Toward New Applications of the Adjoint Sensitivity Tools in Data Assimilation

Abstract

Novel applications of the adjoint-based sensitivity tools are investigated to obtain a priori guidance on the forecast impact of modeling correlated observational errors in a four-dimensional variational data assimilation system (4D-Var DAS) . A synergistic framework is considered that combines a posteriori estimates to the observation error covariance (R) and derivative information extracted from the adjoint-DAS forecast error R-sensitivity (FSR ). It is explained that the FSR approach allows the analysis of structured error correlation models and estimation of their potential impact on reducing the forecast errors. Theoretical aspects are discussed and a proof-of-concept is provided with Lorenz’s 40-variable model. The practical ability to exercise these new adjoint capabilities is shown in experiments performed with the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) and the Navy’s Global Environmental Model (NAVGEM). In particular, the FSR analysis of radiances assimilated from the Infrared Atmospheric Sounding Interferometer (IASI) indicates that modeling inter-channel observation error correlations may provide an increased benefit to the forecasts, as compared with tuning procedures that ignore the error correlations and only adjust the assigned observation error variance parameters.

Appendix

As shown by Desroziers et al. (2005), the error covariance estimates $\widetilde{\mathbf{R}}$ in (3) and $\widetilde{\mathbf{B}}$ in (4) are expressed, respectively, as

$$\begin{aligned} {\widetilde{\mathbf{R}}}= & {} \mathbf{R}\left( \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T} + \mathbf{R}\right) ^{-1}\left( \mathbf{H}{} \mathbf{B}_t\mathbf{H}^\mathrm{T} + \mathbf{R}_t\right) \end{aligned}$$

(39)

$$\begin{aligned} \mathbf{H}{\widetilde{\mathbf{B}}}\mathbf{H}^\mathrm{T}= & {} \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T}\left( \mathbf{H}{} \mathbf{B}\mathbf{H}^\mathrm{T} + \mathbf{R}\right) ^{-1} \left( \mathbf{H}{} \mathbf{B}_t\mathbf{H}^\mathrm{T} + \mathbf{R}_t\right) \end{aligned}$$

(40)

By analogy with (1) and (2), the analysis state associated with the covariancespecification $({\widetilde{\mathbf{B}}},{\widetilde{\mathbf{R}}})$ is expressed as

$$\begin{aligned} \widetilde{\mathbf{x}}^a= & {} \mathbf{x}^b + \widetilde{\mathbf{K}}\left[ \mathbf{y}-\mathbf{h}(\mathbf{x}^b)\right] \end{aligned}$$

(41)

$$\begin{aligned} \widetilde{\mathbf{K}}= & {} \widetilde{\mathbf{B}}{} \mathbf{H}^\mathrm{T}\left( \mathbf{H}{\widetilde{\mathbf{B}}}{} \mathbf{H}^\mathrm{T} + \widetilde{\mathbf{R}}\right) ^{-1} \end{aligned}$$

(42)

By adding (39) and (40), it is noticed that the estimates $({\widetilde{\mathbf{B}}},{\widetilde{\mathbf{R}}})$ are consistent with the innovation error covariance,

$$\begin{aligned} \mathbf{H}{\widetilde{\mathbf{B}}}{} \mathbf{H}^\mathrm{T} + \widetilde{\mathbf{R}} = \mathbf{H}{} \mathbf{B}_t\mathbf{H}^\mathrm{T} + \mathbf{R}_t \end{aligned}$$

(43)

and this property has prompted research on covariance tuning procedures based on the diagnosis estimates (39), (40). However, as explained by Daescu and Langland (2013c), the operator $\mathbf{H}{} \mathbf{K}$ remains invariant when the status quo specification $(\mathbf{B},\mathbf{R})$ is replaced by the estimates $(\widetilde{\mathbf{B}},\widetilde{\mathbf{R}})$,

$$\begin{aligned} \mathbf{H}{} \mathbf{K} = \mathbf{H}\widetilde{\mathbf{K}} \end{aligned}$$

(44)

The relationship (44) was established in Daescu and Langland (2013c) as follows.

$$\begin{aligned} \mathbf{H}{} \mathbf{\widetilde{\mathbf{K}}}&\mathop {=}\limits ^{(42)} \mathbf{H} \widetilde{\mathbf{B}}{} \mathbf{H}^\mathrm{T}\left( \mathbf{H}{\widetilde{\mathbf{B}}}{} \mathbf{H}^\mathrm{T} + \widetilde{\mathbf{R}}\right) ^{-1}\\&\mathop {=}\limits ^{(40)} \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T}\left( \mathbf{H}\mathbf{B}{} \mathbf{H}^\mathrm{T} + \mathbf{R}\right) ^{-1} \left( \mathbf{H}{} \mathbf{B}_t\mathbf{H}^\mathrm{T} + \mathbf{R}_t\right) \left( \mathbf{H}{\widetilde{\mathbf{B}}}{} \mathbf{H}^\mathrm{T} + \widetilde{\mathbf{R}}\right) ^{-1}\\&\mathop {=}\limits ^{(43)} \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T} \left( \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T} + \mathbf{R}\right) ^{-1}\\&\mathop {=}\limits ^{(2)} \mathbf{H}{} \mathbf{K} \end{aligned}$$

From (1), (41), and (44) it follows that

$$\begin{aligned} \mathbf{H}{} \mathbf{x}^a = \mathbf{H}\widetilde{\mathbf{x}}^a \end{aligned}$$

(45)

and therefore, replacing the status quo model $(\mathbf{B}, \mathbf{R})$ by the a posteriori diagnosis model $(\widetilde{\mathbf{B}}, \widetilde{\mathbf{R}})$ has no impact on the observation-space representation of the analysis. In particular, if $\mathbf{H}$ is the identity operator, $\mathbf{H} = \mathbf{I}$, then the covariance specification $(\widetilde{\mathbf{B}}, \widetilde{\mathbf{R}})$ has no impact on the analysis,

$$\begin{aligned} \mathbf{x}^a = \widetilde{\mathbf{x}}^a \end{aligned}$$

(46)

It is also noticed that the equivalence between (5) and (39) is established from the identities

$$\begin{aligned} \mathbf{I} - \mathbf{HK} = \mathbf{R}\left( \mathbf{H}{} \mathbf{B}{} \mathbf{H}^\mathrm{T} + \mathbf{R}\right) ^{-1}; \,\,\,\,\,\, \left( \mathbf{I} - \mathbf{HK_t}\right) ^{-1}{} \mathbf{R}_t = \mathbf{H}{} \mathbf{B}_t\mathbf{H}^\mathrm{T} + \mathbf{R}_t \end{aligned}$$

(47)