1. science
2. mathematics
3. geometry

TOWARDS IMPLEMENTING UNIVERSAL TRANSVERSE MERCATOR
COORDINATES IN GHANA.
John Ayer
Department of Geomatic Engineering
KNUST
Kumasi - Ghana
Abstract.
Over the years, different Nations have defined and used different Spatial Reference
Systems for mapping and spatial data gathering and archiving. In recent times, the
need to share information and also map and manage transboundary resources on
Regional, Continental and Global basis is leading to the adoption of common
reference frame initiatives such as the African Reference Frame (AFREF).
The Universal Transverse Mercator (UTM) coordinate system has been proposed for
some years now for global mapping but its adoption has been slow because it
required new mapping to replace existing maps and map data held by different
Nations over the years. However, its adoption has advantages for data integration.
Besides, recent technological advancements such as the development of Global
Navigation Satellite System (GNSS) which computes coordinates based on WGS84
ellipsoid directly and also provides a program for projecting results in UTM has
provided a further impetus for adopting UTM coordinates.
This paper discusses the different scenarios that could result from the adoption of
UTM and proposes what steps are required for its implementation in Ghana.
Keywords.
Spatial Reference Systems, Universal Transverse Mercator, GNSS, Horizontal
Datums, LWOST.
1
Introduction.
1.1 Reference systems, Datums and Coordinate systems
Spatial reference systems help to integrate data from different sources together such
as images, maps and ground surveys. They also allow one to refer to positions on and
near the Earth’s surface uniquely and unambiguously by allowing the assignment of
unique values to each position in space.
Examples of reference systems include the International Terrestrial Reference System
(ITRS), the Ghana Coordinate System, the Geodetic Reference System of latitude and
longitude definitions using various ellipsoids, Metes and Bounds, and Postal codes
etc.
Datums refer to a set of parameters (values) and control points used to define the
shape of the earth, and the origin and orientation of the coordinate system used by a
country. Ordinarily, the word means facts which are known from which other
conclusions can be drawn. When used with reference systems, it means origin points
with assigned values which could then be used for deriving values of other points.
One may therefore define Horizontal datums for horizontal measurements, Vertical
Datums for height measurements and combined Datums that may combine both
horizontal and vertical measurements.
To understand the concept of datum properly, it must be noted that Location will only
make sense when it is compared to some frame of reference. For example one may
speak of a Building with house number H215 as being opposite the General Post
Office in Accra. This description is relative but in absolute terms, the same building
could be stated as having positional coordinates “05° 35’ 06” N, 00 ° 01’ 09” W” (of
course value depends on ellipsoid!!), or “333382.141 N, 1256469.743 E in Ghana
National Datum (GND)” or “X=6348703.172, Y= -2124.826, Z=616629.997
Cartesian values” or else “618475.6234 N, 830306.7194 E, in UTM Z 30 N”. Thus
the same building has different positional coordinate values depending on the chosen
reference frame.
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1.2 Horizontal Datum
A Horizontal Datum comprises a base reference for horizontal coordinates.
It
includes the position of an initial point of origin and an ellipsoid that models the
surface of the earth in the region of interest. The complete Datum definition consists
of the origin point of the X,Y,Z, axis (three parameters), the orientation of the axes
(three parameters), the ellipsoid used in terms of its shape and size (two parameters)
[Steed, J., 1990].
Why an ellipsoid?
The picture below (figure 1) is that of earth from ERS-1
Figure 1
It could be seen that:
•
The shape is complex and not suitable for any mathematical calculations.
•
The simplified mathematical surface that can approximate to portions of its
surface is the ellipsoid.
•
No single ellipsoid will accurately fit all parts so each region may have
separate best fitting ellipsoid.
•
A global average ellipsoid may be defined.
The reason an ellipsoid is used and not a perfect sphere is because the Earth spins
making it bulge out a little bit at the equator. Since the actual Earth is irregularly
shaped, one can come up with a number of different values for the major radius (a),
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the minor radius (b), and the flattening (f), depending on where they are measured.
[Snyder, J.P.]
1.3 Traditional Horizontal Datums
Many Nations established their own national datums using various national standards
and procedures by calculating ellipsoids that fit well locally. They also established
initial point location and orientations usually with astronomic observations. The
results of the different datums are that, transformations are necessary to convert
between them because ellipsoid shape and size differences result in origin shifts and
rotations. [EPSG, 2005].
To illustrate some limitations to the traditional approach, consider the difference in a
position as defined on the global average datum WGS84, (ellipsoid values Semimajor Axis: a = 6378137 m, Semi-minor Axis: b = 6356752.3142m, Flattening Ratio:
f=(a-b)/a = 1/298.257223563) and the Ghana Horizontal Datum ( a= 6378299.996 m.
b=6356751.69m. and f=1/296)
Positional coordinate values on WGS84, ( N 06° 04' 49.84347" E 0° 03 ' 0.85954" )
Positional coordinate values on Ghana Datum (N 06 ° 04 ' 39.9216 "
E 0 ° 02 '
59.7823 ")
Separation difference between the points on the different datum definitions is 306.6m.
which is more than 1000 feet.
1.4 Vertical Datums
Like horizontal measurements, elevation only has meaning when it is referenced to
some start point. For this reason, different levels of starting references have variously
been used such as the low tide level, the high tide level and the mean sea level which
coincides with a local geoid definition. Fig 2. The Global Positioning System (GPS)
however uses a height above the global ellipsoid.
4
Figure 2: Height references
1.5 Coordinate systems
A Coordinate system is a spatial reference system that allows one to assign unique
values (coordinates) to each location and establish relationships between sets of such
coordinates. It enables the interpretation of a set of coordinates as a representation of
a unique position in a real world space. [Ayer, J. and C. Fosu, 2008]
In Ghana, we have been associated with five (5) different types of coordinates from
Colonial days until now namely:
i. Geodetic or Geographic Coordinate System
This is a Global Coordinate System based on angles defined on the ellipsoidal
Earth surface. It is expressed in longitudes and latitudes. [Survey Records, 1936]
ii. Cartesian coordinate system
This measure position of a point from a defined geocentric origin using three
perpendicular axes in three-dimensional space denoted by X, Y, Z. The Z axis
aligns with the axis of the rotation of the earth. Conversion between geodetic
coordinates and equivalent Cartesian coordinates use standard mathematical
formulations. [Steed, J., 1990]
In Ghana, Cartesian coordinates are not directly used for mapping but these
are also given when a GPS is used.
iii. Projected coordinates
These are planar two dimensional Cartesian-like coordinates that result from
performing a mathematical mapping from a point on the Earth's surface to a
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plane. They relate the coordinates of points on earth’s curved surface with
equivalent values of the same points on a plane or flat surface (MAP).
Many such mathematical mappings are used for particular purposes but the type
used in Ghana and many other parts of the World is known as a Transverse
Mercator Mapping. [Survey Records, 1936]
The Transverse Mercator projection is a cylindrical projection. A scale
distortion is often applied along the central meridian to minimize distortions.
The map projection is itself defined by a set of equations for converting
geographic latitudes and longitudes into projected Eastings and Northings and
the reverse calculation. [Snyder, J. P. 1987]
iv. Local coordinates
In Ghana, local coordinates used to be defined for each town based on a
Cassini projection. Usually, a point close to the centre of the town was chosen
with assigned false origin values of X=10000ft. and Y=10000ft. This together
with azimuth bearing formed the basis for plan mapping and drawing of town
maps. [Survey Records, 1936]
v. The Universal Transverse Mercator Coordinate System
This is a specific implementation of the transverse Mercator projection
designed for world wide mapping in which the world is divided into zones
with each zone projected individually. There are 60 zones, each 6° longitude
wide starting from 180° W eastwards. [Toms R.M, 1996]
The UTM is a standard means of stating any position on earth, by giving two
numbers — the easting or how far East, and the northing or how far north. To
use UTM to specify a position anywhere, one must also state the zone. The
origin for each zone is the intersection of the zone central meridian and the
equator.
In the Northern hemisphere, the northing is zero on the Equator and increases
the farther one moves north from the Equator. The false easting on the zone
central meridian is 500000m. Also, the Scale along the central meridian for
each zone is 0.9996
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In the Southern hemisphere, the false Northing is 10000000m. and the false
Easting is 500000m. [D.M.A., 1990]
1.6 How do we deal with areas of interest that cross UTM zones?
This problem is certain to occur with areas that are wider than 6° of longitude or on
some occasions a narrower area of interest may be bisected by the edge of a UTM
zone
One has to understand that; all features need to be encoded using a consistent
coordinate system so the usual method is to assign the coordinates to the more
predominant zone that contains the majority of the area of interest.
It is also possible to encounter an instance when the area of interest crosses the
Equator. Note that, for Northern zones, the Equator’s Northing is at 0m but for
Southern zones, the Equator’s Northing is at 10,000,000m. Coordinates must be
consistent. But since 0 ≠ 10,000,000, the usual solution is to assign all coordinates to
the Southern zone. [Defence Mapping Agency. (1987)]
2.0 Historical setting- Ghana National Datums for Plan and Height
2.1 National Geodetic framework
Ghana National framework consists of monuments planted at points whose
coordinates are well known. National geodetic controls in Ghana dates back to June
1904 when observations for latitude were done by Captain P. G. Guggisberg from a
pillar established in the house of the then Secretary of Native affairs. This pillar was
connected by traverse to GCS 547 in Accra and longitude for GCS 547 was
determined by telegraphic signal exchange with Cape Town in November 1904 to
establish both latitude and longitude values for GCS 547. The initial pillars, GCS 547
and GCS 121 at Legon were connected by triangulation to obtain their latitudes and
longitudes. [Survey Records, 1936]
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Other series of triangulation nets were established to obtain latitude and longitude of
trigonometric points. The controls in Northern territories and low lying regions were
established by traversing.
The Main Horizontal framework therefore consists of five chains of triangulations and
a number of primary and secondary traverses.
The Calculated best fitting local ellipsoid that fitted the measurements done in the
then Gold Coast was suggested by the British war office using initial measurements.
This ellipsoid has subsequently come to be known as the War Office ellipsoid.
Ghana Horizontal Datum therefore comprises the following:
• Ellipsoid: Ghana War Office.
– Parameters of ellipsoid :
•
a=6378299.996 m.
•
b=6356751.69 m.
• inverse flattening 296
– feet
log
to
meter
conversion
factor
=
0.304799706846.
f −
= 1.484014544967 (Departmental tables, pg. 48)
m
Easting and northing coordinates are derived using a Transverse Mercator
projection. The origin of coordinates is the intersection of latitude 4 degrees 40 (4o
40’ N) minutes North and longitude 1 degree west (1o W). This origin is given
values of X =0.00 ft., Y = 900000.00ft. To avoid distortion at the eastern and
western edges, true distance along the central meridian was decreased by 1/4000.
This is equivalent to a scale of 0.99975 along the central meridian and this secant
implementation of the transverse Mercator meant that scale is correct at
Y=434000 and Y=1366000. ie 466000 feet from the central meridian on either
side. (W 002o 16’ 50” and E 000o 16’ 50”) [Technical Instructions to Cadastral
Branch]
For Ghana then, horizontal datum is defined by:
• Ellipsoid = War office
• Semi major axis a = 6378299.996m.
• Inverse flattening 1/f = 296
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• Latitude of origin 4o 40’ N
• Longitude of origin = 1oW
• False Northing = 0.000ft.
• False Easting = 900000.000 ft.
• Scale along central meridian = 0.99975.
This datum definition is referred to in some publications as Accra Datum.
2.2 Local Town Frameworks
Most Town surveys used to be surveyed based on Local Frameworks determined and
executed to fit the whole town before the establishment of the National geodetic
framework.
All towns have now been connected to the main national framework and formulae for
conversion worked out at the Survey Department (SD) Head office for transformation
between such local values and National values.
Because these cover limited regions of within 100km2, affine transformations were
sufficient. Thus for Accra, the conversion formulae are:
X N = 302406.75 + 0.999829 X L + 0.001355YL
YN = 1154438.16 + 0.999829YL − 0.001355 X L
Local to National conversions for other Towns were derived from the equations:
X N = X o + X L + F0 X L + 2 RX L YL (Y0 − 900000)
YN = Y0 + YL + F0YL + R(YL − X L )(Y0 − 900000)
2
2
Where F0 = scale factor defined for each point and R is a constant taken to
be 1.149 x 10-15
In addition, Cartographic charts are designed for use to avoid manual calculations
by the Survey Department. [Survey Records]
2.3 Ghana Vertical Datum
Level heights in Ghana are based on the Mean Sea Level (MSL) determined in Accra
in 1923 as height datum. Very old lines were based on the Low Water of Ordinary
Spring Tide (LWOST) at Accra or LWOST at Sekondi as Datums.
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Levels based on the LWOST of Sekondi can be converted to MSL by subtracting 3.00
ft. whereas those values based on LWOST of Accra can be converted to MSL by
subtracting 2.85 ft. [Survey Records]
2.4
Leigon Datum
In 1977, the triangulation and traverse framework was re-adjusted together using the
method of variation of Geographical coordinates.
In this new adjustment, it was agreed to hold the existing latitude and longitude values
for GCS 121 (5o 38’ 52.2700” N, 0o 11’ 46.08” W) at Legon as fixed (error free).
Following this adjustment, projected coordinates were computed on the original
Transverse Mercator projection using the same origin of the intersection of latitude 4o
40’N and longitude 1oW to which again false origin values of 0.00 ft. for Northing
and 900000.00 ft. for Easting were adopted.
However ellipsoid proposed for projection was Clarke 1880 because other African
Nations around were already using that ellipsoid and it was thought that this would
enhance future regional mapping. [Survey Records, Aug. 1978]
Although this Re-computation was completed in August 1978, the results were not
implemented or made public. However, it has come to be severally known as “Leigon
datum”, or Ghana metric grid in some publications.
The positional shift between the two different ellipsoidal projected coordinates is on
the average of 25 feet perhaps why it was never used.
3.0 Transformation Procedures for Implementing UTM Coordinates in Ghana
Transformation is the conversion of coordinates from one coordinate system to
another. Transformation involves the conversion of geodetic coordinates from one
geodetic datum to another geodetic datum, usually involving changes in the shape,
orientation, and center position of the reference ellipsoid. [Ayer, J. 2008].
3.1 The Need for Geodetic Transformations
Maps defined on different spheroids, projections, and origin points [datums] can
not be compared or integrated without converting them first into a common datum
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frame. This conversion is often difficult because there are no immediate tools for
computing them. Performing transformations between planimetric coordinates of
different geodetic datums is a complex mathematically involving process. It
requires the selection of an appropriate transformation model which is
computationally demanding so the availability of user-friendly software tools
would be an advantage. Usually transformations are defined between a
local/National datum and WGS84.
Figure 3 below shows various scenarios of such transformations between the
WGS84 Datum and the Ghana datum.
Figure 3 Transformation Scenarios
In Scenario one, a GPS is used to collect data corrected against a reference
Continuous Reference Station (CORS) data in International Terrestrial Reference
Frame (WGS84). Since the result is also required in WGS84, there is no need for
the application of any transformation.
For Scenario two, the GPS data is corrected against a reference CORS data in
WGS84. If the result is required in the Ghana National Datum (GND), then a
transformation is needed from WGS84 to GND.
In Scenario three, measurements are done in GND. If the results are however
required in WGS84, then again a transformation from GND to WGS84 is
required.
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In Scenario four, measurements are done in GND and the results are also required
in GND so here again no transformation is required.
Conclusions and Recommendations
As indicative from above, if any new survey is done using a GPS it is possible to
project results directly to UTM once it is properly referenced to a CORS and there
is no need for any transformation.
Alternatively, UTM controls must be established with GPS and thereafter classical
means may be used and results processed in UTM.
A third option is to use classical means of survey based on existing SD controls
and then transform results to UTM. This third option requires that SD must design
software for doing the transformations. In addition Cartographic charts prepared
by SD can also be used to deduce the transformations.
A forth option is when a GPS is used for Surveying but the results are required in
the GND. This is the only option which may require the use of published SD
transformation parameters as input for GPS processing.
In either way, the major challenge is the conversion of all existing data into the
new UTM grid and this requires the software for direct conversion and the
cartographic charts recommended for option three.
References
1. Ayer, J. 2008, Transformation Models and Procedures for Framework
Integration of the Ghana National Geodetic Network, The Ghana Surveyor,
Journal of the Ghana Institution of Surveyors.
2. Ayer, J. and C. Fosu,2008, Map Coordinate Referencing and the Use of GPS
Datasets in Ghana, Journal of Science and Technology, Vol. 28, No. 1
3. D.M.A., Defense Mapping Agency (1990). Datums, Ellipsoids, Grids and
Grid Reference Systems, DMA Technical Manual 83538.1, Fairfax, Virginia,
USA.
4. Defence Mapping Agency. (1987) Department of defence World Geodetic
System 1984: its definition and relationship with local geodetic systems
12
(second edition). Technical Report no. 8350.2, Defence Mapping Agency,
Washington.
5. EPSG 2005. Coordinate Conversion and Transformation including Formulas.
European Survey and Petroleum Group Guidance Note number 7 part 2.
6. Gold Coast Survey Department Records 1936. Report on the
Chains of
Triangulation surveyed in the Colony
7. Primary and secondary adjustment of TM coordinates for Ghana Aug. 1978.
8. Report on Survey and adjustment of a Network of Precise Traverses measured
between 1924 and 1930 in the Colony.
9. Report on the Secondary Triangulation Surveyed in the Colony prior to 1931.
10. Snyder, J. P. 1987. Map Projections – A Working Manual. USGS Prof. Paper
1395
11. Snyder, J.P., Map Projections used by the U.S. Geological Survey, Geological
Survey Bulletin; 1532, U.S. Government Printing Office, Washington D.C.,
1982.
12. Steed, J., (1990). A practical approach to transformation between commonly
used reference systems. Aust. Surveyor, 35(3), 248-264.
13. Survey Records, Aug. 1978 Primary and secondary adjustment of TM
coordinates for Ghana.
14. Tables for Use in the Survey Department.
15. The Technical Instructions for the Cadastral Branch of the Gold Coast Survey,
1936.
16. Toms R.M, Efficient Transformations from Geodetic to UTM Coordinate
Systems, Fifteenth Workshop on Standards for Interoperability of Distributed
Simulations, Orlando, FL, September 16-20, 1996, August 7, 1996
13