Observation Influence Diagnostic of a Data Assimilation System

Abstract

The influence matrix is used in ordinary least-squares applications for monitoring statistical multiple-regression analyses. Concepts related to the influence matrix provide diagnostics on the influence of individual data on the analysis, the analysis change that would occur by leaving one observation out, and the effective information content (degrees of freedom for signal) in any sub-set of the analysed data. In this paper, the corresponding concepts are derived in the context of linear statistical data assimilation in Numerical Weather Prediction. An approximate method to compute the diagonal elements of the influence matrix (the self-sensitivities) has been developed for a large-dimension variational data assimilation system (the 4D-Var system of the European Centre for Medium-Range Weather Forecasts). Results show that, in the ECMWF operational system, 18 % of the global influence is due to the assimilated observations, and the complementary 82 % is the influence of the prior (background) information, a short-range forecast containing information from earlier assimilated observations. About 20 % of the observational information is currently provided by surface-based observing systems, and 80 % by satellite systems.A toy-model is developed to illustrate how the observation influence depends on the data assimilation covariance matrices. In particular, the role of high-correlated observation error and high-correlated background error with respect to uncorrelated ones is presented. Low-influence data points usually occur in data-rich areas, while high-influence data points are in data-sparse areas or in dynamically active regions. Background error correlations also play an important role: high correlation diminishes the observation influence and amplifies the importance of the surrounding real and pseudo observations (prior information in observation space). To increase the observation influence in presence of high correlated background error is necessary to introduce the observation error correlation but also observation and background error variances must be of similar size. Incorrect specifications of background and observation error covariance matrices can be identified, interpreted and better understood by the use of influence matrix diagnostics for the variety of observation types and observed variables used in the data assimilation system.

Appendix

Influence Matrix Calculation in Weighted Regression Data Assimilation Scheme

Under the frequentist approach, the regression equations for observation

$$\mathbf{y} = \mathbf{H}\boldsymbol{\theta } +\boldsymbol{ \epsilon }_{o}$$

and for background

$$\mathbf{x}_{b} =\boldsymbol{\theta } +\boldsymbol{\epsilon }_{b}$$

are assumed to have uncorrelated error vectors $\boldsymbol{\epsilon }_{\mathbf{o}}$ and $\boldsymbol{\epsilon }_{\mathrm{b}}$, zero vector means and variance matrices R and B, respectively. The $\boldsymbol{\theta }$ parameter is the unknown system state (x) of dimension n. These regression equations are summarized as a weighted regression

$$\mathbf{z} = \mathbf{X}\boldsymbol{\theta } +\boldsymbol{ \epsilon }$$

where $\mathbf{z} ={ \left [{\mathbf{y}}^{T}\mathbf{x}_{b}^{T}\right ]}^{T}$ is (m + n) ×1; $\mathbf{X} ={ \left [{\mathbf{H}}^{T}\mathbf{I}_{n}\right ]}^{T}$ is (m + n) ×n and $\boldsymbol{\epsilon } = {[\boldsymbol{\epsilon }_{o}\boldsymbol{\epsilon }_{b}]}^{T}$ is (m + n) ×1 with zero mean and variances matrix

$$\boldsymbol{\Omega } = \left (\begin{array}{cc} \mathbf{R}& 0\\ 0 &\mathbf{B}\\ \end{array} \right )$$

The generalized LS solution for $\boldsymbol{\theta }$ is BLUE and is given by

$$\boldsymbol{\hat{\theta }}= {({\mathbf{X}}^{T}{\boldsymbol{\Omega }}^{-1}\mathbf{X})}^{-1}{\mathbf{X}}^{T}{\boldsymbol{\Omega }}^{-1}\mathbf{z}$$

(4.34)

see Talagrand (1997). After some algebra this equation equals (4.11). Thus

$$\mathbf{z} = \mathbf{X}\boldsymbol{\hat{\theta }} ={ \left [{\mathbf{H}}^{T}\mathbf{x}_{ a}^{T}\mathbf{x}_{ a}^{T}\right ]}^{T} = \mathbf{X}{({\mathbf{X}}^{T}{\boldsymbol{\Omega }}^{-1}\mathbf{X})}^{-1}{\mathbf{X}}^{T}{\boldsymbol{\Omega }}^{-1}\mathbf{z}$$

and by (4.5) the influence matrix becomes

$$\mathbf{S}_{\mathrm{zz}} = \frac{\partial \mathbf{\hat{z}}} {\partial \mathbf{z}} = \frac{\partial \boldsymbol{\hat{\theta }}} {\partial \mathbf{z}} = \left (\begin{array}{cc} \mathbf{S}_{\mathit{yy}} & \mathbf{S}_{\mathit{yb}} \\ \mathbf{S}_{\mathit{by}} & \mathbf{S}_{\mathit{bb}}\\ \end{array} \right ) = \left (\begin{array}{cc} {\mathbf{R}}^{-1}{\mathbf{HAH}}^{T}&{\mathbf{R}}^{-1}\mathbf{HA} \\ {\mathbf{B}}^{-1}{\mathbf{AH}}^{T} & {\mathbf{B}}^{-1}\mathbf{A}\\ \end{array} \right )$$

where $\mathbf{S}_{yy} = \frac{\partial \mathbf{Hx}_{a}} {\partial \mathbf{y}} ;\mathbf{S}_{yb} = \frac{\partial \mathbf{x}_{a}} {\partial \mathbf{y}} ;\mathbf{S}_{\mathit{by}} = \frac{\partial \mathbf{Hx}_{a}} {\partial \mathbf{x}_{b}} ;\mathbf{S}_{\mathit{bb}} = \frac{\partial \mathbf{x}_{a}} {\partial \mathbf{x}_{b}}$. Note that S _yy = S as defined in (4.4).

Generalized LS regression is different from ordinary LS because the influence matrix is not symmetric anymore. For idempotence, using (4.33) it easy to show that $\mathbf{S}_{\mathrm{zz}}\mathbf{S}_{\mathrm{zz}} = \mathbf{S}_{\mathrm{zz}}.$ Finally,

$$\mathbf{S}_{\mathit{bb}} ={ \mathbf{B}}^{-1}\mathbf{A} = \mathbf{I}_{ n} -{\mathbf{H}}^{T}{\mathbf{R}}^{-1}\mathbf{HA}$$

hence,

$$\mathit{tr}(\mathbf{S}_{\mathit{bb}}) = n -\mathit{tr}({\mathbf{H}}^{T}{\mathbf{R}}^{-1}\mathbf{HA}) = n -\mathit{tr}(\mathbf{S}_{\mathit{ yy}})$$

it follows that

$$\mathit{tr}(\mathbf{S}_{\mathit{zz}}) = \mathit{tr}(\mathbf{S}_{\mathit{yy}}) + \mathit{tr}(\mathbf{S}_{\mathit{bb}}) = n$$

The trace of the influence matrix is still equal to the parameter’s dimension.