Area and volume computation of longitude–latitude grids and three‐dimensional meshes

Abstract

Longitude–latitude grids are commonly used for surface analyses and data storage in GIS. For volumetric analyses, three‐dimensional meshes perpendicularly raised above or below the gridded surface are applied. Since grids and meshes are defined with geographic coordinates, they are not equal area or volume due to convergence of the meridians and radii. This article compiles and presents known geodetic considerations and relevant formulae needed for longitude–latitude grid and mesh analyses in GIS. The effect of neglecting these considerations is demonstrated on area and volume calculations of ecological marine units.

Full text

Transactions in GIS. 2020;00:1–19. wileyonlinelibrary.com/journal/tgis
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© 2020 John Wiley & Sons Ltd
1 | INTRODUCTION
Global and regional datasets are t ypically modeled with some form of spatial par titioning or tessellation of t he area
of interest. For example: digital terrain models (DTMs) or digital elevation models (DEMs), ecological marine units
(EMUs) (Sayre et al., 2017), global grids for population studies (Lloyd, Sorichetta, & Tatem, 2017), spatial variabilit y
of soils (Neteler, 2001), distribution of water masses (Sahlin, Mostafavi, Forest, Babin, & Lansard, 2012), oce-
anic environmental variables (Manley & Tallet, 1990), modeling 3D spatial dynamic fields (Hashemi Beni, Abolfazl
Mostafavi, Pouliot, & Gavrilova, 2011; Ledoux, 2006), and modeling weather and climate data (Armstrong, 2015;
Dobesch, Dumolard, & Dyras, 2013; Dyras et al., 2005). Analyst s can use various tessellation schemes to create
these models, for example Voronoi tessellations (Gold & Condal, 1995; Ledoux, 2006), discrete global grid systems
(DGGSs) (Sahr, White, & Kimerling, 20 03), or hierarchical equal area isolatitude pixelization (HEALPix) (Gorski
et al., 2005), to name a few. However, the most common spatial partitioning scheme is a longitude–latitude grid.
The longitude–latitude partition is a struc tured tessellation composed of orthogonal curvilinear grid cells
bounded by regularly spaced meridians of longitude and parallels of latitude over the Ear th's surface (e.g., Santini,
Taramelli, & Sorichetta, 2010). Such a partition is commonly used for sur face analyses and data storage. For volu-
metric analyses, three-dimensional meshes perpendicularly raised above or below the gridded surface are applied.
Since grid cells are defined with geographic coordinates, due to convergence of the meridians, such cells are not
DOI : 10.1111/tgis .126 36
REVIEW ARTICLE
Area and volume computation of longitude–
latitude grids and three-dimensional meshes
Kevin Kelly | Bojan Šavrič
Environmental Systems Research Institute
Inc., Redlands, CA, USA
Correspondence
Kevin Kelly, Environmental Systems
Research Institute Inc ., 380 N ew York St,
Redlands, CA 92373-8100, USA.
Email: kevin_kelly@es ri.com
Abstract
Longitude–latitude grids are commonly used for surface
analyses and data storage in GIS. For volumetric analyses,
three-dimensional meshes perpendicularly raised above
or below the gridded surface are applied. Since grids and
meshes are defined with geographic coordinates, they are
not equal area or volume due to convergence of the me-
ridians and radii. This article compiles and presents known
geodetic considerations and relevant formulae needed for
longitude–latitude grid and mesh analyses in GIS. The effect
of neglecting these considerations is demonstrated on area
and volume calculations of ecological marine units.

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KELLY and ŠaVRIČ
equal area. Their area decreases as one moves from the equator toward the poles. Similarly, due to convergence
of radii, the volume of a three-dimensional mesh element changes with latitude as well as with height (or depth).
Consequently, surface and volumetric analyses of such grids are affected by the methodologies used to compute
the areas and volumes of these cells and mesh elements.
The most accurate analyses of such gridded data can only be obtained using an ellipsoidal approximation of
the Earth's shape. A more simplistic approximation is a sphere. For example, Florinsky (1998) takes these geodetic
considerations into account and develops methods for calculating local morphometric variables from longitude–
latitude DEM grids. His work is limited to grid cell sizes of no more than 225 km, but is applicable to regions of any
size, even of global extent. Florinsky (2017) quantifies computational errors in local and non-local morphometric
variables that arise when ellipsoidal grids are assumed to be planar. Using sample data taken from DEMs of Mount
Elbrus, Russia, and western Kenya, he finds disparate result s for all variables tested. He finds significant errors
in slope gradients on steep slopes in mountainous areas, as well as for thalwegs in the case of catchment area.
To determine catchment area, a non-local morphometric variable, Florinsky (2017) uses an equation published in
Morozov (1979) to calculate the ellipsoidal area of longitude–latitude grid cells.
The effect of the convergence of meridians and radii can be accounted for using spherical or ellipsoidal equa-
tions to compute the areas of grid cells and volumes of mesh elements in analyses using such grids. However,
some users of these grids project them from either the ellipsoid or the sphere onto a flat plane, such that simple
Cartesian coordinate geometry c an be applied to the area and volume calculations. This approach yields rea-
sonably good precision if the area of interest is small and near the points or lines of zero distortion. Using such
planar equations in a projected grid that covers ver y large regions, or even the whole world, may introduce areal
distortion leading to incorrect results from subsequent analyses. The commonly used Web Mercator projection
(Battersby, Finn, Usery, & Yamamoto, 2014; NGA, 2014) demonstrates this issue markedly; instead of decreasing,
grid cell areas increase as one moves toward the poles.
In a qualitative analysis, Chrisman (2017) discusses how software implementations that only model the Earth
with flat projections introduce distortions ranging from mild to severe. As a possible solution, he suggests provid-
ing corrections for common map projection errors. Inaccuracies from the use of planar equations for area calcula-
tions can be avoided if an equal area map projection is used. Even though relative areas are maintained, additional
pitfalls occur when using this methodology. When an equal area projection does not project meridians and paral-
lels as straight lines, the boundaries of a cell need to be sufficiently densified to maintain their true location and
area in a projected space. Densification is even more impor tant when the grid cell is fur ther away from the points
and lines of zero distor tion, where shape deformation becomes fur ther magnified.
In cases where the Earth's shape can be ignored (e.g., Florinsky, 1998, p. 831 assumes this to be the case when
an area has a maximum diagonal no greater than one-tenth of the Earth's mean radius), or when the area of interest
lies entirely within a 6° meridional zone of the Gauss–Krüger or Universal Transverse Mercator (UTM) projected
coordinate systems (Florinsky, 1998, p. 832), the geodetic considerations presented in this article may not be
needed. However, since the global case is being addressed here, they are essential. These geodetic considerations
can affect spatial analyses where area and volumes are required. As in the examples from the opening paragraph,
spatial analyses involving density (e.g. population density) or global distributions can be impacted when the geo-
detic considerations are ignored.
The goal of this article is to compile and present known geodetic considerations required for longitude–
latitude grid and mesh analyses using GIS, as well as to demonstrate the effec t of neglecting these considerations.
In this article, formulae to calculate grid cell surface area and mesh element volume on the sphere and the ellipsoid
are given. Since web-based GIS applications often use the Web Mercator projection, the areal distortions of grid
cells with respect to this projection will be shown and discussed. The effect on area calculation of non-densified
cell boundaries when an equal area map projection is used will also be presented, as well as the impact of radii
convergence on mesh element volume. We focus only on the geometric approach to area and volume calcula-
tion on the sphere and the ellipsoid. Thus, when discussing volume calculations, differences between physical

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KELLY and ŠaVRIČ
(gravit y-based) heights and geometric (ellipsoidal) heights are not considered, and therefore we neglec t to account
for effects due to deflec tion of the vertical, geoid–ellipsoid separation, dynamic topography, glacial isostatic ad-
justment, and ver tical crustal deformation.
In Section 2, the relevant geodetic considerations are presented, which include formulae to calculate grid cell
surface area on the sphere and the ellipsoid, and to calculate the volume of three geometric types of mesh ele-
ment. Section 3 describes the methods used to analyze the differences in grid cell surface area and mesh element
volume. Following the results in Section 4, the discussion in Section 5 presents some limitations, and the contexts
where a par ticular method is appropriate are suggested. Conclusions are given in Sec tion 6.
2 | GEODETIC CONSIDERATIONS
To extract correct information from a longitude–latitude gridded model, it is important to understand the geomet-
ric representations of the Earth used in their derivation. Earth's shape is complex, and any approximations made to
simplify calculations on it directly affect the accuracy of the gridded model. Undistorted surface area and volume
can only be obtained when computed directly on an ellipsoid of revolution which most closely approximates the
size and shape of the Ear th [e.g. WGS84 (NGA, 2000) or GRS80 (Moritz, 200 0)]. The calculation of an arbitrary
area or volume on the ellipsoid is not trivial. Therefore, more simplistic approximations are available (e.g., spherical
Earth models and methodologies involving projecting the Earth's surface onto a flat plane). This section presents
three approaches for grid surface area and mesh volume c alculation based on a longitude–latitude surface grid.
2.1 | Surface area
Global grid systems are often defined on an m × n longitude–latitude grid or graticule over the globe, where m is
the longitude grid spacing and n is the latitude grid spacing. Such a grid is not equal area due to the convergence
of meridians toward the poles. Moreover, the area of a grid cell differs depending on the surface on which it is
calculated: sphere, ellipsoid, or projected plane.
2.1.1 | Surface area on the sphere
On a sphere, meridians of longitude are great circles that converge toward the poles, meaning that the length of
an arc measured along a parallel of latitude decreases. The implication is that the geometry of each successive grid
cell changes, as does its surface area, which also decreases from equator to pole (Figure 1). At the pole, grid cells
degenerate into non-spherical triangles.
On a sphere, the length of an element of arc along a meridian of longitude is R dφ and an element of arc along
a parallel of latitude is R cosφ dλ (Rapp, 1991; Santini et al., 2010), where R is the radius of the sphere, φ denotes
latitude, and λ denotes longitude. Then the surface area of a longitude–latitude grid cell is given by:
where
𝜑1
,
𝜑2
and
𝜆1
,
𝜆2
are the bounding geocentric latitudes and longitudes, respectively. In Equation (1), the ab-
solute difference in longitude
𝜆2−𝜆1
is in radians.
(1)
A
S=
𝜑
2
∫
𝜑1
𝜆
2
∫
𝜆1
R2cos 𝜑d𝜆d𝜑=R2
(
𝜆2−𝜆1
)(
sin 𝜑2−sin 𝜑1
)

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KELLY and ŠaVRIČ
2.1.2 | Surface area on the ellipsoid
As mentioned previously, Earth's shape is best approximated by an ellipsoid of revolution. In this case, grid cells
appear more flattened and elongated compared to their spherical counterparts. While the calculation of an arbi-
trary area or volume on the ellipsoid is not trivial, it is greatly simplified when longitude–latitude grid cells form
the region. On an ellipsoid of revolution, the surface area of a longitude–latitude grid cell is given by (Bagratuni,
1967; Rapp, 19 91) :
where
e
is the first eccentricity of the ellipsoid, a is it s semi-major axis, b is its semi-minor axis,
𝜑1
,
𝜑2
are the bound-
ing geodetic latitudes, and
𝜆1
,
𝜆2
are the bounding longitudes. In Equation (2), the absolute difference in longitude
𝜆2−𝜆1
is in radians. Morozov's equation to calculate the surface area of a longitude–latitude grid cell (Morozov,
1979), used in Florinsky (2017), is an approximation of Equation (2) above.
An identical result can be obtained by converting an ellipsoid of revolution to an authalic sphere (Snyder, 1987)
and then using Equation (1), where the radius of the sphere
R
is replaced by the radius of the authalic sphere
Rauth
and the bounding geocentric latitudes
𝜑1
and
𝜑2
are replaced by the authalic latitudes
𝛽1
and
𝛽2
. For detailed equa-
tions, see Snyder (1987, p. 16) or Grafarend and Krumm (2006, pp. 76–79).
2.1.3 | Projected surface area
When working with longitude–latitude grids in a projected coordinate system, it is impor tant to understand
the distortion incurred by a map projection. All projections distort the regions they map in one or more ways.
Distortion may be in area, length, angle, or shape. Distortion refers to an untrue representation of these qualities
caused by applying a specific map projection method. An actual map may exhibit combinations of these distor-
tions. However, using a projec ted coordinate system is often required for data display on a map or due to technol-
ogy or sof tware limitations. For examples of detailed assessments of map projec tions and their distortions, see
Richardus and Adler (1972), Pearson (1984), Bugayevskiy and Snyder (1995), and Grafarend and Krumm (2006).
(2)
AE=b2
2

𝜆2−𝜆1
 sin
𝜑2
1−e2sin2𝜑
2
+1
2e
ln
1
+
esin
𝜑2
1−esin 𝜑2

−
sin
𝜑1
1−e2sin2𝜑
1
+1
2e
ln
1
+
esin
𝜑1
1−esin 𝜑1

FIGURE 1 Geometr y of grid cells along a meridional section from the equator to pole showing the latitude
dependence of grid cell area

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KELLY and ŠaVRIČ
Web Mercator projection
When projecting a longitude–latitude grid to a planar surface, it is essential to be aware of the area distortion
caused by the projection. For example, the Web Mercator projection is a popular planar projection implemented in
commercial and open-source datasets, internet-based visualization, and data analysis applications. This projection
covers the Earth from −180° to 180° longitude, and between approximately 85.05° north and south latitude. Web
Mercator equations are formulated on a spherical Earth model, but purport to project data in the WGS84 geodetic
reference system. The projected meridians are vertical, equally spaced parallel lines. They are orthogonal to hori-
zontal straight lines representing parallels of latitude. The projected parallels are increasingly spaced toward the
poles. The projection is non-conformal and non-equal area, which means that areas are distorted (Battersby et al.,
2014; NGA, 2014). The Web Mercator projection greatly distorts area,1 which is of importance when working with
longitude–latitude grids spanning large regions.
To quantify the areal distortion in the Web Mercator projection, we refer to the formulae used to derive the
mapping coordinates (x, y) from ellipsoidal latitude and longitude (φ, λ) (NGA , 2014):
where a = 6,378,137.0 m is the semi-major axis of the WGS8 4 ellipsoid. Equation (3) shows that the Web Mercator
projection uses a sphere for its model of the Earth, the radius of which is equivalent to the semi-major axis of
the WGS84 ellipsoid. Using Equation (3), we can calculate the areal scale factor as (Bugayevskiy & Snyder, 1995;
Snyder, 1987):
where
thus, the areal scale factor becomes:
Since s is independent of longitude, the areal distor tion is rotationally symmetric—it is identical for corre-
sponding grid cells along any parallel of latitude. Using this value of s, one can compute the areal distortion of grid
cells imposed by the projection. At the poles, the areal scale fac tor becomes infinity since the projection cannot
project the poles.
Equal area projections
Aside from the Web Mercator projection, there are several equal area projections which maintain relative areas
on a projected surface. According to Snyder (1987), when mapping an area at large scale, one can select be-
tween cylindrical equal area (for equatorial, predominantly east–west regions), Albers conic (for mid-latitudes,
(3a)
x=a𝜆
y=a
tanh
−1(
sin
𝜑).
(4)
s
=h⋅k⋅sin
𝜋
2
(5a)
h
=sec 𝜑
(
1−e2sin2𝜑
)3∕2(
1−e2
)−1
(5b)
k
=sec 𝜑
√
1−e2sin2
𝜑
(6)
s
=sec2𝜑
(
1−e2sin2𝜑
)2(
1−e2
)−1

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KELLY and ŠaVRIČ
predominantly east–west regions), Lamber t azimuthal equal area (for any polar areas and regions with equal extent
in all direc tions), and transverse cylindrical equal area projections (for predominantly north–south regions along
the selected meridian). For hemispheric extents, Snyder recommends the Lambert azimuthal equal area in oblique
aspect, and any pseudo-cylindrical equal area projection for global extent. Jenny, Šavrič, Arnold, Marston, and
Preppernau (2017) limited the list for global extents to eight projections. If we exclude from their list projections
that map-readers dislike (Šavrič, Jenny, White, & Strebe, 2015), only the Mollweide, Eckert IV, Wagner IV, and
Wagner VII remain.
Notwithstanding the projection selection guidelines of Snyder (1987) and Jenny et al. (2017), the selection
of an equal area projection for longitude–latitude grids depends on the reason for projec ting the grid in the first
place. The requirements of performing analysis or calculating surface area are different from those of displaying
gridded data on a map. When projecting longitude–latitude grids onto an equal area projection, if the purpose is
to calculate grid cell surface area, one needs to consider how the boundaries of a cell are projected onto the plane.
Not all projections suggested by Snyder (1987) and Jenny et al. (2017) project meridians and parallels as straight
lines. Many of them project meridians and parallels as complex curves. Since the surface area on a projection is
calculated using Cartesian coordinate geometry, it is essential that projected cell boundaries reproduce the form
of the curves in projected space. Therefore, the projected boundaries of a cell need to be sufficiently densified.
Cell boundary densification is even more important when the grid cell is farther away from the point s and lines of
zero distortion, where cell deformation becomes magnified further.
Since not all map projec tions support the Ear th’s shape as an ellipsoid of revolution, the projection algorithm
is also a consideration when projecting longitude–latitude grids. While cylindric al equal area, Albers, L ambert
azimuthal (Snyder, 1987), and transverse cylindrical equal area (Snyder, 1985) projections are formulated for an
ellipsoid of revolution, most pseudo-cylindrical projections are formulated only for a spherical Earth model. When
projecting grids onto an equal area projection using spherical equations, the dif ference in the sur face area is sim-
ilar to the difference between the surface area of a sphere versus the surface area of an ellipsoid of revolution.
This issue can be avoided by a process known as “double” projection, that is, first converting geodetic latitudes to
authalic latitudes and then projecting these onto an equal area projection using spherical equations and the radius
of an authalic sphere. For details on authalic sphere equations, see Snyder (1987, p. 16) or Grafarend and Krumm
(2006, pp. 76–79).
2.2 | 3D mesh volume
For three-dimensional analyses when data extends above or below the Earth's surface, volumetric extents must
approximate real Earth volumes. The volume of a three-dimensional mesh element can be calculated based on
a variety of geometric constructions. The simplest, but least representative, of the Earth's actual shape is the
rectangular mesh element. Spherical and ellipsoidal mesh elements provide more realistic representations, but at
the cost of greater complexity. Since the shape of the Earth most closely approximates an ellipsoid of revolution,
the ellipsoidal mesh element best approximates real volumes. Figure 2 illustrates the geometry of these three
mesh types: rec tangular, spherical, and ellipsoidal. The following subsections present geometry and formulae for
calculating the volume of these mesh elements.
2.2.1 | Rectangular mesh element
A rectangular mesh element is defined by perpendicularly raising a base face of a projected longitude–
latitude surface grid above or below the projected surface such that opposite faces of the element are equal

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KELLY and ŠaVRIČ
and parallel (Figure 2a). The rectangular mesh element in Figure 2a illustrates the general case where the
projected surface grid cell boundaries are defined by complex curves. The direction of the sides of the rectan-
gular mesh element, in general, do not coincide with the direction of the radii in a spherical Ear th model, nor
FIGURE 2 In all three mesh elements, the top face is bounded by parallels
𝜑1
and
𝜑2
, meridians
𝜆1
and
𝜆2
,
and prism height is denoted Δh. The top face of the rectangular element (a) lies on the surface of the projection
plane; for the spherical and ellipsoidal mesh elements, the top face lies on the surface of the sphere and the
ellipsoid, respectively. In (b)
R
is the radius of the sphere and in (c)
N
is the prime vertical radius of curvature

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KELLY and ŠaVRIČ
the direc tion of ellipsoidal normals in an ellipsoidal Ear th model. Consequently, the use of rectangular mesh
element s for volumetric analyses fails to account for convergence of Ear th radii. Adjoining elements intersect
toward the bottom and exhibit wedge-like gaps toward the top (Hirt & Kuhn, 2014). In Figure 2a, the top face
is a grid cell on a projected surface and the element extends below the surface to a depth of
Δh
. The volume
computation is face area × Δh.
Since rectangular mesh elements are defined on a projected plane, their volume depends on how their face
surface area is calculated. This means that all considerations presented in Section 2.1 also apply to the volume
calculation of rectangular mesh elements. Undistorted surface area can only be obtained by computing face area
directly on the surface of an ellipsoid or on a projected surface when an ellipsoidal equal area projection is ap-
plied, and face boundaries are satisfactorily densified. Other wise, the convergence of the meridians remains un-
accounted for and surface area inaccuracies propagate into the mesh element volume.
2.2.2 | Spherical mesh element
A simple geometric shape that t akes the spherical shape of the Earth into account is a spherical mesh element.
This is a segment of a sphere defined in geocentric spherical coordinates (Figure 2b). Each element is bounded by
two meridians of longitude
𝜆1
and
𝜆2
, two parallels of latitude
𝜑1
and
𝜑2
, and two concentric spheres of radii r1 and
r2, where Δh = r2‒ r1 (Uieda, Barbosa, & Braitenberg, 2016). Radii are normal to the sur face and converge toward
and intersect at the center of the sphere, ensuring that the volume varies with depth below or height above the
surface. In Figure 2b, the top face is a grid cell on the sur face of the sphere of radius r and the mesh element
extends below the surface to a depth of Δh. The volume of a spherical mesh element is given by (e.g. Moon &
Spencer, 1988):
where the integration limits are shown in Figure 2b, and the absolute difference in longitude
𝜆2−𝜆1
is in radians. In
Equation (7), the upper and lower faces of the mesh element are curved, and all other faces are flat (see Figure 2b).
2.2.3 | Ellipsoidal mesh element
Due to Ear th‘s flattening at the poles, a spherical mesh element only approximates true volume. A more accurate
measure of volume can be obtained using an ellipsoidal mesh element (Figure 2c). Each element is delimited by two
meridians of longitude
𝜆1
and
𝜆2
, two parallels of latitude
𝜑1
and
𝜑2
, and two non-ellipsoidal sur faces at constant
heights (or depths) h1 and h2, where Δh = h2‒h1, and both of which are parallel to the surface of the ellipsoid. If any
of the h variables equal zero, its corresponding surface coincides with the ellipsoidal surface. In Figure 2c, the top
face is a grid cell on the surface of, for example, the WGS8 4 ellipsoid, and the mesh element extends below the
surface to a depth of Δh. Here, the normals to the ellipsoid, defined by the radii of cur vature in the prime vertical
N, vary with latitude and converge toward the center of the ellipsoid, but do not mutually intersect there like they
do for a spherical element. The ellipsoidal normals intersect the z-axis above the equatorial plane for south lati-
tudes and below it for north latitudes. Like spherical mesh elements, ellipsoidal mesh elements ensure the volume
varies with depth below or above the surface. The volume of an ellipsoidal mesh element can be calculated using
the formulation given by Vajda, Vaníĉek, Novák, and Meurers (2004):
(7)
V
S=
𝜑
2
∫
𝜑1
𝜆
2
∫
𝜆1
r
2
∫
r1
r2cos 𝜑drd𝜆d𝜑=1
3

r3
2−r3
1

𝜆2−𝜆1

sin 𝜑2−sin 𝜑1


|
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KELLY and ŠaVRIČ
where the prime ver tical radius of curvature
N
and meridian radius of curvature
M
are given, respectively, by
N=a∕√
1
−e
2sin2
𝜑
,
M
=a
(
1−e2
)
∕
(
1−e2sin2𝜑
)3∕2
, and
𝜆2−𝜆1
is the absolute difference in longitude in radi-
ans. In Equation (8), the upper and lower faces of the mesh element are curved, and all other faces are flat (see
Figure 2c).
Vajda et al. (20 04) show that when the volume integral is carried out in the interior of the ellipsoid, the lower
boundary h1 = hL must satisfy the following condition:
This limitation arises because a parallel surface to the reference ellipsoid at height (or depth) h defines a
non-ellipsoidal surface. The interior limit hL arises from a property of parallel surfaces which states that such
surfaces will have cusps when the distance from the curve is propor tional to the radius of curvature (Hartmann,
2003). Therefore, to avoid discontinuities at the cusps, the depth from the ellipsoid surface to its interior parallel
surface must not exceed hL.
3 | ANALYSIS OF CALCULATION METHODOLOGIES
To assess the effects of the geodetic considerations presented above, EMUs (Sayre et al., 2017) are used as
an example. Area and volume calculation methodologies are assessed separately and presented following a
brief description of the EMUs. The purpose of these analyses is to show the effect of neglec ting the geodetic
considerations.
3.1 | Ecological marine units
EMUs (Sayre et al., 2017) characterize 37 distinct ecosystems of the global ocean derived through statisti-
cal clustering of oceanic observations. Data was aggregated onto a three-dimensional point mesh defined by
equi-angular
1
4
×
1
4
grid cells over the ocean surface and arranged ver tically from the surface to 5,50 0 m depth
with inter vals varying between 5 and 100 m. The geometric structure of the mesh forms a contiguous set of
rectangular mesh elements. Each mesh element is attributed with marine parameters such as: temperature,
salinit y, and oxygen (for a full list of parameters included in EMUs, see Sayre et al., 2017). Figure 3 shows a
cross-section of a mesh element stack. EMUs are referenced to the WGS84 ellipsoid, therefore all analyses in
this article use this ellipsoid.
(8)
V
E=𝜆2−𝜆1
h
2
∫
h1
𝜑2
∫
𝜑1N+hM+hcos 𝜑d𝜑dh
=𝜆2−𝜆1




a21−e2
2h2−h1 sin 𝜑
1−e2sin2𝜑+tanh−1(esin 𝜑)
e𝜑2
𝜑1
+a
2eh2
2−h2
1tan−1esin 𝜑
1−e2sin2𝜑𝜑2
𝜑1
+a1−e2
2h2
2−h2
1sin 𝜑
1−e2sin2𝜑𝜑
2
𝜑
1
+1
3
h3
2−h3
1

sin 𝜑2−sin 𝜑1

h
L
≥
−
(
1−e2
)
Nfor all 0 <
|
𝜑
|≤𝜋
2

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KELLY and ŠaVRIČ
3.2 | Surface area analysis
The surface area of an EMU grid cell on the Earth's sur face dif fers depending on the latitudinal location and the
calculation model. As already mentioned, there are three possible representations of the Earth's surface: ellipsoid,
sphere, and projec ted sur face. In this analysis, the surface area of EMU grid cells was calculated based on the fol-
lowing sur face models:
1. Ellipsoid of revolution.
2. Sphere with ellipsoidal semi-major axis taken as radius.
3. Sphere with an authalic radius.
4. Web Mercator projection.
5. Lambert azimuthal equal area projection (strip of cells along the central meridian).
6. Lambert azimuthal equal area projection (strip of cells away from the central meridian).
FIGURE 3 Cross-section of a columnar stack of rectangular volumetric mesh elements comprising the EMU
three-dimensional point mesh (illustrative and not to scale). The figure also shows the arrangement of the
variable depth zones ranging from 5-m intervals near the surface to 100-m intervals near the deep ocean floor
(after Sayre et al., 2017)

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KELLY and ŠaVRIČ
Since EMU grid cells are longitudinally symmetric, only computed areas along a single meridional strip be-
tween 0º and 0.25º E longitude were considered. Areas computed on the WGS84 ellipsoid using Equation (2) were
taken as true values and differences were calculated between the true values and those from each of the alter-
native surface models listed above. For the sphere, areas were computed on two dif ferent models: one that uses
the semi-major axis of the WGS84 ellipsoid as the radius and another that uses the radius of the corresponding
authalic sphere, since this sphere has the same surface area as that of the WGS84 ellipsoid. In both cases, geodetic
latitudes were used to define the grid cell locations.
For the projected surfaces, two map projections were applied: the commonly used Web Mercator projec-
tion and the Lambert azimuthal equal area projection in equatorial aspect. The latter projec tion was chosen
to illustrate increasing grid cell shape distortion as one moves away from the central meridian. This is because
the Lambert azimuthal projects meridians and parallels as complex curves and not straight lines as in the Web
Mercator projection. This property complicates the problem of area calculation in GIS, since the grid boundaries, if
not sufficiently densified, may introduce additional areal inaccuracy. For this reason, two versions of the Lambert
azimuthal projection were compared to show the effect on area as grid cells move farther away from the point
of zero distor tion. The two meridional strips on the Lambert azimuthal projec tion for which grid cell area was
calculated and compared are: (1) along the central meridian at 0º longitude and (2) along the 170º E meridian. Grid
cell boundaries were not densified. For both projections, equations for conver ting the ellipsoidal surface to the
projected plane were used, since all results are compared to the ellipsoidal surface.
Since the focus of this analysis is to show the effect of neglecting the previously mentioned geodetic
considerations, it does not include an authalic sphere with authalic latitudes because this model reproduces
ellipsoidal area exactly. Similarly, an equal area projection with densified cell boundaries and an ellipsoidal
cylindrical equal area projection without densified cell boundaries also reproduce ellipsoidal area exactly and
are not considered.
3.3 | 3D mesh volume analysis
The volume of a mesh element above or below the Earth's surface will differ depending on the latitudinal location
and the model used to represent it. In this volume analysis, the EMU mesh element s were modeled in 3D space
using the following models:
1. Ellipsoid of revolution.
2. Sphere with semi-major axis taken as radius.
3. Sphere with an authalic radius.
4. Authalic sphere.
5. A rectangular mesh element, where the base face has the ellipsoidal surface area of the EMU grid cell.
Like the area analysis, mesh element volumes were only computed along a single meridional strip bet ween 0º
0◦
and 0. 25º
0.25◦
E longitude. The volume of the complete EMU columnar stack below the ocean surface to a depth
of 5,500 m, as well as the volume of each individual element in the stack, was computed. Volumes computed on
the WGS84 ellipsoid using Equation (8) were taken as true values and differences were calculated between the
true values and those from each of the alternative models listed above. For the rec tangular mesh element, the el-
lipsoidal surface area [Equation (2)] of the EMU grid cell was used as the area of the base face. Using other surfaces
for area calculation would propagate the area difference directly into the volume.

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4 | RESULTS
4.1 | Surface area analysis results
The results of the surface area analysis are summarized in Table 1 and Figure 4. Table 1 shows minimum and
maximum differences bet ween five surface models and the true ellipsoidal area of EMU grid cells along a meridi-
onal strip. Figure 4 illustrates how grid cell area differences change with latitude. Since grid cell area varies with
latitude, the absolute differences in surface area cannot be compared directly. Therefore, differences are reported
TABLE 1 Minimum and maximum EMU grid cell area differences bet ween five surface models and true
ellipsoidal area along a single meridional strip. Units are in ppm, which represents m2/km2 (to obtain values as a
percentage use the conversion 1% = 10,0 00 ppm)
Earth model Min. difference (ppm) Max. difference (ppm)
Sphere ( WGS84 semi-major axis as r adius) −6,694 6,739
Sphere (WGS84 authalic radius) −8,914 4,490
Projected surface
Web Mercator projection 6, 746 ∞
Lambert azimuthal projection (strip along 0 °E) −1, 0 96 −1
Lambert azimuthal projection (strip along 170°E) −25 0 1,185
FIGURE 4 EMU grid cell area differences between different surface models and true ellipsoidal area along
a single meridional strip and their variation with latitude. Units on the ver tical axis are 103 ppm (0.1%), which
represents 103 m2/km2

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KELLY and ŠaVRIČ
in part s per million (ppm, 10−6), where, on the surface, 1 ppm represents 1 m2 per 1 km2. To obtain values as a
percentage, use the 1% = 10,000 ppm. In the results that follow, negative ppm means underestimation of the true
area and positive ppm means overestimation of the true area.
Both spherical models exhibit bounded areal overestimation not exceeding 6,80 0 ppm at the equator and
underestimation by about 9,00 0 ppm at the poles ( Table 1). The area difference decreases toward the pole and
transitions from overestimation to underestimation at EMU grid cells bounded by 45° latitude for the sphere of
radius equivalent to the semi-major axis of the WGS84 ellipsoid. The same transition occurs at EMU grid cells
bounded by 35.25° latitude for the sphere with authalic radius (Figure 4).
Figure 4 shows that the Web Mercator projection overestimates the first EMU grid cell area at the equator
by 6,746 ppm and grows rapidly with latitude. The EMU grid cell beginning at 45° latitude is more than twice its
correct size, the 72.5−72.75° latitude grid cell is more than 10 times larger, and the size of the final cell at the
pole, 89.75−90°, cannot be calculated since Web Mercator projects poles to infinity, therefore, its surface area is
infinitely large (Table 1).
On the surface projected with the Lambert azimuthal equal area projection, differences vary between −1 and
−1,096 ppm for the strip along the central meridian and between −250 and 1,185 ppm for the strip along the 170th
meridian (Table 1). In both cases, the largest differences are observed when EMU grid cells approach the pole
(Figure 4). EMU cells from the equator up to 76.75° latitude along the central meridional strip underestimate area
by less than 1 ppm. EMU cells between 30.5 and 77.75° latitude along the 170th meridian show area differences
smaller than 1 ppm. Area differences transition from under- to overestimation along the 170th meridian at EMU
grid cells bounded by 60° latitude.
4.2 | 3D mesh volume analysis results
Table 2 and Figure 5 show the results of volume analysis of EMU mesh elements along a single meridional strip.
Table 2 shows minimum and maximum differences between four mesh element models and the true ellipsoidal
volume for the complete EMU columnar stack (depth range 0–5,500 m) and for eight EMU mesh elements in the
stack. These eight mesh elements include the first and last elements with depths of 5, 25, 50, and 100 m. Figure 5
illustrates how volume differences of the complete EMU columnar stack change with latitude. Since volume var-
ies with latitude and height (depth), the absolute differences cannot be compared directly. Therefore, these dif-
ferences are repor ted in ppm. For volumes, ppm is given as 103 m3 per 1 km3. To obtain values as a percentage,
use the conversion 1% = 10,000 ppm. In the results that follow, negative ppm means underestimation of the true
volume and positive ppm means overestimation of the true volume.
The spherical model with the WGS84 semi-major axis as its radius exhibits volume overestimation not exceed-
ing 6,800 ppm at the equator and underestimation not exceeding 6,700 ppm at the pole for the complete EMU
stack, as well as for individual EMU mesh elements (Table 2). For the spherical model with the WGS84 authalic
radius, overestimation and underestimation do not exceed 4,50 0 and 9,000 ppm, respectively. The volume differ-
ence decreases between the equator and the pole and transitions from overestimation to underestimation at EMU
grid cells bounded by 45° latitude for the sphere of radius equivalent to the semi-major axis. The same transition
occurs at EMU grid cells bounded by 35.25° latitude for the sphere with authalic radius (Figure 5). Table 2 shows
that the volume difference with depth grows only slightly for both spherical models.
Table 2 also shows that the rectangular element volume approximates the true volume closely for shallow
depths. The volume differences for the first few mesh elements below the surface do not exceed 8 ppm. However,
the volume difference grows as the mesh elements are taken to greater depths. Looking at the complete EMU stack
(Table 2 and Figure 5) as well as individual mesh elements (Table 2), there is very lit tle change in volume difference
with latit ude. Although not o bvious from Table 2 or Figure 5, t he analysis result s showed that for each mesh element,
maximums are always at the EMU cell nearest the equator, and volume difference decreases slowly toward the pole.

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TABLE 2 Minimum and maximum EMU mesh element volume dif ferences between four models and true
ellipsoidal volume along a single meridional strip at various EMU depth ranges. Depth range 0–5,500 represents
the complete EMU columnar stack. Unit s are in ppm, which represents 103 m3/km3 (to obtain values as a
percentage, use the conversion 1% = 10,000 ppm)
Prism model Depth range (m) Min. difference (ppm) Max. difference (ppm)
Rectangular 0–5 ,50 0 859.9 2 2 865.726
0–5 0.781 0 .787
95 –10 0 30. 471 30.677
10 0 –1 25 35.159 35.397
475–500 1 52.371 153.399
500–550 164.093 165.200
1,950–2,000 617. 513 621.680
2,000–2,100 640 .974 645.299
5,400–5,500 1,705.411 1,716.929
Spherical (WGS84 semi-major
axis)
0–5,5 00 −6 , 6 97.1 2 5 6 ,742. 337
0–5 −6,694. 255 6 ,73 9.414
95 –10 0 −6,694.354 6,7 39. 51 5
10 0 –1 25 −6,694.370 6 , 73 9. 5 31
475–500 − 6,694.762 6 , 73 9.9 3 0
500–550 −6,694.801 6,739.970
1,950–2,000 −6,696.316 6 ,741. 513
2,000–2,100 −6,696.394 6, 741. 59 3
5,400–5,500 − 6,699.949 6,745.214
Spherical (WGS84 authalic
radius)
0–5,5 00 −8 , 917. 5 74 4,491.846
0–5 −8,913.754 4,489.898
95 –10 0 −8,913 .886 4 ,4 8 9.9 65
10 0 –1 25 −8,913.907 4 , 489. 976
475–500 −8,914.428 4,490.242
500–550 −8,914.480 4,490.268
1,950–2,000 −8,916.496 4,491.297
2,000–2,100 −8,916.601 4,491.350
5,400–5,500 −8,921. 331 4,493.763
WGS84 authalic sphere 0–5,50 0 −3.857 1.941
0–5 −0.004 0.002
95 –10 0 −0.137 0.069
10 0 –1 25 −0.158 0.079
475–500 −0.684 0.344
500–550 −0.736 0.371
1,950–2,000 −2.77 0 1.394
2,000–2,100 −2.876 1.4 47
5,400–5,500 −7. 6 49 3.850

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KELLY and ŠaVRIČ
Finally, mesh elements modeled with the WGS84 authalic sphere approximate true volumes well, not ex-
ceeding 4 ppm overestimation at the equator and about 8 ppm underestimation at the pole (Table 2). Similar to
rectangular element volume, these mesh elements approximate true volume more closely for shallow depths. The
first few mesh elements below the surface do not exceed ± 4 m3/km3 difference; however, the differences grow
with depth. Volume dif ference also varies with latitude, an effect not well reflected in Figure 5 or Table 2. Figure 6
shows this variation more clearly, illustrating volume differences along a single meridional strip at various EMU
depth ranges. The deeper the range, the larger the absolute volume difference becomes, as well as its variation
with latitude. In all cases, the volume difference is decreasing, and transitions from overestimation to underesti-
mation at EMU grid cells bounded by 35. 25° latitude.
5 | DISCUSSION
In the surface area analysis of EMU grid cells, the Web Mercator projection yields the least accurate measure of
area. Figure 4 shows that areal differences grow rapidly with latitude in Web Mercator. This is because, on this
projected surface, the area distortion increases rapidly from equator to pole and is infinite at the pole (Equation 6).
This distortion propagates directly into surface area calculations. On the Web Mercator projection, EMU grid cells
increase in size and approach infinity near the pole (Table 1), while on a spherical or ellipsoidal Earth surface the
reverse is true, and the size of grid cells decreases with latitude (Figure 4). More generally, these results clearly
show that the Web Mercator projection is not appropriate for area calculations of longitude–latitude grid cells.
Figure 4 shows further that the Lambert azimuthal equal area projection produces an EMU surface area clos-
est to true ellipsoidal values, without the need for cell boundar y densification. The ellipsoidal equal area projec-
tion yields even better results than the spherical models; in fact, spherical surface area differs from its ellipsoidal
counterpart by larger amounts. Figure 4 shows that even when using the authalic sphere, the ellipsoidal equal area
FIGURE 5 Volume differences with latitude for the complete EMU columnar stack bet ween different mesh
element models and true ellipsoidal volume along a single meridional strip. Units are 103 ppm (0.1%), which
represents 106 m3/km3

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KELLY and ŠaVRIČ
projection still yields more accurate area measures. EMU cells bounded by 45° latitude for the sphere of radius
equivalent to the semi-major axis, and cells bounded by 35.25° latitude for the sphere with authalic radius, are
closest to their true values. However, the area for these cells is still less accurate than the area calculated on the
equal area projection. Comparing just sphere models used in the area analysis (Figure 4), a sphere of radius equiv-
alent to the semi-major axis produces slightly more area difference at the equator than the sphere with authalic
radius and slightly less at the pole. The reverse is true for the sphere with authalic radius.
Since there are many other equal area projections, comparisons were also performed using the Albers equal
area conic, sinusoidal, transverse cylindrical equal area, and Wagner VII projections. While these results are not
shown here, the area differences using these projections were still smaller than those of the spheres tested. At
least in the case of EMU grid cells, and possibly for other longitude–latitude grids, an equal area projection using
rigorous ellipsoidal equations results in grid areas closer to the true value than spherical areas, although this may
possibly be due to the small size of the EMU cells analyzed in this work.
In the EMU volume analysis, rectangular mesh elements unexpectedly yield consistently better results than
spheric al ones with radius equivalent to the semi-major axis and the authalic radius (Figure 5). This is because the
correct ellipsoidal surface area is used for the base face, and the difference arising from radii convergence contrib-
utes less than the difference in spherical surface area. The behavior of both spherical models in Figure 5 mimic s
that of the surface area in Figure 4. This suggests that, for shallow depths (heights) such as those analyzed herein,
surface area dif ference dominates over the area difference arising from convergence of radii and propagates
directly into the volume. On the contrary, comparing rectangular mesh elements with spherical ones on the auth-
alic sphere, the results are as expected: the authalic sphere provides bet ter approximation of the true ellipsoidal
volume. This shows that it is crucial to use the correct surface area in volume calculations.
FIGURE 6 EMU mesh element volume differences along a single meridional strip between the WGS84
authalic sphere and true ellipsoidal volume for various EMU depth ranges. Units are in ppm (0.0001%), which
represents 103 m3/km3

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KELLY and ŠaVRIČ
Convergence of radii becomes more impor tant as mesh elements approach the geocenter, as is evident from
Figure 2. Figure 6 illustrates that the difference in volume using an authalic sphere increases with both depth
(height) and latitude, although the magnitude of the difference is smaller than in other approximation models
(Table 2).
While the authalic sphere closely approximates true volumes, at least for shallow depths (height s), its calcu-
lation is not necessarily less computationally burdensome than direct ellipsoidal calculation using Equation (8).
Even though Equation (7) is simpler, one also needs to convert geodetic latitudes to authalic latitudes (see Snyder,
1987, p. 16) to obtain correct volumes, resulting in additional computations for each mesh element. Therefore, it is
advisable to use ellipsoidal equations over authalic sphere ones, which provide only approximate values.
6 | CONCLUSIONS
This article presents the relevant geodetic considerations needed in longitude–latitude grid and three-dimensional
mesh analyses and demonstrates the effec t of neglecting these aspects on global and regional data analyses using
GIS. Previously published formulae to calculate grid cell surface area and mesh element volume on the sphere and
the ellipsoid are compiled and presented. Using EMUs as an example, areas and volumes between these repre-
sentative models are compared, with emphasis on the popular Web Mercator projection and equal area Lambert
azimuthal projection as two cases of a projected surface. As mentioned previously, the most accurate represent a-
tion of terrestrial area and volume results when ellipsoidal Ear th models and equations are used [Equations (2) and
(8)]. Therefore, these equations were used to compute correct values for the analyses.
The results confirm that the Web Mercator projection should not be used for area or volume calculations
because of severe areal distor tion. If a projection must be used, then an ellipsoidal equal area projec tion should
be selected for the calculations. Since this class of projection uses ellipsoidal equations, it yields area and volume
closest to their true ellipsoidal values. Using a spherical model, the area difference is even larger than that of equal
area projection, even when cell boundaries remain un-densified. Therefore, the notion that a spherical model
produces the closest approximation of area is rejected.
For volume calculation at shallow depths (heights), it does not mat ter how radii convergence is modeled. It is
more important to obtain correct surface area, since this propagates directly into the volume and dominates over
the difference arising from convergence of radii. The closest result s to the true ellipsoidal volume can be obtained
on the authalic sphere. However, it is more efficient to use ellipsoidal equations over authalic sphere equations
because the latter are not necessarily less computationally expensive.
It is the authors' hope that this article will help GIS analysts better understand longitude–latitude grids and
three-dimensional meshes and assist them to correctly compute area and volume in their GIS analyses.
ACKNOWLEDGMENTS
The authors would like to thank Charles Preppernau for designing visualizations, David Burrows for verifying
our equations, and Jane Darbyshire for editing the text of this article, all from Environmental Systems Research
Institute Inc. The authors also thank the anonymous reviewers for their valuable comments.
ORCID
Kevin Kel ly https://orcid.org/0000-0001-7487-0015
Bojan Šavrič https://orcid.org/0000-0003-4080-6378
ENDNOTE
1 This can readily be seen when comparing the relative sizes of Greenland and Africa on a Web Mercator projection of
the world. While Greenland appears as large or larger than Africa, Africa is about 14 times larger in realit y.

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How to cite this article: KellyK, Šavrič B. Area and volume computation of longitude–latitude grids and
three-dimensional meshes. Transactions in GIS. 2020;00:1–19. https://doi .org /10.1111/tgis .12636