Are the elements of an averaging kernel matrix always centered on the diagonal, or can they be exclusively off-diagonal in rare cases?

Question

According to Rodgers (2000) equation 2.80

$$ d_s = \mathrm{tr}(\mathbf{A}) $$

where $d_s$ is the number of degrees of freedom for the signal, tr denotes the trace, and $A$ is the averaging kernel matrix.

I'm trying to properly understand this. For example, is it theoretically possible for the averaging kernel to have mainly or even exclusively off-diagonal elements? In practice, this would mean that the retrieved state at height $h$ would correspond to the true state at height $h+1$, if we take remote sounding of an atmospheric column as an example. In such a (admittedly contrived) example, we could get $d_s=0$, although we do have information in the measurement — it's just displaced.

Is this situation possible? If yes, does $d_s$ tell the whole story as for the information content in the measurement?

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^{Clive D. Rodgers, Inverse methods for atmospheric sounding, Theory and Practice. ©2000 World Scientific Publishing Co., London, UK.}

Answer 1

This statement from Geophysical Data Analysis: Discrete Inverse Theory: William Menke should make things clear for you

The diagonal elements of the covariance matrix are a measure of the width of the distribution of the data, and the off-diagonal elements indicate the degree to which pairs of data are correlated.

Regarding the second part of your doubt whether a situation is possible where ds = 0 ; this would imply some error in measurement(for example - some noise) and would not have any implication on the geophysical nature of the data . It would be advisable to take multiple readings at different locations in this kind of situation.

Finally ds provides some insight into the quality of data collected but no true information about the "information content of measurement".