1. Representing the Earth
RG 620
Week 4
My 03, 2013
Institute of Space Technology, Karachi

2. Geodesy Science of measuring the shape of Earth
Science of measuring the size and shape of Earth and its gravity field

3. Modeling the Earth The best model of the Earth is 3D globe
For measuring the Earth, Globes have certain drawbacks
Best model is 3D solid in the same shape as the earth.
Drawbacks:
Globes are large and cumbersome. Difficult to carry around.
Even the largest globe has a very small scale and shows relatively little detail.
Costly to reproduce and update.
Standard measurement equipment (rulers, protractors, planimeters, dot grids, etc.) cannot be used to measure distance, angle, area, or shape on a sphere, as these tools have been constructed for use in planar models.
The latitude-longitude spherical coordinate system can only be used to measure angles, not distances or areas.

4. Modeling the Earth
At any point on Earth there are three important surfaces, the Ellipsoid, the Geoid, and the Earth surface

6. Geoid Definition: “Three-dimensional surface along which
the pull of gravity is constant” OR
“A continuous surface which is perpendicular
at every point to the direction of gravity”
The true shape of Earth is Geoid
The geoidal surface may be thought of as an imaginary sea that covers the entire Earth and is not affected by wind, waves, the Moon, or forces other than Earth’s gravity.
The surface of the geoid is in this way related to mean sea level.
Equipotential surface

7. Geoid
True shape of the Earth varies slightly from the mathematically smooth surface of an ellipsoid
Differences in the density of the Earth cause variation in the strength of the gravitational pull, in turn causing regions to dip or bulge above or below a reference ellipsoid
This undulating shape is called a Geoid
Traditional surveying via leveling measures elevation relative to geoid

8. Ellipsoid Mathematical surface obtained by revolving an
ellipse around earth’s polar axis
Sphere like object where the lengths of all three axes are different

9. Ellipsoid Model of the Earth’s Shape
semi-major axis, radius r1 in the equatorial direction, and the semi-minor axis, the radius r2 in the polar direction
The equatorial radius is always greater than the polar radius for the Earth ellipsoid.
Earth is flattened about 13 miles at poles.
The flattening is the difference in length between the two axes expressed as a fraction or a decimal.

10. Ellipsoid
Different ellipsoid were adopted in various parts of the world
Why difference in Ellipsoidal Estimates?
Because there were different sets of measurements used in each region or continent
These measurements often could not be tied together or combined in a unified analysis
Due to differences in survey methods and data analyses.
A spheroid that best fits one region is not necessarily the same one that fits another region.
Historically, geodetic surveys were isolated by large water bodies. And the scarcity of survey points for many areas were barriers to the development of global ellipsoids.
Methods for computing positions, removing errors, or adjusting point locations were not the same worldwide.
It took time for the best methods to be developed, widely recognized, and adopted.

11. Local or Regional Ellipsoid
Origin, R1, and R2of ellipsoid specified such that separation between ellipsoid and Geoid is small
Examples: Clarke 1880

12. Global Ellipsoid
Global ellipsoid selected so that these have the best fit “globally”, to sets of measurements taken across the globe
Example:
Geodetic Reference System of 1980 (GRS 1980)
World Geodetic System 1984 (WGS84)
It took time for the best methods to be developed, widely recognized, and adopted.

13. Globally Applicable Ellipsoids
Extremely precise measurements across continents and oceans were possible using data derived from satellites, lasers, and broadcast timing signals
This has led to the calculation of globally applicable ellipsoids such as GRS80, or WGS84
Using the wrong ellipsoid can result in errors in geodetic coordinates of the order of hundreds of meters

14. Set of official Ellipsoids
Everest (Sir George) 1830
one of the earliest spheroids; India
r1=6,377,276m r2=6,356,075m f=1/300.8

15. Measuring Heights
Height is measured as a distance from the reference ellipsoid in a direction perpendicular to the ellipsoid
The largest geoidal height is less than the relative thickness of a coat of paint on a ball three meters in diameter.
Even this small geoidal variations in shape must be considered for accurate mapping.

16. Measuring Heights
Orthometric Height/Elevation: Vertical distance above a geoid
Ellipsoidal Height: Heights above the ellipsoid
Geoidal Height/Geoidal Separation: The difference between the ellipsoidal height and orthometric height at any location
Geoidal heights vary across the globe
The absolute value of the geoidal height is less than 100 meters at most of the Earth locations
The geoid is not a mathematically defined surface rather it is a measured and interpolated surface

17. Geoidal heights are positive for large areas near Iceland and the Philippines (A and B, respectively), while large negative values are found south of India (C).

18. Datum
A fixed 3D surface
Provides a frame of reference for measuring locations on the surface of the earth
It defines the origin and orientation of latitude and longitude lines
Examples:
North American Datum of 1983 (NAD 1983 or NAD83),
North American Datum of 1927 (NAD 1927 or NAD27),
World Geodetic System of 1984 (WGS 1984)
Datum represents a reference model of the Earth
An oblate spheroid, that is approximately the size and shape of the Earth.
For NAD27, the USGS decided that Clarke 1866 was a good approximation, and they fixed it at Meade's Ranch, Kansas.
The ideal solution would be a spheroidal model that has both the correct equatorial and polar radii, and is then centered on the actual center of the Earth 18

19. Datum A spheroid model of the Earth is fixed to a base point
Example: For NAD27
Ellipsoid: Clarke 1866
Fixed at Meade's Ranch, Kansas 19

20. Datum Horizontal Datum Vertical Datum Specify the ellipsoid
Specify the coordinate locations of features on this ellipsoidal surface
Vertical Datum
Specify the Geoid –which set of measurements will you use, or which model
Horizontal datums are used for describing a point on the earth's surface, in latitude and longitude or another coordinate system.
Vertical datums measure elevations or depths

21. Datum Many datums have been developed to describe the ellipsoid
Differences between the datums reflect differences in the
control points,
survey methods, and mathematical models and
assumptions used in the datum adjustment
There are hundreds of locally-developed reference datums around the world, usually referenced to some convenient local reference point.
When enough new survey points have been collected, a new datum is estimated (new coordinate locations for all datum points).

22. Local Datum
A local datum aligns its spheroid to closely fit the earth's surface in a particular area
A point on the surface of the spheroid known as the origin point of the datum is matched to a particular position on the surface of the earth
The coordinates of the origin point are fixed, and all other points are calculated from it
The center of the spheroid of a local datum is offset from the earth's center
Not suitable for use outside the area for which it was designed
Examples:
NAD 1927 (designed to fit North America reasonably well)
European Datum of 1950 (ED 1950) (created for use in Europe)

23. Commonly Used Datums North American Datum 1927 (NAD27)
Uses Clarke 1866 spheroid
Fixed at Meade's Ranch, Kansas
Yields adjusted latitudes and longitudes for approximately 26,000 survey stations in the United States and Canada
North American Datum 1983 (NAD83)
Include the large number of geodetic survey points (250,000 stations)
GRS80 ellipsoid was used as reference
NAD83(1986) uses an Earth-centered reference
World Geodetic System of 1984 (WGS84):
Earth-centered datum
Essentially identical to the North American Datum of 1983 (NAD83)
Uses WGS84 ellipsoid
Local ellipsoids may still fit better than WGS84
NAD27 fixed latitude and longitude of a survey station in Kansas.
(1986) placed after and NAD83 designator to indicate the year, or version, of the datum adjustment.
WGS84 developed by the U.S. Department of Defense (DOD)
There are several versions of both NAD83 and of WGS84.
Updated datum known as NAD83(HARN) also known as NAD83(HPGN).
This growing network of satellite observation stations is the basis for newer datums, including NAD83(CORS93), NAD83(CORS94), and NAD83(CORS96). 23

24. Other Datums Bermuda 1957 South American Datum 1969
International Terrestrial Reference Frames, (ITRF)
Source: 24

25. Pre-Satellite Datum Post-Satellite Datum
Large errors (10s to 100s of meters),
Local to continental
Examples: Clarke, Bessel, NAD27, NAD83(1986)
Post-Satellite Datum
Small relative errors (cm to 1 m)
Global
Examples: NAD83(HARN), NAD83(CORS96), WGS84(1132), ITRF99

26. Changing the Datum
The lat/long value of a place on the Earth's surface depends upon the datum
Datum transformation is done to correctly convert data among datums
Changing the datum will change the latitude and longitude of a point on the surface of Earth
Example:
Point: middle of the intersection of Baseline Road and County Line Road near Boulder, Colorado
Location: Latitude and Longitude
NAD27: 40o N, 105o W
NAD83: 39o 59’ 59.97” N, 105o 0’ 01.93” W
Difference between two is 4ft south and 50 ft west
That means latitude/longitude alone does not uniquely describe a location on the surface of the Earth.
Every geopolitical area on Earth has one, or a very few, preferred datums.
GPS receivers calculate their locations referenced to the WGS84 datum

27. Datum Shift

29. Positions on Globe Global Coordinate System

30. Positions on Globe: Lines of Reference
graticules
Figure: 1
1:parallels of latitude (Y)
2:meridians of longitude (X)
3:graticular network
Figure: 3
Figure: 2 30

31. Positions on Globe
Measured by Geographical Coordinates (angles) rather than Cartesian Coordinates
Locations are represented by Latitudes and Longitudes
Latitudes (Y) and Longitudes (X) are angles
Equator  is the reference plane used to define latitude
Prime Meridian is used to define longitude

32. Geographic Coordinate System (GCS)
GCS uses a three-dimensional spherical surface to define locations on the earth.
Latitude and Longitude are angles denoted by ( °,   ',   "  )

33. Latitude and Longitude
The Latitude is measured as the number of degrees from the Equator
The Longitude is measured as the number of degrees from the Prime Meridian
The lines of constant latitude and longitude form a pattern called the Graticule

34. Example: Measuring Lat and Long
Example: In figure
60° E (longitude), 55° N (latitude)
In the image above, the specific point on the surface of the earth is specified by the coordinate (60 °. E longitude, 55 °. N latitude).

35. Latitude “Latitude is the angular distance of any point
on Earth measured north or south of the
Equator in degrees, minutes and seconds”
At poles (North and South Poles) latitudes are 90o North and 90o South
At equator latitude is 0°
The equator divides the globe into Northern and Southern Hemispheres
Each degree of latitude is approximately 69 miles (111 km) (variation because Earth is not a perfect sphere)
90° N
90° S 0°

36. Lines of Equal Latitudes
Lines of constant latitude are called parallels of latitude (horizontal lines)
Parallel lines at an equal distance
On Globe lines of latitude are circles of different radii
Equator is the longest circle with zero latitude also called ‘Great Circle’ (24, miles)
Other lines of latitudes are called ‘Small Circles’
At poles the circles shrink to a point
Circle of Equator is divided into 360 degrees
In figure, lines of Latitude or Parallels

37. Some Important Small Circles
Tropic of Cancer
At 23.5°N of Equator and runs through Mexico, Egypt, Saudi Arabia, India and southern China.
Tropic of Capricorn
At 23.5°S of Equator and runs through Chile, Southern Brazil, South Africa and Australia.
Arctic and Antarctic Circles
At 66° 33′ 39″ N and 66° 33′ 39″ S respectively
Tropical Zone is between tropic of cancer and tropic of Capricorn. The tropic region does not experience seasons and is normally warm and wet throughout the year. Sun directly overhead at least once in a solar year in this region. When sun is directly above Tropic of Cancer summer and winter begins in northern and southern hemisphere respectively. Autumn/summer when sun is at equator and winter and summer in north and south respectively when sun is directly above tropic of Capricorn.

38. Map of the World
Tropical Zone

39. Longitude “Longitude is the angular distance of any point
on Earth measured east or west of the prime
meridian in degrees, minutes and seconds”
Measured from 0° to 180° east and 180° west (or -180°)
The meridian at 0° is called Prime Meridian located at Greenwich, UK
Both 180-degree longitudes (east and west) share the same line, in the middle of the Pacific Ocean where they form the International Date Line
1 degree of Longitude=
69.17 mi at Equator
48.99 mi at 45N/S
mi at 90N/S
W 180°
180° E
Prime
Meridian
No natural reference for longitude
Prime Meridian: This was set by treaty in Before that each major nation had its own zero of longitude
Also values may go from 0 to 360 degrees
Example: 260° (east longitude) is same as -100° or 100° west

40. Lines of Equal Longitude
Lines of Longitude (vertical lines/meridians)
They are also called Meridians
Meridians converge at the poles and are widest at the equator about 69 miles or 111 km apart
On Globe lines of longitude are circles of constant radius which extend from pole to pole
Prime meridian is zero: Greenwich, U.K. (near London)
International Date Line is 180° E&W
Lines of longitudes are on great circle.
In figure, lines of longitude or meridian

41. Prime Meridian
Royal Astronomical Observatory in Greenwich, England E W S

42. Latitude/Longitude Formats
Lat/long coordinates can be specified in different formats:
DD.MM.SSXX (degree, minute, decimal second)
DD.MMXX (degree, decimal minute)
DDXX (decimal degree)
How to convert degree, minute, decimal second format
into decimal degree?
Decimal degree = (Seconds/3600) + (Minutes/60) + Degrees
In class exercise: DD conversion of24° 48' 58” N 66° 59' E
1. degrees-minutes-seconds (DMS),
3. decimal degrees (DD)

43. Quiz 3
DD conversion of 38° 20' 20” N 70° 56' 04” E is _______________ N and ______________ E
The DMS version of ° is _________________ 43

44. References Bolstad Text Book

45. Solution- Quiz 2 (a)
DD conversion of 38° 20' 20” N 70° 56' 04” E is ° N and ° E
The DMS version of ° is 5 ° 14’ 4.416’’ 45