  The study of water bodies such as rivers is an important problem in the
remote sensing community. A meaningful set of quantitative features reflecting
the geophysical properties help us better understand the formation and
evolution of rivers. Typically, river sub-basins are analysed using Cartosat
Digital Elevation Models (DEMs), obtained at regular time epochs. One of the
useful geophysical features of a river sub-basin is that of a roughness measure
on DEMs. However, to the best of our knowledge, there is not much literature
available on theoretical analysis of roughness measures. In this article, we
revisit the roughness measure on DEM data adapted from multiscale
granulometries in mathematical morphology, namely multiscale directional
granulometric index (MDGI). This measure was classically used to obtain
shape-size analysis in greyscale images. In earlier works, MDGIs were
introduced to capture the characteristic surficial roughness of a river
sub-basin along specific directions. Also, MDGIs can be efficiently computed
and are known to be useful features for classification of river sub-basins. In
this article, we provide a theoretical analysis of a MDGI. In particular, we
characterize non-trivial sufficient conditions on the structure of DEMs under
which MDGIs are invariant. These properties are illustrated with some
fictitious DEMs. We also provide connections to a discrete derivative of volume
of a DEM. Based on these connections, we provide intuition as to why a MDGI is
considered a roughness measure. Further, we experimentally illustrate on
Lower-Indus, Wardha, and Barmer river sub-basins that the proposed features
capture the characteristics of the river sub-basin.
