Section 1: Stars and planets have long been used to guide mariners and travellers across oceans and land masses. Today, satellites can be used as reference points in place of stars and a technique known as ‘triangulation’ can be applied to work out location. I’ll first explain how triangulation works in two dimensions using reference points on a flat map.
Section 2: Suppose that I need to find a particular location somewhere in Mali in north-western Africa. I could use major cities as my reference points. Provided that I know the distances from three different reference points, I should be able to draw circles to scale to find the place I am looking for. If the location is 1350 km from Lagos in Nigeria, I can draw a circle centred on Lagos with the radius scaled to 1350 km. The place I am looking for must be on this circle.
Section 3: If the location is also 1580 km from Marrakesh, I can draw another circle but this time centered on Marrakesh with the radius scaled to 1580 km. The place I am looking for is now either at point A or point B, but which one?
Section 4: I need to plot one further circle. If the location is known to be 1310 km from Dakar then the third circle defines the place I am looking for as point B – about 80 km north-east of Timbuktu. This is the basic idea used in triangulation. However, the real world is not flat, and triangulation with satellites or stars is a three-dimensional calculation. I shall explain how three-dimensional triangulation can be used to fix a location on the Earth’s surface, but to do this I shall need to show spheres rather than circles.
Section 5: Each satellite sends out a radio frequency signal, which is picked up by the GPS receiver on the ground. The GPS receiver measures the time it takes for the signal to travel from the satellite to the receiver – the propagation time. The distance from each satellite can be calculated from the known velocity of the signal and the propagation time.
Section 6: Suppose that the GPS receiver measures its distance from satellite A to be 17 500 km. This means that the GPS receiver is somewhere on the surface of a sphere of radius 17 500 km with satellite A at its centre.
Section 7: If the GPS receiver measures its distance from satellite B to be 18 000 km then another sphere can be drawn with a radius of 18 000 km with satellite B at its centre. The intersection of the two spheres traces out a circle, so the GPS receiver’s location could be anywhere on that circle. The distance from a third satellite is needed to narrow the possibilities down.
Section 8: Suppose that the GPS receiver measures the distance from a third satellite, C, as 18 500 km. This can be represented as a sphere centered at C. The intersection of this sphere with the circle created from the measurements from satellites A and B will narrow the position down to two points. In practice one of the two points will not be on the Earth’s surface, so can be disregarded. This leaves just one possible location, which can then be expressed in terms of map coordinates. To provide the required level of measurement accuracy, a fourth satellite is needed.
Section 9: GPS satellites and stars are not fixed reference points because they move in relation to the Earth. GPS receivers need to have information about the satellites’ current positions available to them, to be able to compute the GPS receiver’s location.
