=Paper=
{{Paper
|id=Vol-2323/SKI-Canada-2019-7-6-2
|storemode=property
|title=Research Directions for the rHEALPix Discrete Global Grid System
|pdfUrl=https://ceur-ws.org/Vol-2323/SKI-Canada-2019-7-6-2.pdf
|volume=Vol-2323
|authors=David Bowater,Emmanuel Stefanakis
}}
==Research Directions for the rHEALPix Discrete Global Grid System==
https://ceur-ws.org/Vol-2323/SKI-Canada-2019-7-6-2.pdf
Spatial Knowledge and Information Canada, 2019, 7(6), 2
Research directions for the rHEALPix
Discrete Global Grid System
DAVID BOWATER EMMANUEL STEFANAKIS
Geodesy and Geomatics Engineering Geomatics Engineering
University of New Brunswick University of Calgary
david.bowater@unb.ca emmanuel.stefanakis@ucalgary.ca
data management (Yao and Lin 2018) and
ABSTRACT as the de facto global reference system for
geospatial big data (OGC 2017b).
Discrete Global Grid Systems (DGGSs) are Over the years, several different
important to Digital Earth, the Open DGGSs have been created each with
Geospatial Consortium (OGC), and big data advantages and disadvantages. However,
research. A promising OGC conformant only a subset are deemed appropriate under
quadrilateral-based approach is the the OGC DGGS Abstract Specification (OGC
rHEALPix DGGS. Despite its advantages 2017a). Importantly, an OGC conformant
over hexagonal- or triangular-based DGGSs, DGGS must utilize a method that partitions
little research is being done to explore or the earthβs surface into a uniform grid of
advance our understanding of it. In this equal area cells. Currently, the most popular
paper, we briefly review existing work and method is the Icosahedral Snyder Equal
then present several important directions Area (ISEA) projection and a considerable
for future research related to harmonic amount of research has focused on
analysis, discrete line generation, pure- and hexagonal- or triangular-based DGGSs that
mixed-aperture DGGSs, and DGGS-based adopt this approach.
distance/direction metrics. That being said, quadrilateral-based
DGGSs have several advantages over
1. Introduction hexagonal-or triangular DGGSs, such as
compatibility with existing data structures,
hardware, display devices, and coordinate
A Discrete Global Grid System (DGGS)
systems (Sahr et al. 2003; Amiri et al. 2013).
consists of a hierarchy of discrete global
Moreover, recent work has demonstrated
grids at multiple resolutions. DGGSs
benefits of using a quadrilateral approach in
represent a class of spatial data structures
various domains, such as data transmission
that directly address the earthβs surface via a
and rendering (Sherlock 2017) and point
topologically equivalent approximation such
cloud handling (Sirdeshmukh 2018).
as the sphere or ellipsoid (Sahr and White
The rHEALPix DGGS (Gibb et al.
1998). Since their introduction, DGGSs have
2016) is a promising OGC conformant
slowly gained prominence in the geospatial
quadrilateral-based approach with many
community and in 2015, they were
interesting properties. Despite this, DGGS
identified as the foundation of modern
research remains focused on hexagonal- or
Digital Earth frameworks (Mahdavi-Amiri
triangular-based approaches and little work
et al. 2015). In 2017, they were adopted by
is being done to explore or advance the
the Open Geospatial Consortium (OGC)
rHEALPix DGGS. In this paper we review
with the aim of standardizing the DGGS
existing work and highlight several
model, increasing awareness, and increasing
important directions for future research. We
interoperability between DGGSs (OGC
hope this work will promote the benefits of
2017a). More recently, DGGSs were
suggested as a solution to big spatial vector
2 Research directions for the rHEALPix Discrete Global Grid System
the rHEALPix DGGS and stimulate more
researchers to explore it.
Bowater and Stefanakis (2018a)
2. Review of existing work consider the rHEALPix DGGS from a
Canadian perspective and discuss how
In short, the rHEALPix DGGS is varying cell shape and cell orientation are
constructed by projecting an ellipsoid (e.g. key considerations in the polar region (i.e.,
WGS84) onto the faces of a cube, |π| > 41.9Β°, where π is geodetic latitude).
partitioning each face into square grids, and In addition, they describe how these
then inversely projecting the result back to variations can be avoided or exploited for
the ellipsoid (Figure 1.0). Besides the small regions of interest (e.g., provinces) by
theoretical definition presented in Gibb et rotating the grid cells. In particular, they
al. (2016), there are only three works show how triangular dart cells can be
directly related to the rHEALPix DGGS avoided for regions with longitudinal extent
which we now briefly review. less than approximately 90Β°, and how
north-south aligned quadrilateral grids can
be created in the polar region. This is
important because north-south aligned
quadrilateral grids are familiar to users and
link to similar grids used in remote sensing
and environmental modelling (Gibb 2016).
Bowater and Stefanakis (2018b)
present an open-source web service that
enables users to create grids based on the
rHEALPix DGGS. In their paper, the
authors describe the implementation,
including issues and limitations, and
demonstrate how both discrete global grids
and regional grids can be created. This is
important work because it provides an
easily accessible tool for experimenting with
Figure 1.0. Constructing the rHEALPix DGGS (grid grids based on the rHEALPix DGGS,
resolution 1 is shown) (derived from Gibb 2016).
thereby promoting its use in future
Gibb (2016) advances our research. In addition, it supports
understanding in three ways. Firstly, he interoperability studies with other DGGSs
describes how cell identifiers uniquely which is an important aim of the OGC
address cells across all resolutions of the DGGS Abstract Specification.
rHEALPix DGGS using a cell addressing
scheme that has both space filling and 3. Directions for future work
hierarchical properties. Secondly, he
explores cell adjacency and describes a Evidently, the body of research related to
method to determine cell IDβs of adjacent the rHEALPix DGGS is small. In
cells at any resolution using a combination comparison to hexagonal- and triangular-
of base 3 and base 4 math. Lastly, he based DGGSs, the rHEALPix DGGS is
presents methods to determine DE-9IM largely unexplored which means several
topological relationships (e.g., within, directions exist for future work. In this
contains, and touches) by manipulating cell section, a number of them are presented
IDβs. Importantly, this work shows how cell that we feel deserve attention.
adjacency and topological relationships can Arguably, the most unique property
be efficiently determined using cell IDβs of the rHEALPix DGGS is the distribution of
directly rather than geodetic coordinates. cell nuclei along rings of constant latitude
Research directions for the rHEALPix Discrete Global Grid System 3
(often referred to as isolatitude integer that produces aligned hierarchies.
distribution). Gorski et al. (2005) states that However, if we want to maximize the
this property is essential for computational number of resolutions under a fixed number
speed in operations involving spherical of cells (i.e., to provide a smooth transition),
harmonics, which means the rHEALPix then ππ πππ = 2 is a better approach. In
DGGS is a good choice for applications (e.g., addition, one-to-four refinement is exactly
gravitational field modelling) that involve encoded using 2 bits, as opposed to one-to-
harmonic analysis (Gibb et al. 2016). To our nine refinement which requires 4 bits
knowledge, no other OGC conformant (although only 9 of the 16 possible values
DGGSs have this property. However, it has are actually needed). Note that refinement,
not yet been fully explored. Therefore, sometimes called aperture, simply refers to
future work should explore this property the process of subdividing a cell into smaller
further (e.g., is it possible to determine cells. Therefore, we see two interesting
isolatitude ring directly from cell ID to directions for future work: (i) investigate the
facilitate harmonic computations), and real- ππ πππ = 2 rHEALPix DGGS and make
world applications involving harmonic comparisons with the ππ πππ = 3 approach,
analysis should be investigated. and (ii) explore the possibility of a mixed-
In a DGGS, vector data (i.e., points, aperture rHEALPix DGGS. Unlike pure-
lines, and polygons) are rasterized into cells. aperture DGGSs, mixed-aperture DGGSs
Therefore, if we consider linear features, need not have the same aperture across all
such as roads or rivers, the generation of the resolutions. Therefore, they provide greater
discrete line is an important problem (Du et control over cell area at each resolution and
al. 20108). Although solutions for hexagonal have been successfully implemented for
(Tong et al. 2013) and triangular (Du et al. hexagonal DGGSs (Sahr 2013).
2018) DGGSs have been presented, discrete Our last direction for future research
line generation on the rHEALPix DGGS has is not specific to the rHEALPix DGGS - it is
not been studied. Furthermore, the typical relevant to all DGGSs. Currently, there is no
approach involves solving the problem in way to determine the distance (or direction)
the plane and then projecting the result to between two cell IDs without recourse to
the ellipsoid. But this causes issues in the geodetic coordinates (Sirdeshmukh 2018).
plane when the linear feature intersects Just as a DGGS simplifies integration of
more than one base cell. Moreover, heterogeneous data sets on a global scale, a
regarding the rHEALPix DGGS, square cells DGGS-based distance (or direction) metric
in the plane project to different cell shapes would simplify vector data analysis by
on the ellipsoid. Consequently, the ideal removing the dependence on geodetic
discrete line in the plane may not be so on coordinates and complex ellipsoidal
the ellipsoid. Therefore, an interesting formulae. In this way, DGGSs would
direction for future work is to consider become a more complete solution to all our
discrete line generation in the plane but also geospatial needs. While this may not be
directly on the ellipsoid. possible, future work should attempt to find
It may be generally unknown that out.
the definition of the rHEALPix DGGS
presented in Gibb et al. (2016) describes a 4. Conclusion
general class of DGGSs rather than a single,
unique approach. Specifically, the The rHEALPix DGGS is a promising
definition holds for any integer ππ πππ β₯ 2, quadrilateral-based approach that has
where each planar square is divided into several advantages over hexagonal- or
ππ πππ Γ ππ πππ sub-squares at successive triangular-based DGGSs. However, the body
resolutions. In work thus far, ππ πππ = 3 has of research that explores the rHEALPix
been chosen because it is the smallest DGGS is small. In this paper, we briefly
4 Research directions for the rHEALPix Discrete Global Grid System
reviewed existing work, and then proceeded Gibb, R. G. (2016). The rHEALPix Discrete
to highlight several directions for future Global Grid System. In IOP Conference
work related to harmonic analysis, discrete Series: Earth and Environmental
line generation, pure- and mixed-aperture Science (Vol. 34).
DGGSs, and DGGS-based https://doi.org/10.1088/1755-
distance/direction metrics. We believe these 1315/34/1/012012
areas are interesting, challenging, and
necessary to advance our understanding of Gibb, R., Raichev, A., & Speth, M. (2016).
the rHEALPix DGGS. The rHEALPix Discrete Global Grid
System. doi: 10.7931/J2D21VHM.
Acknowledgements Available online
https://datastore.landcareresearch.co.
This work was funded by the Natural nz/dataset/rhealpix-discrete-global-
Sciences and Engineering Research Council grid-system (accessed on 17 July 2018).
of Canada (NSERC-DG).
Gorski, K. M., Hivon, E., Banday, A. J.,
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