Contrived Zero Observations Method of Combining Individual OPUS Solutions

Contrived Zero Observations Method of Combining Individual OPUS Solutions

Outline:

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 threeGPS vectors from the pseudo fixed network stations to the combined solution. Again, these vectors do not actually exist; they are contrived. Logically this makes sense, 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 matrices of the individual OPUS solutions as the covariance matrices of the contrived zero vectors.

Constrain the individual pseudo stations as fixed network stations.

Solve for the combined solution as in any least squares adjustment.

Below is a side-by-side comparison between the Contrived Zero Observations Method and the Coordinates as Observations Method.

Contrived Zero Observations

Coordinates as Observations

Math Model

Covariance Matrix for Observations

Use the covariance matrices for the individual OPUS solutions as the covariance matrix for the contrived zero observations.

The weight matrix (P) is the inverse of the covariance matrix

Design matrix

The pseudo network stations are constants so they drop out of the design matrix.

Observation Matrix

Misclosure Matrix.

Assume initial estimate of.

Therefore:

Normal Equation Elements:

Normal Equation:

Estimated Parameters:

Math Model

Covariance Matrix of Observations

The weight matrix (P) is the inverse of the covariance matrix.

Design Matrix

Observation Matrix

Misclosure Matrix

Assume initial estimate of

Therefore:

Normal Equation Elements:

Normal Equation:

Estimated Parameters:

Comparing the Contrived Zero Observations Method with the Coordinates as Observations Method:

The design matrices are the same for both methods
The covariance / weight matrices are the same for both methods
The misclosure matrices are the same for both methods.

It is fair to say that both methods will yield the same numerical results. However, the method of using Contrived Zero Observations can be done using any commercial least squares adjustment software that supports the Geodetic 3d Model while using Coordinates as Observations requires a custom adjustment.

Using coordinates as observations the residuals are the corrections to the individual OPUS solutions. Going back to the original model:

X_i being the individual OPUS solutions and X the adjusted combined solution

With the Contrived Zero Observations the residuals are the corrections to the fictional zero observations. The estimate of the combined OPUS position is the individual OPUS solution pseudo station plus the adjusted baseline. Again going back to the original model:

X_i being the individual OPUS solutions, L_b the fictional zero observation and X the adjusted combined solution.

This demonstrates that the residuals, using the Contrived Zero Observations, are equivalent to corrections to the individual OPUS solutions.

Detecting Outliers

When using Coordinates as Observations to combine OPUS solutions, inconsistent individual OPUS solutions are flagged as outliers. With Contrived Zero Observations, the baseline from the outlier OPUS solution will show up as an outlier observation. In addition the baseline magnitude of the adjusted baselines should not be significantly different (in a statistical sense) from zero. That is the range of values within the confidence interval of the adjusted baseline length should include zero.