is a relatively optically thin tracer at these wavelengths
and the size in pixel of the maps is well suitable for this statistical
analysis. We investigate the so-called multifractal spectrum and generalised
fractal dimension of six Hi-GAL regions in the third Galactic quadrant. We
measure systematic variations of the spectrum at increasing wavelength, which
generally correspond to an increasing complexity of the image, and we observe
peculiar behaviours of the investigated fields, strictly related to the
presence of high-emission regions, which in turn are connected to star
formation activity. The same analysis is applied to synthetic column density
maps, generated from numerical turbulent molecular cloud models and from
fractal Brownian motion (fBm), allowing for the confrontation of the
observations with models with well controlled physical parameters. This
comparison shows that, in terms of multifractal descriptors of the structure,
fBm images exhibit a very different, quite unrealistic behaviour when compared
with Hi-GAL observations, whereas the numerical simulations appear in some
cases (depending on the specific model) more similar to the observations.
Finally, the link between mono-fractal parameters (commonly found in the
literature) and multifractal indicators is investigated: the former appear to
be only partially connected with the latter, with the multifractal analysis
offering a more extended and complete characterization of the cloud structure.
