A 4D-Var Analysis System for the California Current: A Prototype for an Operational Regional Ocean Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 342

by solving an equivalent system of linear simultaneous equations. The latter is
achieved iteratively by the direct minimization of using a conjugate gradient

descent algorithm. Similarly, none of the implied matrix multiplications in ( 14.6 )
and ( 14.7 ) are ever performed explicitly, but instead involve the direct integration
of a model. Specifically, G represents an integration of TLROMS sampled at the
appropriate space-time observation points, while G T is the adjoint of TLROMS,
hereafter referred to as ADROMS. The operator G is a linear map from the space
of the control vector to the space of the observations, while G T is a linear map
from the dual of the observation space to the dual of the control vector space. The
prior covariance matrix D is also described by a model using the diffusion operator
approach introduced by Derber and Rosati ( 1989 ). Only the observation error
covariance R is explicitly treated as a matrix since for the applications considered
here it is assumed to have a simple diagonal structure (i.e. spatially and temporally
uncorrelated errors).
ROMS 4D-Var also supports weak constraint data assimilation in which errors in
the model formulation can be admitted in the calculation of the ocean circulation
estimate. However, in the applications presented here, no explicit account is
taken of model errors (i.e. the so-called strong constraint problem), so important
considerations as they relate to the treatment of model errors will not be discussed
further. Full details of weak constraint 4D-Var in ROMS can be found in Moore
et al. ( 2011a ,b).

J

14.2.2 Inner- and Outer-Loops

In general, we are interested in identifying the minimum of J NL in ( 14.2 ), and

in practice this proceeds via a sequence of linear minimizations of in ( 14.5 ).

Each minimization of ( 14.5 ) proceeds iteratively where each iteration is referred
to as an inner-loop. During the first set of inner-loop iterations, G and G T

J

are

linearized about the time evolving prior circulation estimate x b .t/ resulting from

the prior control vector z b .The k th sequence of inner-loops will be represented

in sequel by the superscript k . For the first sequence of inner-loops k D 1 ,and

z 1 has been identified that minimizes

when the increment ı J , a new circulation

estimate x 1 .t/ is computed using the updated control vector z 1 D z 1 ,and z b C ı

a new sequence of inner-loops performed during which G and G T are linearized

about x 1 .t/

. The repeated application of this procedure is equivalent to minimizing
( 14.2 ) using a Gauss-Newton method ( Lawless et al. 2005 ), and the updates of the
circulation x k .t/ about which TLROMS and ADROMS are linearized are referred

to as outer-loops.

14.2.3 Conjugate Gradient Descent and Preconditioning

Following the customary approach adopted in NWP, the minimization of ( 14.5 )
in ROMS is preconditioned by a change of variable, namely D 1=2 ı ı v D z .