How is opacity calculated after relocation

[NeutrinoLiu]

hi, i notice that there is a brief discussion about how opacity is scaled after relocation in your paper.

But it seems that it is still not so obvious. Would you mind elaborating how equation 11 is derived?
further is there any ablation study about the necessity of such scaling for covariance, as shown in equation 9 of your paper. I do understand it's crucial to prevent mcmc process from collapse, but how it matters for optimization in practice?
Thx