Spatial downscaling of rainfall fields is a challenging mathematical problem for which many different types of methods have been proposed. One popular solution consists of redistributing rainfall amounts over smaller and smaller scales by means of a discrete multiplicative random cascade (DMRCs). This works well for slowly varying homogeneous rainfall fields but often fails in the presence of intermittency (i.e., large amounts of zero rainfall values). The most common workaround in this case is to use two separate cascade models, namely one for the occurrence and another for the intensity. In this paper, a new and simpler approach based on the notion of equal-volume areas (EVAs) is proposed. Unlike classical cascades where rainfall amounts are redistributed over grid cells of equal size, the EVA cascade splits grid cells into areas of different sizes, with each of them containing exactly half of the original amount of water. The relative areas of the subgrid cells are determined by drawing random values from a logit-normal cascade generator model with scale and intensity-dependent standard deviation (SD). The process ends when the amount of water in each subgrid cell is smaller than a fixed-bucket capacity, at which point the output of the cascade can be resampled over a regular Cartesian mesh. The present paper describes the implementation of the EVA cascade model and gives some first results for 100 selected events in the Netherlands. Performance is assessed by comparing the outputs of the EVA model to bilinear interpolation and to a classical DMRC model based on fixed grid cell sizes. Results show that, on average, the EVA cascade outperforms the classical method, producing fields with more realistic distributions, small-scale extremes and spatial structures. Improvements are mostly credited to the higher robustness of the EVA model in the presence of intermittency and to the lower variance of its generator. However, both approaches have their advantages and weaknesses. For example, while the classical cascade tends to overestimate small-scale variability and extremes, the EVA model tends to produce fields that are slightly too smooth and block shaped compared to the observations. The complementary nature of the two approaches, and the fact that they produce errors of opposite signs, opens up new possibilities for quality control and bias corrections of downscaled fields.

Stochastic rainfall downscaling algorithms are statistical methods designed to enhance the resolution of coarse-scale rainfall observations for use in hydrological modeling, weather prediction or flood-risk analyses. Their simplicity and low computational cost mean that large ensembles of possible realizations for a single input field can be generated. This leads to a better representation of measurement errors and model uncertainties compared to physical downscaling and a more realistic representation of small-scale variability. However, the statistical nature of the approach means that one needs to find a good balance between model complexity and performance (e.g., the realism of the distributions and spatial patterns that can be reproduced).

Popular statistical downscaling methods for global and regional climate models include various forms of transfer functions and quantile matching

One long-standing and still-unresolved issue of random multiplicative cascade models applied to rainfall concerns the question of how to properly deal
with zero rainfall values. Zeros are fundamentally incompatible with the notion of self-similarity and multiplicative random cascades

Given the numerous challenges mentioned above, there is a strong incentive to design new simple multiplicative cascade models capable of handling
rainfall fields with high levels of intermittency. Particular attention is given to parsimonious models, with a maximum of three parameters, whose values can be
inferred directly from the coarse-scale data. One promising avenue explored in this paper revolves around the notion of equal-volume areas (EVAs),
a natural extension of the interamount times (IATs) concept introduced in the context of time-series analysis by

The rest of this paper is structured as follows: Sect. 2 introduces the new EVA model, including the splitting rule, cascade generator and parameter estimation. In Sect. 3, the potential of the new cascade is demonstrated by applying it to radar rainfall snapshots collected over the Netherlands. First, the parameterization problem is discussed. Then, the performance is evaluated by means of controlled simulation experiments during which 100 high-resolution rainfall fields are aggregated to coarser scales and subsequently downscaled back to their original resolution. Results are compared to two alternative downscaling techniques (i.e., bilinear interpolation with local intensity rescaling and a classical random cascade based on intensity). The advantages and limitations of the model and possible extensions are discussed in Sect. 5, and the conclusions are given in Sect. 6.

Discrete multiplicative random cascades (DMRCs) are statistical downscaling techniques designed to enhance the resolution of a coarse-scale rainfall
field to a desired fine-scale target resolution. For spatial cascades, this is done by successively splitting the dimensions of coarse-scale grid
cells by two (or four, depending on the type of cascade) according to a predefined branching rule. For example, one large 16

As pointed out by

Let

In a classical cascade model, grid cells of area

The random quantities in this case are the rainfall volumes

The main contribution of this paper is to show that many of the issues associated with zero rainfall values can be avoided by adopting a slightly
different representation of rainfall based on the notion of equal-volume areas. In the EVA framework, the scale

Schematic of the branching rules for the classical and equal-volume area (EVA) random cascades. The area is denoted by

Note that by convention, splits always occur perpendicular to the longest grid cell dimension; that is, splitting occurs horizontally if

Finally, note that, by construction, the EVA cascade described above implements an adaptive spatial sampling of the coarse-scale rainfall field, which is very
similar to that of a quadtree

The way grid cells are split at each level plays a crucial role in determining the spatial structure of the downscaled field. Independent of the
used cascade generator, for any weight

Illustration of the splitting rule for a single grid cell (in bold at the center of the figure), with area

The probability distribution of the cascade generator is a crucial component of any discrete multiplicative random cascade

The logit-normal generator model is not necessarily optimal for all types of events and all spatiotemporal scales, but it is a fair enough
approximation of empirical cascade weights to be useful in practice. Moreover, the distribution is continuous, supported over the open unit interval
(0,1) and easy to simulate through its analytical link with the Gaussian distribution. The most important advantage of all, however, lies in the ease
of interpretation of the parameter

Theoretical distribution of the logit-normal cascade weights

Since

Because the amount of water is halved at each split of the cascade, according to Eq. (

In the first case (

Example of an EVA cascade for an 8

Empirical breakdown coefficients for a 4

Therefore, grid cells at subsequent cascade levels will tend to split more and more evenly, eventually converging to a fixed rainfall intensity. In
the second case (i.e.,

An important advantage of the microcanonical model is that the distribution of the cascade weights can be studied directly from the data through the
calculation of empirical breakdown coefficients

In the classical cascade model, no linear interpolation is needed. However, some of the rainfall volumes in the subgrid cells may be zero (i.e., one
size receives all the rain). Such splits are fundamentally incompatible with the logit-normal model prescribed in Eq. (

Once the empirical breakdown coefficients have been determined from the sample, the last step consists of estimating the three model
parameters

While the EVA downscaling technique is the main focus of this paper, two additional spatial downscaling techniques were considered for comparison
purposes. The first is bilinear interpolation, implemented in the function “interp.surface()” of the R package “fields”

The second benchmark is a classical microcanonical discrete multiplicative random cascade based on rainfall intensity as described in
Eq. (

To assess performance, synthetic experiments on high-resolution radar rainfall fields were performed. During these experiments, 100 different
5

Original 1

Summary statistics for the four example events, namely time; proportion of zero rainfall values

Model parameter estimates

Estimated coarse-scale generator parameters

In the following, the cascade generator models for the EVA and classical cascade models (for each of the 100 1

Standard deviation of empirical breakdown coefficients for the 100 radar snapshots in the database as a function of the rainfall intensity

Example of empirical breakdown coefficients

Another important observation that can be made concerns the variance of the generator for the EVA and classical
models. Figure

Figure

Downscaled rainfall fields for events 1–4 and a downscaling factor of 64 (i.e., input resolution of 8

Comparing the outputs of the EVA and the classical cascade, one can see that the EVA cascade tends to produce smoother fields with lower overall
variance and peak intensities. Visually, the fields appear to be in better agreement with the original radar snapshots, both in terms of distribution
and spatial structure (see Sect. 3.3 for more quantitative comparisons). Visually speaking, one of the biggest disadvantages of the EVA cascade appears
to be the fact that the resulting fields look slightly block shaped, with some of the initial coarse-scale grid cells still visible. The block shape can be
attributed to biased parameter estimates

Before moving on to more quantitative assessments, there is another important point that needs to be made here concerning the individual performances
of the two random cascade models. The problem with Fig.

Downscaled rainfall fields for events 1–4 and a downscaling ratio of 64 (i.e., input resolution of 8

Next, the probability distribution functions of the downscaled rainfall rates generated by the random cascades are
assessed. Figure

Observed versus downscaled rainfall rates for the first four events in the database and a downscaling ratio of 64 (i.e., input resolution of 8

Sample variograms of rainfall intensity (given occurrence) for events 1–4 and spatial displacements up to 8

Sample variograms of rainfall occurrence for events 1–4 and spatial displacements up to 8

Overall performance of the random cascade models for 100 high-resolution radar rainfall fields, coarse-scale sample generator estimate and downscaling factor of 64 (i.e., input resolution of 8

First, the rainfall rates generated by the classical random cascade model are analyzed. The distributions appear to be in relatively good agreement with the observations. However, some important discrepancies remain, especially for the very high quantiles. Performance is clearly sensitive to parameterization issues, which vary a lot depending on the type of event and chosen generator model. Homogeneous, low-intensity events such as event 2 are reproduced rather well. But, in events 1 and 4, extremes are clearly overestimated. In fact, in the majority of the 100 considered events, the classical cascade overestimates rainfall extremes when the coarse-scale generator is used. However, there are also a few interesting exceptions to this rule. For example, in event 3, the classical cascade underestimates the 99.9 % quantile compared to the observations. The problem with event 3 is that the rainfall field is highly heterogeneous, consisting of multiple convective and stratiform areas of different sizes, shapes and orientations. Therefore, big local differences in scaling behavior exist within the field, making it hard to derive a meaningful cascade generator model that applies to the entire domain. This is highlighted by the fact that the coarse-scale generator actually produces better results than the fine-scale generator, which is highly unusual and points to serious problems during parameter estimation.

When looking at the results for the EVA model, there appears to be no obvious, substantial improvement in terms of the model's ability to reproduce
higher rainfall rates and small-scale extremes. The only clear advantage, compared to the classical approach, is that the outcomes of the EVA cascade
are more consistent with each other (i.e., they have a lower ensemble spread). However, the downscaled rainfall distributions are clearly too narrow
compared to the observations, meaning that the model underestimates higher rainfall quantiles and small-scale extremes. Still, the underestimation
is much less severe than for bilinear interpolation. The systematic underestimation of higher rain rates is a problem but can be explained by the fact
that the variance of empirical EVA cascade weights for small values of

Figure

Next, the performance of the cascade models as a function of the downscaling ratio is analyzed. Figure

10 %, 25 %, 50 %, 75 %, and 90 % quantiles of the coefficient of determination

10 %, 25 %, 50 %, 75 %, and 90 % quantiles of the coefficient of determination

While this research mainly focused on the description of the EVA cascade model, the underlying generator and its application to a few selected case
studies, there are numerous complementary research lines that can be pursued. One of them revolves around possible ways to overcome biases in cascade
generator parameters and correct for systematic errors as a function of the intermittency and downscaling ratio. Diagnostic tools for detecting
potentially problematic cases based on plausible ranges for each parameter need to be developed. Alternatively, one could apply both an EVA and
a classical cascade and compare the obtained results. If they are wildly inconsistent, the EVA model is likely to be closer to the radar
observations. Another possibility would be to design flexible climatological generator values that can be adjusted depending on rainfall type and
large-scale properties (e.g., intensity, intermittency and range), which is an approach that may be more flexible while limiting sampling issues. Preliminary work
performed within this study (not shown) suggests that this may be promising for larger downscaling ratios as cascade parameters often tend to be
correlated with each other or to large-scale rainfall properties

The second point that is worth discussing concerns the complementary nature of the EVA framework compared to the classical representation in terms of intensity over fixed grid cell sizes. The main advantage of the EVA framework lies in its adaptive sampling strategy. By flipping the problem around and focusing on the areas for fixed amounts of water, rather than the opposite, additional insight into the spatial variability of rainfall within grid cells can be gained. Most importantly, occurrence and intensity are not viewed separately anymore but combined together into a single continuous process. All quantities are strictly positive, which reduces model complexity, improves the scaling and lowers sampling uncertainty. If rainfall fields were perfectly homogeneous and the sensors used to measure them had unlimited precision, the two representations would be equivalent. However, since rainfall fields can be highly variable in space and time, and measurements are affected by sampling uncertainties, one of the two representations is likely to be more appropriate or useful in practice. A better understanding of these cases and how to choose the best framework depending on sampling resolution, intermittency and measurement accuracy is key for improving our understanding of the space–time variability of rainfall and its representation in models.

The third issue that needs to be mentioned relates to the assumption that the cascade generator model is stationary and, in particular, location
invariant (i.e., that the same splitting rules apply to all pixels independent of their location). This may not necessarily be valid for highly
heterogeneous fields, as highlighted by the poor performance and inconsistent behavior of the cascade models during event 3. The key point here is
that there might be specific areas within a rainfall field where the scaling properties are different from the rest (e.g., stratiform versus convective
areas). Similarly, the scaling properties and spatial variability within individual rainfall cells might be very different from the average
variability observed over a large collection of rain cells. Also, elements belonging to larger-scale structures might evolve together in a more
coherent and predictable way than expected based on their size and intensity. One possible solution for overcoming this problem would be to define
multiple local generators instead of a single universal one. But this is a very challenging problem that requires more research, including the ability
to automatically detect strong local variations in scaling properties to help pinpoint problematic regions and come up with a better approach. Also,
the use of multiple generators would require additional model parameters, which is not necessarily desirable and should only be considered when
absolutely necessary (e.g., to account for strong orographic effects). On a more theoretical level, one should also point out that even if
the cascade generator is perfectly stationary, the final disaggregated fields (or time series) obtained after applying the cascade are likely to be
nonstationary with location and time-dependent autocorrelation structures

The fourth point of discussion concerns possible extensions of the EVA model. Similar to classical multiplicative random cascades, the EVA cascade
can be applied to downscale time series, spatial and space–time data. For time series, the equivalent formalism is given by the notion of
equal-volume times, also known as interamount times

One last point that is worth mentioning concerns the computational complexity of the EVA model. One crucial difference between the EVA and the
classical cascade is that the classical cascade stops as soon as the target resolution has been reached. The EVA cascade on the other hand tends to
run over more levels, producing many grid cells that are smaller than the target resolution. The total number of cascade levels and grid cells depends
on (1) the initial rainfall volumes contained in the coarse-scale grid cells and (2) the bucket capacity prescribed by the user. This means that for
large rainfall fields (e.g., several hundreds of km) with high rainfall intensities, the number of generated grid cells can be of the order of several
millions. As a result, both the run time and memory usage will be larger than for a classical cascade. However, there are various ways to limit the
computational burden. The easiest is to stop splitting grid cells once they are about 3–4 times smaller than the target resolution, regardless of how
much water they contain. Similarly, grid cells that are entirely contained within a target resolution pixel do not need to be split up further
(regardless of their size and amount) as these additional splits would not be visible after the resampling anyway. Similarly, there is no need to
split up grid cells once they have converged to a fixed rainfall intensity, i.e., when

A new multiplicative random cascade for downscaling intermittent rainfall fields based on the concept of equal-volume areas (EVAs) has been
proposed. Downscaling experiments on 100 high-resolution radar rainfall snapshots in the Netherlands have shown that, on average, the EVA cascade
outperforms its competitors, both in terms of the reproduced rainfall distributions and spatial structures. Improvements are mainly attributed to the
adaptive sampling strategy in the EVA formalism, which avoids zero rainfall values and leads to more accurate and robust model estimates in the
presence of intermittency. The new proposed logit-normal cascade generator model with scale- and intensity-dependent variance ensures that every grid
cell in the EVA cascade eventually converges to a fixed intensity or a fixed area, putting the new model in the category of bounded microcanonical
cascades. Despite the encouraging results, improvements are not systematic and many challenges remain. The most important is that the EVA cascade
tends to underestimate small-scale extremes, producing fields that are slightly too smooth and block shaped compared to the observations. This is
attributed to biased model parameters and, more generally, to the difficulty of retrieving the true cascade generator from coarse-scale data. The fact
that cascade weights in the EVA framework must be estimated using linear interpolation is also a clear weakness, causing

Apart from introducing a new model, the present study also clearly highlighted the outstanding challenges associated with downscaling intermittent
rainfall fields. The most important issue concerns the estimation of cascade generator models from coarse-scale data. Sensitivity analyses performed
within the framework of this study clearly showed that two of the cascade model parameters (i.e.,

All data used in this study can be downloaded free of charge from the KNMI data center (

The author declares that there is no conflict of interest.

We extend our special thanks to the Royal Netherlands Meteorological Institute (KNMI) for collecting and providing the radar data used in this study.

This research has been supported by the Netherlands Organisation for Scientific Research NWO (grant no. ALWWW.2014.3). The funding for this work has also been provided by Water JPI Europe, an ERA-NET Cofund WaterWorks2014 project, entitled Multi-scale Urban Flood Forecasting (MUFFIN): From Local Tailored Systems to a Pan-European Service.

This paper was edited by Carlo De Michele and reviewed by Katarzyna Siekanowicz and one anonymous referee.