Variance-Covariance matrix
computation in OPUS
Assume a baseline (vector) between two points A and B:
According to this definition:
Every 3D vector has three components, thus:
Let’s assume that 
and 
are, respectively, the
variance-covariance matrices of point B, the baseline 
, and point A. Propagating errors, one can write:
where
J is the Jacobian
and
If we assume 
, then
. Because we also assume without errors the fixed CORS, then 
.
Finally, under the above assumptions we arrived to the expected result that the variance-covariance matrix of point B is equal to the variance-covariance matrix of the baseline.
Because we have three independent determinations of point B:
Propagating errors once more:
Notice that the correlations between baselines are assumed zero.
Then, finally:
Although the correlations between baselines were assumed to be zero at this stage, nevertheless, we know that some correlation must exist considering that the GPS data of the unknown point and its atmospheric conditions are common to the three baselines. To approximate the unknown correlations between baselines we can write:
Propagating errors and replacing the variance-covariance matrices of the points by their baseline counterparts we have:
and finally:
Multiplying the above matrices and simplifying we arrive to the equation used in OPUS:
The values of the covariances 
are given by the
corresponding GFILEs.
T. Soler
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