1. LSU Center for GeoInformatics
GRIDS AND DATUMS
Cliff Mugnier C.P., C.M.S.
LSU Center for GeoInformatics

3. 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

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

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

8. The Longitude Lunatic

9. 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.

10. The Prize - £10,000 Sterling:

11. Inventor of the Chronometer

17. 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.

18. 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.

20. 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.

22. 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.

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

25. Law of Sines

26. 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

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 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

32. Christiaan Huyens invented the pendulum-regulated clock.

33. Sir Isaac Newton

34. French Meridian Arcs First expedition was to Lapland (left)
Took about two years
Second expedition was to Quito, Peru (right)
Took about nine years

35. 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

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 meters
b = 6,356,583.6 meters

37. U. S. Ellipsoids, continued
GRS 80 / WGS 84
a = 6,378, meters
b (GRS 80) = 6,356, meters
b (WGS 84) = 6,356, meters
Defined the gravity field differently
NAD 83 was the same as WGS84, has changed
centimeter/millimeter level

38. 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

39. 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.

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 secret
mapping (unauthorized) in China is now espionage!

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

43. a

44. Radii of curvature

45. Relationship between φλh and XYZ

46. 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

47. Helmert transformations, II
Three parameter “Molodensky:”

48. 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.

49. Datum Transformations, I
Be aware of (in)accuracies
DMA/NIMA published error estimates on the values in TR (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!

50. 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

51. 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)

53. 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.

54. 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

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 gravity
Equipotential 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 heights H h
(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. model
GEOID99 – decimeter level, 1-minute grid
GEOID03 – decimeter level, 1-minute grid
10 cm absolute, local is closer to 1 cm relative

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

59. 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

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 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.)

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. River
1878 – USC&GS Transcontinental Levels
1880 – Memphis Datum connects to Cairo
Sea Level Datum - First continental VERTICAL datum in the world
26 Tide Gauges for U.S. - Pensacola & Galveston based on full Metonic Cycles
See: 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 unknown
Delta Survey Datum of
Old Memphis Datum of
Old Cairo Datum of
New Memphis Datum of
Mean Gulf Level Datum (Prelim.) of
Mean Gulf Level Datum (Adopted) of 1899*
New Cairo Datum of
Mean Low Gulf Datum of 1911 *

68. Kilometers of Leveling Number of Tide Stations
Year of Adjustment
Kilometers of Leveling
Number of Tide Stations
21,095 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 1929
adjusted forward in time to 1955
tied to Morgan City & Mobile (‘29)
tied to Norco well (‘29 value)
tied to ‘63 lines
1973 Federal Register: SLD’29 changed to NGVD 1929
tied to Index, AR & Logtown, MS
NGVD29 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 1955
tied to Morgan City & Mobile (‘29)
tied to Norco well (‘29 value)
1968 – tied to ’63 lines
1976 – 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 adjustment
1994 – 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, 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.”

72. 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

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 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

76. 7 August 1985 Letter of Frederick M. Chatry

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

78. FG-5 Absolute Gravity Meter (±1μgal)

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

80. Absolute Gravity Observed in New Orleans:
March, ,316, mgals
Sept., ,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 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

82. Absolute Gravity Observed in New Orleans:
Nov., ,316, mgals
Aug., ,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 + MSSH
Hence 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 University
Southeastern Louisiana Univ.
LSU
McNeese State Univ.
Venice-Boothville H.S.
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)
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

87. MAP PROJECTIONS

88. 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.

89. Normal Mercator projection

90. 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

91. 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

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

95. Conformal Latitude ( χ )

96. Isometric Latitude ( τ )

97. Equivalent Spheres

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

99. Lambert Conformal Conic basic mapping equations:

100. Gauss-Krüger Transverse Mercator

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 )

102. QUESTIONS?