The document discusses calculating the average combined scale factor and convergence angle for state plane coordinate systems. It describes accessing a tool on the NGS website to input latitude and longitude values for project corners to determine the grid scale factor and gamma angle. The average combined scale factor is calculated by multiplying the elevation factor by the average grid scale factor. The convergence angle is also averaged from the four corners to allow converting between grid and geodetic bearings.

1. Surv 222 - Geodesy/State Plane STATE PLANE COORINATES Computing Project Average Combined Scale Factor and Convergence angle (Gamma)

2. Purpose To reduce ground measurements to project grid values by reducing to the ellipsoid and then scaling to the grid equivalent (two-step procedure) To calculate the value of convergence (Gamma angle) to allow determination of true north from grid north observations Surv 222 - Geodesy/State Plane

3. Features Calculations are made on NGS site using Geodetic Tool Kit, State Plane Conversion Routine Go to NGS site at: http://www.ngs.noaa.gov/ Select Geodetic Tool Kit (See slide) from left-hand side of menu Select ‘State Plane Coordinates’ (See Slide) Input the values for Latitude & Longitude for the four corners of perimeter rectangle encompassing project site Either select zone or let software determine zone Hit submit and program will process the data providing grid scale factor and convergence (gamma angle) for position Surv 222 - Geodesy/State Plane

4. NGS SITE Geodetic Tool Kit Surv222 Geodesy/State Plane

5. Conversion Program State Plane Coordinates Surv 222 - Geodesy/State Plane

6. Geodetic to SPC Surv 222 - Geodesy/State Plane

7. Conversion Output Grid Scale Factor & Convergence (Gamma Angle) Surv 222 - Geodesy/State Plane

8. Surv222 - Geodesy/State Plane Reduction of Ground Surveyed Distance to the Ellipsoid Derived by averaging the elevations throughout the survey area. This is an approximate value and should have 50% of area above and 50% below mean elevation, or alternatively on the perimeter if that is where you will be surveying, such as the perimeter of a section. Start by averaging the elevations at the top left, middle and right corners of mapping area; then middle left, middle of map and middle right; ending with lower left, lower middle and lower right.

9. Surv 222 - Geodesy/State Plane Determination of Elevation Factor Calculate the elevation factor using the formula (Ra/Ra+H+N) Where: H (average orthometric height above the geoid); N (geoid separation); Ra=20,902,000’ (Mean radius of the Earth) From Ghilani & Wolf Elementary Surveying, Section 20.8.1 Grid Reduction of distances. Since Geoid separation is around -20 meters (65’) in Washington, this component can be dismissed as insignificant in this calculation. Simplified, this means for local surveys, such as section line retracements that the ratio is simply 20,902,000/Mean elevation of site (20,902,000+average ortho height)

10. Determination of Elevation Factor From Ghilani & Wolf Elementary Surveying, 12th Edition Section 20.8.1 Surv 222 - Geodesy/State Plane

11. Combined Average Scale Factor Final step of calculation Multiply elevation factor x average grid factor Example 0.9999427 x 0.999921 = .9998637(CSF) Average the convergence angle at each of the 4 corners of the survey project area to use for converting grid bearing to geodetic bearings. The sign of the convergence angle is always from Grid azimuth to geodetic Azimuth. Example: Grid Azim.= N 01°48’15”W-01°56’53”(convergence) =N 00°08’38” E (Geodetic Azimuth) Surv 222 - Geodesy/State Plane