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also possible but are less amenable to a hydrological interpretation). For two stores in series,
the individual components are multiplied together so that
b
z
−
ı
1
b
z
−
ı
1
u
t
Q
t
=
(B4.1.13)
−
a
z
−1
−
a
z
−1
or
b
o
Q
t
=
a
2
z
−2
u
t
−
ı
(B4.1.14)
1
−
a
1
z
−1
−
ı
For two stores in parallel, two first-order components are added such that
b
b
;
a
1
=
a
+
a
;
a
2
=−
a
a
; and
ı
ı
+
where
b
o
=
=
b
z
−
ı
1
u
t
b
z
−
ı
Q
t
=
+
(B4.1.15)
−
a
z
−1
1
−
a
z
−1
or, in the case where both components have the same time delay
ı
b
1
z
−1
b
o
+
Q
t
=
a
2
z
−2
u
t
−
ı
(B4.1.16)
1
−
a
1
z
−1
−
b
;
b
1
=−
b
a
+
b
a
;
a
1
=
a
a
.
By extension, a general higher order linear transfer function model may be written
b
+
a
+
a
;
a
2
=−
where
b
o
=
b
o
+
b
1
z
−1
+···+
b
m
z
−
m
Q
t
=
a
n
z
−
n
u
t
−
ı
(B4.1.17)
1
−
a
1
z
−1
−
a
2
z
−2
+···+
This is the general form of linear transfer function that forms the basis of the TFM package
described in Appendix A. The Nash cascade discussed in Section 2.3 is one specific form
of the general linear model. Combinations of linear storage elements were also investigated
for rainfall-runoff modelling in the 1960s by Diskin and Kulandaiswamy (see Chow, 1964,
pages 14.31-14.33; Chow and Kulandaiswamy, 1971,; Diskin and Boneh, 1973). The specific
case of a second-order parallel model in hydrological applications has been discussed by
Young (1992).
B4.1.3 Model Structure Identification
Given an input-output data set, there is then the problem of identifying an appropriate transfer
function model structure, that is finding the best values of
(
m, n, ı
)
and the corresponding
coefficients. There may be no unique answer to this problem, partly because the model will
always be an approximation to what may really be a complex nonlinear relationship between
the inputs and outputs, partly because there will always be some degree of error associated
with the input and output data sets, and partly because the real-time delay might not be a
constant or might not be precisely an integer number of time steps. It is often found that there
are several model structures that give acceptably accurate simulations after calibration of the
a
and
b
coefficients, so some way of evaluating these structures is required.
There are two considerations in making such an evaluation. One is how well the model
fits the data, i.e. goodness of fit. A commonly used goodness-of-fit index is the coefficient of
determination or proportion of the variance in the data explained by the model (introduced
