5.2. The Sensitivity
Function
133
parameters and inputs, the coefficients and the inputs for the response must
be determined with much greater accuracy. A sensitivity analysis of a
river basin model thus will reveal how accurate the parameters and inputs
must be. There is no point in meticulously obtaining data for relationships
that can be shown to play little part in the final analysis, and vice versa.
If a river basin is composed of a number of subsystems, each representing
a somewhat arbitrarily defined region or activity in the total river basin,
it is possible to examine the effect of input and parameter perturbations
both on the subsystem outputs and on the total system outputs. Because
each equation or relationship in a subsystem model also contains one or
more parameters or coefficients, it is also desirable, in the search for appropriate parameter values, to be able to evaluate how sensitive portions of
the system may be to uncertainties in the parameters or disturbances in
the inputs.
To sum up, a sensitivity analysis on a river basin model can be used to
(1) prevent placing undue emphasis on any portion of the real basin, either
in the planning or execution stage; (2) help determine the scope of the
model of the system in light of the existing or future criteria for evaluation;
and (3) aid in developing and comparing alternate courses of action.
Thus a sensitivity analysis can make the planning for a regional river basin
more economic in terms of time, money, and effort.
5·2· The Sensitivity Function
In this section we define single- and multiparameter sensitivity functions. Keep in mind that analytical evaluation of the functions is not particularly easy, but, on the other hand, that numerical evaluation of the
functions is quite time consuming. If a function / depends on a single
parameter x y we write the function as / = /(x) to abbreviate the notation,
and define the sensitivity of / with respect to χ as
£/=
(df/f)/(dx/x)
= (dlnf)/(d\nx)
(5.1)
The first ratio indicates that the sensitivity represents the fractional change
in / divided by the fractional change in x } while the second ratio indicates
that we are concerned with relative changes rather than absolute changes
in defining the sensitivity. The sensitivity is evaluated at nominal or design
values of x } and for reasonably small changes in χ the fractional change in
f(x) is
df/f=
S/dx/x
(5.2)
Function
133
parameters and inputs, the coefficients and the inputs for the response must
be determined with much greater accuracy. A sensitivity analysis of a
river basin model thus will reveal how accurate the parameters and inputs
must be. There is no point in meticulously obtaining data for relationships
that can be shown to play little part in the final analysis, and vice versa.
If a river basin is composed of a number of subsystems, each representing
a somewhat arbitrarily defined region or activity in the total river basin,
it is possible to examine the effect of input and parameter perturbations
both on the subsystem outputs and on the total system outputs. Because
each equation or relationship in a subsystem model also contains one or
more parameters or coefficients, it is also desirable, in the search for appropriate parameter values, to be able to evaluate how sensitive portions of
the system may be to uncertainties in the parameters or disturbances in
the inputs.
To sum up, a sensitivity analysis on a river basin model can be used to
(1) prevent placing undue emphasis on any portion of the real basin, either
in the planning or execution stage; (2) help determine the scope of the
model of the system in light of the existing or future criteria for evaluation;
and (3) aid in developing and comparing alternate courses of action.
Thus a sensitivity analysis can make the planning for a regional river basin
more economic in terms of time, money, and effort.
5·2· The Sensitivity Function
In this section we define single- and multiparameter sensitivity functions. Keep in mind that analytical evaluation of the functions is not particularly easy, but, on the other hand, that numerical evaluation of the
functions is quite time consuming. If a function / depends on a single
parameter x y we write the function as / = /(x) to abbreviate the notation,
and define the sensitivity of / with respect to χ as
£/=
(df/f)/(dx/x)
= (dlnf)/(d\nx)
(5.1)
The first ratio indicates that the sensitivity represents the fractional change
in / divided by the fractional change in x } while the second ratio indicates
that we are concerned with relative changes rather than absolute changes
in defining the sensitivity. The sensitivity is evaluated at nominal or design
values of x } and for reasonably small changes in χ the fractional change in
f(x) is
df/f=
S/dx/x
(5.2)