Rainfall generator for the Meuse basin

Transcription

1 KNMI publication; IV Rainall generator or the Meuse basin Description o year simulations R. Leander and T.A. Buishand De Bilt, 2008

2 KNMI publication = KNMI publicatie; IV De Bilt, 2008 PO Box AE De Bilt Wilhelminalaan 10 De Bilt The Netherlands Telephone +31(0) Teleax +31(0) Authors: Leander, R. Buishand, T.A. KNMI, De Bilt. All rights reserved. No part o this publication may be reproduced, stored in retrieval systems, or transmitted, in any orm or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission in writing rom the publisher.....

3 KNMI-publication; 196-IV Rainall generator or the Meuse basin Description o year simulations R. Leander, T.A. Buishand De Bilt, 2008

4 KNMI-publication; 196-IV De Bilt, 2008 PO Box 201, 3730 AE De Bilt The Netherlands Wilhelminalaan 10 Telephone Teleax Author(s): R. Leander, T.A. Buishand c KNMI, De Bilt. All rights reserved. No part o this publication may be reproduced, stored in retrieval systems, or transmitted, in any orm or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission in writing rom the publisher.

5 Rainall generator or the Meuse basin Description o year simulations R. Leander, T.A. Buishand KNMI-publication; 196-IV Work perormed under contracts RI-2726A and RI-4750 to Ministry o Transport, Public Works and Water Management, Institute or Inland Water Management and Waste Water Treatment RIZA, P.O. Box 17, 8200 AA Lelystad (The Netherlands). Telephone ; Teleax

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7 Summary : The sensitivity o simulated maxima o 10-day precipitation to the set o historical base years used or resampling has been assessed. Eight subsets o 33 years were extracted rom the base period (with the exclusion o 1940). The raction o days with 10 mm o precipitation or more in the winter hal-year and the mean winter precipitation in these subsets were both either high or low, near the edge o the 95%-conidence region o their expected values. Furthermore, the subsets diered with respect to the inclusion or exclusion o the years 1995 or With each subset a year simulation was perormed and the quantiles o the simulated 10-day winter maxima were considered. In addition to this, a year simulation was run in which the entire base period was used. This simulation serves as a reerence and is assumed to be comparable with earlier simulations. It was ound that or return periods up to ive years the inluence o 1995 or 1984 on the 10-day maxima was negligible compared to that o the mean winter precipitation and the raction o winter days with 10 mm o precipitation or more. For longer return periods, especially the inluence o the winter o 1995 was noticeable. The 10-day precipitation with a return period o 1250 year in the eight simulations spanned a range rom 165 to 210 mm ( 24%). The inluence o the repetition o historical days in the simulation was investigated, using a modiied algorithm in which such repetition was excluded. This resulted in slightly lower 10-day maxima (or a return period o 1250 years, the dierences are within 5%). 1 Introduction 5 2 Setup o the simulations Construction o subsets o the historical period Selection o eight subsets or simulation Results 8 4 Conclusions 13 Acknowledgements 13 Reerences 14 Appendix A Conidence region or the mean o a bivariate normal variate 15

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9 1 Introduction The GRaDE instrument (Generator o Rainall and Discharge Extremes), currently under development or the river Meuse, aims at the estimation o the quantiles o extreme river discharges or return periods in the order o 1000 years. This is achieved by simulating long sequences o daily discharge. The GRaDE instrument comprises a weather generator, producing long sequences o daily precipitation and temperature or the dierent subbasins and a rainall-runo model or the entire basin. Both components have a number o uncertain elements aecting the reliability o the estimated discharge quantiles. This study ocusses on the weather generator, which is based on k-nn resampling o historical daily records o precipitation and temperature. An important source o uncertainty is related to the selection o the historical period rom which data are resampled. It was suspected that the simulation o extreme multi-day amounts was sensitive to certain historical years containing consecutive periods o very wet days. The inluence o these years on the generated precipitation sequences could in turn have consequences or the simulation o extreme discharges. Thereore, year simulations were perormed using dierent subsets o the historical data or resampling. Section 2 o this report describes the setup and the choice o the simulations. In Section 3 the results are discussed. A summary is presented in the inal Section 4. 5

10 2 Setup o the simulations The simulations conducted in this study are essentially based on the same model and data as those discussed in Leander and Buishand (2004) and Leander et al. (2005). In these studies two dierent base periods or resampling were considered. The longer o the two spanned the years , (except or the year 1940 or which insuicient data were available) In the simulations based on this period the resampling algorithm was driven by data rom a set o stations in the basin (7 or precipitation and 2 or temperature). In a second stage the areal precipitation o the subbasins was resampled rom the available records o areal precipitation A secondary nearest-neighbour step was used to substitute resampled days rom beore 1961 by their closest analogue in the period rom 1961 onward. The analogy between historical days was based on the station data. The same procedure was ollowed or the temperature o 11 stations in and around the Meuse basin, using data or the period In the current study the same historical period ( ) is considered. For the purpose o hydrological modelling, the area-average temperature or each subbasin was derived rom the simulated station data by means o Thiessen interpolation. The simulated daily potential evapotranspiration (PET) or each subbasin was derived rom the simulated daily temperature using a linear relation between PET and daily temperature (Leander and Buishand, 2007). The secondary nearest-neighbour step was not only used or the simulation o new sequences, but also or the artiicial extension o the sequence o historical basin-average precipitation backwards in time rom 1961 to 1930 (except or 1940). That constructed sequence is used to estimate two characteristics o winter precipitation and or comparison o the simulated 10-day maxima. 2.1 Construction o subsets o the historical period The eect o the historical period was investigated by resampling rom subsets o the historical period. In the context o this study it was desirable to redeine a year as a 365-day period centred around the winter hal-year, because o the emphasis on extreme multiday precipitation in winter. In this deinition consecutive winter months are kept together within a year. Thence, year i consists o the last 182 days o the calendar year i 1 and the irst 183 days o calendar year i (enclosing the entire winter o year i). The historical period o the calendar years , except or 1940, thus contains 66 complete winters ( and ). From these, subsets o 33 (winter hal-)years were selected randomly without replacement, 14 rom the period and 19 rom the period This procedure was repeated many times, resulting in an ensemble o subsets o historical years. It must be noted that this ensemble does not represent all possible combinations o historical years, since the number o years rom beore and ater 1961 is ixed. These numbers are chosen such that the raction o years or which the secondary nearest-neighbour step has to be applied is the same as in the entire data set. This setup is similar to that in applications o the bootstrap to time series o unequal length (GREHYS, 1996). 6

11 2.2 Selection o eight subsets or simulation For each subset o 33 years in the ensemble, the raction o winter days with 10 mm o precipitation or more and the mean daily winter amount o precipitation P were determined, shown in Fig The dashed ellipse in this igure delineates a 95%-conidence region or the expectations o and P, derived rom the entire record o 66 years, under the assumption that the pairs (, P) have a bivariate normal distribution. The univariate normality o the individual variables was conirmed by Q-Q plots. Details on the construction o the conidence region can be ound in Appendix A. Based on earlier simulations, it was suspected that particular days in January 1995 had a considerable inluence on the simulated 10-day extremes (though this inluence did not appear in the modelled extreme discharges rom the simulations). Furthermore, it emerged rom the simulations without 1995 that the winter o 1984 has some inluence. The ensemble o subsets was thereore sorted into our groups. The subsets in group 1 included both 1995 and 1984, those in group 2 and 3 respectively contained either 1995 or 1984 and group 4, inally, consisted o all subsets which had neither 1995 nor From each group two subsets were selected, one with low values o and P (denoted as a ) and one with high values o and P (denoted as b ). The relevant locations in the (, P)-plane relative to the 95%-conidence region are shown in Fig The eight cases are clustered our by our near the edge o the region and will be reerred to as 1a, 1b, 2a,...,4b b 2.75 P [mm/day] a [%] Figure 2.1: Mean daily precipitation amount P versus raction o days with 10 mm o precipitation or more in the winter hal-year or all sets in the ensemble (black dots). The eight sets o historical years selected or simulation are marked with boxes. The dashed ellipse marks the 95%-region or the expectations o and P, based on the extended 66-year historical record. 7

12 3 Results With each o the eight selected subsets o historical years, a year simulation was perormed. In addition to this, a year simulation based on the complete historical period was available, which is used here as a reerence. This simulation is expected to be comparable to the earlier 3000-year Sim30 simulations with a 121-day moving window (Aalders et al., 2004). Figure 3.1 compares the Gumbel plots o the simulated winter maxima o 10-day basinaverage precipitation or the dierent cases with the Gumbel plot or the extended historical series. For return periods up to ive years, the plots or the simulations with the cases b (i.e. those located in the upper right extent o the ellipse in Fig. 2.1) are close together and are systematically above the historical quantiles. For those o the cases a (i.e. in the lower let extent o the ellipse) the opposite holds. The dierence between a and b is roughly in the order o 10%. In this regime the winters o 1984 and 1995 are ar less inluential on the simulated quantiles than the mean precipitation and the occurrence o daily precipitation amounts 10 mm. For longer return periods the inluence o 1995 (and to a lesser extent 1984) is noticeable as the plots o the cases a as well as those o the cases b start to diverge (though all the b-cases result in higher quantiles than their corresponding a-cases). At return periods o approximately 100 years, the plots o cases 1a and 2a cross those o cases 3b and 4b. This implies that beyond return periods o 100 years the inclusion o certain historical years in (or the exclusion rom) a subset has more inluence than the aorementioned average characteristics o the subset. Furthermore, the plots o the b-cases suggest a larger spread than those o the a-cases, which implies that the inluence o 1984 and 1995 also depends on the average characteristics o the set. In all simulations 10-day amounts above the largest historical maxima o 164 mm (1995) were ound. However, in all simulations (including the reerence simulation) the rate o occurrence o this value is less than expected rom the historical record, except in those or the cases 1b and 2b. These are the two cases that are relatively wet and contain the winter o Apparently, or that historical event to be simulated with the same requency as it is associated with in the historical record, the subset o historical years should be relatively wet on average as well as contain the winter o The simulated 10-day maxima or case 3a show the largest deviation rom the Gumbel plot o the historical maxima. The signiicance o this deviation was investigated by generating 1500 bootstrap samples o 66 maxima. Pointwise 95%-conidence intervals were determined or the Gumbel plot o these maxima, delineated by the solid lines in Fig Even though this simulation is associated with the lowest plot in Fig. 3.1, the 95%-conidence intervals derived rom it enclose most o the historical quantiles. Table 3.1 lists the empirical quantiles o the 10-day precipitation maxima, obtained rom the dierent simulations corresponding to return periods o respectively 2, 200, 500 and 1250 years. The simulations with cases b all result in a 2-year event (median) that is on average 10% higher than those or the cases a. The 2-year event rom the reerence simulation is ound right in between cases a and b. For return periods o 500 and 1250 years the range o the quantiles or the our b-cases is roughly twice as large as the range 8

13 10-day winter maximum [mm] Re 1a,b 2a,b 3a,b 4a,b mm Return period T [year] Standardized Gumbel variate [-] Figure 3.1: Extreme 10-day winter precipitation rom resampling subsets a (solid), subsets b (dashed) and the entire historical period ( Re, bold solid). Each o the simulations had a total length o years. The ordered 10-day maxima in the extended historical series are indicated by the year o occurrence. The largest 10-day maximum (164 mm in January 1995) is accentuated. 180 Return period T [year] day winter maximum [mm] Standardized Gumbel variate [-] Figure 3.2: Ordered 10-day precipitation maxima in the extended historical series (winter hal-year) and pointwise 95%-conidence intervals or the Gumbel plot o 10-day maxima o a 66-year sequence, obtained by bootstrapping 1500 sequences o 66 maxima rom the simulation or case 3a. 9

14 Table 3.1: Simulated quantiles corresponding to return periods o resp. 2, 200, 500 and 1250 years or 10-day winter maxima (in mm). The results are shown or the reerence simulation and or each o the eight selected cases 1a to 4a and 1b to 4b. The last two columns indicate whether the winter o 95 or 84 was included in the subsets. In the inal row the range o values rom respectively the a-cases and the b-cases is given or each return period. 2-yr 200-yr 500-yr 1250-yr 2-yr 200-yr 500-yr 1250-yr Re a b a b a b a b or the our a-cases. The dierence between the simulations with and without 1995 is larger than the eect o the mean precipitation. For the 1250-year return period the entire range spanned by the eight simulations runs rom 165 upto 210 mm (compared to 183 mm or the reerence simulation). Because o special interest in the return period o 1250 year, the 10-day event which roughly corresponds with this return period (the 16th largest) was investigated more closely in each o the our simulations b. In each o these simulations this event was located and dissected into daily values. These are listed in Table 3.2, along with their historical year and calendar day. In particular in the 2b simulation a considerable contribution o the winter o 1995 is seen (mainly January 27 and 29), but also an isolated large daily amount (42.3 mm) rom a day in February Repetition o the same historical day on two consecutive days occurs in all our simulations. Especially the degree o repetition in the 3b- and 4b-simulations is noteworthy (the slight dierences among amounts obtained rom the same historical day are the result o the standardisation o historical amounts and the transormation o standardised amounts back to their original scale). The mechanism responsible or this phenomenon in the simulations is not yet entirely clear. To assess its eect on the simulated 1250-year event, the simulations or cases 1b to 4b were repeated with a slightly modiied resampling algorithm in which the sampling o the same historical day on two consecutive days in the simulation was prohibited. From these simulations the 16th largest 10-day events were extracted in the same way and their daily amounts are listed in Table 3.3. These events are slightly, but systematically, smaller than those obtained with the original algorithm. The decrease is in the range o 2.5-5%. The relative eect o the modiication o the algorithm is largest in the 4b-simulation, probably due to three days with more than 40 mm in the original simulation. Though direct repetition o historical days is prevented, multiple occurrences o certain historical days within a single 10-day stretch, however, are not. Figure 3.3 presents the changes in the Gumbel plots due to the modiication o the algorithm. 10

15 Table 3.2: Daily values o basin-average precipitation in the 16th largest 10-day event (winter hal-year) or each o the simulations or cases 1b, 2b, 3b and 4b. The historical years (YY ) and calendar days (ddd) rom which the daily amounts were sampled are also given (notation: YY.ddd). The last row reports the 10-day totals or each simulation. Sim 1b Sim 2b Sim 3b Sim 4b P [mm] hist. date P [mm] hist. date P [mm] hist. date P [mm] hist. date Table 3.3: Same as in the Table 3.2, except that the resampling algorithm prevents the selection o the same historical day on consecutive days in the our simulations. Sim 1b Sim 2b Sim 3b Sim 4b P [mm] hist. date P [mm] hist. date P [mm] hist. date P [mm] hist. date

16 10-day winter maximum [mm] b 2b 3b 4b 5 Return period T [year] Standardized Gumbel variate [-] Figure 3.3: Gumbel plots o the winter maxima o 10-day precipitation rom the simulations or cases 1b to 4b perormed with the original (dashed) and the modiied algorithm (solid). 12

17 4 Conclusions The sensitivity o the simulation o extreme winter maxima o 10-day precipitation to the selection o historical years was investigated or the Meuse basin. The emphasis was on the basin-average precipitation and the historical period was considered. Simulations with a length o years were perormed with various subsets o historical years rom this period (the years being centred around the winter hal-year by deinition). The selection o the subsets or resampling was based on two average properties (the raction o days with 10 mm o precipitation or more in the winter hal-year and the mean winter precipitation) and the inclusion or the exclusion o the winters o 1995 and The Gumbel plots o the 10-day maxima or the simulations were compared. For return periods up to ive years, where the eect o individual years was negligible, the inluence o the average characteristics o the subsets was particularly noticeable. Over the entire range o return periods, a higher mean precipitation and a higher raction o days with 10 mm o precipitation or more leads to larger simulated 10-day maxima. At long return periods the plots start to diverge, in particular those with and without the winter o The range o 1250-year return values or the eight simulations runs rom 165 to 210 mm. In spite o deviations between the Gumbel plots o the simulations and the empirical quantiles rom the historical records, the pointwise 95%-conidence intervals or the most deviant simulation still enclosed most o the historical maxima. Beside the simulations with the subsets, a reerence simulation was perormed based on the entire historical period. The Gumbel plot o the 10-day precipitation maxima rom this simulation approximately alls halway the range spanned by the subsets, which shows that the inluence o 1995 on the maxima reduces as the number o years taken into account increases. The inluence o the winter o 1995 is also seen in the daily amounts rom which large 10-day maxima (1250-year events) are ormed. In the simulations in which this winter is taken into account, the largest 10-day events generally contain many historical values originating rom January The year 1984 has less inluence. It was discovered that in the simulated sequences some historical days are repeatedly drawn on consecutive days. This might be considered less realistic. The inluence o this phenomenon on the 10-day maxima was thereore investigated by repeating our simulations with a modiied algorithm, in which such repetition o historical days was not allowed. The inluence on the composition o the typical 1250-year event as well as the entire Gumbel plot was assessed. It was ound that the eect o the altered algorithm on the 1250-year event was within 5%. For short and moderate return periods no eect on the 10-day maxima was ound. Acknowledgements The work was perormed in co-operation with the Institute or Inland Water Management and Waste Water Treatment (RIZA), Lelystad. The meteorological data were kindly provided by Météo France and the Royal Meteorological Institute o Belgium (RMIB). 13

18 Reerences Aalders, P., Warmerdam, P. M. M. and Tors, P. J. J. F., Rainall Generator or the Meuse Basin: 3000 year discharge simulations in the Meuse basin. Report No. 124, Sub-department Water Resources, Wageningen University, Wageningen. GREHYS, Inter-comparison o regional lood requency procedures or Canadian rivers. Journal o Hydrology 186, Leander, R. and Buishand, T. A., Rainall Generator or the Meuse Basin: Development o a multi-site extension or the entire drainage area. KNMI-publication 196-III, KNMI, De Bilt. Leander, R., Buishand, T. A., Aalders, P. and de Wit, M. J. M., Estimation o extreme loods o the river Meuse using a stochastic weather generator and a rainallruno model. Hydrological Sciences Journal 50(6), Leander, R. and Buishand, T. A., Resampling o regional climate model output or the simulation o extreme river lows. Journal o Hydrology 332, Morrison, D. F., Multivariate statistical methods. McGraw-Hill, New York. 14

19 Appendix A Conidence region or the mean o a bivariate normal variate = (X 1, X 2 ) T has a bivariate normal distri- We assume that the two-dimensional vector X bution. The density o X is given by ( where µ ( x ) = 1 2π Σ exp 1/2 1 2 ( x µ ) T Σ 1 ( x µ ) = (µ 1, µ 2 ) T is the vector o means and Σ Σ ( ) σ 2 = 1 γ γ σ2 2. ), (A.1) is the covariance matrix given by (A.2) It is easy to show that the inverse covariance matrix has the orm ( ) Σ 1 1 σ 2 = 2 γ σ1σ γ 2 γ σ1 2. (A.3) The α% conidence region o µ can be determined rom the sample mean x = (x 1, x 2 ) T and the sample covariance matrix C, i.e. the matrix in which the elements o Σ are replaced by their sample estimates. This region is deined by the inequality (Morrison, 1976): N( µ x ) T C 1 ( µ 2(N 1) x ) N 2 F α,2,n 2, (A.4) where F α,2,n 2 denotes the α%-point o the F-distribution with degrees o reedom 2 and N 2. The conidence region appears as a tilted ellipse in the x -plane, the tilt and elongation being determined by the covariance matrix C. In the case o and P this matrix contains the sample variances and covariance o the amount o precipitation and the raction o days with 10 mm o precipitation or more in each winter hal-year. 15

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