Fine-scale rainfall observations for modelling exercises are often not
available, but rather coarser data derived from a variety of sources are
used. Effectively using these data sources in models often requires the
probability distribution of the data at the applicable scale. Although
numerous models for scaling distributions exist, these are often based on
theoretical developments, rather than on data. In this study, we develop a
model based on the

Rainfall is one of the most important drivers of hydrological processes and
is an important data source for hydrological modelling. These models
typically operate on a spatial scale of less than 100 km

A basic rainfall model, graphically illustrated. The left-hand side of the image is the dressing procedure, whereas the right-hand side is the generation.

Whenever suitable data are not available, the scaling behaviour in rainfall
can be exploited to yield a statistical estimate of the rainfall at a finer
scale. At a very basic level, this behaviour leads to a cascade of scales,

The above multifractal models can be employed to simulate (non-zero) rainfall
according to a few parameters (i.e. they are universal; see Sect.

In previous investigations, imperfect scaling has been studied by fitting and
refitting various cascade models and studying the dependence of the
parameters on coarse-scale intensity and other variables (e.g.

We start by explaining the simulation of rainfall (Sect.

Scale-invariant processes and their generation are perhaps easiest understood
in the context of discrete-in-scale (discrete) cascades

The field described above is multifractal and no longer has a single fractal
dimension, but rather an infinity of fractal dimensions, each associated with
a specific singularity. Evidently, this is problematic, requiring and
infinity of parameters to describe the behaviour. In practice, these cascades
converge to a universal multifractal if the increments

Fields simulated with the above method are “conservative”: they are the
direct outcome of multiplicative cascades and the realizations themselves are
scale invariant. However, for most observed rainfall fields only the
fluctuations of the field scale, i.e.

A further convenient way to diagnose whether measured fields are
non-conservative is the relation to the slope of the Fourier power spectrum.
The power spectrum of scaling fields behaves as

Multifractal fields generated with the above model produce non-zero values
everywhere, and thus they are only appropriate to simulate regions where it
is raining everywhere. To overcome this, rainfall is often assumed to be the
result of two separate processes, one to determine where it is raining, the
support of the rainfall field, and another to determine the observed rain
rates. Several different methods of introducing zero values have been
proposed in literature, generally there are those which simulate a separate
(mono-)fractal rainfall support (e.g.

A log-transformed rainfall field, together with the radius of reliable observations (circles) at 60 and 180 km.

Evidently, the above fields are simulated only to a finite scale. In
contrast, if the observed fields were simulated according to such a model,
they would be developed to an infinite scale and then integrated back up by
the radar. This distinction is referred to as a dressed cascade, i.e. it has
been developed to an infinite scale and then integrated back up. Fields
simulated only to a finite scale, without integration, are referred to as
bare cascades with fields in between being partially dressed. This difference
is shown in Fig.

The data for this study were acquired by a C-band weather radar near
Wideumont, Belgium, operated by the Belgian Royal Meteorological Institute
(RMI). This installation covers a circular area with a radius of 240 km,
producing a multilevel scan every 5 min. The region covered includes
coastal landscapes to the west, and a low mountain range, the Ardennes, to
the east with land cover mostly composed of forests, urban development and
agriculture. The entire region has a temperate climate and receives about
800 mm of rain annually, almost uniformly distributed throughout the year

The actual 5 min radar images are taken from large events during 2009,
with 9 winter storms and 17 summer storms. These images were extracted from a
6-month time series during which larger storm episodes were selected to
ensure sufficient data. These images correspond to the basic 5 min
interval images; however, to reduce the data load, we opted to use only the
first image of each hour. The images used were not aggregated in order to
retain the basic spatial scaling behaviour as well as to avoid ripple effects

The raw radar data are produced by a 5-elevation scan performed every 5 min. Measurements are collected up to 240 km with a resolution of 250 m
in range and 1

As with all weather radars, not all measurements are suitable for
quantitative analysis. Firstly, the radar cannot accurately measure rain
rates below

The power spectra of all rainstorms, up to a range of 180 km from
the radar, for images which have more than 10 % active pixels and storms with
at least 10 valid images. The storm spectra are found by averaging together
the spectra of each of the images. The number at the end of each line is the
slope

We analysed the rainfall fields both individually and for each of the storms
(by averaging the power spectra of each image in the storm), prior to any
changes made to the image, i.e. the raw fields

Furthermore, the summer storms tend to have a

A boxplot of the non-conservation parameter

To find corroboration for the slopes with

The empirical moments

The moment scaling functions of the rainfall images and storms

Instead of the regular double trace moment, the fields were analysed using
the weighted multifractal analysis (WMA) (see

The averages taken in upscaling are only over raining pixels.

Each pixel has a weight associated with the fraction of rainy pixels within the disjoint boxes at the finest-scale level.

The empirical function

The results of this analysis are shown in Fig.

The difference between the parameter

As mentioned in the introduction and Sect.

The empirical means of the increments, averaged over all images, and its fit. The error bars denote the 25th and 75th percentiles. There does appear to be some steady behaviour, but it appears highly complex, with relatively small values suggesting that the mean might be sufficient to model its behaviour.

The empirically fitted

The empirical

The difference between the probability of dry pixels as predicted, and as observed.

There are a number of different parametrizations available for the

When

The weighted histograms of rainfall field increments at a range of scales. The lines are the fitted distributions without any preset parameters.

The

The starting point for any analysis is the rainfall intensity field

The weighted histograms of a rainfall field at a range of scales. The lines are the fitted distributions without any preset parameters. Note that the observed tail is a result of the local normalization, not a natural feature of the rainfall field.

Subsequently, the data were coarse-grained for analysis. This scaling was done
using a moving average (low-pass) filter using a box with sides of length

The resulting set of rainfall images

The correlations of all scales for each of the storms. Note that almost all rainfall fields exhibit correlations, and that almost all of them are positive.

Moreover, as mentioned earlier, the parameter

The difference between the relative error of the distribution without correlation and that with correlation propagated from the largest scale. The inclusion of correlations leads to small, but consistent, improvements (i.e. negative values). It is immediately clear that the coarsest scale is not well captured by the functions, evidenced by the large relative error.

Besides the basic parameters of the distribution, we are also interested in
establishing whether or not the fields and their increments are actually
i.i.d.. A simple test would be to use the correlation to assess
whether or not these distributions are uncorrelated. However, the

The relative errors of the mean, shape and correlation functions.
All functions appear to behave relatively stable throughout winter and summer,
with the exception of

The basis of the correlation value of

The parameter for the mean of the increment, shown as boxplots for each storm. Summer storms clearly have lower increments.

The parameters for the

The relationship between the shape parameters of the rainfall field and its
increments,

The parameters for the correlation of the increment, shown as boxplots for each storm. The functions for the correlation appear stable throughout winter and summer and have a high intercept, but a low slope suggesting that the correlation is somewhat the same for all scales.

To investigate the behaviour of the scaling of the

Finally, the number of dry pixels are modelled based on the fractal box-counting dimension

The difference between the relative error of the distribution without correlation, and that with correlation propagated from the largest scale. The inclusion of correlations leads to small, but consistent, improvements (i.e. negative values). It is immediately clear that the coarsest scale is not well captured by the functions, evidenced by the large relative error.

The assumption that the both the distribution of

After appropriate normalization, the distribution of the increments

In Fig.

The functions

The analyses confirm the common finding that summer storms tend to be more energetic with higher variances and higher mean rainfall. Moreover, summer storms appear to exhibit a smaller decrease in correlations, resulting in a stronger correlation at the lower-scale levels.

Figure

In this paper, we investigated the scaling behaviour of the distributions of rainfall. To this end, a novel scaling model was introduced that only relies on the basic assumptions regarding the cascade structure responsible for the fractal nature of rainfall. Furthermore, this framework is based on direct empirical comparison with the observed distributions. In contrast, most previous work relied on theoretical considerations and indirect use of the scaling distributions. Therefore, this framework allows for a more direct and empirical investigation into the scaling behaviour of rainfall, and provides a more adaptable framework to be used for practical purposes.

Rainfall was found not to be the result of an i.i.d. cascade, but
rather of a cascade where the distribution changed and the increments are
dependent on their coarse-scale parents. The changes in distribution, as
described by the shape parameter of the

The correlations found in the cascade were positive for almost all storms,
and were shown to depend only on the large-scale values and not on the
season. However, these correlations were clearly dependent on the scale of
averaging, where larger scales resulted in larger correlations, up to the
point were scaling became erratic. These dependencies have also been observed
by other authors in time series

The inclusion of correlations into the distributional model showed only
moderate improvements, in part due to the small magnitude of the scale
parameters where the correlations were found. Nonetheless, the deviation
from identical distributions, as evidenced through the change in

In future research, the full dependence structure will need to be evaluated
to allow for a more accurate representation of the dependence between scale
levels and their increments. This will allow for a deeper investigation into
this aspect of imperfect scaling and possibly a better way of representing
the scaling behaviour. Moreover, it was observed that local trends were
present in all rainfall images, not only in the mean of the field, but also
in the correlations, this will need to be investigated further. Finally, the
difference with respect to the scaling behaviour, between convective and
stratiform storms, will need further investigation, using a classification
algorithm such as the Steiner algorithm

We wish to thank the Special Research Fund (B.O.F.) of Ghent University and the Flanders Research Fund (FWO, grant number: G.0837.10) for funding this research. We would like to thank A. Langousis and A. Seed for a fruitful discussion and their insightful comments. Edited by: A. Bárdossy