# Erie Canal GPS Benchmarks prove the Earth is not Flat

Monday, August 29, 2022 - 23:55 | Author: wabis | Topics: FlatEarth, Geodesy, Analysis
The Erie Canal is a historic canal in upstate New York that runs east-west between the Hudson River and Lake Erie. When built, the 584 km canal was the second-longest in the world. The overall elevation difference is about 172 m. In this post I show how we can use the location data of GPS stations along the Erie Canal to measure the drop due to earths curvature.

## Expected versus Measured Drop

We can use the GPS Data of some stations along the Erie Canal to measure the locations of the stations in Earth Centered Earth Fixed ECEF cartesian coordinates. With such coordinates we have the locations of the GPS stations in space with respect to the center of the earth, the origin of the ECEF coordinate system. This locations have nothing to do with the shape of the earth. They are pure vectors that can be used to get the shape of the ground they are placed on.

As each location is given as the coordinates of a vector, we can use vector algebra to calculate the drop of one station with respect to the horizontal plane of the other station. If the earth is flat then this drop will be zero and we get only the elevation difference between the stations.

In the following analysis I used the raw GPS Data of station P0 = AH9234 at Buffalo at one end of the canal and station P18 = MZ0796 at Troy at the other end of the canal. But we can use any pair of stations to do the same measurements and calculations.

PID Location Measured Drop Expected Drop ΔDrop
AH9234 Buffalo 14,064.251 m 14,064.059 m 0.192 m
MZ0796 Troy 14,387.343 m 14,387.364 m −0.021 m

The table above shows that the Measured Drop matches the Expected Drop of the WGS84 globe model to cm accuracy!

The earth is measurably not flat.

## GPS Data

Below is a list of GPS Control Benchmarks along the Erie Canal that provide Geodetic data like the ECEF cartesian coordintes X,Y,Z, Elevation, Ellipsoid height and Geoid height. Using the Control Benchmarks coordinates it is possible to calculate the earth curvature drop and the various radii of the earth. A useful tool to calculate some ellipsoid radii is the WGS84 Distance, Azimuth and Radius Calculator.

Some Benchmarks are not located very near the water, so their elevation may be some meters above the water elevation. Such Benchmarks have the Name in parenthesis.

To make calculations like curvature drop and radii use the Geo-Data Visualisation and Calculator App. The Benchmark data are listet in the table at the bottom of the App. Because the Benchmarks are not located at the water surface, I measured the nearest water elevation in Google Earth. It is displayed in the table on the App page in the column labeled Additional Data.

PID Name USGS State/County Elev HEll HGeoid X Y Z Data Sheet
AH9234 906 3020 H BUFFALO NY/ERIE 176.5 141.322 -35.146 902080.915 -4593625.452 4317598.521 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AH9234
DE7803 (TOWN OF AMHERST MONUMENT 62) CLARENCE CENTER NY/ERIE 175.8 140.375 -35.386 912110.030 -4578411.782 4331533.017 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=DE7803
AE2178 LOCKPORT LOCKPORT NY/NIAGARA 181.763 146.085 -35.675 913179.466 -4568967.492 4341212.269 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AE2178
OG0490 (MEDINA) KNOWLESVILLE NY/ORLEANS 196.456 160.890 -35.571 939643.563 -4560522.245 4344443.747 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OG0490
OF1042 L 129 ROCHESTER WEST NY/MONROE 168.024 132.581 -35.441 996723.410 -4554321.684 4338223.528 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OF1042
DF5882 ROCHESTER OPS CNT CORS ARP PITTSFORD NY/MONROE 172.544* 137.142 -35.402 1000295.368 -4555059.500 4336643.828 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=DF5882
DI0614 PITTSFORD CORS ARP PITTSFORD NY/MONROE 148.889 113.478 -35.355 1007672.284 -4554836.560 4335145.129 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=DI0614
AE2168 LOCK 29 MACEDON NY/WAYNE 136.200 101.098 -35.106 1029955.952 -4552039.040 4332843.226 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AE2168
DI0626 WATERLOO CORS ARP SENECA FALLS NY/SENECA 144.3191 109.959 -34.360 1064476.707 -4557029.058 4319337.374 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=DI0626
NB2147 (TT 40 R) BURDETT NY/SCHUYLER 225.727 192.564 -33.171 1071592.861 -4591438.100 4281348.164 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=NB2147
AB3841 (LANSING) LUDLOWVILLE NY/TOMPKINS 118.6 85.324 -33.357 1095086.670 -4577565.021 4290031.370 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AB3841
AB3847 (WEEDSPORT) WEEDSPORT NY/CAYUGA 121.940 87.742 -34.202 1083941.240 -4541153.405 4331088.553 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AB3847
AB3840 LOCK 24 BALDWINSVILLE NY/ONONDAGA 115.0 81.075 -34.037 1100875.832 -4528281.869 4340212.925 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AB3840
OF1307 (FULTON) FULTON NY/OSWEGO 134.995 100.575 -34.415 1093018.104 -4516439.598 4354447.421 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OF1307
AB3837 (CENTRAL SQ) CENTRAL SQUARE NY/OSWEGO 135.139 101.270 -33.872 1115086.608 -4514620.921 4350762.282 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AB3837
OE0979 (G 34 RESET) CANASTOTA NY/MADISON 131.867 98.937 -32.929 1148304.377 -4522709.112 4333799.143 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OE0979
OE1686 (U 465) ROME NY/ONEIDA 158.598 126.181 -32.394 1175554.457 -4507489.155 4342324.091 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OE1686
OE1692 (Z 465) UTICA WEST NY/ONEIDA 130.332 98.158 -32.222 1183787.456 -4510086.628 4337379.662 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=OE1692
AA7945 UTICA UTICA EAST NY/ONEIDA 132.559 100.567 -32.006 1192050.672 -4509945.418 4335280.551 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AA7945
DI0464 HERKIMER CORS ARP HERKIMER NY/HERKIMER 126.896 95.284 -31.600 1209173.580 -4511392.211 4329062.840 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=DI0464
AA7916 GPS 2G93020 RANDALL NY/MONTGOMERY 89.351 58.419 -30.939 1258876.263 -4502846.227 4323770.294 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=AA7916
NA1829 T 444 AMSTERDAM NY/MONTGOMERY 84.663 53.753 -30.908 1277707.613 -4500143.208 4321071.065 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=NA1829
MZ1442 (UNION RM 2) SCHENECTADY NY/SCHENECTADY 83.470 52.199 -31.263 1296985.618 -4502681.363 4312729.432 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=MZ1442
MZ07963 WATERFORD RM 2 TROY NORTH NY/SARATOGA 10.855 -20.4412 -31.296 1317679.236 -4498882.876 4310328.535 https://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?PidBox=MZ0796

1) Elevation is calculted using the following equation from the data in the data sheet. Elevations without superscript1 are taken from the data sheet (ORTHO HEIGHT).

 (1) Elev = HEll − HGeoid

2) Ellipsoid height HEll is calculated from Elevation Elev and Geoid height HGeoid as follows:

 (2) HEll = Elev + HGeoid

3) Classic Horz and Vert Control used, where X,Y,Z are calculated using the WGS84 Calculator from Latitude, Longitude and Elevation. All other Controls used are GPS sites, which natively provide X,Y,Z coordinates.

Elevation Profile Erie Canal 1832

In elevation profiles like the image above the curvature of the earth is straightened, because it is irrelevant for local elevation measurements. This does not mean that the earth is flat as the measurements on this page prove.

## Measured Drop

Given 2 location vectors we can use the following vector equation to calculate the measured drop $Z_m$ at the far location with respect to the horizontal plane at the observer location:

(3)
where'
 $Z_{m,far}$ ' =' 'measured drop at the far location $\vec P_{obs}$ ' =' 'GPS vector of the observer station $\vec P_{far}$ ' =' 'GPS vector of the far station $\hat u_{obs}$ ' =' 'unit up-vector at the observer location, which is perpendicular to the ellipsoid surface at $P_{obs}$

Note: due to the ellipsoidal shape of the earth the up vector $\hat u_{obs}$ is not parallel to the vector $\vec P_{obs}$ except at the equator and the poles.

We can use the Geo-Data Visualisation and Calculator App to get the measured drops as calculated above. The drop values are displayed at Hlvl:

(4)
(5)

This are the measured geometrical drops of a far station from the plane that is tangent at the observer station using the raw GPS coordinates of the stations. The different drops at the 2 locations is due to the elevation differences of the GPS receivers and the ellipsoidal shape of the earth (rather than a perfect sphere).

## Expected Drop

I use the math of the WGS84 globe model to calculate the expected drop. This model uses a reference ellipsoid as the first approximation of the shape of the earth. To calculate the expected drop $Z_e$ at the far station, the directional radius of curvature of the ellipsoid at the observer station is used, although the curvature is not quite constant between the stations. But it can be shown that this approach is accurate enough for this calculation.

(6)
where'
 $Z_{e,far}$ ' =' 'expected drop at the far station, assuming the earth is the reference ellipsoid of WGS84 $R_{obs}$ ' =' 'directional radius of curvature of the reference ellipsoid at observer station in the direction to the far station $d$ ' =' 'distance between the stations along the ellipsoid surface $\Delta H_{ell}$ ' =' '$H_{ell,0} - H_{ell,18}$ = difference between the ellipsoid heights of the 2 stations. Use positive value for drop at station 18, negative for drop at station 0.

The radius of curvature of an ellipsoid at a certain location depends on the direction it is measured. The radii $R_{obs}$ at each of the stations and the surface distance $d$ is calculated with the WGS84 Distance, Azimuth and Radius Calculator, by entering the coordinates of the 2 stations and then setting the ellipsoid heights to 0.

(7)
(8)
(9)

To get the drop between the stations we have to take the ellipsoid heights $H_{ell,i}$ into account. I get the ellipsoid height from the data sheets of the benchmarks:

(10)

Now I have all values to calculate the expected drops at each station using the equation (6):

(11)
(12)