## Effect of Boat Speed on GPS Position Accuracy

jimh
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### Effect of Boat Speed on GPS Position Accuracy

Effect of Boat Speed on GPS Position Accuracy

A boater recently commented about the speed of his boat while a GPS position solution was being obtained, possibly implying that the boat speed might influence accuracy of the deduced position from GPS. There is little influence on the speed of the GPS receiver on Earth on a slowly moving boat in the position solution using GPS because the other element in any range solution used to establish that position, the GPS satellite, is in very rapid motion relative to the receiver.

A GPS satellite has an orbital height above Earth of about 12,550-Miles.

The GPS satellite orbital period is about 12-hours.

From the above three factors we can deduce the approximate speed of travel of a GPS satellite. We assume a circular orbit and use the Earth radius at the equator. We calculate the circumference of the orbital path in miles and divide by the time in hours to travel that path:

`Earth radius + orbital height = radius of orbital path3,963 + 12,550 = 16513-miles radiusCircumference of circle = 2 × π × radiusCircumference of orbital path = 2 × π × 16513-miles radiusCircumference of orbital path = 103,754-milesSpeed = Distance ÷ TimeSpeed = 103,754-miles ÷ 12-hoursSpeed = 8,464-miles-per-hour`

If a boat is moving at a slow speed, say 8.4-MPH, the satellites are moving one-thousand times faster. On that basis, any motion on the boat does NOT have an effect on the solution of the range from the satellite to the GPS receiver.

In GPS position finding, the method used is to measure a range, in this case the range from the GPS receiver to the satellite transmitting the signal. Modern GPS receivers calculate a new position once per second or faster. We can then look at how much change in location will occur in a period of one second.

We know the satellite is traveling at 8646-mles-per-hour. In one second the satellite movement will be

`8646-miles/1-hour × 1-hour/3600-seconds × 5280-feet/1-mile = 12,680-feet`

If the boat travels at 8.646-miles-per-hour, then in one second the boat will travel

`8.646-miles/1-hour × 1-hour/3600-seconds × 5280-feet/1-mile = 12.68-feet`

jimh
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Joined: Fri Oct 09, 2015 12:25 pm
Location: Michigan, Lower Peninsula
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### Re: Effect of Boat Speed on GPS Position Accuracy

A further aspect of the relative motion of a boat and a GPS satellite can be considered as follows:

The boat is on the curved spherical surface of the Earth and traveling at slow speed. The satellite is traveling a circular orbital path around the Earth moving 1,000-times faster. The method of finding position using GPS uses the measurement of the distance between the receiver and the satellite.

At some instant a GPS satellite is directly overhead of the GPS receiver on the boat. Thus at t0 the distance to the satellite is the difference between the radius of the Earth, 3963-miles, and the radius of the orbit, 16,513-miles, or simply the orbital height of the GPS satellite, 12,550-miles.

At one second later, T1, we look at the change in distance (range) from satellite to GPS receiver. The satellite moves 12,680-feet in its circular orbital path. The receiver moves 12.68-feet in its circular path on the Earth surface. We know the direction of movement of the receiver and the satellite are not particularly correlated. For the worst case, the paths will be collinear and in opposite directions. So we could say the satellite goes 12,680-feet East and the receiver goes 12.68-feet West. Now we compute the new path distance between them. The distance from the Earth surface to the satellite has not changed. It is in orbit, and it is still 12,550-miles or 66,264,000-feet above the Earth surface. The receiver just moved 12.68-feet away from where it was on the Earth surface. Now we can compute the new path easily if we assume that for this analysis the Earth is flat, and the receiver has just moved 13-feet from the spot where the satellite is now directly overhead. We can compute the new distance to the satellite as being the hypotenuse of a right triangle with sides of 66,264,000-feet and 13-feet. The hypotenuse is then the square root of the sum of the squares, or, for all intents and purposes, still 66,264,000-feet.

If the boat path and satellite path were collinear and correlated, the boat would move in a direction that would tend to keep it the satellite directly overhead, and the path distance would change even less.

jimh
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Joined: Fri Oct 09, 2015 12:25 pm
Location: Michigan, Lower Peninsula
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### Re: Effect of Boat Speed on GPS Position Accuracy

Another case to explore for relative motion between a GPS satellite and a receiver on the surface of the Earth is the situation where the satellite has just appeared on the horizon or is just about to set on the horizon of the receiver. In that case, the path distance to the satellite from the receiver can be calculated as follows:

1. assume the satellite is on the geometric horizon, that is, at elevation 0-degrees; generally such an orbital location is not actually usable for GPS position finding, but we will examine this situation anyways
2. the satellite remains at a distance from the Earth center equal to the Earth radius plus its orbital height, or 16,513-miles
3. The receiver on the Earth's surface remains at a distance from the center of the earth equal to the Earth radius, or 3,963-miles

The above describes a right triangle with an hypotenuse of 16513-miles and one side of 3,963-miles. We find the length of the other side, the distance from the satellite to the receiver, using the familiar rules for a right-triangle: 16,030-miles, a shorter distance than the zenith path by 483-miles.

Since a 360-degree orbital period will be 12-hours, then one-quarter of an orbit, that is the 90-degrees from being on the horizon to being overhead, will take 3-hours. The path distance from the receiver to the satellite then changes 483-miles in 3-hours, for an apparent speed of motion in the path distance to a receiver of 161-MPH. This is rate of change in the path distance of 236-feet-per-second.