Recent Applications in Representer-Based Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 283

objective function corresponds to an extremum of the original objective function.
When nonlinearity is present, the incremental objective function is used to build
an iterative solver for the original, nonlinear, data assimilation problem. In this
article we assume that some linearization strategy has been selected, e.g., the tangent
linearization proposed in Courtier et al. ( 1994 ) or the bounded iterate strategy of
Bennett and Thorburn ( 1992 ), so that the so-called inner loop solver must minimize
a strictly quadratic objective function. Henceforth, we shall restrict our attention to
the objective function,

f f b / T F 1 . f f b /

J Πx .t 0 /; f D .

.t 0 / x b / T B 1 . .t 0 / x b /

C . x x (12.4)

/ T R 1 . C . y Hx y Hx /;

2 R m n is a linear approximation to the operator

where the matrix H ,and

inhomogeneities resulting from the linearization have been absorbed into x b , f b ,
and y .
There are practical considerations which make the implementation of W4D-Var
considerably more complex than 4D-Var for realistic models. The first issue is the
dimensionality of the unknown vectors, which has consequences for the design
and implementation of solvers for minimizing

H

J . Assuming the state vector x .t/

is of dimension T

is the cardinality of the time interval under consideration. The dimension of the
space-time covariance matrix F is formally the square of this. The second key
issue is scientific, and relates to the determination of the error covariances B and
F . Quantitative estimation of these objects requires vast amounts of data which are
rarely available; in practice they are often parameterized in terms of a spatially-
or temporally-varying variance function, and a set of correlation scales for the
orthogonal coordinate directions.
Here we review recent developments associated with the application of
representer-based solvers ( Bennett 1992 ) to 4D-Var and W4D-Var problems, an
approach which is the foundation for the so-called dual form of variational data
assimilation ( Courtier 1997 ). Recall that the minimizer of the objective function is
the solution to 2 r J .

n , then the model forcing f may be as large as T n ,where

x / D 0 ;applieditto( 12.1 ) yields,

B 1 C H T R 1 H x D H T R 1 y C B 1 x b ; . / (12.5)

where uniqueness is assured provided that B is of full rank. Equivalently, the
solution can be expressed as the sum of the background and a linear combination of
representer functions x D x b C BH T x , yielding the equation for the dual variables x ,

HBH T C R

/ x D y Hx b :

.

(12.6)