For several hydrological modelling tasks, precipitation time series with a
high (i.e. sub-daily) resolution are indispensable. The data are, however,
not always available, and thus model simulations are used to compensate. A
canonical class of stochastic models for sub-daily precipitation are Poisson
cluster processes, with the original Bartlett–Lewis (OBL) model as a
prominent representative. The OBL model has been shown to well reproduce
certain characteristics found in observations. Our focus is on
intensity–duration–frequency (IDF) relationships, which are of particular
interest in risk assessment. Based on a high-resolution precipitation time
series (5

Precipitation is one of the most important atmospheric
variables. Large variations on spatial and temporal scales are
observed, i.e. from localized thunderstorms lasting a few tens of
minutes up to mesoscale hurricanes lasting for days. Precipitation on
every scale affects everyday life: short but intense extreme
precipitation events challenge the drainage infrastructure in urban
areas or might put agricultural yields at risk; long-lasting extremes
can lead to flooding

Risk quantification is based on an estimated frequency of occurrence for
events of a given intensity and duration. This information is typically
summarized in an intensity–duration–frequency (IDF) relationship

Due to a limited availability of observed high-resolution records with
adequate length, simulations with stochastic precipitation models are used to
generate series for subsequent studies

Due to the high degree of simplification of the precipitation process, known
drawbacks of the OBL model include the inability to reproduce the proportion
of dry periods, as reported by

The OBL model and IDF relationships are of
particular interest to hydrological modelling and impact
assessment. In the following, we address three research questions
by means of a case study:

Is the OBL model able to reproduce the intensity–duration relationship found in observations?

How are IDF curves affected by very rare extreme events which are unlikely to be reproduced with the OBL model in a reasonably long simulation?

Is the parametric extension to the GEV a valid approach to obtain IDF curves?

Section

From early radar-based observations of precipitation, a hierarchy of
spatio-temporal structures was suggested by

The idea of modelling rainfall with stochastic models has existed since

Similarly to various others studies

Simulations with the OBL model are in continuous time on the level of storms
and cells. We aggregate the resulting cell rainfall series to hourly time
series. Figure

Scheme of the OBL model. A similar scheme can be found in

Example realization of the OBL model. The top panel shows
the continuously simulated storms and cells by the model. In the
middle panel the cell intensities are combined with a step
function. The bottom panel shows the aggregated artificial
precipitation time series. Parameters used:

An alternative to the Bartlett–Lewis process is the Neyman–Scott process

Due to known drawbacks of the OBL model, several improvements and extensions
have been made in the past:

Parameter estimation for the OBL model is by no means trivial. The canonical
approach is a method-of-moment-based estimation

It turns out that

A few tests indicate that the symmetric
version is robust and faster in the sense that fewer iterations are
needed to ensure convergence into the global minimum (not shown).
Numerical optimization techniques based on gradient calculations,
e.g.

Following studies by

Models of this type suffer from parameter non-identifiability, meaning
that qualitatively different sets of parameters lead to minima of the
objective function with comparable values

During this work the authors developed and published the

IDF curves show

However, the data points for different durations are dependent (as they are derived from the same underlying high-resolution data set by aggregation), and thus the i.i.d. assumption required for maximum-likelihood estimation is not fulfilled. Consequently, confidence intervals are not readily available from asymptotic theory; however, they can be estimated by bootstrapping.

A precipitation time series from the station

Minimizing the symmetric objective function (Eq.

OBL model parameter estimates for all months of the year
obtained from the Berlin-Dahlem precipitation time series. Top: cell cluster
generation rate

With the OBL model parameter estimates (Table

Comparison of statistics derived from the observational
record (red dots) and 1000 simulated time series (box plots):

An important aspect for hydrological applications is the model's ability to reproduce extremes on various temporal scales. This behaviour is investigated in the next section with the construction of IDF curves.

Monthly block maxima for every month in the year are drawn for various
durations (1, 3, 6, 12, 24, 48, 72, 96

IDF curves for Berlin-Dahlem obtained from observation are shown as
dotted lines in Fig.

IDF curves obtained via dd-GEV for

Analogously, IDF curves are derived from 1000 simulations of the OBL model
precipitation series; see Sect.

Relative differences between observed and simulated return levels
obtained with including the third moment (red) and with using the probability
of zero rainfall (blue) in parameter estimation for

We interpret the different behaviour for short durations (flattening versus continuation of the straight line) for summer (July) and the remaining seasons as a result of different mechanisms governing extreme precipitation events: while convective events dominate in summer, frontal and thus more large-scale events dominate in the other seasons.

As an example, we show segments of time series including the maximum
observed/simulated rainfall in July for durations 1, 6, and 24

Parts of the observed and simulated rainfall time series corresponding to the
extreme events for the three different durations are shown in the left and
right column, respectively. Additionally, the middle column shows the
simulated storms and cells generating this extreme event in the artificial
time series. As an example, we only show one single model simulation. Visual
inspection of several other simulated series support the main features. For
all durations, the extremes are a result of a single long-lasting cell with
high intensity. In contrast to an analysis based on the random parameter BL
model

Visualization of July extremes as observed
(

For January, IDF curves from observations and OBL model simulations exhibit
large discrepancies: for all durations, the 0.99 quantile (100-year return
level) is above the range of variability from the OBL model, and the
0.5 quantile (2-year return level) is below for small durations. This
implies that the shape of the extreme value distribution characterized by
the scale

We furthermore find that the OBL model is generally able to reproduce the
observed seasonality in IDF parameters; see Fig.

Seasonality of IDF model parameters estimated directly from the Berlin-Dahlem series (blue line), and estimated from 1000 OBL model simulations (red). The red shadings give the range of variability (5 to 95 %) from the 1000 simulations with the median as solid red line.

The convective cold front passage of Kyrill accounted for a maximum
intensity of 24.8

Return period for the event Kyrill as estimated from the observational time series with this particular event left out and included for parameter estimation.

For this data set, we estimate the OBL model parameters and again simulate
1000 time series with these new parameters. The simulated time series were
also reduced in length by 1 year, containing 12 years of rainfall in total.
From those precipitation time series, we constructed the dd-GEV IDF curves;
see Fig.

dd-GEV IDF curves for

In the frame of a model-world study, long time series simulated with the OBL
model can be used to investigate the adequacy of the dd-GEV model conditional
on the simulated series. To this end, we compare the resulting IDF curves to
a GEV distribution obtained for various individual durations. The basis is a
set of 1000-year simulations with the OBL model with parameters optimized for
Berlin-Dahlem. For a series of this length, we expect to obtain quite
accurate (low variance) results for both the dd-GEV IDF curve and the GEV
distributions for individual durations. However, sampling uncertainty is
quantified by repeatedly estimating the desired quantities from 50
repetitions. The resulting dd-GEV IDF curves are compared to the individual
duration GEV distribution in Fig.

dd-GEV IDF curves for

For most durations in January and July, the dd-IDF curves are close to the
quantiles of the individual duration GEV distributions. Notable differences
appear for small durations and large quantiles (return levels for long return
periods); particularly in January, the dd-GEV IDF model overestimates the
10-year and 100-year return levels (duration of 1

The original version of the Bartlett–Lewis rectangular pulse (OBL) model has been optimized for the Berlin-Dahlem precipitation time series. Subsequently, IDF curves have been obtained directly from the original series and from simulation with the OBL model. The basis for the IDF curves has been a parametric model for the duration dependence of the GEV scale parameter which allows a consistent estimation of one single duration-dependent GEV using all duration series simultaneously (dd-GEV IDF curve). Model parameters for the OBL model and the IDF curves have been estimated for all months of the year and seasonality in the parameters is visible. Typical small-scale convective events in summer and large-scale stratiform precipitation patterns in winter are associated with changes in model parameters.

We have shown that the OBL model is able to reproduce empirical statistics used
for parameter estimation; mean, variance, and autocovariance of
simulated time series are in good agreement with observational values,
whereas the probability of zero rainfall is more difficult to
capture

With respect to the first research question posed in the introduction, we
have investigated to what extent the OBL model is able to reproduce the
intensity–duration relationship found in observations. We have shown that
they do reproduce the main features of the IDF curves estimated directly from
the original time series. However, a tendency to underestimate return levels
associated with long return periods has been observed, similar to

Furthermore, IDF curves for January show a strong discrepancy between the OBL
model simulations and the original series. We have hypothesized and have
investigated that this is due to the Berlin-Dahlem precipitation series
containing an extreme rainfall event associated with the winter storm
Kyrill passing over Berlin on 18–19 January 2007. This
event was very rare, in the sense that on short timescales (e.g. 1 and
3 h) such an event has probability to occur only once in a period larger
than 1000 years on average. This addresses the second research question: how
are IDF curves affected by very rare extreme events which are unlikely to be
reproduced with the OBL model for a reasonably long simulation? Having
excluded the year 2007 from the analysis, the aforementioned discrepancy in
January has disappeared. We conclude that an extreme event which is rare
(return period of 23 000 years) with respect to the timescales of
simulation (

The third question addresses the validity of the duration-dependent
parametric model for the GEV scale parameter which allows a consistent
estimation of IDF curves. For a set of long simulations (1000 years) with the
OBL model, the comparison of IDF curves with the duration-dependent GEV
approach with quantiles from a GEV estimated from individual durations
suggests a systematic discrepancy associated with the flattening of the IDF
curve for short durations. Quantiles from individual durations are smaller
for short durations than in the dd-GEV approach IDF curves, which is a
challenge for the latter modelling approach. However, instead of altering the
duration-dependent formulation of the scale parameter

We have not found the OBL model producing unrealistically high precipitation
amounts, as discussed for the random-

In summary, the OBL model is able to reproduce the general behaviour of extremes across multiple timescales (durations) as represented by IDF curves. Very rare extreme events do not have the potential to change the OBL model parameters but they do affect IDF statistics, and consequently modify the previous conclusion for these cases. A duration-dependent GEV is a promising approach to obtain consistent IDF curves; its behaviour at small durations needs further investigation.

The Berlin–Dahlem rainfall time series
used in this publication has been published at PANGAEA:

Boundary constraints used in OBL model parameter estimation.

In the estimation of OBL model parameters we limited the parameter space by using boundary constraints. Lower and upper parameter limits have been set in a physically realistic range; see Table 1. For those parameter ranges, numerical optimization mostly converged into a global minimum. No constraints are applied in the model variant with the third moment implemented in the OF.

Using a Latin hypercube approach, we generated 100 different sets of
initial guesses for the parameters used in the numerical optimization
of the symmetrized objective function, Eq. (

Optimum of estimated OBL model parameters for individual months
of the year for the Berlin-Dahlem precipitation series and
corresponding value of the objective function

Figure

Relative differences (Eq.

The project was funded by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114. Henning W. Rust was supported by Freie Universität Berlin within the Excellence Initiative of the German Research Foundation. Edited by: Carlo De Michele Reviewed by: Reik Donner and three anonymous referees