When you've been involved with adjusting surveying networks as long as I have, you develop a feel, and a deep respect, for measurements and their accuracy estimates. You also see a variety of examples of how not to do adjustments. So here is this little article, which simply expresses my view of the importance of not editing the measurements, and the importance of not editing the structure of the covariance matrix of the measurements. |

A measurement is the result of a measurement process. The result of this process is a set of one or more measured values, as well as the covariance matrix of those values. In the case of only one measured value, the covariance matrix is a single variance. Some measurements are very simple, but others are very complex.

An example of a simple measurement is a distance between two points measured with a steel tape marked off in millimetres. One has a feel for the accuracy of this type of measurement, and it can be calculated from a set of repeated measurements using standard least squares estimates (i.e. the standard calculation of the mean and variance).

An example of a more complex measurement is a 3D coordinate difference between two points, measured with two GNSS receivers, for example, which is a process that calculates estimates of the coordinate differences by processing other geometrical and physical measurements using detailed mathematical models.

The covariance matrix of such a derived set of measured coordinate differences is a full three-by-three symmetric, positive-definite matrix. The structure of this covariance matrix gives us an estimate of the structure of the variance of the coordinate difference measurements in all directions. The covariance matrix, a result of a complex process, gives us a measure of the closeness to reality of the measured (estimated) coordinate differences.

Obviously, if we change the structure of the covariance matrix, we are changing the structure of the estimated accuracy of the coordinate difference measurements in various directions, depending of the changes made. Scaling, on the other hand, is valid because the structure of the matrix is not changed, and it is also how we estimate the actual accuracy of the measurements through the least squares adjustment process.

It is therefore important to respect measured values as well as their covariance matrix. As it is invalid to change, willy-nilly, the numeric values of the measurements, it is also invalid to change the relative values of the elements of the covariance matrix.

So, if we edit our measured values or their covariance matrices (other than scaling), are we actually using real measurements in our adjustments?

An example of a simple measurement is a distance between two points measured with a steel tape marked off in millimetres. One has a feel for the accuracy of this type of measurement, and it can be calculated from a set of repeated measurements using standard least squares estimates (i.e. the standard calculation of the mean and variance).

An example of a more complex measurement is a 3D coordinate difference between two points, measured with two GNSS receivers, for example, which is a process that calculates estimates of the coordinate differences by processing other geometrical and physical measurements using detailed mathematical models.

The covariance matrix of such a derived set of measured coordinate differences is a full three-by-three symmetric, positive-definite matrix. The structure of this covariance matrix gives us an estimate of the structure of the variance of the coordinate difference measurements in all directions. The covariance matrix, a result of a complex process, gives us a measure of the closeness to reality of the measured (estimated) coordinate differences.

Obviously, if we change the structure of the covariance matrix, we are changing the structure of the estimated accuracy of the coordinate difference measurements in various directions, depending of the changes made. Scaling, on the other hand, is valid because the structure of the matrix is not changed, and it is also how we estimate the actual accuracy of the measurements through the least squares adjustment process.

It is therefore important to respect measured values as well as their covariance matrix. As it is invalid to change, willy-nilly, the numeric values of the measurements, it is also invalid to change the relative values of the elements of the covariance matrix.

So, if we edit our measured values or their covariance matrices (other than scaling), are we actually using real measurements in our adjustments?