Projection Computations in PLSS Applications

**Raymond J. Hintz
Boardman Hall
Department of Surveying Engineering
University of Maine
Orono, Maine 04469
(207) 581-2189**

**Jerry L. Wahl &
Corwyn J. Rodine
USDI-BLM Eastern States Office
Division of Cadastral Surveys
Alexandria, Virginia**

The efficient automation of reduction of survey measurements to a state plane grid system in a PC environment has been available for a number of years. The validity of this procedure in surveys covering small areas has been fully documented. This paper looks at attempting to quantify when the limits of a projection system are reached due to a survey covering a large area. The effect of different amounts and quality of elevation information will also be examined. This limit will examined in two ways. The first method will be in the examination of a large survey having its measurements reduced to state plane and then processed through a least squares analysis. The same data will then be processed through a completely geodetic least squares analysis. Comparisons will be made between both real and fictitious data sets. The second method will examine computational Public Land Survey functions such as proportioning and subdivision in both the state plane and the full geodetic computational framework. Again both real and fictitious data sets will be used to examine if and when a projection system approach can lead to problems.

Survey data is measured on the real earth. Since the real earth is an undulating surface, horizontal position (latitude, longitude) requires the reduction of horizontal distance measurements to the ellipsoid surface using the so-called elevation factor.

**Traditional surveys can be processed in one of four ways:**

(1) **Assumed coordinates** - No reduction of measurements is made
at all. Unfortunately this assumes the earth is flat, i.e., meridians do
not converge converge toward the poles and latitudinal lines are not curved.
While possibly applicable for surveys where only very relative distances
and bearings are required, it is very unsuitable for a survey within the
USPLS where bearings are mean geodetic, and thus comparisons will not be
made to this type of coordinate reduction system. It is the opinion of
the authors that this assumption of coordinates is directly related to
the proliferation of survey software systems which make this assumption.
One should realize assuming a geodetic coordinate origin for a survey (scaled
from a USGS quadrangle map) is no more difficult than assuming plane coordinates
of 10000 N, 10000 E.

(2) **State plane or UTM coordinates** - This paper will assume that
proper software has been developed for reduction of data to the state plane
grid using individual scale factors, convergence angles, T-t (second term)
corrections, and some form of elevation reduction (Hintz, 1989). It will
later examined how varying levels of elevation information will affect
final results in both real and fictitious data sets.

(3) **Geodetic coordinates** - If there is "round-off"
in the reductions based on state plane coordinates, differences will be
found between the coordinates generated in (2) and computations performed
in full latitude / longitude.

(4) **Geocentric coordinates** - Due to the impact the Global Positioning
System (GPS) has had on the surveying community, it is possible to reduce
even conventional survey measurements in this type of coordinate system.
Obviously upon completion of the data processing final coordinates will
be converted back to both geodetic and state plane values.

The elevation reduction is of special importance because the product of a cadastral type survey is not specifically elevation. Comparisons will be made based on the following elevation information:

- One overall mean project elevation for a job.
- Individual elevation differences generated by reducing slope distance and zenith angles to an elevation component will be analyzed in a one-dimensional least squares. The case of measured height of instruments (hi) / height of targets (ht) and assumed hi's / ht's will both be considered.
- Performed a simultaneous three-dimensional least squares analysis of both situations defined in (2).

The purpose of this multitude of comparisons is to study if there is any danger in using state plane coordinates (with proper reductions) in a cadastral survey specific to the USPLS. It will also investigate the necessity and completeness of the elevation information - project mean vs. individual elevations with assumed hi's & ht's vs. individual elevations with measured hi's & ht's. This article is not designed to be a theoretical discussion of roundoff in the state plane reductions. Instead it is a series of practical examples indicating if and when problems occur.

All software used in this study has been the product of cooperative
work between the Bureau of Land Management (BLM) and the Department of
Surveying Engineering at the University of Maine. The survey software developed
for the dependent resurvey is known as Cadastral Measurement Management
(CMM) (*Hintz, 1991; Wahl, 1992*) and the survey data used in the
study was primarily collected using the Cadastral Electronic Field Book
(CEFB) (*Wahl, 1991; Hintz, 1992*).

The focus of this study is not a geodetic control survey. The product of a cadastral survey within the USPLS is concerned with the relative changes in latitude and longitude between stations (not their absolute values).

The first data set (Antitem) was collected by a BLM field crew from the Eastern States Office. The site was the Antitem battlefield in southern Maryland. The site only varies in elevation by approximately 100 ft. The data set encompasses an area 2 miles E-W by 2 miles N-S. HI's and ht's were measured for all data. Inadvertently the project elevation used was within 20 ft. of the highest elevation derived from the survey.

The second data set (G786) is in western Montana, and was collected by a field crew from the Montana BLM Trout Creek project office. It is very rugged terrain and varies in elevation from 2200 to 3200 ft. The selected average project datum was 2800 ft. The area covers 2 miles E-W by 3 miles N-S. HI's and ht's were not measured and thus are assumed at project default values.

In both cases error estimates of measurements for the least squares analysis were derived from the repetition errors resulting from the field measurements plus a user defined "add-on" which models error (such as setup error) which is not modeled by repetition. Multiple azimuths from solar observations were included. Only one horizontal control station and one benchmark were used so that error in control coordinates would not contribute to any comparative analyses.

Comparing the coordinate results of the state plane least squares analysis to the full geodetic mode generated final horizontal coordinates to better than 0.001 ft. This proves for typical survey data sets the state plane reductions result in favorable results to a full geodetic solution well beyond any survey precision level.

**Comparing the project elevation** results to the results from a
2D adjustment with individual elevations derived from a 1D adjustment of
elevation differences resulted in a maximum coordinate comparison of 0.017
ft. Of more interesting analysis is that stations in that general area
all shifted about the same amounts. To confirm this adjusted geodetic mean
bearings and ground distances were compared for all traverse lines. In
this comparison the maximum bearing shift was 5 seconds (on a 110 ft line
so very negligible) and the maximum adjusted distance change was 0.002
ft.

The conclusion to be drawn from this is that for conventional survey measurements the correct use of state plane reductions results in a comparison to the full geodetic process which is several magnitudes beyond one's measuring ability in surveying. The use of a project elevation in relatively flat terrain causes no apparent error when compared to results from a fully elevation controlled system. Finally, the measurement of hi's and ht's results in no significant improvement in horizontal positions since in this case even using individual elevations was "overkill" in production of final results. The specific conclusion to be drawn is that measurement of hi's and ht's is only required in a cadastral survey if the elevations of corners to a survey precision is required (a very unusual requirement).

The next comparison was of the results of the 2D adjustment with full elevation control derived from a 1D vertical adjustment (referred to subsequently as a 2D/1D approach) to horizontal coordinate results of a full 3-D least squares analysis. The explicit difference is that in the 3-D approach slope distances and zenith angles are treated as measurements. In the 2D/1D approach these measurements are reduced to horizontal and vertical components, and analyzed separately.

The maximum coordinate difference between the two analysis techniques was 0.022 ft. Again this shift was common to all coordinates in this general vicinity. In comparing adjusted geodetic mean bearings and ground distances of the traverse legs the maximum differences were 3 seconds and 0.004 ft. respectively. While again these differences are negligible part off the difference is attributable to an error estimate of an elevation difference in a 1D analysis is not directly comparable to treating slope distances and zenith angles as separate measurements, each with unique error estimates, in the 3D analysis.

Since the comparison of project elevation to individual elevation results was so remarkable the comparison of measured hi's/ht's results to assumed hi's/ht's results was deemed unnecessary.

Due to the undulating terrain of this project it was felt this would be an excellent test of the errors that result from a use of an overall average project elevation instead of individual elevations.

The coordinate comparison yielded a maximum difference of 0.11 ft. which is an accumulation of the error due to a single project mean elevation. Again the shifts were very consistent in a specific area of the traverse.

When adjusted geodetic mean bearings and ground distances of the traverse lines were compared the maximum differences were 5 seconds and 0.022 ft. respectively. The 5 seconds resulted in a ground distance of 0.001 ft as it was on a relatively short line. As expected, the maximum distance comparison was on a relatively long line (1100 ft.) which was at the lower end of the elevations for the project (2200 ft vs. a project mean elevation of 2800 ft.). Since this job was collected with CEFB, it should be noted that the effort to obtain least squares adjusted elevations using the 1D technique is less than one minute of additional effort in computer use. The conclusion in this comparison is that while even with 1000 ft. of elevation change the resultant changes were essentially negligible, the use of a data collector enables nearly effortless generation of reasonable elevations for the reduction to the ellipsoid without measurement of hi's and ht's.

The data was also subjected to a full 3-D least squares analysis in spite of the fact that the assumption of default hi's and ht's creates systematic errors in the data. To accommodate this factor the zenith angles were assigned unusually large error estimates. This forced the brunt of the systematic errors into the residuals in the zenith angles and not into the slope distances. The comparisons were dramatically worse as the worst coordinate comparison was 1.11 ft.

In comparing adjusted mean bearings and ground horizontal distances of traverse lines the worst comparisons were 68 seconds and 0.71 ft. respectively. The worst distance comparison had the largest zenith angle residual of 2400 seconds (40 minutes) in the 3D least squares analysis. It is obvious the major contributor to this residual is the assumed hi and ht.

The conclusion drawn from this comparison is in a 3D analysis large errors in the vertical component can affect the horizontal results. If one does not measure hi's and ht's in a survey (which has been proven to be not necessary by the Antietam results) one should not perform a 3D analysis of the data as the systematic errors in the assumed hi's and ht's can affect the quality of your horizontal results. In this situation the 2D/1D approach is preferred as it does not allow the vertical systematic errors directly harm the horizontal results, and the elevations derived from the 1D analysis are precise enough for the reduction of measurements to the ellipsoid.

The horizontal adjustments were both run in state plane and full geodetic mode. Again the maximum error in the coordinate comparisons were less than 0.001 ft.

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