Geoscience Reference
In-Depth Information
then it can be verified that
e.1/M T H T C e.2/.M 2 / T H T
r x.0/ f.x.0// D 1
2
(2.76)
0
a
and
0
a 2
,weget
M T H T D
.M 2 / T H T D
But since
0
ae.1/ C a 2 e.2/
r x.0/ f D 1
2
(2.77)
this in turn implies that
f.x.0//
is a constant with respect to the first component of
x.0/
. Hence, the initial condition
x.0/
cannot be recovered from z
.1/
and z
.2/
2.4.2
Observer-Based Nudging: Linear Dynamics
Let
x.k C 1/ D M x.k/
(2.78)
where
x.k/
is the true linear dynamical state with true initial condition
x.0/
and
z
.k/ D H x.k/ C V.k/
(2.79)
.M;H/
the observations. It is assumed that the pair
is observable (See Sect. 2.4.1 ).
Let the observer be given by ( Luenberger 1964 , 1971 ) the dynamics
x.k C 1/ D Mx.k/ C G.
.k/ Hx.k//
z
(2.80)
G 2 R n m . The idea of the observer is that the observer state
x.k/
where
is an
x.k/
estimate of the true state
. In the parlance of meteorology this observer is
called the nudged dynamics ( Anthes ( 1974 )) and the matrix
G
is called the nudging
coefficient.
To analyze the behavior of ( 2.80 ), using ( 2.79 ) and simplifying, we obtain
e.k/ D x.k/ x.k/
(2.81)
Subtracting ( 2.78 ) from ( 2.80 ), using ( 2.81 ) and simplifying, we obtain
e.k C 1/ D .M GH/e.k/ C GV.k/
(2.82)
Since it is given that the pair
.M:H/
is observable, by the fact 2.1 in Sect. 2.4.1 ,
G 2 R n m such that
there exists a matrix
.M GH/
is a Hurwitz matrix. Then
setting
A D .M GH/
, from ( 2.82 ) we obtain
 
Search WWH ::




Custom Search