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How does GPS work? Know the maths behind it.

At over 20,000 kilometers above sea level(20,180 kilometers (12,540 miles)) is a constellation of satellites, each orbiting Earth every 11 hours and 58 minutes. These satellites are continuously beaming data down to us on earth, which in turn is received by devices such as your phone or navigational units in your cars, allowing you to see where you are on the planet. There are a lot of misconceptions about how GPS’s actually work, an example being that your phone and the GPS satellites are both talking to each other. So let’s get down to it: how do GPS’s work?

image to say let's begin
GPS stands for Global Positioning System, which works through trilateration, not triangulation or multilateration, which is commonly misconceived. There are many different types of navigational satellite systems from countries across the world, but the most popular and commonly used system is Navstar, which is the USA system. There are however Russian, Indian, Chinese, and European equivalent systems, although the Indian and Chinese systems sit in a geosynchronous orbit above their countries which means they are not worldwide systems.

The Navstar system, which is simply referred to as GPS, is what we will be focusing more on – although most phones and devices tend to have the capabilities to use both GPS and GLONASS. GPS satellites are set up in such a way that from almost anywhere on the surface of Earth you should have a direct line of sight of at least four GPS satellites. This is quite important on the basis that GPS point positioning requires at least four satellites to calculate three position coordinates and the clock deviation.

man using GPS in Mobile
As GPS units are receivers, there needs to be something sending some sort of signal to devices such as your phone to receive. Each GPS satellite broadcasts a navigational message towards Earth which contains an extremely accurate timestamp (obtained through atomic clocks on-board the satellites), and the satellites also broadcast their position at the time of broadcast, with all GPS signals broadcasting at 1.57542 GHz (L1 signal) and 1.2276 GHz (L2 signal). These two bits of information allow you to begin to work out your position on Earth.

With the satellites all sending exceptionally accurate time down to Earth, your phone or GPS receiver can compare the difference of time between the signal being sent and received to work out the distance between you and the satellite. By multiplying this time difference with the speed of light (as the signal is sent as the speed of light), you can get the distance you are from the satellite. As the satellites are also sending whereabouts they are, you can begin to draw spheres around the satellites, with you being somewhere on the outside border of the sphere.

A satellite image
As we introduce more GPS satellites into the mix we begin to get closer to where we are. By calculating the time differences between these satellites we move from having no idea where we are, to be able to pinpoint where we are, typically down to five to ten minutes on average, with the potential error being around 15 meters.  

There are a lot of factors that escalate the potential error, but the most significant is due to the ionosphere, a part of the upper atmosphere extending from 60 km to 2,000 km, where free elections occur frequently enough to have an appreciable influence on the proportion of electromagnetic waves passing through this layer. This error is substantially smaller when satellites are directly overhead, compared to being larger the closer satellites are to the horizon relative to you as the path between you and the GPS satellites goes through more of the atmosphere compared to being directly overhead. Even things such as small variations in the atomic clocks found onboard these satellites can cause major errors.

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A clock error of 1 nanosecond can translate through to a 1 foot or 30-centimeter miscalculation. Then the theory of relativity kicks in with the atomic clocks (which are moving at GPS orbital speeds) ticking ever so slightly slower than clocks that are stationary on the ground which translates into a 7 microsecond a day delay. General relativity then kicks in with the effect of the gravitational frequency shift being far greater than the 7 microsecond per day delay due to the velocity relative to Earth. As the theory states that a clock that is closer to an object will be slower than a clock further away, the atomic clocks onboard the GPS’s are faster by about 45.9 microseconds per day.

Combining the 7 microsecond a day delay due to the satellite’s velocity relative to Earth and the GPS’s being further away from Earth, this adds up to a 38-microsecond delay, which if left uncorrected would translate through to a 10 km/day pseudo-range error, rendering GPS’s invalid from the get-go if this was not to be taken into consideration. This is compensated by the GPS’s clock frequencies being slightly slowed down from 10.23MHz to 10.22999999543 MHz to cancel out the effects of relativity. So next time you turn on your GPS, you should realize just how much physics and maths is going into you finding your own location.

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6 thoughts on “How does GPS work? Know the maths behind it.”

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