A Question of Adequacy of Observations in Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 113

x.k/ D x.k 1/.1 C T t/ C C T t
D x.k 1/ .1 c/ C c

(5.2)

where is a nondimensional exchange coefficient (D0.25) based on typ-

ical values of parameters involved in the air/sea interaction [See Liu et al. ( 1992 )].
A closed form solution to ( 5.2 )is:

c D C T t

x.k/ D .1 c/ k .x.0/ / C

(5.3)

5.2.3 Sensitivities

The solution ( 5.3 ) is nonlinear in the elements of control. The associated sensitivi-
ties, again nonlinear in control, are given by:

@x.k/
@x.0/

D .1 c/ k

@x.k/
@

D Œ1 .1 c/ k

(5.4)

@x.k/
@

D k

10 .1 c/ k 1 Œx.0/

D 10c

where . This change of control-element variable is a form of precondi-

tioning that leads to faster convergence of the optimization process described below
[See Gill et al. ( 1981 ) for a discussion of preconditioning]. The true control vector
is taken to be

Y D Œx.0/;; D Œ1 ı C;11 ı C;2:5

while the incorrect control

Y 0

D Œx 0 .0/; 0 ; 0 D Œ2 ı C;10 ı C;3:0 is . Thus, the difference between true and

Y Y 0

D Œ 1 ı C; C 1 ı C; 0:5 erroneous control is given by . Entries in Table 5.1

exhibit the time evolution of the three sensitivities for correct and incorrect control
(to be used in the numerical experiments). Under the assumption that true elements
of control are unknown in practice, incorrect sensitivities are used to initiate the
variational data assimilation process.

5.2.4 Cost Function for Data Assimilation

We assume that observations of air temperature are made at a subset of the points

along the trajectory displayed in Fig. 5.1 . Indices associated with these observation
points are represented by the sequence f i W 1 ; 2 ;:::; M g. With z

M

. i / representing

the observation at point i , the cost function takes the form