Stochastic versus Deterministic Approaches - Environmental Modelling: Finding Simplicity in Complexity - page 137

10 0

Mean estimate
Deterministic k

15

10 − 1

10

10 − 2

5

10 − 3

10 − 4 0

0 1000 2000

distance to river [m]
(a)

3000 4000 5000 0 1000 2000

distance to river [m]
(b)

3000 4000 5000

1

cdf
pdf

15

0.8

0.6

10

0.4

5

0.2

0 0

0 1000 2000

distance to river [m]
(c)

3000 4000 5000 0 5 10 15

hydraulic head [m]
(d)

Figure 8.4 Monte Carlo analysis of the problem illustrated in Figure 8.2 when the hydraulic conductivity k is the unique source of
uncertainty. (a) example of a random hydraulic conductivity field, (b) comparison between the mean hydraulic head of 2000 Monte
Carlo simulations (plain line) and the deterministic head distribution obtained with a constant mean deterministic k (dotted line),
(c) results of the Monte Carlo procedure: mean hydraulic head as a function of x (plain line), two individual simulations of h (- -),
and the envelope of the 95% confidence interval (dash-dotted lines), (d) probability density distribution (dashed line) and
cumulative probability density distribution (solid line) of the hydraulic head at the location of the building (x = 2000m).

The ensemble of the 2000 simulated head distributions
is the main result of the Monte Carlo analysis. In practice,
one is often not interested in the results of all the simula-
tions separately but in the statistics of the ensemble. The
first thing that can be done is to calculate the statistical
moments of the ensemble, such as mean and variance,
and plot them as a function of space (and/or time, if
necessary). The result is displayed in Figure 8.4b and c
where the horizontal line depicts the ensemble average
(mean) of all simulated hydraulic head distributions as
a function of the distance to the river. One can observe
there that the local fluctuations of h have been smoothed
out due to the averaging. The dotted lines in Figure 8.4c
depict the mean plus or minus twice the standard devi-
ation of the numerical results (i.e., the 95% confidence
interval), which allows us to see rapidly the envelope in
which most of the simulations fall (more than 95% if
the distribution is Gaussian). Thus, this is a clear illus-
tration of the amount of uncertainty on the hydraulic
head at any location in the domain. In simple terms, the

larger is the envelope, the larger is the uncertainty of the
solution.
A very important result arising from Monte Carlo
analysis is that the mean hydraulic head (the same can be
made for the median or the mode) can be significantly
different from the hydraulic head computed with the
same deterministic model using the mean value of the
hydraulic conductivity, assumed to be constant in space
(Figure 8.4b). One can observe that the mean hydraulic
head is about twice as high as the value computed by the
deterministic model.
Another way to look at the same results is to analyse
the probability distribution of hydraulic head at a loca-
tion of interest. For example, the hydraulic head at the
future location of the building ( x 2000). The deter-

ministic model assuming a mean and constant hydraulic
conductivity (Equation 8.11) estimated at that location a
single hydraulic head value of 3.17 m. Instead, the Monte
Carlo approach allows us to compute the histogram of
all possible values at that location. From that histogram,

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