Included in the OPUS extended output is the Variance-Covariance (VCV) matrix for the solution. The VCV matrix can serve as the Stochastic Model for a least squares adjustment to combine multiple OPUS solutions. Using the VCV matrix multiple OPUS solution coordinates can be treated as observations in a least squares adjustment.
For this exercise three independent OPUS solutions were obtained for CORS NLIB. The OPUS reports for days 95, 96 and 97 are linked to this post. The VCV matrix for the solution is found directly below the OBS BY SATELLITE VS. BASELINE section of the report. Strictly speaking these VCV matrices are referenced to ITRF00, the SV ephemeris reference frame. However the difference between an ITRF00 VCV matrix and the same matrix transformed to NAD83 is less than the precision reflected in the number of digits in the VCV matrix. These three individual solutions were combined into a single solution with a VCV for the combined solution.
What follows is a general discussion. For a much more detailed derivation using the actual data Click here.
This adjustment uses the Observation Model. In the Observation model the adjusted measurements are a function of the parameters being sought in the adjustment [La=F(Xa)]. In this case the parameters are the coordinates of the combined OPUS solutions. The observations are the individual OPUS solutions. The functional model is that the combined solution should equal the individual solutions. With one OPUS solution there is exactly enough information to determine a solution. That is, with one OPUS solution the combined solution is exactly the individual solution. With two or more OPUS solutions there is an overdetermined situation. In that case we set up a least squares adjustment to determine the most probable solution of the combined coordinates. For an adjustment combining three individual OPUS solutions, the VCV matrix of the observations used in the adjustment is created by combining the individual 3x3 matrices into a 9x9 matrix. The individual OPUS solutions are assumed to be independent, uncorrelated, observations. This is reasonable as they are observed using independently collected data at different times under different environmental conditions. This condition results in a block diagonal VCV matrix:sXX1 sXY1 sXZ1 0 0 0 0 0 0 sXY1 sYY1 sYZ1 0 0 0 0 0 0 sXZ1 sYZ1 sZZ1 0 0 0 0 0 0 0 0 0 sXX2 sXY2 sXZ2 0 0 0 0 0 0 sXY2 sYY2 sYZ2 0 0 0 0 0 0 sXZ2 sYZ2 sZZ2 0 0 0 0 0 0 0 0 0 sXX3 sXY3 sXZ3 0 0 0 0 0 0 sXY3 sYY3 sYZ3 0 0 0 0 0 0 sXZ3 sYZ3 sZZ3
The adjustment results in the best estimate for the combined OPUS solutions. Along with the estimated coordinates the adjustment yields a 3x3 VCV matrix for the solution and the Variance of Unit Weight. The Variance of Unit Weight provides an indication of the consistency of the original error estimates used in the adjustment. In this adjustment the Variance of Unit Weight was close to 53, meaning that the original VCV matrix should be scaled by that amount before the adjustment to be consistent with the precision of the original measurements.
Using this method is different than simply meaning the individual solutions. Using the VCV matrices of the individual solutions takes into account the reliability of the individual solutions and the correlations between the coordinates of the daily solutions. It also yields a better estimate of the reliability of the solution and the original data. Extending this method would allow one to determine outliers in the individual OPUS solutions.
This adjustment process requires a custom adjustment routine. In this case, the adjustment was implemented using MathCAD. In a future post commercial least squares adjustment software will be used to yield the same results by manipulating the input.