Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 230

Substituting for r i from ( 10.3 ), using vector notation rather than i indices, such

G N / T , the cost function is given by that G D . G 1 ; G 2 ;:::;

J D 1 / C 1

2

/ T R 1 . z T P 1 z

2 .

d Gz d Gz : (10.5)

The minimal solution for ( 10.5 ) is the analysis increment, z a , that yields

@ J =@ z D

0 ,givenby

z a D G T R 1 G P 1 1 G T R 1 d

C ; (10.6)

where the transpose to the tangent-linear integration, G T , is the adjoint model
integrated backwards over

. Equation ( 10.6 ) is the solution to the 4D-Var
problem in model space. The analysis increment is the perturbation applied to the
initial conditions and forcing such that the residuals ( 10.3 ) are minimized. This is
the form used in the incremental scheme shown by Courtier et al. ( 1994 ) and used
by the European Centre for Medium-range Weather Forecasting (ECMWF) as well
as in the ocean as shown in studies such as Weaver et al. ( 2003 ), Powell et al. ( 2008 ),
and Broquet et al. ( 2009 ). Simplifying ( 10.6 ) by replacing G and G T with H and
H T , respectively, results in the solution to 3D-Var, which ignores the time-dependent
dynamics of the system.
The solution ( 10.6 ) may be rearranged using the Woodbury Identity ( Golub and
Van Loan 1989 ) such that the minimization is performed in the data space to yield

.t i ;t 0 /

PG T GPG T R 1 d

z a D C : (10.7)

This is used by the Physical-space Statistical Analysis System (PSAS) ( Courtier
1997 ) and “Representer” ( Chua and Bennett 2001 ; Bennett 2002 ) methods. These
data-space methods have been used successfully for research in both atmospheric
( Chua et al. 2009 ) and oceanic applications ( Di Lorenzo et al. 2007 ; Muccino et al.
2008 ; Kurapov et al. 2007 ). For this discussion,

PG T GPG T R 1

K D C ; (10.8)

will be referred to as the “Kalman Gain Matrix.” The total solution to the data-space
minimization procedure is given by x a D x b C Kd D x b C z a .
Due to the large size of the typical geophysical problem, one cannot solve for z a
explicitly, and the solution is found iteratively via a conjugate-gradient algorithm.
This minimization iteration to find an approximation to ( 10.7 ) is referred to as
the “inner-loop.” The estimate of z a is revealed at the conclusion (determined by
convergence or reaching a maximum number of inner-loops) of the inner-loops. If
the problem is linear, then the global minimum is reached when the inner-loops
converge.