factor to the variance of the observed values of
the output variable (WR and HCR).
A simple method for assessing the model
sensitivity to natural variations of environmental
factors used as input ones was developed. In
Eq. (7.2), the criterion A is proposed to check the
adequacy of calculation methods and/or models
by comparing the patterns of the observed and
calculated data. The heart of evaluating the sensitivity of mathematical models is criterion FS,
which characterizes the sensitivity directly to
natural variations of environmental factors. FS is
close in meaning to the known “percentage of
explained variance” and is calculated by the
formula
FS ¼ A
0
ð Þ
2 À A
ð Þ
2 ¼
ðS
0
dif Þ
2 À ðS dif Þ
2
2ðS obs Þ
2
¼
2ðS fac Þ
2
2ðS obc Þ
2
¼
ðS fac Þ
2
ðS obc Þ
2
ð7:6Þ
where FS is the model sensitivity to the target
input factor; A is calculated from (7.2); A' is
A value obtained from (7.2) by using the randomly mixed values of the input factor instead of
initially ordered ones; in this case, the randomly
mixed pattern has a former statistical distribution
and variance; S dif
ð Þ
2 is the variance for the difference between the calculated and observed data
patterns (WR or HCR); S
0
dif
À Á 2 is a similar variance for the difference between the calculated
and observed output variable after the substitution the randomly mixed values of the input
factor; S fac
ð Þ
2 is the contribution of input factor
variations to the variance of the model output
variable (calculated WR/HCR); S obs
ð
Þ
2 is the
variance of the observed output variable.
It should be pointed out that FS could be also
expressed as a function of RSR. Taking into
account the abovementioned equality, RSR =
A
ffiffi ffi
2
p
, we have FS ¼ ðRSR
0
Þ
2 À ðRSRÞ
2
h
i
=2.
Fig. 7.8 The total ion runoff (g/s  10
4
) as a function of
hypothetically different lateral slopes of the basin and
precipitation (at mean values of other environmental
factors) for the upper reaches of the Katun river: a, b, c,
d are four hydrological seasons (see Fig. 7.6)
7 System-Analytical Modeling of Water Quality …
95
the output variable (WR and HCR).
A simple method for assessing the model
sensitivity to natural variations of environmental
factors used as input ones was developed. In
Eq. (7.2), the criterion A is proposed to check the
adequacy of calculation methods and/or models
by comparing the patterns of the observed and
calculated data. The heart of evaluating the sensitivity of mathematical models is criterion FS,
which characterizes the sensitivity directly to
natural variations of environmental factors. FS is
close in meaning to the known “percentage of
explained variance” and is calculated by the
formula
FS ¼ A
0
ð Þ
2 À A
ð Þ
2 ¼
ðS
0
dif Þ
2 À ðS dif Þ
2
2ðS obs Þ
2
¼
2ðS fac Þ
2
2ðS obc Þ
2
¼
ðS fac Þ
2
ðS obc Þ
2
ð7:6Þ
where FS is the model sensitivity to the target
input factor; A is calculated from (7.2); A' is
A value obtained from (7.2) by using the randomly mixed values of the input factor instead of
initially ordered ones; in this case, the randomly
mixed pattern has a former statistical distribution
and variance; S dif
ð Þ
2 is the variance for the difference between the calculated and observed data
patterns (WR or HCR); S
0
dif
À Á 2 is a similar variance for the difference between the calculated
and observed output variable after the substitution the randomly mixed values of the input
factor; S fac
ð Þ
2 is the contribution of input factor
variations to the variance of the model output
variable (calculated WR/HCR); S obs
ð
Þ
2 is the
variance of the observed output variable.
It should be pointed out that FS could be also
expressed as a function of RSR. Taking into
account the abovementioned equality, RSR =
A
ffiffi ffi
2
p
, we have FS ¼ ðRSR
0
Þ
2 À ðRSRÞ
2
h
i
=2.
Fig. 7.8 The total ion runoff (g/s  10
4
) as a function of
hypothetically different lateral slopes of the basin and
precipitation (at mean values of other environmental
factors) for the upper reaches of the Katun river: a, b, c,
d are four hydrological seasons (see Fig. 7.6)
7 System-Analytical Modeling of Water Quality …
95