3 Object Space Coordinate Systems Orderly arrangement for displaying locationsMapping requires interpolation in-between known control pointsHistorical origins at observatoriesPrecise location observed astronomicallyBasis for a datum definition
7 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
9 Measuring LongitudeRelative 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.
17 Systematic errors in historical data Longitude errors 5-7x larger than latitude errorsBiases often due to different time-keepingRotations are gravity-relatedFrench navigators once sailed between Caribbean islands 7 times with different chronometers and then averaged the results.
18 EphemerisAstronomical almanac of predicted positions for heavenly bodiesCountries had Royal Astronomers with observatories in capitol citiesDatum origins were mainly at observatoriesNew England Datum origin was at the U.S. Naval Observatory in Washington, D. C.
20 Datum Origin PointObservations based on time-keeping at the observatoryOne known point measured over decadesAstronomic position:Φo based on vertical angle to PolarisΛo zero longitude is the observatory pierαo azimuth from Polaris (or mire) to another point.
26 Historical Distance Tools Wooden rods or staffsMagnolia wood boiled in paraffinGlass rods (encased in wood boxes)Platinum caps (expansion same as glass)Metal chains made of “links”Gunter’s chain = 66ft = 100 linksLength increases due to repeated use
27 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.
31 Shape of the EarthPendulum clock’s rate varies at different latitudes.Sir Isaac Newton concluded that the Earth is an oblate ellipsoid of revolution.Equatorial axis is largerC. F. Cassini de Thury disagreed – it’s a prolate ellipsoid of revolution.Polar axis is larger
32 Christiaan Huyens invented the pendulum-regulated clock.
34 French Meridian Arcs First expedition was to Lapland (left) Took about two yearsSecond expedition was to Quito, Peru (right)Took about nine years
35 Ellipsoids Published by individuals for local regions Everest 1830 Bessel 1841Clarke 1858, 1866, 1880Hayford 1906/Madrid/Helmert 1909/InternationalRecent ones are by committees
36 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 metersb = 6,356,583.6 meters
37 U. S. Ellipsoids, continued GRS 80 / WGS 84a = 6,378, metersb (GRS 80) = 6,356, metersb (WGS 84) = 6,356, metersDefined the gravity field differentlyNAD 83 was the same as WGS84, has changedcentimeter/millimeter level
38 Survey Orders 4th Order – ordinary surveying 3rd Order – Topographic/Planimetric mapping, control of aerial photography2nd Order – Federal / State, multiple county or Parish control1st Order – Federal primary controlZero Order – Special Geodetic Study Regions
39 Triangulation Primary triangulation is North – South Profile of the ellipse is North – SouthProfile of a circle is East – WestBaselines control the scale of the networkLaPlace stations control azimuth and the correction for deflection of the vertical where Latitude and Longitude are observed astronomically.
40 Datums and control points I. Traditional Military Secrets - WWII Nazis:
41 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 secretmapping (unauthorized) in China is now espionage!
42 Geocentric Coordinate System Originally devised for use in astronomy3D Cartesian Orthogonal CoordinateX-Y-Z right-handedUnits are in meters
46 Helmert transformations, I Select common points in the two datumsCalculate the Geocentric coordinate differences and average them:Use for several counties or for a small nation
47 Helmert transformations, II Three parameter “Molodensky:”
48 Survey of IndiaSoutheast Asia:Vietnam, Lao, Cambodia, Myanmar, Malaysia, Indonesia, Borneo, etc.Bangladesh, India, Pakistan, Afghanistan, Iran, Iraq, Trans-Jordan, SyriaIndian Datum 1916, 1960, 1975, etc.
49 Datum Transformations, I Be aware of (in)accuraciesDMA/NIMA published error estimates on the values in TR (now obsolete)Lots of control points used = small errorsOne or two points used = ±25 meters in each component which equates to ~ 43 meters on the ground!
50 Datum Transformations, II LTCDR Warren Dewhurst modeled the NAD27-NAD83 for his dissertation3 maps – one each of: Δφ, Δλ, ΔhFirst Order Triangulation stations (280,000)Two coordinate pairs at each stationSurface of Minimum CurvatureNADCON grids
51 Transformation accuracies For the United States:Three parameters: regionally – ±3 to ±5 metersThree parameters: local county – ±2 metersNADCON: ±0.5 metersHARN: ±0.1 meter (±5 inches)Seven Parameters: local county: ±0.1 meterMilitary MREs (multiple regression equations)Not for Theater Combat Operations (indirect fire)
53 GPSFor 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.
54 ITRFxx International Terrestrial Reference Frame xx = yearPublished by International Earth Rotation Service (IERS)Keeps track of the Earth’s wobbleIncludes continental drift informationCompares Atomic Clocks around the world
55 Elevations and height Mathematical equation which models the geoid Geoid – an imaginary surface where no topography exists and the oceans are only subject to gravityEquipotential surface (gravity potential is constant)Not smooth because of composition of the Earth
56 Geoid models Spherical harmonics (polynomials) Models the relationship between geoidal and ellipsoidal heightsHh(Topography)H = geoid height (elevation)h = ellipsoid height (GPS “vertical”)
57 GEOIDS EGM96 – 360 degree/order, 15 minute grid GEOID96 – meter level NGS, U.S. modelGEOID99 – decimeter level, 1-minute gridGEOID03 – decimeter level, 1-minute grid10 cm absolute, local is closer to 1 cm relative
58 Elevations versus heights Elevation benchmarks do not record ellipsoid heightsElevations are based on the tidesLocal mean sea level
59 Tides Diurnal = Gulf of Mexico northern coast One high/low tide cycle per daySemidiurnal = East & West U.S. coastsTwo high/low tide cycles per dayHigh tide is 11 minutes later each dayAffected by storms, geology, variation of the Earth’s density, wobble of Earth & Moon, the planet Venus
61 Effects on the Tides Chandler motion (1880) – migration of the poles Great Venus term(+ Sun + Moon)Perturbations and nutations of the axes
62 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)
63 Leveling and datums Based on gravity Theodolite – measures solid anglesHorizontal 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 improvedNeed to specify name AND the date of a datum (e.g., NAD 1927, NAD 1983, ED50, ED75, etc.)
64 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
65 Historical Leveling in the U.S. 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.
66 1871 – “Old” Cairo Datum (+300 ft) General Survey of the Miss. River1878 – USC&GS Transcontinental Levels1880 – Memphis Datum connects to CairoSea Level Datum - First continental VERTICAL datum in the world26 Tide Gauges for U.S. - Pensacola & Galveston based on full Metonic CyclesSee: Mugnier, Clifford J., DATUMS OF THE LOWER MISSISSIPPI VALLEY, SURVEYING AND MAPPING, March 1979, Volume XXXIX, No. 1, pp. 49‑60.
67 Vertical Datum (to) NAVD88 Ellet Datum of unknownDelta Survey Datum ofOld Memphis Datum ofOld Cairo Datum ofNew Memphis Datum ofMean Gulf Level Datum (Prelim.) ofMean Gulf Level Datum (Adopted) of 1899*New Cairo Datum ofMean Low Gulf Datum of 1911 *
68 Kilometers of Leveling Number of Tide Stations Year of AdjustmentKilometers of LevelingNumber of Tide Stations21,095 31,789 38,359 46,468 75,159 (U.S.) 31,565 (Canada) 21 (U.S.) 5 (Canada)Precursors to the Sea Level Datum of 1929 (later re-named National Geodetic Vertical Datum of 1929).
69 History of Levels in New Orleans 1935 – WPA local adjustment to SLD 1929adjusted forward in time to 1955tied to Morgan City & Mobile (‘29)tied to Norco well (‘29 value)tied to ‘63 lines1973 Federal Register: SLD’29 changed to NGVD 1929tied to Index, AR & Logtown, MSNGVD29 Epochs in Southeast Louisiana:adjustment based on SLD 1929 (Tied to Harahan Junction, City Park, and Ft. McComb Chef Menteur Bridge).adjusted forward in time to 1955tied to Morgan City & Mobile (‘29)tied to Norco well (‘29 value)1968 – tied to ’63 lines1976 – tied to Index, AR. and Logtown, MS.1984 – Orleans and Plaquemines Parishes tied to Waggaman & Rigolets ’76 values.1986 – Jefferson and St. Bernard Parishes tied to ‘84 values.NAVD88 Epochs in Louisiana:1992 – Original adjustment1994 – Observations in Orleans Parish– Adjustment to GPS observations to validate NGS TR 50 subsidence rates – 85 Benchmarks for ALL of South Louisiana.
70 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, later,In 1976, Don Eames of the New Orleans District, Corps of Engineers traveled to Rockville, MD to “hire” NGS to do a leveling project that was to be paid for by the New Orleans District Corps of Engineers. NGS leveled from Index, Arkansas through Simmsport to Morgan City and to New Orleans and then to Logtown, Mississippi. The Chief of Vertical Network Adjustment Section/Branch at the time was Dr. Muneendra Kumar. He came down to Baton Rouge and met at the USGS offices in Baton Rouge to announce that the apparent subsidence in metro New Orleans was so severe that he could NOT adjust the levels to fit. Dr. Kumar stated that if he constrained to any previous published elevations in New Orleans, the adjustment would be beyond the allowable range of First Order Level closures! Don Eames and I (at that meeting in the Fall of 1978) suggested that he perform a “Free Adjustment” based on the successful ties to benchmarks that appeared to be stable – those in Simmsport, LA., thus producing what was later termed the “Simmsport Datum of 1978.”
71 Surprise! There’s subsidence way down yonder … Allowable misclosure for 530 kilometers of levels was 92 mm,Actual misclosure was 86 mm, butToo much error manifested in Metro New Orleans to close locally.1978 National Geodetic Survey changes name from SLD 1929 to NGVD 1929Let’s do a “Regional Paper Adjustment.”
72 1982-83 NGS Regional Adjustment of South Louisiana Catastrophic Floods of :May 3, 1979; April 12-13, 1980Orleans, Jefferson, and Plaquemines Parishes funded NGS to re-observe BMs.Corps of Engineers concerned with the “NGS FREE ADJUSTMENT”Deep casement marks introduced
74 Class “A” Benchmark Triplet (S-369, T-369, U-369) in Jefferson Parish, LA. Class “A” Triplets are three deep-casement sleeved benchmarks installed for the purpose of being highly resistant to local differential subsidence due to Holocene muck deposits. Total perimeter distance to level through all three and close is less than 1.0 mile such that a Surveyor with an Automatic Level and Philadelphia rod can close better than ±0.05 ft vertically, if NO differential subsidence has occurred among the three benchmarks since publication. These have been installed in strategic locations throughout Jefferson, St. Bernard, Orleans, and Plaquemines Parishes in the middle 1980s. Original installation specifications by Cliff Mugnier.
75 Local Governments fund Geodetic Surveys in 1986-88 Jefferson Parish Benchmark SystemEntire East BankMetro West Bank and south to LafitteRelative gravity observed at ~350 benchmarksSt. Bernard Parish Benchmark SystemIHNC to ReggioRelative gravity observed at ~100 benchmarks
79 A μgal is one-millionth of a gal! The acceleration due to gravity at the Earth's surface is 976 to 983 gal, depending on the latitude and the ellipsoid heightA μgal is one-millionth of a gal!(That’s nine significant figures.)
80 Absolute Gravity Observed in New Orleans: March, ,316, mgalsSept., ,316, mgals(-0.91 centimeters per year)The Earth’s gravity increases as one gets closer to the pole, such as with subsidence.
81 1993 Adjustment by NGS for Subsidence Zone Elevations Last visit to New Orleans for the centuryNational Geodetic Survey loses funding for Long Line Leveling CrewGPS Constellation continues to growDefense Mapping Agency downgrades security classification on the GEOID
82 Absolute Gravity Observed in New Orleans: Nov., ,316, mgalsAug., ,316, mgals(-0.91 centimeters per year)
83 Defense Mapping Agency awards $1,000,000 contract to re-compute the GEOID Contract went to Dr. Richard Rapp, Department of Geodetic Science, Ohio State University.
84 The EGM96 surface was enhanced using local gravity and terrain data to create a regional geoid model (USGG2003). The difference between the enhanced EGM96 surface and that implied by the GPSBMs serving as control data are significant. Not all problems are related to a bias difference, significant trends occur along the coasts.The question is, which is more correct? Another line of evidence would be need to ascertain this – namely a comparison with the actual sea surface and a modeled mean dynamic topography: GEOID + MDT + MSSHHence comparison of the gravimetric geoid and MDT models at coastal stations (TBMs) might resolve the datum question in an absolute sense.This seems to offer some hope of deriving a seamless and consistent set of gravity across the region. These gravimetric geoid values may be directly compared to the NAVD 88 datum at tidal bench marks to estimate the magnitude of error in NAVD 88 – ASSUMING the derived gravimetric geoid agrees well with the MDT and tide models as well as the observed lidar sea surface heights.
86 Absolute Gravity Observations In 2002:UNO (5th time)Stennis Space Center (2nd)Loyola UniversitySoutheastern Louisiana Univ.LSUMcNeese State Univ.Venice-Boothville H.S.CocodrieOakdale H.S.LSU AlexandriaOld River Aux. Control StructureNicholls State Univ.Univ. of Louisiana in LafayetteNorthwestern State Univ.Sicily Island H.S.LSU ShreveportLouisiana 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)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. StationLamar Univ. in BeaumontUniv. of Mississippi in Hattiesburg
88 Mercator projection Gerhard Kramer = Gerhardus Mercator Published his atlas in 1569Straight line on his projection has a constant compass bearingCalled a “loxodrome” or a “rhumb line”Fundamental equation is the basis for the most important class of projections for large- and medium-scale mapping.
90 Classes of Map projections Conformal (orthomorphic) – maintains shapes and preserves anglesEqual Area (authalic) maintains areasAzimuthal (from the Classical Greeks) – used in undergraduate classes and for logosAphylactic (none of the above) – Polyconic; Europeans used the Polyhedric and the Cassini-Soldner
91 Classes, continuedAzimuthal – all directions from center of projection are correctGnomonic – all straight lines are great circlesConformal – 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
92 Projections, continued f (φ , λ) → (x, y)Graticule – network of Latitude and Longitude linesGrid – network of (x,y) lines
93 Developable surfaces Cylinder Cone Plane Complex figure (aposphere – shaped like a turnip)
94 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
101 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 )