Modelling extreme rainfalls using a modi®ed random pulse
Bartlett±Lewis stochastic rainfall model (with uncertainty)
David Cameron
a,*, Keith Beven
b, Jonathan Tawn
ca
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 rep-resentation of storm inter-event arrival times is an im-portant 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,
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 dura-tion. Further statistical distributions are used to rep-resent the attributes of the raincells occurring within the rainstorm. The birth and decay of each raincell is ap-proximated as a ``pulse'' (often assumed to be rectan-gular) 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 ap-proach is (intuitively) physically reasonable. Indeed,
after a pulse-based modelÕs parameters have been
opti-mised 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 dem-onstrated 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 ex-treme rainfall amounts can be exex-tremely important in
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*
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@lancas-ter.ac.uk (J. Tawn).
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 dierent types of raincells [14,16] and/or dierent representations of raincell in-tensity [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 dierent data sets (which are assumed to be representative of the dif-ferent types of raincells or raincell intensities [14]). This may be very dicult 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 re-gions. Consequently, there may be signi®cant uncer-tainty associated with the parameter values of, and the simulations obtained from, the model of interest. This uncertainty is rarely addressed in this speci®c hydro-logical 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 disti-bution 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 dicult stochastic rainfall modelling chal-lenge. 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 rain-fall 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 SEA-MAX 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
rep-resentation 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 likeli-hood ®tting of a generalised extreme value (GEV) dis-tribution to the observed SEAMAX series of interest, and the subsequent calculation of pro®le likelihood con®dence limits for quantiles (see [11]). The distribu-tion funcdistribu-tion of the GEV is de®ned as:
F y exp ÿ1s yÿl=fÿ1=s; 1
where F(y) is a non-exceedance probability,f, sand l
the scale, shape and location parameters andyis a given
SEAMAX rainfall amount. The shape parameter f
al-lows for three dierent 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 offwill 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).
appro-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 reparamete-risation and maximisation of the GEV likelihood. By accounting for the presence of asymmetry in the likeli-hood 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 ob-served 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 rain-fall simulation, was used in this study. This section be-gins with a brief outline of the original random pulse Bartlett±Lewis model. The modi®ed model is then de-scribed. The parameter estimation procedures used for ®tting the new model are detailed in Section 4.
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 ratejg
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
inde-pendent exponentially distributed random variables
with parameterg. For each storm,g is randomly
sam-pled from a gamma distribution with the parameters a
(shape) and 1/m(scale). The raincell depth is assumed to
be exponentially distributed with the parameterlx.
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
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
accumu-lations 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 ob-served 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 SEA-MAX data, it fails to reproduce the 1 h SEASEA-MAX rainfalls adequately. In other words, the heavy-tailed nature of the observed 1 h maxima data cannot be ad-equately 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 expo-nential and generalised pareto distributed raincell inten-sities
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 modify-ing the RPBLM, we generalise the distribution of rain-cell intensities from exponential in a dierent 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 clas-si®cation of a given raincell as being of low or high in-tensity is made via the sampling of an initial depth from
an exponential distribution (with parameter lx, Section
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 aboveu) is resampled from a
gen-eralised pareto distribution (GPD). The GPD has the distribution function:
F x 1ÿ 1 n xÿu=rÿ1=n; n60;
F x 1ÿexpÿ xÿu=r; n0; 2
where F(x) is a non-exceedance probability, n a shape
parameter, u (the intensity threshold) a location
parameter,x)uan exceedance (wherex>u), andris a
scale parameter.
The GPD was selected for its ¯exibility. The shape
parameter n allows for three dierent 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 distri-bution, 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 ex-treme 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 as-sumed 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 cor-relation [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:
R1nE Yh i 1i; varhYi 1i; covhYi 1;Yi 11i;
p 1; covhYi 6;Yi1 6i; p 24o; 3
whereEis the mean,Yi htheith value in the continuous
time-series ofh hourly rainfall depths, var the variance,
cov the covariance, andp(h) is the proportion of dry h
hour time intervals within the continuous rainfall time-series. 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].
Table 1 details the parameter set obtained using this approach. An examination of the parameter values in-dicates 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
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 suc-cessfully in rainfall-runo and stochastic rainfall mod-elling [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
ac-cepts the existence of multiple acceptable (or
behav-ioural) 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 dierent 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 evalu-ated, 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, / andlx) 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 fol-lowing 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
uni-form 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
time-series (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 ob-served series 1 h SEAMAX rainfall amounts. The
Table 1
The ®rst six parameters estimated for the RPBLGDPM
Parameter Value
Bartlett±Lewis parameter values identi®ed by Onof and Wheater [26]a
A M J J A S
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
adandqare parameters of the gamma distribution used to represent
second was used in order to maintain a consistency with the assumption that the GPD parameters have a mini-mal impact upon the simulated standard rainfall statis-tics (Section 4). These criteria are described in Appendices A.1 and A.2, respectively.
Following the application of the two criteria, resam-pling of the GPD parameter space was conducted in order to provide a suciently large sample of behav-ioural 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 evalu-ation 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 eect, and it is the model realisation as a whole which is evaluated. This eect 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 thel(q) constraint
(Ap-pendix A.1), 1320 simulations (of the initial sample of 5000) were retained. Of these, 765 simulations were
re-tained on the basis ofPAE(Appendix A.2). The system
of evaluation is therefore eective in rejecting non-behavioural simulations. Interestingly, when considered
independently, the parameter ranges of the r and n
parameters associated with the 1000 behavioural simu-lations 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
exam-ination of the 1 h SEAMAXL(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 eect in de®ning the acceptability of each model simulation. (Incidentally, because of the realisation eect, and
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 behav-ioural 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) likeli-hood weighted uncertainty bounds, are shown. The corresponding 24 h SEAMAX rainfall totals are illus-trated 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 dierent short duration extreme rainfall simulations; see [11] for a consideration of dierent RPBLGM parameterisations.)
Encouragingly, there is also a reasonable agreement
between the RPBLGPDMÕs 95% uncertainty bounds
0 5 10 15
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 SEA-MAX 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 gener-alised pareto distribution (GPD) to represent high in-tensity raincell depths. These high inin-tensity raincells make a noticeable dierence 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 parametersk,a,m,j,/, andlx
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 realisa-tions).
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 Oce 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.
–2 –1 0 1 2 3 4 5
0 10 20 30 40 50 60
ev1 reduced variate
rainf
all [mm]
1 2 5 10 25 50 100 T [yrs]
–2 –1 0 1 2 3 4 5
0 20 40 60 80 100 120 140 160 180 200
ev1 reduced variate
rainf
all [mm]
1 2 5 10 25 50 100 T [yrs]
(a) (b)
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 SEA-MAX 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 X
44
i1
ÿlogfs ÿ1=ssÿ1 log1ss
yiÿls=fs ÿ 1ss yiÿls=fsÿ1=ss; A:1
where fs, ss and ls are the scale, shape and location
parameters of the GEV distribution ®tted to the
simu-lated series, and yi is a SEAMAX amount extracted
from the GEV distribution ®tted to the observed series.
A simulation is retained asbehaviouralif:
Dpa6TD; A:2
where D is the deviance calculated between the
maxi-mum value ofl(q) in the sample of 5000 parameter sets
(l(P)), and the value of l(q) for a given parameter set
(pa), as:
Dpa2l p ÿl qpa A:3
and TD is a threshold deviance of 6.25 obtained from
the v2 distribution at 3 d.f. (for the GEV) and
prob-ability levelP0.9 (see [10] for a further description of
this procedure and the choice of thresholds).
A.2. Reproduction of standard rainfall statistics
The simulations which were retained as behavioural
under the l(q) constraint (Appendix A.1) were also
evaluated in terms of their ability to reproduce the
ob-served series values ofEYi hand varYi h(Section 4.1)
at thehhourly timescales of 1 and 24. This was done in
order to maintain a consistency with the assumption
that the GPD parameters have a minimal impact upon the standard rainfall statistics (Section 4). For each statistic, the percentage absolute error (PAE) was cal-culated. It is de®ned as:
PAE100jstatsimÿstatobs=statobsj; 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 calcu-lated 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
eval-uation 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
prob-ability). 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 simula-tions 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 likeli-hoods of 0.025, 0.50, and 0.975. This allowed 95% un-certainty bounds, in addition to a median simulation, to be derived.
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