K. Reduction Example: SPC
1. Preliminary Calculations
a. Survey Data
This chapter covers measurements reduction for an SPC conic zone. Distances and coordinates will be in survey feet.
The survey data and SPC grid information for point Jerry are summarized in Figure K1 and Table K1.
Figure K1 Survey Data 
Table K1
Orthometric Heights


Point 
H, ft 
Jerry 
1177 
A1 
1805 
B7 
1340 
b. Approximate Coordinates
Compute approximate SPC coordinates of A1 and B7.
Point A1
Point B7
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.
3. Grid Direction
To convert the geodetic azimuth of line JerryA1 to grid, we need to know:
 where the two meridians are with respect to each other, and
 if the arctochord correction should be applied
a. Meridians
Convergence is the angle from Geodetic N to Grid N. Jerry's SPC convergence is +0°11'03.9". Because it is positive, Geodetic N is west of Grid N, Figure K4.
Figure K4 Geodetic and Grid North 
Another way to determine the meridians' relationship is by comparing longitudes of Jerry and the CM. This requires knowledge of the zone parameters. The CMs of all three Wisconsin SPC zones are at 90°00'00"W longitude. Jerry's longitude is 89°43'53.76413"W which places it east of the CM. Because Jerry is east of the CM, Geodetic N west of Grid N, Figure K5.
Figure K5 Longitudes 
b. ArctoChord
The arctochord correction for line JerryA1 can be computed using either Equation H9 or H10. We'll use both to see if there's a difference.
Equation H9  
Equation H10 
Both equations require N_{o} which is the north coordinate of the CP at the CM. Even though a constant, it is not given as a zone parameter and must be computed using the projection equations. NCAT does not provide this information but the NAD 83 Coordinate Conversion workbook does.
Wis South SPC Zone N_{o} = 510,708.62 ft = 155,664.30 m
(1) Equation H9
Because Equation H9 is set up for metric coordinates,we'll need to convert point A1.
Equation H9 in terms of point identifiers:
Coordinate differences:
Substituting into the correction equation:
(2) Equation H10
Equation H10 in terms of point identifiers is:
This coordinates may be metric or Imperial. We'll stick to feet.
This equation requires r_{o} which, like N_{o}, is constant for for a zone but not given as a defining parameter. NCAT does not provide this information but the NAD 83 Coordinate Conversion workbook does.
For Wis SPC South zone, r_{o} = 20,920,156.06 ft.
Coordinate differences:
Substituting into the correction equation:
(3) Application
Both equations yield the same result (surprise, surprise).
Is the arctochord correction significant? That's up to the surveyor and the type of project. Even though it's less than a second, we'll apply it as it may affect rounding.
To compute the Grid Azimuth from Geodetic, convergence, and arctochord correction, use Equation H7.
Equation H7 
A sketch can also be used to determine how to apply the correction. Because the North coordinates of Jerry and A1 are less than N_{o}, the line is south of the CP which makes it concave northerly toward the CP, Figure K6.
Figure K6 Line JerryA1 Concavity 
The arctochord correction computed with Equations H9 and H10 is negative making it an angle to to the left which Figure K6 supports. Figure K7 shows all the elements to convert the Geodetic Azimuth to Grid.
Figure K7 Grid Azimuth 
Using Jerry's convergence and a negative arctochord correction, the Grid Az of JerryA1 is:
4. Ground to Grid Angle
To convert angle B7JerryA1 to grid, we need the arctochord correction for lines JerryA1 and JerryB7. We have the former, now must compute the latter. Line JerryB7 is longer and oriented more eastwest than JerryA1 so the magnitude of its arctochord correction should to be larger.
We've shown that both arctochord equations give the same results so we'll use Equation H10.
The correction for JerryB7 is larger than JerryA1's as expected.
At Jerry the correction to A1 is negative (angle left) and to B7 positive (angle right). The angle and corrections are shown in Figure K7.
Figure K7 Angle Corrections 
The grid angle B7'JerryA1', β_{G} , is:
The user must decide if combined bascksight and foresight arctochord correction is significant.
5. Summary
This example looked at a few different SPC grid reduction methods. Depending on the accuracy needs of the survey, simple methods may be sufficient instead of more complicated (and time consuming) ones. The decision should be based on examination of worsecase scenarios.
If the two lines of the example survey represent the extreme situations, we determined that
 distance reduction  Jerry's grid scale could be used for the entire project but each line should use its own average elevation
 arctochord correction  was not significant enough to affect directions or angles.