L. Reduction Example: UTM
1. Preliminary Calculations
a. Survey Data
This chapter covers measurements reduction for a UTM cylindric zone.,
The survey data and UTM grid information for point Jerry are summarized in Figure L1 and Table L1.
Figure L1 Survey Data 
Table L1 Orthometric Heights 

Point 
H, ft 
Jerry  1177 
A1 
1805 
B7 
1340 
b. Lengths, Coordinates and Orthometric Heights
Convert lengths and orthometric heights to meters; compute approximate UTM coordinates of A1 and B7.
Point A1
Point B7
Point Jerry
From its datasheet: orthometric height is 358.6 m.
2. Ground Distance to Grid
a. Ground to Ellipsoid
Figure L2 Ground to Geodetic 
We computed geodetic distances in Chapter K  those can be converted to meters to use here.
However, we'll do the computations from scratch using Equation H2 to.
Jerry's 33.902 meter geoid height will be used as a project average.
Equation H2
(1) Geodetic distance JerryA1
(2) Geodetic distance JerryB7
b. Ellipsoid to Grid
Figure L3 Geodetic to Grid 
We'll use and compare the same reductions for UTM as we did for SPC. Table L2 shows the grid scale at each point and how they were obtained.
Table L2 Grid Scales 

Point 
Scale 
Source 
A1  1.00020 6404  software 
JerryA1 midpoint  1.00020 9086  Equation H4 
Jerry  1.00021 177  datasheet 
JerryB7 midpoint  1.00021 5856  Equation H4 
B7  1.00020 8953  software 
Jerry's CF from the datasheet is 1.00016 082.
Because the process is similar to SPC grid reduction, computations are not shown but results are summarized in Tables L3 and L4.
Table L3Line JerryA1 Grid Reduction ComparisonsD_{E} x k k
Grid Dist, m
Diff, m
Wtd Ave k 1.00020 9086 1305.3993
 
Average k 1.00020 9087
1305.3993
0.0000
Jerry's k 1.00021 177
1305.4028
+0.0035
Combined Factor D_{H} x CF 1.00016 082
1305.4000
+0.0007
Unlike SPC reduction, using Jerry's scale as a project average resulted in the largest differences. Average line scale worked well. Jerry's CF reduction had small (inconsequential?) differences. Using CF reduction could be sufficient if the elevations of points A1 and B7 represent the range across the project.
Table L4Line JerryB7 Grid Reduction ComparisonsD_{E} x k k
Grid Dist, m
Diff, m
Wtd Ave k 1.00021 4025
1568.8175
 
Average k 1.00021 0362
1568.8175
0.0000
Jerry's k 1.00021 177
1568.8111
0.0064
Combined Factor D_{H} x CF 1.00016 082
1568.8173
0.0002
3. Grid Direction
To convert the geodetic azimuth of line JerryA1 to grid, in addition to convergence we need to know:
 where the two meridians are with respect to each other, and
 if the arctochord correction is significant
a. Meridians
Convergence is the angle from Geodetic N to Grid N. Jerry's UTM convergence is 1°51'37.6". Because it is negative, Geodetic N is east of Grid N, Figure L4.
Figure L4
Geodetic and Grid North
We can also determine the relationships by comparing longitudes. Jerry's coordinates are in UTM Zone 16 whose CM is at 87° W longitude. Jerry is west of the CM at 89°43'53.76403" W longitude, Figure L5, placing Geodetic N east of Grid N
Figure L5
Longitudes
b. ArctoChord
The arctochord correction for line JerryA1 can be computed using either Equation H11 or H12. We'll use both to see if there's a difference.
Equation H11 Equation H12
Both equations require E_{o} which is a defining zone parameter: it is the east coordinate of the CM (false easting). E_{o} is 500,000.00 m for all UTM zones.
(1) Using Equation H11
Equation H11 is set up for metric coordinates so we don't need to convert any points. Set up in terms of point identifiers:
Coordinate differences:
Substituting into the correction equation:
(2) Using Equation H12
Equation H12 in terms of point identifiers is:
This equation requires r_{o} which 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 UTM Zone 16, r_{o} = 6,354,209.61 m
Substituting coordinate differences into the correction equation:
(3) Application
As expected (with fingers crossed) both equations yield the same result.
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
Another way to determine the correction's behavior is by examining the line's position relative to the CM. In a transverse cylindric projection the projected line is concave toward the CM. Line JerryA1 is west of the CM so it is concave to the east, Figure L6.
Figure K6
Line JerryA1 Concavity
Putting the pieces together, Figure K7, it's apparent that the correction's value must be added in Equation H7.
Figure K7
Grid Azimuth
Using Jerry's convergence and the 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. Although line JerryB7 is longer it is oriented more eastwest than JerryA1 so the magnitude of its arctochord correction should be smaller..
Since we've already shown that Equations H11 and H12 give that same results, we'll use the latter to compute the correction.
As expected, because of the line orientation, its arctochord correction is smaller.
Figure K8 is a sketch of the angle with all parts in place.
Figure K8 Angle Correction 
The grid angle B7'JerryA1', β_{G} , is:
The total effect of the backsight and foresight arctochord corrections amounts to only 00.20" which may be small enough to disregard.
5. Summary
In this example, we looked at a few different grid reduction methods. Depending on the accuracy needs of the survey, simple methods may be fine to use 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 probably not significant enough to affect directions or angles.