Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue2.2000:

Advances in Water Resources 24 (2001) 203±211
www.elsevier.com/locate/advwatres

Modelling extreme rainfalls using a modi®ed random pulse
Bartlett±Lewis stochastic rainfall model (with uncertainty)
David Cameron a,*, Keith Beven b, Jonathan Tawn c
a

Water Resources, The Environment Agency, Tyneside House, Skinnerburn Road, Newcastle Business Park, Newcastle Upon Tyne NE4 7AR, UK
b
Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LA1 4YQ, UK
c
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

Abstract
A modi®ed random pulse Bartlett±Lewis stochastic rainfall model is used for extreme rainfall simulation. The model features the
use of a generalised pareto distribution (GPD) to represent the depths of high intensity raincells. A point rainfall record, obtained
from a UK site is used to test the model. Parameter estimation is carried out using a two-stage approach based on the generalised
likelihood uncertainty estimation (GLUE) methodology. This procedure acknowledges the limited sample of the extreme rainfall
data in the observed record in conditioning individual realisations of the random storm model. The extreme rainfall simulations
produced using the model are shown to compare favourably with the site's observed series seasonal maxima. A comparison of the

modelled extreme rainfall amounts, with those obtained from a direct statistical analysis of the data, is also conducted. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Rainfall; Extreme; Stochastic rainfall modelling; Uncertainty

1. Introduction
The simulation of continuous rainfall time-series is
currently an important area of hydrological research,
particularly within the context of ¯ood estimation. This
includes the topics of design storm evaluation and ¯ood
frequency estimation by continuous rainfall-runo€
modelling [3,4,7±10,12,23]. In both cases, the quality of
the resulting estimates is dependent upon the accurate
simulation of the extreme rainfall characteristics of the
site of interest. In addition, in the latter case, the representation of storm inter-event arrival times is an important control upon the antecedent soil moisture
conditions simulated by the rainfall-runo€ model. The
task of continuous rainfall simulation has often been
approached through the use of a stochastic rainfall
model (a model which operates through the generation
of random rainstorms). One type of stochastic rainfall
model which is currently popular is the pulse-based

model (e.g., the Neyman±Scott model, [14±17,

*

Corresponding author. Fax: +44-191-203-4004.
E-mail addresses: david.cameron@environment-agency.gov.uk (D.
Cameron), k.beven@lancaster.ac.uk (K. Beven), j.tawn@lancaster.ac.uk (J. Tawn).

22,23,25,30,35]; and the Bartlett±Lewis model, e.g.,
[24,26±28,30±32,35,36]).
This type of model typically utilises independent, or
dependent, variables in order to characterise a random
storm event in terms of its inter-arrival time and duration. Further statistical distributions are used to represent the attributes of the raincells occurring within the
rainstorm. The birth and decay of each raincell is approximated as a ``pulse'' (often assumed to be rectangular) of individual intensity and duration. The total
storm intensity at a given timestep is obtained through
the summation of the intensities of each raincell active at
that timestep.
One of the attractions of pulse-based modelling is
that, through the direct simulation of raincells, the approach is (intuitively) physically reasonable. Indeed,
after a pulse-based modelÕs parameters have been optimised upon a rainfall data series, that model can often

adequately reproduce many of the properties of that
data series (including dry periods). This has been demonstrated many times [16,17,24,26,27,32]. However, the
capability of pulse-based models for extreme rainfall
simulation has often been found to be less clear-cut,
particularly with respect to the extreme rainfalls of short
duration (e.g., 1 h maxima, [11,17,24,27,36]). These extreme rainfall amounts can be extremely important in

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PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 4 2 - 7

204

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211

controlling the ¯ood response of small- to medium-size
catchments. Consequently, further model development
has been conducted in order to tackle this problem
through the incorporation of di€erent types of raincells
[14,16] and/or di€erent representations of raincell intensity [17,27].
A key challenge in using these revised models lies in

the estimation of their parameters. For example,
parameter estimation may require the subdivision of the
available observed data series into several di€erent data
sets (which are assumed to be representative of the different types of raincells or raincell intensities [14]). This
may be very dicult to achieve objectively. Indeed, use
of the available sample of observed extreme rainfall data
within the parameter estimation procedure is often
conducted under the assumption that the sample is an
adequate representation of the underlying population of
extreme rainfall events [27]. It is quite likely that this
assumption is incorrect, particularly in data-limited regions. Consequently, there may be signi®cant uncertainty associated with the parameter values of, and the
simulations obtained from, the model of interest. This
uncertainty is rarely addressed in this speci®c hydrological context.
A recent study by Cameron et al. [11] evaluated three
stochastic rainfall models (two pro®le-based models,
and Onof and WheaterÕs [27] gamma version of the
random pulse Bartlett±Lewis model, the RPBLGM)
using point raingauge data from three independent sites
in the UK. Although providing good simulations of the
seasonal extreme rainfall totals of 24 h duration and

standard rainfall statistics at each site, the RPBLGM
was found to underestimate the observed seasonal
maxima of 1 h duration. This problem was related to the
in¯exibility of the tail of the gamma distribution used to
represent raincell intensity within the model. This distibution has a medium, rather than heavy, tail.
In what follows, we describe a new version of the
random-pulse Bartlett±Lewis model which has been
developed for extreme rainfall simulation. Using point
rainfall data from a UK site, we utilise an uncertainty
framework in order to explore parameter estimation for
the component of the model used in extreme rainfall
simulation. This approach acknowledges the limited
representativeness of the observed extreme rainfall data
sample. A comparison of the resulting extreme rainfall
estimates with those obtained from an analysis of the
data (using standard extreme value methods) is also
conducted.

2. Rainfall data
Forty-four years (1949±1993) of hourly point rainfall

data were obtained from the Elmdon raingauge
(Birmingham, England), and the summer half-year data

(April±September) extracted. In a previous study,
Cameron et al. [11] found the reproduction of the short
duration extreme characteristics of this data to be a
particularly dicult stochastic rainfall modelling challenge. The Elmdon summer data therefore represents a
reasonable benchmark for evaluating the performance
of the new variant of the Bartlett±Lewis model.
Following many other stochastic rainfall modelling
studies [1,11,27,32,35,36], this paper focuses upon the
extreme rainfall totals of 1 h and 24 h duration. The
seasonal maxima (SEAMAX) appropriate to those two
durations were extracted from the observed series rainfall in a manner consistent with an annual maximum
method of analysis. An examination of the continuous
hourly rainfall series also indicated that the 24 h SEAMAX rainfall totals generally did not occur within the
same 24 h period as the 1 h SEAMAX rainfall amounts.
The two SEAMAX rainfall series are therefore largely
independent. Onof and WheaterÕs [27] random pulse
Bartlett±Lewis gamma model (RPBLGM) is capable of

adequately reproducing this independence (see [11]).
A siteÕs observed SEAMAX series is only one representation of a very large number of possible
SEAMAX series of equal record length for that site. It is
therefore useful to have an approximate guide to those
other possible series, particularly when considering the
performance of a stochastic rainfall model. In this study,
this guide was obtained through the maximum likelihood ®tting of a generalised extreme value (GEV) distribution to the observed SEAMAX series of interest,
and the subsequent calculation of pro®le likelihood
con®dence limits for quantiles (see [11]). The distribution function of the GEV is de®ned as:
F …y† ˆ exp …ÿ‰1 ‡ s…y ÿ l†=fŠ

ÿ1=s

†;

…1†

where F(y) is a non-exceedance probability, f, s and l
the scale, shape and location parameters and y is a given
SEAMAX rainfall amount. The shape parameter f allows for three di€erent shapes of the distribution. On an

extreme value probability plot (with the standard
Extreme Value I, or Gumbel, reduced variate on the
x-axis), for example, a negative value of f will produce a
convex plot, a positive value will yield a concave plot
(and indicates a heavy tailed distribution), and a zero
value produces a linear plot. All of the extreme value
probability plots displayed in this paper follow this
convention (return periods, T, in years, are also included
for convenience).
In many cases of maximum likelihood estimation, if a
parameter value (or a quantile of speci®ed return level)
is plotted against likelihood, the resulting likelihood
surface (or pro®le) will be asymmetrical, with a maximum at the maximum likelihood. This information is
utilised in the calculation of pro®le likelihood con®dence
limits. In this study, 95% pro®le likelihood con®dence
intervals were calculated for the GEV quantiles appro-

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211

priate to the return period levels of the observed

SEAMAX rainfall totals of interest. For each of the
quantiles, the procedure [2,18] entailed the reparameterisation and maximisation of the GEV likelihood. By
accounting for the presence of asymmetry in the likelihood surface, this method produces con®dence limits
which are perhaps more reliable than those which are
based upon the symmetrical con®dence limits centred on
the maximum likelihood estimate of the quantile with
their width determined by standard error estimates. It
was assumed that the pro®le likelihood 95% con®dence
limits provided a reasonable guide to the spread of the
numerous possible observed record length SEAMAX
rainfall series for the duration that they had been
calculated for.
For the 1 and 24 h durations, it was therefore possible
to assess the performance of the new random pulse
Bartlett±Lewis model against the observed data (and its
corresponding GEV ®t), and against other possible observed series (as represented by the pro®le likelihood
con®dence limits).

3. The stochastic rainfall model
A new version of the random pulse Bartlett±Lewis

model, which was modi®ed for enhanced extreme rainfall simulation, was used in this study. This section begins with a brief outline of the original random pulse
Bartlett±Lewis model. The modi®ed model is then described. The parameter estimation procedures used for
®tting the new model are detailed in Section 4.

205

RPBLGM underestimates the extreme rainfalls of short
duration.
In the case of the Elmdon summer data, Cameron
et al. [11] used the RPBLGM to produce multiple,
random, simulations of observed series length with an
hourly timestep. The model was driven by a single
parameter set (which comprised of seven RPBLGM
parameters with values estimated under Onof and
WheaterÕs [26,27] procedures). For SEAMAX accumulations of 1 and 24 h, the spread of the modelled extreme
rainfall data was quanti®ed through the calculation of
2.5%, median, and 97.5% model simulations directly
from each simulated quantile.
Figs. 1(a) and (b) illustrate the results obtained for
the 1 and 24 h SEAMAX data, respectively. The observed series (circles), the GEV ®t and associated pro®le

likelihood 95% con®dence limits (solid lines; Section 2),
and the pointwise median (dotted line), 2.5%, and 97.5%
(dashed lines) model simulations, are shown. From these
®gures, it can clearly be seen that, although the
RPBLGM provides good simulations of the 24 h SEAMAX data, it fails to reproduce the 1 h SEAMAX
rainfalls adequately. In other words, the heavy-tailed
nature of the observed 1 h maxima data cannot be adequately reproduced through the multiple, random
combinations of raincell intensitites of medium-tailed
gamma distribution origin. This limitation of the
RPBLGM provided the impetus for the further model
development presented here.
3.2. The random pulse Bartlett±Lewis model with exponential and generalised pareto distributed raincell intensities

3.1. The random pulse Bartlett±Lewis model
Full details of the Bartlett±Lewis pulse model are
provided in [19,26,27,30±32], so only a brief summary is
given here.
The random pulse Bartlett±Lewis model (or
RPBLM) describes the random arrival of storms via a
Poisson process (governed by the parameter k). Each
storm origin is followed by a Poisson process of rate jg
of cell origins; the process of new cell origins terminates
after a time that is exponentially distributed with
parameter /g. The durations of the raincells are independent exponentially distributed random variables
with parameter g. For each storm, g is randomly sampled from a gamma distribution with the parameters a
(shape) and 1/m (scale). The raincell depth is assumed to
be exponentially distributed with the parameter lx .
Onof and Wheater [27] also describe a version of the
RPBLM (the RPBLGM) which features gamma (rather
than exponential) distributed raincell intensities in order
to improve the modelÕs capability for extreme rainfall
simulation. However, recent studies [11,36] have shown
that, at particular sites in Belgium and the UK, the

One approach to improving the quality of the
RPBLMÕs extreme rainfall simulations is to assume that
there are two or more classes of raincell (representing,
e.g., convective and stratiform rainfall), with each class
of raincell possessing its own set of parameters (such as
raincell inter-arrival time, duration and depth). This
approach has previously been implemented successfully
for a Neyman±Scott rainfall model [14,16]. However, if
the only available observed data are, say, a continuous
hourly raingauge record (rather than extensive radar
data), then there may be problems with respect to
parameter estimation under this procedure. In modifying the RPBLM, we generalise the distribution of raincell intensities from exponential in a di€erent way.
We assume that raincell depth can be of either low
intensity or high intensity. The storm and raincell arrival
and duration processes are modelled in an identical
manner to the RPBLM. During a model run, the classi®cation of a given raincell as being of low or high intensity is made via the sampling of an initial depth from
an exponential distribution (with parameter lx , Section
3.1). If that depth is less than a threshold (u), then the

206

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211

Fig. 1. Comparison of the observed and RPBLGM simulated SEAMAX rainfalls for the Elmdon summer season. (a) 1 h duration; (b) 24 h duration.
Circles ± observed SEAMAX rainfalls. Solid line ± observed series GEV distribution and resulting pro®le likelihood 95% con®dence limits. Dotted
line ± median model simulation. Dashed lines ± 2.5% and 97.5% model simulations. (Adapted from [11].)

raincell is of low intensity, its initial depth is retained,
and the model proceeds as per the RPBLM. If the depth
exceeds u, then the raincell is of high intensity and its
depth (which is still above u) is resampled from a generalised pareto distribution (GPD). The GPD has the
distribution function:
F …x† ˆ 1 ÿ …1 ‡ ‰n…x ÿ u†=rŠ†ÿ1=n ;
F …x† ˆ 1 ÿ exp‰ÿ…x ÿ u†=rŠ;

n 6ˆ 0;

n ˆ 0;

…2†

where F(x) is a non-exceedance probability, n a shape
parameter, u (the intensity threshold) a location
parameter, x ) u an exceedance (where x > u), and r is a
scale parameter.
The GPD was selected for its ¯exibility. The shape
parameter n allows for three di€erent shapes of the
distribution. On an extreme value probability plot, for
example, a negative value of n will produce a convex
plot, a positive value will yield a concave plot, and a
zero value reduces the GPD to the exponential distribution, producing a linear plot. In traditional extreme
event frequency analysis, the GPD can be used to
model peaks over threshold (POT) data. When coupled
with a Poisson distribution for the number of extreme
events per year, the GPD is equivalent to the use of the
GEV to describe annual maxima data (censored at the
threshold, u).
The modi®ed model therefore allows enhanced extreme rainfall simulation while keeping the number of
additional parameters required to a minimum (two:
r, n).

4. Parameter estimation
The new Bartlett±Lewis model (here termed the
RPBLGPDM) has a total of eight parameters (k, a, m, j,
/, lx , r, n). The threshold, u, must also be selected.

Parameter estimation is therefore a challenge. In this
paper, we adopt a two-stage approach to parameter
estimation (whereby the Bartlett±Lewis parameters, k, a,
m, j, /, lx , are ®tted ®rst, followed by the GPD
parameters). In this approach, it is assumed that, since
the GPD parameters …r; n† are only appropriate to the
simulation of extreme rainfalls, they should only have a
minimal impact upon the standard statistics of the
simulated continuous rainfall time-series. It is also assumed that the standard statistics of the (relatively large)
available observed rainfall data sample (but not the
extremes) can be reproduced using a single, acceptable,
``regular'' Bartlett±Lewis parameter set (k, a, m, j, /, lx )
for continuous rainfall simulation.
4.1. Stage one
The parameters k, a, m, j, /, lx are estimated using
the iterative moment ®tting procedure of Onof and
Wheater [26±29]. This procedure requires the use of a
small, representative, set of observed rainfall properties,
which are subject to minimal sampling error and correlation [32]. Following a period of sensitivity testing,
Onof and WheaterÕs [26], R1 set of observed rainfall
properties was selected. This set is de®ned as:
i
h
i
n h
h
i
…1†
…1†
…1†
…1†
R1 ˆ E Yi ; var Yi ; cov Yi ; Yi‡1 ;
h
i
o
…6†
…6†
p…1†; cov Yi ; Yi‡1 ; p…24† ;
…3†
…h†

where E is the mean, Yi the ith value in the continuous
time-series of h hourly rainfall depths, var the variance,
cov the covariance, and p(h) is the proportion of dry h
hour time intervals within the continuous rainfall timeseries. A dry interval is de®ned as a period with zero
rainfall. Further details of this moment ®tting technique,
including those of the objective function, are supplied in
[26±28].

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211
Table 1
The ®rst six parameters estimated for the RPBLGDPM
Parameter

Value

k
a
m
j
/
lx

0.02
3.02
0.61
0.19
0.04
2.30

Table 2
Bartlett±Lewis parameter values identi®ed by Onof and Wheater [26]a
k
a
m
j
/
lx

A

M

J

J

A

S

0.02
3.85
1.02
0.60
0.09
0.96

0.02
2.76
1.17
0.26
0.02
2.74

0.02
2.53
0.28
0.29
0.04
2.28

0.01
2.29
0.25
0.04
0.01
4.19

0.01
2.88
0.69
0.04
0.01
3.44

0.02
2.48
0.44
0.42
0.07
1.73

a

Only monthly (April (A)±October (O)) parameter values are available
for the summer season from this source.

Table 3
RPBLGM parameter values identi®ed by Cameron et al. [11] for the
Elmdon summer seasona
Parameter

Value

k
a
m
j
/
d
q

0.02
2.82
0.56
0.13
0.03
0.46
1.16

a

d and q are parameters of the gamma distribution used to represent
raincell intensities (they therefore replace the lx parameter of the
RPBLM).

Table 1 details the parameter set obtained using this
approach. An examination of the parameter values indicates that they are similar to those identi®ed in the
earlier studies of Onof and Wheater [26,27] and
Cameron et al. [11], Tables 2 and 3. Indeed, when the
parameter set in Table 1 was used to drive the RPBLM
(as a test run) the reproduction of the observed standard
rainfall properties was of the same quality as in those
earlier studies.
4.2. Stage two
Several possible procedures, including Bayesian
methods (through the use of Markov Chain Monte
Carlo simulation, e.g., [13,33]), could be used to estimate
the GPD parameters. The approach adopted here
acknowledges the limited representativeness of the
observed series extreme data sample. It also attempts to
quantify the uncertainty associated with the GPD
parameter estimates and the resulting extreme rainfall

207

simulations via the generalised likelihood uncertainty
estimation (GLUE) framework of Beven and Binley [6].
GLUE is a Bayesian Monte Carlo simulation-based
technique, developed as an extension of Spear and
HornbergerÕs [34] Generalised Sensitivity Analysis.
Freer et al. [21] describe the rationale of the GLUE
methodology within the context of Bayesian statistics.
The GLUE approach has previously been applied successfully in rainfall-runo€ and stochastic rainfall modelling [5,10,20,21]. This provided the impetus for using
GLUE within this study.
The GLUE methodology rejects the concept of a
single, global optimum parameter set and instead accepts the existence of multiple acceptable (or behavioural) parameter sets (the equi®nality concept of Beven
[5]). The operation of the GLUE procedure features the
generation of very many parameter sets from speci®ed
ranges using Monte Carlo simulation. The performance
of individual parameter sets is assessed via likelihood
measures which are used to weight the predictions of the
di€erent parameter sets. This includes the rejection of
some parameter sets as non-behavioural.
The GLUE approach therefore requires the selection
of those parameters which will be generated and evaluated, and those which will be held constant (if any).
Since the main area of interest in this study is extreme
rainfall simulation, only the two GPD parameters
(r and n) were varied, with the other RPBLGPDM
parameters (k, a, m, j, / and lx ) and the threshold (u)
held constant (note that this approach does not preclude
the possibility of a future GLUE analysis of all of the
RPBLGPDM parameters).
The values of the parameters k, a, m, j, /, and lx
detailed in Table 1 were used. After an initial period of
testing, a threshold (u) value of 10 mm was set. This
threshold results in an average of ®ve exceedances per
summer season. It was selected on the basis that it
permitted enhanced extreme rainfall simulation while
minimising the impact of the high intensity raincells
upon the modelled standard rainfall statistics. The following procedure was utilised to identify behavioural
parameter sets and simulations.
Five thousand GPD parameter sets (consisting of r
and n), were initially generated from independent uniform distributions over a broad range of parameter
values (with a range of 0.01±15.00 for r and )1.00 to
1.00 for n). A single continuous hourly rainfall timeseries (of observed record length) was produced for each
parameter set using the full RPBLGPDM. Each of these
simulations featured the use of the GPD parameter set
in combination with the other, ®xed, parameters. The
runs were conducted using a 20 processor parallel Linux
PC cluster at the University of Lancaster, UK. Each
simulation was then evaluated using two criteria. The
®rst assessed the quality of the modelling of the observed series 1 h SEAMAX rainfall amounts. The

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211

second was used in order to maintain a consistency with
the assumption that the GPD parameters have a minimal impact upon the simulated standard rainfall statistics (Section 4). These criteria are described in
Appendices A.1 and A.2, respectively.
Following the application of the two criteria, resampling of the GPD parameter space was conducted in
order to provide a suciently large sample of behavioural simulations. As per Cameron et al. [10], a sample
size of 1000 was assumed to be adequate. Likelihood
weighted uncertainty bounds were then calculated from
the 1000 behavioural simulations for the SEAMAX
rainfall totals of 1 and 24 h duration, (Appendix A.3). On
extreme value probability plots, these bounds were
compared with the corresponding observed SEAMAX
series, their GEV ®ts, and the GEVÕs pro®le likelihood
95% con®dence limits (which had been calculated
through a direct statistical analysis of the data, Section 2).
It is important to recognise that, in the above evaluation procedure, the acceptability of a given model
simulation does not depend solely upon the generated
GPD parameters. The timings, durations, and intensities
of the raincells which are spawned over the course of the
simulation are also very signi®cant. Taken together with
the GPD parameters, these factors contribute to an
important model realisation e€ect, and it is the model
realisation as a whole which is evaluated. This e€ect is
handled naturally within the GLUE methodology which
accepts that there be many models and parameter sets
that are consistent with the set of available observations.

5. Results and discussion
Following the application of the l(q) constraint (Appendix A.1), 1320 simulations (of the initial sample of
5000) were retained. Of these, 765 simulations were retained on the basis of PAE (Appendix A.2). The system
of evaluation is therefore e€ective in rejecting nonbehavioural simulations. Interestingly, when considered
independently, the parameter ranges of the r and n
parameters associated with the 1000 behavioural simulations were found to be equally as broad as their initial
sampling ranges (Section 4). Indeed, a scatterplot of r
against n for these behavioural simulations (Fig. 2)
indicates that, although there is an important interaction
between these two parameters, there are acceptable
parameter combinations located across a very wide
range of the parameter space. Furthermore, an examination of the 1 h SEAMAX L(q) likelihoods (associated
with the sample of 1000 behavioural simulations; see
Appendix A.3) determined that the ``best'' L(q) values
were located right across that range. These ®ndings
highlight the importance of the model realisation e€ect
in de®ning the acceptability of each model simulation.
(Incidentally, because of the realisation e€ect, and

1

0.8

0.6

0.4
gpd shape parameter

208

0.2

0

–0.2

–0.4

–0.6

–0.8

–1

0

5

10

15

gpd scale parameter

Fig. 2. Scatterplot of the GPD scale (r) and shape (n) parameters
associated with each of the 1000 behavioural RPBLGPDM simulations. (The likelihoods associated with these simulations are illustrated
in Fig. 3.)

consequently, the broad range of good model ®ts across
the behavioural parameter space, it is not possible to
produce useful contour plots of the GPD two-parameter
space.)
The likelihood weighted RPBLGPDM results for the
1 h SEAMAX series (obtained from the 1000 behavioural model simulations) are depicted in Fig. 3(a). The
observed series (circles), the GEV ®t and associated
pro®le likelihood 95% con®dence limits (solid lines), and
the median (dotted line) and 95% (dashed lines) likelihood weighted uncertainty bounds, are shown. The
corresponding 24 h SEAMAX rainfall totals are illustrated in Fig. 3(b).
Figs. 3(a) and (b) illustrate the RPBLGPDMÕs ability
to reproduce the SEAMAX extremes of the observed
rainfall series. These ®gures show a comparison of the
95% likelihood weighted uncertainty bounds (calculated
under the GLUE procedure, Section 4.2 and Appendix
A.3) with the GEV pro®le likelihood 95% con®dence
limits. From Fig. 3(a), it can be seen that the
RPBLGPDM successfully simulates the observed 1 h
SEAMAX series. The 95% uncertainty bounds
``bracket'' the observed data, and the median simulation
lies close to that data. This result is an improvement
upon the previous best-®t performance of the RPBLGM
for the summer season (Fig. 1(a), [11]). (Although it
should be remembered that the RPBLGM results were
obtained using a single parameter set, however, it is
unlikely that a GLUE analysis of the RPBLGM would,
within the limitations of that modelÕs structure, produce
signi®cantly di€erent short duration extreme rainfall
simulations; see [11] for a consideration of di€erent
RPBLGM parameterisations.)
Encouragingly, there is also a reasonable agreement
between the RPBLGPDMÕs 95% uncertainty bounds

209

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211
60
T [yrs]

1

2

5

10

25

50

200

100

T [yrs]

1

2

5

10

25

50

100

180

50
160

140

120
rainfall [mm]

rainfall [mm]

40

30

100

80

20
60

40

10
20

(b)

(a)
0
–2

–1

0

1
2
ev1 reduced variate

3

4

5

0
–2

–1

0

1
2
ev1 reduced variate

3

4

5

Fig. 3. Comparison of the observed and RPBLGPDM simulated SEAMAX rainfalls for the Elmdon summer season. (a) 1 h duration; (b) 24 h
duration. Circles ± observed SEAMAX rainfalls. Solid line ± observed series GEV distribution and resulting pro®le likelihood 95% con®dence limits.
Dotted line ± median model simulation (calculated under the GLUE procedure). Dashed lines ± 95% uncertainty bounds (calculated under the
GLUE procedure).

and the pro®le likelihood 95% con®dence limits
(although there are some variations at plotting positions
of greater than approximately 2.25, or a return period
level of 10 yr). These results indicate that the model
simulations are consistent with both the observed data,
and the other possible 1 h SEAMAX series with
observed record length.
Similar results were also obtained for the 24 h SEAMAX rainfall totals (Fig. 3(b)). However, at plotting
positions of greater than approximately 1.5 (or a 5 yr
return period level), the median model simulation and
97.5% uncertainty bound indicate higher rainfall
amounts than those suggested by the observed series and
97.5% con®dence interval. These ®ndings are similar to
those obtained for this site in the earlier RPBLGM
study of Cameron et al. [11] (see also Fig. 1(b)).

6. Conclusions
This paper has explored the use of a modi®ed version
of the Bartlett±Lewis pulse model for extreme rainfall
simulation for a UK site (44 summer half-year data at
Elmdon, Birmingham). The greater part of the model is
identical to that of the random pulse Bartlett±Lewis
model (RPBLM) used by Onof and Wheater [26]. The
modi®cation to the model features the use of a generalised pareto distribution (GPD) to represent high intensity raincell depths. These high intensity raincells
make a noticeable di€erence to the distribution of 1 h
extreme values.
The model is termed the RPBLGPDM and parameter
estimation is carried out using a two-stage process. The
®rst stage estimates the parameters k, a, m, j, /, and lx
via an iterative moment-®tting procedure (as was the

case for the RPBLM, e.g., [26]). The GPD threshold, u,
is then ®xed and the two GPD parameters (r and n)
estimated using the generalised likelihood uncertainty
estimation (GLUE) approach of Beven and Binley [6].
Following the simulation of numerous continuous
hourly rainfall time-series, the GLUE procedure is also
used to identify behavioural model simulations (which
consist of acceptable GPD parameter sets operating in
combination with favourable random model realisations).
The RPBLGPDMÕs ability to reproduce the observed
series 1 h seasonal maxima (SEAMAX) for the summer
season at Elmdon is superior to the earlier versions of
the model [11]. The modelÕs reproduction of the 24 h
SEAMAX totals is reasonably consistent with that of
earlier versions of the Bartlett±Lewis model (e.g., Onof
and WheaterÕs [27] gamma raincell depth version). The
simulations also compare favourably with a statistical
analysis of the extreme value data alone.

Acknowledgements
The authors wish to thank the Meteorological Oce
for access to the Elmdon raingauge data. Very grateful
thanks are due to Christian Onof and Howard Wheater
for access to, and aid with, the exponential and gamma
raincell intensity versions of the Bartlett±Lewis model.
Stuart Coles is also thanked for comments upon
parameter estimation for the gpd raincell intensity
version of the Bartlett±Lewis model. The comments of
two anonymous referees contributed to the clarity of the
®nal manuscript. David Cameron's contribution to this
work was carried out at Lancaster University under the
NERC CASE studentship GT4/97/112/F.

210

D. Cameron et al. / Advances in Water Resources 24 (2001) 203±211

Appendix A

A.1. Reproduction of 1 h SEAMAX rainfall amounts
The ®t of the simulated 1 h SEAMAX rainfall
amounts to those of the observed data was assessed
using a procedure similar to that described in [10]. This
procedure assumes that, since the observed 1 h SEAMAX rainfall amounts are adequately represented by a
GEV distribution (Section 2), then a behavioural
RPBLGPDM simulation is one which yields a 1 h
SEAMAX GEV ®t which is close to that of the observed
data.
The approach therefore entails the ®tting of a GEV
distribution to the observed SEAMAX data (Section 2)
and (independently) to each simulated SEAMAX series.
The non-exceedance probabilities of the observed 1 h
SEAMAX series (without GEV ®t) are calculated, and
the corresponding rainfall amounts extracted from the
observed series GEV ®t. The evaluation consists of the
calculation of the goodness of ®t of a given simulated
series GEV ®t to those 44 rainfall quantities. This is
conducted using a log likelihood measure (l(q)), as:
l…q† ˆ

44
X

ÿ log fs ‡ …ÿ1=ss ÿ 1†
log ‰1 ‡ ss

iˆ1


…yi ÿ ls †=fs Š ÿ ‰1 ‡ ss …yi ÿ ls †=fs Š

ÿ1=

ss ;

…A:1†

where fs , ss and ls are the scale, shape and location
parameters of the GEV distribution ®tted to the simulated series, and yi is a SEAMAX amount extracted
from the GEV distribution ®tted to the observed series.
A simulation is retained as behavioural if:
Dpa 6 TD;

…A:2†

where D is the deviance calculated between the maximum value of l(q) in the sample of 5000 parameter sets
(l(P)), and the value of l(q) for a given parameter set
(pa), as:
Dpa ˆ 2‰l…p† ÿ l…q†pa Š

…A:3†

and TD is a threshold deviance of 6.25 obtained from
the v2 distribution at 3 d.f. (for the GEV) and probability level P ˆ 0.9 (see [10] for a further description of
this procedure and the choice of thresholds).

that the GPD parameters have a minimal impact upon
the standard rainfall statistics (Section 4). For each
statistic, the percentage absolute error (PAE) was calculated. It is de®ned as:
PAE ˆ 100 j‰statsim ÿ statobs Š=statobs j;

…A:4†

where stat is the statistic of interest, and sim and obs are
the simulated and observed series, respectively.
A simulation was de®ned as behavioural if the PAE
was less than or equal to 10% for each statistic of interest.
This acceptance threshold was based upon the range of
statistics obtained from the production of multiple
RPBLM realisations using the parameters in Table 1.
The performance of the RPBLGPDM for the Elmdon
summer data is therefore very similar to that of the
RPBLM with respect to the standard rainfall statistics.

A.3. Calculation of likelihood weighted uncertainty
bounds
Likelihood weighted uncertainty bounds were calculated from the 1000 behavioural simulations for the
SEAMAX rainfall totals of 1 and 24 h duration. In the
former case, the l(q) values calculated during the evaluation were used. In the latter, further l(q) values were
calculated between the observed 24 h SEAMAX rainfall
totals and those associated with each of the 1000
behavioural simulations. In each case, the exponential of
l(q) was taken in order to yield the likelihood measure
L(q) (where each likelihood is equivalent to a probability). A standard procedure [10,21] was then used to
calculate the uncertainty bounds independently for each
duration.
This procedure involved the rescaling of the L(q)
likelihood weights over all of the behavioural simulations in order to produce a cumulative sum of 1.0. A cdf
of rainfall estimates was constructed for each SEAMAX
amount of the duration of interest using the rescaled
weights. Linear interpolation was then used to extract
the rainfall estimate appropriate to cumulative likelihoods of 0.025, 0.50, and 0.975. This allowed 95% uncertainty bounds, in addition to a median simulation, to
be derived.

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…h†
served series values of E‰Yi Š and var‰Yi Š (Section 4.1)
at the h hourly timescales of 1 and 24. This was done in
order to maintain a consistency with the assumption

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