- Email: [email protected]
Estimation of regional intensity-duration-frequency curves for extreme precipitation
e>
Pergamon
Wal. Sci Tech. Vol. 37, No. II. pp. 29-36,1998.
IAWQ = 1998 Published byElsevier Science Ltd. Printed inGreat Britain.All righu reserved
PH: 80273-1223(98)00313-8
0273-1223198 $19'00 + 0'00
ESTIMATION OF REGIONAL INTENSITY-DURATION-FREQUENCY CURVES FOR EXTREME PRECIPITATION H. Madsen*, P. S. Mikkelsen**, D. Rosbjerg* and P. Harremoess" * Department ofHydrodynamics and Water Resources, Technical University of Denmark; Building 115, DK-2800Lyngby, Denmark ** Department ofEnvironmental Science and Engineering, Technical University of Denmark; Building 1 J5, DK-28oo Lyngby, Denmark
ABSTRACf Regional estimation of extreme precipitation from a high resolution rain gauge network in Denmark is considered. The appliedextremevalue modelis basedon the partialduration series(POS)approachin which all events above a certain threshold level are model1ed. For a preliminary assessment of regional homogeneity and identification of a proper regional distribution Lmoment analysis is applied. To analyse the regional variability in more detail, a generalised least squares regression analysis is carried out that relatesthe PDS model parameters to climaticand physiographic characteristics. The analysis revealsthat the mean annual number of extreme events varies significantly within the region. and a large part of this variability can be explained by the mean annual rainfal1. The mean value of the exceedance magnitudes can be assumed constant for intensities with durations less than one hour. For larger durations a pronounced metropolitan effect is evident, the mean intensities in the Copenhagen area being significantly larger than found in the rest of the country. With respect to second and higher order moments the region can be considered homogeneous for intensities with durations less than 24 hours. A regional parent distribution is identified as the generalised Pareto distribution. ~ 1998 Published by Elsevier Science Ltd. AI1 rights reserved
KEYWORDS Extreme rainfalls; partial duration series; regional estimation; L-moments; regression analysis INTRODUCfION In 1979 a new system of high resolution automatic rain gauges with specific relevance for urban hydrologic applications was introduced in Denmark. At present, 90 stations have been connected to the system. and the longest records consist of more than 18 years of data. Details of the measurement system and the initial quality control of the data can be found in Jergensen et at. (1998). Different statistical models have been applied for analysing the data. Ambjerg-Nielsen et at. (1994) elaborated the traditional non-parametric approach by using a bootstrap method to quantify the estimat ion uncertainty. Madsen et al. (1994) and Mikkelsen et al. (1996) applied a parametric model based on the 29
30
H. MADSEN et al.
partial duration series (PDS)approach. The preliminary studies revealed a significant geographical variation of extreme rainfalls. Assuming this variation to be purely stochastic. Madsen et al. (1994) formulated a regional model that provides an estimate of the T-year event as well as an estimate of the associated uncertainty which includes both sampling uncertainty and regional variability. Arnbjerg-Nielsen et al, (1996) analysed the potential of including physiographic and climatic information to describe the regional variability. They found that such covariate information was able to sufficiently describe the variability of extremeevents only for very small return periods. In 1996the DanishWater Pollution ControlCommittee initiated a projectto establishing new guidelines for use of historical rainfall data from the automatic rain gauge system in urban storm drainage analysis (Mikkelsen et al.; 1998). An important element in this project has been to identify and model the regional variability of extreme rainfall statistics. The basis for the modelling has been the regional PDS model introduced by Madsen et 01. (1994) and Mikkelsen et al. (1996). including recent developments by Madsen and Rosbjerg (199780 b). The present paper describes the regional modelling approach and summarises the main results.
EXTREME VALUE MODEL For the regional analysis. 41 stationswith morethan 10yearsof data havebeen included. corresponding to a total of about 650 station years. The raw data consistof one-minute rain intensities of individual events.The preliminary separation of rain events is defined as periods exceeding one hour without precipitation. The maximum rain intensity of individual rain events (hereafter for simplicity denoted intensity) for given durationis definedas follows. For durations less than or equal to one hour. the preliminary separation of rain events is used to define the intensity. For largerdurations. the definition of independent rain events is related to the duration. In this case the intensity is calculated for rain events that are separated by dry periods of lengthslargerthan the consideredduration.
=
The intensity for given duration is described by a stochastic variable Z with observations ('jo i 1,2...,m) where m is the total number of rain events in the historical time series. The first step in the modelling of extreme rainfalls is selection of the extreme population. In the PDS model (also denoted the peak over threshold (Pan model) all events above a predefined threshold level '0 are considered. i.e, the PDS is definedas (X Z - zOo Z> zo I withobservations {Xi = ' j - ZOo i = 1.2....,n,J where n (<< m) is the PDS sample size.
=
In PDS modelling it is generally assumed that the occurrence of exceedances can be described by a Poisson process with constant or one-year periodic intensity. Hence. the number of threshold exceedances can be described by a Poissondistribution with intensity. A., that equals the expectednumberof annualexceedances. For modelling of the exceedance magnitudes a statistical distribution is adopted. In the basic PDS model a simple one-parameter exponential distribution was applied. Differenttwo-parameter distributions have been proposed. including the gamma. Weibull, Log-normal. and generalised Pareto distributions. Denoting by F(x) the cumulative distribution function of the exceedances, the T-year event is given by
(l)
An estimateof the T-yearevent is obtainedfrom (I) by inserting estimates of the PDS model parameters. An estimateof the Poisson parameteris given as the averagenumberof observedexceedances per year. i.e, A. = nit. For estimation of the parameters of F(x). the method of L-moments is adopted. i.e. the parameters are estimated from the sample mean value and higher order sample L-moments (see description of L-moments below). Partial duration series for the 41 stationswere defined by using the same threshold level at all stations.The threshold level was chosen on the basis of a preliminary sensitivity analysis of regional average extreme
Regional estimation of extreme precipitation
31
valuecharacteristics as a function of the threshold level,implying a regional average numberof exceedances per year in the range 2.5-3.2 for the analysed variables.
L-MOMENT ANALYSIS Evaluation of the regional variability of extreme rainfall statistics is based on the PDS parameters defined above; that is, the Poisson parameter, the meanvalue of the exceedances, and higher order moments. In this section homogeneity of higher order moments and the closely related selection of a regional parent distribution is described using the theoryof L-moments. In the following section a general methodology for assessment of regional homogeneity and for estimation of regional statistics for all PDS parameters is described. L-moments are defined as linear combinations of expected values of order statistics (Hosking, 1990). The first L-moment 0"1) is the mean value identical to the first ordinary moment: the secondL-moment O"V is a measure of dispersion analogous to the variance; the third L-moment (A3> is a measure of skewness; and the fourth L-moment (A,,> is a measure of kurtosis. L-moment ratiosare definedas, respectively, L-coefficient of variation: L-CV = A,P'''I' L-skewness: L-CS = AfA.2, and L-kurtosis: L-KUR = A4~' For estimation of Lmoments, unbiased estimates of probability weighted moments are applied. L-moment ratio estimates are nearly unbiased and are relatively insensitive to outliers; properties that are extremely important in regional studies. For a visual judgement of regional homogeneity and identification of a proper regional distribution, an Lmoment diagram is constructed where L-moment ratio estimates for the regional stations are compared to the theoretical relationships for a number of candidate distributions (see Figure I). The dispersion of the sample points in this diagram is a measure of regional heterogeneity. In addition, the cloud of points indicates which distribution is the most appropriate. Since L-moment ratio estimates are nearly unbiased, half of the samplepointsare expectedto lie abovethe theoretical curve and half to lie below. The following candidate distributions are considered: exponential (EXP), gamma (GAM), Weibull (WEI), log-normal (LN),and generalised Pareto(GP). 0.70
e
At-site estimates Weighted average
•
--GP 0.60
- - - LN ----GAM ~----'WEI
•
~
-'
EXP
0.50
·it·.'
";;'0
.......'/ . r
0.40 .'
..
.'
/ /
r
0.30 +-:.,...,;.,.......-,.;:....-.............,r'-r--.................-....................,..-.-............-.......--..-.--. 0.30 0.40 0.20 0.50 0.60 0.10 L-5kewness Figure I. L-moment diagram for the IG-minutes rainintensity. At-site L-moment ratioestimates from41 stanons and the record-length-weighted average compared to thetheoretical relationships for a number of candidate distributions.
For a more formal evaluation of regional homogeneity Hosking and Wallis(1993) proposeda heterogeneity measure where the observed dispersion of L-CV is compared to the expecteddispersion in a homogenous
32
H. MADSEN et al.
region (i.e. where the variability is caused by sampling uncertainty only) . Hosking and Wallis ( 1993) also propo sed a goodness-of-fit measure based on L-moment statistics. This measure, however, spec ifically applies to 3-parameter distribut ions for modelling annual maximum ser ies. For discriminating between the EXP distribution and the different 2-parameter distributions in PDS modelling, the goodne ss-of-fit measure has been refo rmulated. For the EXP distribution the measure is based on the distance between the regional average L-CY estimate (record-length-weighted average) and the theoretic al L-CY (= 0.5) . To quantify the significance of this distance, it is related to the sampling uncertainty of the regional L-CV estimate determin ed on the basis of Monte Carlo simulations. Similarly, for 2-parameter distributions the goodne ssof-fit measure is based on the distance between the regional L-CS estimate and the theoretical counterpart, related to the sampling uncertainty of the L-CS estimate. The bias correction used in Hosking and Walli s' (1993) goodness-of-fit measure is not applied since L-CY and L-CS esti mates have negligible biases. Table I . Heterogeneity measure Hand goodness-of-fit measure G for the gamma (GAM), Weibull (WEI), log-normal (LN), generali sed Pareto (GP), and exponential (EXP ) distributions. Shading for I < H < 2 indicate regions that are possibly heterogeneous. Shading for IG I < 1.96 indicates that the distribution cannot be rejected at a 5% level of significance Duration
H
10 min 30 min 60 min 3h 6h 12 h 24 h 48 h 48h (+23127 )
0.5 -1.2 0.2 -0.8 - 1.7 -0.9 -1.2 1.2 0.4
GG.Ht
GW fJ
Gt.N
GGt'
Gsxr
-1.7 0.5 0.2 -1.6 -1.6 -1.2 - 1.0 -0.7 -0.8
7.6 9.2 8.7 6.4 6.4 6.2 6.6 7.8 7.7
-0.7 2.6 1.9 -0.5 0.6 0.4 0.9
-2.7 -6.0 -4.7 -3.2 -6.2 -4.5 -5.6 -4.9 -5.3
1.1 1.1
Th e heterogeneity and goodne ss-of-fit measure s for the analy sed intensities are shown in Table I. The region can be considered acceptably homogeneous if the heterogeneity measure H < I, possibly heterogeneous if I < H < 2, and defin itely heterogeneous if H > 2. Thus, for intensities with duration s d :::; 24 hours, the region can be conside red homogeneous. For d = 48 hours, the region is possibly heterogeneous. In this case, howe ver , one of the statio ns (no. 23127 ) is sign ificantly different from the group as a whole, and by excluding this station the remaining 40 stations form a homogeneous group . Hence, in general it can be concluded that ext reme rainfall intensities are homogeneous with respect to second and higher order moment s. The goodne ss-of-fit measure G is approximately normally distributed, and hence the acceptance of a hypothesised distribution can be evalu ated by comparing G with quantiles in the standard normal distribution. The LN and EXP distributions are rejected for all 8 analysed variable s, the GAM distribution is rejected for 6 variables, whereas the WEI and GP distribution s are generally accepted (see Table 1). Analy sis of the Lmoment diagrams reveals that the GP distribution generally should be preferred, since the cloud of Lmoment estimates in the diagrams is overall better described by the GP theoretical line (see Figure 1 as an example). Thu s, it is concluded that extreme rainfall intens ities can be described by a parent GP distribution. For the analysed variables the regional average L-CY and L-CS estimates are shown in Figure 2. For all durations. the regional parent is a heavy tailed distribut ion. correspond ing to a negative shape parameter in the GP distribution .
Regional estimation of extreme precipitation
0.56
•
We ighted average
/
- - GP
/' 30min . :
.
·· ··· · WEI
0.54
•
.. /
- - - LN - . . - GAM
0.55
'
EXP
/
60 min • . . .
> o ...:.
0.53
/
/
0.52
33
/
'
• 48 h /
24 h
• • 12 h
. :'
. :
0.51 0.50 / /
0.49 0.32
0.34
0.36
0.38
0.40
0.42
0.44
L-skewness Figure 2. Regional weighted average L-momenl statistics for PDS of intensities with different durations compared to the theoretical relationships for a number of candidate distributions.
REGRESSION ANALYSIS To investigate the regional variability of all PDS parameters, a regression analysis is carried out. This analysis has a twofold purpose . First, the regression model is applied for assessment of the regional heterogeneity of the PDS parameters. Secondly, for parameters that show a significant regional variability, the regression model is adopted to evaluate the potential of describing the variability from physiographic and climatic characteristics. The regression model is based on a generalised least squares (GLS) estimation method that explicitly accounts for sampling uncertainty and intersite dependence. Denote 8 j by an estimate of a PDS parameter at station no. i. The following linear relationship is considered
IJ, =Po+ fpkA"
TE, + 0,
(2)
,
where A jk are the climatic and physiograph ic characteristics, Pk are the regression parameters, £ j is a random sampling error, and OJ is the residual model error. To evaluate (2), the covariance structure of the sampling errors must be known. The sampling error variances of the PDS parameters can be determined using asymptotic theory or Monte Carlo simulations (Madsen and Rosbjerg, 1997a, b). In Mikkelsen et al. (1996) a method is described for estimation of the intersite correlation of, respectively, the annual number of exceedances and the concurrent exceedance magnitudes in the PDS. Due to the usually large sampling uncertaint ies of the correlation coefficient estimates, a smoothing is performed relating the intersite correlation to the distance between stations. For the Poisson parameter, the intersite correlation showed no significant spatial structure and a constant value of the correlat ion coefficient was applied in the regress ion analysis . For the exceedance magnitudes, a significant spatial correlation structure was observed. In this case the intersite correlation decreases for increasing distance between stations . Moreover, the correlation structure was seen to depend strongly on the considered duration, the correlation being larger for larger durations. This is due to the fact that extreme intensities for large durations are mainly caused by moving frontal rain systems with a large spatial extent, whereas extreme intensities for small durations are caused by convective rain cells with a limited spatial extent.
34
H. MADSEN et al,
Having quantified the covariance structure of the sampling errors, the GLS regression parameters and the residual model error variance can be estimated. The resulting GLS model provides an estimateof the PDS parameter as well as an estimate of the associated uncertainty. This uncertainty is a combination of, respectively, sampling errors corrected for intersite correlation and residual model error (see details in Madsen and Rosbjerg (l997b». A special case of the regression model, referred to as the regional meanmodel, is obtained by including only the intercept ~o in (2). The regional mean model is applied for evaluating the regional heterogeneity of the PDS parameter. If the regression analysis provides an estimate of the residual model error variance equal to zero, the observed regional variability can be explained purelyby sampling errors,and hence the regioncan be considered homogeneous. On the other hand. if the estimate of the residual model error variance is larger than zero, the regional variability is significant. and the full regression model (2) is adopted for modelling this variability. In the regression analyses the following climatic and physiographic characteristics were considered: meanannualrainfall, altitude, distance fromthe coast, and geographical position. For the Poisson parameter, the GLS regression analysis revealed a significant regional variation for all variables. A significant part of this variability can be explained by the mean annual rainfall (MAR). In this case, the Poisson parameter is an increasing function of MAR, i.e, the largerMAR the more extremeevents are observed. For large durations, MAR explains about 70-80% of the regional variability of the Poisson parameter, whereas for smallerdurations only about 15-30% is explained. Inclusion of other characteristics in the regression equation did not improve the regional modelling. For the mean value of the exceedances, the region can be considered homogeneous (residual model error variance equal to zero) for smalldurations d S I hour. For largerdurations, a significant metropolitan effect was observed, the stations in the Copenhagen area having significantly largerextreme intensities than in the rest of the country. This effect was also observed by Ambjerg-Nielsen et al. (1996) and can probably be ascribed to smog and local heating. In this case the GLS regional mean model is applied to stations in the two subregions (I) the Copenhagen area, and (2) the rest of the country. The mean valueof the exceedances in subregion (2) can be considered homogeneous, whereas subregion (I) has a significant variability. Furthermore. since the stations in subregion (I) are situated in a small area, the intersite correlation is large, and hencethe resulting uncertainty is significantly largerin subregion (I) than in subregion (2). Inclusion of climatic and physiographic characteristics in the regression equation. for subregion (I) was not able to explain the regional variability. For second and higher order moments, the regression analysis was generally consistent with the L-moment analysis described above; that is, the region can be considered homogenous for all variables except for duration d =48 hours. The considered climatic and physiographic characteristics werenot able to explainthe regional variability of this variable, and hencethe regional meanmodel was applied for all variables. ESTIMAnON OF INTENSITY-DURATION-FREQUENCY CURVES For GP distributed exceedances, the regional T-year eventestimate is givenby, cr. (I) , ft=
I
-- - 2
(3)
t1
where A. is the regional estimateof the Poisson parameter, I! is the regional estimate of the mean value,and 1C is the estimateof the shape parameter determined on the basisof the regional estimate of L-CV. An estimate of the uncertainty of the regional T-yeareventestimate is approximately given by
(4) where Ai are functionsof the estimated PDS parameters (see Madsen and Rosbjerg (I 997a,b) for details).
Regional estimation of extreme precipitation
35
Basedon (3)-(4) the T-yearevent and the associated uncertainty can be estimatedat an arbitrary location in the region. The procedure can be summarised as follows. 1. 2.
3. 4.
Based on an estimate of MAR at the site in question, the Poisson parameter and the associated uncertainty are estimated fromthe regression equation. For intensities withsmalldurations, d S I hour.the meanvalueof the exceedances and the associated uncertainty are obtained from the regional mean model covering the whole region. For larger durations, the regional mean model for one of the two subregions is adopted, depending on the location of the site in question (in the Copenhagen areaor not). The regional estimateof L-CVand associated uncertainty is obtained from the regional meanmodel. The regional PDS parameter estimates andestimates of the uncertainties are inserted in (3)-(4).
An example of an estimated intensity-duration-frequency curve is shown in Figure 3.
T-year event estimate
-
- _••• -68%
~"
<~' ' ' ' ~I:-':I-$-;.+L-+:-+,-<-j+! ----+--1 : -
confidence limits
I ,:
I
10 ""' _
.
""""'-
0.1 +--
'. '- .~
~.
:
'~, i l
Ii : I:
I
i
I
I:
I ; ! I I ij , J I i ! ! I I' : I I ill ; I ! I II : : I I ' I , I --l----:..---;.............-+--~----+---+--~-'-+-<'-'-- --f-- __i__-+--+--l-~
10
100
1000
10000
Duration [min) Figure3. Intensity-duration-frequency curvefor a location in the Copenhagen area witha meanannual rainfallof 600 mm.The 68%confidence interval corresponds to ± one standard deviation of the regionalT-yearevent estimate. Return periodTis givenin [yearsI.
CONCLUSIONS A regional model for estimation of extreme precipitation has been introduced. The model consists of the following elements: (I) the PDS model for definition of extreme events, (2) L-moment analysis for identification of a regional parent distribution and preliminary assessment of regional homogeneity, and (3) GLS regression analysis of the PDS parameters for quantifying the regional heterogeneity and modelling the variability fromclimaticand physiographic characteristics. The regional model has been applied to rainfall data from a high resolution rain gauge network in Denmark. The L-moment analysis revealed that for all variables the region can be considered homogenous with respect to second and higher order moments of the exceedance magnitudes. Furthermore, the generalised Pareto distribution was found appropriate as a regional parent. The GLS regression analysis revealed that a largepart of the regional variability of the mean annual number of threshold exceedances in the PDS can be explained by the mean annual rainfall; that is, for larger mean annual rainfall the moreextremeevents are observed. The mean valueof the exceedance magnitudes can be assumed constant in the region for durations d S I hour. For larger durations, a significant metropolitan
36
H. MADSEN et al.
effect was observed, the mean intensities in the Copenhagen area beingsignificantly larger than in the rest of the country. For secondand higherorder moments a regional mean model was adopted. Based on the regional GLS models for the PDS parameters, an estimate of the T-year event and the associated uncertainty can be found at an arbitrary site in the region.
REFERENCES Ambjerg-Nielsen. K.• Harremoes, P. and Spliid, H. (1994). Non-parametric statistics on extremerainfall. Nordic Hydrol.• 15(4), 261-266. Ambjerg-Nielsen. K.• Harremoes, P. and Spliid, H. (1996). Interpretation of regional variation of extreme values of point precipitation in Denmark. Atmos. Res., 42, 99-111. Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. RoyalStatist. Soc. B. 52(1), 10.5-124. Hosking. J. R. M. and Wallis. J. R. (1993). Some statistics useful in regional frequency analysis. Water Resour. Rtf.• 29(2),211281.Correction. Waltr Resour. Rtf.• 31(1),2.51,199.5. Jergensen, H. K., Rosenern, 5.• Madsen, H. and Mikkelsen, P. S. (1998) Quality control of rain data used for urban runoff systems. Wat. Sci. Tech., 37(11),this issue. Madsen. H.• Rosbjerg, D. and Harremoes, P. (1994). PDS-modelling and regional Bayesian estimation of extreme rainfalls. NordicHydrol.• 25(4),279-300. Madsen, H. and Rosbjerg, D. (1997a). The partialduration series methodin regional index-flood modeling. Water Resour. Res.• 33(4),137-146. Madsen. H. and Rosbjerg, D. (1991b). Generalized least squaresand empirical Bayesestimation in regional partialdurationseries index-flood modeling. WaterResour. Res.• 33(4), 711-781. Mikkelsen. P. 5., Madsen, H., Rosbjerg. D. and Harremoes, P. (1996). Properties of extremepoint rainfall III: Identification of spatial inter-site correlation structure. Atmos. Res.• 40. 71-98. Mikkelsen, P. S., Madsen, H., Ambjerg-Niclsen, K., Jargensen, H. K., Rosbjerg, D. and Harremoes, P. (1998). A rationale for using local and regional point rainfall data for design and analysis of urban storm drainage systems. Wat. Sci. Tech., 37(11),this issue.
Pergamon
Wal. Sci Tech. Vol. 37, No. II. pp. 29-36,1998.
IAWQ = 1998 Published byElsevier Science Ltd. Printed inGreat Britain.All righu reserved
PH: 80273-1223(98)00313-8
0273-1223198 $19'00 + 0'00
ESTIMATION OF REGIONAL INTENSITY-DURATION-FREQUENCY CURVES FOR EXTREME PRECIPITATION H. Madsen*, P. S. Mikkelsen**, D. Rosbjerg* and P. Harremoess" * Department ofHydrodynamics and Water Resources, Technical University of Denmark; Building 115, DK-2800Lyngby, Denmark ** Department ofEnvironmental Science and Engineering, Technical University of Denmark; Building 1 J5, DK-28oo Lyngby, Denmark
ABSTRACf Regional estimation of extreme precipitation from a high resolution rain gauge network in Denmark is considered. The appliedextremevalue modelis basedon the partialduration series(POS)approachin which all events above a certain threshold level are model1ed. For a preliminary assessment of regional homogeneity and identification of a proper regional distribution Lmoment analysis is applied. To analyse the regional variability in more detail, a generalised least squares regression analysis is carried out that relatesthe PDS model parameters to climaticand physiographic characteristics. The analysis revealsthat the mean annual number of extreme events varies significantly within the region. and a large part of this variability can be explained by the mean annual rainfal1. The mean value of the exceedance magnitudes can be assumed constant for intensities with durations less than one hour. For larger durations a pronounced metropolitan effect is evident, the mean intensities in the Copenhagen area being significantly larger than found in the rest of the country. With respect to second and higher order moments the region can be considered homogeneous for intensities with durations less than 24 hours. A regional parent distribution is identified as the generalised Pareto distribution. ~ 1998 Published by Elsevier Science Ltd. AI1 rights reserved
KEYWORDS Extreme rainfalls; partial duration series; regional estimation; L-moments; regression analysis INTRODUCfION In 1979 a new system of high resolution automatic rain gauges with specific relevance for urban hydrologic applications was introduced in Denmark. At present, 90 stations have been connected to the system. and the longest records consist of more than 18 years of data. Details of the measurement system and the initial quality control of the data can be found in Jergensen et at. (1998). Different statistical models have been applied for analysing the data. Ambjerg-Nielsen et at. (1994) elaborated the traditional non-parametric approach by using a bootstrap method to quantify the estimat ion uncertainty. Madsen et al. (1994) and Mikkelsen et al. (1996) applied a parametric model based on the 29
30
H. MADSEN et al.
partial duration series (PDS)approach. The preliminary studies revealed a significant geographical variation of extreme rainfalls. Assuming this variation to be purely stochastic. Madsen et al. (1994) formulated a regional model that provides an estimate of the T-year event as well as an estimate of the associated uncertainty which includes both sampling uncertainty and regional variability. Arnbjerg-Nielsen et al, (1996) analysed the potential of including physiographic and climatic information to describe the regional variability. They found that such covariate information was able to sufficiently describe the variability of extremeevents only for very small return periods. In 1996the DanishWater Pollution ControlCommittee initiated a projectto establishing new guidelines for use of historical rainfall data from the automatic rain gauge system in urban storm drainage analysis (Mikkelsen et al.; 1998). An important element in this project has been to identify and model the regional variability of extreme rainfall statistics. The basis for the modelling has been the regional PDS model introduced by Madsen et 01. (1994) and Mikkelsen et al. (1996). including recent developments by Madsen and Rosbjerg (199780 b). The present paper describes the regional modelling approach and summarises the main results.
EXTREME VALUE MODEL For the regional analysis. 41 stationswith morethan 10yearsof data havebeen included. corresponding to a total of about 650 station years. The raw data consistof one-minute rain intensities of individual events.The preliminary separation of rain events is defined as periods exceeding one hour without precipitation. The maximum rain intensity of individual rain events (hereafter for simplicity denoted intensity) for given durationis definedas follows. For durations less than or equal to one hour. the preliminary separation of rain events is used to define the intensity. For largerdurations. the definition of independent rain events is related to the duration. In this case the intensity is calculated for rain events that are separated by dry periods of lengthslargerthan the consideredduration.
=
The intensity for given duration is described by a stochastic variable Z with observations ('jo i 1,2...,m) where m is the total number of rain events in the historical time series. The first step in the modelling of extreme rainfalls is selection of the extreme population. In the PDS model (also denoted the peak over threshold (Pan model) all events above a predefined threshold level '0 are considered. i.e, the PDS is definedas (X Z - zOo Z> zo I withobservations {Xi = ' j - ZOo i = 1.2....,n,J where n (<< m) is the PDS sample size.
=
In PDS modelling it is generally assumed that the occurrence of exceedances can be described by a Poisson process with constant or one-year periodic intensity. Hence. the number of threshold exceedances can be described by a Poissondistribution with intensity. A., that equals the expectednumberof annualexceedances. For modelling of the exceedance magnitudes a statistical distribution is adopted. In the basic PDS model a simple one-parameter exponential distribution was applied. Differenttwo-parameter distributions have been proposed. including the gamma. Weibull, Log-normal. and generalised Pareto distributions. Denoting by F(x) the cumulative distribution function of the exceedances, the T-year event is given by
(l)
An estimateof the T-yearevent is obtainedfrom (I) by inserting estimates of the PDS model parameters. An estimateof the Poisson parameteris given as the averagenumberof observedexceedances per year. i.e, A. = nit. For estimation of the parameters of F(x). the method of L-moments is adopted. i.e. the parameters are estimated from the sample mean value and higher order sample L-moments (see description of L-moments below). Partial duration series for the 41 stationswere defined by using the same threshold level at all stations.The threshold level was chosen on the basis of a preliminary sensitivity analysis of regional average extreme
Regional estimation of extreme precipitation
31
valuecharacteristics as a function of the threshold level,implying a regional average numberof exceedances per year in the range 2.5-3.2 for the analysed variables.
L-MOMENT ANALYSIS Evaluation of the regional variability of extreme rainfall statistics is based on the PDS parameters defined above; that is, the Poisson parameter, the meanvalue of the exceedances, and higher order moments. In this section homogeneity of higher order moments and the closely related selection of a regional parent distribution is described using the theoryof L-moments. In the following section a general methodology for assessment of regional homogeneity and for estimation of regional statistics for all PDS parameters is described. L-moments are defined as linear combinations of expected values of order statistics (Hosking, 1990). The first L-moment 0"1) is the mean value identical to the first ordinary moment: the secondL-moment O"V is a measure of dispersion analogous to the variance; the third L-moment (A3> is a measure of skewness; and the fourth L-moment (A,,> is a measure of kurtosis. L-moment ratiosare definedas, respectively, L-coefficient of variation: L-CV = A,P'''I' L-skewness: L-CS = AfA.2, and L-kurtosis: L-KUR = A4~' For estimation of Lmoments, unbiased estimates of probability weighted moments are applied. L-moment ratio estimates are nearly unbiased and are relatively insensitive to outliers; properties that are extremely important in regional studies. For a visual judgement of regional homogeneity and identification of a proper regional distribution, an Lmoment diagram is constructed where L-moment ratio estimates for the regional stations are compared to the theoretical relationships for a number of candidate distributions (see Figure I). The dispersion of the sample points in this diagram is a measure of regional heterogeneity. In addition, the cloud of points indicates which distribution is the most appropriate. Since L-moment ratio estimates are nearly unbiased, half of the samplepointsare expectedto lie abovethe theoretical curve and half to lie below. The following candidate distributions are considered: exponential (EXP), gamma (GAM), Weibull (WEI), log-normal (LN),and generalised Pareto(GP). 0.70
e
At-site estimates Weighted average
•
--GP 0.60
- - - LN ----GAM ~----'WEI
•
~
-'
EXP
0.50
·it·.'
";;'0
.......'/ . r
0.40 .'
..
.'
/ /
r
0.30 +-:.,...,;.,.......-,.;:....-.............,r'-r--.................-....................,..-.-............-.......--..-.--. 0.30 0.40 0.20 0.50 0.60 0.10 L-5kewness Figure I. L-moment diagram for the IG-minutes rainintensity. At-site L-moment ratioestimates from41 stanons and the record-length-weighted average compared to thetheoretical relationships for a number of candidate distributions.
For a more formal evaluation of regional homogeneity Hosking and Wallis(1993) proposeda heterogeneity measure where the observed dispersion of L-CV is compared to the expecteddispersion in a homogenous
32
H. MADSEN et al.
region (i.e. where the variability is caused by sampling uncertainty only) . Hosking and Wallis ( 1993) also propo sed a goodness-of-fit measure based on L-moment statistics. This measure, however, spec ifically applies to 3-parameter distribut ions for modelling annual maximum ser ies. For discriminating between the EXP distribution and the different 2-parameter distributions in PDS modelling, the goodne ss-of-fit measure has been refo rmulated. For the EXP distribution the measure is based on the distance between the regional average L-CY estimate (record-length-weighted average) and the theoretic al L-CY (= 0.5) . To quantify the significance of this distance, it is related to the sampling uncertainty of the regional L-CV estimate determin ed on the basis of Monte Carlo simulations. Similarly, for 2-parameter distributions the goodne ssof-fit measure is based on the distance between the regional L-CS estimate and the theoretical counterpart, related to the sampling uncertainty of the L-CS estimate. The bias correction used in Hosking and Walli s' (1993) goodness-of-fit measure is not applied since L-CY and L-CS esti mates have negligible biases. Table I . Heterogeneity measure Hand goodness-of-fit measure G for the gamma (GAM), Weibull (WEI), log-normal (LN), generali sed Pareto (GP), and exponential (EXP ) distributions. Shading for I < H < 2 indicate regions that are possibly heterogeneous. Shading for IG I < 1.96 indicates that the distribution cannot be rejected at a 5% level of significance Duration
H
10 min 30 min 60 min 3h 6h 12 h 24 h 48 h 48h (+23127 )
0.5 -1.2 0.2 -0.8 - 1.7 -0.9 -1.2 1.2 0.4
GG.Ht
GW fJ
Gt.N
GGt'
Gsxr
-1.7 0.5 0.2 -1.6 -1.6 -1.2 - 1.0 -0.7 -0.8
7.6 9.2 8.7 6.4 6.4 6.2 6.6 7.8 7.7
-0.7 2.6 1.9 -0.5 0.6 0.4 0.9
-2.7 -6.0 -4.7 -3.2 -6.2 -4.5 -5.6 -4.9 -5.3
1.1 1.1
Th e heterogeneity and goodne ss-of-fit measure s for the analy sed intensities are shown in Table I. The region can be considered acceptably homogeneous if the heterogeneity measure H < I, possibly heterogeneous if I < H < 2, and defin itely heterogeneous if H > 2. Thus, for intensities with duration s d :::; 24 hours, the region can be conside red homogeneous. For d = 48 hours, the region is possibly heterogeneous. In this case, howe ver , one of the statio ns (no. 23127 ) is sign ificantly different from the group as a whole, and by excluding this station the remaining 40 stations form a homogeneous group . Hence, in general it can be concluded that ext reme rainfall intensities are homogeneous with respect to second and higher order moment s. The goodne ss-of-fit measure G is approximately normally distributed, and hence the acceptance of a hypothesised distribution can be evalu ated by comparing G with quantiles in the standard normal distribution. The LN and EXP distributions are rejected for all 8 analysed variable s, the GAM distribution is rejected for 6 variables, whereas the WEI and GP distribution s are generally accepted (see Table 1). Analy sis of the Lmoment diagrams reveals that the GP distribution generally should be preferred, since the cloud of Lmoment estimates in the diagrams is overall better described by the GP theoretical line (see Figure 1 as an example). Thu s, it is concluded that extreme rainfall intens ities can be described by a parent GP distribution. For the analysed variables the regional average L-CY and L-CS estimates are shown in Figure 2. For all durations. the regional parent is a heavy tailed distribut ion. correspond ing to a negative shape parameter in the GP distribution .
Regional estimation of extreme precipitation
0.56
•
We ighted average
/
- - GP
/' 30min . :
.
·· ··· · WEI
0.54
•
.. /
- - - LN - . . - GAM
0.55
'
EXP
/
60 min • . . .
> o ...:.
0.53
/
/
0.52
33
/
'
• 48 h /
24 h
• • 12 h
. :'
. :
0.51 0.50 / /
0.49 0.32
0.34
0.36
0.38
0.40
0.42
0.44
L-skewness Figure 2. Regional weighted average L-momenl statistics for PDS of intensities with different durations compared to the theoretical relationships for a number of candidate distributions.
REGRESSION ANALYSIS To investigate the regional variability of all PDS parameters, a regression analysis is carried out. This analysis has a twofold purpose . First, the regression model is applied for assessment of the regional heterogeneity of the PDS parameters. Secondly, for parameters that show a significant regional variability, the regression model is adopted to evaluate the potential of describing the variability from physiographic and climatic characteristics. The regression model is based on a generalised least squares (GLS) estimation method that explicitly accounts for sampling uncertainty and intersite dependence. Denote 8 j by an estimate of a PDS parameter at station no. i. The following linear relationship is considered
IJ, =Po+ fpkA"
TE, + 0,
(2)
,
where A jk are the climatic and physiograph ic characteristics, Pk are the regression parameters, £ j is a random sampling error, and OJ is the residual model error. To evaluate (2), the covariance structure of the sampling errors must be known. The sampling error variances of the PDS parameters can be determined using asymptotic theory or Monte Carlo simulations (Madsen and Rosbjerg, 1997a, b). In Mikkelsen et al. (1996) a method is described for estimation of the intersite correlation of, respectively, the annual number of exceedances and the concurrent exceedance magnitudes in the PDS. Due to the usually large sampling uncertaint ies of the correlation coefficient estimates, a smoothing is performed relating the intersite correlation to the distance between stations. For the Poisson parameter, the intersite correlation showed no significant spatial structure and a constant value of the correlat ion coefficient was applied in the regress ion analysis . For the exceedance magnitudes, a significant spatial correlation structure was observed. In this case the intersite correlation decreases for increasing distance between stations . Moreover, the correlation structure was seen to depend strongly on the considered duration, the correlation being larger for larger durations. This is due to the fact that extreme intensities for large durations are mainly caused by moving frontal rain systems with a large spatial extent, whereas extreme intensities for small durations are caused by convective rain cells with a limited spatial extent.
34
H. MADSEN et al,
Having quantified the covariance structure of the sampling errors, the GLS regression parameters and the residual model error variance can be estimated. The resulting GLS model provides an estimateof the PDS parameter as well as an estimate of the associated uncertainty. This uncertainty is a combination of, respectively, sampling errors corrected for intersite correlation and residual model error (see details in Madsen and Rosbjerg (l997b». A special case of the regression model, referred to as the regional meanmodel, is obtained by including only the intercept ~o in (2). The regional mean model is applied for evaluating the regional heterogeneity of the PDS parameter. If the regression analysis provides an estimate of the residual model error variance equal to zero, the observed regional variability can be explained purelyby sampling errors,and hence the regioncan be considered homogeneous. On the other hand. if the estimate of the residual model error variance is larger than zero, the regional variability is significant. and the full regression model (2) is adopted for modelling this variability. In the regression analyses the following climatic and physiographic characteristics were considered: meanannualrainfall, altitude, distance fromthe coast, and geographical position. For the Poisson parameter, the GLS regression analysis revealed a significant regional variation for all variables. A significant part of this variability can be explained by the mean annual rainfall (MAR). In this case, the Poisson parameter is an increasing function of MAR, i.e, the largerMAR the more extremeevents are observed. For large durations, MAR explains about 70-80% of the regional variability of the Poisson parameter, whereas for smallerdurations only about 15-30% is explained. Inclusion of other characteristics in the regression equation did not improve the regional modelling. For the mean value of the exceedances, the region can be considered homogeneous (residual model error variance equal to zero) for smalldurations d S I hour. For largerdurations, a significant metropolitan effect was observed, the stations in the Copenhagen area having significantly largerextreme intensities than in the rest of the country. This effect was also observed by Ambjerg-Nielsen et al. (1996) and can probably be ascribed to smog and local heating. In this case the GLS regional mean model is applied to stations in the two subregions (I) the Copenhagen area, and (2) the rest of the country. The mean valueof the exceedances in subregion (2) can be considered homogeneous, whereas subregion (I) has a significant variability. Furthermore. since the stations in subregion (I) are situated in a small area, the intersite correlation is large, and hencethe resulting uncertainty is significantly largerin subregion (I) than in subregion (2). Inclusion of climatic and physiographic characteristics in the regression equation. for subregion (I) was not able to explain the regional variability. For second and higher order moments, the regression analysis was generally consistent with the L-moment analysis described above; that is, the region can be considered homogenous for all variables except for duration d =48 hours. The considered climatic and physiographic characteristics werenot able to explainthe regional variability of this variable, and hencethe regional meanmodel was applied for all variables. ESTIMAnON OF INTENSITY-DURATION-FREQUENCY CURVES For GP distributed exceedances, the regional T-year eventestimate is givenby, cr. (I) , ft=
I
-- - 2
(3)
t1
where A. is the regional estimateof the Poisson parameter, I! is the regional estimate of the mean value,and 1C is the estimateof the shape parameter determined on the basisof the regional estimate of L-CV. An estimate of the uncertainty of the regional T-yeareventestimate is approximately given by
(4) where Ai are functionsof the estimated PDS parameters (see Madsen and Rosbjerg (I 997a,b) for details).
Regional estimation of extreme precipitation
35
Basedon (3)-(4) the T-yearevent and the associated uncertainty can be estimatedat an arbitrary location in the region. The procedure can be summarised as follows. 1. 2.
3. 4.
Based on an estimate of MAR at the site in question, the Poisson parameter and the associated uncertainty are estimated fromthe regression equation. For intensities withsmalldurations, d S I hour.the meanvalueof the exceedances and the associated uncertainty are obtained from the regional mean model covering the whole region. For larger durations, the regional mean model for one of the two subregions is adopted, depending on the location of the site in question (in the Copenhagen areaor not). The regional estimateof L-CVand associated uncertainty is obtained from the regional meanmodel. The regional PDS parameter estimates andestimates of the uncertainties are inserted in (3)-(4).
An example of an estimated intensity-duration-frequency curve is shown in Figure 3.
T-year event estimate
-
- _••• -68%
~"
<~' ' ' ' ~I:-':I-$-;.+L-+:-+,-<-j+! ----+--1 : -
confidence limits
I ,:
I
10 ""' _
.
""""'-
0.1 +--
'. '- .~
~.
:
'~, i l
Ii : I:
I
i
I
I:
I ; ! I I ij , J I i ! ! I I' : I I ill ; I ! I II : : I I ' I , I --l----:..---;.............-+--~----+---+--~-'-+-<'-'-- --f-- __i__-+--+--l-~
10
100
1000
10000
Duration [min) Figure3. Intensity-duration-frequency curvefor a location in the Copenhagen area witha meanannual rainfallof 600 mm.The 68%confidence interval corresponds to ± one standard deviation of the regionalT-yearevent estimate. Return periodTis givenin [yearsI.
CONCLUSIONS A regional model for estimation of extreme precipitation has been introduced. The model consists of the following elements: (I) the PDS model for definition of extreme events, (2) L-moment analysis for identification of a regional parent distribution and preliminary assessment of regional homogeneity, and (3) GLS regression analysis of the PDS parameters for quantifying the regional heterogeneity and modelling the variability fromclimaticand physiographic characteristics. The regional model has been applied to rainfall data from a high resolution rain gauge network in Denmark. The L-moment analysis revealed that for all variables the region can be considered homogenous with respect to second and higher order moments of the exceedance magnitudes. Furthermore, the generalised Pareto distribution was found appropriate as a regional parent. The GLS regression analysis revealed that a largepart of the regional variability of the mean annual number of threshold exceedances in the PDS can be explained by the mean annual rainfall; that is, for larger mean annual rainfall the moreextremeevents are observed. The mean valueof the exceedance magnitudes can be assumed constant in the region for durations d S I hour. For larger durations, a significant metropolitan
36
H. MADSEN et al.
effect was observed, the mean intensities in the Copenhagen area beingsignificantly larger than in the rest of the country. For secondand higherorder moments a regional mean model was adopted. Based on the regional GLS models for the PDS parameters, an estimate of the T-year event and the associated uncertainty can be found at an arbitrary site in the region.
REFERENCES Ambjerg-Nielsen. K.• Harremoes, P. and Spliid, H. (1994). Non-parametric statistics on extremerainfall. Nordic Hydrol.• 15(4), 261-266. Ambjerg-Nielsen. K.• Harremoes, P. and Spliid, H. (1996). Interpretation of regional variation of extreme values of point precipitation in Denmark. Atmos. Res., 42, 99-111. Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. RoyalStatist. Soc. B. 52(1), 10.5-124. Hosking. J. R. M. and Wallis. J. R. (1993). Some statistics useful in regional frequency analysis. Water Resour. Rtf.• 29(2),211281.Correction. Waltr Resour. Rtf.• 31(1),2.51,199.5. Jergensen, H. K., Rosenern, 5.• Madsen, H. and Mikkelsen, P. S. (1998) Quality control of rain data used for urban runoff systems. Wat. Sci. Tech., 37(11),this issue. Madsen. H.• Rosbjerg, D. and Harremoes, P. (1994). PDS-modelling and regional Bayesian estimation of extreme rainfalls. NordicHydrol.• 25(4),279-300. Madsen, H. and Rosbjerg, D. (1997a). The partialduration series methodin regional index-flood modeling. Water Resour. Res.• 33(4),137-146. Madsen. H. and Rosbjerg, D. (1991b). Generalized least squaresand empirical Bayesestimation in regional partialdurationseries index-flood modeling. WaterResour. Res.• 33(4), 711-781. Mikkelsen. P. 5., Madsen, H., Rosbjerg. D. and Harremoes, P. (1996). Properties of extremepoint rainfall III: Identification of spatial inter-site correlation structure. Atmos. Res.• 40. 71-98. Mikkelsen, P. S., Madsen, H., Ambjerg-Niclsen, K., Jargensen, H. K., Rosbjerg, D. and Harremoes, P. (1998). A rationale for using local and regional point rainfall data for design and analysis of urban storm drainage systems. Wat. Sci. Tech., 37(11),this issue.