To show how equation differencing works to account for receiver clock offset error we will first go back to post III and apply equation differencing to the situation discussed in that post where no errors existed. In post III there were 4 satellites observed at 4 times, producing 16 equations to be solved for 7 unknowns. The unknowns being the three receiver coordinates and the 4 integers associated with the initial observations of the 4 satellites. One can also solve for the receiver coordinates after perfoming equation differencing using the 16 equations described in post III. Consider one of the 4 observation times, say time, T3. Using the shorthand notation given in post V the equations for time T3 can be written:

WxNis + WxRCis3 = XYZis3 (1)

WxNit + WxRCit3 = XYZit3 (2)

WxNiu + WxRCiu3 = XYZiu3 (3)

WxNiv + WxRCiv3 = XYZiv3 (4)

Now pick a satellite, say satellite s, and call it the reference satellite. Then subtract equations (2), (3), and (4) from equation (1). This gives:

Wx(Nis-Nit) + Wx(RCis3-RCit3) = (XYZis3-XYZit3) (5)

Wx(Nis-Niu) + Wx(RCis3-RCiu3) = (XYZis3-XYZiu3) (6)

Wx(Nis-Niv) + Wx(RCis3-RCiv3) = (XYZis3-XYZiv3) (7)

Proceeding simularly for the other three times, T1, T2, and T4, will lead to three similar equations for each time. Thus after differencing there will be a total of 12 equations with 6 unknowns. The unknowns will be the receiver coordinates, Xi, Yi, and Zi, and the three integer differences, (Nis-Nit), (Nis-Niu), and (Nis-Niv). With 12 equations and 6 unknowns, the unknowns can be computed.

Now assume a time varying offset of the receiver clock relative to the satellite clocks; it being assumed that the satellite clocks remain sychronized with one another. This time varying offset will cause the carrier phase observations by the receiver to be different from the RC values needed in equations (1) through (7) above, which are needed to compute the receiver coordinates. What we are talking about is a time dependent error associated with the receiver clock and independent of what satellite is being observed at a particular time. So, if there are four satellites being observed at the same time, the receiver clock error will be the same for all of these observations. In the situation discussed above where there are four observation times, there will be four receiver clock unknowns, with the same unknown entering into the observations of all four satellites at a given time. For the times T1,T2, T3, and T4 call these receiver clock errors RTEi1, RTEi2, RTEi3, and RTEi4, where RTE stands for Receiver Time Error. Since these errors are not satellite dependent there is no satellite index associated with them. Now the phase observation of a satellite at a time epoch will be the sum of the RC value it would observe if there were no error plus the relevant RTE error term. So. for example, at the times T1,T2,T3, and T4, the satellite phase observations (which we will represent by SPO) for satellite, s, would be:

SPOis1 = RCis1 + RTEi1 (8)
SPOis2 = RCis2 + RTEi2 (9)
SPOis3 = RCis3 + RTEi3 (10)
SPOis4 = RCis4 + RTEi4 (11)

and the observations for the other three satellites at time, T3, would be:

SPOit3 = RCit3 + RTEi3 (12)
SPOiu3 = RCiu3 + RTEi3 (13)
SPOiv3 = RCiv3 + RTEi3 (14)

Now considering time, T3, subtract equations (12), (13), and (14) from equation (10). This gives:

(SPOis3-SPOit3) = (RCis3+RTEi3) - (RCit3+RTEi3) (15)

or since the RTEi3 terms cancel one another this gives

(SPOis3-SPOit3) = (RCis3-RCit3) (16)

Proceeding similarly for satellites u and v we will get

(SPOis3-SPOiu3) = (RCis3-RCiu3) (17)

and

(SPOis3-SPOiv3) = (RCis3-RCiv3) ((18)

There will be equations such as (16)<

This is why we need four SV's? Three will not solve for the clock offset without leaving a large margin for error. (an expanded point of intersection on a graph of time related positions or a large possible matrix). Or the matrix will show many possible solutions...

If we expand and add 5 or more SV's, then this clock offset can be more accurately modeled with least squares internal to the process? Yes?

dp

The situation with carrier phase is different than with range. With carrier phase the clock offset induced error is eliminated by differencing, not by solving in a least squares solution. So the clock offset error is accounted for equally as well whether you have 2 satellites or 5 satellites. After differencing you have as unknowns the 3 coordinates of the receiver and one less integer difference unknown than there are satellites being used. For each measurement epoch you have one less equation than there are satellites. So with 2 satellites there would 4 unknowns and 1 equation for each epoch and you could, in theory get a solution with 4 epochs of data. With 3 satellites there would be 5 unknowns and 2 equations for each epoch, so you would need 3 epochs to get a solution. With 4 satellites, 6 unknowns and 3 equations per epoch so in theory 2 epochs would be needed. With 5 satellites, 7 unknowns and 4 equations per epoch, thus again 2 epochs required. And as you add more satellites you add one more unknown and one more equation per epoch and always only need two epochs.

In real life of course the more satellites you have the better the satellite geometry for any given length of observing and this geometry is important when solving for the coordinates and integers given the existance of random errors in the real world. In the early days of GPS we got good results using three satellites, but we had to observe a lot longer to get the same accuracy as you get today with 5 or 6 satellites.