# Rainy Lake Experiment: GPS Measurements

## Accuracy of common GPS

Common GPS receivers e.g. in your smartphone have an accuracy of only about 15 m horizontally and even worse vertically. So how can cm accuracy be achieved using GPS?

A GPS receiver uses the positions of at least 4 satellites to calculate its position. The calculated position accuracy depends on many factors: how many satellites can be received, their location distribution in space, the accuracy of their clocks, accuracy of the transmitted satellite position data (ephemeris) and atmospheric delays.

## Improving GPS accuracy with Differential GPS

The error factors are common for all receivers in the same vicinity. So if we have a receiver at an exactly known position, it can calculate its position vector from the GPS signals and take the difference to its known exact position to calculate the position error vector. This position error vector can then be transmitted to other GPS receivers which can correct their position vectors accordingly. This method is called Differential GPS.

There are multipe methods to get the position error.

We can use our own base station. That is a special GPS receiver placed at an exactly known position, capable of calculating the position error and transmiting it to other receivers in real time. You can get better than 1..5 m accuracy with this method.

We can use Satellite-based augmentation system (SBAS) to get the position error. Corrections are computed from ground station observations and then uploaded to geostationary satellites. This data is then broadcast to GPS devices which are equipped with corresponding receivers. Wide Area Augmentation System WAAS, EGNOS, and MSAS are examples of satellite-based augmentation systems. George's Magellan SporTrak Pro is a WAAS enabled GPS receiver.

The most accurate positions can be achieved by post-processing the vectors from the GPS receivers. Post-processing methods take place upon return to the office rather than in the field to take advantage of base station data available on the internet. Base station files are posted on the internet daily or hourly. The most accurate positions can be achieved with post-processed carrier phase differential GPS correction: 1..30 cm!

National Geodetic Survey NGS manages a network of Continuously Operating Reference Stations (CORS) that provide Global Navigation Satellite System (GNSS) data consisting of carrier phase and code range measurements in support of three dimensional positioning, meteorology, space weather, and geophysical applications throughout the United States, its territories, and a few foreign countries.

The accuracy of GPS can be further improved by using Survey grade GNSS receivers that are capable of receiving multiple GPS satellite systems like Navstar and GLONASS simultaneously.

## GPS Vectors

Throughout this report I call the locations directly measured using survey grade GNSS receivers and expressed in ECEF Cartesian coordinates GPS Vectors, to emphasize the fact, that this vectors have no relation to the shape of the earth.

## Coordinate Systems

A position in 3D space can be represented by a vector with 3 components, called the coordinates of the vector. There are many coordinate systems to represent the same 3D vector. All have in common, that a certain vector points from an origin to the same location in space. The length of the vector is the same in every coordinate system. Likewise is the distance from one vector to another the same in every coordinate system. Only the coordinate values are different in each coordinate system. To uniquely define a vector, you have to state in which coordinate system the coordinates of the vector are expressed.

The choice of the coordinate system used depends on the application. Common coordinate systems are:

• Cartesian Coordinate Systems like the Earth Centered Earth Fixed (ECEF) (x,y,z) coordinates used internally by GPS/GNSS for all calculations
• Spherical Coordinate Systems used in spherically symmetric applications where the vector components are represented with 2 angles and a radius (φ,θ,r)
• Geodetic (Ellipsoidal) Coordinate Systems used in geodesy where the vector components are represented as latitude, longitude and ellipsoid height (φ,λ,h)
• Cylindrical Coordinate Systems used in cylindrically symmetric applications like the Flat Earth where the vector components are represented with 2 angles and a height or latitude, longitude and height above the zero plane (φ,λ,h)

Note: although Geodetic and Cylindrical Coordinates both have a height component h, their value for a certain GPS Vector are not the same, because in Geodetic coordinates height is measured from the surface of an ellipsoid, while in Cylindrical (Flat Earth) coordinates height is measured from the base plane through the origin.

It is possible to transform a vector from one coordinate system into another system without changing the location the vector is pointing to and without changing the length of the vector or the distance between different vectors. See WGS84 Coordinate System for the transformation between ECEF Cartesian coordinates and Geodetic Ellipsoidal coordinates.

Projections on the other hand change the vectors, their length and the distances between them. If you have a vector, say in the Geodetic coordinate system of the Globe model, and simply use the same coordinate values in the Cylindrical coordinate system of the Flat Earth model, you are making a projection which changes the length of all vectors and the distances between them. So the Flat Earther claim, that Geodetic coordinates work the same on the Cylindrical coordinate system of the Flat Earth, is not true. That's the reason why all distances on the Flat Earth model are not the same as on the Globe model. The real distances are only correct on the Globe model.

## Vectors in ECEF Cartesian Coordinates

GPS Vectors: The locations calculated by GNSS receivers and error corrected with any of the methods mentioned at Improving GPS accuracy with Differential GPS are vectors in ECEF Cartesian (x,y,z) coordinates, not in geodetic coordinates latitude, longitude and ellipsoid height or geoid elevation.

GNSS receivers calculate the ECEF Cartesian coordinates (GPS Vectors) from the measured distances to multiple satellites with exactly known positions in space (in about 20,200 km altitude for the Navstar GPS system) using Multilateration. GNSS receivers can export the measured distances to the satellites and the vectors calculated from them in the RINEX data format for later processing on a PC.

A GPS Vector has its origin at the mass center of the earth and points to a certain location in space which may lie on the surface of the earth or anywhere above it. The ECEF Cartesian coordinate system is fixed to the earth. As the earth rotates, the coordinate system rotates with it.

So each measured target GPS Vector points to a position in 3D space, independent of the shape of the earth. Some satellites and the ISS, use the same coordinates gathered with GNSS receivers on board to calculate their positions in space [1]. They don't use Geodetic coordinates for their trajectory calculations.

For example, the water level GPS Vector of the observer location has the following (x,y,z) coordinates in the ECEF Cartesian coordinate system: (-243772.154, -4217822.457, 4762585.891). The length of this vector can be calculated using Pythagoras: l = √ x2 + y2 + z2 = 6,366,449.207 m. This is approximately the mean radius of the earth.

Note: because the location is not at sea level and the earth is not a perfect sphere but in the first approximation an Ellipsoid and more accurate a Geoid, the radius of curvature at any location on the surface of the earth diverges slightly from the length of the vector from the center of the earth to that location.

## Geodetic Coordinate System

The measured GPS Vectors in ECEF Cartesian coordinates are only transformed into Geodetic Ellipsoidal coordinates (latitude,longitude,elevation) for geodesy, mapping and navigation. Most calculations like GPS/GNSS internal calculations, trajectory calculations or air and space navigation calculations are done in ECEF Cartesian coordinates, because vector calculations are especially easy to carry out.

Note: Transforming a vector into another coordinate system is not a projection onto a Globe, Flat Earth or map. Coordinate System Transformations are not projections. Coordinate System Transformations retain the vector pointing to a certain location in space. Coordinate System Transformations retain the length of vectors and the distance between vectors. Projection Transformations do not retain the vectors. Distances between vectors do change. That's the reason why distances on the flat earth AE map and other projections of the whole earth onto a plane map are completely wrong and why distances on flat maps of small parts of the earth are only approximately correct.

## GPS Vectors and the Computer Model

All positions that are used in the Computer Model of the Rainy Lake Experiment are derived from the measured GPS Vectors in ECEF Cartesian coordinates, see Measuring the Targets.

Walter Bislin converted the GPS Vectors into local coordinates required by his Computer Model using his WGS84 Calculator. The WGS84 Calculator can also transform the GPS Vectors from the ECEF Cartesian coordinate system into the Geodetic coordinate system and vice versa. The transformations from the WGS84 Calculator into latitude, longitude and ellipsoid heights confirm the same data also provided by Jesse Kozlowski.

## Visualizing GPS Vectors

All target locations were measured using Differential GPS (DGPS). DGPS are enhancements to the Global Positioning System (GPS) which provide improved location accuracy from the 15-meter nominal GPS accuracy to about 1-3 cm in case of the best implementations. A Base Station or network of Base Stations calculates differential corrections for its own location and time and this correction signal is then transmitted to the mobile GNSS receiver which corrects its measurements accordingly, see Improving GPS accuracy with Differential GPS.

Jesse Kozlowski measured the GPS Vectors to the locations of each target and the observer locations with his Differential GPS Equipment as described at Measuring the Targets. All data is provided in ECEF Cartesian coordinates (GPS Vectors) plus Elevation (see Obtaining Elevations).

So we have a set of vectors from the center of the earth to the locations of all targets and observation locations in a coordinate system that has no relation to the shape of the earth and is not affected by refraction or perspective.

We can now import the GPS Vectors into a 3D software like the GNSS Data Viewer, written by Walter Bislin. The Viewer displays the GPS Vectors as white markers and allows to make calculations between the data points. The software allows to import and overlay an image onto the display to see how observations match the measured GPS Vectors.

Img 23: Overlay GPS Vectors with image of Bedford targets → App
Img 24: Overlay GPS Vectors with image of Tangent targets → App

If we look along the points at a shallow angle we can recognize that the Bedford target, ice and water level GPS Vectors do not lie on a plane but curve down. The radius of the curvature can be calculated with the software, see Measuring the Radius of the Earth.

We can also recognize that the targets on the images appear as higher than the GPS Vectors, as farther away they are from the observer. This is due to refraction and the images are consistent with the calculations of Apparent Lift due to Refraction. We can calculate the refraction from the deviation of the target images from the corresponding GPS Vectors, see Measuring Refraction.

## Measuring the Radius of the Earth

In the GNSS Data Viewer we can select 3 Bedford target markers of the same height or 3 water level markers and calculate the radius of the earth from the arc that this markers span. The ECEF Cartesian coordinates of all data points are listed in the Table at the bottom of the GNSS Data Viewer.

Img 25: Measuring the radius of the earth from 3 data points → App

In this example, the calculated radius of the earth from 3 selected markers is R3pt = 6025 km. That is only 363 km or 5.4% too less.

The distance between the data points is too small to get an accurate value for the radius of the earth with this method. A height deviation of only 1 cm results in a change of the calculated radius of Rerr = 71.1 km/cm. The height variation of the selected points in this example is dHel = 18 cm which results in an uncertainty for the radius of about ±9 · 71.1 km = ±640 km.

The exact radius of the earth depends on the location and the measuring direction, because the earth's surface is in a first approximation an Ellipsoid. The exact radius of the Ellipsoid at this location in the direction of the targets is calculated as Re12 = 6388 km.

We also have to consider that sea level of the earth forms a Geoid. The surface of the Rainy Lake is curved more than the reference Ellipsoid of the earth. The Ellipsoid height of the last water level position is about 24 cm less than the Ellipsoid height of the water level at the observers, see GNSS Ellipsoid and Height Data, due to variations in the gravitational field of the earth. So we can expect to measure a too small radius of the earth from the data points.

The GNSS Data Viewer calculates the radius as described in Method to calculate the Radius of the Earth from 3 Points.

Why are all data points sitting near each other in the Data Viewer when they are more than 1 km appart?

This is a highly zoomed in view almost perfectly along the targets. So due to perspective compression we can clearly see that the water level points are curving over a horizon. Water and ice level of the last 2 targets are beyond and below the horizon, indicated by brown horizontal ground lines. Targets nearer than the horizon are indicated by green ground lines.

P1 is the observation point for the Bedford targets. The thick glowing red lines lie on the eye level plane of the reference point Pref. Due to the curvature of the earth the points P1 and P2 lie below this plane. Remember, all data points are Vectors in ECEF Cartesian Coordinates that are independent of the shape of the earth. The points of equal height above the surface show a curve.

This can only be if the surface of the earth is curved. If the earth were flat then all points with the same height would lie on a plane.

For more examples of perspective compression see Comparison of Globe and Flat-Earth Model Predictions with Reality.

## References

Does GPS work on the ISS?; Robert Frost, Specialist on GPS use for space navigation
https://www.quora.com/Does-GPS-work-on-the-ISS
 About Walter Bislin (wabis) Rights Public Domain
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