The treatment of input data uncertainty in hydrologic models is of crucial importance in the analysis, diagnosis and detection of model structural errors. Data reduction techniques decrease the dimensionality of input data, thus allowing modern parameter estimation algorithms to more efficiently estimate errors associated with input uncertainty and model structure. The discrete cosine transform (DCT) and discrete wavelet transform (DWT) are used to reduce the dimensionality of observed rainfall time series for the 438 catchments in the Model Parameter Estimation Experiment (MOPEX) data set. The rainfall time signals are then reconstructed and compared to the observed hyetographs using standard simulation performance summary metrics and descriptive statistics. The results convincingly demonstrate that the DWT is superior to the DCT in preserving and characterizing the observed rainfall data records. It is recommended that the DWT be used for model input data reduction in hydrology in preference over the DCT.

Rainfall uncertainty is the biggest obstacle hydrologists face in their
pursuit of accurate, precise and timely streamflow forecasts

The propagation of input errors in rainfall runoff modeling impedes the
hydrologic community's ability to validate model structural error. Despite
the vast amount of literature on rainfall measurement, estimation,
statistical analysis

Recent advancements in computational power as well as remote sensing have led
to considerable improvements in availability and quality of hydrological
observations

The advancement of computational
power has also aided the search for hydrological model parameters that
optimally simulate hydrological observations. These approaches initially
focused on finding only the global optimum values of the parameters for a
given objective function

Fourier transforms use sinusoidal functions to represent the spectral
component of an input signal; thus, a periodic signal could be represented
using a smaller number of Fourier coefficients than the number of input data
points. A pitfall of the Fourier transform is that it represents the spectral
components of a signal, without any indication of the time localization of
those specific spectral components. In order to account for this, the
windowed Fourier transform (WFT), sometimes referred to as the short-time
Fourier transform, segments the signal into discrete time windows before
performing the Fourier analysis. A major drawback to this approach is that
the uncertainty principle of signal processing imposes a limitation on the
time and frequency resolutions that can be obtained for a given signal. As a
response to this,

Wavelet analysis was first introduced to the geophysical sciences by

Traditionally, transform coefficients are the result of a convolution operation on an input signal. However, the aim of model input data reduction is to estimate these transform coefficients. Hence, they shall be referred to as transform parameters from herein. This paper provides novel theoretical and numerical comparisons of the DCT and DWT in a hydrological context. The ability of both transforms to reproduce key components of hydrological data sets is investigated. The extent to which each transform can reproduce hydrologic data using a decreasing number of parameters will serve as a metric upon which their ability to be used as a tool for model input data reduction for hydrological data will be evaluated. To address the requirements for hydrologic model input data reduction, this paper details (i) theoretical differences between the DCT and DWT, (ii) methodologies to reduce input rainfall to parameters and (iii) an evaluation of the proposed methodologies using several simulation performance summary metrics.

For this study, model input data reduction theory is introduced using a
lumped conceptual watershed model. Consider a nonlinear model,

If the traditional hydrological perspective in which the inputs

Given a daily rainfall data record with

Sparse transforms convey large amounts of data using fewer parameters than data points in the observed signal. An input rainfall signal can be reduced to sparse transform parameters. Doing so allows multiple rainfall observations to be modified using a single parameter. Some or all of these transform parameters can be altered before the transform is inverted to produce a new input signal for streamflow simulation and posterior analysis. The use of sparse transforms to represent input time series enables input uncertainty to be explored in great detail. The ability of discrete wavelet and Fourier transformations to reduce hydrological input data to a set of parameters for uncertainty estimation is compared using theoretical and analytical methods.

Wavelet and Fourier transforms are invertible transforms in which a forward
convolution operation can be used to decompose a signal into various
components. Similarly, a backwards deconvolution operation can be applied to
retrieve the original signal. Fourier-based transforms decompose signals into
frequency components and are best used for regular time-invariant signals
that do not exhibit time-specific information. Alternatively, wavelet-based
transforms decompose signals into frequency and time components. The
advantage of using wavelet functions to transform data is that time-specific
information about when higher frequency components occur can be preserved. To
obtain time-specific information, Fourier-based transforms can be applied
over pre-specified temporal windows. Yet, this approach is limited by the
uncertainty principle of signal processing. The uncertainty principle of
signal processing imposes a lower limit on obtainable resolutions in the
time–frequency domain such that

Applying the uncertainty principle of signal processing (Eq.

Considering that there is no time–frequency window that is able to obtain
limitless resolution in both the time and frequency domains, it is clear that
an alternative solution must be found. Wavelet transforms can be used to
decompose a signal into different levels that consist of different time and
frequency resolution windows. Thus, the wavelet transform is able to be
configured to simultaneously obtain high levels of resolution in both the
time and frequency domains. For a more detailed discussion on wavelets and
sparse transforms, the reader is referred to

The DCT

Using the pyramid algorithm, depicted in Fig.

A schematic showing the pyramid algorithm used to decompose and
downsample

This study utilizes data from the Model Parameter Estimation Experiment
(MOPEX) data set. The 10 years of rainfall data spanning the 1990s for 438
catchments in the United States of America (USA) are used to compare the
suitability of the DWT and DCT to represent rainfall time series. The
catchments used in this study were chosen to ensure they had sufficient rain
gauge density and represented a range of catchment sizes and climates.
Rainfall for the Leaf River catchment (Collins, Mississippi), a catchment
that is frequently used for hydrological studies

This experiment does not involve the use of any hydrological models. Due to this and the nature of the transforms, there are no calibration and evaluation periods. A major use of both the DWT and DCTs has been in image compression; consequently, the observed input signals were compressed and decompressed using a methodology similar to that used in image compression. In order to determine which transform's parameters are able to effectively store the most hydrological input data, both DWT and DCT parameters will be compressed to varying extents for the MOPEX rainfall time series.

The process undertaken involves a number of steps. Firstly, before any
compression is applied, the original rainfall signal for a given catchment is
transformed into DCT and DWT parameters using Eqs. (

To provide a meaningful comparison between the DCTs' and DWTs' ability to
reproduce different rainfall time series with an increasing POP, a number of
simulation performance summary metrics are used. Following

Figure

By comparing the reconstructed DWT and DCT signals, using 20 POP and the
observed rainfall signal as a reference, a histogram for the NSE is shown for
all catchments in Fig.

Empirical plots showing the relationship between RSR and the POP used for reconstructing an input rainfall signal using the DWT and DCT. The three catchments, from the top to the bottom of the figure, represent the smallest, largest and mean rainfall volumes throughout the 1990s for the MOPEX data set.

Histogram representing the reconstructed DWT (dark bins) and DCT (clear bins) NSE when compared to the observed rainfall signal. Rainfall is reconstructed after the input signal is compressed to 20 POP. Each frequency count represents a catchment from the MOPEX data set.

Figure

The mean and standard deviation (SD) of NSE for the DWT and DCT using a different POP.

Comparative plots of RSR for the DCT and DWT using a different POP. Each data point represents a catchment.

The bias, variance and skewness observed in the reconstructed signals for
each catchment are shown in Fig.

Bias and normalized variance and skewness of the reconstructed DWT and DCT signals for each catchment using a different POP.

The normalized kurtosis and PE for all catchments using different
POPs are shown in Fig.

Figure

Normalized kurtosis of the reconstructed DWT and DCT signals and percentage PE for the reconstructed DWT and DCT signals for each catchment using a different POP.

As the bias for the DWT is consistently close to zero, the use of the DWT for
rainfall input data reduction is likely to be beneficial for hydrologic
studies that have short time steps and involve rainfall as an input. Whilst
modification of the DWT parameters may slightly overestimate input rainfall,
it is not as significant as the consistent underestimation of input rainfall
by the DCT. The diminishing ability of both the DWT and DCT to match the
input rainfall signal variance indicates that both transforms smooth out
input data towards the mean. This behavior is more significant for the DCT
than the DWT. Consequently, when used as a technique for input data
reduction, the DWT will reconstruct temporal variances better than the DCT.
The increased skewness for the reconstructed DWT signals compared to the
observed input signals indicates that there is an increased reconstruction of
low-magnitude rainfall events. On the contrary, the decreased normalized
skewness for the reconstructed DCT signals indicates that a number of the
low-magnitude rainfall events are tending to be reconstructed towards the mean.
The kurtosis results shown in Fig.

Comparison of the reconstructed DCT and DWT signal for the Leaf River (Collins) catchment using 20 POP.

Panel

Whilst it is important that rain gauges measure high-magnitude rainfall
events with accuracy and precision, it is also important that low-magnitude
rainfall events are recorded. Consequently, when evaluating the merits of the
DCT and DWT to reconstruct rainfall it would be prudent to analyze the
frequency in which each transform is either unable to reconstruct a rainfall
event or erroneously constructs a rainfall event. Table

Due to rapid increases in rainfall intensity, high-magnitude rainfall events
tend to have high-frequency components. In Fig.

The mean and standard deviation (SD) for the number of missed rainfall events for the DWT and DCT using a different number of parameters.

Succinct descriptions of the DCT and DWT were provided to determine the suitability of each transform to be used as a tool for hydrologic model input data reduction. Due to their different construction, each transform provides different possibilities for use in model input data reduction. Since it is infeasible to estimate all transform parameters, the modeller could choose to estimate high- or low-frequency parameters of the DCT. This would result in minimal control of the temporal component being modified. Due to the multi-level decomposition of an input signal into high- and low-frequency parameters by the DWT, the modeller is able to specify the estimation of both time and frequency components. Hence, portions of the input data record can be targeted for estimation. The use of the DWT as a hydrologic model input data reduction technique allows the modeller more flexible options. A comparison of the DWTs' and DCTs' ability to reconstruct MOPEX rainfall data using standard simulation performance summary metrics, descriptive statistics and peak errors was then made, and it was found that the DWT is most efficient at preserving high-magnitude and transient rainfall events. Thus, it is recommended that the DWT be used as a model input data reduction technique for hydrologic studies that have short time steps and involve rainfall as an input. Considering that the bias for the reconstructed DWT rainfall signal is consistently lower than that of the reconstructed DCT signal and that the skewness, kurtosis and variance are also closest to the input rainfall signal, it is recommended that the DWT also be used as a model input data reduction technique for hydrologic studies that have long time steps with rainfall as an input.

All data were obtained from publicly available data sets
(

AW conducted the experimental work, contributed towards the theory and wrote the paper. JW and DR assisted in the writing process. VP contributed towards the theory and assisted in the writing process.

The authors would like to extend their gratitude to Jasper Vrugt, Hamid
Bazargan and the anonymous reviewers for their
comments and recommendations. This work was supported
by the Multi-modal Australian Sciences Imaging and Visualisation Environment
(MASSIVE) (