Documentation for ETKI

docs/src/ensemble_kalman_inversion.md

@@ -174,9 +174,9 @@ Here, ``\kappa_*`` is a limiting condition number, ``\mu_{s,1}`` is the largest
 
 # [Ensemble Transform Kalman Inversion](@id etki)
 
-Ensemble transform Kalman inversion (ETKI) is a variant of EKI based on the ensemble transform Kalman filter ([Bishop et al., 2001](http://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2)). It is a form of ensemble square-root inversion, and was previously implemented in [Huang et al., 2022](http://doi.org/10.1088/1361-6420/ac99fa). The main advantage of ETKI over EKI is that it has better scalability as the observation dimension grows: while the naive implementation of EKI scales as ``\mathcal{O}(p^3)`` in the observation dimension ``p``, ETKI scales as ``\mathcal{O}(p)``.
+Ensemble transform Kalman inversion (ETKI) is a variant of EKI based on the ensemble transform Kalman filter ([Bishop et al., 2001](http://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2)). It is a form of ensemble square-root inversion, and was previously implemented in [Huang et al., 2022](http://doi.org/10.1088/1361-6420/ac99fa). The main advantage of ETKI over EKI is that it has better scalability as the observation dimension grows: while the naive implementation of EKI scales as ``\mathcal{O}(p^3)`` in the observation dimension ``p``, ETKI scales as ``\mathcal{O}(p)``. This, however, refers to the online cost. ETKI may have an offline cost of ``\mathcal{O}(p^3)`` if ``\Gamma`` is not easily invertible; see below.
 
-The major disadvantage of ETKI, as currently implemented, is that it cannot be used with localization or sampling error correction. ETKI also requires the inverse observation noise covariance, ``\Gamma^{-1}``. In typical applications, when ``\Gamma`` is diagonal, this will be cheap to compute; however, if ``p`` is very large and ``\Gamma`` has non-trivial cross-covariance structure, computing the inverse may be prohibitively expensive.
+The major disadvantage of ETKI is that it cannot be used with localization or sampling error correction. ETKI also requires the inverse observation noise covariance, ``\Gamma^{-1}``. In typical applications, when ``\Gamma`` is diagonal, this will be cheap to compute; however, if ``p`` is very large and ``\Gamma`` has non-trivial cross-covariance structure, computing the inverse may be prohibitively expensive.
 
 ## Using ETKI