GRIDS AND DATUMS

Cliff Mugnier C.P., C.M.S.

LSU Center for GeoInformaticscjmce@LSU.edu

Object Space Coordinate Systems

• Orderly arrangement for displaying locations– Mapping requires interpolation in-between

known control points

• Historical origins at observatories

• Precise location observed astronomically– Basis for a datum definition

Historical maps

• Reasonably accurate in North-South direction

• East-West distorted due to systematic errors in timekeeping– (Pendulum clocks don’t work onboard ships).

Latitude (Φ) and Longitude (Λ)

• Latitude (Φ) is measured positive north of the equator, negative south of the equator.– Can be determined very accurately with

astronomical techniques.

• Longitude (Λ) is measured east and west from a chosen (Prime) meridian.– Time-based measurement

The Longitude Lunatic

Measuring Longitude

• Relative calculation based on distance from zero meridian.

• Chronometer – navigation instrument with known (and constant) error rate.

• Lunar Distances could find Longitude.

• Moons of Jupiter could find Longitude.

The Prize - £10,000 Sterling:

Inventor of the Chronometer

Systematic errors in historical data

• Longitude errors 5-7x larger than latitude errors

• Biases often due to different time-keeping• Rotations are gravity-related

• French navigators once sailed between Caribbean islands 7 times with different chronometers and then averaged the results.

Ephemeris

• Astronomical almanac of predicted positions for heavenly bodies

• Countries had Royal Astronomers with observatories in capitol cities

• Datum origins were mainly at observatories

• New England Datum origin was at the U.S. Naval Observatory in Washington, D. C.

Datum Origin Point

• Observations based on time-keeping at the observatory

• One known point measured over decades• Astronomic position:

– Φo based on vertical angle to Polaris– Λo zero longitude is the observatory pier– αo azimuth from Polaris (or mire) to another

point.

Classical Astro Stations

• 12 sets of directions• 2 nights of observation• 1 day of computation• Determination of:• Φ, Λ, α, (ξ, η)• Positional accuracy of

~ 100 meters.

Surveying and Mapping

• Interpolate, not extrapolate

• Set control points along the perimeter– Interpolate for interior positions

• Create baselines and work outward

Cliff Mugnier - LSU C4G

Chain of Quadrilaterals computed with the Law of Sines

Φo, Λoφ, λ

φ, λ φ, λ

φ, λφ, λ

φ, λ φ, λ

φ, λ φ, λ

Law of Sines

sin sin sin

a b c

A B C

Historical Distance Tools

• Wooden rods or staffs– Magnolia wood boiled in paraffin

• Glass rods (encased in wood boxes)– Platinum caps (expansion same as glass)

• Metal chains made of “links”– Gunter’s chain = 66ft = 100 links– Length increases due to repeated use

Baselines

• Use baselines and trigonometry to calculate other positions

• Used to form a triangulation “chain”

• With one known length and known interior angles of a triangle, we can calculate the positions of other points with the Law of Sines.

Shape of the Earth

• Pendulum clock’s rate varies at different latitudes.

• Sir Isaac Newton concluded that the Earth is an oblate ellipsoid of revolution.– Equatorial axis is larger

• C. F. Cassini de Thury disagreed – it’s a prolate ellipsoid of revolution.– Polar axis is larger

• Christiaan Huyens invented the pendulum-regulated clock.

Sir Isaac Newton

French Meridian Arcs

Ellipsoids

• Published by individuals for local regions

• Everest 1830

• Bessel 1841

• Clarke 1858, 1866, 1880

• Hayford 1906/Madrid/Helmert 1909/International

• Recent ones are by committees

U. S. Ellipsoids

• Used Bessel 1841 through the Civil War (1860s)

• Clarke 1866 (used for 100+ years)– COL. Alexander Ross Clarke, R.E., used Pre-

Civil War triangulation arcs of North America.• a = 6,378,206.4 meters

• b = 6,356,583.6 meters

U. S. Ellipsoids, continued

• GRS 80 / WGS 84– a = 6,378,137.0 meters– b (GRS 80) = 6,356,752.314 14 meters– b (WGS 84) = 6,356,752.314 24 meters

• Defined the gravity field differently• NAD 83 was the same as WGS84, has

changed • centimeter/millimeter level

Survey Orders

• 4th Order – ordinary surveying• 3rd Order – Topographic/Planimetric

mapping, control of aerial photography• 2nd Order – Federal / State, multiple county

or Parish control• 1st Order – Federal primary control• Zero Order – Special Geodetic Study

Regions

Triangulation

• Primary triangulation is North – South– Profile of the ellipse is North – South– Profile of a circle is East – West

• Baselines control the scale of the network– LaPlace stations control azimuth and the

correction for deflection of the vertical where Latitude and Longitude are observed astronomically.

Datums and control points I.

• Traditional Military Secrets - WWII Nazis:

Datums and control points II.

• Datum ties done via espionage & stealth.

• The Survey of India is military-based and data is/was denied to its own citizens.

• South America–triangulation data along borders is commonly a military secret.

• China and Russia–ALL data still secret

mapping (unauthorized) in China is now espionage!

Geocentric Coordinate System

• Originally devised for use in astronomy

• 3D Cartesian Orthogonal Coordinate

• X-Y-Z right-handed

• Units are in meters

aaaaaaaaaaaaaaa

Radii of curvature

2 2

a,

e sin 2 2

a,

e sin

2 2

a,

e sin

2 2

a

e sin

2 2

a

e sin

2

2 2 3

(1 )

(1 sin )

a e

e

Relationship between φλh and XYZ

2

cos cos

cos sin

1 sin

X h

Y h

Z e h

Helmert transformations, I

• Select common points in the two datums• Calculate the Geocentric coordinate

differences and average them:

• Use for several counties or for a small nation

, ,X Y Z

Helmert transformations, II

• Three parameter “Molodensky:”

2 2

' '

' '

' '

a a

e e

i i i i

XX U

Y Y V

h Z W hZ

Survey of India

• Southeast Asia:Vietnam, Lao, Cambodia, Myanmar, Malaysia, Indonesia, Borneo, etc.

• Bangladesh, India, Pakistan, Afghanistan, Iran, Iraq, Trans-Jordan, Syria

• Indian Datum 1916, 1960, 1975, etc.

Datum Transformations, I

• Be aware of (in)accuracies

• DMA/NIMA published error estimates on the values in TR 8350.2 (now obsolete)– Lots of control points used = small errors– One or two points used = ±25 meters in each

component which equates to ~ 43 meters on the ground!

2 2 2error x y z

Datum Transformations, II

• LTCDR Warren Dewhurst modeled the NAD27-NAD83 for his dissertation

• 3 maps – one each of: Δφ, Δλ, Δh

• First Order Triangulation stations (280,000)– Two coordinate pairs at each station

• Surface of Minimum Curvature

• NADCON grids

Transformation accuracies

• For the United States:– Three parameters: regionally – ±3 to ±5 meters– Three parameters: local county – ±2 meters– NADCON: ±0.5 meters– HARN: ±0.1 meter (±5 inches)– Seven Parameters: local county: ±0.1 meter

• Military MREs (multiple regression equations)– Not for Theater Combat Operations (indirect fire)

GPS

• For single-frequency consumer-grade receivers using the broadcast ephemeris: will yield accuracies of ~ 4-5 meters at present. (Compare to 100 m Astro position)

• For dual-frequency receivers using post-processing with the precise ephemeris: will yield accuracies of ~ 1 centimeter or less.

ITRFxx

• International Terrestrial Reference Frame– xx = year

• Published by International Earth Rotation Service (IERS)– Keeps track of the Earth’s wobble– Includes continental drift information– Compares Atomic Clocks around the world

Elevations and height

• Mathematical equation which models the geoid• Geoid – an imaginary surface where no

topography exists and the oceans are only subject to gravity

• Equipotential surface (gravity potential is constant)

• Not smooth because of composition of the Earth

Geoid models

• Spherical harmonics (polynomials)

• Models the relationship between geoidal and ellipsoidal heights

H = geoid height (elevation)h = ellipsoid height (GPS “vertical”)

h H(Topography)

GEOIDS

• EGM96 – 360 degree/order, 15 minute grid• GEOID96 – meter level

– NGS, U.S. model

• GEOID99 – decimeter level, 1-minute grid– NGS, U.S. model

• GEOID03 – decimeter level, 1-minute grid– 10 cm absolute, local is closer to 1 cm relative– NGS, U.S. model

Elevations versus heights

• Elevation benchmarks do not record ellipsoid heights

• Elevations are based on the tides– Local mean sea level

Tides

• Diurnal = Gulf of Mexico northern coast– One high/low tide cycle per

day• Semidiurnal = East & West U.S.

coasts– Two high/low tide cycles per

day• High tide is 11 minutes later each

day• Affected by storms, geology,

variation of the Earth’s density, wobble of Earth & Moon, the planet Venus

Effects on the Tides

• Chandler motion (1880) – migration of the poles

• Great Venus term– (+ Sun + Moon)

• Perturbations and nutations of the axes

Types of Tidal Datums

• Mean Higher High Water (MHHW)

• Mean High Water (MHW)

• Mean Tide Level (MTL)

• Diurnal Tide Level (DTL)

• Mean Sea Level (MSL)

• Mean Low Water (MLW)

• Mean Lower Low Water (MLLW)

Leveling and datums

• Based on gravity• Theodolite – measures solid angles• Horizontal positions have errors because of

gravity effects that are unknown (deflection of the vertical)

• Thus, each country has more than one classical datum as technology has improved

• Need to specify name AND the date of a datum (e.g., NAD 1927, NAD 1983, ED50, ED75, etc.)

Local Mean Sea Level

• 18.67-year Metonic cycle• To determine “local mean sea level,”

observe tides for at least one Metonic cycle.• Every 5 years = new tidal Epoch (based on

a running average).• New epochs are published by the

International Hydrographic Organization (IHO), Monaco

Historical Leveling in the U.S.

• 1850 - Congress directed Charles Ellet to make a complete survey of the Ohio & Mississippi Rivers, & Capt. Humphreys, Corps of Engineers, started a separate report of the survey of the Mississippi River Delta. The flood of 1858 was used as the “plane of reference” – The Delta Datum of 1858.

• 1871 – “Old” Cairo Datum (+300 ft)

• 1876 - General Survey of the Miss. River

• 1878 – USC&GS Transcontinental Levels

• 1880 – Memphis Datum connects to Cairo

• 1929 - Sea Level Datum - First continental VERTICAL datum in the world

• 26 Tide Gauges for U.S. - Pensacola & Galveston based on full Metonic Cycles

Vertical Datum (to) NAVD88

• Ellet Datum of 1850 unknown• Delta Survey Datum of 1858 +0.86• Old Memphis Datum of 1858 - 8.13• Old Cairo Datum of 1871 - 21.26• New Memphis Datum of 1880 - 6.63• Mean Gulf Level Datum (Prelim.) of 1882 + 0.318• Mean Gulf Level Datum (Adopted) of 1899* 0.000• New Cairo Datum of 1910 - 20.434• Mean Low Gulf Datum of 1911 * - 0.78

Year of Adjustment

Kilometers of Leveling

Number of Tide Stations

19001903190719121929

21,09531,78938,35946,468

75,159 (U.S.)  31,565 (Canada)

5889

        21 (U.S.)              5 (Canada)

History of Levels in New Orleans

• 1935 – WPA local adjustment to SLD 1929

• 1951 - adjusted forward in time to 1955

• 1955 - tied to Morgan City & Mobile (‘29)

• 1963 - tied to Norco well (‘29 value)

• 1969 - tied to ‘63 lines– 1973 Federal Register: SLD’29 changed to NGVD 1929

• 1976 - tied to Index, AR & Logtown, MS

1976-77 NGS Leveling (funded by Corps of Engineers)

• Start at Index, Arkansas,

• through Simmsport, LA to:

• Morgan City & Baton Rouge, both then to:

• New Orleans, thence to:

• Venice, LA (spur) and Logtown, MS to close the line.

• … and $1,500,000.00 later,

Surprise! There’s subsidence way down yonder …

• Allowable misclosure for 530 kilometers of levels was 92 mm,

• Actual misclosure was 86 mm, but

• Too much error manifested in Metro New Orleans to close locally.

• 1978 National Geodetic Survey changes name from SLD 1929 to NGVD 1929

• Let’s do a “Regional Paper Adjustment.”

1982-83 NGS Regional Adjustment of South LouisianaCatastrophic Floods of :

May 3, 1979; April 12-13, 1980

• Orleans, Jefferson, and Plaquemines Parishes funded NGS to re-observe BMs.

• Corps of Engineers concerned with the “NGS FREE ADJUSTMENT”

• Deep casement marks introduced

Local Governments fund Geodetic Surveys in 1986-88

• Jefferson Parish Benchmark System– Entire East Bank– Metro West Bank and south to Lafitte– Relative gravity observed at ~350 benchmarks

• St. Bernard Parish Benchmark System– IHNC to Reggio– Relative gravity observed at ~100 benchmarks

7 August 1985 Letter of Frederick M. Chatry

North American Vertical Datum of 1988

Actual published data available starting in 1990

No data available for South Louisiana (Crustal Motion Area)

FG-5 Absolute Gravity Meter(±1μgal)

• The acceleration due to gravity at the Earth's surface is 976 to 983 gal, depending on the latitude and the ellipsoid height

• A μgal is one-millionth of a gal!– (That’s nine significant figures.)

Absolute Gravity Observed in New Orleans:

March, 1989 979,316,847.7 gals

Sept., 1991 979,316,854.2 gals

(-0.91 centimeters per year)

1993 Adjustment by NGS for Subsidence Zone Elevations

• Last visit to New Orleans for the century

• National Geodetic Survey loses funding for Long Line Leveling Crew

• GPS Constellation continues to grow

• Defense Mapping Agency downgrades security classification on the GEOID

Absolute Gravity Observed in New Orleans:

Nov., 1993 979,316,856.3 gals

Aug., 1994 979,316,860.6 gals

(-0.91 centimeters per year)

Defense Mapping Agency awards $1,000,000 contract to re-

compute the GEOID

Absolute Gravity Observations

• In 2002:– UNO (5th time)– Stennis Space Center (2nd)– Loyola University– Southeastern Louisiana Univ.– LSU– McNeese State Univ.– Venice-Boothville H.S.– LUMCON @ Cocodrie– Oakdale H.S.– LSU Alexandria– Old River Aux. Control Structure– Nicholls State Univ.– Univ. of Louisiana in Lafayette– Northwestern State Univ.– Sicily Island H.S.– LSU Shreveport– Louisiana Tech Univ.

• In 2006:– UNO (6th time)– Stennis Space Center (3rd)– Loyola University (2nd)– Southeastern Louisiana Univ. (2nd)– LSU (2nd)– McNeese State Univ. (2nd)– Venice-Boothville H.S. (2nd)– LUMCON @ Cocodrie (2nd)– Oakdale H.S. (2nd)– LSU Alexandria (2nd)– Old River Aux. Control Structure (2nd)– Nicholls State Univ. (2nd)– Univ. of Louisiana in Lafayette (2nd)– Grand Isle U.S.C.G. Station– Lamar Univ. in Beaumont– Univ. of Mississippi in Hattiesburg

MAP PROJECTIONS

Mercator projection

• Gerhard Kramer = Gerhardus Mercator• Published his atlas in 1569• Straight line on his projection has a constant

compass bearing– Called a “loxodrome” or a “rhumb line”

• Fundamental equation is the basis for the most important class of projections for large- and medium-scale mapping.

Normal Mercator projection

Classes of Map projections

• Conformal (orthomorphic) – maintains shapes and preserves angles

• Equal Area (authalic) maintains areas• Azimuthal (from the Classical Greeks) –

used in undergraduate classes and for logos• Aphylactic (none of the above) – Polyconic;

Europeans used the Polyhedric and the Cassini-Soldner

Classes, continued

• Azimuthal – all directions from center of projection are correct

• Gnomonic – all straight lines are great circles• Conformal – 99% of large scale mapping world-

wide: UTM, State Plane, etc.• Trinidad & Tobago offshore oil leases use the

Cassini-Soldner (from Colonial usage)Projections, continued

Projections, continued

• f (φ , λ) → (x, y)

• Graticule – network of Latitude and Longitude lines

• Grid – network of (x,y) lines

Developable surfaces

• Cylinder

• Cone

• Plane

• Complex figure (aposphere – shaped like a turnip)

Types of Ellipsoidal Latitudes

• Conformal Latitude* (χ )

• Isometric Latitude ( τ )

• Authalic Latitude* ( β )

• Geocentric Latitude ( ψ )

• Rectifying Latitude* ( ω )

• Parametric Latitude ( θ ) * associated Equivalent Sphere

Conformal Latitude ( χ )

1 2 3 4C sin 2 sin 4 C sin 6 C sin8C

2 4 6 8

1

5 3 281C

2 24 32 5760

e e e e

4 6 8

2

5 7 697C

48 80 11520

e e e

6 8

3

13 461C

480 13440

e e

8

4

1237C

161280

e

Isometric Latitude ( τ )

tan4 2

Ln

Equivalent Spheres

R R

2 4 65n 81 325R a(1-n) 1+

4 64 256

n n

2 4 6e 17 67R a 1-

6 360 3024

e e

2

2

1 (1 )

1 (1 )

en

e

Zones, Grids, and Belts

• ZONE – Lambert Conformal Conic

• GRID – Cassini-Soldner (aphylactic)

• BELT – Transverse Mercator (conformal)

Lambert Conformal Conic basic mapping equations:

sinX r FE

cosoY r r FN

Gauss-Krüger Transverse Mercator

33 2

55 3 2 2 2 2 4

77 2 4 6

cos cos tan3!

F.E. cos 4 1 6 tan 1 8 tan 2 tan tan5!

cos 61 479 tan 179 tan tan7!

oX m

2 4 32 2

4 2 3 26 5

2 2 2 4

8 72 4 6

cos cos4 tan

2! 4!

8 11 24 tan 28 1 6 tancosF.N. sin

6! 1 32 tan 2 tan tan

cos1385 3111tan 543tan tan

8!

oY m M

Corrections for systematic errors

• Sea Level – to correct a distance at some altitude back to the surface of the ellipsoid:→ Ellipsoidal dist. = Surface dist. × (Rφ÷[Rφ+h])

• Grid distance vs. Geodetic (True) distance:→ Grid dist = Geodetic dist. × Scale factor (m)

• Grid azimuth vs. Geodetic azimuth:→ Geod. az = Grid az – Convergence angle ( γ )

QUESTIONS?