Geoscience Reference
In-Depth Information
and analogous equations for
i
3 and 4. These constitute a system of four linear equa-
tions for the four unknown values of
u
i
, which can readily be solved.
The technique may also be used in the solution of even more complex cases; various
algorithms involving operations of matrix transposition and inversion are available and
can be found in textbooks on numerical analysis.
=
Transform methods of identification
Beside the direct solution methods of Equations (12.9) and (12.11), numerous other
methods are available to derive optimal
u
i
values. In several of these, by using a different
formulation or a transformation of the original functions
y
(
t
),
x
(
t
), and
u
(
t
), a simpler,
usually algebraic, relationship between the three is obtained, which is more tractable for
computation than the convolution integral (12.2) or (12.9). In the method of moments,
the functions are characterized by their moments. The optimal
u
i
values, or the optimal
constants in the function, that is used to describe
u
(
t
), are determined such that the
moments of the calculated output function are equal to the moments of the observed
output function
y
(
t
). In principle, this method is based on the application of the theorem
of moments, as given by Equations (A22) and (A28). In harmonic analysis, the functions
are described as Fourier series expansions. The optimal
u
i
values or the constants in
u
(
t
),
are determined such that the constants of the Fourier series expansion of the calculated
output are exactly the same as those of the observed output. In Fourier and Laplace
transform methods, the functions are formulated, respectively, in the frequency domain
and in the
s
(i.e. the Laplace transform) domain.
A review of the application of many of the identification methods investigated in
earlier years in catchment hydrology has been presented by Dooge (1973). In practice,
however, some of these techniques of direct identification from available data result in
response functions, which can be quite sensitive to small errors in the measurements,
exhibiting such “non-physical” features as severe oscillations or negative values. Ways
of coping with such problems have been discussed by, among others, Neuman and de
Marsily (1976) and Singh (1976).
12.2.2
More concise parameterizations by linear runoff routing
The data needed to derive the unit hydrograph for a given basin are not always available.
Therefore, it should be no surprise that over the years many attempts have been made
to develop methods enabling the prediction of this unit response function from basin
characteristics. The goal of these studies was to derive unit hydrographs for ungaged
watersheds from maps and from some other readily available physical attributes. In one
class of methods, empirical equations and empirical curves were used to describe the
unit hydrograph, with parameters in terms of basin characteristics. However, because of
their strictly empirical nature, their applicability tends to be limited to the region where
they were developed, and they will not be considered any further here.
In another class of methods, which are of a more fundamental interest, various theoret-
ical forms of the response function were proposed by postulating different combinations
of model elements, which replicate the most important flow mechanisms to transform
