Environmental Engineering Reference
In-Depth Information
shown (e.g., Degroot and Schervish, 2002) that the pdfs
for
x
,
p
X
(
x
), and
y
,
p
Y
(
y
), are related by
and (4) errors in using the selected equations to describe
the process being modeled. This section addresses the
first two sources of uncertainty. The third source of error
can be identified using numerical experiments in which
the model's numerical solution is compared with results
from accurate analytical solutions. The fourth source of
error is related to the validity of the model itself, and
such errors can only be estimated by comparing model
predictions with direct measurements of those same
quantities in simple cases where all other uncertainties
are negligible.
Consider the typical case in which the calculated
quantity
Y
is being estimated using
N
measured vari-
ables
X
i
,
i
= 1,
N
. Functionally, this may be written as
−
1
dx
dy
dg
( )
y
p y
( )
=
p x
( )
=
p g
(
−
1
( ))
y
(11.53)
Y
X
x
dy
where the superscript −1 indicates the inverse function.
The direct propagation of probability densities, there-
fore, is restricted to functions that are monotonic
increasing or decreasing or that can be separated in
corresponding monotonic branches so that the inverse
functions can be derived. In case no analytic solutions
for inverse functions can be derived, semi-analytical
approximations can be applied.
Applications of analytical probability models can be
found in Kunstmann and Kastens (2006) for the Theis
and Penman-Monteith equations, and Behera et al.
(2006) for the estimation of pollutant loads in the runoff
from urban catchments. Analytical probability models
are typically used for preliminary planning and design,
with more complex models used for detailed design
applications.
Kunstmann and Kastens (2006) used analytical prob-
ability models to show that an
a priori
assumption of
the type (e.g., normal, log-normal, and Weibull) of
dependent-variable pdf is not possible in some cases,
since it depends on the location in parameter space and
the specific values in the input pdf. This observation
sounds a note of caution is assuming that observed
variables always have the same type of probability
distribution.
Y
= φ(
X
(11.54)
where
X
is a vector having elements
X
i
. If the true
values of the measured quantities are denoted by 〈
X
i
〉,
then, assuming the model is valid, the true value of
Y
,
〈
Y
〉, is given by
Y
= φ(
X
(11.55)
where 〈
X
〉 is a vector having elements 〈
X
i
〉. The error in
Y
, Δ
Y
, is then given by
∆
Y Y Y
=
−
=
φ
(
X
)
−
φ
(
X
)
(11.56)
Using a multidimensional Taylor expansion,
ϕ
(
X
) can
be expressed in the following form:
11.6.4 First-Order Uncertainty Analysis
N
∂
∂
φ
The error in a measured variable is defined as the dif-
ference between the measured value and the true value
of the variable. The error can usually be assumed to be
random, and characterized by a probability distribution.
Also, in cases where measured data are used to calculate
quantities that are not directly measurable, calculated
quantities deviate from their true values, with errors
whose probability distributions depend on the probabil-
ity distributions of the errors in the measured data. The
objective of uncertainty analysis is to estimate the
statistics of quantities that are calculated from uncertain
data. Whenever mathematical models are used to esti-
mate quantities from measured data, uncertainty in the
calculated quantities may occur from four possible
sources (Fiering and Kuczera, 1982; Jackson, 1975; Lund,
1991): (1) uncertainty in the values of parameters and
constants in the model equations; (2) uncertainty in the
measured or estimated data; (3) errors introduced by
the numerical method that solves the model equations;
∑
X
φ
(
X
)
=
φ
(
X
)
+
(
X
−
X
)
i
i
i
X X
=
i
=
1
N
N
2
1
2
∂
φ
∑
∑
+
(
X
−
X
)(
X
−
X
)
i
i
j
j
!
∂
X X
∂
i
j
i
=
1
j
=
1
X X
=
+
R
(11.57)
where the remainder,
R
, involves higher-order products
of the errors, (
X
i
− 〈
X
i
〉). If the measurement errors are
small, then higher-order terms are much smaller than
lower-order terms, and retaining only first-order terms
in Equation (11.57) results in the following approximate
relationship
N
∂
∂
φ
∑
X
φ
(
X
)
=
φ
(
X
)
+
(
X
−
X
)
(11.58)
i
i
i
X X
=
i
=
1
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