Copulas have already proven their flexibility in rainfall modelling. Yet, their use is generally restricted to the description of bivariate dependence. Recently, vine copulas have been introduced, allowing multi-dimensional dependence structures to be described on the basis of a stage by stage mixing of 2-dimensional copulas. This paper explores the use of such vine copulas in order to incorporate all relevant dependences between the storm variables of interest. On the basis of such fitted vine copulas, an external storm structure is modelled. An internal storm structure is superimposed based on Huff curves, such that a continuous time series of rainfall is generated. The performance of the rainfall model is evaluated through a statistical comparison between an ensemble of synthetical rainfall series and the observed rainfall series and through the comparison of the annual maxima.

Rainfall serves as an important base for many studies involving hydrological
applications including flood risk estimation, the design of hydraulic
structure and urban drainage systems or the evaluation of hydrological
effects of climate change. Ideally, one should then have extensive observed
rainfall time series at hand, both in time and space and at different timescales. Therefore, several rainfall modelling approaches have been proposed
during the last decades (e.g.

The variables that characterize a storm, i.e. the storm intensity, duration
and volume, mostly exhibit some kind of mutual dependence: a long storm
duration is more likely to be associated with a low storm intensity than with
a high one. It is therefore of utmost importance to construct joint
probability distribution functions whenever frequency analysis studies, e.g.
to analyse extremes, need to be carried out. Yet, the marginal probability
distribution functions of these storm variables usually do not exhibit the
same type of parametric distribution and are largely
skewed

Copulas are functions that couple the marginal distribution functions of the
random variables into their joint distribution function and therefore
describe the dependence structure between these random variables

This paper explores how a point-scale rainfall model can be constructed using
multivariate copulas, in order to incorporate all relevant dependences
between the storm variables of interest. The application of multivariate
copulas in hydrology is, in contrast to the application of bivariate copulas,
a less explored domain. Some applications can be found in the modelling of
trivariate rainfall

The model that is developed in this paper consists of two submodels. In a
first submodel, the vine copula model, 3- and 4-dimensional vine
copulas are used to describe the dependence between the storm duration, storm
volume, the interstorm period following the storm and, in case a
4-dimensional vine copula is used, also the dry fraction within the storm.
In a second submodel, the intrastorm-generating model, the intrastorm
variability is obtained based on Huff curves

A vine copula mixes (conditional) bivariate copulas stage by stage in order
to build a high-dimensional copula, i.e. the full density function is
decomposed into a product of low-dimensional density functions. Consider the
case of two random variables

A bivariate copula or a 2-copula is a function

for all

for all

Hierarchical nesting of bivariate copulas in the construction of a 3-dimensional vine copula through conditional mixtures.

Similarly, for a random vector

Figures

In practice, the bivariate copulas in a higher tree of the vine copula (e.g.

Hierarchical nesting of bivariate copulas in the construction of a 4-dimensional vine copula through conditional mixtures.

A general simulation algorithm is presented next, borrowed from the theory on
conditional mixtures. The literature on vine copulas reports very similar
simulation algorithms

The time series used in this paper for fitting the model consists of a
105-year 10 min rainfall record of Uccle, Belgium. These data were obtained
by a Hellmann–Fuess pluviograph, installed in and operated by the Royal
Meteorological Institute at Uccle near Brussels, Belgium

Observed probability of

A kernel-smoothed distribution function was fitted to the observed values of

As the storm characteristics

Huff curves for the second-quartile autumn storms. The 10 % (lower) and 90 % (upper) percentile curves are given.

By examining the storm characteristics of the historical time series, it is
observed that some storms have internal dry 10 min intervals while others
have not. It was decided to fit, for each season, a 4-dimensional vine
copula to the values of

Contour plots of the empirical (dotted lines) and the fitted Frank copulas (solid lines) for the
different trees in the 3-dimensional vine copula for season 1. Bivariate copulas between

The ordering of the copulas in the vine copula, i.e. the selection of a
D-vine, is based on the values of Kendall's tau as listed in
Table

Correspondence between the observed and simulated pairwise dependences
among

Parameters of the bivariate Frank copulas in the construction of the 3- and 4-dimensional vine copulas.

Contour plots of the empirical (dotted lines) and the fitted Frank copulas (solid lines) for
the different trees in the 4-dimensional vine copula for season 1. Bivariate copulas between

The inverse CDFs are then used to transform simulated uniformly distributed values in

In order to employ Huff curves in the disaggregation of the rectangular
pulses, a random quartile group is first assigned based on the probabilities
of occurrence of the four quartile groups, as indicated in
Table

The internal storm structure is then generated as follows. Firstly, time
intervals having zero rainfall are randomly assigned within the storm such
that the sampled value of

Secondly, the cumulative storm depths are randomly selected. This procedure
calculates the normalized cumulative depth at the end of a time interval,
i.e. at time instant

Time instant

Time instant

Time instant

Illustration of the generation of an internal storm structure. The part of the Huff curve that is
already generated (up to time instant a) is indicated by a thick solid line. The value at time instant b needs
to be determined. Four sampling strategies are possible: sampling in between two consecutive wet periods (case 1;

Based on the historical time series, it was observed that the increment in
cumulative storm depth between two subsequent time instants in a Huff curve
is not uniformly distributed (this observation was neglected in

Probability of a storm to belong to a certain quartile group.

Illustration of the procedure to sample the storm depth at the next time instant (b). First, the minimal and maximal increment in percentage of storm depth at time instant b are determined (top panel). Then, the corresponding sampling range in the CDF of normalized increments is defined based on the minimal and maximal increment in percentage of storm depth derived from the top panel (bottom panel).

Comparison of the empirical cumulative distributions of the yearly statistics of the observed time series (black line) and the bundle of empirical cumulative distributions of synthetic time series generated by means of the copula-based model (grey) at a 10 min (a) and a 1 h aggregation level (b).

It is common to validate the performance of a model through comparing
statistics of one modelled time series to those calculated on the observed
time series. However, given that the model has a stochastic nature, the
statistics of the simulated time series will show some variability. To
account for these stochastic effects, the model described in the previous
section is employed to generate an ensemble of 100 time series of 105 years
of 10 min rainfall (i.e. similar to the length of the observed time series).
In order to evaluate whether the model performs well in the reproduction of
aggregated rainfall statistics, the 100 time series are furthermore regarded
as equally probable realizations and the statistics are calculated on a
yearly basis. The traditional first- and second-order statistical moments
(i.e. mean and variance), autocorrelation (AC) at different time lags and the
zero depth probability (ZDP) are calculated along with the third-order
central moment (skewness). These statistics are calculated on a yearly basis
for each ensemble member at aggregation levels of 1/6, 1, 3, 6, 12 and 24 h. Thus, for an aggregation level, 100

Comparison of empirically derived annual maxima related to the empirical return periods for different aggregation levels on the observed (black asterisks) and ensemble of synthetic time series generated by means of the copula-based rainfall model (grey asterisks).

Figure

As simulated time series are often used to simulate extreme discharges

This study is the first in its kind in which a continuous stochastic rainfall
generator is developed that uses vine copulas to describe the storms and
their arrival process. The internal storm structure is based on the concept
of Huff curves, while the fraction of dry periods within the storm is
determined by the copulas. The main advantage of this approach is that the
model is completely data driven and is easier to calibrate than other
rainfall generators such as the commonly used Modified Bartlett–Lewis model
as, once the structure of the vine copula is determined, the calibration is
reduced to estimating the parameters of the bivariate copulas. It should,
however, be noted that we have at our disposal an exceptionally long time
series of rainfall data on the basis of which the vine copulas are
determined. If one would follow the same approach and search for the
best-fitting copula family on a more commonly shorter time series of e.g.

The model applies 3- and 4-dimensional vine copulas to describe the dependence between the different storm characteristics. The 3-dimensional vine copulas are employed to describe the seasonal dependence between storm duration, storm volume and the interstorm period for storms that have no dry fraction within the storm. The 4-dimensional vine copulas are employed to describe the seasonal dependence between these storm characteristics and the dry fraction within the storm. These vine copulas were fitted to the observed storm characteristics of a 105-year time series of 10 min rainfall. Because of its frequent successful application in hydrological applications, the Frank copula family was chosen to be used within the vine copulas. On the basis of these vine copulas, values of these four storm characteristics were drawn, representing the external storm structure, ensuring a time series of 105 years of rectangular rainfall pulses. According to their seasonal probability of occurrence, storms with zero dry fractions were sampled from the 3-dimensional vine copulas. The internal storm structure of the rectangular pulses is superimposed based on Huff curves, which were identified on the basis of the observed time series, leading to the generation of continuous 10 min rainfall time series. In the generation of the internal storm structure, it is ensured that the fraction of dry periods within the storm as drawn from the vine copulas, is maintained. The internal storm structures are furthermore generated according to the probability of occurrence of the quartile storms in the observed time series, and the season in which they occur. In order to determine the difference of cumulative storm depths in the internal storm structure, the empirical cumulative PDF of increments between two subsequent wet periods in the storm is employed. In this way it is guaranteed that smaller increments occur more often than larger increments, as was observed in the measured time series.

In order to evaluate the performance of the rainfall model, an ensemble of 100 time series of ca. 105-year 10 min rainfall was generated, such that stochastic effects were accounted for. The results show that the copula-based rainfall model represents the mean value of the time series well, whereas the other statistics are either represented (fairly) well, over- or underestimated, depending on the aggregation level. A second evaluation of the generated ensemble encompassed the calculation of the annual maximum series, for different aggregation levels. It was observed that the annual maxima simulated by means of the copula-based model were larger than the observed maxima for an aggregation level of 10 min, and the moderate return period of the 24 h aggregation level. For aggregation levels of 1–12 h and the smaller and larger return periods of an aggregation level of 24 h, a good correspondence between the simulated and observed extremes was observed. Future research will reveal whether the representation of the ZDP statistic for larger aggregation levels by the copula-based model can be improved by better selecting the internal dry storm periods. The performance of the copula-based model will also be compared to state-of-the-art stochastic rainfall generators. Also, it should be investigated whether including other bivariate copula families in the vine copulas can further improve the performance of the vine-copula-based model.

This research has been performed in the framework of projects G.0837.10 and G.0013.11 granted by the Research Foundation Flanders. Edited by: C. De Michele