Chapter 4

The Geoid and Different Height

Systems
 The Geoid and Different Height
Systems
• The geoid can be understood as the physical shape of
the Earth.
• A reference ellipsoid can be interpreted as the
mathematical shape of the Earth, and a normal
ellipsoid is the mathematical and physical shape of the
Earth.
• The reference ellipsoid or normal ellipsoid is a close
approximation of the geoid.
• The shape of the Earth studied in geodesy primarily
refers to the shape of the geoid.
• The geoid also serves as a reference surface for height
determination of a given point on the Earth’s surface.
 4.1 Gravity Potential of the Earth and Geoid
4.1.1 Gravity and Gravity Potential
• Newton’s law of universal gravitation,
– any two bodies in the universe, possessing mass, exert
gravitational attraction on each other, which will thus create a
gravitational field around the point mass.
• F is used to represent this attractive force.
• The force F is directly proportional to the product of
their masses m and m’ and inversely proportional to the
square of the distance r between them, and can be
expressed as:

G (gravitational constant) =
 4.1.1 Gravity and Gravity…
• In geodesy, the particle of mass m is referred to
as the attracting mass, while the other particle of
mass m’ is the attracted mass, the mass of which
is used as a unit, i.e., m’ = 1.

• The Earth can be regarded as a body constituted

by infinite number of continuous point masses.
• The attraction that the Earth has exerted on the
unit point mass F is the integral:
 4.1.1 Gravity and Gravity…

• The attraction that the Earth has exerted on

the unit point mass F is the integral:

• where dm is the differential mass element of the Earth and r

represents the position vector between dm and the attracted
mass, which is a variable of integration; the integral area is
the total mass of the Earth.
• Due to the rotation of the Earth, every point on the Earth
experiences an inertial centrifugal force P .
 4.1.1 Gravity and Gravity…

• The attracting mass

m and the attracted
• where ρ denotes the vertical distance
mass m0
vector between the unit point mass and
the spin axis of the Earth and ω
represents the angular velocity vector
of the Earth rotation, which can be
determined precisely using
astronomical methods. Fig 1.The attracting mass m and the
attracted mass m’
• Its value is ω 7.292115 10⁻⁵ rad/s.
• P is perpendicular to the axis of
rotation and is directed against the spin
axis.
 4.1.1 Gravity and Gravity…

• The force of gravity of the Earth g ! is the resultant

of the gravitational force acting upon a unit point
mass and the centrifugal force of the Earth, namely:

• The gravity acceleration is measured in centimeters

per second squared (cm/𝑠 2 ), known as gal (after
Galileo; symbol Gal) in geodesy.
 4.1.1 Gravity and Gravity…

• Gravitational force, centrifugal force, and gravity

force all have their corresponding potential
functions.
• The function of gravitational potential is a numeric function
with respect to the variables of coordinate axes x, y, and z. Its
partial derivatives with respect to the three coordinate axes
correspond to the components Fx, Fy, Fz of the gravitational
force F ! in these three directions respectively, namely:
• As shown in Fig.1, m is the mass of the gravitationally
attracting body at the point (0,0,0); m0 is the attracted
mass at the point (x, y, z); the distance r between them is
given by:
𝑥 2 + 𝑦2 + 𝑧2 = 𝑟2
• Gravity Potential of the Earth and Geoid

Apparently,
• This gives the components of the gravitational force F ! along the
three coordinate axes.
• This indicates that the numeric function V is the gravitational
potential function of a point mass.
 4.1.1 Gravity and Gravity…

• assume that the unit point mass m0 moves from point B1

(distance r1) to point B2 (distance r2); then the work
(energy transfer) done by the gravitational force is:

– where dr denotes the displacement in the direction of the force.

• The potential difference between the two points is the
energy needed to move the point mass from the point of
lower potential to that of higher potential.
• If the potential value at point B1 is zero, then the
potential of a point equals the energy needed to move the
point mass from B1 to this point.
 4.1.1 Gravity and Gravity…
• Particle systems consist of a large number of point masses,
and the gravitational potential is the sum of the gravitational
potentials of the masses 𝑚1 𝑚2 ,… 𝑚𝑛

• The mass is continuously distributed within the body, and

hence it only requires conversion of the sum of (4.9) to an
integral to obtain the formula for gravitational potential of
the body:

• where dm denotes the differential mass element at the point (a, b, c)

• r = (𝑥 − a)2 + (𝑥 − b)2 +(𝑥 − c)2
 4.1.1 Gravity and Gravity…

• The centrifugal force or acceleration is given by:

 4.1.2 Earth Gravity Field Model
• The precise knowledge of the Earth’s gravity field
is fundamental to provide information about
the Earth’s shape, its interior and fluid envelope.
• It is essential for many Earth system research areas.
– such as quantifying mass distribution and mass transport,
monitoring oceanic transport, continental hydrology, ice
mass balance and sea level, and dynamics of mantle and
crust.
• global gravity field models are indispensable for
an improved understanding of the Earth system and
the interactions between its sub-components.
• relevant for geosciences, such as geophysics, glaciology,
hydrology, oceanography, and climatology.
Geoid from combined global gravity field model
EIGEN-6C4
Types of global gravity field models
1. satellite-only and combined models
• Satellite-only gravity field models are solely
determined by observed orbit perturbations
(e.g. CHAMP, GRACE, GRACE-Follow-On) and by
satellite gravity gradiometry (GOCE),
• Combined models are computed by adding
terrestrial, airborne, ship and altimetric gravity
observations to the satellite data.
 Types of global gravity…
2. Static and time-variable gravity field models.
• Time variable gravity field models can only be
determined from satellite data as they perform
repeated observations depending on the chosen orbit
and the satellite lifetime (e.g. time series of monthly
models).
• All other data have been collected over decades and
one hardly can define the observation epoch.
Therefore, combined models can always be regarded
as static (averaged) gravity field models.
 Types of global gravity…
• Altimetry satellites as well as terrestrial, marine
and airborne gravity measurements are used for the
computation of high-resolution combined gravity
field models with significant better accuracy and
spatial resolution.
• Many institutions and agencies worldwide are
computing such gravity field models for different
applications by applying different mathematical
approaches and analysis strategies.
 International Association of Geodesy
(IAG) Services
• Within the International Gravity Field
Service (IGFS) of IAG, the International Centre for
Global Earth Models (ICGEM), provides the
scientific community with a state-of-the-art archive
of static and temporal global gravity field models of
the Earth.
 International Association of Geodesy…
• ICGEM
– collects temporal and topographic global gravity field
models.
– Validates the models, and made these models publicly
available in a standardized format with digital object
identifiers (DOIs) assigned through GFZ Data Services.
– provides a web interface to calculate gravity field
functionals on freely selected grids or user-defined
coordinates, as well as a 3-D interactive visualization
service for these functionals using static and time variable
gravity field models.
– the users can perform quality checks of the models by
comparing them with other models in terms of
visualization and with respect to geoid undulations .
Source: http://icgem.gfz-potsdam.de/tom_longtime
 4.1.3 Level Surface and the Geoid
• The geoid is the equipotential surface, which
approximately coincides with the mean sea level (MSL) in
the ocean and its extension under the continents.
• The geoidal body closely approximates the natural surface
of the Earth.
• The geoid serves as
– a reference surface for height determination of a given
point on the Earth’s surface while studying the shape
of the Earth’s surface.
– It is also employed as the reference surface for
reduction of the astronomical longitude, latitude,
azimuth, and the values of gravity.
 4.1.3 Level Surface…

• The geoid is an irregular curved surface.

• The geoid, everywhere perpendicular to the
direction of the plumb line, is correspondingly an
irregular curved surface with slight undulations.
• The MSL is not the level (equipotential) surface, for
many factors can exert influence on the oceans such
as temperature and pressure variations, salinity,
winds, currents, rotation of the Earth, etc.
• It is measured value using tide gauges in different
countries or areas also varies.
 4.1.3 Level Surface…
• If a certain equipotential
surface is chosen as the
standard sea level, then
separation between MSL
and the standard sea level is
referred to as the sea
surface topography or sea
surface slope.
• The rise and fall is about 1–
2 m on a global scale
• MSL is not an equipotential
surface.
https://journals.lib.unb.ca/index.php/ihr/arti
cle/viewFile/23325/27100
 4.2 Earth Ellipsoid and Normal Ellipsoid
4.2.1 Earth Ellipsoid
• The ellipsoid of rotation that represents the Earth’s
shape and size is referred to as the Earth ellipsoid,
shortened to ellipsoid.
• The Earth ellipsoid is specified by four parameters:
– the semi major axis a and flattening f that represent
geometric properties of the Earth; ….
– the total mass M of the ellipsoid, which represents
the physical properties of the Earth; and
– the angular velocity ω of the ellipsoid rotating
around its minor axis
 4.2.1 Earth Ellipsoid…

• The four geometric and physical parameters of the

Earth ellipsoid have been calculated using data from
global terrestrial geodetic measurements and
satellite geodetic surveys .
• The plane that contains the rotation axis (minor
axis) of the reference ellipsoid is called the geodetic
meridian plane.
– It is the intersection of the plane containing the rotation
axis with the surface of the ellipsoid.
 4.2.1 Earth Ellipsoid…
• The plane through the center of the ellipsoid and
perpendicular to the axis of rotation is the Earth’s
equatorial plane.
• The equator is the intersection of the equatorial plane
with the ellipsoid.
• A parallel circle (parallel line) is an intersection of the
plane parallel to the equator with the ellipsoid, also
termed circle of latitude.
• The northernmost point N of the spin axis on the Earth
is the North Pole, lying diametrically opposite the
South Pole, S.
 4.2.1 Earth Ellipsoid…
• To facilitate the study of gravity and the gravity
field, the Earth ellipsoid is introduced, which is
called the normal ellipsoid.

Table 1 Earth ellipsoid parameters (semimajor axis a,

flattening f, gravitational constant X total mass GM, and
angular velocity ω)

Fig.4 Earth ellipsoid

 4.2.2 Normal Ellipsoid and Normal Gravity
• The normal ellipsoid is an imaginary rotational
ellipsoid with regular shape and homogeneous
mass distribution that satisfies certain conditions.
• It is the regular shape of the geoid and is used to
represent the ideal body of the Earth.
• The gravity field generated by the normal ellipsoid
is termed the normal gravity field.
• Corresponding gravity, gravity potential, and level
surface are called normal gravity, normal gravity
potential, and the spheropotential surface
(spherop), respectively.
 4.2.2 Normal Ellipsoid and…
The normal gravity field is a close approximation to the
actual Earth’s gravity field. In order to narrow down the
difference between the two, we select the normal ellipsoid in
accordance with the requirements below:
1. The spin axis of the normal ellipsoid coincides with the
Earth’s axis of rotation, and with equivalent angular
velocity.
2. The center of the normal ellipsoid is at the Earth’s center
of mass. The coordinate axis coincides with the Earth’s
principal axis of inertia.
3. The total mass of the normal ellipsoid is equal to that of
the actual Earth.
4. The sum of squares of the deviations of the geoid from
the normal ellipsoid is the least.
 4.2.2 Normal Ellipsoid and…

• The normal ellipsoid is defined by these four basic

parameters: semimajor axis of the ellipsoid a,
flattening f, are specify the geometric shape of the
ellipsoid and
• total mass M of the ellipsoid, and the angular
velocity ω of the ellipsoid identify the physical
properties of the ellipsoid.
 4.2.2 Normal Ellipsoid and…
• The normal ellipsoid is regular, so obviously its
gravitational potential is independent of λ and is only
a function of ρ and θ .
• The gravitational potential of the normal ellipsoid is
symmetric with respect to the equator.
• Find the cosine of θ and 180 θ for the two points that
are symmetric to the equator with opposite signs.
• Therefore, in the spherical harmonics series expansion
of the Earth’s gravitational potential, there are only
even zonal harmonics.
 4.2.3 Disturbing Potential

• After introducing the normal ellipsoid, there exist two

values of gravity potential for any arbitrary points on
the Earth: the true (measured) potential W of the real
Earth and the normal gravity potential U.
• There is a difference in value between the two
potentials.
• Subtracting the normal gravity potential from the
true potential W of the real Earth, the result is defined
as T, the disturbing potential.
T =W + U
 4.2.3 Disturbing…
• Earth’s gravity potential is equal to the sum of the normal
gravity potential and the disturbing potential.
• T = VE - VN + QE - QN:
o The subscript E is for the actual Earth, and the subscript N for the normal
ellipsoid.
o When the normal ellipsoid is chosen, its axis of rotation is made coincident
with the spin axis of the actual Earth and with equal angular velocity; thus
QE = QN.
o T = VE - VN:
o the disturbing potential can be interpreted as the difference in
gravitational potential caused by the differences in mass distribution
between the Earth and the normal ellipsoid
o The mass difference between the Earth and the normal ellipsoid is called
the disturbing mass
4.3 Height Systems
 4.3.1 Requirements for Selecting Height Systems

• The height of a point on the Earth’s surface is

geometrically defined as the distance from the point
along the reference line to the reference surface.
• Different reference lines or reference surfaces for heights
will constitute different height systems.
• Essentially, there are two classes of height system:
– ones that ignore the Earth’s gravity field and thus use straight-
line paths; and
– those that are naturally linked to the equipotential
surfaces and plumblines of the Earth’s gravity field
and thus follow curved paths.
 4.3 Height Systems…

• For the height system to be chosen, the following

requirements need to be fulfilled.
1. To represent the position of a point, the height of the
point is required to be unambiguous and independent of
the leveling path.
2. In practice, when converted to the adopted height system,
the corrections to the measured height differences for
points in a limited area should be very small so that they
can possibly be ignored while dealing with low-order
leveling data
 4.3 Height Systems…

3.From the geometric problem-solving perspective,

the ellipsoidal height is the sum of the measured
height and the geoid height; thus it requires that the
adopted height system should make the method for
determining the difference between the geoid and the
reference ellipsoid (normal ellipsoid) sufficiently
rigorous and convenient, as well as practical.
 4.3 Height Systems…
4. From the physical problem-solving perspective, the
chosen height system is also required to ensure that the
height of each point on the same level surface be
equivalent as much as possible.
• This is because the leveling data is actually used to
determine the physical problem of the relative position
of the Earth’s natural surface and the level surface of
the real gravity field, which is essential in avoiding the
“water runs uphill” phenomenon in engineering
application.
• In practice, choosing the most appropriate height
system requires compromise according to the different
requirements of the application
 4.3.2 Non-uniqueness of Leveled
Height
• Historically, the most commonly used technique for
the practical determination of heights is spirit
levelling.
• This technique measures the (geometrical) height
difference between two points (staves), where the
reference surface is the local horizon defined by the
set-up of the levelling instrument.
• Both staves and the levelling instrument are aligned
with the direction of the local plumbline (specifically,
the gravity vector) at each respective point.
 Height Systems Not Related to Gravity: Ellipsoidal
Heights

• It is conceptually simpler to first deal with purely

geometrical height systems, where the heights are
measured along straight lines.
• The most common geodetic height system not
directly related to gravity is the ellipsoidal height
system.
 Height Systems Not Related to Gravity…

• The ellipsoidal height is a straight-line distance reckoned

along the ellipsoidal normal from the geometrical surface
of a reference ellipsoid to the point of interest.
• The geometrical surface of the ellipsoid provides the
height reference surface by definition, on which the
ellipsoidal heights are zero.
• ellipsoidal height of a point is a function of the location,
orientation, size and shape of the reference ellipsoid used.
• the same point can have different ellipsoidal heights on
different ellipsoids
 Height Systems Related to Gravity: Natural or
Physical Heights
• These height systems come in several forms,
depending principally on the treatment of gravity and
thus the curved path over which the one-dimensional
metric distance (height) is defined.
• They also depend on the choice of the reference
surface used, though this is not as noticeable as it is
for the ellipsoidal heights (e.g., maximum differences
of ~2 m).
 Height Systems Related to Gravity…

• Geopotential Numbers Strictly, all natural or

physical height systems must be based on
geopotential numbers C.
• It is the difference between the Earth’s gravity
potential at the point of interest W and that on the
reference geopotential surface chosen W0 (i.e.,
C=W-W0).
 Height Systems Related to Gravity…

The Dynamic Height System

• The dynamic height system is most closely related to
the system of geopotential numbers.
• Dividing the geopotential number by a constant
gravity value (for a certain region, or even globally)
yields the dynamic height
 Height Systems Related to Gravity…
Orthometric Height
• Orthometric height (Ho) is the length between the
geoid (reference surface) and a point on the Earth’s
surface measured along the plumb line.
 Height Systems Related to Gravity…
Normal Height
• If the normal height of each surface point is HN,
measuring HN downward along the plumb line
(the normal gravity line in fact) results in the
corresponding points of each surface point.
• A continuous curved surface as the reference
surface for normal heights can be formed by
connecting these corresponding points.
• It is also called the quasi-geoid because of its
close approximation to the geoid.
• Therefore, the so-called normal height system
is the height system with the quasi-geoid as its
reference surface.
• The normal height of a surface point is the
distance from this point to the quasi-geoid
along the plumb line (the normal gravity line).
 Chapter 5
Reference Ellipsoid and the Geodetic Coordinate
System
5.1 Fundamentals of Spherical Trigonometry
Reading Ass.
 5.2 Reference Ellipsoid
• Reference ellipsoid used as the reference surface for
geodetic surveying computations.
• Conventional terrestrial surveys can only determine
directions, distances, and astronomical azimuths
between points on the Earth’s surface.
• To obtain coordinates of the horizontal control
points, a series of computations need to be carried
out .
 Reference Ellipsoid…
• The reference surface that fits for the geodetic
surveying computations should satisfy the following
three conditions:
1. The reference surface should be a curved surface that approximates
the physical shape of the Earth, so that the corrections for reduction
of the terrestrial observations are small.
2. The curved surface should be a mathematical surface on which
computations are easily performed so as to assure the possibility of
calculating coordinates through observational quantities.
3. The positions of the curved surface relative to the geoid should be
fixed so as to establish the one-to-one correspondence between the
points on the Earth’s surface and those on the reference surface.
 Reference Ellipsoid…
• The North Pole bulges out by 16 m and the South
Pole is depressed by approximately 16 m when the
geoid is compared with a properly defined ellipsoid.
• 21.4 km of difference between the Earth’s equatorial
radius and the polar radius .
 Reference Ellipsoid…
• The intersection line between the geoid and the
equatorial plane is not a perfect circle, but more
closely approximates an ellipsoid.
• The major axis of the ellipsoid on the equator is at
15⁰ west longitude.
• The difference between the semimajor axis (equatorial
radius) and the semiminor axis (polar radius) is 69.5
m.
• The equatorial flattening is 1:91,827, which is
approximately one three-hundredth of the polar
flattening
 Reference Ellipsoid…
• As a result, the “pear-shaped” sphere is a
mathematical surface that is an approximation to the
true shape of the Earth.

Fig. 1 Intersection line between the geoid

and the meridian plane (L =90:) Fig. 2 Intersection line between the geoid
and the equatorial plane
 Reference Ellipsoid…
• When a set of ellipsoidal parameters or an
Earth ellipsoid is selected, its location relative
to the geoid should be determined, namely to
complete the orientation of the ellipsoid.
• Therefore, the corresponding relationship
between the Earth’s surface and the ellipsoid
can be established to reduce the observations
from the terrestrial geodetic control network
to the ellipsoid.
 Reference Ellipsoid…
• The reference ellipsoid is the Earth ellipsoid
with defined parameters and orientation.
• The terrestrial observations in the geodetic
control network need to be reduced to the
reference ellipsoid and computations are to be
performed on this surface.
• Hence, the reference ellipsoid becomes the
reference surface for surveying computations.
 Reference Ellipsoid…
• The points from the physical surface of the Earth
are projected directly onto the ellipsoid along the
ellipsoidal normal.
• As a result, the ellipsoidal normal becomes the
datum line for surveying computations.
• The reference ellipsoid has defined the geodetic
coordinate system
 Reference Ellipsoid…
• The reference ellipsoid has played prominent roles in
surveying and mapping, as follows:
1. Used as the reference surface for the determination
of the horizontal coordinates (geodetic longitude
and latitude) and the geodetic height of a point on
the Earth’s surface.
2. Is the reference surface to describe the shape of
the geoid
3. Serve as the reference surface for map projection
 Reference Ellipsoid…
• To study global geodetic problems, there needs to be
a reference ellipsoid that best fits the geoid
throughout the entire Earth.
• Its center must coincide with the center of the
Earth.
• If the study is conducted both geometrically and
physically, then the general Earth ellipsoid can be
defined as the normal ellipsoid that best represents
the shape of the geoid.
 Reference Ellipsoid…
• The normal ellipsoid is the reference surface for
studying the Earth’s gravity field in physical
geodesy.
• The reference ellipsoid, on the other hand, is the
reference surface for studying geodetic computations
in geometric geodesy.
• Practically, due to the same mathematical properties,
the normal ellipsoid can be used as both the physical
and mathematical reference surface in geodesy
 5.2.2 Geometric Parameters of the
Reference Ellipsoid
• The six commonly used geometric parameters
in the Earth ellipsoid are as follows:
 Earth’s flattening
WGS 84 ellipsoid:
• a = 6,378,137m
• b = 6,356,752.3m
• equatorial diameter = 12,756.3km
• polar diameter = 12,713.5km
• equatorial circumference = 40,075.1km
• surface area = 510,064,500km2

• The flattening ranges from 0 to 1. A flattening value of 0

means the two axes are equal, resulting in a sphere. The
flattening of the earth is approximately 0.003353
61
 5.3. Which Spheroid to use?
• Hundreds have been defined depending
upon:
– Available measurement technology
– Map extent
– Country, Continent or Global
– Area of the globe
• e.g North America, Africa

62
 Which Spheroid …
• There are now two Ellipsoids/Spheroids most
commonly used to describe the shape of the Earth:
• The first was determined by the International
Association of Geodesy (IAG) is the Geocentric
Reference System 1980, or GRS 80
• The second was determined by the US Defence
Department and is known as the World Geodetic
System 1984, WGS 84 (a=6,378,137 b=6,356,752.31)

63
 5.4. Fitting an Ellipsoid/Spheroid to the Earth
• In Geodetic terms every point on the surface of
the Earth‘s Geoid is defined by 3 values.
• The most common method Geodesists use
involves measuring the distance from the
centre of the Earth to the point on the surface
of the Ellipsoid/Spheroid. This is called an
Earth-centred Cartesian Coordinate System.
• Three values are recorded (x, y and z), and
there are no angles - only distance.

64
 Cont‘d
• Another way is to use latitude, longitude and
ellipsoidal/spheroidal height (the height
above or below the datum‘s ellipsoid/spheroid
surface). The first two are angles and the
third is a distance.
• It is important to note that if a point is
identified using either system it is possible to
rigorously convert one to the other - provided
they use the same datum.

65
Fit…Cont‘d

66
Cont’d

67
 5.5. Datum:
• A datum is a system which allows the location of
latitudes and longitudes (and heights) to be
identified onto the surface of the Earth – i.e onto
the surface of a ‘round‘ object.
• A spheroid only gives you a shape—a datum
gives you locations of specific places on that
shape. Hence, a different datum is generally
used for each spheroid
• Two things are needed for datum: spheroid and
set of surveyed and measured points
68
 Datum: Ethiopia
• The Ethiopian Datum of 1936 was established
by the Italians at the West End of Metahara
Base (10,083.560 m) where Φ0 = 8°53′22.53″±
0.18″N, Λ0 = 39°54′24.99″ East of Greenwich,
the reference azimuth to Mont Fantalli was
α0 = 13°05′21.97″+ 0.43″, and the presumed
ellipsoid of reference was the International
1924 where a = 6,378,188 m and 1/f = 297.

69
 Blue Nile Datum of 1958
• The Blue Nile River Basin Investigation Project
was funded by the United States, and the
geodetic work was performed by the U.S. Coast
& Geodetic Survey. The origin of the geodetic
work was in southern Egypt near Abu Simbel,
south of Lake Nasser, at station Adindan where
Φ0 = 22°10′07.1098″N, Λ0 = 31°29′21.6079″ East
of Greenwich, the deflection of the vertical ζ =
+2.38″ and η = –2.51″, and the ellipsoid of
reference was the Clarke 1880 (modified) where
a = 6,378,249.145 m and 1/f = 293.465.

70
 Cont’d
• The Blue Nile Datum of 1958 appears to be
the established classical datum of Ethiopia
and much of North Africa.
• Adindan is the name of the origin, it is not
the name of the datum; a most common
mistake found in many ―reference works.‖

71
 Cont’d
• The Ethiopian Transverse Mercator grid is
based on a central meridian where λ0 =
37°30′E, scale factor at origin where mo =
0.9995, False Easting = 450 km, and False
Northing = 5,000 km.

72
 Common Datums:
• Previously, the most common spheroid was Clarke 1866; the
North American Datum of 1927 (NAD27) is based on that
spheroid, and has its center in Kansas.
• NAD83 is the new North American datum (for Canada/Mexico
too) based on the GRS80 geocentric spheroid. It is the official
datum of the USA, Canada and Central America
• World Geodetic System 1984 (WGS84) is a newer
spheroid/datum, created by the US DOD; it is more or less
identical to Geodetic Reference System 1980 (GRS80).
• The GPS system uses WGS84.

73
 6. Coordinate systems

• A coordinate system is a reference system used to represent

the locations of geographic features, imagery, and
observations such as GPS locations within a common
geographic framework.

• Coordinate systems enable geographic datasets to use

common locations for integration.

74
 6.1. Type of Coordinate systems

1. A global or spherical coordinate system such as latitude-

longitude. These are often referred to as geographic coordinate
systems.
2. A projected coordinate system based on a map projection such
as transverse Mercator, Albers equal area, or Robinson, all of
which (along with numerous other map projection models)
provide various mechanisms to project maps of the earth's
spherical surface onto a two-dimensional Cartesian coordinate
plane. Projected coordinate systems are sometimes referred to as
map projections. 75
 6.1.1. Geographic coordinates

• Spherical Co-ordinates
– To locate the position of places on a spherical earth we depend
on geographic coordinates.
– To know whether a map is representing the position of places
exactly or not, we need a frame of reference.
– The most common frame of reference is the system of
geographic coordinates
– These coordinates are related to the earth’s axis of rotation and
plane of the equator.
 Cont’d
• Geographic coordinate Systems (GCS) defines location on
the earth using a three dimensional spherical surface.
– They are networks of parallels and meridians

– These networks of parallels and meridians are known as graticules

• The rings around the earth parallel to the equator are called
parallels of latitude
– Which help to measure angular distance north or south of the equator
 Cont’d
• Lines of parallels of latitude run east-west but measure north-
south distances from the equator.

• Starting with 0° at the equator, the parallels of latitude are

numbered to 90° north and south.

• The distance of a point north or south of the equator is known

as its latitude.
 Cont’d
• A second set of rings around the globe that crosses at right
angles to lines of latitude and passing through the poles are
known as meridians of longitude or simply meridians.

• Meridians of longitude measures angular distance east or west

of the prime meridian.

• Lines of longitude (meridians) run north-south but east-west

distances from the prime meridian are measured between them.

• Starting with 0° at the prime meridian, longitudes are measured

both east and west around the world.
 Cont’d
• Lines east of the prime meridian are numbered to 180° and identified as
east longitude; lines west of the prime meridian are numbered to 180° and
identified as west longitude.

• The direction E or W must always be given

Origin of Geographic coordinates Network of latitudes and Longitudes ( Graticules)
 Basic Properties of the Earth’s Graticules

 All latitudes are parallel and are equally spaced along

meridians.

 The ground distance covered by one degree of latitude is about

111 kilometers

 One minute is equal to 1.85 km (111/60) while one second is

equal to about 30 meters (1.85/60).

 They run in west east directions and measure angular distances

north or south of the equator ranging from 00 to 900.
 Cont’d
 Equator divides the earth into two halves (Northern and Southern Hemisphere).

 Latitudes (except 900 lat.) seem as concentric circles when they are viewed from the
poles.
 Cont’d
 The only great circle of all parallels of latitude is equator, while
others are small circles. Therefore the circumference of any latitude
will be

C=2∏r x cos θ;
Θ = represents degree latitude.
r = represents radius of the earth

Example
The circumference of the 00 latitude (Equator) is

= 2x3.14x 6378.5x cos θ = 40057 km

Calculate the circumfrence of 600 and 850 of latitude?

 Cont’d
• Meridians are equally spaced along a given parallel, but

different along other parallels.

• Meridians and parallels intersect at right angles.

• Thus quadrilaterals formed by the same two parallels and the

same longitudinal dimensions have the same areas.

• Therefore area scale and distance scale are uniform.

• Thus, the distance of one degree of longitude varies from approximately

111 km at the Equator to 0 km at the poles. At any latitude, the distance of

1 degree of longitude can be calculated by multiplying the ground distance

of 1 degree of longitude at the Equator times the cosine of the latitude.

10 longitude= 111 x Cos θ; θ represents latitudinal location

 Cont’d
Example

• At 00 latitude the distance of one degree longitude is

• 111xcos θ = 111km
Question

1. At 600 latitude the distance of one degree longitude will be?

---------------------------------------------------------
-
 Distance Calculation using Spherical Coordinates

• Emphasizes on how distances are measured using spherical

coordinates.

• The great circle distance between two points is often difficult

to measure on a globe.

• Great circle or arc distances (shortest distance) can be

calculated easily; given the latitudinal and longitudinal
location of two points or places is known, using the following
formula from spherical trigonometry:
 Cont’d
Cos D = ( sin a )(sin b) + (cos a)(cos b)(cos P)

where: D is the angular distance between points A and B

a is the latitude of point A

b is the latitude of point B

P is the longitudinal difference between points A and B

 6. 1. 2. Projected Coordinate system(UTM)

• On a global map, UTM lines are straight.

• UTM is a type of projection, calculated to make a flat map of
the round Earth.
• UTM zones are numbered east to west and lettered north to
south.
• Each zone is equivalent to 6o.
• The measurement unit is meters.

89
UTM

90
 Each coordinate system is defined by

• Its measurement framework which is either geographic (in

which spherical coordinates are measured from the earth's
center) or planimetric (in which the earth's coordinates are
projected onto a two-dimensional planar surface).
• Unit of measurement (typically feet or meters for projected
coordinate systems or decimal degrees for latitude-
longitude).

91
 Coordinate system…cont‘d

• The definition of the map projection for projected

coordinate systems.

• Other measurement system properties such as a spheroid of

reference, a datum, and projection parameters like one or
more standard parallels, a central meridian, and possible
shifts in the x- and y-directions.

92
• Brainstorming
• How Geoid, Ellipsoid, Spheroid and Datum are
related?

93
 6.1.3. vertical coordinate systems

• The vertical coordinate system defines the origin for height

and depth values.

• Without the correct vertical coordinate system defined, you

may see your data incorrectly shifted vertically, this is
especially relevant when working with datasets with different
vertical coordinate systems.
 Vertical Datum Definition

 Vertical datum is defined by the surface of reference – geoid or

ellipsoid
 An access to the vertical datum is provided by a vertical control
network (similar to the network of reference points furnishing the
access to the horizontal datums)
 Vertical control network is defined as an interconnected system
of bench marks
 Why do we need vertical control network?
• to reduce amount of leveling required for surveying job
• to provide backup for destroyed bench marks
• to assist in monitoring local changes
• to provide a common framework
 6.2. Map Projections
• Mathematical method for systematically
transforming a 3-D earth into a 2-D map.
• Three traditional types:
– cyllindrical
– conical
– planar (azimuthal-zenithal)
• Newer Mathematical Projections
– Robinson

96
97
98
99
100
101
Grid North = very close to true north. Used
to place grids on maps for archaeology,
mines, artillery targeting.

102
103
Mercator’s Navigation
Technique

Gnomonic Projection shows great

circles as straight line.
Mercator Projection shows
constant compass headings
(azimuth) as straight lines.

Rhumb
Lines

104
105
 6.2.1 Process of Map Projection

• the process of map projection is accomplished in three specific

steps:
a) ellipsoidal or spherical surfaces are used to represent the surface of
the Earth.
b) curved reference surfaces are then projected on a map formed into a
cylinder, cone or flat plane reduced in size.
c) each point on the reference surface of the Earth with geographic
coordinates may be transformed to set of Cartesian coordinates or
map coordinates representing positions on the map plane.
– The systematic transformation of curved surfaces into flat plane is
called map projection.
106
The process of representing the Earth on a flat map.
107
108
• Hundreds of map projections are developed in order to
accurately represent a particular map or to best suit a
particular type of map.
• Map projections are typically classified according to the
geometric surface from which they are derived: cylinder,
cone or plane.
• The three classes of map projections are respectively
cylindrical, conical and azimuthal.

109
• A map projection is a mathematically described technique
of how to represent the Earth‘s curved surface on a flat
map.

• Mapping onto a 2D mapping plane means transforming

each point on the reference surface with geographic
coordinates (lat,long) to a set of Cartesian coordinates (x,y)
representing positions on the map plane (figure below).

110
Example of a map projection where the reference surface with
geographic coordinates (f,l) is projected onto the 2D mapping plane with
2D Cartesian coordinates (x, y).
111
 6.2. Classification of map projections

• Map projections can be described in terms of their:

• class (cylindrical, conical or azimuthal),

• point of secancy (tangent or secant),

• aspect (normal, transverse or oblique), and

• distortion property (equivalent, equidistant or

conformal).

112
i. Classes of map projections
– Cylindrical, conical and azimuthal.

– The Earth's reference surface projected on a map wrapped around

the globe as a cylinder produces a cylindrical map projection.

– Projected on a map formed into a cone gives a conical map

projection.

– When projected directly onto the mapping plane it produces an

azimuthal (or zenithal or planar) map projection.

– The figure below shows the surfaces involved in these three classes of
projections.

113
The three classes of map projections: cylindrical, conical and azimuthal. The projection
planes are respectively a cylinder, cone and plane.

114
ii. Point of secancy (tangent or secant),
– The planar, conical, and cylindrical surfaces in the figure above are
all tangent surfaces; they touch the horizontal reference
surface in one point (plane) or along a closed line (cone and
cylinder) only.
– Another class of projections is obtained if the surfaces are chosen to
be secant to (to intersect with) the horizontal reference surface;
illustrations are in the figure below.
– the reference surface is intersected along one closed line (plane) or
two closed lines (cone and cylinder).
– Secant map surfaces are used to reduce or average scale
errors because the line(s) of intersection are not distorted
on the map.
115
Three secant projection classes

116
iii. Aspect (normal, transverse or oblique)
– Projections can also be described in terms of the direction of the
projection plane's orientation (whether cylinder, plane or cone) with
respect to the globe. This is called the aspect of a map projection.
– The three possible apects are normal, transverse and oblique.
– In a normal projection, the main orientation of the projection surface
is parallel to the Earth's axis (as in the figures below for the cylinder
and the cone).
– A transverse projection has its main orientation perpendicular to the
Earth's axis.
– Oblique projections are all other, non-parallel and non-
perpendicular, cases. The figure below provides two examples.

117
A transverse and an oblique map projection.

118
• The terms polar and equatorial are also used. In a polar azimuthal
projection the projection surface is tangent or secant at the pole.

• In an equatorial azimuthal or equatorial cylindrical projection, the

projection surface is tangent or secant at the equator.

119
iv. distortion property (equivalent, equidistant or conformal)
– The distortion properties of map are typically classified according to
what is not distorted on the map:

– In a conformal (orthomorphic) map projection the angles between

lines in the map are identical to the angles between the original
lines on the curved reference surface. This means that angles (with
short sides) and shapes (of small areas) are shown correctly on the
map.

120
• In an equal-area (equivalent) map projection the areas in the map are
identical to the areas on the curved reference surface (taking into account
the map scale), which means that areas are represented correctly on
the map.
• In an equidistant map projection the length of particular lines in
the map are the same as the length of the original lines on the
curved reference surface (taking into account the map scale).
• A particular map projection can have any one of these three
properties. No map projection can be both conformal and
equal-area.
• A projection can only be equidistant (true to scale) at certain places or in
certain directions.

121
V) Name of Inventor
• Another descriptor of a map projection might be the name of the
inventor (or first publisher) of the projection, such as Mercator,
Lambert, Robinson, Cassini etc., but these names are not very helpful
because sometimes one person developed several projections, or several
people have developed similar projections.

• For example J.H.Lambert described half a dozen projections. Any of

these might be called 'Lambert's projection', but each need additional
description to be recognized.

122
• Based on these discussions, a particular map projection can be classified.
• An example would be the classification ‗conformal conic projection
with two standard parallels‘ having the meaning that the
projection is a conformal map projection, that the intermediate surface
is a cone, and that the cone intersects the ellipsoid (or sphere) along two
parallels; i.e. the cone is secant and the cone‘s symmetry axis is parallel
to the rotation axis.
• This would amount to the projection of the figure above (conical
projection with a secant projection plane).

123
• Other examples are:
– Polar stereographic azimuthal projection with secant projection
plane;

– Lambert conformal conic projection with two standard parallels;

– Lambert cylindrical equal-area projection with equidistant

equator;

– Transverse Mercator projection with secant projection plane.

124