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GRIDS AND DATUMS Cliff Mugnier C.P., C.M.S. LSU Center for GeoInformatics cjmce@LSU.edu

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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

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Historical maps Reasonably accurate in North-South direction East-West distorted due to systematic errors in timekeeping –(Pendulum clocks don’t work onboard ships).

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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

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The Longitude Lunatic

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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.

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The Prize - £10,000 Sterling:

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Inventor of the Chronometer

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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.

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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.

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Datum Origin Point Observations based on time-keeping at the observatory One known point measured over decades Astronomic position: –Φ o based on vertical angle to Polaris –Λ o zero longitude is the observatory pier –α o azimuth from Polaris (or mire) to another point.

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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.

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Surveying and Mapping Interpolate, not extrapolate Set control points along the perimeter –Interpolate for interior positions Create baselines and work outward

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Law of Sines

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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

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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.

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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

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Christiaan Huyens invented the pendulum- regulated clock.

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Sir Isaac Newton

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French Meridian Arcs

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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

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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

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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

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Survey Orders 4 th Order – ordinary surveying 3 rd Order – Topographic/Planimetric mapping, control of aerial photography 2 nd Order – Federal / State, multiple county or Parish control 1 st Order – Federal primary control Zero Order – Special Geodetic Study Regions

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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.

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Datums and control points I. Traditional Military Secrets - WWII Nazis:

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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!

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Geocentric Coordinate System Originally devised for use in astronomy 3D Cartesian Orthogonal Coordinate X-Y-Z right-handed Units are in meters

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Radii of curvature

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Relationship between φλh and XYZ

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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

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Helmert transformations, II Three parameter “Molodensky:”

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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.

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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!

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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

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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)

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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.

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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

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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

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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)

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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

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Elevations versus heights Elevation benchmarks do not record ellipsoid heights Elevations are based on the tides –Local mean sea level

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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

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Effects on the Tides Chandler motion (1880) – migration of the poles Great Venus term –(+ Sun + Moon) Perturbations and nutations of the axes

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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)

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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.)

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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

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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.

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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

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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

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Year of Adjustment Kilometers of Leveling Number of Tide Stations 1900 1903 1907 1912 1929 21,095 31,789 38,359 46,468 75,159 (U.S.) 31,565 (Canada) 5 8 8 9 21 (U.S.) 5 (Canada)

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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

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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,

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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.”

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1982-83 NGS Regional Adjustment of South Louisiana Catastrophic 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

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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

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7 August 1985 Letter of Frederick M. Chatry

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North American Vertical Datum of 1988 Actual published data available starting in 1990 No data available for South Louisiana (Crustal Motion Area)

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FG-5 Absolute Gravity Meter (±1μgal)

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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.)

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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)

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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

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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)

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

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Absolute Gravity Observations In 2002: –UNO (5 th time) –Stennis Space Center (2 nd ) –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 (6 th time) –Stennis Space Center (3 rd ) –Loyola University (2 nd ) –Southeastern Louisiana Univ. (2 nd ) –LSU (2 nd ) –McNeese State Univ. (2 nd ) –Venice-Boothville H.S. (2 nd ) –LUMCON @ Cocodrie (2 nd ) –Oakdale H.S. (2 nd ) –LSU Alexandria (2 nd ) –Old River Aux. Control Structure (2 nd ) –Nicholls State Univ. (2 nd ) –Univ. of Louisiana in Lafayette (2 nd ) –Grand Isle U.S.C.G. Station –Lamar Univ. in Beaumont –Univ. of Mississippi in Hattiesburg

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MAP PROJECTIONS

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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.

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Normal Mercator projection

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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

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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

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Projections, continued f (φ, λ) → (x, y) Graticule – network of Latitude and Longitude lines Grid – network of (x,y) lines

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Developable surfaces Cylinder Cone Plane Complex figure (aposphere – shaped like a turnip)

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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

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Conformal Latitude ( χ )

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Isometric Latitude ( τ )

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Equivalent Spheres

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Zones, Grids, and Belts ZONE – Lambert Conformal Conic GRID – Cassini-Soldner (aphylactic) BELT – Transverse Mercator (conformal)

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Lambert Conformal Conic basic mapping equations:

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Gauss-Krüger Transverse Mercator

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Corrections for systematic errors Sea Level – to correct a distance at some altitude back to the surface of the ellipsoid: → Ellipsoidal dist. = Surface dist. × (R φ ÷[R φ +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 )

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QUESTIONS?

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