Incorrect Latex equations for inversion docs.

[odunbar]

Issue

The equation on the EKI page is actually incorrect. and does not reflect the exact algorithm coded.

EnsembleKalmanProcesses.jl/docs/src/ensemble_kalman_inversion.md

Line 29 in bb04532

\tag{2} \theta_{n+1}^{(j)} = \theta_{n}^{(j)} - \dfrac{\Delta t_n}{J}\sum_{k=1}^J \left \langle \mathcal{G}(\theta_n^{(k)}) - \bar{\mathcal{G}}_n \, , \, \Gamma_y^{-1} \left ( \mathcal{G}(\theta_n^{(j)}) - y \right ) \right \rangle \theta_{n}^{(k)} ,

Two things are at play.

• Though not present in the loss function (or continuous time formulations), the C^gg terms weighting the data-misfit is important and implemented in the discrete time algorithm. it should appear, and is missing.
• The expression above does not usefully convey the algorithm itself,

Possible solution

It would be better to reexpress the algorithm equations as in more recent literature, in discrete time. E.g as expressed here https://arxiv.org/pdf/2010.13299