Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 41

.q W q C n 1/ 2 R Nm denotes the column vector of
observations on the l. h. s. of ( 2.66 )and

To simplify the notation, z

denotes the column
vector of observation noise in the second term on the r. h. s. of ( 2.66 ). Consequently
( 2.66 ) becomes

V.q W q C N 1/

z .q W q C N 1/ D H.0 W N 1/x.q/ C V.q W q C N 1/ (2.67)

mathematically, observability relates to solving the linear least squares problem
( 2.67 )for .

From the standard linear least squares theory (Chap. 5 , Lewis et al. ( 2006 )), the
best

x.q/

x.q/ is the one that minimizes

f.x.q// D 1

/ 1 e.q W q C N 1/ >

2 <e.q W q C N 1/;.I ˝ R

(2.68)

where

e.q W q C N 1/ D z .q W q C N 1/ H.0 W N 1/x.q/ (2.69)

I 2 R N N ,andR2

is the vector of residuals, I ˝ R is the Kronecker product of

R m m .
It can be verified (Chap. 5 , Lewis et al. ( 2006 )) that the minimizer is given by

x ls .q/ D H T .0 W N 1/.I ˝ R / 1 H.0 W N 1/ 1

H T .0 W N 1/.I ˝ R

.q W q C N 1/ / 1 z (2.70)

Hence the solution exists and is unique if the observability matrix

O N D H T .0 W N 1/.I ˝ R/ 1 H.0 W N 1/ (2.71)

N X

.M k 1 / T .H T R 1 H/M k 1

D

k D 0

is nonsingular. A necessary and sufficient condition for O N to be nonsingular is that

the matrix ( Bernstein ( 2009 ))

2 3

H
HM
HM 2
HM N 1

4 5

2 R Nm n H.0 W N 1/ D (2.72)