Combining OPUS Solutions Using Contrived Zero Observations

Using Coordinates as Observations is a rigorous least squares method of combining the coordinate values of multiple OPUS solutions. However I am not aware of any commercial least squares software that can be set up to use this method. The method described below produces results very consistent with the method of using Coordinates as Observations and can be employed using commercial least squares adjustment software. This method, dubbed Contrived Zero Observations, is a kludge; it does however have a sound mathematical basis. For this discussion Columbus Network Adjustment Software will be used but any adjustment software that uses the Geodetic 3d Model should work.

What follows below is a general discussion. For the mathematical basis including a detailed derivation using three stations Click Here .

The general outline for using Contrived Zero Observations is as follows:

Create pseudo network stations with the coordinates of the individual OPUS solutions. These stations are called pseudo network stations as in reality they are all physically the same station. For this exercise, however treat them as three individual stations.
Contrive three [0, 0, 0]^T GPS vectors from the pseudo fixed network stations to the combined solution. Again these vectors do not actually exist; they are contrived. Logically this has validity, as the three stations are physically the same point so a vector between them and the adjusted position should have zero magnitude.
Use the covariance from the individual OPUS solutions as the covariance matrix of the contrived zero vectors.
Constrain the individual OPUS solutions as fixed network stations.
Solve for the combined solution as in any least squares adjustment.

This exercise uses the same three independent OPUS solutions for CORS NLIB used for the Coordinates as Observations exercise so the results are directly comparable. The OPUS reports for days 95, 96 and 97 are linked to this post.

The covariance matrices of the individual OPUS solutions are used as the covariance matrix of the contrived GPS vectors. The covariance matrix for the solution is found directly below the OBS BY SATELLITE VS. BASELINE section of the report. These covariance matrices are for the OPUS point solution, not a GPS vector, however for this adjustment they are treated as if they apply to the contrived [0, 0, 0]^T GPS vector. Strictly speaking these covariance matrices are referenced to ITRF00, the SV ephemeris reference frame. However the difference between an ITRF00 covariance matrix and the same matrix transformed to NAD83 is less than the precision reflected in the number of digits in the covariance matrix.

In practice, when computing the adjustment, Columbus Network Adjustment Software issued an error when three contrived [0, 0, 0]^T GPS vectors are used in the adjustment. While setting up the adjustment Columbus determines the approximate position of the stations to be adjusted using the fixed stations and the observations. Columbus halted when the first fixed station and the station to be adjusted had the same approximate coordinates. This was remedied by changing one of the components of the first contrived GPS vectors to 0.00001 meters, that is the vector was entered as [0.00001, 0, 0]^T. That was enough to differentiate between the fixed station and the station to be adjusted. The other two vectors were left as [0, 0, 0]^T.

Performing the adjustment while fixing the three pseudo network stations resulted in an a posteriori variance of unit weight of 52.91 (Std Dev of Unit Weight 7.27) verses 52.97 (Std Dev of Unit Weight 7.28) for the method of Coordinates as Observations. After scaling the observation covariance matrix by 52.91 and re-running the adjustment the variance of unit weight became one as expected. The coordinates resulting from the least squares adjustment are identical to the Coordinates as Observations method to the 0.1 millimeter magnitude. A link to the Columbus Report is provided here. The (n, e, u) reference frame covariance matrix of the combined solution is slightly different than that obtained using coordinates as observations. This is possibly due to using 0.00001 for the vector magnitude rather than zero and different computational method between MathCAD and Columbus. The differences in the covariance matrix are not enough to affect the standard deviation of the coordinates which are the same as those computed using Coordinates as Observations to 0.01 mm magnitude.

When using Coordinates as Observations, an inconsistent OPUS solution will be flagged as an outlier. Using Contrived Zero Observations, the vector from the inconsistent OPUS solution is flagged. A further check on the validity of the adjustment can be had by examining the magnitude and confidence interval of the adjusted vectors. Looking at the distance errors section of the report:

AT                 TO    Distance   Stan Dev
OPUS095          NLIB      0.0131     0.0158
OPUS096          NLIB      0.0031     0.0131
OPUS097          NLIB      0.0122     0.0160

For each adjusted vector the confidence interval surrounding the adjusted distance includes zero. Based on this information one cannot say, with 95% certainty that the vector does not in fact have zero length.

NGS included the extended output at the request of the survey community. These posts highlight some of the usefulness of this additional information. Hopefully these post will encourage others to mine this additional information for additional uses and share those uses.