ADVANCED DIGITAL EARTH
A. Earth Geometry and Geodesy Outline
• Representa on of Earth (La tude and Longitude) • Determining La tude • Time Zones and the Longitude Problem • Maps • Map Projec ons • Coordinate Systems • Map Scale • Georeferencing • Earth Shape: Sphere and Ellipsoid, Datum and Geoid
Representa ons of Earth
What if you wanted to travel from San Diego to London, England? What path would you take? Which direc on yields the shortest distance? The shortest distance between any two points is a straight line. But how do you draw a straight line on a globe? For that ma er, how can you navigate upon a sphere, at all? Certainly, traveling to London, England is not as simple as saying, “Take a right turn at the Atlan c Ocean.” The way that we have overcome this apparent problem is with the geographic grid. The geographic grid divides the world into both northern/ southern components (la tude) and eastern/western components (longitude). The range of la tude is from 0° (the Equator) to 90° North and South La tude. The range of longitude is from 0° (the Prime Meridian) to 180° (the Interna onal Date Line). Lines of la tude are known as parallels. Lines of longitude are known as meridians. Every place on Earth has a geographic grid coordinate. For example, San Diego, California is located at 32° North La tude, 117° West Longitude. Diagram of La tude
90 degrees North Lat. = North Pole
90 degrees
0 degrees Lat. = Equator
45 degrees
45th Parallel 45 degrees South La tude North Pole 180°Longitude = Interna onal Date Line 90° West Longitude
180°
90°
90° East Longitude
0° Longitude = Prime Meridian View from above the North Pole (NP) 180 Long 135 West Long 135 East Long
Diagram of Longitude 90 West Long NP 90 East Long
45 West Long 45 East Long 0 Long Please click on the following link to watch a video lecture that describes and explains La tude and Longitude: h ps://www.youtube.com/watch?v=9vfQM_M1Pec
Determining La tude The use of the North Celes al Pole star for naviga on purposes has been used for thousands of years. In par cular, the posi on of the North Star (Polaris) will tell you your la tude (whereas a good clock will help you to determine your longitude).
Finding Polaris
Little Dipper
Big Dipper Time Zones There are 24 me zones in the world; each me zone is separated by 15° (the world turns 360° per 24 hours = 15° rota on per hour) (see Figure 3 on next slide). If you are west of the Prime Meridian (west of 0° Longitude), you are behind GMT (Greenwich Mean Time). If you travel east of the Prime Meridian, you are ahead GMT (remember, the Sun rises in the east and sets in the west due to Earth’s easterly – or, counterclockwise -- rota on). In the example below, the me for the person at the Prime Meridian is solar noon. For the person standing in the east, the me is a ernoon (no ce the Sun is in the western, se ng sky). For the person standing in the west, the me is before noon (no ce the Sun is in the eastern, rising sky).
Direc on of Sun (rises in the east and sets in the west)
West Prime Meridian East
If you cross the IDL (Interna onal Date Line = 180°) from the Western Hemisphere into the Eastern Hemisphere, you lose a day (set your calendar ahead a day). If you cross the IDL from the Eastern Hemisphere into the Western Hemisphere, you gain a day (set your calendar back a day). Figure 3 World Poli cal Time Zones (adopted from Encarta Encyclopedia, 2000)
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12
As you can see from the above me zone map, the 15° rule is not always followed! Why not? Please click on the following link to watch a video lecture that describes and explains Time Zones: h ps://www.youtube.com/watch?v=OXuc84bP90k
Early Voyage Problem: Longitude! How do you find East/West? Key = Time
Just as you use longitudinal distance to calculate me, you can also use me difference to calculate your longitude! The rela onship between me and longitude (1 hour = 15°) is the key. If you know the difference in me between two places, and you know the longitude of one place, you can calculate the longitude of the other. For example, imagine you are a sailor and you need to know where you are (you don’t have a GPS unit!). You find your la tude by no cing the posi on of the North Star above your horizon. To determine longitude, however, you must use a clock. Assume you leave San Diego. You set your clock to San Diego me and set sail. Eventually, you want to determine your loca on. On your boat, you no ce when the shadow is shortest on the mast (the Sun is at its highest point in the sky). It is solar noon. You check your clock. It reads 8 am. If it is noon for you and 8 am in San Diego, you are ahead of San Diego me (you are east of San Diego) by 4 hours. 4 x 15 = 60°. Therefore, you are located 60° east of San Diego, which means you are at 120° - 60° = 60° West Longitude: In the Early days of sea voyages, determining longitude was a difficult task. Why? How do you read a pendulum clock at sea! A contest was created by England with the following results. We can thank John Harrison for our wristwatch.
Chronometer #1: Finished in 1735 AD
Chronometer #4: Finished in 1760 AD Please click on the following link to watch a video lecture that describes and explains The Longitude Problem: h ps://www.youtube.com/watch?v=s4hsI3Cz2tg
Maps
The map is the primary tool of the geographer….It can * fit in a pocket * give loca on * provide area data * be thema c (such as weather maps, veg maps, soil maps…)
Mapping: Projec ons and Coordinates • There are many reasons for wan ng to project the Earth’s surface onto a flat plane, rather than deal with the curved surface. – The paper used to output GIS maps is flat. – Flat maps are scanned and digi zed to create GIS databases. – The Earth has to be projected to see all of it at once. – It’s much easier to measure distance on a plane. Projec ons (cont.) • A transforma on of the spherical or ellipsoidal earth onto a flat map is called a map projec on. • The map projec on can be onto a flat surface or a surface that can be made flat by cu ng, such as a cylinder or a cone. • If the globe, a er scaling, cuts the surface, the projec on is called secant. Lines where the cuts take place or where the surface touches the globe have no projec on distor on.
From: h p://na onalatlas.gov/ar cles/mapping/a_projec ons.html Projec ons (cont.) • Projec ons can be based on axes parallel to the earth's rota on axis (equatorial), at 90 degrees to it (transverse), or at any other angle (oblique). • A projec on that preserves the shape of features across the map is called conformal. • A projec on that preserves the area of a feature across the map is called equal area or equivalent. • No flat map can be both equivalent and conformal. Most fall between the two as compromises. • To compare or edge-match maps in a GIS, both maps MUST be in the same projec on. Distor ons
• Any projec on must distort the Earth in some way • Two types of projec ons are important in GIS – Conformal property: Shapes of small features are preserved: anywhere on the projec on the distor on is the same in all direc ons. – Equal area property: Shapes are distorted, but features have the correct area. – Both types of projec ons will generally distort distances. Map Projec ons
Fla en half of a rubber ball? No. Instead, features are projected onto one of three “developable” surfaces.
Planar: a map projec on Cylindrical: a map projec on where the Conic: a map projec on where the earth’s surface is resul ng from the earth’s surface is projected onto a tangent projected onto a tangent or secant cone, which is conceptual projec on of the or secant cylinder, which is then cut then cut from apex to base and laid flat earth onto a tangent or lengthwise and laid flat secant plane
h p://www.na onalatlas.gov/ar cles/mapping/a_projec ons.html#two
21 Conformal Projec on
• Cylindrical projec on
• Parallels and meridians at right angles. • Angles and shapes of small objects preserved (at every point, east–west scale same as north–south scale). • The size/shape/area of large Example: Mercator projec on (1569) objects distorted. used for nau cal purposes (constant • Seldom used for world maps. courses are straight lines).
22 Equivalent Projec on
• Conic projec on
• Preserves accurate area. • Scale and shape are not preserved.
Example: Albers Equal Area standard projec on for US Geological Survey, US Census Bureau
23 Compromise Projec ons
• Neither equivalent nor conformal. • Meridians curve gently, avoiding extremes. • Doesn’t preserve proper es, but looks right.
Example: Robinson projec on (1961) • Good compromise projec on for viewing en re world.
• Used by Rand McNally and the Na onal Geographic Society.
24 Cylindrical Projec ons
• Conceptualized as the result of wrapping a cylinder of paper around the Earth. • The Mercator projec on is the best- known cylindrical projec on. – The cylinder is wrapped around the Equator. – The projec on is conformal. • At any point scale is the same in both direc ons. • Shape of small features is preserved. • Features in high la tudes are significantly enlarged. Conic Projec ons
• Conceptualized as the result of wrapping a cone of paper around the Earth. – Standard Parallels occur where the cone intersects the Earth. – The Lambert Conformal Conic projec on is commonly used to map North America. – On this projec on lines of la tude appear as arcs of circles, and lines of longitude are straight lines radia ng from the North Pole. The Universal Transverse Mercator (UTM) Projec on
• A type of cylindrical projec on. • Implemented as an interna onally standard coordinate system. – Ini ally devised as a military standard. • Uses a system of 60 zones. – Maximum distor on is 0.04%. • Transverse Mercator because the cylinder is wrapped around the Poles, not the Equator. UTM: Zones are each six degrees of longitude, numbered as shown at the top, from W to E Implica ons of the Zone System
• Each zone defines a different projec on. • Two maps of adjacent zones will not fit along their common border. • Jurisdic ons that span two zones must make special arrangements. – Use only one of the two projec ons, and accept the greater- than-normal distor ons in the other zone. – Use a third projec on spanning the jurisdic on. – E.g. Italy spans UTM zones 32 and 33. When the Correct Projec on is Important
• Small-scale maps – Comparing shapes, areas, distances, or direc ons of map features. – Natural appearance desired.
New York New York
Los Angeles Los Angeles Los Angeles
Projec on: Mercator Projec on: Albers Equal Area Distance: 3,124.67 miles Distance: 2,455.03 miles
Actual distance: 2,451 miles 30 When Projec on is not Important • Many business, policy, and management applica ons. • On large-scale maps. – Error is negligible.
31 Coordinate Systems
• A coordinate system is a standardized method for assigning codes to loca ons so that loca ons can be found using the codes alone. • Standardized coordinate systems use absolute loca ons. • A map captured in the units of the paper sheet on which it is printed is based on rela ve loca ons or map millimeters. • In a coordinate system, the x-direc on value is the eas ng and the y-direc on value is the northing. Most systems make both values posi ve. Coordinate Systems for the US • Some standard coordinate systems used in the United States are – geographic coordinates. – universal transverse Mercator system. – military grid. – state plane. • To compare or edge-match maps in a GIS, both maps MUST be in the same coordinate system. UTM Coordinates
• In the N Hemisphere define the Equator as 0m N. • The central meridian of the zone is given a false eas ng of 500,000 mE. • Eas ngs and northings are both in meters allowing easy es ma on of distance on the projec on. • A UTM georeference consists of a zone number, a hemisphere, a six-digit eas ng and a seven-digit northing. – E.g., 14, N, 468324E, 5362789N UTM zones in the lower 48 Military Grid Coordinates State Plane Coordinates
• Defined in the US by each state. – Some states use mul ple zones. – Several different types of projec ons are used by the system. • Provides less distor on than UTM. – Preferred for applica ons needing very high accuracy, such as surveying. Map Scale • Map scale is based on the representa ve frac on, the ra o of a distance on the map to the same distance on the ground. • Most maps in GIS fall between 1:1 million and 1:1000. • A GIS is scaleless because maps can be enlarged and reduced and plo ed at many scales other than that of the original data. • To compare or edge-match maps in a GIS, both maps MUST be at the same scale and have the same extent. • The metric system is far easier to use for GIS work. Scale of a baseball earth
• Baseball circumference = 226 mm. • Earth circumference approximately 40 million meters. • Scale is : 1:177 million. Length of the Equator at Scale
Rep. Frac on Map Distance Ground Distance 1:400 Million 0.10002 cm 0.328 feet 1:40 Million 1.0002 cm 3.28 feet 1:1 Million 40.008 cm 131 feet 1:100,000 400.078 cm 1,312 feet 1:24,000 1,666.99 cm 5,496 feet (1.036 miles) 1:1,000 40,007.8 cm 131,259 feet (24.86 miles) Georeferencing
• Is essen al in GIS, since all informa on must be linked to the Earth’s surface. • The method of georeferencing must be: – Unique, linking informa on to exactly one loca on. – Shared, so different users understand the meaning of a georeference. – Persistent through me, so today’s georeferences are s ll meaningful tomorrow. Uniqueness
• A georeference may be unique only within a defined domain, not globally. – There are many instances of Springfield in the U.S., but only one in any state. – The meaning of a reference to London may depend on context, since there are smaller London’s in several parts of the world. Georeferences as Measurements
• Some georeferences are metric. – They define loca on using measures of distance from fixed places. • E.g., distance from the Equator or from the Greenwich Meridian. • Others are based on ordering. – E.g. street addresses in most parts of the world order houses along streets. • Others are only nominal. – Placenames do not involve ordering or measuring. The Spheroid and Ellipsoid Because of the combined effects of gravita on and rota on, the Earth’s shape is roughly that of a sphere slightly fla ened in the direc on of its axis. For that reason, in cartography the Earth is approximated by an oblate spheroid instead of a sphere. The current World Geode c System (WGS) model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles.
The word spheroid originally meant an approximately spherical body and that is how it is used in some older papers on geodesy. In order to avoid confusion, spheroid should be defined as an ellipsoid of revolu on (if that is the intended meaning). • The sphere is about 40 million meters in circumference. • An ellipsoid is an ellipse rotated in three dimensions about its shorter axis. • The earth's ellipsoid is only 1/297 off from a sphere. • Many ellipsoids have been measured, and maps based on each. Examples are WGS84 and GRS80. The History of Ellipsoids
• Because the Earth is not shaped precisely as an ellipsoid, ini ally each country felt free to adopt its own as the most accurate approxima on to its own part of the Earth. • Today an interna onal standard has been adopted known as WGS 84. – Its US implementa on is the North American Datum of 1983 (NAD 83). – Many US maps and data sets s ll use the North American Datum of 1927 (NAD 27). – Differences can be as much as 200 m. La tude and the Ellipsoid
• La tude is the angle between a perpendicular to the surface and the plane of the Equator. • WGS 84 – Radius of the Earth at the Equator 6378.137 km. – Fla ening 1 part in 298.257. The Datum
• An ellipsoid gives the base eleva on for mapping, called a datum. • Examples are NAD27 and NAD83. • The geoid is a figure that adjusts the best ellipsoid and the varia on of gravity locally. • It is the most accurate method, and is used more in geodesy than GIS and cartography. GeoidEarth Models and The geoid is the shape that the surface of the oceans would take under the influence of Earth's gravita on and rota on alone, in the absence of other influences such as winds and des.
The surface of the geoid is higher than the reference ellipsoid wherever there is a posi ve gravity anomaly (mass excess) and lower than the reference ellipsoid wherever there is a nega ve gravity anomaly (mass deficit). The differences in gravity arise from the uneven distribu on of mass in the Earth.