1. STATSPUNE 1 Are our dams over-designed? S.A.ParanjpeA.P.Gore

2. STATSPUNE 2 Dams Desirable effects Irrigation Electricity Undesirable effects Displacement of people Submergence of valuable forest Long gestation period Large investment S.A.ParanjpeA.P.Gore

3. STATSPUNE 3 One factor affecting cost- strength required A large dam must be able to withstand even a rare but heavy flood How big a flood to assume in dam design? PMF- Probable maximum flood PMP- Probable maximum precipitation S.A.ParanjpeA.P.Gore

4. STATSPUNE 4 P.C.Mahalnobis Floods in Orissa Work half a century ago Engineers: Build embankments to avoid flood (caused by rise in riverbed) PCM: (1923) no noticeable rise in riverbed Build dams upstream (Hirakud) North Bengal: build retarding basins to control flood PCM(1927): Rapid drainage needed (Lag correlation between rainfall and flood) S.A.ParanjpeA.P.Gore

5. STATSPUNE 5 Indian Institute of Tropical Meteorology, Pune 1 day PMP atlas for India (1989) Higher PMP  costlier Dam Our finding: PMP calculations of IITM - Overestimates may lead to costlier dams (avoidable) S.A.ParanjpeA.P.Gore

6. STATSPUNE 6 How to calculate PMP? X: Max rainfall in a day in one year (at a location) P(X > X T ) =1/T Then X T is called T year return period value of rainfall (convention in hydrology) X 100 : value of one day max rainfall exceeded once in 100 years X 100 is considered suitable PMP for ‘minor dams’. Major dams: X 10000 as PMP T= 10,000 years. S.A.ParanjpeA.P.Gore

7. STATSPUNE 7 How to estimate X T ? Data: daily rainfall records for 90 years. Y i = maximum rainfall in a day in year i Y 1, Y 2, ….Y 90 available Est(X 90 ) = max(Y 1, Y 2, ….Y 90 ) Good enough for minor dams. What about X 10000 ? Purely empirical approach - inadequate. Model based approach needed. S.A.ParanjpeA.P.Gore

8. STATSPUNE 8 Model based approach Extreme value: Gumbel distribution used commonly f(x) = 1/ . exp( - (x-  )/  –exp((x-  )/  )) Estimate ,  by maximum likelihood Test goodness of fit by chi-square. Data available -358 stations Fit good at 86% stations (  =0.05), 94% stations (  =0.01) If fit is good –obtain 10 -4 upper percentile of the fitted distribution -- use as estimate of X 10000 S.A.ParanjpeA.P.Gore

9. STATSPUNE 9 Gumbel fit rejected at 1% S.A.ParanjpeA.P.Gore

10. STATSPUNE 10 Hershfield method: a) X 1,X 2, ….,X n annual one day maxima at a station for n years K = (X max – av(X n-1 ))/S n-1 av(X n-1 ): average after dropping X max b)K m = largest K over all stations in a locality X PMP = av(X n ) + K m S n How does this method compare with model based method? S.A.ParanjpeA.P.Gore

11. STATSPUNE 11 X:Hershfield, Observed highest :10,000 year value(model based) Hershfield Method overestimates PMP S.A.ParanjpeA.P.Gore

12. STATSPUNE 12 Stability of model based estimate If the new estimator is volatile i.e. has large standard error and is unstable, it may not be usable. Computing standard errors -analytically intractable Simulation study carried out design:for each station generate 100 samples each of size 100 compute competing estimates empirical mean and sd Zone-wise comparison S.A.ParanjpeA.P.Gore

13. STATSPUNE 13 Homogeneous rainfall zones in India S.A.ParanjpeA.P.Gore

14. STATSPUNE 14 Comparative volatility of estimators ZoneModel based estimator(cm) Hershfield estimator(cm) meansdmeansd 730.812.5449.94.16 141.223.5963.015.29 243.823.6562.315 354.164.2764.715.92 436.903.0146.323.93 543.073.6555.364.50 637.153.0034.313.12 Proposed estimator more stable S.A.ParanjpeA.P.Gore

15. STATSPUNE 15 Further work Gumbel model fitted to rainfall data from 358 stations Acceptable fit – 299 stations What about remaining stations? Alternative models: log-normal, gamma, Weibull, Pareto Which model gives good fit to data? How robust is the resulting estimate? Pareto does not fit any data set. S.A.ParanjpeA.P.Gore

16. STATSPUNE 16 S.A.ParanjpeA.P.Gore

17. STATSPUNE 17 S.A.ParanjpeA.P.Gore

18. STATSPUNE 18 Fitting alternative models StationBest fit distribution Percentile(cm).001 |.0001 NagpurLog- normal20.7729.88 Bankuragamma29.8936.08 LucknowWeibull22.2124.36 BaramatiGamma17.4120.41 Estimates based on Gumbel were robust What about the above? S.A.ParanjpeA.P.Gore

19. STATSPUNE 19 Simulation study One station (Pune) All models Data 1901- 1990 Av(X (n) ) = 7.09 cm sd(X (n) ) = 2.64 cm Parameter of each model Chosen such that Mean, sd match with Observed values Sample size 100 S.A.ParanjpeA.P.Gore

20. STATSPUNE 20 Parameters chosen and true quantiles DistributionParametersQuantiles(mm).001 |.0001 Gammaa= 7.21b=9.84180.88213.0 Weibulla=3.76b=77.36129.4139.7 Log-normal  =4.2  =0.36 136.6195.9 a: shape parameter,b: scale parameter Bias and MSE stabilized at 2000 samples S.A.ParanjpeA.P.Gore

21. STATSPUNE 21 Results of simulation study (10,000 simulations) DistributionReturn period(T) True value estimateRMSE Gamma1000180.99182.071.54 10000213.01213.761.54 Log- normal 1000136.62135.658.60 10000195.92194.3016.63 Weibull1000129.40129.805.14 10000139.77140.146.20 Estimates are stable in these distributions as well Gamma model performs better than others. S.A.ParanjpeA.P.Gore