Environmental Engineering Reference
In-Depth Information
model geometry, initial and boundary conditions, and so
forth, are not well known or, in the best case, are never
known exhaustively. Still, it would also be a waste not
to take into account the physics. Stochastic methods can
be regarded as a tool to combine physics, statistics and
uncertainty within a coherent theoretical framework. On
the one hand, the unknown parameters are described by
statistical distributions. On the other hand, the differ-
ent variables describing the problem are related to each
other and to the (uncertain) model parameters through
(deterministic) physical laws. The resulting models take
the form of stochastic partial differential equations.
Coming back to the simple groundwater example, the
recharge
r
and the inflow
q
may not be known accurately.
Consequently, these two parameters can be viewed as ran-
dom functions that vary in space and/or time but that have
some statistical properties (such as a mean, variance and
covariance) that can be inferred from samples. Consider-
ing them as random functions implies that the hydraulic
head
h
is also a random function, related to
r
and
q
through equation (8.3) and the corresponding boundary
conditions. In that case, Equation (8.9) is interpreted as
a stochastic differential equation because it relates ran-
dom functions and not simply spatio-temporal functions.
Under certain simplifying assumptions, one can derive
also analytically the statistics of the random function
h
.
For example, if we assume that
k
is known and constant in
space (even though both assumptions are, indeed, incor-
rect), Equation (8.11) holds and the expected value E(
h
)
of the hydraulic head can be expressed as:
This interesting expression shows that the variance is
zero (there is no room for uncertainty) in the vicinity of
the river (
x
=
0). This interpretation makes sense because
the boundary condition there states that the hydraulic
head in the aquifer is equal to the water elevation in
the river
h
r
. Thus, there is no uncertainty regardless
of those of uncertain parameters (unless the boundary
condition
h
r
is also considered as a random function).
As expected, the uncertainty increases with distance from
the river. This approach allows us to plot the uncertainty
bounds corresponding to the expected value plus/minus
two times the standard deviation (Figure 8.3a). These are
the so-called 95% confidence intervals.
Alternatively, one can address the uncertainty by apply-
ing the deterministic solution in Equation (8.5) using
extreme values of the unknown parameters
p
and
r
.To
that end, we assume that the only uncertain parame-
ter is now recharge and that it can be represented by a
Gaussian distribution with known mean (100 mm a
−
1
,
the deterministic value used before) and standard devia-
tion (i.e. the square root of the variance, 10 mm a
−
1
in
this case). Under such an assumption, recharge values lie
in the corresponding 95% confidence interval [80, 120]
mm a
−
1
. Solving the deterministic equation (8.11) with
these two extreme values yields two solutions for the
hydraulic head that depict an envelope of possible values
of
h
(Figure 8.3a, outer dashed lines). However, this is not
a good option because this envelope defining the uncer-
tainty is much larger than the one defined by the stochastic
model (Figure 8.3a (left), dashed lines). This difference
increases with the variance of the unknown parameter.
Another argument to defend the use of stochastic mod-
els lies in the fact that the stochastic formulation of the
problem allows us to obtain the full-range distribution of
the hydraulic head (i.e. the complete pdf) that accounts
for possible correlations between the different sources of
uncertainty (Figure 8.3b (right)).
In most cases it is not possible to derive simple
expressions such as Equations (8.12) and (8.13). Several
alternative methods exist to obtain exact or approximate
statistical relations between the variables and the param-
eters governing a stochastic partial differential equation.
In a broad sense, these methods consist of obtaining
expressions for the first statistical moments of the vari-
able of interest and relating them to the moments of the
input variables. Under certain circumstances, it is pos-
sible to obtain directly the expression of the pdf of the
variable of interest. This approach is used for example
in fluid mechanics for turbulent flow (Jenny
et al
., 2001)
but, most generally, approximate expressions are derived
E
q
+
rL
k
x
h
r
r
2
k
x
2
E
[
h
(
x
)]
=
−
+
+
E
[
q
]
+
E
[
r
]
L
k
x
E
[
r
]
2
k
x
2
=
+
+
h
r
(8.12)
This simple equation states that we can directly estimate
the expected value of the hydraulic head at any point
of the domain
E
[
h
(
x
)] if the expected values of recharge
E
[
r
] and inflow
E
[
q
] are known. Note that Equations
(8.5) and (8.6) are identical but for the fact that the
meaning of the intervening parameters has changed, i.e.,
the deterministic values of
r
and
q
have been replaced by
their expected values. Following the same logic, one can
compute the variance of the hydraulic head at any point
in the domain:
L
2
2
x
h
(
x
)
r
q
σ
=
−
σ
+
σ
L
−
2
cov (
r
,
q
)
x
k
2
x
+
(8.13)
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