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

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?

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

• I realize inversion is very important in the Earth Sciences, but wouldn't this question be better placed in a maths forum? – winwaed Jun 16 '14 at 20:29
• @winwaed I'd expect it's too applied for Mathematics. I don't know if it would fit at Physics either. Let's discuss it at meta. – gerrit Jun 16 '14 at 20:34
• Why the downvote? If you believe it to be off-topic, please do not downvote, but vote to close instead. If you believe it is a bad question, please leave a comment explaining why. – gerrit Jun 17 '14 at 14:20
• (I did not downvote it.) – winwaed Jun 17 '14 at 14:32
• @winwaed I didn't think you were, I was addressing whoever did. – gerrit Jun 17 '14 at 14:37

## 1 Answer

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".