Table 4:

Divergence decomposition of the EnKF with LPRM retrievals according to Equation [14]. These divergences are from the empirical conditional in Equation [12] to the specified hybrid conditionals, as described by Equation [13]. All divergence metrics are reported as a fraction of the entropy of evaluation data, H(Z).

Interpretation	Prior	Likelihood	Info. Loss
Total EnKF divergence	$\mathcal{N}\left[\overline{X_t},\overline{Q_t}\right]$	$\mathcal{N}\left[X_t,R\right]$	4.08
Total effects of model prior	$\mathcal{N}\left[{\overline{X}}_t,\overline{Q_t}\right]$	p(yt|xt, Zt)	2.55
Effects of Gaussianity in the prior	$\mathcal{N}\left[\widehat{{\mu }_t},\widehat{{\sigma }_t}\right]$	p(yt|xt, Zt)	0.06
Effects of ensemble variance	$\mathcal{N}\left[\widehat{{\mu }_t},\overline{Q_t}\right]$	p(yt|xt, Zt)	1.62
Effects of linearity (identity) mean	$\mathcal{N}\left[{\overline{X}}_t,{\widehat{\sigma }}_t\right]$	p(yt|xt, Zt)	0.14
Total effects of retrieval operatora	p(Zt|xt)	$\mathcal{N}\left[Z_t,R\right]$	1.17
Effects of Gaussianity in retrieval operator	p(Zt|xt)	$\mathcal{N}\left[{\widehat{\mu }}_t,{\widehat{\sigma }}_t\right]$	0.08
Effects or prescribed retrieval variance	p(Zt|xt)	$\mathcal{N}\left[\widehat{{\mu }_t},R\right]$	1.91
Effects of linearity (identity) mean	p(Zt|xt)	$\mathcal{N}\left[Z_t,{\widehat{\sigma }}_t\right]$	0.06

^aThe retrieval operator is often called an ‘observation operator’ in data assimilation literature.