HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-22-727-2018Censored rainfall modelling for estimation of fine-scale extremesCrossDaviddavid.cross12@imperial.ac.ukOnofChristianWinterHugohttps://orcid.org/0000-0001-5871-5461BernardaraPietroDepartment of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UKEDF Energy R&D UK Centre, Interchange, 81–85 Station Road, Croydon, CR0 2RD, UKCEREA, EDF R&D – ENPC, 6 quai Watier, 78400 Chatou, FranceDavid Cross (david.cross12@imperial.ac.uk)26January201822172775621July20173December201723November20177September2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://hess.copernicus.org/articles/22/727/2018/hess-22-727-2018.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/22/727/2018/hess-22-727-2018.pdf
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.
Introduction
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 mm. Fine-scale rainfall is highly
intermittent (starting and stopping with high frequency), yet a tipping bucket gauge can only make a recording when the
bucket is full. The limitations of tipping bucket measurements at fine temporal scales have long been understood
(Goldhirsh et al., 1992; Nystuen et al., 1996; Yu et al., 1997), although the first formal estimation of sampling error
was performed by Habib et al. (2001). In this study, the authors investigate the ability of tipping bucket gauges to
capture the temporal variability of fine-scale rainfall at 1, 5, and 15 min
scales using tipping bucket measurements simulated from high-resolution
optical rain-gauge observations. The authors show that for the lowest
rainfall intensities
(< 5 mmh-1) the mean relative error of the tipping bucket gauge at the 5 min resolution is +3.5 %,
with corresponding SD just under +30 % for a bucket volume of 0.254 mm. Larger errors are obtained for the
1 min resolution. They also show that increasing the bucket volume to 0.5 mm significantly increases the spread
of the sampling error for low observations at the 5 min resolution. The observed record comprised mainly convective storm
events which are typical for Iowa in the US where the data were collected, although the error estimates are significant
and present compelling evidence of the impact of sampling error on fine-scale
low-intensity rainfall observations.
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.
for a review). Despite this, little progress has been achieved. To test our hypothesis, a simple approach is proposed in
which low observations for fine-scale data are censored from the models in calibration. For a given temporal resolution,
a censor amount is set. Rainfall below the censor is set to zero and rainfall over the censor is reduced by the censor
amount. This focusses model fitting on the heavier portion of the rainfall record at fine temporal scales, and reduces
rainfall intensity at coarser scales. The aim is to investigate whether
existing mechanistic models can be used as simulators
of fine-scale storm events by changing the data and not the model, thereby reducing the impact of low observations and
sampling error on fine-scale extreme rainfall estimation.
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. , we outline the main mechanistic model developments for improved representation of extremes. The
censored modelling approach for the estimation
of fine-scale extremes is described in Sect. . Model structure and
selection are explained in Sect. , and the data and fitting
methodology are presented in Sect. . Results
are given in Sect. together with validation analysis. Discussion on the results and censor selection are given in
Sect. . Section provides further discussion and outlines our main conclusions and thoughts for future
research.
Mechanistic model developments
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 η) Bartlett–Lewis (MBL) model (Rodriguez-Iturbe et al., 1987) with the
gamma distribution to improve the estimation of extremes. Despite the good fit to hourly extremes cited by Onof and
Wheater (1994b), subsequent studies have continued to show underestimation at hourly and sub-hourly scales (Verhoest
et al., 1997; Cameron et al., 2000a; Kaczmarska et al., 2014).
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 (ξ), the GP converges to
one of three forms: upper-bounded (ξ<0), exponential (ξ=0), and
Pareto
(ξ>0). In the last case, we have a distribution with a heavier tail than the exponential or the gamma. Because the GP
distribution is a model for threshold exceedance, the authors specify a threshold below which the MBL model with
exponential intensity distribution is used to simulate rain-cell depth, and
above which the Pareto distribution is used. This is justified on the
assumption that the rain-cell depth may be of either high or low intensity.
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 η) Neyman–Scott rectangular pulse (NSRP) model
(Rodriguez-Iturbe et al., 1987) by allowing rain-cell intensity and duration
to be drawn from more than one pair of exponential distributions. The new
model, termed the Generalised NSRP model (GNSRP), leads to a significant
increase in parameterization over the original NSRP model, although
the author presents an intelligent way to simplify calibration by relating model parameters to harmonic signals. While
improvement is achieved in the fit to hourly extremes, the performance of the model in replicating other important
statistics is not presented, in particular autocorrelation and the proportion of dry periods. Both of these properties are
addressed by Rodriguez-Iturbe et al. (1987, 1988) for the Bartlett–Lewis model with the inclusion of a “high frequency
jitter” and randomization of the rain-cell duration parameter η.
Entekhabi et al. (1989) present a randomized
version of the Neyman–Scott model with significant improvement in the fit to dry periods. However, because no analytical
expression was available for the proportion of dry periods, this statistic
was not used in model fitting, and other model parameters were not allowed to
vary from storm to storm with randomization. Consequently, while the MBL and
the GNSRP models each allow rain-cell intensity and duration to be drawn from
more than one pair of distributions, the MBL structure
is preferred because it has fewer parameters.
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 η) NSRP model, analytical equations for skewness of the aggregated rainfall
depth are presented and used in fitting the models. Empirical analysis showed that including skewness in the fitting
statistics improved the estimation of Gumbel distribution parameters from simulated maxima when compared with parameters
obtained from observed annual maxima.
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 X and zero
duration, the summation of pulses giving the aggregated time rainfall intensity. Considered initially to offer a more
suitable representation of rainfall at sub-hourly scales than rectangular pulses, Kaczmarska et al. (2014) found that the
best-performing Bartlett–Lewis Instantaneous Pulse (BLIP) model effectively
generated rectangular pulses when depth X was kept constant and cell
duration η was randomized. Because of the very large number of pulses
generated within cells, the authors noted that this model structure imposes
the “most extreme form of dependence” – Kaczmarska
et al. (2014, p. 1977). Consequently, the authors developed a new rectangular pulse model in which both η
and μx are randomized (BLRPRX), which was found to perform
equally as well as the randomized version of the BLIP model but without
additional parameterization.
Censored modelling for fine-scale extremes
Despite the model improvements outlined in Sect. , there is an
ongoing tendency for stochastic mechanistic models
to underestimate extremes at hourly and sub-hourly scales. Consequently, the practitioner is required to employ additional
methods for better extreme value performance, including disaggregation
(Koutsoyiannis and Onof, 2000, 2001; Onof
et al., 2005; Onof and Arnbjerg-Nielsen, 2009; Kossieris et al., 2018) and model fitting with more information about the
variability of precipitation (Kim et al., 2013a). We propose a censored approach to mechanistic rainfall modelling for
improved estimation of fine-scale extremes by
focussing model fitting on the heavy portion of the rainfall time series. The
aim of this research is to investigate whether mechanistic models can be used
as simulators of fine-scale design storm events
to reduce the impact of low observations on the estimation of fine-scale extremes. In this approach, rainfall below a low
censor is set to zero and rainfall over the censor is reduced by the censor amount. The effect is to generate
a time series of heavy rainfall based on the observed record in which the
proportion of dry periods is increased and
rainfall amounts are reduced.
Example censoring applied to 15 min rainfall data at Atherstone in 2005. Arbitrary censors of 0.25 and 0.5 mm are applied to demonstrate the effect of censoring on the rainfall record.
Figure 1 shows two arbitrary censors applied to 15 min data at Atherstone in 2005 (refer to Sect. for
a description of the data). The left plot shows the uncensored rainfall, and the two right plots the change in rainfall
with increasing censors. The reduced rainfall amounts are shown on the secondary y axes. It can be seen from these plots
that the minimum recorded rainfall is 0.2 mm, which corresponds to
the tip volume of the tipping bucket rain
gauge. Compared with higher rainfall amounts this volume is recorded with very high frequency throughout the year at the
15 min resolution.
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 (mm) and apply it to the observed rainfall time series
by setting rainfall amounts below the censor to zero, and reducing rainfall amounts over the censor by the censor amount.
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.
Model structure and selection
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 Δt. Most models assume rectangular pulses to describe
rainfall amount and duration, although alternative patterns have included a Gaussian distribution (Northrop and Stone,
2005) and instantaneous pulses (Cowpertwait et al., 2007, 2011; Kaczmarska et al., 2014). In this latter formulation,
pulses are assumed to arrive according to a Poisson process within cells, with each pulse representing an amount with zero
duration. The continuous-time rainfall is therefore the summation of all pulse amounts in time Δt.
Rainfall generation mechanism for mechanistic stochastic models with
rectangular pulses. Panel (a) shows the arrival of storms and cells.
Rain-cell intensity defines the height of each cell (X), and duration the
length (L). Panel (b) shows the unobserved continuous-time
rainfall time series derived from the superposition of cells shown
in (a).
Model parameters for the original and two randomized Bartlett–Lewis rectangular pulse models (BLRP, BLRPRη, and BLRPRX) and the original Neyman–Scott rectangular pulse model (NSRP).
UnitsBLRPNSRPBLRPRηBLRPRXStorm arrival rateh-1λλλλCell arrival rateh-1ββ{β} 1{β}Ratio of cell arrival rate to cell duration–––κ=β/ηκ=β/ηMean cell depthmmh-1μxμxμx{μx}Ratio of mean cell depth to cell durationmm–––ι=μx/ηRatio of SD to the mean cell depth–r=σx/μxr=σx/μxr=σx/μxr=σx/μx
Expected square of the cell depth 2
mm2h-2{μx2}{μx2}{μx2}{μx2}
Expected cube of the cell depth for inclusion of skewness in the objective function 2
hααStorm duration parameterh-1γ–{γ}{γ}Ratio of storm duration to cell duration–φ=γ/ηφ=γ/ηMean number of cells per storm––μc––Number of parameters: exponential cell intensity–5566Number of parameters: gamma cell intensity–6677
1 Parameters in curly brackets {} are not included in the objective function (see Sect. ).2 For the two-parameter gamma cell depth distribution, the expected
square and cube of the cell depth (μx2 and μx3) are
calculated from the SD (σx) and mean (μx) of the cell depth.
In practice it is the ratio of these (r) which is parameterized, enabling
calculation of μx2 and μx3. For both the exponential and
gamma distributions, μx2=f1μX2 and
μx3=f2μX3 where f1=1+r2 and
f2=1+32+2r4. Because the exponential distribution is a special
case of the gamma distribution where r is equal to 1,
μx2=2μx2 and μx3=6μx3. Therefore it is not
necessary to parameterize r for the exponential distribution, meaning the
exponential versions of these models require one parameter less with r set
to 1 in calibration.
In the original form of the model, storms arrive according to a Poisson process with rate λ, and terminate after
an exponentially distributed period with rate γ. The arrival of rain cells within storms follows a clustering
mechanism which defines a secondary Poisson process with rate β. Two clustering mechanisms are specified by
Rodriguez-Iturbe et al. (1987): the first is the Neyman–Scott mechanism in which the time intervals between storm and
cell origins are assumed to be independent and identically distributed random variables; the second is the Bartlett–Lewis
mechanism in which the time intervals between successive cell origins are independent and identically distributed random
variables. In each case, the time intervals are assumed to be exponentially
distributed. Rain-cell profiles are
rectangular with heights X for amounts, and lengths L for durations. Both X and L are assumed to be independent of
each other and follow exponential distributions with parameters 1/μx and η respectively. Figure 2 shows
a graphical illustration of the continuous-time rainfall generation process.
Table 1 sets out the model parameters for the original and two randomized
Bartlett–Lewis rectangular pulse models (BLRP, BLRPRη, and BLRPRX),
as well as the original Neyman–Scott rectangular pulse model (NSRP).
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 η (Rodriguez-Iturbe et al., 1988; Onof and Wheater, 1993,
1994b) with a gamma distribution allows all storms to be drawn from different
distributions. Because rain-cell durations are assumed to be exponentially
distributed, rain cells with high values of η are more likely to be shorter in duration, and those with low values
of η will typically have longer durations. Additionally, the rate at
which rain cells arrive, and the storm durations,
are defined in proportion to η by keeping the ratios β/η and γ/η constant (equal to κ
and φ respectively). This means that, typically, shorter storms will
comprise shorter rain cells with shorter rates
of arrival and the opposite for longer storms, which is characteristic of the differences between convective and
stratiform rainfall.
The modified (random η) Bartlett–Lewis model (see BLRPRη in Table 1) of Onof and Wheater (1993, 1994b) is
the most parsimonious of the model structures able to accommodate multiple storm types comprising a minimum of
six parameters for the exponential version. The modified (random η)
Neyman–Scott model has the same number of
parameters as the modified Bartlett–Lewis model, but because there is no evidence that it has any advantage over the
latter, it is excluded from this study. The updated random η
Bartlett–Lewis model with mean cell depth μx also randomized (see
BLRPRX in Table 1) requires fewer parameters than its instantaneous
pulse counterpart and the same
number of parameters as the modified BLRPRη model. Structurally, it is identical to the modified model,
although μx is also allowed to vary randomly between storms by keeping the ratio ι=μx/η constant.
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 (BLRPRη), and the linear
random parameter model with randomized μx (BLRPRX). Hereafter,
these models are referred to as BL0, BL1, and BL1M respectively. For the
models used in this
study, it is assumed that rain cells start at the storm origin to prevent the simulation of empty storms which can occur
with the Bartlett–Lewis clustering mechanism if the first rain cell starts after the end of the storm.
Data and model fittingData selection
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 years from 1967 to 2015, with one notable period of missing data from
January 1974 to March 1975. The reason for the missing data is unknown, although it is not expected to affect model
fitting and the estimation of extremes. This site was selected from all TBRs in the Environment Agency's Midlands Region
on the basis that the number of Environment Agency quality flags highlighted as “good” in the record is greater than
90 %, and the number of “suspect” flags less than 10 % (92.3 and
6.7 % respectively). Between 8 February 1981 and 20 November 2003 the
gauge resolution is 0.5 mm. Before and after this period it is
0.2 mm. In the period before 8 February 1981, the TBR record includes
a number of observations of 0.1 mm at precisely 09:00:00. It is
assumed that these are manual observations to correct the rain-gauge totals
to
match with check gauge totals following quality checks of the data.
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 mm. The duration
is 69 years from 1931 to 1999, and the data are aggregated to a minimum temporal resolution of 5 min. These sites
are selected to represent rainfall in different geographical regions obtained using different measurement
techniques. Figure 3 shows the locations of these two gauges.
Parameter estimation
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:
S(θ|t)=∑i=1kωi[ti-τi(θ)]2. The reader is referred to Wheater et al. (2007b;
Appendix A) for a detailed explanation of the fitting methodology.
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
θ where t=(t1…tk)′ is a vector of k
observed summary statistics, and
τ(θ)=(τ1(θ)…τk(θ))′ is
a vector of k expected summary statistics which are functions of θ=(θ1…θp)′,
i.e. of the vector of p model parameters for which analytical expressions are available. The ith summary statistic is
weighted according to the inverse of its observed variance ωi=1/var(ti) where var(ti) is the
ith diagonal element of the estimated covariance matrix of the observed summary
statistics, Σ^. While this weighting is not optimal, it provides a reasonable approximation to the
optimal weights for the GMM giving robust estimation of the parameter standard errors (Chandler et al., 2010). Other
weights can be applied allowing the user to influence the dominance of specific rainfall properties, although for unbiased
estimates of the summary statistics the weights must be independent of the model parameters and the data (Wheater
et al., 2007b).
Typically, the vector of observed summary statistics t comprises the mean,
variance, auto-correlation, and proportion of
dry periods for temporal scales between 1 and 24 h. Prior to model fitting and to allow for seasonality, summary
statistics are calculated for each month over the record length and pooled between months. For each month, the pooled
statistics are used to estimate the covariance matrix of model parameters required for parameter uncertainty estimation,
and the mean of the monthly statistics. Therefore 12 parameter sets are obtained for the whole year.
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.
Experimental design
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 mm respectively. Initial experiments with the coefficient of skewness and proportion of dry periods
included in model fitting for censored data were limited by the inability to
obtain well-identified parameters for some or
all months. While good model fits were obtained for some low censors, extreme value estimation continued to be
underestimated. On the basis that censoring is a new approach to enhance the estimation of rainfall extremes, skewness is
not considered to be an important fitting statistic for censored simulations. Furthermore, because censored models cannot
be used to generate continuous time series of the sort which may be used for
hydrological modelling, the proportion of dry
periods is also considered to be unimportant for censoring. Consequently for censored model calibration, the choice of
fitting statistics is reduced to the 1 h mean, the coefficient of variation
and lag-1 autocorrelation of the rainfall depths at the censor resolution,
and the 6 and 24 h resolutions. Again, to ensure well-identified model
parameters for the Atherstone dataset, it was necessary to extend the choice
of fitting statistics to include the 1 h statistics for 5 min simulations.
This was neither necessary for 15 and 60 min simulations at Atherstone nor
the Bochum dataset.
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
α is constrained to a fixed value in
calibration and simulation. The gamma shape parameter α is an insensitive model parameter and can take any value
within a very large range without significant impact on the estimation of summary statistics or extremes. For the BL1
model, parameterization without an upper bound on α often results in
poor identifiability with parameter estimates in the thousands to tens of
thousands. For the BL1M model, α is typically better identified than
for BL1, with
a tendency to move towards the lower boundary. In order to avoid having infinite skewness, α must be greater than 4
for the BL1 model and 1 for the BL1M model (see Kaczmarska et al., 2014, and
references therein for a discussion of these
criteria). Therefore, by fixing α at 100 for the BL1 model and 5 for the BL1M model, the number of parameters to be
identified for these models is reduced by one. All models are fitted using the exponential distribution for mean cell
depth. This further reduces the number of model parameters to be fitted for
both uncensored and censored models;
therefore,
in all cases the ratio of SD to the mean cell depth (r=σx/μx) is fixed at 1. Sensitivity of the gamma shape
parameter (α) and its impact on extreme estimation is investigated in Appendix A. Fitted model parameters for the
BL1M model are presented in Appendix B for 5 and 15 min rainfall at both sites for uncensored and censored rainfall using
censors selected in Sect. (Table 2).
ResultsExtreme value estimation
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 years in duration are
simulated using parameters randomly sampled from the
multivariate normal (MVN) distribution of model parameter estimators. This allows model parameter uncertainty to be
represented in the spread of the MVN extreme value estimates (hereafter
referred to as MVN realizations), covering the full range of observations.
Extreme value estimation up to the 1000-year return level is also
provided to indicate the potential magnitude of rarer events. For this
extrapolation, extremes are estimated from one realization using the
mean of the MVN distribution of parameter estimators (hereafter referred to as the optimal estimates). To ensure stability
of the extreme value estimates up to approximately the 1000-year
return level, simulations have been extended to
10 000 years.
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 years in duration with the optimal
parameter set. Upper limits on censoring were identified when model
parameterization noticeably deteriorated, resulting in the means of the MVN
realizations deviating away from the optimal. Results presented are limited
to the four highest censors with well-identified model parameters, together
with 95 % simulation bands. The simulation bands show the range of
extreme value estimation between the 2.5th and 97.5th percentiles of the
100 MVN realizations for each
simulated data point.
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 mm at Bochum,
and between 0.6 and 0.75 mm at Atherstone. For all three models there is slight underestimation of extremes higher
than approximately the 10-year return period, although the BL1M model
accurately estimates the highest observed
extreme at both sites. At the 15 min scale, convergence at Bochum has occurred for censors between 1.0 and
1.3 mm, while at Atherstone convergence has occurred for censors between 0.6 and 0.9 mm. As for the 5 min
resolution models, the BLIM model appears to perform slightly better than the BL0 and BL1 models, resulting in improved
estimation of the highest observed extremes and elevated estimates of the
1000-year return period rainfall at both
sites. At the 60 min resolution, there is good convergence in estimation for all three models at Bochum, and the BL1M
model at Atherstone. However, extreme value estimation with the BL0 and BL1 models at Atherstone is more widely spread
across the applied censors. For the BL0 and BL1 models, the 0.2 mm censor results in much lower estimates than the
three higher censors, although the means of the MVN realizations for the 0.6
and 0.8 mm censors are starting to deviate away from the optimum
realization. For the BL1M model, there is good convergence between the
optimal realizations with each censor, although the means of the MVN
estimates for the 0.6 and 0.8 mm censors have significantly
deviated from the optimum.
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 mm censors (see Fig. 6)
diverge from the optimum because of the generation of unrealistic extremes. This divergence is also observable in the
larger spread of 95 % simulation bands over 100 realizations. While it
has been possible to fit the model, Fig. 7
shows that as censoring has increased to 0.8 mm, confidence intervals on model parameters have widened for several
months of the year, notably January, February, and June. When sampling from
the MVN distribution in simulation, these large
confidence intervals mean that there is a high chance of sampling parameters which deviate significantly from the mean of
the distribution, thereby giving rise to a wide spread in extreme value
estimates. These large confidence intervals indicate that the confidence in
parameter estimation decreases with higher censors, and consequently the
model error is
too large for the reliable simulation of extremes.
Validation
The rainfall extremes presented in Sect. have been generated mechanistically using model parameters derived
from central moments of the censored rainfall time series. While censored
models cannot be used to simulate the whole
rainfall hyetograph, it is important to ensure that the process by which the extremes are estimated is
reliable. Therefore, model performance is validated in the usual way for this class of model by comparing the analytical
summary statistics under the model parameters with the observations – here the observations are censored. The lowest
censors presented in Figs. 4–6 are selected for validation. No distinction is made between models in this choice,
although it is recognized that there is some variation in the extreme value
performance of specific censors between model
types. See Table 2 for censor selection at each site and scale.
Replication of fitting statistics
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 Y axes are
optimized at each site and scale; therefore, the reader is advised to pay
careful attention to the scales when comparing
summary statistics.
Censor selection for model validation.
5 min15 min60 minBochum0.5 mm1.0 mm1.0 mmAtherstone0.6 mm0.6 mm0.2 mm
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.
Replication of statistics not used in fitting
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. , may be more prone to
short intense downpours at fine temporal scales. This suggests that the censored models may be more effective at
simulating the heavier short-duration rainfall characteristic of summer
convective storms than the longer-duration low-intensity rainfall
characteristic of winter storms.
Discussion on censor selection
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 (> 1.0 and > 0.6 mm) and poorly identified
(> 1.5 and > 1.0 mm) censored model parameters. The spread of
the individual realizations is also shown with solid lines (mvn. RL).
Stability of confidence intervals
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 mm respectively) and extremes from higher censors (1.5 and 1.0 mm respectively).
Simulation bands on extreme value estimates for Bochum 15 min rainfall
obtained with censors from 1.0 to 1.3 mm, and for Atherstone with
censors from 0.6 to 0.9 mm (Fig. 5), are broadly stable and
unchanging. This is indicative that parameterization across each model
variant and censor is good, enabling robust estimation of extremes. As the
censor
at Bochum is increased to 1.5 mm (Fig. 10a–c), there is a noticeable increase in the upper limit on the
simulation bands and the means of the MVN realizations have diverged, leading
to overestimation of the extremes. Increasing
the censor at Atherstone to 1.0 mm has resulted in very significant widening of the simulation bands and
divergence of the means of the MVN realizations (Fig. 10d–f). In each case,
this divergence results from the generation of unrealistic extreme value
realizations which are shown in Fig. 10 (light grey lines).
Fitted model parameters for the BL1M model with a 1.5 mm
censor applied to Bochum 15 min data.
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 α/ν, κ and φ fitted to Bochum 15 min data with
a 1.5 mm censor. Parameters λ and ι are well identified with tight confidence brackets around the
optimum, while r and α are fixed; therefore, these parameters are not
shown. Confidence intervals on α/ν, κ and φ are very
large in the winter months, indicating that the identifiability of these
parameters has
deteriorated. When sampling from the MVN distribution for model parameter estimators in simulation, these
large uncertainties give rise to poor extreme value estimation. The same behaviour was observed for the BL1M model at
Atherstone for 60 min data, as shown in Fig. 7.
Variation in extreme value estimation with censors for 15 min data
at Bochum and Atherstone for two annual return periods: 10 and
25 years. Plots show the optimal realizations (opt. AM), the means of
the MVN realization (mvn. AM), and the spread of the individual realizations
(mvn. RL) with box plots.
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-year return
periods.
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 mm censors for Bochum and Atherstone
respectively. The plots for Bochum also show a large spread in the extreme
realizations, with a 1.4 mm censor for the BL1M model, suggesting
that parameter identifiability is deteriorating at this censor.
At Atherstone, there is clear evidence of convergence in estimation between censors 0.5–0.9 mm. However,
convergence is less obvious at Bochum. At Bochum, there is continual improvement in extreme value estimation with the
increasing censors, although there is a perceptible reduction in improvement with each successive increase in censor. For
censors of 0.7 mm and above, all model realizations bracket the
observed extremes (horizontal dashed blue line),
which is also true for censors above 0.5 mm at Atherstone. Therefore, ranges may be identified at both sites for
censors, which may be considered to give satisfactory estimation of extremes:
0.7–1.3 mm at Bochum and
0.5–0.9 mm at Atherstone.
How much rainfall to censor?
In Sect. we identify plausible censor ranges based on parameter stability and convergence of extreme value
estimation. However, this does not address the question of how much rainfall
to censor. Because extremes are generated mechanistically, we want to
simulate the storm event hyetograph; therefore, it is in our interest to keep
the censor low in
relation to the rainfall depth profile. The most basic check is that the minimum observed extreme (here designated as the
smallest annual maxima) is greater than the censor being used. This is true for all the sites and scales investigated in
this study, with the lowest observed annual maxima of 1.6 mm occurring at the 5 min scale in Atherstone. However,
this significantly exceeds the maximum censor applied to 5 min data at Atherstone, 0.75 mm (see
Fig. 4); therefore, it is unlikely that a well-parameterized model would be
achieved.
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. for 15 min
rainfall are shown and highlighted in green. The ECDF plots are truncated at
the 99th percentile to aid comparison of the applied censors; therefore, the
maximum rainfall is highlighted in red text on the right of each plot. For
all censors, their rainfall percentile values
are shown with the colour matching the plotted lines.
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 mm removes in excess of 98 and 96 % of the
above zero rainfall from Bochum and Atherstone respectively. At the 15 min scale, the selected censors of 1.0 and
0.6 mm remove in excess of 96 and 81 % respectively. These percentiles are high and support the hypothesis that
mechanistic models may be poor at estimating fine-scale extremes because the training data are dominated by low
observations.
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 mm from February 1981 to November 2003, and
0.2 mm before and after these dates. The stepped nature of the plots at Atherstone highlights that the selected
censor quantiles (blue) are just greater than the 0.5 mm quantiles. We also know from Fig. 12 that a censor of
0.5 mm for 15 min rainfall at Atherstone would give very similar extreme value estimation to the selected
0.6 mm censor (highlighted in green in the ECDF plot, Fig. 13). This
implies that to improve the estimation of fine-scale extremes at Atherstone,
it has been necessary to remove all observations which correspond to the
gauge
resolution.
Proportion of maximum rainfall and number of independent peaks per year for the selected censors given in Table 2.
Scale (min)BochumAtherstoneProportion of maximum rainfall53.0 %5.7 %153.6 %3.5 %Number of independent peaks/year55327154665
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 mm for Bochum 15 min rainfall. For
a standard peaks-over-threshold extreme value analysis, the threshold is
typically set so that between three and five independent
peaks per year remain in the partial duration series. Using a temporal separation of 48 h to define independence, the
number of peaks per year retained after censoring is between 27 and 65 (Table 3). The actual number of peaks retained for
fitting the Bartlett–Lewis models is greater than this because serial
dependence in the rainfall time series is
simulated with mechanistic modelling. While it is possible to estimate return levels for serially dependent extremes using
extreme value theory, the analysis set out in Fawcett and Walshaw (2012) demonstrates that estimating the extremal index is non-trivial and can be subjective.
Further discussion and conclusions
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-year return period, estimation
is broadly equivalent
across all models. For rarer events, the BL1M model appears to perform better than the other two at the 5 and 15 min
scales at Bochum and Atherstone by accurately estimating the highest
observations at those scales. This improvement over
the BL0 and BL1 models is significant in the event that extreme rainfall estimation is required beyond the range of
observations. This is demonstrated in all four cases (5 and 15 min scales at
Bochum and Atherstone) with the higher estimation of extremes at the
1000-year return level by the BL1M model compared with the other two.
Below approximately the 25-year return period the differences in
extreme rainfall estimation are so small that it is not
possible to single out any one model as having the best overall performance, although for increasingly rare events the
results suggest a preference for the BL1M model. This result supports the findings reported by Kaczmarska et al. (2014)
that the dependence structure between rain-cell amounts and duration in the
BL1M model is beneficial in estimating
fine-scale extremes.
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 (ξ>0). This curvature is more pronounced for the BL1M
model,
which would be consistent with a higher positive shape parameter. While extreme value theory encompasses a range of
distributions characterized by the sign of the shape parameter, Koutsoyiannis
(2004a) argues that rainfall extremes naturally follow the Fréchet
distribution for annual maxima (equivalent to the GEV with ξ>0),
supported by
empirical evidence in Koutsoyiannis (2004b). The positive growth in extremes observed in our results is consistent with
this hypothesis, and suggests that important information about the distribution of extremes is captured in the full storm
profile hyetograph over the low censor. Further research is required to
investigate the theoretical link between
mechanistic model parameters and their extreme value performance.
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 α in the
randomized models, and increasing the data for parameterization by widening
the fitting window to 3 months has enabled the models to be fitted
successfully to censored observations. It is likely that these aides to
parameterization are necessary because censoring truncates the
statistical distribution of the training data. The analytical solutions in the models do not make this
assumption; therefore, a mismatch between the training data and the models
arises with censoring. At low censors, truncation is minor
and the analytical solutions in the models are able to make reasonable estimates of the fitting statistics. However, as
the censor increases and the mismatch grows, a point is reached at which the
analytical solutions are no longer able to estimate the fitting statistics,
causing deterioration in parameter identifiability.
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
z=F{λ,μx,δx,δc,μc,δs} for the original Bartlett–Lewis model (see Appendix B
for definitions of mechanistic model parameters). Future research is required to link model parameters to environmental
covariates and spatial information. The latter may follow the regionalization
methodology of Kim et al. (2013b).
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.
Bartlett–Lewis model parameter sensitivity and impact on extreme value estimation
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 α=100 and
α=5 for the BL1 and BL1M models respectively.
To demonstrate the insensitivity of α for the randomized
Bartlett–Lewis models, the BL1 and BL1M models were
fitted to Bochum 15 min rainfall with changing constraints on α. The models were fitted using the 1 h mean and
the 0.25, 6, and 24 h coefficient of variation, skewness coefficient, and
lag-1 autocorrelation. For the BL1 model, α is constrained between 4.1
(lower bound) and 5, 10, 25, 50, 75, and 100 (upper bounds). For the BL1M
model, α is constrained between 5, 10, 25, 50, 75, and 100 (lower
bounds) and infinity (upper bound). For the BL1
and BL1M models, α converges on the upper and lower bounds respectively, although because α is not held
fixed, parameter uncertainty is estimated. Parameter ranges are presented in
the parallel coordinate plots in Fig. A1 by
sampling 1000 parameter sets from the multivariate normal distribution of model parameter estimators for
4.1 <α< 106. The parameter sets corresponding to α=100 and α=5 are shown for the
BL1 and BL1M models respectively with dashed magenta lines.
Fitted gamma distributions for the cell duration parameter η
for the BL1 and BL1M models with α=5, 50, 100, and 1000. Plots show
the equivalent normal distributions fitted to the mean and SD of the gamma
distributions. The range of exponential distributions for the cell duration
parameter η is obtained by sampling 500 η values from the fitted
gamma distributions. The exponential distributions for the mean of the fitted
gamma distributions are also shown.
The parallel coordinate plots clearly show the insensitivity of α compared with the other model
parameters. When α is constrained with upper and lower bounds of
between 25 and 100 for the BL1 and BL1M models
respectively, α is poorly identified and can take any value over a very large range (see Fig. A1). When α is
constrained with upper and lower bounds of less than 25 for the BL1 and BL1M
models respectively, the identifiability
of α is improved. This insensitivity results from the shape of the fitted gamma distribution used to sample η
shown in Fig. A2.
Sensitivity of extreme value estimation to choice of α for
the randomized Bartlett–Lewis models.
As α increases the gamma distribution converges on the
normal
distribution and becomes increasingly
flat. Therefore, for high values of α, the probability of randomly sampling anywhere within the distribution is
greater compared with low values of α. For α≥50, the gamma distribution is approximately normal and the
range of η values which may be randomly sampled by both models is always
large, resulting in a narrow range of potential exponential distributions from which to sample L
where L is the cell duration. This impacts the estimation
of extremes as shown in Fig. A3. Figure A3 shows extreme rainfall estimates from the BL1 and BL1M models with α
fixed at 5, 50, and 100. For α≥50, extreme rainfall estimation by
both models is identical. For α=5,
the BL1 model estimates lower extremes than with higher α values, while the BL1M model gives improved estimation of
the growth curve of extremes. Because of this combination of parameter insensitivity and relative performance in the
extremes, α is fixed at 100 and 5 for the BL1 and BL1M models respectively.
Fitted model parameters
Tables B1–B4 show fitted model parameters for the BL1M model (BLRPRX in Table 1) for 5 and 15 min rainfall at Bochum
and Atherstone with uncensored and censored data. Censored model parameters correspond to the censors selected in
Table 2. Additionally, Tables B1–B4 show the objective function value, Smin, for the fitted parameter set, as
well as mechanistic model parameters defined by Wheater et al. (2007b) which are listed
below.
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.
Acknowledgements
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
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