# 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