Reliable estimation of rainfall extremes is essential for drainage system design, flood mitigation, and risk quantification. However, traditional techniques lack physical realism and extrapolation can be highly uncertain. In this study, we improve the physical basis for short-duration extreme rainfall estimation by simulating the heavy portion of the rainfall record mechanistically using the Bartlett–Lewis rectangular pulse (BLRP) model. Mechanistic rainfall models have had a tendency to underestimate rainfall extremes at fine temporal scales. Despite this, the simple process representation of rectangular pulse models is appealing in the context of extreme rainfall estimation because it emulates the known phenomenology of rainfall generation. A censored approach to Bartlett–Lewis model calibration is proposed and performed for single-site rainfall from two gauges in the UK and Germany. Extreme rainfall estimation is performed for each gauge at the 5, 15, and 60 min resolutions, and considerations for censor selection discussed.

Extreme rainfall estimation is required for numerous applications in diverse disciplines ranging from engineering and hydrology to agriculture, ecology, and insurance. It facilitates the planning, design, and operation of key municipal infrastructure such as drainage and flood defences, as well as scenario analysis for climate impact assessment, and hazard risk modelling. Extremes are usually estimated using frequency techniques and intensity duration frequency curves. However, these methods are highly dependent on the observed rainfall record, which may not be characteristic of the extreme behaviour.

In this study we improve the physical basis of short-duration extreme rainfall estimation by simulating the heavy portion of the observed rainfall time series. Traditional approaches to extreme value estimation rely on sampling extremes from the observed record. However, rainfall observations present various problems for the practitioner. They are often not available at the location of interest, they are typically short in duration, and they may not be available at the temporal scale appropriate for the intended use. These difficulties, together with the necessity to obtain perturbed time series representative of future rainfall, have motivated the development of stochastic rainfall generators since the earliest such statistical models developed by Gabriel and Neumann (1962). The reader is referred to Waymire and Gupta (1981), Wilks and Wilby (1999), and Srikanthan and McMahon (2001) for detailed reviews of early developments in rainfall simulation.

The principle of rainfall simulation is to replicate statistical properties of the observed record such that multiple realizations of statistically identical rainfall may be synthesized (Richardson, 1981). Various methods of simulation exist, and there have been several attempts in the literature to categorize the different approaches. Aside from dynamic methods used in numerical weather prediction models, Cox and Isham (1994) suggest that statistical simulation methods may be broadly categorized as either purely statistical or stochastic, while Onof et al. (2000) further categorize stochastic methods as either multi-scaling or mechanistic. The latter of these differ from other statistical approaches because rainfall synthesis follows a simplified representation of the physical rainfall-generating mechanism. Through the clustering of rain cells in storms, the unobserved continuous-time rainfall is constructed by superposition, enabling the synthetic rainfall hyetograph to be aggregated to whatever scale is desired (Kaczmarska et al., 2014). Because of this simplified process representation, mechanistic model parameters have physical meaning, which makes this class of model particularly appealing in the context of extreme value estimation.

When no likelihood function can be formulated (Rodriguez-Iturbe et al., 1988; Chandler, 1997), mechanistic models are typically calibrated using a generalized method of moments (Wheater et al., 2007a) with key summary statistics at a range of temporal scales such as the mean, variance, autocorrelation, and proportion of dry periods. Performance is assessed on the ability of the models to reproduce the calibration statistics as well as others not used in calibration including central moments and extremes. Since their inception in the late 1980s by Rodriguez-Iturbe et al. (1987, 1988), numerous studies have demonstrated the ability of these models to satisfactorily reproduce observed summary statistics (see Cowpertwait et al., 1996; Verhoest et al., 1997; Cameron et al., 2000a, b; Kaczmarska et al., 2014; Wasko and Sharma, 2017; and Onof et al., 2000, for a review). However, these studies have also shown that mechanistic models tend to underestimate rainfall extremes at the hourly and sub-hourly scales, which limits their usefulness (see Verhoest et al., 2010, and references therein).

We hypothesize that stochastic mechanistic pulse-based models may be poor at estimating fine-scale extremes because the training data, and calibration method, are dominated by low-intensity observations. Mechanistic stochastic models are fitted to the whole rainfall hyetograph, including zeroes, aggregated to a range of temporal scales. Typically, the range of scales used varies from hourly to daily, although implicit in most studies is the assumption that scales required in simulation should be within the range of scales used in calibration. Hence, if the intention of the model is to simulate 15 min rainfall, the training data should include 15 min observations. As the temporal resolution of rainfall data becomes finer, the distribution of rainfall amounts becomes more positively skewed. Primarily, this is because of the increased proportion of dry periods, but also the higher proportion of low-intensity events characteristic of fine-scale rainfall. Because the calibration method uses central moments to fit model parameters, the greater skewness at finer temporal scales makes it difficult to obtain a good fit to extremes at these scales.

In addition to the dominance of low observations, the estimation of fine-scale extremes may be further undermined by
operation and sampling errors. This is particularly true of tipping bucket gauges where measurement precision at fine
temporal scales is limited to the bucket volume, typically 0.2 or 0.5

Significant effort has been made since the late 1980s to improve the performance of mechanistic rainfall models through
structural developments, with substantial focus on the improved representation of fine-scale extremes (see Sect.

The choice of models is limited to those within the Bartlett–Lewis family of models which conform to the original concept of rectangular pulses developed by Rodriguez-Iturbe et al. (1987). Preference is given to the most parsimonious model variants on the basis that having fewer parameters improves parameter identifiability and reduces uncertainty. The Neyman–Scott family of models is excluded on the understanding that the clustering mechanisms of both model types perform equally well (Wheater et al., 2007a), and there is no evidence that randomization of the Neyman–Scott model (Entekhabi et al., 1989) has any advantage over its Bartlett–Lewis counterpart.

In Sect.

Attempts to improve the estimation of fine-scale extremes for point (single-site) rainfall using mechanistic models have focused on changing the model structure. Several authors have cited significant improvement (Cowpertwait, 1994; Cameron et al., 2000b; Evin and Favre, 2008), although increased parameterization and limited verification with real data have meant that most changes have not been widely adopted. An early criticism of the original mechanistic models presented by Rodriguez-Iturbe et al. (1987) is that the exponential distribution applied to rainfall intensities is light-tailed. This choice is consistent with the observation that rainfall amounts, which in the model are obtained through the superposition of such cells, are approximately gamma distributed (Katz, 1977; Stern and Coe, 1984).

On the basis that the gamma distribution gives more flexibility in generating
rain-cell intensities, Onof and Wheater
(1994b) reformulate the modified (random

In an extension of this approach, Cameron et al. (2000b) replace the exponential distribution in the MBL model with the
generalized Pareto (GP) distribution for rain cells over a high threshold.
Depending on the value of the shape parameter (

The authors present a calibration strategy in which they first fit the MBL model with exponential cell depths to the whole rainfall record using the method of moments from Onof and Wheater (1994b). Generalized likelihood uncertainty estimation (Beven and Binley, 1992) is then used to find behavioural parameterizations of the Pareto scale and shape parameters for rain-cell depths over the threshold – the location parameter being fixed at the threshold value. The central assumption of this model is that the Pareto scale and shape parameters for cell depths over the threshold will have “minimal impact on the standard statistics of the simulated continuous rainfall time-series” (Cameron et al., 2000b, p. 206). The validity of this assumption is disputed by Wheater et al. (2007a), who argue that the MBL model should be fitted to rainfall coincident with rain cells below the threshold, but point out that this is “impossible since cell intensities are not observed” (Wheater et al., 2007a, p. 16).

The model framework of Cameron et al. (2000b) differs from that of the MBL gamma model of Onof and Wheater (1994a) and is essentially the nesting of two models. The authors present significant improvement in the estimation of hourly extremes and show good agreement with generalized extreme value (GEV) estimates. However, because the underlying process of continuous-time rainfall is unobserved, the authors are forced to implement a calibration strategy which limits the impact on standard rainfall statistics – an approach which is undesirable (Wheater et al., 2007a). Furthermore, the framework appears to be an analogue of the N-cell rectangular pulse model structure initially developed by Cowpertwait (1994) for the Neyman–Scott model, and later incorporated into the Bartlett–Lewis models by Wheater et al. (2007a). Regardless of their relative performance, the large number of parameters required for these models is undesirable on the basis that more parameters reduce parameter identifiability and increase parameter uncertainty.

In an earlier study, Cowpertwait (1994) differentiates between light and heavy rain cells in a modified version of the
original (fixed

In a later study, Cowpertwait (1998) hypothesized that including higher-order
statistics in the fitting routine for
mechanistic rainfall models would give a better fit to the tail of the empirical distribution for rainfall
amounts. Focussing on the original (fixed

A criticism of the rectangular pulse model structure by Evin and Favre (2008) is that it assumes independence between rain-cell intensity and duration. Following previous attempts to link the two variables (Kakou, 1997; De Michele and Salvadori, 2003; Kim and Kavvas, 2006), Evin and Favre (2008) present a new NSRP model in which the dependence between rain-cell depth and duration is explicitly modelled using a selection of copulas. While the authors are not primarily motivated to improve the estimation of rainfall extremes, good estimation of fine-scale extremes is achieved. However, the manner in which the results are presented makes interpretation and comparison with other studies difficult. In the first instance, the extreme performance of all models is almost entirely indistinguishable, indicating that no overall improvement is achieved. Secondly, monthly annual extremes are presented at hourly and daily scales but without clearly stating which month in the year. Despite this, it is likely that monthly extremes will have lower variability than those taken from the whole year, and hence model performance is likely to be better. On the basis of the results presented, it is not clear that explicitly modelling dependence between rain-cell depth and duration with copulas offers any discernable benefit over the original model structure.

Theoretically, copulas offer an attractive framework for modelling the dependence structure between rainfall intensity and duration. However, the obvious mechanism for building copula dependence into mechanistic rainfall models is at the rain-cell level as per Evin and Favre (2008). This approach draws upon the intuition that, just as for the rainfall amounts of storm events, rain-cell amounts may be correlated with their duration. Such intuition follows earlier studies into the dependence structure between rainfall intensity and duration (Bacchi et al., 1994; Kurothe et al., 1997) – although as stated by Vandenberghe et al. (2011, p. 14), “it is not very clear in which way this modelled dependence at cell level alters the dependence between the duration and mean intensity of the total storm”.

In recent years, renewed focus on estimating rainfall extremes at hourly and sub-hourly scales has led to the development
of a new type of mechanistic rainfall model based on instantaneous pulses (Cowpertwait et al., 2007; Kaczmarska, 2011). In
this model structure, rectangular pulses are replaced with a point process of instantaneous pulses with depth

Despite the model improvements outlined in Sect.

Example censoring applied to 15 min rainfall data at Atherstone in 2005. Arbitrary censors of 0.25 and 0.5

Figure 1 shows two arbitrary censors applied to 15 min data at Atherstone in 2005 (refer to Sect.

Censored rainfall synthesis is a method for estimating sub-hourly to hourly extremes. Because observations below the
censor are omitted from model fitting, censored model parameters are
scale-dependent and can only be used to simulate
storm profiles above the censor at the same scale as the training data. It is the ability to simulate the heavy portion of
storm profiles which enables extreme rainfall estimation. The basic procedure is as
follows.

For the chosen temporal resolution, select a suitable censor (

Fit the mechanistic rainfall model to the censored rainfall by aggregating the censored time series to a range of temporal scales and calculating summary statistics as necessary for model fitting.

Simulate synthetic rainfall time series at the same resolution as the training data in Step 1 and sample annual maxima.

Restore the censor to the simulated annual maxima and plot against the observed maxima.

Mechanistic point process rainfall models, first developed by
Rodriguez-Iturbe et al. (1987), exist in various forms, although all models
are formulated around two key assumptions about the rainfall-generating
process. Firstly, rainfall is
assumed to arrive in rain cells following a clustering mechanism within storms. Secondly, the total rainfall within cells
is represented by a pre-specified rainfall pattern which describes the
rain-cell duration and amount. The continuous-time
rainfall is the summation of all rainfall amounts in time

Rainfall generation mechanism for mechanistic stochastic models with
rectangular pulses. Panel

Model parameters for the original and two randomized Bartlett–Lewis rectangular pulse models (BLRP, BLRPR

Expected square of the cell depth

Expected cube of the cell depth for inclusion of skewness in the objective function

Gamma scale parameter for

Gamma shape parameter for

In the original form of the model, storms arrive according to a Poisson process with rate

The original BLRP and NSRP models with exponential cell depth distributions are the most parsimonious, each having only five parameters (see Table 1). A limitation of these models is that their simplicity implies all rainfall – stratiform, convective, and orographic – has the same statistical properties. On the assumption that rainfall may derive from different storm types, in particular convective and stratiform, it is physically more appealing to allow the statistical composition of rainfall models to vary between storms.

Two different approaches have been developed to accommodate the simulation of different rainfall types with rectangular
pulses. For the Neyman–Scott model, concurrent and superposed processes have
been developed in generalized (Cowpertwait,
1994) and mixed (Cowpertwait, 2004) rectangular pulse models respectively. Both models enable explicit simulation of
multiple storm types, although their increased parameterization and consequent impact on parameter identifiability means
that it is undesirable to simulate more than two storm types. For the
Bartlett–Lewis model, randomization of the rain-cell duration
parameter

The modified (random

Because the Neyman–Scott and Bartlett–Lewis clustering mechanisms are
considered to perform equally well, model selection
is limited to the most parsimonious model structures within the Bartlett–Lewis family of models: the original model
(BLRP), the linear random parameter model (BLRPR

Estimation of fine-scale extremes with censored rainfall simulation is performed on two gauges: Atherstone in the UK and
Bochum in Germany. Atherstone is a tipping bucket rain gauge (TBR) operated
and maintained by the Environment Agency of
England. The record duration is 48

Plan showing the location of the UK and German rain gauges used in this study.

Bochum is a Hellmann rain gauge operated and maintained by the German
Meteorological Service. It uses a floating pen
mechanism to record rainfall on a drum or band recorder with a minimum gauge resolution of 0.01

Model fitting is performed in the R programming environment (R Core Team, 2017) using an updated version of the MOMFIT
software developed by Chandler et al. (2010) for the UK Government Department for the Environment, Food and Rural Affairs
(DEFRA) FD2105 research and development project (Wheater et al., 2007a, b).
In this software, parameter estimation is performed using the generalized
method of moments (GMM) with a weighted least squares objective function:

The GMM is preferred for mechanistic rainfall models because the complex dependency structure and marginal distribution of
aggregated time series make it very difficult to obtain a useful likelihood
function (Rodriguez-Iturbe et al., 1988). In this procedure, the difference
between observed and expected summary statistics of the rainfall time series
at a range of temporal scales is minimized, giving an optimal parameter set

Typically, the vector of observed summary statistics

Model parameters are estimated using two minimization routines. First, Nelder–Mead optimizations are performed on random perturbations around user-supplied parameter values to identify promising regions of the parameter space. Following a series of heuristics to identify the best-performing parameter set, random perturbations around these values are used as new starting points for subsequent Newton-type optimizations. The parameter set with the lowest objective function is the best-performing and selected for that month. Following the approach employed by Kaczmarska (2013) to obtain smoothly changing parameters throughout the year, this two-step optimization is only applied to one month. Subsequent parameter estimation is based on a single Newton-type optimization using the previous month's estimate as the starting point. Testing of this approach has shown that when the parameters are well identified, the same seasonal variation is achieved regardless of the starting month. The sampling distribution of the estimators resulting from the GMM minimization routine is approximately multivariate normal (MVN). The optimal parameter set is estimated by the mean of this distribution, and the covariance matrix is estimated from the Hessian of the least squares objective function S (Wheater et al., 2007b). The MVN distribution of model parameter estimators is used to estimate 95 % confidence intervals for the parameter estimates. On occasions that the model parameters are poorly identified, it may not be possible to calculate the Hessian of the objective function, preventing the estimation of parameter uncertainty.

To test the effectiveness of censored rainfall modelling for the estimation of fine-scale extremes, the approach is
applied to three temporal scales: 5, 15, and 60 min. At each scale, the
three selected Bartlett–Lewis models are fitted
to both datasets with incrementally increasing censors. The minimum increments applied at each resolution are 0.05,
0.10,
and 0.20

For all simulations the fitting window is widened to 3 months; hence, for any given month the models are fitted to data for that month, together with the preceding and following months. This approach is used to increase the data available for fitting the models when censoring on the basis that censoring removes data which would otherwise be used in fitting. Tests have shown that widening the fitting window from 1 to 3 months has the effect of smoothing the seasonal variation in model parameters and improving parameter identifiability. There is also negligible impact on the estimation of summary statistics and extremes under the model parameters.

For the two randomized models, BL1 and BL1M, the gamma shape parameter

Rainfall extremes are estimated from the models by sampling annual maxima directly from simulations. For each model fitted
to uncensored data, 100 realizations of 100

Extreme value estimation at 5 min resolution. Optimal realizations (opt. AM) are shown with solid lines and the means of the MVN realization (mvn. AM) are shown with dashed lines. Simulation bands (SBs) are shown with filled polygons.

Extreme value estimation at 15 min resolution. Optimal realizations (opt. AM) are shown with solid lines and the means of the MVN realization (mvn. AM) are shown with dashed lines. Simulation bands (SBs) are shown with filled polygons.

Extreme value estimation at 60 min resolution. Optimal realizations (opt. AM) are shown with solid lines and the means of the MVN realization (mvn. AM) are shown with dashed lines. Simulation bands (SBs) are shown with filled polygons.

Extreme value estimation for the censored calibrations is shown in Figs. 4–6
for the 5, 15, and 60 min temporal resolutions
respectively. The top three plots in each figure show the results for Bochum, and the bottom three plots the results for
Atherstone, with observed and simulated annual maxima plotted using the Gringorten plotting positions. All plots show the
equivalent extreme value estimates obtained without censoring by simulating
one realization of 10 000

All censored models have significantly improved the estimation of extremes at each site and scale with very good estimation by all three model variants, particularly at the 5 and 15 min scales. At these scales, the estimation of extremes with the four censors presented has approximately converged on the observations. At the 60 min scale there is notable improvement in the estimation of extremes, with some convergence in estimation with increasing censors, although there is continued underestimation of the observed. The 95 % simulation bands for all censored models broadly bracket the observations and are largely unvaried with increasing censors, other than with the BL1M model at the 60 min resolution.

At the 5 min scale, estimation has converged on the observations with censors between 0.5 and 0.65

Comparison of censored BL1M model parameters for Atherstone 60 min data. Optimal parameter estimates (params.) are shown with dot-dashed lines, and parameter uncertainty is represented with 95 % confidence intervals (CIs).

The means of the MVN realizations for the BL1M model at Atherstone with the
0.6 and 0.8

The rainfall extremes presented in Sect.

Figure 8 shows the seasonal variation in mean, coefficient of variation, and
lag-1 autocorrelation for all three models at Atherstone with the selected
censors in Table 2. Comparable performance is achieved with the models for
Bochum and hence
these results are not presented. The plots show the estimated summary statistics calculated using the optimum parameter
estimates, together with 95 % simulation bands obtained by randomly sampling 100 parameter sets from the multivariate
normal distribution of model parameters. Summary statistics are estimated under the model for all 100 parameter sets and
simulation bands generated for the range of estimates between the 2.5th and
97.5th percentiles. Because models are fitted
over 3-monthly moving windows, estimated summary statistics are compared with
summary statistics for censored observations for
the same periods. Fitting statistics for the 6 and 24 h scales are not shown. The limits on the vertical

Censor selection for model validation.

Seasonal variation in mean, coefficient of variation, and lag-1 autocorrelation for selected censors at Atherstone, observed vs. estimated.

All models perform very well with respect to replicating the summary statistics used in fitting, with the 95 % simulation bands comfortably bracketing the observations. The estimated summary statistics are very close to the observed ones, with all models performing equally well. The seasonal variation in mean monthly rainfall varies between scales because there is a higher proportion of low observations at short temporal scales removed by the censors. The greater prominence in seasonal variation shown in plots a and b indicates that the summer months (approx. April–October) are more prone to short intense bursts of rain, and the winter months longer periods of low-rainfall intensity. This is consistent with there being more convective rainfall in the summer, and stratiform rainfall in the winter. The plots in Fig. 8 demonstrate that the models are able to reproduce the censored fitting statistics, confirming the reliability of the process.

Seasonal variation in the proportion of wet periods for selected censors, observed vs. estimated.

A consequence of censoring is that it truncates the thin tail of the rainfall amount distribution, which significantly changes its shape. Because this truncation is not replicated in the analytical equations of the models used in this study, the models are not expected to be able to reproduce skewness well. Therefore this statistic is excluded from validation. Conversely, censoring is not expected to significantly impact the ability of the models to estimate the proportion of wet periods. Despite this, censoring significantly changes this statistic at fine temporal scales. Figure 9 shows the seasonal variation in the proportion of wet periods for all three models at both sites with the selected censors in Table 2.

The ability of the models to reproduce the proportion of wet periods is generally good, although there is a tendency for
all models to overestimate this statistic at both sites. At the 5 min resolution for Bochum, the 95 % simulation
bands comfortably bracket the observations between the months of May and
October, although there is overestimation in the
other months and for all months at the 15 and 60 min scales. At Atherstone, there is good representation of the
proportion of wet periods at the 15 min scale, but overestimation at the 5
and 60 min scales. Generally, there is
very slightly better agreement in the summer months which, as highlighted in Sect.

The censors selected for validation in Table 2 were chosen based on their extreme value performance. For the estimation of extremes at other locations, it would be preferable to have a set of heuristics to guide censor selection. The following discussion of extreme value estimation performed in this study is intended to guide practitioners in the application of censored modelling.

Change in 95 % simulation bands (SBs) and means of the MVN
realizations (mvn. AM) for Bochum and Atherstone 15 min data with
well-identified (

Upper limits on censoring were identified where model parameters were either poorly identified or the means of the MVN
realizations deviated significantly from the observations. The onset of this
effect was observed in Fig. 6 for estimation
of hourly extremes at Atherstone with the BL1M model. Figure 10 shows the change in 95 % simulation bands and the
means
of the MVN realizations obtained from censored models with well-identified
and poorly identified parameters for 15 min data at
Bochum and Atherstone. The comparison is made between extremes for the selected censors given in Table 2 (1.0 and
0.6

Simulation bands on extreme value estimates for Bochum 15 min rainfall
obtained with censors from 1.0 to 1.3

Fitted model parameters for the BL1M model with a 1.5

While it has been possible to fit models to data with these high censors, examination of the parameter estimates and
associated uncertainty reveals that parameter identifiability is reducing. Figure 11 shows the seasonal variation in
estimates for the BL1M model parameters

Variation in extreme value estimation with censors for 15 min data
at Bochum and Atherstone for two annual return periods: 10 and
25

With the upper bound on censoring identified, the obvious question is how to
identify a lower bound. The results presented
in Figs. 4–6 suggest that there is convergence in the estimation of extremes with increasing censors. If so, when is the
onset of convergence? Figure 12 shows the change in extreme value estimation with censor for 15 min rainfall at Bochum
(top plots) and Atherstone (bottom plots) for 10- and 25-

At both locations, divergence in the means and spread of the MVN realizations shown in Fig. 12 is easily
identified with the very large box-plot whiskers at 1.5 and 1.0

At Atherstone, there is clear evidence of convergence in estimation between censors 0.5–0.9

In Sect.

Empirical cumulative distribution function plots for Bochum and Atherstone rainfall aggregated to 5 and 15 min temporal resolutions. The plots are limited to the 99th percentile rainfall and show the rainfall quantiles corresponding to the optimum censors used in the estimation of extremes in Figs. 4–6.

Figure 13 shows the empirical cumulative distribution function (ECDF) plots for the above zero rainfall records at Bochum
and Atherstone aggregated to 5 and 15 min resolutions. All the censors used for the estimation of fine-scale extremes in
Figs. 4–6 are shown, with the top three censors highlighted in magenta. The
censors selected for model validation (Table 2) are highlighted in blue, and
the lower limits on censors identified in Sect.

It can be seen from Fig. 13 that a substantial proportion of the above zero rainfall record is masked from the models with
censoring. At the 5 min scale, the selected censor of 0.5 and 0.6

A striking difference in the ECDF plots for the two locations is the smoothness of the curves. The stepped nature of the
Atherstone plots is very pronounced and reflects the resolution of the gauge:
0.5

Proportion of maximum rainfall and number of independent peaks per year for the selected censors given in Table 2.

While the proportion of rainfall observations removed prior to model fitting is large – over 90 and 80 % for 5 and 15 min rainfall from Bochum and Atherstone respectively – comparison with the maximum rainfall amounts and an assessment of the number of independent peaks over the censor demonstrate that the censors are low. Table 3 shows the proportion of maximum rainfall and the number of independent peaks per year for the selected censors given in Table 2. The number of peaks over the censors are estimated using a temporal separation of 48 h to define independence.

The proportion of the maximum observed rainfall is less than 6 % in all
cases, which is very low considering that the
maximum recorded rainfall across both sites and scales is just 27.9

The estimation of rainfall extremes presented in this study using censored rainfall simulation is highly promising and offers an alternative to frequency techniques. The estimation of extremes at sub-hourly scales has far exceeded expectations, with all three models giving a very high level of accuracy across a range of censors. However, censoring uses rainfall models in a way they were never previously intended. Rainfall models have invariably been used for simulation of long-duration time series across a range of scales for input into hydrological and hydrodynamic models. Censored rainfall synthesis cannot be used in this way because only the heavy portion of the hyetograph is simulated.

The success of this research is to broaden the scope of mechanistic rainfall modelling and ask new questions of it. Mechanistic models and related weather generators are very powerful at simulating key summary statistics for a range of environmental variables. An area where these models have consistently underperformed is the estimation of fine-scale extremes. Efforts to improve extreme value estimation at fine temporal scales have focussed on structural developments. But those developments have always been undertaken in the context of rainfall time-series generation. Continued underestimation at fine temporal scales has given rise to the notion that rectangular pulse models are potentially “unsuitable for fine-scale data” (Kaczmarska et al., 2014, p. 1985).

For effective scenario planning with hydrological models, good reproduction of rainfall time series is necessary, with accurate estimation of key summary statistics. However, for assessment of extremes and estimation of storm profiles, good replication of rainfall central moments is arguably less important. The ability of the censored models to adequately reproduce the central moments used in calibration was checked to ensure that the process by which the extremes are constructed is reliable. Because rainfall over the censor is by definition coincident with rainfall below the censor, the censored models can be used to estimate uncensored extremes by simply restoring the censor to the estimates.

Extreme rainfall estimation with censoring across all models, scales, and
sites is significantly improved on that without censoring as shown in
Figs. 4–6. Up to approximately the 25-

In all three models, there is a slight upward curvature in the Gumbel
plotting of extremes, which is consistent with the
GEV and GP distributions taking a positive shape parameter (

The results presented in this paper show that censored rainfall modelling has worked for single-site data from two very different locations, and recorded using different gauging techniques. Consistency in the relative magnitude of selected censors identified at each location, and the stability of estimation across a range of censors, give confidence in the approach and support the original hypothesis. It is an obvious limitation of censoring that it cannot be used to obtain time series of synthetic rainfall, as is the principal application of mechanistic rainfall models. However, the intention of this research was to investigate whether mechanistic models could be used for estimation of fine-scale extremes as an alternative to frequency techniques. The accuracy of estimates for sub-hourly rainfall extremes using all three model variants is very good, although the BL1M model appears to outperform the other two models at both sites for the 5 and 15 min scales by accurately predicting the highest observed extreme.

Reducing parameterization by fixing the gamma shape parameter

A principal goal of this research was to improve the physical basis of
short-duration extreme rainfall estimation. This
has been achieved by simulating storm profiles mechanistically in a way which mimics the phenomenology of rainfall
generation. This has given rise to extreme rainfall estimation which may be described as a function of a set of model
parameters with physical meaning, e.g. the extreme rainfall quantile

Further research is also required to investigate the potential for incorporating censored modelling into a multi-model approach for synthetic rainfall generation. This may take the form of simulating the rainfall below the censor using a bootstrapping approach similar to that in Costa et al. (2015), or continuous simulation of uncensored rainfall with replacement of storms simulated using the censoring approach.

The Atherstone tipping bucket rain-gauge dataset was obtained directly from the Environment Agency for England, UK. The data are not publicly accessible because they are used by the Environment Agency for operational purposes, but can be obtained for non-commercial purposes on request. The Bochum dataset was obtained directly from Deutsche Montan Technologie and was recorded by the Emschergenossenschaft/Lippeverband in Germany. The data are not publicly accessible because they belong to the Emschergenossenschaft and Lippeverband public German water boards and are used for operational purposes.

Parallel coordinate plots for the two randomized Bartlett–Lewis
rectangular pulse models, BL1 and BL1M. Plots show the range of Jan parameter
values for uncensored models fitted to Bochum 15 min rainfall. The dashed
magenta lines show the parameter sets corresponding to

To demonstrate the insensitivity of

Fitted gamma distributions for the cell duration parameter

The parallel coordinate plots clearly show the insensitivity of

Sensitivity of extreme value estimation to choice of

As

Tables B1–B4 show fitted model parameters for the BL1M model (BLRPR

BL1M model parameters for the Bochum 5 min data.

BL1M model parameters for the Atherstone 5 min data.

BL1M model parameters for the Bochum 15 min data.

BL1M model parameters for the Atherstone 15 min data.

DC designed the experiments, carried them out, and prepared the manuscript. CO, HW, and PB supervised the work and reviewed the manuscript preparation. The contribution by PB was made when he was employed at the EDF Energy R&D UK Centre.

The authors declare that they have no conflict of interest.

David Cross is grateful for the award of an Industrial Case Studentship from the Engineering and Physical Sciences Research Council and EDF Energy. The Environment Agency of England is gratefully acknowledged for providing the UK rainfall data, and Deutsche Montan Technologie and Emschergenossenschaft/Lippeverband in Germany are gratefully acknowledged for providing the Bochum data. We thank the two anonymous referees and the editor for their helpful comments which have improved this paper. Edited by: Demetris Koutsoyiannis Reviewed by: two anonymous referees