ADVANCED DIGITAL EARTH

A. Earth Geometry and Outline

• Representaon of Earth (Latude and ) • Determining Latude • Time Zones and the Longitude Problem • Maps • Map Projecons • Coordinate Systems • Map Scale • Georeferencing • Earth Shape: and Ellipsoid, Datum and Geoid

Representaons of Earth

What if you wanted to travel from San Diego to London, England? What path would you take? Which direcon 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 maer, 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 Atlanc 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 (latude) and eastern/western components (longitude). The range of latude is from 0° (the ) to 90° North and South Latude. The range of longitude is from 0° (the ) to 180° (the Internaonal Date Line). Lines of latude 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 Latude, 117° West Longitude. Diagram of Latude

90 degrees North Lat. =

90 degrees

0 degrees Lat. = Equator

45 degrees

45th Parallel 45 degrees South Latude North Pole 180°Longitude = Internaonal 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 Latude and Longitude: hps://www.youtube.com/watch?v=9vfQM_M1Pec

Determining Latude The use of the North Celesal Pole star for navigaon purposes has been used for thousands of years. In parcular, the posion of the North Star (Polaris) will tell you your latude (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° rotaon 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 -- rotaon). In the example below, the me for the person at the Prime Meridian is solar . For the person standing in the east, the me is aernoon (noce the Sun is in the western, seng sky). For the person standing in the west, the me is before noon (noce the Sun is in the eastern, rising sky).

Direcon of Sun (rises in the east and sets in the west)

West Prime Meridian East

If you cross the IDL (Internaonal Date Line = 180°) from the into the , you lose a (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 Polical 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: hps://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 relaonship 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 latude by nocing the posion 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 locaon. On your boat, you noce 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: hps://www.youtube.com/watch?v=s4hsI3Cz2tg

Maps

The map is the primary tool of the geographer….It can * fit in a pocket * give locaon * provide area data * be themac (such as weather maps, veg maps, soil maps…)

Mapping: Projecons and Coordinates • There are many reasons for wanng 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 digized 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. Projecons (cont.) • A transformaon of the spherical or ellipsoidal earth onto a flat map is called a map projecon. • The map projecon can be onto a flat surface or a surface that can be made flat by cung, such as a cylinder or a cone. • If the globe, aer scaling, cuts the surface, the projecon is called secant. Lines where the cuts take place or where the surface touches the globe have no projecon distoron.

From: hp://naonalatlas.gov/arcles/mapping/a_projecons.html Projecons (cont.) • Projecons can be based on axes parallel to the earth's rotaon axis (equatorial), at 90 degrees to it (transverse), or at any other angle (oblique). • A projecon that preserves the shape of features across the map is called conformal. • A projecon 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 projecon. Distorons

• Any projecon must distort the Earth in some way • Two types of projecons are important in GIS – Conformal property: Shapes of small features are preserved: anywhere on the projecon the distoron is the same in all direcons. – Equal area property: Shapes are distorted, but features have the correct area. – Both types of projecons will generally distort distances. Map Projecons

Flaen half of a rubber ball? No. Instead, features are projected onto one of three “developable” surfaces.

Planar: a map projecon Cylindrical: a map projecon where the Conic: a map projecon where the earth’s surface is resulng from the earth’s surface is projected onto a tangent projected onto a tangent or secant cone, which is conceptual projecon 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

hp://www.naonalatlas.gov/arcles/mapping/a_projecons.html#two

21 Conformal Projecon

• Cylindrical projecon

• 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 projecon (1569) objects distorted. used for naucal purposes (constant • Seldom used for world maps. courses are straight lines).

22 Equivalent Projecon

• Conic projecon

• Preserves accurate area. • Scale and shape are not preserved.

Example: Albers Equal Area standard projecon for US Geological Survey, US Census Bureau

23 Compromise Projecons

• Neither equivalent nor conformal. • Meridians curve gently, avoiding extremes. • Doesn’t preserve properes, but looks right.

Example: Robinson projecon (1961) • Good compromise projecon for viewing enre world.

• Used by Rand McNally and the Naonal Geographic Society.

24 Cylindrical Projecons

• Conceptualized as the result of wrapping a cylinder of paper around the Earth. • The Mercator projecon is the best- known cylindrical projecon. – The cylinder is wrapped around the Equator. – The projecon is conformal. • At any point scale is the same in both direcons. • Shape of small features is preserved. • Features in high latudes are significantly enlarged. Conic Projecons

• 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 projecon is commonly used to map North America. – On this projecon lines of latude appear as arcs of circles, and lines of longitude are straight lines radiang from the North Pole. The Universal Transverse Mercator (UTM) Projecon

• A type of cylindrical projecon. • Implemented as an internaonally standard coordinate system. – Inially devised as a military standard. • Uses a system of 60 zones. – Maximum distoron 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 Implicaons of the Zone System

• Each zone defines a different projecon. • Two maps of adjacent zones will not fit along their common border. • Jurisdicons that span two zones must make special arrangements. – Use only one of the two projecons, and accept the greater- than-normal distorons in the other zone. – Use a third projecon spanning the jurisdicon. – E.g. Italy spans UTM zones 32 and 33. When the Correct Projecon is Important

• Small-scale maps – Comparing shapes, areas, distances, or direcons of map features. – Natural appearance desired.

New York New York

Los Angeles Los Angeles Los Angeles

Projecon: Mercator Projecon: Albers Equal Area Distance: 3,124.67 Distance: 2,455.03 miles

Actual distance: 2,451 miles 30 When Projecon is not Important • Many business, policy, and management applicaons. • On large-scale maps. – Error is negligible.

31 Coordinate Systems

• A coordinate system is a standardized method for assigning codes to locaons so that locaons can be found using the codes alone. • Standardized coordinate systems use absolute locaons. • A map captured in the units of the paper sheet on which it is printed is based on relave locaons or map millimeters. • In a coordinate system, the x-direcon value is the easng and the y-direcon value is the northing. Most systems make both values posive. 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 easng of 500,000 mE. • Easngs and northings are both in meters allowing easy esmaon of distance on the projecon. • A UTM georeference consists of a zone number, a hemisphere, a six-digit easng 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 mulple zones. – Several different types of projecons are used by the system. • Provides less distoron than UTM. – Preferred for applicaons needing very high accuracy, such as surveying. Map Scale • Map scale is based on the representave fracon, the rao 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 ploed 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. Fracon 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 essenal in GIS, since all informaon must be linked to the Earth’s surface. • The method of georeferencing must be: – Unique, linking informaon to exactly one locaon. – Shared, so different users understand the meaning of a georeference. – Persistent through me, so today’s georeferences are sll 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 locaon 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 gravitaon and rotaon, the Earth’s shape is roughly that of a sphere slightly flaened in the direcon of its axis. For that reason, in cartography the Earth is approximated by an oblate spheroid instead of a sphere. The current World Geodec 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 revoluon (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, inially each country felt free to adopt its own as the most accurate approximaon to its own part of the Earth. • Today an internaonal standard has been adopted known as WGS 84. – Its US implementaon is the North American Datum of 1983 (NAD 83). – Many US maps and data sets sll use the North American Datum of 1927 (NAD 27). – Differences can be as much as 200 m. Latude and the Ellipsoid

• Latude 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. – Flaening 1 part in 298.257. The Datum

• An ellipsoid gives the base elevaon for mapping, called a datum. • Examples are NAD27 and NAD83. • The geoid is a figure that adjusts the best ellipsoid and the variaon 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 gravitaon and rotaon alone, in the absence of other influences such as winds and des.

The surface of the geoid is higher than the wherever there is a posive gravity anomaly (mass excess) and lower than the reference ellipsoid wherever there is a negave gravity anomaly (mass deficit). The differences in gravity arise from the uneven distribuon of mass in the Earth.