A Weak Constraint 4D-Var Assimilation System for the Navy Coastal Ocean Model Using the Representer Method - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 364

15.3.2 Adjoint Derivation

Once the linear perturbation model was obtained, the adjoint model was derived by
transposition of the perturbation model as follows for both sigma layers and z-levels:

D n C 1
u

n C 1
u

D 0

u G n

2t
x

n C 1
z u

D n C 1
z u

z u / n 1 u n 1 C

.

h

/ n C 1 i 2

u

y

. z u

1 2t

x

G D G C

u

(15.6)

/ n C 1

u

y

. z u

u n 1
.

n 1 D n 1

/ n C 1

C

z u z u

z u

/ n 1 z u

C .

n 1
u

D n 1
u

/ n C 1

.

z u

and

D n C 1
u

n C 1
u

D 0

2t

G D G C

z u x

u

y

u

n 1
u

D n 1
u

C (15.7)

i a where denotes the adjoint variable associated with the state variable a at the

is a temporary variable. In ( 15.6 )and( 15.7 ) it is assumed
that the adjoint variables have been initialized at a prior time level. In practice, the
model is usually computer programmed by subroutines, with individual terms of
the model equations computed in separate subroutines. Similarly, the linearization
and the adjoint derivation were carried out one subroutine at a time, and care was
taken to ensure that symmetry between the linearized subroutine and its adjoint
was preserved. The entire linearized model was obtained once every subroutine was
linearized, and the entire adjoint was obtained with individual adjoint subroutines
appearing in reverse order as compared to the linearized model.
In practice, both the linearized and adjoint models were obtained with the help of
the Parametric Fortran compiler (PFC). Parametric Fortran is an extension of Fortran
that supports defining Fortran program templates by allowing the parameterization

time level i ,and