More frequent is the well-understood case in which the physical quantities are obtained as a result of a minimization, and the terms of the inverse of the covariance matrix are related to the curvature of at its minimum:

In most cases one determines independent values of physical quantities with the same detector, and the correlation between them originates from the detector calibration uncertainties. Frequentistically, the use of () in this case would correspond to having a ``sample of detectors'', each of which is used to perform a measurement of all the physical quantities.

A way of building the covariance matrix from the direct measurements
is to consider the original measurements and the calibration
constants as a common set of independent and uncorrelated
measurements, and then to calculate corrected values that take into
account the calibration constants.
The variance/covariance propagation will automatically provide the full
covariance matrix of the set of results.
Let us derive it for two cases that occur frequently, and then
proceed to the general case.