Environmental Engineering Reference
In-Depth Information
2. The Results
In the article [2] a simple formula for mean square error (
msqe
) was presented to
explain the advantage of the multi-model ensemble system for long-term ozone
simulations in comparison with a single model approach, namely:
1
2
2
msqe
=
(
+
)
σ
+
b
(1)
m
The formula (1) is valid under the assumption that statistical distributions of
model results and observations are independent and identical. The latter assumption
is obviously a simplification as the models differ in terms of used parameterizations
or numerical concept. Nevertheless it clearly demonstrates that
msqe
for the
ensemble (after bias correction) is always less than that of any single model (m = 1).
Starting from this, we would like to give a slightly deeper look at the analysis
based on statistical characteristics of
msqe
. We will focus on simple one-
dimensional case i.e. when the simulation results can be described by a
scalar-
valued random variable
for each model (for instance when multi-model ensemble
is applied at a single point in space and time), however analogous formulas can be
obtained for the multi-dimensional case i.e. when the problem is described by a
vector-valued random variable
[1].
Let us then assume that by using a set of
m
atmospheric dispersion models we
get
m
simulations of the concentration of an unspecified substance at some point
in space and time together with measurements. Both model data and observations
are characterized by some estimation of uncertainty, which we formally represent
by
random variables x
j
,
j = 1
,…,
m
, where
x
j
corresponds to data produced by
model
j
, and analogously for the observations by
random variable
y
. Each
x
j
has
statistical distributions characterized by probability density functions (pdf) with
bias and variance. We consider two cases: when the models are not correlated (i.e.
E{x
1
, x
2
} = E{x
1
}E{x
2
}
where
E{}
is the expectation value) or correlated.
We assume that the multi-model ensemble results are represented by a
normalized linear combination of model results which we denote by
X
, namely:
∑
∑
T
X
=
α
j
x
,
α
=
1
α
=
[
α
,...,
α
]
(2)
j
j
1
m
j
j
Assuming that the models and measurements are un-correlated (i.e. random
variables
X
and
y
are un-correlated) we can express
msqe
as:
2
2
msqe
=
V
(
X
−
y
)
+
(
E
(
X
−
y
))
=
V
(
X
)
+
V
(
y
)
+
b
,
b
=
E
(
X
−
y
)
(3)
where
b
is the ensemble bias and
V
stands for the variance. Furthermore
information on model biases allows eliminating bias term from (3). Then one can
