Correcting a GPS receiver's clock

One of the most significant error sources is the GPS receiver's clock. Because of the very large value of the speed of light, c, the estimated distances from the GPS receiver to the satellites, the pseudoranges, are very sensitive to errors in the GPS receiver clock; for example an error of one microsecond (0.000 001 second) corresponds to an error of 300 metres (980 ft). This suggests that an extremely accurate and expensive clock is required for the GPS receiver to work. Since manufacturers prefer to build inexpensive GPS receivers for mass markets, the solution for this dilemma is based on the way sphere surfaces intersect in the GPS problem.
Diagram depicting satellite 4, sphere, p4, r4, and da

It is likely that the surfaces of the three spheres intersect, since the circle of intersection of the first two spheres is normally quite large, and thus the third sphere surface is likely to intersect this large circle. It is very unlikely that the surface of the sphere corresponding to the fourth satellite will intersect either of the two points of intersection of the first three, since any clock error could cause it to miss intersecting a point. However, the distance from the valid estimate of GPS receiver position to the surface of the sphere corresponding to the fourth satellite can be used to compute a clock correction. Let \scriptstyle r_4 denote the distance from the valid estimate of GPS receiver position to the fourth satellite and let \scriptstyle p_4 denote the pseudorange of the fourth satellite. Let \scriptstyle da \,=\, r_4 \,-\, p_4. \scriptstyle da is the distance from the computed GPS receiver position to the surface of the sphere corresponding to the fourth satellite. Thus the quotient, \scriptstyle b \,=\, da / c\ , provides an estimate of

(correct time) − (time indicated by the receiver's on-board clock),

and the GPS receiver clock can be advanced if \scriptstyle b is positive or delayed if \scriptstyle b is negative. However, it should be kept in mind that a less simple function of \scriptstyle da may be needed to estimate the time error in an iterative algorithm as discussed in the Navigation section.

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