How GPS receivers calculate coordinates
The software in a GPS receiver uses trilateration to compute its current coordinates. This is a technique of trigonometry that uses measured distances from points of known location to determine the unknown location. (This is different to triangulation, which relies on taking bearings from points of known location.)
This exercise uses a map of the UK to illustrate the principle of trilateration. A suitable map scale would be 1; 3 000 000, with tracing paper taped over the area from Middlesborough to the Isle of Wight.
For a description of how distances from satellites are measured by the GPS receiver, see the GPS pass notes.
The imagined situation for this exercise is that of a space alien beamed down to the UK with a map, but not knowing where on the map it has landed. The first clue is in the form of a road sign that reads "Birmingham 70 miles". Using the map and a pair of compasses, the alien can draw a circle with a radius of 70 miles, centred on Birmingham. Any point on the circumference of that circle fits with the information that it is 70 miles from Birmingham.
Using the scale of the map, or by calculation, set a pair of compasses to radius 70 miles and draw a circle with its centre at Birmingham. List some of the possible locations which fit if this information is true.
The alien finds a second road sign saying "York 100 miles". Draw a circle (or part of) centred at York with a radius of 100 miles. Note that only two points on the map fit the information about the distances to York and Birmingham. This is where the circles intersect.
In our imagined scenario, the space alien finds another road sign with the information "London 80 miles". A circle centred on London, radius 80 miles, coincides with one of the two possible locations identified in the previous stage.
Discuss how you would now estimate the alien's location, given that the 3 circles do not intersect at exactly the same point.
Discuss possible sources of error in this example (e.g. inaccuracies in road measurements and rounding errors; differences between distances as road miles and as straight line measurements).
In GPS measurements, errors may arise from a number of sources such as: differences between the actual orbital position of the satellite and the one it reports in its signal; clock errors; delay of the radio signal as it passes through the ionosphere and troposphere of the Earth's atmosphere; signal "noise" generated at the receiver; multi path errors (where a version of the radio signal also reaches the receiver after bouncing off the satellite's solar panels or objects on the ground such as water, landscape or buildings. Each of these sources can introduce errors of up to 2.5 metres. Use of additional data, known as differential GPS, can reduce or eliminate most of this error, but cannot address receiver noise and multi path errors.
A GPS receiver uses mathematical formulae to compute its position. In the imagined example, the calculation would be based on the formula for a circle:
(x - long) 2 + (y - lat) 2 = r 2
For the first road sign, the coordinates of Birmingham (52.48 N, 1.91 W) would be substituted for "lat" and "long" respectively, and "r" would be given the value of 70 miles, giving an equation describing the first circle drawn. Similar equations could be generated for the circle from York (53.96 N 1.11 W) and London (51.52 N, 0.10 W). This gives three simultaneous quadratic equations with two unknowns, which can be solved for one solution. (The latitude and longitude of Peterborough are 52.59 N and 0.25 W).
GPS receivers trilaterate in three dimensions. They require data from four satellites, generating four simultaneous equations with three unknowns.