2. Ground Distance to Grid
a. Ground to Ellipsoid
Going from ground to the ellipsoid is independent of the grid system.
Figure K2 Ground to Geodetic 
Use Equation H2 to reduce ground to the ellipsoid
Equation H2
Te equation uses a line's average elevation and the average geoid height to determine the geodetic length on the ellipsoid.
R_{E} is 20.906 x 10^{6} ft.
Jerry's geoid height is 33.902 meters computed using the GEOID18 model. It is negative because the geoid is below the ellipsoid in Wisconsin.
Since we're working in feet, the geoid height must be converted from meters:
Geoid height generally does not vary significantly over an area of this size. Since Jerry is centrally located, 111.2 ft can be used as a project average.
(1) Geodetic distance JerryA1
(2) Geodetic distance JerryB7
b. Ellipsoid to Grid
Figure K3 Geodetic to Grid 
To go from ellipsoid to grid, the geodetic distance multiplied by the grid scale factor, k, Equation H3.
Equation H3
The grid scale in Equation H3 can be:
Jerry's scale used for the entire project
Average of the endpoint scales for each line
A weighted average scale, Equation H4, for each line based on scales at the mid and endpoints
Equation H4
We'll compute the grid distance each way and compare their results.
(1) Using Jerry's scale: k=0.99996 957.
Line
Grid Distance
JerryA1: JerryB7:
(2) Average scale for each line
Using software, scale at points A1 and B7 can be determined using their approximate coordinates. NGS's NCAT can be used for this as well as the NAD 83 Coordinate Conversion workbook.
Point 
Scale 
A1 
0.99996 8421 
B7 
0.99996 8894 
Multiplying each line's geodetic distance by its average scale:
Line 
Average k 
Grid Dist 
JerryA1 
0.99996 8996  4281.768 
JerryB7 
0.99996 9232  5145.760 
(3) Weighted average scale for each line
Use the same software to determine the scale at each line's midpoint; midpoint coordinates are the average of the endpoint coordinates.
Line 
Midpoint k 
JerryA1 
0.99996 8991 
JerryB7 
0.99996 9230 
Multiplying each line's geodetic length by its weighted average scale:
Line 
Weighted k 
Grid Dist 
JerryA1 
0.99996 8992  4281.768 
JerryB7 
0.99996 9231  5145.760 
(4) Combined Factor
Grid distance can also determined from multiplying ground distance by Jerry's Combined Factor: 0.99991 863, Equation H6.
Equation H6 
This simplified method does not require computing geodetic distances  the entire project is scaled by a single CF.
Line 
Grid Dist 
JerryA1 

JerryB7 
(5) Comparing results
Theoretically, the weighted average scale gives the best grid distance. In Table K2 it is used as the base to which the others are compared.
Table K2 Grid Reduction Comparisons 

JerryA1  JerryB7  
D_{E} x k 
Grid Dist, ft 
Diff, ft 
Grid Grid, ft 
Diff, ft 
Wtd Ave k 
4281.768 
  
5145.760    
Average k 
4281.768  0.000  5145.760  0.000 
Jerry's k 
4281.771 
+0.003 
5145.761 
+0.001 
Combined Factor  
D_{H}xCF 
4281.762 
0.006 
5145.781  +0.021 
The largest differences are for the CF grid distances. Considering there is a 255 foot elevation variation across the project, that's not surprising. The other differences are 0.003 ft or less. An acceptable level of accuracy might be achieved with Jerry's scale factor for all lines  it simplifies computations somewhat although using line averages increases the accuracy without too much more effort.
Depending on accuracy requirement, a worsecase scenario should be examined. Consider a line furthest from the control in direction of scale variation and/or at largest elevation difference. Any method acceptable for that line will be acceptable for the entire project.