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

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

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.

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

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

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

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

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.

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.

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

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.

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

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

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.

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

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).

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

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

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

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).

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)

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

Other Datums Bermuda 1957 South American Datum 1969 International Terrestrial Reference Frames, (ITRF) Source: http://maic.jmu.edu/sic/standards/datum.htm 24

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

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

Datum Shift

Positions on Globe Global Coordinate System

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

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

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

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

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).

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°

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’ (24,901.55 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

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.

Map of the World Tropical Zone

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 0.0 mi at 90N/S W 180° 180° E Prime Meridian No natural reference for longitude Prime Meridian: This was set by treaty in 1878. 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

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

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

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)

Quiz 3 DD conversion of 38° 20' 20” N 70° 56' 04” E is _______________ N and ______________ E The DMS version of 5.23456° is _________________ 43

References Bolstad Text Book http://courses.washington.edu/gis250/lessons/projection/

Solution- Quiz 2 (a) DD conversion of 38° 20' 20” N 70° 56' 04” E is 38.33889 ° N and 70.93444 ° E The DMS version of 5.23456° is 5 ° 14’ 4.416’’ 45