Maybe we can separately estimate "sigma" from ice cores, from tree rings, etc? Each of these methods will have their own systematic errors, but perhaps "sigma" could be more robust to those. For example, if all ice core data is systematically 5 C below all tree ring data, this will not affect the esimate of "sigma".

My analysis is here,
https://chaource.dreamwidth.org/134632.html
https://chaource.dreamwidth.org/136180.html

Also
https://chaource.dreamwidth.org/tag/climate