A basic hypothesis is proposed: given that wavelet-based analysis has
been used to interpret runoff time-series, it may be extended to
evaluation of rainfall-runoff model results. Conventional objective
functions make certain assumptions about the data series to which they
are applied (e.g. uncorrelated error, homoscedasticity). The difficulty
that objective functions have in distinguishing between different
realizations of the same model, or different models of the same system,
is that they may have contributed in part to the occurrence of model
equifinality. Of particular concern is the fact that the error present
in a rainfall-runoff model may be time dependent, requiring some form
of time localization in both identification of error and derivation of
global objective functions. We explore the use of a complex Gaussian
(order 2) wavelet to describe: (1) a measured hydrograph; (2) the same
hydrograph with different simulated errors introduced; and (3) model
predictions of the same hydrograph based upon a modified form of
TOPMODEL. The analysis of results was based upon: (a) differences in
wavelet power (the wavelet power error) between the measured hydrograph
and both the simulated error and modelled hydrographs; and (b) the
wavelet phase. Power difference and wavelet phase were used to develop
two objective functions, RMSE(power) and RMS(phase), which were shown
to distinguish between simulated errors and model predictions with
similar values of the commonly adopted Nash-Sutcliffe efficiency index.
These objective functions suffer because they do not retain time,
frequency or time-frequency localization. Consideration of wavelet
power spectra and time- and frequency-integrated power spectra shows
that the impacts of different types of simulated error can be seen
through retention of some localization, especially in relation to when
and the scale over which error was manifest. Theoretical objections to
the use of wavelet analysis for this type of application are noted,
especially in relation to the dependence of findings upon the wavelet
chosen. However, it is argued that the benefits of localization and the
qualitatively low sensitivity of wavelet power and phase to wavelet
choice are sufficient to warrant further exploration of wavelet-based
approaches to rainfall-runoff model evaluation.