
200 Modeling and Inverse Problems in the Presence of Uncertainty
Thus taking K different sampling maps in Ω
C
, represented by the 1 × n-
dimensional matrices C
k
, k = 1, 2 , . . . , K, we construct the discrete distr ibu-
tion on Ω
K
C
=
K
O
i=1
Ω
C
(the k-fold cro ss products of Ω
C
)
P
S
=
K
X
k=1
∆
C
k
, (6.11)
where ∆
a
represents the Dirac measur e with atom at a. Using P
S
in (6.10),
we obtain the GFIM for multiple discrete observation methods taken contin-
uously over [t
0
, t
f
] given by
F (P
1
, P
S
, θ
0
) =
Z
t
f
t
0
K
X
k=1
1
σ
2
(t, C
k
)
∇
T
θ
(C
k
x(t; θ
0
)) ∇
θ
(C
k
x(t; θ
0
)) dP
1
(t)
=
Z
t
f
t
0
K
X
k=1
1
σ
2
(t, C
k
)
∇
T
θ
x(t; θ
0
)C
T
k
C
k
∇
θ
x(t; θ
0
)dP
1
(t)
=
Z
t
f
t
0
K
X
k=1
∇
T
θ
x(t; θ
0
)C
T
k
1
σ
2
(t, C
k
)
C
k
∇
θ
x(t; θ
0
)dP
1
(t)
=
Z
t
f
t