2-D vs. 3-D Fixes

(an article by Ed Weston*)

Most receivers calculate a horizontal position after they acquire three satellites. To do this they must make presumptions on their altitude. Usually the last calculated altitude is used. If the altitude assumed by the receiver is in error, the accuracy of the horizontal position is impacted. For example, if the receiver assumed mean sea level and it was at 10,000 feet, a horizontal error of more than a mile could result. On the other hand, if the assumed value is very accurate the horizontal error may be less than if four satellites are tracked.

I’m going to try to present a simplified description of the GPS position calculation procedure to help the reader understand why the above is true. If a receiver is tracking four satellites, it is able to make simultaneous pseudo-range measurements from each. Each pseudo-range can be thought of as being equal to the range to the satellite plus the offset in the receiver’s clock, scaled to distance units. Since the receiver can calculate the satellite’s position from the Navigation Message, there are four unknowns in the pseudo-range equations: the three components of the receiver’s position, and the offset in the receiver’s clock. To determine its location it then must solve the four equations for the four unknowns. The receiver uses suitable algorithms to perform this arithmetic [Note for the more mathematically inclined. You’ve noticed that the equations are non-linear which makes the task non-trivial. If a rough estimate of position is available a Taylor series expansion about the estimated position is generally used. The partial derivatives of the pseudo-range wrt to receiver position are equal to the satellite-to-receiver unit line of sight vector.]

If the receiver “knows” one of the three position components, then measurements from only three satellites are required to solve for the other two components and the receiver clock offset (3 equations and 3 unknowns). If, for example, the altitude is known perfectly, this is equivalent to tracking a satellite directly above (or below) the user, and having a perfect pseudo-range measurement to it. Since a real fourth satellite will never provide a perfect pseudo-range, the three-satellite solution will be better than the four-satellite solution (assuming similar geometry quality, i.e., PDOP). If on the other hand the assumed altitude is in error, then the two sides of the equations will be equal only if a compensating error is added to the horizontal position components and clock offset.

How does a receiver calculate its position if more than four satellites are tracked? Well, without getting into the mathematics it will be a little difficult to explain, but I’ll try. This description will be conceptual, that is should not be taken literally. As was described above, with four satellites (and good geometry) the position (and necessarily the clock offset) can be calculated. If a measurement from a fifth satellite is available, the receiver can calculate what the measurement should be (expected pseudo-range is range plus clock offset, based on the four satellite solution). The difference between the actual pseudo-range and the expected pseudo-range indicates whether the fifth satellite “agrees” with the four-satellite position solution. That is, if the difference is zero the five-satellite solution is the same as the four-satellite solution. If it is non-zero, then the value is used to adjust the four satellite position (and clock offset) to produce the five satellite position. How is the adjustment done? Well mathematically, linear algebra techniques are used to produce a least square solution. Conceptually think about it like this. You guess a position (and clock) solution, then use it to calculate the five expected pseudo-range values as described above. You then square each of the pseudo-range differences (expected minus actual pseudo-range) and sum them up. This is a measure of the quality of the five-satellite solution. You try many different position solutions, calculating the quality measure for each. The one with the smallest quality measure is your least squares five satellite position. A similar technique is used to calculate a least squares solution for more than five satellites.

* Ed Weston is a GPS analyst and has worked on GPS for DOD since 1975.