Observation Influence Diagnostic of a Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 93

Since V and R are positive definite covariance matrices, the matrix ( V C R is

positive definite as well. In fact by definition for a non-zero vectors z with real
entries the quantity z T

/

z D z T Vz C z T Rz 0 .

Let's consider the following theorem: If D is positive definite matrix then D 1 is
positive definite and defining
D 1 Df • ij g ;

. V C R / >

D Dfd ij g we have: • ii 1= d ii where the equality holds if and only

if d i 1 D D d ii 1 D d iiC 1 D D d in D 0 .

V C R) 1 Df • ij g are then larger than the
diagonal elements of ( V C R ). Moreover, if V Dfv ij g and R D diag

The diagonal elements of D 1 D .

. r i / we obtain

1
v ii C r i

ı ii (4.19)

0
@

1
A

0
: : :
r i
: : :
0

/ 1 R is

And since the i -diagonal element of ( V C R .ı i1 ;:::;ı in / D ı ii r i

r i
v ii C r i

ı ii r i (4.20)

From ( 4.18 ) considering that the product of two positive definite matrix is still a
positive definite matrix

r i
v ii C r i D

v ii
v ii C r i <1

0<S ii D 1 ı ii r i 1 (4.21)

( 4.21 ) proves that the diagonal elements of the influence matrix for the weighted
regression DA scheme are bound between (0,1).

4.3.3 Toy Model

Let's assume a simplified model with two observations, each coincident with a point
of the background - that is H D I 2 . Assume the error of the background at the two

1˛
˛1

, with variance

,thatisB D ¢ b b

locations have correlation ˛ ,andthat

1ˇ
ˇ1

with variance

similarly R D o ¢ o

and correlation “ . For this simple case S

is obtained from ( 4.14 )