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Home , Ellipsoid, Geodesy, Geoid, Height, Surveying, Vertical position

, Coordinate Systems, Objectives and Projections • Introduce the main concepts of geodesy – coordinate systems, datums, , and • Describe map projections and coordinate systems we use in GIS •Two key questions –what is the of the , and where are things on the earth?

Our first references were local, relative, and Even today, addresses are qualitative. qualitative locations ASSIGNED by a government body, but how do we know the quantitative location?

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1 ..with a coordinate “address,” also assigned by governments ....but, how is Lat/Lon defined, and or , and what is this WGS84 business? Coordinates Differ Because:

1) Our best estimates of where things are have evolved over (pre‐ and post‐ (Geodesy)

2) We use different methods to “project” our data from the curved Earth to a flat map surface (Map Projections, next lecture)

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Describe, as quantitatively and Geodesy: Why all the confusion? All measurements are relative to some reference, and our precisely as possible, where A and B best estimates of the reference have changed (better are measurements, better models).

We started with, visualize the World, and still use 2‐ dimensional systems of coordinates. This won’t work for long , or the whole Earth.

The Earth’s shape is approximately spherical, but irregular. We didn’t know this at first, and we’ve since discovered relative positions are changing ().

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2 y

For most the reaction is to: Cartesian Coordinates (right angle) • Establish an origin (starting reference, or Two , X and Y, reference frame) or 90o x • Identify, through measurements, the E and N locations of important “things” using this reference frame Two‐dimensional system most often used with All positions are relative to the origin and • Projected coordinates ( and GIS) principal directions within this reference frame

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Geodesy - science of measuring • Establish a reference frame • Identify, through measurements, the locations of the size and shape of the Earth important “things” using this reference frame Datum - a reference surface e.g., a site datum - a reference height against which are measured (235,78) p3 p1

(Ferris State/ACSM)

(10,30) p2

h1 (0,0)11 11 h2 h3 Datum height

3 Except for the Earth, our standard locations Datum Example were first, local “fixed” objects (large rock outcrops, placed monuments,

then, estimated imaginary lines on the surface (Greenwich , the )

then, the center of of the Earth, and standard axes

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Datum Evolution Two Kinds of

Pre 1500’s –Plane Surveying 1500’s to > We can assume the underlying Earth (our 1980s reference surface) is flat, with topographic changes over this flat base post 1980s > Only valid over “short distances”

– Geodetic Surveying > Takes Earth’s curvature into account - our reference surface is curved > But the calculations become more complicated

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4 Plane Surveying is Based on Horizontal and Surveying Methods Evolved Through Time Vertical Measurement 1600s Horizontal measurements required “leveling”

– either raising the end of the measuring rods to maintain a level measurement, or 1700s

- measuring angles and calculating the horizontal distance 1800s

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1800’s - early 1900‘s, improved precision, but same methods, over much larger scope and distances

1600’s - scales and optics

19 20 Wikimedia commons USGS

5 21 22 21 UNH Dimond Library USGS

Earth Curvature Distortion

How Big a Difference?

Curved Distance, Straight line Angle, q Difference S Distance, C

1 second 30.8195 m 30.8195 m 0.0000 m

1 minute 1,894.5 m 1,894.5 0.0000 m

30 minutes = 0.5 degree 55.475 km 55.475 0.1760 m

1 degree 110.95 km 110.95 km 1.408 m

23 5 degrees 554.75 km 554.75 km 176.0 m

6 At some point, we have to move from a plane to a globe. What is our reference “surface” for a globe?

It depends When we thought the World was a , we invented a spherical system, with spherical references

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Measuring Earth Size, Shape, and Location Horizontal location on a sphere, using Spherical of Points is Complicated Coordinates The Earth Isn’t Round • Use angles of to define a directional vector The Continents are Moving • Use the length of a vector originating near the center to The Poles Wander define the location on the surface

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7 deduced the Earth’s Earth’s is Flattened - an Ellipsoid circumference to Two radii: within a few r1, along semi-major (through Equator) percent of our r2, along semi-minor (through poles) present estimate

30 Photobucket

Contradictory Measurements, Cleared Up by the French

No one could agree on the right size

- Differening estimates of r1 and r2?

- Different countries adopted different “standard” ellipsoids

8 Different Countries Adopted Different Ellipsoids as Datums Local or Regional Ellipsoid Why? Origin, R1, and R2 of ellipsoid specified such that separation •Surveys rarely spanned continents between ellipsoid and is small

•Best fit differed These ellipsoids have names, e.g., Clarke 1880, or Bessel •Nationalism

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Global Ellipsoid

Selected so that these have the best fit “globally”, to sets of measurements taken across the globe.

Generally have less appealing names, e.g. WGS84, or ITRF 2000

9 Different Ellipsoids Transforming Between in Different Countries Reference Surfaces There are locally “best fit” ellipsoids

geoid Ellipsoid B We can transform positions from one ellipsoid to another via mathematical operations

e.g., an origin shift, and mapping

Ellipsoid from one surface to the next A

Geocentric Ellipsoid and Coordinate System •3‐D Cartesian Z system •Origin (0,0,0) at the Earth center of mass •Best globally fit X , e.g., Y WGS84, used for global coordinate systems

10 The Earth’s True Shape a GEOID Why?

Law of :

Force = g x Massa x massb

r2

Larger mass, stronger pull, and the Earth’s mass isn’t uniform throughout, e.g., near mountains, subsurface ore, deep mantle density

variation42

11 The Geoid is a measured surface Definition of the Geoid A Geoid is the surface Gravimeter to a plumb line, and for which the pull of gravity is a given value

Note, there are at least three definitions of vertical •A plumb direction •The center of the celestial disk •Perpendicular to the adopted ellipsoid CIRES, U. of Colorado, and Gerd Steinle-Neumann, U. Bayerisches

The Geoid is a measured surface Separation between the Geoid and best-fitting global ellipsoid averages about 30 meters – this “undulation” is always below 100 meters Gravimeter on a Satellite

12 We have three surfaces to keep track of “Horizontal” Position at each point on Earth Our position on 1.the ellipsoid the surface of the 2.the geoid, and 3.the physical surface Earth is defined by a / pair on a surface specified ellipsoid ellipsoid (also known as a spheroid) geoid

Heights are usually specified Spheroidal Height

relative to the Geoid

Heights above the geoid are orthometric heights These are the heights usually reported on topographic or other maps Earth Surface

Heights above the ellipsoid are spheroidal heights Ellipsoid Sometimes used to define , but usually to specify geoid-ellipsoid separation Geoid

13 Defining a Datum Specifying a Horizontal Datum

Horizontal Datum •Measure positions Specify the ellipsoid (celestial observations, Specify the coordinate locations of features surveys, satellite on this ellipsoidal surface tracking) • Adjust measurements to account for geoid, Specify the ellipsoid determine position on Specify the Geoid – which set of adopted ellipsoid measurements will you use, or which model

Specifying a Horizontal Datum Defining the Horizontal Datum Horizontal datums are realized through large A horizontal datum is a reference survey networks ellipsoid, plus a precisely-measured set of points that establish locations on 1. Define the shape of the Earth (the ellipsoid) the ellipsoid. 2. Define the location of a set of known points – control points – for which the position on the ellipsoid is precisely known These points define the reference 3. This is the reference surface and network surface against which all horizontal against which all other points will be measured positions are measured

14 Datum, Survey Network Astronomical Observations Were Historically: • Network Accurate, but Time‐Consuming • Astronomical observation • Intermittent baselines • Multiple, redundant angle measurements

Why these technologies? • Easy to measure angles • Difficult to measure distance accurately • Time consuming to measure point position accurately

Horizontal Surface Measurements: Horizontal Surface Measurements Compensation Bars

15 Triangulation Triangulation

S1 B S3 c If we measure the initial baseline length A, a and measure the angles a, b, and c, we are S1 B S3 c With the length C known, angles then able to calculate the lengths B and C: a e e, f, and g may then be measured. sine (a) The law of sines may be used with by the law of sines, _ = the now known distance C to A B sine (b) C calculate lengths E and F. sine (b) sine (c) A F Successive datum points may then = A and = A C sine (a) sine (a) be established to extend the network using primarily b angle measurements. b S2 f E g S2 S4

Horizontal Survey Benchmarks Survey Network, 1900

(from Schwartz, 1989)

16 1920s of 1927, 26,000 measured points, Clarke 1866 1950s spheroid, fixed starting point in Kansas

NAD27 survey network 1990s

Survey NAD83 North American Datum 1983 Network, 1981 Successor to NAD27. 250,000 measurements points GRS80 Ellipsoid, No fixed stations

(from Schwartz, 1989) (from Schwartz, 1989)

17 North American Datum of 1927, 26,000 measured points, Clarke 1866 NAD83 spheroid, fixed starting point in Kansas North American Datum 1983 Successor to NAD27.

250,000 measurements points GRS80 Ellipsoid, No fixed stations

NAD27 survey network (from Schwartz, 1989)

Datum “Adjustment” Two Main Classes of Datums Pre-satellite: e.g., Clarke, Bessel, A datum adjustment is a calculation of the coordinates of each – this is how we NAD27, NAD83(1986) specify the “reference surface” •large errors (10s to 100s of meters), • local to continental Not straightforward, because of: Post-satellite: e.g., NAD83(HARN), Errors in surface measurements from 800 BC to 1940 AD NAD83(CORS96), WGS84(1132), ITRF99 •small relative errors (cm to 1 m) After 1960, errors in assumptions came to dominate •global

18 Our Estimate of Locations Change There are Three Main Families of Datums

19 NAD27 NAD27 and NAD83

Summary Vertical Datums Like horizontal, but referenced to standard •The Earth’s shape is approximated by an and established using vertical leveling ellipsoid Two major vertical datums, • Horizontal positions defined relative to the ellipsoid North American Vertical Datum of 1927 (NAVD29), •A datum is a reference surface, a and an update, realization of the ellipsoid, against which North American Vertical Datum of 1988 (NAVD88) locations are measured • Multiple datums, improved through time

20 Summary Cartesian Coordinates (right angle) 90o • Ellipsoid is an approximation –the shape is Two dimensions, X and Y, or in better characterized by a geoid three dimensions, the X, Y, and Z y •Heights measured relative to the geoid Two-dimensional system most •Estimates of geoid changes through time often used with •Sets of horizontal, vertical points and Projected coordinates x reference surfaces are datums z Three dimensional system used x with Geocentric coordinates

y

Spherical north pole Greenwich meridians 90 meridian Coordinates 67.5

45 •Use angles of rotation to define a directional 22.5

5

. e

2 d

vector 2 -45

-67.5 - 0 45

22.5 67.5 u

t 0 i

•Use the length of a t equator a vector originating l -22.5 near the ellipsoid -45 center to define the location on the -67.5 parallels surface south pole -90

longitude

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