290
8. Applications of Groundwater Quality Models
parameters. By solving the management problem for each realization, a great
number of realizations of management decisions are obtained, then the
expected value and the variance of the optimal decision can be calculated,
and consequently, its confidence interval can be estimated (see Section 7.4.4).
Another approach is to formulate the management problem into a stochastic, rather than deterministic problem. Wagner and Gore1ick (1987)
modified deterministic constraint (8.3.43) into the following stochastic
constraint:
Prob{Zl :5: C} ~ n,
(8.3.46)
that is, when the model parameters are uncertain, the probability of the
variable Zl satisfying constraints Zl :5: C must not be smaller than n, where
o < n < 1 is the preset reliability level. If Zl is normally distributed, Eq.
(8.3.46) can be rewritten as
C - m[ZlJ > F-l(n)
a[ZlJ -
,
(8.3.47)
where m[ZlJ and a[ZlJ are the mathematical expectation and standard
deviation of Zl' respectively, and F-l(n) is the value ofthe standard normal
cumulative distribution corresponding to the re li ability level n. After Eq.
(8.3.47) is used to replace Eq. (8.3.43), the previous management problem can
be expressed as
(8.3.48)
subject to constraints:
(8.3.49)
and
(8.3.50)
The first term on the left-hand side of Eq. (8.3.49) is the deterministic part of
the constraint, while the second term is an additional demand created by the
uncertainty of the simulation model. When there is no uncertainty on Zl' the
latter is equal to zero. Otherwise, when the standard deviation of Zl becomes
larger, or a higher reliability level is required (n becomes larger), the value of
the term will be larger and the constraints will be more difficult to satisfy.
m[ZlJ and a[ZlJ in Eq. (8.3.49) can be obtained by solving the water quality
equation with stochastic parameters (see Section 7.4). Once m[ZlJ and
a[ZlJ are obtained, the method for solving the stochastic management problem will be the same as that of the deterministic one.
Kaunas and Haimes (1985), and Massmann and Freeze (1987a, b) considered the efTects of inaccurate dispersion coefficients on the management decisions from the point of view of a multi-objective management, and estimated
the risk of the decisions. Wagner and Gorelick (1989) further considered the
solution of water quality management and the requirement for observation
8. Applications of Groundwater Quality Models
parameters. By solving the management problem for each realization, a great
number of realizations of management decisions are obtained, then the
expected value and the variance of the optimal decision can be calculated,
and consequently, its confidence interval can be estimated (see Section 7.4.4).
Another approach is to formulate the management problem into a stochastic, rather than deterministic problem. Wagner and Gore1ick (1987)
modified deterministic constraint (8.3.43) into the following stochastic
constraint:
Prob{Zl :5: C} ~ n,
(8.3.46)
that is, when the model parameters are uncertain, the probability of the
variable Zl satisfying constraints Zl :5: C must not be smaller than n, where
o < n < 1 is the preset reliability level. If Zl is normally distributed, Eq.
(8.3.46) can be rewritten as
C - m[ZlJ > F-l(n)
a[ZlJ -
,
(8.3.47)
where m[ZlJ and a[ZlJ are the mathematical expectation and standard
deviation of Zl' respectively, and F-l(n) is the value ofthe standard normal
cumulative distribution corresponding to the re li ability level n. After Eq.
(8.3.47) is used to replace Eq. (8.3.43), the previous management problem can
be expressed as
(8.3.48)
subject to constraints:
(8.3.49)
and
(8.3.50)
The first term on the left-hand side of Eq. (8.3.49) is the deterministic part of
the constraint, while the second term is an additional demand created by the
uncertainty of the simulation model. When there is no uncertainty on Zl' the
latter is equal to zero. Otherwise, when the standard deviation of Zl becomes
larger, or a higher reliability level is required (n becomes larger), the value of
the term will be larger and the constraints will be more difficult to satisfy.
m[ZlJ and a[ZlJ in Eq. (8.3.49) can be obtained by solving the water quality
equation with stochastic parameters (see Section 7.4). Once m[ZlJ and
a[ZlJ are obtained, the method for solving the stochastic management problem will be the same as that of the deterministic one.
Kaunas and Haimes (1985), and Massmann and Freeze (1987a, b) considered the efTects of inaccurate dispersion coefficients on the management decisions from the point of view of a multi-objective management, and estimated
the risk of the decisions. Wagner and Gorelick (1989) further considered the
solution of water quality management and the requirement for observation