Position calculation introduction

To provide an introductory description of how a GPS receiver works, error effects are deferred to a later section. Using messages received from a minimum of four visible satellites, a GPS receiver is able to determine the times sent and then the satellite positions corresponding to these times sent. The x, y, and z components of position, and the time sent, are designated as \scriptstyle\left[x_i,\, y_i,\, z_i,\, t_i\right] where the subscript i has the value 1, 2, 3, or 4. Knowing the indicated time the message was received \scriptstyle\ {t}_{r}, the GPS receiver computes the transit time of the message as \scriptstyle\left ( {t}_r-t_i\right ) . A pseudorange, \scriptstyle p_i \triangleq \left ( {t}_r-t_i\right )c, is computed as an approximation of the distance from satellite to GPS receiver.

A satellite's position and pseudorange define a sphere, centered on the satellite, with radius equal to the pseudorange. The position of the receiver is somewhere on the surface of this sphere. Thus with four satellites, the indicated position of the GPS receiver is at or near the intersection of the surfaces of four spheres. In the ideal case of no errors, the GPS receiver would be at a precise intersection of the four surfaces.