Write draft of methods

[iangrooms]

First draft of the methods section is done; please leave comments. I still need to expand the appendix describing the filterSpec function though. So I'll leave this issue open until that's finished.

[iangrooms]

Just realized that I wanted to add a figure showing the Fourier transform of the boxcar kernel, the Gaussian, and the "taper/sharp" one for comparison, so still not quite done.

[iangrooms]

There's a lot of comments on the overleaf draft about whether this filter commutes with derivatives. I think I'll aim to add an appendix on the subject, but for now I've prepared some notes showing that it does not commute with derivatives in the presence of boundaries, even if you use Aluie's treatment of boundaries, except in a few special cases. The notes are here. (Link has edit privileges)

I think it probably does commute on the full sphere as long as the operators are appropriately defined, but I'm still trying to find solid math references so that I can be 100% sure.

[rabernat]

Note that the second option breaks conservation unless you integrate over the \emph{extended} domain.

Just wanted to note that this is what Aluie recommends and does in his paper. Here he invokes the uncertainty principle: you cannot know precisely BOTH the location and the scale-dependence of a quantity at the same time. This implies that the very notion of boundaries becomes fuzzy. After filtering, it no longer makes sense to define the coastline with a sharp curve. Rather, the boundary bleeds over a finite-size region. I guess this is what you mean by "extended domain"?

[iangrooms]

Yes, that's what I meant. If someone wants to use Aluie's convolutions with his treatment of boundaries that's fine, just a caveat about the correct way to do it. The more salient point of the notes is that our filters do not commute, even when you use Aluie's treatment of boundaries.