The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters

Abstract

Adjoint techniques are effective tools for the analysis and optimization of the observation performance on reducing the errors in the forecasts produced by atmospheric data assimilation systems (DASs). This chapter provides a detailed exposure of the equations that allow the extension of the adjoint-DAS applications from observation sensitivity and forecast impact assessment to diagnosis and tuning of parameters in the observation and background error covariance representation. The error covariance sensitivity analysis allows the identification of those parameters of potentially large impact on the forecast error reduction and provides a first-order diagnostic to parameter specification. A proof-of-concept is presented together with comparative results of observation impact assessment and sensitivity analysis obtained with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System – Accelerated Representer (NAVDAS-AR) and the Navy Operational Global Atmospheric Prediction System (NOGAPS).

Appendix

All vectors are represented in column format and the superscript ^T denotes the transposition operator. The elementwise (Hadamard) product of two vectors $\mathbf{u} \in {\mathcal{R}}^{n}$ and $\mathbf{v} \in {\mathcal{R}}^{n}$ is denoted u ∘ v and is the vector $\mathbf{w} \in {\mathcal{R}}^{n}$ with entries defined as

$$\mathbf{w} = \mathbf{u} \circ \mathbf{v},\,\,w_{i} = u_{i}v_{i},\,i = 1 : n$$

(9.60)

For two matrices of the same order $\mathbf{X},\mathbf{Y} \in {\mathcal{R}}^{n\times m}$

$$\langle \mathbf{X},\mathbf{Y}\rangle _{{\mathcal{R}}^{n\times m}} = Tr\left (\mathbf{X}{\mathbf{Y}}^{\mathrm{T}}\right ) = Tr\left ({\mathbf{X}}^{\mathrm{T}}\mathbf{Y}\right )$$

(9.61)

denotes the Frobenius inner product that is expressed in terms of the matrix trace operator Tr. Given the vectors $\mathbf{u} \in {\mathcal{R}}^{n}$, $\mathbf{v} \in {\mathcal{R}}^{m}$, and the matrix $\mathbf{X} \in {\mathcal{R}}^{n\times m}$,

$$\langle \mathbf{u},\mathbf{X}\mathbf{v}\rangle _{{\mathcal{R}}^{n}} ={ \mathbf{u}}^{\mathrm{T}}\mathbf{X}\mathbf{v} =\langle \mathbf{u}{\mathbf{v}}^{\mathrm{T}},\mathbf{X}\rangle _{{ \mathcal{R}}^{n\times m}}$$

(9.62)

Given a functional $e : {\mathcal{R}}^{n\times m} \rightarrow \mathcal{R}$ of matrix argument $\mathbf{X} \in {\mathcal{R}}^{n\times m}$, the sensitivity of e with respect to X is the matrix of the first order partial derivatives denoted as

$$\frac{\partial e} {\partial \mathbf{X}} = \left [ \frac{\partial e} {\partial X_{i,j}}\right ]_{i=1,n;j=1,m} \in {\mathcal{R}}^{n\times m}$$

(9.63)

The first order variation δe induced by a variation δ X is expressed as

$$\delta e = \left \langle \frac{\partial e} {\partial \mathbf{X}},\delta \mathbf{X}\right \rangle _{{\mathcal{R}}^{n\times m}} = Tr\left [ \frac{\partial e} {\partial \mathbf{X}}{(\delta \mathbf{X})}^{\mathrm{T}}\right ]$$

(9.64)

For a nonsingular matrix $\mathbf{X} \in {\mathcal{R}}^{n\times n}$, the first order variation δ X ^− 1 in the inverse matrix X ^− 1 induced by a variation δ X is expressed as

$$\delta {\mathbf{X}}^{-1} = -{\mathbf{X}}^{-1}\delta \mathbf{X}{\mathbf{X}}^{-1}$$

(9.65)