RTK GPS is a two step operation. The first step is to compute the integers. The second step is to compute the coordinates of points with a roving receiver. This post will cover computing the integers under the nearly perfect conditions as discussed in the thread of 10/31 below. This amounts to using multiple satellites to obtain a static solution as discussed in the thread of 11/03. The only difference is that what one is interested in out of this solution is the integers rather than station coordinates. So this post serves two purposes, to describe getting static coordinates using multiple satellites and getting integers from the same type of solution.

Assume 4 satellites, satellites s,t, u, and v, observations at 4 times, T1, T2, T3, and T4, and the use of the indexing nomenclature discussed in the post of 11/3. Call the station where the reciever is located station I. In the RTK context this can be thought of as the initialization station where the rover is placed to compute integers.

Using the 11/3 indexing we can write equations (6) and (7) of the 10/31 post for satellite s at times T1 and T2 (with a slight rearranging of the terms on the left hand side of the equations) as

WxNis + Wx FCis = [(Xs1-Xi)^2+(Ys1-Yi)^2+(Zs1-Zi)^2]^(1/2) (1)
and
WxNis + Wx(FCis+CCis12) = [(Xs2-Xi)^2+(Ys2-Yi)^2+(Zs2-Zi)^2]^(1/2) (2)

To make equations (1) and (2) above simpler to write we will use XYZ with the correct indexing to represent the terms on the right hand side. Then (1) and (2) can be written:

WxNis + WxFCis = XYZis1 (3)
and
WxNis + Wx(FCis+CCis12) = XYZis2 (4)

The other two equations for satellite s at times T3 and T4 would be

WxNis + Wx(FCis+CCis13) = XYZis3 (5)

and
WxNsi + Wx(FCis+CCis14) = XYZ1s4 (6)

There would be four similar equations for each of the other satellites, t, u and v with s replaced by t,u and v as appropriate in equations (3) through (6). So for four observing times of the four satellites there would be 16 equations.

On the left hand side of equations (3) through (6) the terms FCis, (FCis+CCis12). (FCis+CCis13), and (FCis+CCis14) are the observations by the receiver of the carrier phase signal from satellite s. There are similar observed quantities in the equations for the other satellites. On the right hand side of the 16 equations there would be satellite coordinates in each equation appropriate for the satellite and time associated with that equation. These coordinates would be known from the orbit information. There would then be 7 unknowns in the 16 equations: the three coordinates of point I, (Xi,Yi,Zi), and the 4 integers, Nis, Nit, Niu, and Niv. These unknowns could be computed from the 16 equations using least squares.

Thinking in terms of getting a static solution for the coordinates for I one's work would be complete. Thinking in terms of RTK one would have computed the needed integers and would be ready to move the receiver to other stations to get their coordinates.

The next post will discuss how to use the integers that have been computed to get coordinates at RTK points from observations of multiple satellites at a single time epoch. Keep in mind we are still talking about a situation where clock time offsets and most error sources are absent.

Please keep it going, Mr. G.