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THE GEOID

The Kentucky Single Zone State Plane Coordinate System Bryan W. Bunch, PLS, PG Geoprocessing Specialist III Kentucky NREPC-OIS-GIS 500 Mero Street 14 th FL CPT 502-564-5174 bryan.bunch@mail.state.ky.us. THE GEOID.

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THE GEOID

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  1. The KentuckySingle Zone State Plane Coordinate SystemBryan W. Bunch, PLS, PGGeoprocessing Specialist IIIKentucky NREPC-OIS-GIS500 Mero Street 14th FL CPT502-564-5174bryan.bunch@mail.state.ky.us

  2. THE GEOID • geoid: The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level. • The Geoid is a representation of the earth based on an equipotential gravimetric surface resulting in an irregular and non-mathematical approximation of the earth’s size and shape relative to a base of reference approximating global mean sea level. • All points on the surface of the Geoid experience the same magnitude of gravimetric force.

  3. http://www.csr.utexas.edu/grace/gallery/animations/world_gravity/world_gravity_wm.htmlhttp://www.csr.utexas.edu/grace/gallery/animations/world_gravity/world_gravity_wm.html

  4. THE GEOID and MEAN SEA LEVEL SOURCE: National Geodetic Survey http://www.ngs.noaa.gov/GEOID/geoid_def.html

  5. DATUMS • A datum is a reference ellipsoid that is defined and oriented so as to achieve a best fit approximation of the Geoid either in part or in whole. • The North American Datum (NAD) has been defined by two different ellipsoids, the Clarke ellipsoid of 1866, which was oriented to best fit the North American continent and is the basis of NAD27, and the Global Reference System ellipsoid of 1980 (GRS80) which is a globally defined ellipsoid and the basis of NAD83.

  6. THE GEOID and ELLIPSOID SOURCE: National Geodetic Survey http://www.ngs.noaa.gov/GEOID/geoid_def.html

  7. DATUMS: NAD’27 and NAD’83 CLARKE 1866 Ellipsoid (NAD’27) GRS80 Ellipsoid (NAD’83) Earth MassCenter Approximately236 meters GEOID Source: National Geodetic Survey

  8. COMPARISON OF NORTH AMERICAN DATUMS NAD 27 NAD 83 ELLIPSOID CLARKE 1866 GRS80 a = 6,378,206.4 m a = 6,378,137 m 1/f = 294.9786982 1/f = 298.257222101 DATUM POINT Triangulation Station EARTH MASS CENTER MEADES RANCH, KANSAS ADJUSTMENT 25k STATIONS 250k STATIONS Several Hundred Base Lines Approx. 30k EDMI Base Lines Several Hundred Astro Azimuths 5k Astro Azimuths Doppler Point Positions VLBI Vectors BEST FITTING North America World-Wide Source: National Geodetic Survey

  9. z DATUM 1 P Z f 1 x l 1 y DATUM 2 f 2 Y l 2 X GEOGRAPHIC POSITION (Lat-Long) General Variation Between Datums

  10. 1927 North American Datum To place on the predicted North American Datum 1983 move the projection lines 4 meters south and 6 meters west as shown by dashed corner ticks. GEOGRAPHIC POSITION (Lat-Long) Variation Between Datums: NAD27 vs NAD83

  11. MAPPING PROJECTIONS • A mapping projection is a geometric tool that allows a portion of a spherical surface to be represented on a two dimensional surface such as a flat sheet of paper or computer screen in a spatially consistent manner. • A state plane coordinate system is a specialized mapping projection that allows direct conversion between spherical geographic coordinates of latitude () and longitude (), and rectangular Cartesian coordinates of northing (y) and easting (x).

  12. MAPPING PROJECTIONS (cont) • A state plane coordinate system allows the utilization of plane surveying methods and techniques to measure small portions of the earth with minimal distortions as opposed to the more rigorous and spatially accurate methods of geodetic surveying. • It is much simpler to represent and work with engineering and mapping data on a planar surface than it is a spherical surface

  13. Transverse Secant Cylinder (Mercator) Coaxial Secant Cone (Lambert) COMMON MAPPING PROJECTION SCHEMES

  14. BASIC ELEMENTS OF A CONIC PROJECTION Cone (Projection Surface) Ellipsoid (Datum)

  15. LAMBERT CONIC PROJECTION(Northern Hemisphere) North StandardParallel Polar Axis Central Meridian South StandardParallel Parallel ofGrid Origin(Base Parallel)

  16. LAMBERT CONE TO GRID PROJECTION 6,000 7,000 8,000 12,000 13,000 14,000 9,000 10,000 15,000 5,000 11,000 Apex of Cone 16,000 6,000 North Standard Parallel South Standard Parallel 15,000 5,000 ConvergenceAngle 14,000 4,000 NORTHING ( y ) 13,000 3,000 12,000 2,000 Central Meridian 11,000 1,000 Parallel ofGrid Origin 10,000 0 -4,000 -3,000 -2,000 2,000 3,000 4,000 -5,000 -1,000 0 1,000 5,000 EASTING ( x ) FALSE NORTHING = 10,000 FALSE EASTING = 10,000

  17. E1 < G1 < S1 S1 E = Distance on ellipsoidG = Distance on gridS = Distance on surface SF = Grid Scale Factor = Geodetic latitude G1 G2 < E2 < S2 S2 E1 E2 SF>1 Ellipsoid SF=1 G2 Topographic Surface(Ground) SF<1 Axis of Rotation North Standard Parallel Projection Grid E3 < S3 < G3 SF=1 G3 South Standard Parallel S3 SF>1 E3 Equatorial Plane LAMBERT GROUND, GRID, and ELLIPSOID

  18. Ground to Ellipsoid Distortion • Occurs when the surface of the earth does not coincide with the surface of the ellipsoid. These distortions are compensated by elevation factors derived from surveyed or estimated elevations (orthometric heights), and observed or estimated differences between the geoid and ellipsoid (geoid height). Orthometric height is usually measured by differential leveling methods or estimated from USGS topographic maps. Geoid height can be obtained by direct GPS observation on a vertical control monument or utilizing GEOID99, a computer program and model available as a free download from the National Geodetic Survey.

  19. TopographicSurface N > 0 h N < 0 H N R = Radius of ellipsoid in prime vertical at position latitude () H = Ground height above geoid (orthometric height) N = Geoid height (determined by GEOID99) h = Ellipsoid height Ellipsoid R Geoid N < 0 ELEVATION CORRECTION FACTOR (EF)

  20. Ellipsoid to Grid Distortion • Occurs where positions on the ellipsoid must be ‘warped’ to fit their corresponding positions on the planar surface of the grid. This results in positions on the ellipsoid lying between the standard parallels being contracted to fit onto the projection plane, and positions lying outside the standard parallels being expanded. Distortions do not occur for positions lying on the standard parallels where both surfaces intersect and are coincidental with one another.

  21. Scale factor: The grid scale factor (SF) is the ratio between a distance represented on the grid and its corresponding value on the ellipsoid. It is greater than one for regions lying outside the standard parallels, less than one for the region lying between the standard parallels, and exactly one for positions lying on the standards parallels. • Correcting from ground to grid: Correcting overall ground to grid distortion is achieved by computing a correction factor (CF) that is the product of the grid scale factor (SF) and the elevation factor (EF). This is multiplied by the ground distance to achieve a corresponding distance on the grid. Thus: CF = EF x SF Grid distance = ground distance x correction factor

  22. E1 < G1 < S1 S1 E = Distance on ellipsoidG = Distance on gridS = Distance on surface SF = Grid Scale Factor = Geodetic latitude G1 G2 < E2 < S2 S2 E1 E2 SF>1 Ellipsoid SF=1 G2 Topographic Surface(Ground) SF<1 Axis of Rotation North Standard Parallel Projection Grid E3 < S3 < G3 SF=1 G3 South Standard Parallel S3 SF>1 E3 Equatorial Plane LAMBERT GROUND, GRID, and ELLIPSOID

  23. KENTUCKY SPCS – NORTH AND SOUTH ZONES NORTH ZONE SOUTH ZONE FIGURE 2

  24. RATIONALE FOR A SINGLE ZONE • Advances in Geographic Information Systems, computer hardware and software, and network technology have made it feasible to create, manage, access, view, query, and analyze high resolution datasets covering large geographic areas. • Analyzing raster data on statewide basis or over a large geographic region such as watershed or highway corridor within the state is practical only when all data is presented in the same coordinate system or reference frame. • A uniform coordinate system will facilitate non-vector data integration and viewing on statewide basis, and eliminate extra cost and effort needed to maintain the current overlap areas between the North and South Zones. For example, overlaps added approximately 17% to statewide DEM contract.

  25. KENTUCKY NORTH AND SOUTH ZONEDRG OVERLAP and DUPLICATION

  26. RATIONALE FOR A SINGLE ZONE (CONT) • It is often confusing for surveyors and other professionals working with multiple state plane and UTM zones to determine which zone and datum applies for a given project location, especially if no metadata is available or given. • The constraints that gave rise to the necessity for limiting ellipsoid to grid distortions no longer exist due to advances in surveying and mapping technology.

  27. KENTUCKY PROJECTIONSUTM Zones 16 & 17Transverse Mercator (Secant Cylinder) UTM Zone 17 UTM Zone 16

  28. State PlaneNorth Zone State Plane South Zone KENTUCKY PROJECTIONSNorth and South State PlaneLambert Conformal Conic (Secant Cone)

  29. FISCAL ISSUES • Adopting a single zone system will reduce cost over time by working with and maintaining data in one zone as opposed to two. • Existing software across a broad range of disciplines does not provide economical solutions for displaying and manipulating vector and raster data simultaneously in multiple zones. • There will be a one time data conversion and migration cost, which will be absorbed by long term savings.

  30. KENTUCKY SINGLE ZONE PROJECTION

  31. CONSTANTS DEFINING THE KENUCKY SINGLE ZONE PROJECTION The Kentucky Single Zone Projection was developed to cover the entire state in a balanced manner that attempts to minimize ground to grid distortions and convergence angles. • Datum = NAD83 / GRS80 Ellipsoid • Central Meridian = 85° 45’ West Longitude • Parallel of Grid Origin (Base Parallel) = 36° 20’ North Latitude • Southern Standard Parallel = 37° 05’ North Latitude • Northern Standard Parallel = 38° 40’ North Latitude • False Northing = 1,000,000 meters (3,280,833.333 feet*) • False Easting = 1,500,000 meters (4,921,250.000 feet*) *U.S. Survey Feet (1 USFt. = 1200 / 3937 meter)

  32. COORDINATE SPACE COMPARISON 1,500,000 m 1,250,000 m NAD'83 SINGLE ZONE 1,000,000 m NORTHING 750,000 m NAD'83 SOUTH ZONE 500,000 m NAD'83 NORTH ZONE 250,000 m NAD'27 NORTH ZONE NAD'27 SOUTH ZONE 0 m 0 m 500,000 m 750,000 m 250,000 m 1,000,000 m 1,500,000 m 1,750,000 m 1,250,000 m 2,000,000 m EASTING

  33. C o u n t y B o u n d a r y D i s t o r t i o n R a t i o s B e t t e r t h a n 1 : 2 5 , 0 0 0 1 : 1 5 , 0 0 0 - 1 : 2 5 , 0 0 0 1 : 1 0 , 0 0 0 - 1 : 1 5 , 0 0 0 1 : 7 , 5 0 0 - 1 : 1 0 , 0 0 0 1 : 5 , 0 0 0 - 1 : 7 , 5 0 0 W o r s e t h a n 1 : 5 , 0 0 0 Distribution of Ground to Grid Distortion Ratiosby Quad for Worst Case SituationsSouth Zone Projection

  34. C o u n t y B o u n d a r y D i s t o r t i o n R a t i o s B e t t e r t h a n 1 : 2 5 , 0 0 0 1 : 1 5 , 0 0 0 - 1 : 2 5 , 0 0 0 1 : 1 0 , 0 0 0 - 1 : 1 5 , 0 0 0 1 : 7 , 5 0 0 - 1 : 1 0 , 0 0 0 1 : 5 , 0 0 0 - 1 : 7 , 5 0 0 W o r s e t h a n 1 : 5 , 0 0 0 Distribution of Ground to Grid Distortion Ratiosby Quad for Worst Case SituationsSingle Zone Projection

  35. National Geodetic Survey, Retrieval Date = AUGUST 12, 2002 GZ0132 *********************************************************************** GZ0132 CBN - This is a Cooperative Base Network Control Station. GZ0132 DESIGNATION - I75 L 83 GZ0132 PID - GZ0132 GZ0132 STATE/COUNTY- KY/MADISON GZ0132 USGS QUAD - RICHMOND NORTH (1993) GZ0132 GZ0132 *CURRENT SURVEY CONTROL GZ0132 ___________________________________________________________________ GZ0132* NAD 83(1993)- 37 51 34.68569(N) 084 19 59.19617(W) ADJUSTED GZ0132* NAVD 88 - 286.293 (meters) 939.28 (feet) ADJUSTED GZ0132 ___________________________________________________________________ GZ0132 X - 497,891.230 (meters) COMP GZ0132 Y - -5,017,564.276 (meters) COMP GZ0132 Z - 3,893,310.905 (meters) COMP GZ0132 LAPLACE CORR- 0.42 (seconds) DEFLEC99 GZ0132 ELLIP HEIGHT- 253.79 (meters) GPS OBS GZ0132 GEOID HEIGHT- -32.51 (meters) GEOID99 GZ0132 DYNAMIC HT - 286.087 (meters) 938.60 (feet) COMP GZ0132 MODELED GRAV- 979,903.0 (mgal) NAVD 88 GZ0132 GZ0132 HORZ ORDER - B GZ0132 VERT ORDER - SECOND CLASS 0 GZ0132 ELLP ORDER - FOURTH CLASS I . . . GZ0132; North East Units Scale Converg. GZ0132;SPC KY1Z - 1,170,338.983 1,624,669.125 MT 0.99990498 +0 52 11.7 GZ0132;SPC KY1Z - 3,839,687.15 5,330,268.62 sFT 0.99990498 +0 52 11.7 GZ0132;SPC KY S - 670,320.267 624,679.514 MT 0.99998738 +0 51 33.4 GZ0132;SPC KY S - 2,199,209.08 2,049,469.37 sFT 0.99998738 +0 51 33.4 GZ0132;SPC KY N - 39,921.624 492,686.218 MT 1.00001792 -0 03 06.1 GZ0132;SPC KY N - 130,976.19 1,616,421.37 sFT 1.00001792 -0 03 06.1 GZ0132;UTM 16 - 4,193,593.825 734,612.496 MT 1.00027800 +1 38 15.0 KY1Z = Kentucky Single Zone http://ngs.state.ky.us

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