Environmental Engineering Reference
In-Depth Information
2.4 Peculiarities of Dual Estimates and Sensitivity Formulas
We now discuss the main features of the dual estimate (
2.25
), or its simplification
(
2.26
), and show the usefulness of the adjoint estimates in the study of sensitivity
of mean concentration
J
i
(ˆ)
to variations in the discharge rates and positions of the
sources as well as in the initial distribution
0
of nutrient.
In environmental monitoring, the adjoint estimate (
2.25
) is a good complement
to the direct mean concentration estimate
J
i
(ˆ)
ˆ
(
r
)
. One can use either direct or adjoint
estimates depending on the specific situation. Assume, for example, that the mean
concentration
J
i
(ˆ)
of a nutrient is monitored in
N
ecologically important zones
ʩ
i
of domain
D
. If the number of zones
N
is large enough then it is
better to solve problem (
2.4
)-(
2.11
) and use direct estimate of
J
i
(ˆ)
(
i
=
1
,...,
N
)
in each zone. On
the other hand, if number
N
is rather small then it is more effective and economical
to solve adjoint problem (
2.18
)-(
2.24
) and use adjoint estimate (
2.25
). Unlike the
direct mean concentration estimate of nutrient, the adjoint estimate (
2.25
) permits to
explicitly evaluate the contribution of each source to value
J
i
(ˆ)
.
In the case of invariable emission rates
(
Q
j
(
t
)
=
Q
j
)
, evaluation (
2.26
) becomes
even simpler:
N
J
i
(ˆ)
=
Q
j
w
ij
,
(2.33)
j
=
1
where
T
w
ij
=
g
i
(
r
j
,
t
)
dt
.
(2.34)
0
Each weight
w
ij
depends only on the adjoint solution and characterizes the con-
tribution of the source with emission rate
Q
j
to the mean concentration
J
i
(ˆ)
in
ʩ
i
.
What is then the basic difference between the direct and adjoint estimates of the
mean concentration of nutrient
J
i
(ˆ)
? The direct estimate, relating to the solution
ˆ(
, but depends on
the discharge rates
Q
j
and position
r
j
of sources, and also on the initial distribution of
nutrient
r
,
t
)
of problem (
2.4
)-(
2.11
), is independent of a concrete zone
ʩ
0
in
D
. For this reason such a estimate is preferable if one needs to know
the concentration of a substance in many zones of
D
, or in each point of
D
ˆ
(
r
)
.
However, in themodel sensitivity study, this approach requiresmuch computing time,
because the solution
×
(
0
,
T
)
of problem (
2.4
)-(
2.11
) must be recalculated whenever
new values of the parameters
Q
j
,
r
j
or
ˆ(
r
,
t
)
0
ˆ
(
r
)
are used. Unlike it, the solutions of
0
adjoint problem
g
i
(
r
j
,
t
)
depend on
ʩ
i
zone, but are independent of
Q
j
,
r
j
or
ˆ
(
r
)
.
In the adjoint evaluation (
2.25
),
g
i
(
serves as the weight function characterizing
the model response to these three parameters. Since the problem is linear, Eq. (
2.25
)
leads to the main sensitivity formula
r
j
,
t
)
T
N
0
ʴ
J
i
(ˆ)
=
g
i
(
r
j
,
t
)ʴ
Q
j
(
t
)
dt
+
g
i
(
r
,
0
)ʴˆ
(
r
)
dr
(2.35)
0
D
j
=
1
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