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Digital Elevation Models: Terminology and Definitions

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 remote sensing
Article
Digital Elevation Models: Terminology and Definitions
Peter L. Guth 1, * , Adriaan Van Niekerk 2 , Carlos H. Grohmann 3 , Jan-Peter Muller 4 , Laurence Hawker 5 ,
Igor V. Florinsky 6 , Dean Gesch 7 , Hannes I. Reuter 8 , Virginia Herrera-Cruz 9 , Serge Riazanoff 10 ,
Carlos López-Vázquez 11 , Claudia C. Carabajal 12 , Clément Albinet 13 and Peter Strobl 14

1 Department of Oceanography, US Naval Academy, Annapolis, MD 21402, USA

2 Department of Geography and Environmental Studies, Stellenbosch University,
Stellenbosch 7600, South Africa; avn@sun.ac.za
3 Institute of Energy and Environment, University of São Paulo, São Paulo 05508-010, Brazil; guano@usp.br
4 Mullard Space Science Laboratory, Department of Space & Climate Physics, University College London,
Holmbury St Mary, Surrey RH5 6NT, UK; j.muller@ucl.ac.uk
5 School of Geographical Sciences, University of Bristol, Bristol BS8 1SS, UK; laurence.hawker@bristol.ac.uk
6 Institute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics, Russian Academy
of Sciences, 142290 Pushchino, Russia; iflor@mail.ru
7 U.S. Geological Survey, Earth Resources Observation and Science Center, Sioux Falls, SD 57198, USA;
gesch@usgs.gov
8 Eurostat, European Commission, L-2920 Luxembourg, Luxembourg; hannes.reuter@ec.europa.eu
9 Airbus Defence and Space, 88039 Friedrichshafen, Germany; virginia.herrera@airbus.com
10 VisioTerra, 14 rue Albert Einstein, Champs-sur-Marne, 77420 Paris, France; serge.riazanoff@visioterra.fr
11 LatinGEO Lab IGM+ORT, Universidad ORT Uruguay, Montevideo 11100, Uruguay;



carloslopez@uni.ort.edu.uy

 12 NASA Goddard Space Flight Center Geodesy and Geophysics Laboratory, SSAI Inc., Mail Code 61A,
Citation: Guth, P.L.; Van Niekerk, A.; Greenbelt, MD 20771, USA; claudia.c.carabajal@nasa.gov
13 ESA—European Space Agency-Via Galileo Galilei, 1, 00044 Frascati, Italy; Clement.albinet@esa.int
Grohmann, C.H.; Muller, J.-P.;
14 European Commission, DG Joint Research Centre, 21027 Ispra, Italy; Peter.STROBL@ec.europa.eu
Hawker, L.; Florinsky, I.V.; Gesch, D.;
* Correspondence: pguth@usna.edu; pguth@verizon.net
Reuter, H.I.; Herrera-Cruz, V.;
Riazanoff, S.; López-Vázquez, C.;
Abstract: Digital elevation models (DEMs) provide fundamental depictions of the three-dimensional
Carabajal, C.C.; Albinet, C.; Strobl, P.
shape of the Earth’s surface and are useful to a wide range of disciplines. Ideally, DEMs record the
Digital Elevation Models:
Terminology and Definitions. Remote
interface between the atmosphere and the lithosphere using a discrete two-dimensional grid, with
Sens. 2021, 13, 3581. https:// complexities introduced by the intervening hydrosphere, cryosphere, biosphere, and anthroposphere.
doi.org/10.3390/rs13183581 The treatment of DEM surfaces, affected by these intervening spheres, depends on their intended
use, and the characteristics of the sensors that were used to create them. DEM is a general term,
Academic Editor: Tomaž Podobnikar and more specific terms such as digital surface model (DSM) or digital terrain model (DTM) record
the treatment of the intermediate surfaces. Several global DEMs generated with optical (visible and
Received: 28 July 2021 near-infrared) sensors and synthetic aperture radar (SAR), as well as single/multi-beam sonars and
Accepted: 30 August 2021 products of satellite altimetry, share the common characteristic of a georectified, gridded storage
Published: 8 September 2021
structure. Nevertheless, not all DEMs share the same vertical datum, not all use the same convention
for the area on the ground represented by each pixel in the DEM, and some of them have variable data
Publisher’s Note: MDPI stays neutral
spacings depending on the latitude. This paper highlights the importance of knowing, understanding
with regard to jurisdictional claims in
and reflecting on the sensor and DEM characteristics and consolidates terminology and definitions of
published maps and institutional affil-
key concepts to facilitate a common understanding among the growing community of DEM users,
iations.
who do not necessarily share the same background.

Keywords: DEM; topography; elevation; surface; modeling; geomorphometry

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.
This article is an open access article
1. Introduction
distributed under the terms and
conditions of the Creative Commons Digital topography, expressed as a digital elevation model (DEM), has been embraced
Attribution (CC BY) license (https:// by a wide variety of disciplines and communities including geomorphometry, cryospheric
creativecommons.org/licenses/by/ and soil sciences, precision agriculture, defense, sport, tourism, telecommunications, land
4.0/). planning, hydrology, natural hazards, remote sensing, and even video games. In addition

Remote Sens. 2021, 13, 3581. https://doi.org/10.3390/rs13183581 https://www.mdpi.com/journal/remotesensing

Remote Sens. 2021, 13, 3581 2 of 19

to the users, the data producers include scientists, government mapping agencies, and
commercial vendors. These user communities and data producers do not read the same
literature and consequently many have adopted various definitions for digital topography
and other digital representations (see Section 3.1) of topographic surfaces, as well as their
derivatives. This paper aims to consolidate the definitions as agreed upon by partici-
pants of the 2019 Joint Research Centre (JRC) meeting and the Digital Elevation Model
Intercomparison eXperiment (DEMIX) working group [1], which began work in 2020.
The paper starts with a definition of surface types. We separate physical phenomena
into six distinct categories, known in geospatial sciences as “spheres” resulting in a set of
complete, mutually exclusive object classes based on observable properties of the contained
matter (e.g., lithosphere, hydrosphere, and atmosphere). The defined spheres have surfaces
(or boundary layers) along which they interface with other spheres in their surround-
ings. These surfaces are defined as “real surfaces” in accordance with Florinsky [2] and
categorized according to the adjacent spheres.
These physical “real” surfaces that can be referenced by elevation have limited prac-
tical value if they are too complex for rigorous mathematical handling [3]. This is over-
come by simplifying and mathematically representing “real” surfaces as “topographic”
surfaces [2]. These topographic surfaces have been chosen to be the underlying concept
of every DEM and its immediate derivatives. Mathematical methods for calculating mor-
phometric variables such as slope and aspect are collated and scrutinized according to the
established definitions and translated into algorithms suitable for DEMs, including their
extension to grids in latitude and longitude, taking into account the differences in data
spacing with non-rectangular pixels.
The term “model” within DEM has multiple meanings, and different communities
interpret it differently. The two most distinct interpretations consist of a data model in
raster form, and a mathematical model. The natural world is a physical terrain, which can
be conceptualized and abstracted into a nominal terrain that can be modeled either as an
empirical grid, or as a formal mathematical model.
In a computer science or data science view, “model” refers to a data storage model,
and in the case of a DEM refers to a grid, matrix, or array data structure, all of which
have the same meaning for different user communities. A DEM in this context is a very
convenient and efficient way of storing and analyzing digitized elevation data as opposed
to storing the elevations as lists of vectors/coordinates.
A data elevation model merely represents the terrain, with no assumptions about its
mathematical properties. The first developments of digital terrain models defined them
in 1958 as “a statistical representation of the continuous surface of the ground by a large
number of selected points with known xyz coordinates” [4]. These were not mathematical
models, since they made no assumptions about the nature of the surface, but soon after
their introduction they were being used to help create physical 3D terrain models [5].
Purely physical terrain models have a long history, and they have a scale based on the
actual detail incorporated into the model. The original digital terrain model definition
would comprise rasters (grids), Triangular Irregular Networks (TIN), and point clouds.
A mathematical elevation model consists of a series of analytical expressions to repre-
sent a surface, with coefficients, and these expressions can be integrated and differentiated
rigorously [2,6–8]. Widely used mathematical models with similarities to elevation mod-
els include the World Magnetic Model (WMM) [9] and the Earth Gravitational Model
(EGM) [10], and consist of coefficients and source code to compute magnetic and gravity
values at any desired location up to a resolution defined by the coefficients. These models,
which are not DEMs because they do not represent the Earth’s surface, have a scale, de-
termined by knowledge of the relevant geophysical field, and the number of coefficients
chosen for the representation which determine the level of detail and smoothing.
Some users prefer to consider a DEM as a mathematical model because such repre-
sentations are favorable for interpolation, generalization and denoising, and computation
of derivative geomorphometric variables [2]. These mathematical modeling approaches
Remote Sens. 2021, 13, 3581 3 of 19

have not been widely adopted, mainly because they are not easily implemented in most
available GIS software.
The term DEM has been accepted across a wide range of disciplines as a GIS raster
format for elevation, and its use of “model” agrees with common meanings of model. If a
fully mathematical model of topography becomes available, with both data and software
to manipulate it, a new name will be required, but it is premature to provide a name for
something that does not exist and would only confuse DEM users in many disparate fields.

2. Spheres and Interfaces

For the purpose of defining different surfaces that can be represented by elevation
models, we introduce the concept of “spheres” as a collective of (physical) matter which
share specific properties [11]. On planet Earth and some other solid celestial bodies,
two of these spheres can be considered ubiquitous and persistent, meaning they exist at
any geographical location and at any point in time (at least for the purpose of terrestrial
elevation models). These are:
• Lithosphere: The rigid outer layer of planet Earth and other solid celestial bodies.
Although not consistently depicted as such in literature, for the purposes of elevation
models, the lithosphere is considered to include soils (“pedosphere”).
• Atmosphere: The layer of gases, commonly known as air, including suspended liquid
and solid particles known as aerosols (including dust, clouds, snow, and ice).
Because of their ubiquity and relative permanence, the upper boundary of the litho-
sphere and the lower boundary of the atmosphere can be defined as unique, globally con-
tiguous surfaces, with the caveat that caves and overhanging cliffs need to be generalized.
The following spheres are temporary and local, i.e., they may or may not appear at
any given location or point in time. For the purpose of elevation models, they are only
considered where they occur between the upper surface of the lithosphere and the lower
surface of the atmosphere (and have interfaces with at least two other spheres).
• Hydrosphere: The masses of liquid water, such as oceans, lakes, and rivers;
• Cryosphere: The masses of frozen water, such as sea ice, glaciers, and snow cover;
• Biosphere: The masses of living organisms, such as vegetation and animals, including
dead but still connected parts such as trunks or branches;
• Anthroposphere: The masses consisting predominantly of matter processed by hu-
mans, such as wood, bricks, concrete, asphalt, glass, or plastics.
Digital topography records the surface separating the lithosphere from the atmosphere
(or space for atmosphere-free solid celestial bodies). Challenges arise when intermedi-
ate layers such as the hydrosphere, cryosphere, biosphere and anthroposphere intervene
(Figure 1 and Table 1), and may not be handled consistently even within a single elevation
model. For example, ETOPO1 [12] removes sea ice from the oceans but leaves the conti-
nental ice sheets for Greenland and Antarctica. The oceanic hydrosphere is removed, and
some lakes (e.g., the Caspian Sea and the North American Great Lakes), but not the lakes
in the East African Rift Valleys. The treatment of biosphere and anthroposphere add to
the challenge of picking the desired surface and accurately recording it. In addition, the
particular technology used to collect the topographic data also influences which surface is
being captured. For example, radar signals might experience some degree of penetration
within the vegetation canopy, depending on the band or wavelength they are acting in;
laser altimeters can measure distinct intermediate surfaces in the vegetation structure;
while the raw data from stereophotography represent the top of canopy.
Any DEM must balance the ideal surface that users need with the limitations of the
measurement system and the first observable surface that can be detected. A DEM will
always be an abstraction of the real surface, and in some cases will approximate a ground
surface without man-made features such as buildings. Removal of the hydrosphere and
cryosphere may or may not be possible, or desirable, depending on the end use of the
Remote Sens. 2021, 13, 3581 4 of 19

DEM. Derived grids, such as ice thickness or vegetation canopy height, are not DEMs but
fill many specialized needs.

Figure 1. Possible interfaces and intermediate layers between the lithosphere and atmosphere.

A DEM requires a coordinate system and a reference frame, with horizontal, vertical,
and temporal components, which need to be specified in the metadata. Datums are defined
at different scales (global, regional, national, or local) and timeframes, with each datum
ideally assigned a unique European Petroleum Survey Group (EPSG) code. The horizontal
datum, generally WGS84 or an equivalent, determines how the latitude and longitude
coordinates map to the Earth’s surface. The vertical datum defines the 0 elevation, and can
be in terms of an ellipsoidal or geoidal (mean sea level) reference frame, which can differ
by up to about 100 m (see Figure 2). Geoidal datums can be global, such as EGM2008 [10],
continental, national, or local. The temporal component reflects when the data were
acquired, which can be almost instantaneous (e.g., the Shuttle Radar Topography Mission,
SRTM, was collected in less than two weeks), or the collection can be over a significant
period of time during which the Earth’s surface could have changed. Over time, the land
surface can change through natural or human activity and even plate tectonics, which
creates measurable displacements.

Figure 2. Vertical datums used as the starting point for elevation, and the resulting height measures.

Previous documents summarizing topographic surfaces, and the resulting definitions

for digital topography, include [2,3,13–17] and form the background for our definitions. A
large majority of the participants in the 2019 JRC meeting and the 2020 DEMIX exercise
agreed with the definitions below, treating DEM as a general term, as does the general
public (e.g., [18]), commercial data providers, and mapping agencies in Western Europe
who supply both digital surface models (DSM) and digital terrain models (DTM) products
(e.g., United Kingdom, Norway, Netherlands, Denmark). These definitions seek to provide
clear explanations that match current usage as much as possible. They match the usage
for the Shuttle Radar Topography Mission (SRTM) DEM ([19], 752,000 results on Google
and 50,600 on Google Scholar in April 2021), which is actually an SSG (sensor surface
grid) that approximates a DSM (see definitions in Section 3.3). The use of the term DEM
does not match usage by the U.S. Geological Survey (USGS), which produces a DTM
but calls it a bare-earth DEM [20], or the Spanish Instituto Cartográfico Nacional, which
includes lidar point clouds within the DEM types [21]. From these semantic differences,
it is clear that providers and users of DEMs must clearly understand what the datasets
intend to represent.
Remote Sens. 2021, 13, 3581 5 of 19

3. Glossary of Terms
3.1. Basic Geometric Definitions
• Height: Distance of a point from a chosen reference surface positive upward along a
line perpendicular to that surface (Figure 2) [22]. A height below the reference surface
will have a negative value. Without one of the specific descriptors below, height is
an informal term which will most often be interpreted as an orthometric height or
the vertical size of a feature such as a tree or a building. To avoid ambiguity, the
description must include the reference surface, since even minor differences in these
surfaces can have a significant impact on analysis.
– Orthometric height (H): The distance from the reference geoid or mean sea level
to the point;
– Ellipsoidal height (h): The distance from the designated ellipsoid to the point;
– Geoid height: The distance from the designated ellipsoid to the reference geoid
which can be positive or negative with a magnitude up to about 100 m.
• Elevation: Informal equivalent to height, which will most often be interpreted as an
orthometric height.
• Depth: Distance below the surface of a body of water, with an implicit negative sign,
referenced to mean sea level or a local lake or river datum. When the body of water is
the ocean, the depth is also an elevation but with an explicit negative sign.
• Surface: A surface in the context of topography is a geographic feature that marks the
(uppermost) boundary layer (in the gravitational direction) between two spheres as
defined earlier. Figure 1 shows several types of real surfaces for the Earth’s lithosphere,
hydrosphere, and cryosphere. These real surfaces are too complex for rigorous
mathematical treatment because they are not smooth and regular [2,3], they are
therefore approximated by topographic surfaces.
• Topographic surface: The topographic surface is a closed, oriented, continuously
differentiable, two-dimensional manifold (S) in the three-dimensional Euclidean space
(E3 ). Five key characteristics (constraints in a mathematical sense) of topographic
surfaces include: (1) single-valued (caves and overhanging cliffs are not allowed); (2)
smooth, with the topographic surface having derivatives of all orders; (3) uniform
local gravity, approximated by a plane; (4) planar size limitedness, so that Earth
or planetary curvature can be ignored in computations; and (5) scale dependence
(non-fractality and any fractal component is noise) [2,16].
• Grid: A network composed of two or more sets of curves in which the members of
each set intersect the members of the other sets in an algorithmic way [22]. The curves
partition a space into grid cells. A grid can be understood as a regular network of
grid nodes (points at which curves intersect) or a mesh of grid cells (areas which
are enclosed by curves). In practical terms, a grid is an efficient way for storing and
accessing digital data.
• Grid spacing: The horizontal distance of neighboring samples in a grid. These are
most commonly in either meters or arc seconds, and generally but not always the
same in the x and y directions.
• Spatial resolution of gridded data: The horizontal dimensions of the smallest feature
detectable by the sensor and modified after the gridding procedure, generally given
in meters.
• Sparse grid: A grid in which not all nodes or cells have values attached to them.
These missing values or “voids” need to be filled using interpolation or be treated
separately in grid operations.
• Area-based grid: In this type of grid sampling, the values stated are representative for
the entire area of the grid cell to which they refer (see Figure 6A). They can generally
be assumed to be close to the median or (weighted) average of the original distribution
of values within a given cell and its immediate surroundings. In this case, the spatial
extent of the measurement is on the order of sampling distance or even slightly larger
Remote Sens. 2021, 13, 3581 6 of 19

(“oversampling”, as shown in Figure 3). This is usually the case for DEMs based
on technologies such as InSAR and photogrammetric techniques, including Satellite
Pour l’Observation de la Terre (SPOT)-derived DEMs, and all the DEMs discussed in
Table 1. As we will discuss in Section 4, the sampling strategy must be differentiated
from the grid storage format.
• Point-based grid: In this type of grid sampling, the values stated are only represen-
tative for the grid node to which they are associated (see Figure 6B). Point-based
grids could be based on ground surveys, but this is no longer a common production
method for DEMs.
• (Geo)Rectified grid: Grid for which there is an affine transformation between the
grid coordinates and the coordinates of an external coordinate reference system [23].
If the coordinate reference system is related to the Earth by a datum, the grid is a
georectified grid.
• Pixel reference point: The single point that can represent the pixel for DEM manip-
ulation. For point-based grids, this is the point, while for area-based grids, it is the
pixel centroid.
• Irregular networks: These are networks which do not qualify as grids because they
lack algorithmic regularity. An example is a TIN, which can be constructed from any
set of nodes (points) as they, for example, result from data collection with a random
distribution of ground survey points, topographic cross-sections, or a single-beam
bathymetric survey. They can be used to produce regular grids by interpolation
or extrapolation.
• TIN: Triangular or triangulated irregular network. Triangles connect discrete sampled
points on the surface. This creates a single-valued surface, and the density of the
triangles can vary with terrain complexity and slope.
• (Geospatial) Point cloud: Dataset with X and Y coordinates, Z (height) values, and
possibly other attributes. X and Y coordinates in point clouds are irregular, i.e.,
they do not fulfill the criteria of the nodes in a grid. For some sensors, the point
cloud can be used to create an SSG, while for others, it can create a DSM, DTM,
and intermediate surfaces (see Section 3.3). With lidar data, the point cloud is often
distributed separately from DEMs, but for other sensors, it generally remains only
with the data producer. The point cloud is not a DEM.
• Mesh: A collection of vertices, edges, and faces used in computer graphics and solid
modeling. A TIN is a mesh, but more complex polygons can also be used. If the mesh
vertices do not lie on a regular grid, the mesh is not a DEM.
• Contours: A vector representation of topography with isolines of constant elevation.
The contour interval can change with terrain complexity and slope.
• Tile: A rectangular representation of geographic data, often part of a set of such ele-
ments, covering a tiling scheme and sharing similar information content and graphical
styling. Tiles are mainly used for fast transfer and easy display at the resolution of a
rendering device [24]. Tile boundaries are usually parallels and meridians, similar to
the map quadrangles used for paper maps from national mapping agencies. Distribu-
tion files are named for the tile, and generally use the SW corner location in the DEM.
For the quasi-global DEMs, the tile size is usually 1◦ × 1◦ .
• DEM bounding box: The smallest rectangle that will contain all pixel reference
points in the DEM in a point-based grid, and all the pixel areas in an area-based grid.
Some software, notably GDAL, adds a 1⁄2 pixel buffer to create the bounding box for
point-based DEMs such as SRTM.
Remote Sens. 2021, 13, 3581 7 of 19

Figure 3. (A) SRTM sampling for 100 cells with the values actually representing a slightly larger area,
and resampling to 300 cells by either averaging (B) or thinning (C).

3.2. Topographic Grid Surface Definitions

• DEM (digital elevation model): General term for a digital representation of eleva-
tions (or height) of a topographic surface in form of a georectified point-based or
area-based grid, covering the Earth or other solid celestial bodies. Currently most
common DEMs use rectangular grids (“arrays”) and raster image file storage formats.
Alternative structures for digital topography, such as triangulated irregular networks
(TINs), contours, and point clouds are not DEMs as defined here because they are
not grids.

3.3. Definitions of Specific (Topographic) Surfaces

• DSM (digital surface model): A DEM that records the lower boundary of the atmo-
sphere (and either the lithosphere, hydrosphere, cryosphere, biosphere or anthropo-
sphere) (Figure 4).
• DTM (digital terrain model): A DEM that records the boundary between the litho-
sphere and the atmosphere, without biosphere and anthroposphere, also called a
“bare-earth” DEM. The treatment of hydrosphere, cryosphere, and voids (e.g., ex-
cluded buildings, water and trees) must be specified and clearly localized, e.g., by
respective masks (Figure 4).
• Subaerial Topography: The boundary between the lithosphere and the atmosphere
where there is no hydrosphere; most often represented as a DTM.
• Submarine (or underwater or submerged) topography: The boundary between litho-
sphere and hydrosphere. It can be recorded as a positive depth from a starting datum,
or an absolute elevation. If the datum and signs are correctly interpreted, the subma-
rine topography is continuous with subaerial topography
• Bathymetry: Can refer to either submarine topography or depth.
• TBDEM (topographic-bathymetric DEM): A DEM that is a merge of subaerial to-
pography and bathymetry that represents a seamless surface across the land–water
interface (shoreline).
• BDEM/DBM (bathymetric DEM/digital bathymetric model): A DEM that only
records depths. This term is optional, as nothing in the definition of a DEM pre-
cludes it only recording underwater features.
• SSG (sensor surface grid): A DEM created by a sensing system (e.g., lidar, optical
imagery, radar, or sonar), which often records elevations between those of a DSM and
a DTM. Modeling and removing vegetation and/or buildings can create a DSM, DTM,
or NVS (non-vegetated surface) from the SSG.
• PDEM (planetary DEM): A DEM of a solid planet, moon, asteroid, or other celestial
body. For PDEM, the reference datum can use a spheroid, ellipsoid, tri-axial ellipsoid
or other geometry depending on the shape of the planetary object.
• Hybrid DEMs: Extracting the height of different elements for particular user needs;
examples include:
Remote Sens. 2021, 13, 3581 8 of 19

– NVS (non-vegetated surface): A DSM that excludes the biosphere (and isolated
overhanging man-made features such as power lines) while maintaining the
anthroposphere. Removing elements of a DSM creates prominent voids and arte-
facts whose values need to be interpolated, and consequently adds uncertainty
to the studied processes. NVS might better reflect the needs of some applications,
and it can also be easier to compute because it merely selects the lowest point in
each pixel and does not require building/vegetation classification and hypothe-
sizing a surface below. The ability to create an NVS will depend on the collection
method and data resolution [25] (Figure 4). For instance, removing vegetation
can reveal archeological structures hidden by vegetation, and archeologists have
called this a Digital Feature Model (DFM) [26].
– NUS (non-urbanized surface): A DSM that excludes the anthroposphere but
includes the biosphere (mainly as a closed vegetation canopy). An example
application is the creation of surfaces used for the orthorectification of high to
very high resolution satellite images [27]. The top-reflective height information
of a DSM in the anthroposphere can lead to distortions in the rectified satel-
lite images. The transformation of these regions to a bare-ground-like height
information ensures a better interpretation of the high to very high resolution
satellite images to the disadvantage of high geolocation accuracy of the top of
high urban elements. For dense canopy areas in the image, this effect is of less
importance and therefore conserving the canopy top surface favors a more real
representation including high geolocation accuracy.

Figure 4. (A) Terrain being represented by a (B) digital surface model (DSM), and digital terrain
model (DTM), and (C) a non-vegetated surface (NVS).

3.4. Definitions of Terrain Representations That Are Not DEMs

DEMs can be used to generate a range of other layers that do not represent elevations
and thus are not DEMs. Examples of such DEM derivatives include:
• nDSM: Normalized DSM, the difference between a DSM and a DTM. This surface rep-
resents the heights of objects on the Earth’s surface such as buildings and vegetation;
• CHM: Canopy height model, the height of the vegetation above the ground. Except
for features such as power lines, it is the difference between the DSM and the NVS;
• DHM: Digital height model represents the height of features (objects) such as vegeta-
tion and artificial structures;
Remote Sens. 2021, 13, 3581 9 of 19

• Water depth in rivers, lakes, or oceans as the difference between the DSM over water
bodies (hydrosphere) and the bathymetric surface (lithosphere);
• Ice thickness of glaciers or ice shields as the difference between the DSM over ice
bodies (cryosphere) and the subglacial topography (lithosphere);
• Geomorphometric surfaces, such as slope, aspect, several types of curvature, and
hill-shading;
• Landscape parameters such as drainage basin area and upslope contributing area.

4. Point-Based Versus Area-Based DEMs

Point-based or area-based DEMs refer to several very different areas: (1) how the
raw data are sampled, (2) how the data are stored, and (3) how the data are displayed.
Disentangling these three can be complex.

4.1. DEM Sampling Methods

In area-based sampling, each point in the DEM is collected such that it is representative
(e.g., weighted mean, median) for an interval (or cell) that has about the same nominal
extent (usually slightly larger) as the interval between adjacent samples. This is done
to prevent undersampling, which otherwise could have repercussions on the ability to
resample or interpolate the data.
Except for NASADEM, all of the DEMs in Table 1 are stored in the GeoTIFF format;
100 grids all use area-based sampling and the elevations from the radar or optical sensors
reflect a single, integrated elevation over the sampling area. The 300 grids vary in their
sampling strategy; some average a number of 100 cells, and others take the central value so
that the points in common to the 100 and 300 grids will have the same elevation (Figure 3).
Different versions of the SRTM dataset have used both approaches to resampling.
Point-based techniques, such as using lidar to create a grid with spacing much larger
than the point footprint, have many options on picking the point elevation to use:
• Point elevation closest to the nominal center coordinates of the pixel;
• The minimum or maximum elevation within the pixel;
• Mean or median of a small number of the closest points to the nominal coordinates.
As the size of the region increases, this approaches the area-based technique;
• Inverse distance weighted mean of the local points within the pixel;
• Value at a specified percentile of the points within the pixel.
OpenTopography [28] implements a number of these algorithms, and the chosen
algorithm can greatly affect the resulting DEM, as can any filtering to selectively choose
points. Figure 5 compares area-based and point-based approaches to creating a 90 m DEM,
and the effect of shifting the pixel’s starting location.
For the area-based approach (Figure 5B), the means remain centered in the point cloud
and a bilinear interpolation between the center point of the pixel produces a smooth profile
that is not greatly affected by shifts of the starting location for the grid. For the point-based
approach (Figure 5C), where the profile points represent the lidar value closest to the pixel
center, the starting location shifts make a much larger difference in the interpolated profiles.
The point-based algorithms might best be employed to create a DEM with a spacing that
places a relatively small number of points in each pixel (the pixels in Figure 5 have tens
of thousands of points per pixel), and then downscaling the DEM. In this case, a DSM or
DTM might be easier to create.
To distinguish between these sampling concepts the common GeoTIFF specification [29]
includes a GTRasterTypeGeoKey (#1025), which may be set to either RasterPixelIsArea (the
default), or RasterPixelIsPoint. Unfortunately, nothing prevents the providers of such files
to use an inappropriate flag; for example, legacy reasons force the DGED standard [30] to
use RasterPixelIsPoint irrespective of whether this corresponds to the nature of the data.
Remote Sens. 2021, 13, 3581 10 of 19

Figure 5. Lidar point cloud from a university campus in Brazil. (A) Location of topographic profiles.
(B) Area-based, which depicts the average elevation in the pixel. (C) Point-based DEM representation,
which uses a single point elevation for the pixel (here the closest value to the 90 m pixel center).
Remote Sens. 2021, 13, 3581 11 of 19

4.2. Data Storage and Image Metadata

Figure 6C shows the difference between the elevation storage in the GeoTIFF files
for the ASTER, ALOS, and SRTM 100 DEMs (Copernicus DEM is stored the same way as
SRTM). For the purpose of visualization, the figure shows 3000 pixel sizes in a 1◦ tile. The
red rectangle shows the extent of the tile (usually named for the latitude and longitude of
the SW corner), with the other corners 1◦ away. The gray star symbols, computed from
the GeoTIFF file, show the RasterPixelIsPoint positions for the SRTM and Copernicus
elevations. The gray rectangles show the RasterPixelIsArea areas for the ASTER elevations.
Because the corner coordinates of edge pixels occur outside the nominal tile boundaries,
the centroids of the pixels occur at the same place as the SRTM RasterPixelIsPoint positions
and ASTER elevations can in principle be compared directly with SRTM and Copernicus
elevations as the latter in reality are not point but area-based elevations, even though they
are stored as RasterPixelIsPoint. Although ASTER and SRTM use different parts of the
electromagnetic spectrum, their sensors sample the same areas, slightly larger than the
100 pixels. ASTER, SRTM, and Copernicus DEMs also duplicate a row or column with each
neighboring tile. For the illustrative 3000 pixels in Figure 6C, there will be 3 columns and
3 rows of data; for the actual 100 pixels, the 1◦ tile will have 3601 × 3601 elevations.
The ALOS pixels in Figure 6C are shown in black, with the black star for the center
point displaced by a half pixel in both the x and y directions from the corresponding points
in the other DEMs. The tile has one fewer column and one fewer row compared to the other
DEMs, which means that the 1◦ ALOS tiles have 3600 × 3600 100 pixels. Direct comparisons
of single elevations with the other DEMs are difficult, because the ground area sampled by
ALOS is one quarter of the sampling area for each of 4 adjacent pixels in the other DEMs.

Figure 6. (A) GeoTIFF RasterPixelIsArea DEM where the grid nodes sit at the corners of the sampling
area. (B) GeoTIFF RasterPixelIsPoint DEM where the grid nodes coincide with the sampling points
which likely are at the center of the sampling area. (C) Difference between the elevation storage in
the GeoTIFF files for the ASTER GDEM and ALOS AW3D30 DEMs.

The difference between point and area-based approaches to DEM storage is largely
convention as defined by the Digital Terrain Elevation Data (DTED) standard from the
U.S. military [31]. DTED and the formats that have followed it repeat the edge rows and
columns in adjacent cells to accommodate 3601 × 3601 elevations in each 1◦ tile and use
the RasterPixelIsPoint model. As noted above, ASTER uses RasterPixelIsArea, but through
an alternative selection of the coordinates for each pixel, it has the same effective locations
as the RasterPixelIsPoint DEMs. The area-based ALOS DEM has 3600 × 3600 elevations,
which results in a negligible 0.06% saving in storage, but requires a half-pixel shift to align
with ASTER, SRTM and Copernicus data. These pseudo point-based DEMs could drop
adherence to DTED standards [31] and store only 3600 × 3600 elevations by eliminating
the top row and rightmost column and software should correctly handle them, except for
Remote Sens. 2021, 13, 3581 12 of 19

formats such as the SRTM HGT files which have no internal metadata about the number
of rows and columns and rely on software to recognize the simple format which has the
3601 × 3601 size hard-wired into the definition.

5. DEM Altering Operations

The first steps in DEM editing attempt to remove or correct sensor-related effects,
such as data voids or water related artefacts, and will commonly be done by the data
producer [17,32]. Some DEMs have metadata at the pixel level to record the source data or
edit history. These layers can greatly assist in interpretation and analysis, and users should
be aware to look for them and be cautioned that some secondary sources of DEMs strip out
these metadata layers, which can more than double the size of the DEM. Evaluations of
DEM quality should avoid areas that were changed to fill voids or from building or water
edits, although, it would be desirable to report the percentage of the area filled with voids
or building/water edits.
DEMs can be modified for particular purposes such as geomorphometry [33], sink/pit
removal and hydrological flow enforcement, or flattening of water bodies or for the pro-
duction of derivative products (e.g., from DSM to DTM) [17]. While the modifications will
improve the DEM for a particular purpose, they could degrade its performance for other
purposes by discarding valid measurements in favor of adhering to a particular landscape
model. Removal of karst sinkholes to create an artificial drainage network would be one
example that many DEM users would not want.
Users must be aware that resampling a DEM from one grid (e.g., geographic/unprojected)
to another grid (e.g., projected) almost always will result in artefacts due to pixel shifting
and cell size adjustments which require interpolation. Choosing the right interpolation
technique must be done with great care and knowledge of the specificity of the original
data and the resampling process.

6. Quasi-Global DEMs at Intermediate Scales

Table 1 lists 8 quasi-global DEMs approaching 1 and 3 arc second grid spacing; these
are commonly and informally called 30 m or 90 m (sometimes 100 m) DEMs. Copernicus 1
and 3 arc second DEMs cover the entire land surface of the Earth; SRTM and NASADEM
cover about 80% of the Earth’s land area; the others cover more of the high latitude areas. All
of these stop at the hydrosphere or cryosphere, so they do not include subglacial topography
or the land surface below oceans, rivers, or lakes. At coarser scales, ETOPO1 (1 arc
minute, [12]), SRTM 30+ (30 arc seconds, [34]), and SRTM 15+ (15 arc seconds, [35]) merge
various DEMs on land with the best bathymetry, generally derived from radar altimetry.

Table 1. Free Quasi-global DEMs at 1 and 3 arc second scales.

Vertical Longitudinal
DEM Spacing Primary Source Producer Precision Grid Storage Acquired
Datum Spacing
SRTM (v3) Orthometric
100 , 300 C band radar NASA [36,37] Integer RasterPixelIsPoint Constant 2000 (11 days)
[19] EGM96
ASTER
Stereo NIR NASA/METI Orthometric
GDEM (v3) 100 Integer RasterPixelIsArea Constant 2000–2013
imagery [37,41] EGM96
[38–40]
ALOS World
Stereo pan Orthometric
3D AW3D30 100 JAXA [43] Integer RasterPixelIsArea Variable 2006–2011
imagery EGM96
v3.2 [42]
NASADEM Reprocessed C Orthometric Integer or
100 NASA [37,45] RasterPixelIsPoint Constant 2000 (11 days)
[44] band radar EGM96 floating point
Copernicus
X band radar,
DEM GLO30 ESA/Airbus Orthometric
100 , 300 Edited Floating point RasterPixelIsPoint Variable 2011–2015
and GLO90 [46,47] EGM2008
WorldDEM
[23]
TanDEM-X Ellipsoidal
300 X band radar DLR [50] Floating point RasterPixelIsPoint Variable 2011–2015
DEM [48,49] WGS84
Radar + Stereo Orthometric
MERIT [51] 300 Univ. Tokyo [52] Floating point RasterPixelIsArea Constant 2000–2013
pan imagery EGM96
All are WGS84 horizontal datum; All use the default TIFF orientation (code 274 = 1), with the first points in the
file in the NW corner. The DTED format starts with the SW corner. Copernicus 100 , 300 are available at both DTED
and DGED formats; All name tiles for the SE corner (USGS NED/3DEP names for the NW corner); All have been
reprocessed, and most have voids filled with other DEMs; All of these DEMs use area-based sampling, but the
grid storage specifying the extent of each pixel’s area used both point and area conventions (Figure 6).
Remote Sens. 2021, 13, 3581 13 of 19

7. Implications for Applications

DEMs are regular grids, with two main types. Most local and regional DEMs use
a projected coordinate system; they are plane square grids with square pixels, based
on a map projection such as the Universal Transverse Mercator (UTM) or a Lambert
Conformal Conic. Manipulation of these grids uses the projected coordinates, which form
a Cartesian reference system. Almost all quasi-global, global, and continental DEMs use
a geographic coordinate system (latitude/longitude) or spheroidal equal angular grid.
This is an equirectangular projection; the pixels in these DEMs are spheroidal trapeziums
and differ slightly from rectangles in terms of their linear dimensions in meters. At
finer mapping scales, with DEM grid spacing up to 1500 pixels, the differences between
the spheroidal trapezium shapes of pixels and rectangles are insignificant for almost all
computations. GIS software should use geodetic formulas to account for differences in grid
spacing and correctly display the DEM in a cartographically reasonable way and correctly
compute derivatives such as the slope, but this is problematic in some software.
The quasi-global DEMs suffer severe geographic distortion near the poles, in common
with any cylindrical map projection. Various solutions have been proposed, with grids
using other geometric forms [53,54] or selecting a map projection to minimize distortion
and breaking it into continental regions [55]. The polar community has long faced this
problem, and uses the polar stereographic projection to produce DEMs [56]. This approach
requires distinct grids, with a break between the polar grids and those for low latitudes,
perhaps with some overlap. New geometries for the grids require adoption by mainstream
GIS software, as well as the creation of these data grids for such GIS software.
The best choice of DEM depends on the application and the characteristics of what is
available. For land cover mapping, having both a DSM and a DTM allows the creation of an
nDSM or CHM, which estimates land cover height (e.g., vegetation canopy heights), which
can be utilized in classification techniques. Rectification of satellite imagery ideally needs a
DEM which contains the vegetation coverage, since optical images map the surface of the
canopy, while the inclusion of the anthropogenic elements might add artifacts to the results
depending on the grid spacing of both the image and the DEM. Hybrid DSM/DTM can be
produced in those cases [57]. Telecommunication applications with line-of-sight also need
a DSM. Hydrological applications prefer a DTM, which may require drainage enforcement
but depending on the scale, they might combine it with locally added buildings from a 3D
product or from a DSM with finer grid spacing.
Resolution in the geosciences refers to the real-world dimension of the smallest observ-
able feature. In a DEM, this could be a small canyon, a small ridge, a boulder. Sampling
theory, often named after Shannon, Nyquist, or Whittaker, states that the smallest features
that reliably can be resolved in the DEM will have linear dimensions at least twice the grid
spacing [58,59] and [2] (Section 3.3, pp. 104–105). In that sense, grid spacing sets a lower
limit to the identifiable object within that grid. The actual spatial resolution of a DEM is
initially limited by the imaging and processing systems (and by the atmospheric clarity,
especially for optical systems). The primary goal of resampling to a grid (gridding or
“pixelization”) during DEM production is to not waste that spatial resolution by choosing a
cell size (and grid spacing) that is too large. The secondary goal of pixelization in DEM
production is to not produce wasteful volumes of data by selecting values of grid spacing
size that excessively oversamples the inherent spatial resolution of the imaging system.
For area-based sampling, signal theory recommends the sampling area (over which values
are integrated) to be slightly bigger (1.2–1.4 times) than the grid spacing (or cell size, see
Section 4 and Figure 3), as this slight oversampling helps in avoiding artefacts due to
aliasing. Consequently, well-produced DEMs have a grid spacing size that is not much
different (only slightly smaller than) the spatial resolution of the measurement device.
However, grid spacing, and with it pixel size, can never be assumed to equal the spatial
resolution of a DEM, and later resampling to smaller grid spacings via interpolation will
always cause pixel size and DEM spatial resolution to further diverge.
Remote Sens. 2021, 13, 3581 14 of 19

Although smaller pixel size does not always equate to higher spatial resolution,
smaller pixel-sized DEMs are generally preferred for most applications as such datasets can
potentially accommodate finer detail. However, there is often a trade-off between pixel size
and data usability (storage, management and analyses). Resampling options can upscale
(create a new DEM with smaller spacing, but it will generally be more generalized than
a native DEM at that resolution, and the increased detail may be illusory) or downscale
(the new DEM has larger spacing and less detail). Downscaling loses the extremes in the
DEM, both the high topography and the valleys. Recognizing the significant challenges
resulting from this loss, some of the DEMs provide auxiliary information such as the
minimum, median, and maximum elevations in each cell of the averaged DEM (e.g., Global
Multi-resolution Terrain Elevation Data (GMTED2010) [57]). Others provide indication
of certain classes, e.g., water, which can then be used within a resampling process that
respects the water border lines [60]:
• Downscale by thinning. This ensures that the elevations at common locations in grids
with different pixel sizes have the same elevation;
• Downscale by averaging. If the DEM is area-based, this preserves the statistical
sampling integrity of the data, but at the cost of generalization;
• Reinterpolation using various techniques, such as bilinear interpolation, bicubic
interpolation, or kriging. This allows creating any desired grid size, even smaller but
smoothed pixels compared to the original DEM. It can also reproject the data to a
new map projection. Some redistributions of the global DEMs use reinterpolation to
simplify and standardize their data handling.
The larger the change in scale, the more severe the changes in the DEM and the greater
the importance of selecting an appropriate method.
Figure 7 shows the changes in a DEM, in this case a DSM that averages the surface, as
the grid spacing increases. The large grid sizes produce greater averaging, losing the high
and low elevations and driving the elevations toward the mean.

Figure 7. Stepwise nature of DEM grids with increasing grid spacing (schematic representation).
These might be (A) 1–2 m grid spacing from lidar, (B) 30 m (1 arc second) or (C) 90 m (3 arc second),
where one elevation (a) represents the entire cell and (b) depends on the measurement method.

Comparing the global DEMs must consider that they may sample different areas, and
store the points at different locations on the ground. Direct comparison is challenging, and
the necessity of interpolation to compare elevations may affect the results. As shown in
Remote Sens. 2021, 13, 3581 15 of 19

Figure 6C, the ALOS pixels sample an area that includes 1⁄4 of each of 4 pixels in the other
grids. This makes a direct point-to-point comparison impossible, and requires interpolation
in either the ALOS grid, the other grids, or both. Additionally, the differences in date
should always be considered. Figure 8 shows a comparison of 5 global DEMs to a lidar
point cloud from Bled, Slovenia, along an east-west profile. ASTER, SRTM, NASADEM,
and Copernicus DEM all have elevations at the same locations, and the figure shows how
the elevations vary. ALOS has a half pixel shift in both the latitude and longitude directions;
the east-west shift is obvious with the points being at different locations along the longitude
axis. The shift in the north-south direction appears with the differences in the point cloud
most clearly seen on the castle on the left side of the profile.
Some of the DEMs, as noted in Table 1, change the longitudinal grid spacing at higher
latitudes. Users must be aware of this, as it may make merging DEMs with different
grid spacing difficult. Near 60◦ latitude, nominal 100 DEMs can have 1 × 100 , 1 × 1.500 , or
1 × 200 grid spacings, and average computed slopes vary systematically as the grid spacing
changes. Some distributions of these DEMs also reinterpolate them to preserve the 1 × 100
spacing at all latitudes.

Figure 8. Profiles along a one arc second east-west slice through lidar point cloud in Bled, Slovenia
and the points from (A) 4 global DEMs with common pixel representation, and (B) ALOS which has
half pixel offsets compared to the others. As shown in Figure 6C, the area sampled by each ALOS
pixel covers 1⁄4 of the area of each of 4 pixels in the other DEMs. In this profile, half of the lidar points
in the two profiles are the same, but the other half are to the north or south of the common points.

8. Data Quality
Users must assess the quality of a DEM [61], particularly when multiple comparable
DEMs cover the region of interest, such as the 1 arc second DEMs discussed in Section 6. Re-
cent reviews [62,63] highlight the challenges which have led to a variety of inconsistent, ad
Remote Sens. 2021, 13, 3581 16 of 19

hoc approaches. The assessment can use robust quantitative [64] or qualitative visual [65]
methods, and can use external reference data or consider the internal consistency and char-
acteristics of the DEM. In making any comparisons, the user must ensure that the horizontal
and vertical datums of the DEMs to be compared match, that the pixel representations
match, that geolocation errors do not introduce a horizontal shift, and considerations such
as how a geodetically measured benchmark elevation should correspond with a DEM
elevation representing a 1 arc second pixel. Results comparing the same two DEMs can
lead to opposite rankings for the DEM in floodplains [66] and mountainous areas [67].

9. Conclusions
This paper provides an overview of fundamental concepts and terminology relating
to DEMs. It consolidates the findings of the Digital Elevation Model Intercomparison
eXperiment (DEMIX) working group established in 2020. One of the aims of DEMIX is to
guide end-users in selecting the most appropriate DEM for their specific application and to
highlight the main characteristics that should be considered in the selection process, as they
can influence the interpretation of the results. In addition, it aims to find consensus and
reduce ambiguities among DEM-related terms. Many of the terms defined in this paper are
well known and frequently used by geospatial practitioners. Others are more obscure and
there is often disagreement among experts about their meaning.
Given that DEMs are in essence data models to digitally represent topographic sur-
faces, we started with an overview of the different interfaces between the lithosphere and
atmosphere, as well as the hydrosphere, cryosphere, biosphere and anthroposphere. Due
to limitations in measurement technologies, the interfaces between these “spheres” are
not consistently represented in DEMs and users are advised to take these differences into
consideration. A major source of confusion among users is that DEMs can (among others)
represent DTMs, DSMs, and NVSs, or even combinations of these. This paper clearly
defined and illustrated each of these surfaces to help users understand how they differ and
how the selection of a particular DEM may impact their application. We explained that
most DEMs are sensor surface grids (SSGs), created by a sensing system (e.g., lidar, optics,
radar, or sonar), and that these often record elevations between those of a DSM and a DTM.
Users should also be aware that DEMs use different reference frames (e.g., horizontal
and vertical datums), that can have a significant impact on applications, especially if
multiple DEMs are combined, or when DEMs are used along with other geospatial data.
Scale is another important consideration, as elevation posting interval (pixel size) is not
necessarily an effective measure of how much topographic detail is contained in a DEM,
but the pixel size does limit the potential spatial resolution [9,68]. The technology used and
the scale at which the measurements were taken should rather be considered. One should
also consider how elevations in a particular DEM are sampled (point-based or area-based)
and resampled (nearest-neighbor, bilinear or cubic), as this may result in misalignment
with other data. To demonstrate, we summarized and compared free global DEMs at 1 and
3 arc second grid spacing. The paper concluded with a synopsis of what users should look
out for when selecting a DEM for their application.

Author Contributions: Conceptualization, A.V.N., C.H.G., D.G., I.V.F., J.-P.M., L.H., P.L.G., P.S.;
writing—original draft preparation, A.V.N., I.V.F., P.L.G.; writing—review and editing, A.V.N., C.A.,
C.C.C., C.H.G., C.L.-V., D.G., H.I.R., I.V.F., J.-P.M., L.H., P.L.G., P.S., V.H.-C., S.R.; visualization, C.H.G.,
P.L.G.; supervision, P.L.G., P.S.; project administration, P.L.G., P.S.; funding acquisition, J.-P.M., P.S.
All authors have read and agreed to the published version of the manuscript.
Funding: Partial funding from the STFC MSSL Consolidated Grant ST/K000977/1.
Data Availability Statement: Lidar data from Brazil provided by the City of São Paulo City Hall,
available at http://geosampa.prefeitura.sp.gov.br/ (accessed on 1 July 2021). Lidar data from
Slovenia lidar is available at http://gis.arso.gov.si/evode/profile.aspx?id=atlas_voda_Lidar@Arso
(accessed on 1 July 2021).
Remote Sens. 2021, 13, 3581 17 of 19

Acknowledgments: The authors thank UCL Library Open Access for their support. CHG is sup-
ported by CNPq (#304413/2018-6, #423481/2018-5) and FAPESP (#2019/26568-0). Any use of trade,
firm, or product names is for descriptive purposes only and does not imply endorsement by the
U.S. Government. Suggestions by associate editor Tomaž Podobnikar, reviewer Robert Crippen, an
internal USGS reviewer, and three anonymous reviewers improved the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.

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