Design of the Wisconsin County Coordinate System: Some Current Issues Al Vonderohe SIAC Presentation - October 27, 2004

Outline Review: Datums / Map Projections Need for Wisconsin County Coordinate Systems (WCCS) Original Design Methodology Current Issues / WLIA Task Force on Wisconsin Coordinate Systems Methodology for Possible Re-Design Initial Results

Ellipsoid Meridian of Longitude Prime Meridian Parallel of Latitude

Rotate about minor axis to generate oblate spheroid.

How do we choose the size and shape of the ellipsoid to use? Best fit to geoid. The geoid is a gravitational equipotential surface corresponding to “mean sea level”. Ellipsoid Used for Current National Geodetic Datum (NAD83) is Named “GRS 80”: –a = m –b = m

Computational and Visualization Problem Latitude / Longitude are Angular, not Rectangular Coordinates. Ellipsoid Surface Cannot be Cut and Laid Flat. Latitude / Longitude Must be Projected to a “Developable” Surface to Obtain Rectangular Coordinates.

Developable Surfaces

One Way to Conceptualize “Projection” Points on the ellipsoid are projected to the projection surface by straight lines from the center of the ellipsoid. Note scale factor and how it varies across the projection surface. “Secant” projection.

Lambert Projections

Scale varies north- south. Good for areas with east-west extents. Projection Parameters: 0 (longitude of central meridian) Central Meridian ( 0 )  1,  2 (latitudes of standard parallels) 11 22  0, X 0,Y 0 (latitude, false easting, false northing of the coordinate origin) X Y 00 XoXo YoYo Alternative to  1,  2 is  0,k 0 (latitude and scale factor at central parallel).

Lambert Projections Scale varies north- south. Good for areas with east-west extents. Projection Parameters: 0 (longitude of central meridian) Central Meridian ( 0 )  1,  2 (latitudes of standard parallels)  0, X 0,Y 0 (latitude, false easting, false northing of the coordinate origin) X Y 00 XoXo YoYo Alternative to  1,  2 is  0,k 0 (latitude and scale factor at central parallel). 0k00k0  0,k 0 can be computed from  1,  2.

Transverse Mercator Projections

Scale varies east-west. Good for areas with north-south extents. Projection Parameters: 0 (longitude of central meridian) k 0 (scale factor along central meridian)  0, X 0,Y 0 (latitude, false easting, false northing of the coordinate origin) X Y 00 XoXo YoYo 0,k 0

Design of Map Projections Design means choosing the projection parameters so that the projection meets specific objectives across specific geographic extents. The design objective is often having the scale factor fall within specific bounds. For example, the State Plane Coordinate System was designed so that the scale factor is no larger than 1:10,000 anywhere within a zone.

State Plane Coordinate Zones North Central South

What is This Scale Factor? It is the ratio of distances on the map projection surface to distances on the ellipsoid surface. It is not the ratio of distances on the map projection to distances on the ground. To convert ground distances to map projection distances an additional factor must be taken into account.

Scale Transforms Ellipsoid Distances to Map Projection Distances We know this can be done, but how do we get ellipsoid distances in the first place?

How Do We Get Ellipsoid Distances in the First Place? R / (R + N + H) is called the “ellipsoid factor”. We then have to multiply S by the scale factor to get a map projection distance (M). So M = D * (ellipsoid factor)* (scale factor). Both ellipsoid and scale factors vary with position.

There are actually four surfaces! Map Projection Surface Distances on property maps and construction stakeout are measured here. GIS Spatial databases and design drawings are developed here.

Wisconsin County Coordinates Property maps have tens of thousands of ground-level distances on them. –Too difficult to convert to existing map projections for GIS spatial database development. Design drawings have tens of thousands of map projection distances on them. –Too difficult to convert to ground distances for construction stakeout.

Wisconsin County Coordinates Solution: Develop map projections that are even more localized than state plane coordinate projections. –There should be no significant differences between ground distances and map projection distances, so that ground distances can be used directly in spatial databases and design distances can be used directly for stakeout. –How?

The ellipsoid was enlarged to the mean elevation of the terrain for the local area. –a local = a GRS80 + (N + h) local –b local = b GRS80 + (N + h) local Ellipsoid factor = 1. Projection parameters chosen so that scale factor is –1:50,000 in urban areas. –1:30,000 in rural areas. Differences between ground and map projection distances are negligible in local area. Enlarged Local Ellipsoid Surface Local Map Projection Surface (Secant)

Wisconsin County Coordinate Systems Design funded by WisDOT 59 separate map projections (Lambert and Transverse Mercator) and coordinate systems for the state. 72 counties – some have shared projections.

Wisconsin County Coordinate Systems

Many counties are using these systems in development of their GIS spatial databases. –Some have officially adopted them through ordinances. WisDOT uses the systems for all infrastructure design and construction.

Emerging Issue Enlarging the ellipsoid has the mathematical effect of modifying the underlying geodetic datum. This has caused difficulties in both the vendor and user communities. –Vendors want to support WCCS, but there is complexity. –Most of the user community doesn’t have a clue about datums and map projections.

WLIA Task Force The WLIA Task Force on Wisconsin Coordinate Systems was formed early this year to address this and other issues associated with location referencing in Wisconsin. A question that emerged: Can the WCCS be re-designed so that: 1.There is no need to change the ellipsoid from GRS 80. That is, there will be one datum for all projections. 2.Coordinate differences between the existing and re- designed systems will be within negligible bounds. In this way, legacy databases and records will not have to be modified.

Leave the ellipsoid where it is and enlarge only the map projection surface. This way, the ellipsoid factor and the scale factor are nearly inverses of one another and their product = 1. Map Projection Surface

Approach to Lambert Re-Design Two strategies: 1.Make the original and re-designed map projection surfaces be identical in three- dimensional space. –This will cause the latitude of the central parallel (  0 ) to change. –Challenge: Finding  0. 2.Hold  0 constant. –This will cause the original and re-designed map projection surfaces to be dissimilar. –Challenge: Finding k 0.

Approach to Strategy 1 Work in geocentric coordinates (3D rectangular). Use analytical geometry. Find equations of the line that is the projection of the central meridian. Find the point of tangency between GRS 80 ellipsoid and a line parallel with the above line. Convert X,Y,Z of this point to ,,h.  is the latitude of the central parallel.

Geocentric / Geodetic Coordinates Geocentric coordinates are based upon a 3D right-handed system with origin at ellipsoid center, XY plane is the equatorial plane, +X axis passes through = 0 , +Y axis passes through = 90  E. For any point, there are direct and inverse transformations between X,Y,Z and ,,h.

Approach to Strategy 1 Profile through GRS80, enlarged ellipsoid, and original map projection surface at 0 : GRS 80 Enlarged ellipsoid Map projection surface Geodetic coords =  1, 0, 0; Compute X,Y,Z Geodetic coords =  2, 0, 0; Compute X,Y,Z Compute equations for this line. Parallel Find X,Y,Z of point of tangency. Transform to  0, 0, h. NOTE: Two sets of geodetic coordinates; one set of geocentric coordinates.

Approach to Strategy 1 To find k 0 : GRS 80 Enlarged ellipsoid Map projection surface Compute perpendicular distance between these two lines. D where R is the radius in the meridian of GRS 80 at  0. Ellipsoid Normal Minor Axis R

Approach to Strategy 1 There will be discrepancies because the two ellipsoids do not have the same shape. Compute best fit translation in Y (change in false northing) and scale from sets of coordinates of points in both the original and re-designed systems. –Points should be well-distributed across geographic extent. Apply these best fits to final re-designed parameters.

Dane County Test of Lambert Methodology (Strategy 1)  X = m;  Y = 0.000m  X = m;  Y = m  X = m;  Y = m  X = m;  Y = m  X = m;  Y = 0.000m

Approach to Transverse Mercator Re-Design Hold all parameters initially constant except k 0. Compute new k 0 in manner similar to that for Lambert re-design. Compute best fits for translation in Y (false northing) and scale. Apply best fits to final parameters. NOTE: Cannot hold map projection surface identical because the 2 cylinders have different shapes.

Lincoln County Test of Transverse Mercator Methodology  X = m;  Y = m  X = m;  Y = m

Conclusions Under the re-design, all WCCS would have a single, common datum based upon the GRS 80 ellipsoid. Initial tests indicate that WCCS can be re- designed to within 5mm or better. The WLIA Task Force has deemed 5mm to be a negligible difference. The WLIA Task Force is recommending re- design.