Comment on paper draft

[kdrushka]

"Scale-Aware Along-Track SLA Variance" doc in overleaf

[rabernat]

I would like to use tracked changes to make suggestions on this paper, but I notice it has been written with a free version of overleaf. https://www.overleaf.com/project/5f8f0594cf685a00011a9b5b

I'd be happy to copy the paper over to my paid account so we can use tracked changes and other paid features. What do you think?

[jakesteinberg]

I think using track changes would be helpful. Seeing that I use overleaf a lot for multiple projects, I'd be happy to upgrade my account.

[jakesteinberg]

Hey Ryan, Thanks for adding that section in the manuscript last Friday. I wanted to reach out to again clarify what we have talked about a few times now. My question is that in equations you added, you write the filtering operator $$\langle \rangle$$ as acting on \eta^2....(i.e. $$\langle \eta^2 \rangle$$) In my experience, and like you mentioned when we last spoke about this, the filtered squared quantity does not really change with filter scale. I thought that the variance at scales larger than $l$ should be written as $$\langle \eta \rangle^2_l$$. With the square outside the angle brackets? Am I confusing this again? Thanks.

[rabernat]

Very good point Jake. I'm looking into this today. I made have made a mistake in my haste to get this done.

[rabernat]

Jake you were definitely right about this. I got confused when translating this into the multi-band framework. The key point, derivable from the mathematical properties of convolution is that the filters are conservative, meaning that

$$ \int \langle \eta^2 \rangle dx = \int \eta^2 dx $$

for any filter. So it is pointless to try to decompose variance that way!

I have rewritten it with the square outside of the convolution, and I think it makes more sense now. A "band" of variance is defined as

$$ \taun(\eta^2) = \langle \eta \rangle{\elln}^2 - \langle \eta \rangle{\ell_{n-1}}^2 $$

With the highest band as

$$ \tauN(\eta^2) = \eta^2 - \langle \eta \rangle{\ell_{N-1}}^2 $$

The only ambiguity is whether you also want to do smoothing on the first term on the RHS, i.e.

$$ \tauN(\eta^2) = \langle \eta^2 \rangle{\ell{N-1}} - \langle \eta \rangle{\ell_{N-1}}^2 $$

This makes sense when you only have one filter (like in the bar-prime notes), but I'm not sure it's necessary once we move into the spectral view. Do you have thoughts on this?

[rabernat]

p.s if you do it in the second way, then you should probably also do the band pass as

$$ \taun(\eta^2) = \langle \langle \eta \rangle{\elln}^2 \rangle{\ell{n-1}} - \langle \eta \rangle{\ell_{n-1}}^2 $$

which looks pretty ugly.

[jakesteinberg]

Thanks for revisiting the idea and rewriting a bit of that section. Your comments make sense and I think the way you have it written now works (as a side note I'll probably go back and switch \eta for u). Regarding the choice of whether or not to filter that first term $$ \langle \eta^2 \rangle{\ell{N-1}} ... I'll compare the two cases. I think I would leave it when applying only one filter (like in the bar/prime paper), but then in the spectral view follow the notation you've just updated (eq. 11 I believe). Thanks again! Hope you have a good weekend.