Recent Applications in Representer-Based Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 289

In summary, the GCR algorithm has been found useful for data assimilation
solvers based on the representer expansion. Being applicable to non-symmetric
linear systems, the solver is more tolerant of symmetry errors in the adjoint
model, such as are present when the continuous adjoint equations are discretized.
The GCR solver is currently being used for a variety of weak-constraint ocean
data assimilation problems, and it has been implemented within the IOM data
assimilation software system ( Bennett et al. 2008 ; Muccino et al. 2008 ).

12.3 Diagnosis of Error Variances

The preceding analysis of the solver performance and interpretation in terms of
explained variance is contingent upon having correct descriptions of the model
and observation error covariances. Validation of B and R is thus of paramount
importance. This section outlines the posterior diagnosis strategy of Desroziers and
Ivanov ( 2001 ) for validating the errors B and R , with application to a large-scale
operational weather analysis system, the Naval Research Laboratory Atmospheric
Variational Data Assimilation System-Accelerated Representer, or (NAVDAS-AR;
Xu et al. 2005 ; Rosmond and Xu 2006 ).

12.3.1 Notation and Background Materials

First, recall some established results using the notation employed here. It may
be shown ( Lorenc 1986 ) that the analysis x a , the minimizer of the objective
function ( 12.8 ), is given by

x a D x b C K y Hx b /; . (12.16)

where K denotes the so-called Kalman gain ,

K D BH T . HBH T C R

/ 1 :

(12.17)

J

At this optimum, the value of the objective function is given by Bennett ( 1992 ),

x a / D d T D 1 d

J . ; (12.18)

where D D HBH T C R denotes the stabilized representer matrix ,and d D y Hx b
denotes the innovation vector . If the background and observation errors are correctly
modeled by B and R , it may be shown that the minimum value of J is a chi-squared

random variable with m degrees of freedom ( Bennett 1992 ),

x a / gD E f m gD m; E f J . (12.19)