DATA CORRECTIONS FOR REAL WORLD SURVEYING APPLICATIONS

Prepared by: Devon McDonough

SET 490 – Senior Project in Surveying May 14th, 2014

Abstract: The intent of this report is to provide an example of a real world application of grid-to-ground geodetic corrections that are already calculated in today’s world by Global Positioning System (GPS) instrument software. Surveyors today take this technology for granted and often overlook the importance of these calculations. The general goal of this report is to show how grid scale and azimuth corrections can impact a physical survey. There comes times where large scale surveys must be conducted using trigonometric leveling (due to lack of GPS satellite communication) and corrections must be applied to the observations to compensate for the curvature of the Earth and the state plane coordinate system’s scaling when converting from ground to grid coordinates.

2 | P a g e

Table of Contents Introduction ………………………………………………………………………………………………………………………………Page 3 History of the Parcel ………………………………………………………………………………………………………………….Page 3

Figure 1 ………………………………………………………………………………………………………………………..Page 4 Concepts and Basis for Calculations …………………………………………………………………………….……………Page 5

Figure 2…………………………………………………………………………………………………………………………Page 5 Figure 3 ……………………………………………………………………………………………………….……………….Page 6 Figure 4 ……………………………………………………………………………………………………….……………….Page 7

Results ………………………………………………………………………………………………………………………………..……Page 8 Table 1 ………………………………………………………………………………………………………..……………….Page 8 Table 2 ………………………………………………………………………………………………………..……………….Page 9 Figure 5 …………………………………………………………………………………………………….………………….Page 9 Table 3 ………………………………………………………………………………………………………..……………….Page 10

Conclusion ………………………………………………………………………………………………………………….………….…Page 11 Appendix ………….………………………………………………………………………………………………………………….……Page 12 References ……….……………………………………………………………………………………………………………….………Page 13

3 | P a g e

Introduction: With the advent of modern technologies, most work that a land surveyor conducts today is measured electronically and corrections are instantaneously calculated and quickly harvested from the instrument being operated. Surveyors tend to take these technologies for granted, as the evolution of electronics and the accessibility to vast amounts of data have greatly streamlined and simplified the role of the professional. However, the corrections applied to observations still may have to be processed manually when these modern measuring devices fail to yield accurate and reliable values. Grid Scale Factors (GSF) and conversions from Grid Azimuth to Geodetic Azimuths are two of the essential correction factors a surveyor would have to apply to any field observations. Most professionals rarely apply these corrections because of the size of the projects the survey work is being completed for. For small boundary surveys, for example one of less than 5 acres, the correction factors that would be applied to the observations would be inconsequential because the curvature of the Earth is so minute that over such a short distance it would not have a noticeable effect any of the measurements. For large scale surveys, such as the parcel observed in this report (in which case is almost 1,000 acres), must have corrections applied to the observations because the curvature of the Earth will significantly affect any ground observations. The parcel under analysis is approximately 970 acres in size and is located in Byram Township, Sussex County, New Jersey (known from here on as “the parcel”) and is currently owned by the Boy Scouts of America. Upon a site visitation on May 5th, 2014, the parcel it was observed to be bounded to the south by Interstate Highway Route 80 and to the east by Sussex County Route 604 as well as having several lakes, streams and mountains on site. This project will observe the correction factors as they are applied to two boundary lines of this specific parcel, as some of the property boundaries are in excess of 3,000 feet long and the elevation changes on site are well over 500 feet. The extreme separation of property corners, both horizontal and vertical, will show the effect of the curvature of the Earth and the importance of the GSF that a smaller parcel would not. History of the Parcel: Also known as Block 378, Lot 1 and Block 384, Lot 1 in Byram Township, the parcel is a densely forested camp ground for the Boy Scouts organization. The land is in the process of various building upgrades to compliment several cabins and utility buildings already constructed on site. An original survey map was provided by the owner, dated April 1946 by Snook and Hardin, which indicated that all of the six main property corners were monumented, either by stone heaps or concrete monuments. Prior to the site visit, an examination of site elevations was conducted using the 2011 Tranquility Quadrangle map produced by the United States Geological Survey, which determined that there was an elevation change of approximately 500 feet in various locations throughout the site (National Geodetic Survey). Both the map provided and the description from Deed Book 868 Page 398, dated May 1st, 1970 (Sussex County Clerk's Office), were in agreement with both the metes and bounds and the calls for the monumentation at the property corners. Since the survey conducted by Snook and Hardin in 1946, the New Jersey Department of Transportation (NJDOT) purchased right-of-way for the construction of Interstate Route 80, which then divided the main parcel into smaller portions and demolished the most southerly corner, as seen in Figure 1.

4 | P a g e

Figure 1: Map of the Boy Scouts Property showing Interstate Route 80 (thick dark line from top to

bottom) and Sussex County Route 604 (thin dark line left to right on the bottom of the page) Using NJDOT maps and baselines, the approximate location of the parcel was determined (as the deed and survey did not reference state plane coordinates for any corners or monuments). Once approximate coordinates were determined, and by using a handheld GPS unit, the existing monuments were searched for and subsequently found prior to any traversing and static GPS observations (see Figures A and B in the Appendix for examples of the monumentation found). Due to the dense tree cover and remote locations of the property corners, observations using a Real-Time Kinematic (RTK) GPS were unavailable to be conducted. The other options presented were traversing using trigonometric leveling or by conduction static GPS observations at each corner. The most time and cost effective of the two in this situation was static GPS observations. The traversing was not a valid option as the rugged topography and long distances between corners would have consisted of surveying for several days or possibly weeks. Instead, the static observations consisted of setting up a GPS unit on the corners for approximately a half hour, until the data collected was sufficient enough for a reliable reading. It is these numbers that are automatically corrected by the GPS receiver.

5 | P a g e

Concepts and Basis for Calculations: The first calculation under review is that for the reduction using the Grid Scale Factors. For this example, two boundary lines were used: Line 1, the longest on site, which runs from the westerly corner to northerly corner and Line 2, which runs from the same northerly corner to the most easterly corner, as seen in Figure 2.

Figure 2: Analysis for the Grid Scale factors include Line 1 (black) and Line 2 (blue)

(Source: Google Maps) At each corner, there is a different Grid Scale Factor (GSF). In order to apply a scale factor to a line, especially one the length of Line 1, the following formula should be used:

Eq. 1:

Where: KGSF = GSF of a line between points A and B KA = GSF at Corner A KB = GSF at Corner B KM = GSF at line’s midpoint

6 | P a g e

The value of K1 will result in the GSF for Line 1, which can also be approximated by just the average value of the scales at each corner. This correction is for distortion due to the projection of the ellipsoid onto the state plane grid. In New Jersey, the value of K can also be calculated by using the following equation (New Jersey Department of Transportation):

Eq. 2: Where: E = Given NJSPCS83 easting where the GSF is to be calculated, in meters KGSF = GSF at the given NJSPCS83 easting Next is the elevation factor, which reduces the ground readings onto that of the ellipsoid. This is described by the following formula:

Eq. 3:

Where: Sgeodetic = Geodetic distance D = Horizontal distance (measured) R = Mean Radius of the Earth H = Mean elevation (refer to Figure 3) N = Mean geoid height (refer to Figure 3)

Figure 3: The Relationship of the Ground Surface Compared to the Ellipsoid & Geoid Surfaces (Fraczek)

7 | P a g e

Figure 4: Grid Scale Factor (GSF) is used to reduce the ellipsoidal distance to the grid, whereas the

Elevation Factor is used to reduce ground distances onto the ellipsoid. Together, the equations reduce ground distances onto the grid (New Jersey Department of Transportation)

To transform the ground distance to that of the grid, the ellipsoidal distance (Sgeodetic) is multiplied by the ground scale factor which produces the grid distance, as seen in Eq. 4.

Eq. 4: When converting to grid coordinates from ground coordinates, it is important for the surveyor to know the conversion from grid azimuth to geodetic azimuth. Eq. 5 shows the relationship between the two azimuths:

Eq. 5:

Where: AZgrid = Grid azimuth AZgeodetic = Geodetic azimuth γ = Meridian convergence (t-T) = Arc to chord correction The grid azimuth can be determined by calculating the inverse between two points on a grid. The geodetic azimuth, if not calculated from the grid azimuth, is often given by NGS or by GPS observations. Datasheets from NGS will also include the meridian convergence (New Jersey Department of Transportation). The key element for the conversion from geodetic to grid azimuths is the (t-T) correction. The reason for the (t-T) correction is that the line from one point to another is in reality an arc on the ground (a curved object) and not a straight line as assumed on the grid. Table 1 shows the approximate

8 | P a g e

(t-T) correction for various distances from the central meridian in New Jersey. The further the distance is from the central meridian (where there are no corrections necessary), the higher the correction that is to be applied when converting from geodetic azimuth to grid azimuth.

Average Distance of the line from the Central Meridian (m)

ΔN 150,000 100,000 50,000 0

2 km 0.8" 0.5" 0.3" 0"

5 km 1.9" 1.3" 0.6" 0"

10 km 3.8" 2.5" 1.3" 0"

20 km 7.6" 5.1" 2.5" 0"

Table 1: The Magnitude for Correction from the Central Meridian for New Jersey (New Jersey Department of Transportation)

To calculate the (t-T) correction, which will be in seconds of arc, Eq. 6, 7 & 8 below must be used:

Eq. 6:

Eq. 7:

Eq. 8:

Where: (t-T)” = Arc to chord corrections (seconds of arc) N1 = Northing of point 1 N2 = Northing of point 2 E1 = Easting of point 1 E2 = Easting of point 2 E0 = 150,000 meters As with the Grid Scale Factors, the correction for azimuths can be neglected for small surveys, as the effect is minimal. Also, the sign of the correction is related to the north azimuth (New Jersey Department of Transportation). Results: Modern GPS instruments already factor in the corrections for angles and scale factors. When traversing using a total station and reflector, or a theodolite and chains during the early days of surveying, the curvature of the Earth and Azimuth corrections are not automatically applied to the observations. When the Grid Scale Factor is introduced into Eq. 4, the measured ground distance is reduced to that on the grid. As previously mentioned, this correction is insignificant to short boundary lines however measurements similar to Lines 1 and 2 will be corrected by a considerable amount, as seen in Table 2:

9 | P a g e

Corrections for Lines 1 and 2

Line 1 Line 2

Ground Distance (m): 2,499.36 1,449.87

Grid Distance (m): 2,499.02 1,449.68

Difference (m): 0.34 0.19

Table 2: The ground-to-grid corrections for Lines 1 and 2 calculated using Eq. 1 and Eq. 2 (See tables

below for actual values used)

Though 0.34 and 0.19 meters may seem hardly noticeable on the overall scale of the property, with the difference on a length of several kilometers, any area calculations will be skewed and, if the survey is traversed on the ground, it may unintentionally create gores and overlaps in parcels due to the line being too short. This is especially the case if there are no calls for monuments at the corners. As the boundary line becomes longer, the difference will grow higher, and when retracing surveys that have boundary lines several miles long, this can cause confusion when searching for an iron pipe, where one may think is in the correct location (using the grid distance), where in reality it is a meter or two further away because the grid distance was not scaled to that of the ground distance.

Figure 5: Concept of Ground versus Grid Coordinates (Ghilani)

As can be seen, the grid distance is considerably less than the ground distance. Figure 5 shows a representation of why this occurs. The ground distance is measured on the surface of the Earth, which has various changes in elevations from one point to another and it also does not account for the curvature of the Earth. By projecting it onto the Earth, the new grid distance is absolute and requires no compensation for the curvature of the Earth or changes in elevation. It is strictly a flat working surface that is slightly distorted so that elevation changes

10 | P a g e

Grid Scale Factor for Line 1 from Corner A to Corner B (West-East Line)

KA = 0.999907360 (Grid scale factor at Corner A) Km = 0.999906745 (Grid scale factor at the line's mid-point)

KB = 0.999906130 (Grid scale factor at Corner B) K1 = 0.999906745 (Grid scale factor of Line 1)

Grid Scale Factor for Line 2 from Corner B to Corner C (North-South Line)

KB = 0.999906130 (Grid scale factor at Corner B) Km = 0.999906020 (Grid scale factor at the line's mid-point)

KC = 0.999905910 (Grid scale factor at Corner C) K2 = 0.999906020 (Grid scale factor of Line 2)

Elevation Factor for Line 1 from Corner A to Corner B (West-East Line)

D = 2,499.360 (Measured Horizontal Distance, in meters) H = 304.648 (Mean Elevation, above MSL, in meters) N = -32.8835 (Mean Geoid Height, in meters)

R = 6,372,000 (Mean Radius of the Earth, in meters) S1 = 2,499.253 (Geodetic Distance, in meters)

Elevation Factor for Line 2 from Corner B to Corner C (North-South Line)

D = 1,449.870 (Measured Horizontal Distance, in meters) H = 277.825 (Mean Elevation, above MSL, in meters) N = -32.8595 (Mean Geoid Height, in meters)

R = 6,372,000 (Mean Radius of the Earth, in meters) S2 = 1449.814 (Geodetic Distance, in meters)

For the azimuth corrections, the (t-T) values will be the focus of the following calculations as typically all other measurements are known values received by the NGS. The coordinates for each corner (in meters) is necessary to perform these calculations, as seen in Table 3.

Coordinates (NJSPCS 83)

NA = 232,661.91

EA = 125,547.15

NB = 233,968.46

EB = 127,677.58

NC = 232,578.02

EC = 128,088.16

Table 3: New Jersey State Plane Coordinates (1983) for Corners A, B & C

11 | P a g e

(t-T) Correction for Line 1 from Corner A to Corner B (West-East Line)

(t-T)" = -0.0776 ΔN12 = 1,306.55 ΔE12 = -23,387.64 E0 = 150,000.00

(t-T) Correction for Line 2 from Corner B to Corner C (North-South Line)

(t-T)" = -0.0781 ΔN23 = 1,390.44 ΔE23 = -22,117.13 E0 = 150,000.00

As seen in the calculations above for corners A, B and C, the (t-T) correction is relatively small. This is due to the proximity to the Central Meridian for New Jersey and the length of the lines, as the parcel is approximately 22 kilometers west of the Central Meridian (where there are no corrections needed) and the boundary lines are less than 3 kilometers in length. Though a correction of -0.07 seconds may not appear to be significant, over long distances this correction can alter an azimuth project by several feet. Conclusion: Large scale surveys produced using modern instruments can quickly be processed since corrections are often implemented in the instrument software. When this technology cannot be used, as seen with the Boy Scouts of America property, manual corrections need to be applied in order to work in a grid coordinate system. These simple corrections may not appear to be significant, but to a surveyor locating the boundary on the ground and determining where the proper corners are, differences of just fractions of an inch are important to note. As noted earlier in the report, these correction factors are often not applied to ordinary surveys, surveys where the traverses are small and the scale factor has no significance on the outcome of the data. However, with the introduction of large elevation differences and long boundary lines, the curvature of the Earth and the varying azimuths can distort surveys when projected from the ground. The calculations in the Results section for Lines 1 and 2 clearly show the significance of these corrections. It is important to note that the ground measurements cannot assume that trigonometric leveling is the exact elevation difference between two points. With short distances, this may be an acceptable practice, but the longer the traverse, the more correction that will be needed for the curvature of the Earth.

12 | P a g e

Appendix

Figure A: Concrete Monument Found Using Rough GPS Coordinates

Figure B: Stone Heap Found Using Rough GPS Coordinates

13 | P a g e

References Fraczek, Witold. Mean Sea Level, GPS, and the Geoid. July 2003.

<http://www.esri.com/news/arcuser/0703/geoid1of3.html>.

Ghilani, Charles. Where Theory Meets Practice: Grid versus Ground. September 2013.

<http://www.profsurv.com/magazine/article.aspx?i=71419>.

National Geodetic Survey. Computation of Geoid12 Geoid Height. n.d. <http://www.ngs.noaa.gov/cgi-

bin/GEOID_STUFF/geoid12_prompt1.prl>.

—. Geodetic to SPC. n.d. <http://www.ngs.noaa.gov/cgi-bin/spc_getpc.prl>.

—. National Geodetic Survey Data Explorer. n.d. <http://geodesy.noaa.gov/NGSDataExplorer/>.

—. SPC to Geodetic. n.d. <http://www.ngs.noaa.gov/cgi-bin/spc_getgp.prl>.

New Jersey Department of Transportation. Chapter 2: Control Surveys and State Plane Coordinate

Systems. 6 March 2007.

<http://www.state.nj.us/transportation/eng/documents/survey/Chapter2.shtm>.

Surveying. 6 November 2010. <http://dc141.4shared.com/doc/mZkuhtIP/preview.html>.

Sussex County Clerk's Office. "AiLIS Public Inquiry." n.d. Public Inquiry.

<http://sussex.landrecordsonline.com/>.

Wolf, Paul R. and Charles D. Ghilani. Elementary Surveying: An Introduction to Geomatics. Eleventh.

Upper Saddle River: Pearson Prentice Hall, 2006. Print.