MAP PROJECTIONS MODULE 1: SPATIAL DATA MANAGEMENT Arjumand Zaidi Diploma Course on “Flood Forecasting and Flood Hazard Management” November, 2017 – May, 2018 Arjumand Zaidi Contact: arjumand.uspcasw@faculty.muet.edu.pk US – Pakistan Center for Advanced Studies in Water

Earth Model 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.

Converting the 3D Model to 2D Plane

Projecting Earth's Surface into a Plane

Map Projection Notice how the continents look stretched or squashed depending on the projection 5

Map Projection 6

Map Projection All flat maps are distorted to some degree Can not be accurately depicted on 2-D plane In the graphic above data near the poles is stretched . Different projections have different spatial relationships between regions There is always a distortion in 1 or 2 of its characteristics when projected to a 2-D map 7

Source: Text book

Map Projection Classification Based on Distortion Characteristics Based on Developable Surface 9

1. Base on Distortion Characteristics Map Projection Classification 1. Base on Distortion Characteristics The 4 basic characteristics of a map likely to be preserved / distorted depending upon the map projection are: Conformity Area Distance Direction In any projection at least 1 of the 4 characteristics can be preserved (but not all) Only on globe all the above properties are preserved Conformal: feature outlines look same on map as they look on earth. 10

Distortion Transfer of points from the curved ellipsoidal surface to a flat map surface introduces Distortion Distortions are unavoidable when making flat maps. Distortion may take different forms in different portions of the map. In one portion of the map features may be compressed and exhibit reduced areas or distances relative to the Earth’s surface measurements, while in another portion of the map areas or distances may be expanded. At few locations distortion may be zero. Distortion is usually less near the point or line of intersections, where the map surface intersects the imaginary Globe. Distortion usually increases with increasing distance from the intersection points or lines.

Map Projection Each type of projection has its advantages and disadvantages Choice of a projection depends on Application – for what purposes it will be used Scale of the map For example, a projection may have unacceptable distortions if used to map the entire country, but may be an excellent choice for a large-scale (detailed) map of a county. 12

Map Projection 2- based on developable surface A developable surface is a simple geometric form capable of being flattened without stretching Map projections use different models for converting the ellipsoid to a rectangular coordinate system Example: conic, cylindrical, plane  Each cause - distortion in scale and shape miscellaneous =which include special cases not falling into the other three categories. 13

Transverse Cylindrical Oblique Cylindrical Secant Cylindrical

Summary of Projection Properties 15

Where at Map there is Least Distortion? The lines where the cylinder is tangent or secant are the places with the least distortion.

Summary – Map Projection Portraying 3-D Earth surface on a 2-D surface (flat paper or computer screen) Map projection can not be done without distortion Some properties are distorted in order to preserve one property In a map one or more properties but NEVER ALL FOUR may be preserved Distortion is usually less at point/line of intersections of map surface and the ellipsoid Distortion usually increases with increase in distance from points/line of intersections

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

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 Local or Regional Ellipsoid: Origin, R1, and R2 of ellipsoid specified such that separation between ellipsoid and Geoid is small Example: Clarke 1880 Global Ellipsoid: Global ellipsoid selected so that these have the best fit “globally”, to sets of measurements taken across the globe Example: World Geodetic System 1984 (WGS 84) 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.

Set of official Ellipsoids ARC/INFO supports 26 different spheroids. Check: Everest (Sir George) 1830 one of the earliest spheroids; India a=6,377,276m b=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.

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). This figure depicts positive geoidal heights in lighter tones (geoid above the ellipsoid) and negative geoidal heights in darker tones.

Datum A fixed 3D surface It defines the origin and orientation of latitude and longitude lines 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 28

Commonly Used Datums North American Datum 1927 (NAD27) Uses Clarke 1866 spheroid Fixed at Meade's Ranch, Kansas World Geodetic System of 1984 (WGS84): Earth-centered datum Uses the position of GPS Uses WGS84 ellipsoid Google Earth uses the WGS84 geodetic datum 29

Local ellipsoids may still fit better than WGS84 NAD27 fixed latitude and longitude of a survey station in Kansas Source: http://maic.jmu.edu/sic/standards/datum.htm 30

Datum Shift

Does Datum change over time? WGS 84 geodetic latitude and longitude of any location on earth is constantly changing primarily due to the motion of the location’s tectonic plate Datum revision or Realization NAD 83 (2011), WGS 84 (G1762). While the definition of WGS84 ellipsoid hasn't changed, the location of its center has changed slightly because it is intended to coincide with the center of mass of the Earth. As more and more precise measurements become available the datum changes. To denote the different flavors of the same datum (called realizations), it is customary to add a year or some other number to distinguish between them. Periodic worldwide adjustments of WGS 84 geodetic latitude and longitudes due to tectonic plate motion are made by using International Terrestrial Reference Frame (ITFR) position updates.

Coordinates without a specified datum, are vague….. http://www.pdhcenter.com/courses/l117/l117content.pdf

Reference: David Conner National Geodetic Survey, 2003

Coordinate Systems After projection, it is necessary to set up a coordinate system on the map that will allow a point to be described in X-Y space (or northing and easting) To describe a location in a universally understandable manner a grid system is necessary For a useful grid it is necessary for it to define an origin and a uniform grid spacing There are several types of Coordinate System to represent the Earth’s surface Uniform Grid Spacing means the distance between grid lines should remain constant. X-Y: ordered pairs. Projection of points from the Earth’s surface onto a reference ellipsoid and finally onto flat maps. The problem is often illustrated by trying to flatten part of an orange peel. The orange peel stands in for the surface of the Earth. A small part, say a square a quarter of an inch on the side, can be pushed flat without much noticeable deformation. But when the portion gets larger problems appear. A portion of the Earth can be projected directly onto the flat plane. In fact this is the typical method for establishing an independent local coordinate system. 37

Coordinate Systems The method of projection, onto a simple flat plane, is based on the idea that a small section of the Earth, as with a small section of the orange, conforms so nearly to a plane that distortion on such a system is negligible Mapping a considerable portion of the Earth using a large number of small individual planes Offers the convenience of working in plane Cartesian coordinates and still keep distortion at manageable levels Note: when these planes are brought together they cannot be edge-matched accurately http://www.pdhcenter.com/courses/l117/l117content.pdf They cannot be joined properly along their borders.

Coordinate Systems Geographic UTM Some commonly used Coordinate Systems are: Geographic Latitude and Longitudes are used UTM Shape is preserved and precise measurements in meter Geographic: One of the most common CS in use. UNIVERSIAL TRANSVERSE MERCATOR GRID COORDINATES ARE A DIRECT MATHEMATICAL CONVERSION FROM LATITUDE AND LONGITUDE TO A CARTESIAN NORTHING AND EASTING (Y & X) COORDINATE SYSTEM 39

Geographic Coordinate System (WGS84 datum) Scale, distance, area, and shape are all distorted with the distortion increasing toward the poles.

Universal Transverse Mercator Global coordinate system Globe is divided into narrow longitudinal zones Best used for north-south oriented areas (little distortion in this direction) Relatively undistorted regions created by changing the orientation of the cylinder slightly These regions are called UTM zones Each zone is six degrees of longitude wide Total _?_____zones Error is less than 0.04% Total 60 zones. 1 km = 0.4 m error 41

Universal Transverse Mercator These zones are numbered from west to east Zone 1 begins at the International Date Line (1800 W), Zone 2 at 174°W and extends to 168°W Each Zone is further divided into Eastern and Western halves by drawing a center line called Central Meridian Zones are further split north and south of the equator Zone 60: starts at 174 degrees E and extends to International Date Line 10-19: UTM zones for the lower 48 contiguous states of the United States of America 42

Universal Transverse Mercator At equator a zone is about 40,000/60 = 667 Km wide Any point can be described by ‘Easting’ and ‘Northing’ values Easting is the distance to the "false easting", which is uniquely defined in each UTM zone The equator is used as the northing origin for all north zones (northing value of zero) South zones have a false northing value added to ensure all coordinates within a zone are positive For UTM south zones, the northing values at the equator are set to equal 10,000,000 meters 40,000 = circumference of earth 43

UTM – Easting and Northing 44

Source: Text book

Universal Transverse Mercator Important thing to remember Coordinate values are discontinuous across UTM zone boundaries, therefore, analyses are difficult across zonal boundaries

UTM Zones - Pakistan

UTM – Finding Grid Zone for any Latitude Take west longitude as (-) negative and east longitude as (+) positive Add 180 and divide by 6 Round off the resultant value to the next higher number For example, Denver, Colorado is near 105° W. Longitude, -105°. -105° + 180° = 75° 75°/ 6 = 12.50 Round up to 13 Hyderabad Sindh longitude is around 68.3694° E Zone ? Zone = 42 49

Quiz E = 500,000 + 100,000 = 600,000 N = 10,000,000 – 1,000,000 = 9,000,000 156 W 153 W 150 W

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