Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 39

and

x.k C 1/ D x f .k/ C GΠ.k/ h.x f .k// z

Define an innovation

.k/ h x f .k/ 2 R m d.k/ D z (2.56)

and define

d.1 W N/ D d T .1/;d T .2/;:::;d T .N/ T 2 R Nm

(2.57)

Define

x.0/ x b .0/ T

B 1 x.0/ x b .0/

J b .x.0// D 1
2

(2.58)

J n .x.0/;G/ D 1

2 d T .1 W N/G T .P f / 1 Gd.1 W N/

(2.59)

x b .0/

Clearly, J b .x.0// measures the weighted squared distance between x.0/ and

and is called the nudging term that measures the weighted square of the

model error term in ( 2.55 ), and

J n .x.0/;G/

Rm is the model error covariance matrix
computed using the standard method used in the Kalman filter literature ( Lewis
et al. ( 2006 )).
Vidard et al. ( 2003 ) then pose the estimation problem as one of minimizing

P f 2 R

Nm

Q 3 .x.0/;G/ D J 2 .G/ C J b .x.0// C J n .x.0/;G/ (2.60)

where is defined in ( 2.22 ) and the nudged dynamics in ( 2.56 )isusedasa

strong constraint. This minimization is again solved by invoking the standard adjoint
method [ Lewis et al. ( 2006 )].
Following the arguments at the end of Sect. 2.3.1 , it can be readily verified that
the error vector e(1:N) which is a part of

J 2 .G/

J 2 .G/ in ( 2.22 ) is temporally correlated.

Hence, the correct formulation J 2 .G/ in ( 2.60 ) must be replaced by J 3 .G/ in ( 2.33 ).

Similarly, it can be verified that the innovation vector d.1 W N/ is also temporally

correlated and the in ( 2.60 ) by a similar correct form of the functional

that takes the serial correlation of

J n .x.0/;G/

W 2 R Nm Nm be the

d.1 W N/ into account. Let

serial correlation of d.1 W N/ . Then define

J 3 .x.0/;G/ D 1

2 d T .1 W N/G T W 1 Gd.1 W N/

(2.61)

Hence the correct formulation is to minimize

Q 3 .x.0/;G/ D J 3 .G/ C J b .x.0// C J 3 .x.0/;G/ (2.62)

W

We leave the computation of the elements of as an exercise to the reader.