IEEE Geoscience and Remote Sensing Magazine - December 2020 - 139

example that I can quote here is that " we must keep in mind
that shape matters " [56].
DEM users are often unaware of the technicalities of
the model's generation, provenance, and error characteristics. This information is often not available or unfamiliar. For example, the SRTM DEM measures the electrical height of the surface as seen by the radar, not the
ground or canopy top measured by other techniques. In
addition, DEM analysis tools are often used without a
deep understanding of the methods employed. As a case
in point, the methods described in this article were developed by specialists and require some dedication to
understand and apply properly. For geoscientists, it is
a challenge of the utmost importance to decipher the
geometric, topologic, and morphologic relevance data
concealed by DEMs, and the geoscientist cannot expect
the algorithm developer approaching the analysis from
a mathematical perspective to provide a solution for any
DEM or geographical context. Therefore, it is imperative
that geoscientists have some facility with the theory, algorithm development, and computer code generation to
collaborate with and guide the specialists in their sophisticated endeavor.
This element is presently missing from the geoscience curriculum. It is challenging to educate budding
young geoscientists to become mathematical geoscientists because, in many cases, the mathematical underpinnings needed are not prerequisites. Undergraduate
and postgraduate curricula on mathematical knowledge,
suc h as t hree- c redit-hour courses in mathematical

granulometries. From these results, various topologic
and morphologic quantities can be retrieved to estimate
the complexity measures, which are scale-invariant but
shape-dependent characteristics [50]. These measures
can characterize several terrestrial phenomena and processes quantitatively in terms of deriving power laws
that exhibit scale invariance but shape dependence [26].
These two illustrative examples were selected to prompt
the reader to realize that DEM is, indeed, a great resource
that can be further processed and enriched by a large
number of mathematical approaches and transformations, unfamiliar to geoscientists, to unravel its hidden
geometric, morphologic, and topologic information.
DEMs provide a source data with which one can demonstrate, by mimicking the natural process of landscape
evolution, the effects of systematic and unsystematic forces that act in a nonlinear fashion to alter the landscape.
Such a study would be useful for generating basin-specific attractors that are process specific and thus helpful for
understanding the basin dynamics. In geosciences, several phenomena (e.g., watersheds, stream networks, and
elevation structures) and processes (e.g., tectonic, hydrological, and geomorphological) are interdependent. Such
interdependencies are described by structure-function
relationships. An interesting example is the shape of a
basin and its response to perturbations. These phenomena and processes might diverge at different scales and in
the case of various forms. However, it was found through
fractal geometry that many Earth phenomena and processes are scale invariant but shape dependent [26]. An

(a)

(b)

(c)

FIGURE 1. (a) Random topography (synthetic DEM) simulated via morphological reconstruction. This DEM contains rich topologic, morphologic, and geometric information, including naming peaks, pits, global and local minima and maxima, contours, crenulations, unique
valley-ridge connectivity networks, topographic depressions, flat regions, and mountain objects. (b) Two such unique features, namely the
ridge and valley connectivity networks, mapped from the DEM in (a) via morphological skeletonization. (c) The subwatersheds partitioned
from (a) and (b). The hierarchical watershed segmentation of the DEM shows partitioned watersheds of several orders. Note that there are
two fifth-order watersheds and five, 10, 36, and 86 fourth-, third-, second-, and first-order watersheds, respectively, in the sixth-order basin.
This hierarchical partitioning was done via the powerful mathematical morphological approach called watershed transformation. For details
on the mathematics involved in this illustration, see [50].
DECEMBER 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

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IEEE Geoscience and Remote Sensing Magazine - December 2020

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