Geoscience Reference
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Then the observation
z
.j/
is given by
.j/
D
u
T
.jt/
C
V.j/
D
u
T
.0/
C
V.j/
z
(2.112)
V.j/
N.0;
2
/
where
is the Gaussian noise affecting the observations and
t
is
the time discretization used in the Euler scheme.
2.5.1
Analysis of Forward Nudging
Iterating (
2.107
), it can be verified that the forward solution of (
2.107
) at anytime is
given by
k
X
.k/
D
.1
g/
k
u
.1
g/
j
z
u
.0/
C
g
.k
1
j/
(2.113)
j
D
0
where
u
is the arbitrary initial condition used to start the forward run.
Substituting (
2.112
)into(
2.113
)weget
.0/
u
.N/
D
DPF
C
SPF
(2.114)
where the deterministic part,
DPF
,isgivenby
N
X
DPF
D
.1
g/
N
u
u
T
.0/
.1
g/
j
.0/
C
g
j
D
0
D
u
T
.0/
C
.1
g/
N
u
.0/
u
T
.0/
(2.115)
.0/
u
T
.0/
where
Œ
u
is the error in the initial condition. Similarly, the stochastic
part is
N
X
.1
g/
j
V.k
1
j/
SPF
D
g
(2.116)
j
D
0
whose mean is zero and the variance is given by
2
Var.SPF/
D
g
2
1
.1
g/
2N
1
(2.117)
1
.1
g/
2
Combining (
2.115
)and(
2.117
), it follows that
u
.N/
is a Gaussian random variable
whose mean is
DPF
and variance is given by
Var.SPF/
:
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