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 341

where M b .t i ;t 0 / denotes the linear operator (also sometimes referred to as the

resolvent or propagator matrix) that advances the initial increment forward in

time, and the subscript b indicates that the linearization is about the prior circulation
x b .t/

ı x .t 0 /

subject to the prior forcing and prior open boundary conditions f b and b b .

Since ı x .t i / also depends on ı f .t/ and ı b .t/ , it is sometimes more convenient to

express TLROMS as:

ı x .t i / D M b .t i ;t 0 /ı z (14.4)

where ı z is the vector of control increments composed of ı x .t 0 / ,and ı f and ı b at all

times in the interval is the linear operator that maps a

control vector increment into a state vector increment.
Using the incremental approximation, the cost function can be re-expressed as:

Œt 0 ;t N . Therefore, M b .t i ;t 0 /

z T D 1 ı

T R 1 .

J D ı z C . d G ı z / d G ı z / (14.5)

y o where d is referred to as the innovation vector, and G represents

the convolution in time of

D H. x b /

M b with H ,where H is the tangent linearization of the

observation operator H . Since the constraints in ( 14.5 ) are linear in ı z , a unique

global minimum value of exists. The vector of control increments that yields the

optimal circulation estimate will be denoted as

J

ı z a and the vector of the total control

vector as z a D z b C ı z a . At the minimum of ( 14.5 ) the gradient is @J=ı z D 0 ,and

the optimal control vector increment is given by z a D Kd where K is referred to as

the Kalman gain matrix. The Kalman gain matrix can be expressed in two equivalent
forms as:

ı

D 1 C G T R 1 G / 1 G T R 1 K D . (14.6)

K D DG T GDG T / 1 : . C R (14.7)

Equation ( 14.6 ) is referred to as the primal form, and corresponds to the case
where the minimum of z directly in control

space. Conversely, ( 14.7 ) is referred to as the dual form, and corresponds to the
case where the minimum of

J in ( 14.5 ) is found by searching for ı

z in observation space.
Both approaches yield the same optimal circulation estimate, as demonstrated in
ROMS by Moore et al. ( 2011b ). The main advantage of the dual formulation over
the primal formulation is that the control vector can be expanded in the former
to include corrections for model error without any increase in the dimension of
the matrix inverse in ( 14.7 ). Until recently the dual formulation was known to
suffer from poor convergence properties ( El Akkraoui and Gauthier 2010 ; Moore
et al. 2011b ) making it difficult to use for large problems. However, Gratton and
Tshimanga ( 2009 ) have shown that the same rate of convergence of the primal and
dual formulations toward the minimum of

J is found by searching for ı

can be guaranteed by using a restricted
preconditioned conjugate gradient method (RPCG), as confirmed recently by Gurol
et al. ( 2013 ) in two complex ocean general circulation models, including ROMS.
It is important to note that while ( 14.6 )and( 14.7 ) are written in matrix notation,
the matrix inverse in each case is never directly evaluated, but is instead computed

J