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Regional frequency analysis of hydrological droughts Henrik Madsen DHI Water & Environment

Regional frequency analysis of hydrological droughts Henrik Madsen DHI Water & Environment. About. 1/1.

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Regional frequency analysis of hydrological droughts Henrik Madsen DHI Water & Environment

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  1. Regional frequency analysis • of hydrological droughts • Henrik Madsen • DHI Water & Environment

  2. About 1/1 • This PowerPoint includes a self-guided tour on regional frequency analysis of hydrological droughts. It has been prepared as part of the text book Hydrological Drought - Processes and Estimation Methods for Streamflow and Groundwater, Chapter 6. • To navigate through this presentation different options are available: • 1. To move forward or backward following the chronological order of the presentation use the Arrow Buttons in the lower right corner. • 2. To move to a specific category use the Category Buttons in the lower panel. • 3. Main categories may be divided into subjects. In this case you can move to a subject using the Subject Buttons to the left. • Within each category or subject page numbers are shown in the upper right corner. Introduction Index Method Regional Procedure Application Example References

  3. Introduction 1/1 • Main objectives of regional frequency analysis: • 1. To reduce the sampling uncertainties in the estimation of extreme drought events by combining streamflow records at different sites in a region that can be assumed to have similar drought characteristics (space substitutes time). • 2. To provide the basis for estimation of extreme drought events at ungauged sites by relating drought statistics with catchment characteristics. • Motivation: • The mean value at a site can usually be estimated adequately even if the available record is short. Second and higher order moments, however, have large sampling uncertainties. Regional data are applied to obtain more reliable estimates of these statistics. Introduction Index Method Regional Procedure Application Example References

  4. Index method 1/3 • Assumptions: • 1. Product moment ratios or L-moment ratios of order 2 and higher (coefficient of variation (CV), skewness (CS), kurtosis) are constant in the region. • 2. Data at the different sites in the region follow the same statistical distribution except for scale. Introduction Index Method Regional Procedure Application Example References

  5. Index method 2/3 • L-moment approach (Hosking & Wallis, 1997): • 1. At each site calculate L-moment ratio estimates (L-CV, L-CS and L-Kurtosis). • 2. Calculate regional record-length-weighted average L-moment ratio estimates. • 3. Based on the regional L-moment ratio estimates determine the parameters of the normalised regional distribution. • 4. Calculate the normalised regional T-year event using • where l(s) is the average annual number of drought events (at- site or regional estimate), and F-1(.) is the inverse of the cdf of the normalised regional distribution Introduction Index Method Regional Procedure Application Example References

  6. Index method 3/3 • 5. Calculate the T-year event estimate of the drought characteristic at an arbitrary site by multiplying the at-site mean value estimate m(s) with the regional normalised quantile • An estimate of the associated uncertainty (variance) is given by • where Var{m(s)} is the variance of the at-site mean value estimate and Var{zT(s)} is the variance of the normalised quantile estimate Introduction Index Method Regional Procedure Application Example References

  7. Regional procedure - outline 1/2 Outline • Building of the regional model includes: • 1. Grouping of sites into homogeneous (or fairly homogeneous) regions. • 2. Determination of a regional distribution in each of the defined regions. • 3. Estimation of regional parameters (L-Cv and L-Cs) and associated uncertainties. • 4. Determination of regression model that relates the mean value with catchment characteristics. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  8. Regional procedure - outline 2/2 Outline • Tools available for the regional analysis: • L-moment analysis • - testing of regional homogeneity • - determination of a regional distribution • Generalised least squares (GLS) regression • - estimation of regional parameters and associated uncertainties • - testing of regional homogeneity • - determination of regression model • Split sample grouping • - grouping of sites into homogenous regions L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  9. L-moment analysis 1/11 Outline • Objective: • L-moment analysis is applied for testing regional homogeneity and determination of a regional distribution. • L-moment ratio diagram: • For a visual judgement an L-moment ratio diagram is constructed. In an L-moment ratio diagram sample estimates of the L-moment ratios L-Cv, L-Cs and L-kurtosis are compared to the theoretical relationships for a range of probability distributions. • L-moment statistics: • For a more formal evaluation Hosking and Wallis (1993) proposed a test statistic for regional homogeneity and a goodness-of-fit statistic for determination of a regional distribution. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  10. L-moment ratio diagram 2/11 Outline • L-moment ratio diagram illustrating the relationship between L-skewness and L-kurtosis for the generalised Pareto (GP), generalised extreme value (GEV), log-normal (LN), gamma (GAM), Weibull (WEI), Gumbel (GUM) and exponential (EXP) distributions. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  11. L-moment ratio diagram 3/11 • L-moment ratio diagram illustrating the relationship between L-Cv and L-skewness for the two-parameter generalised Pareto (GP), log-normal (LN), gamma (GAM), and Weibull (WEI) distributions and the one-parameter exponential (EXP) distribution. Outline L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  12. Test of regional homogeneity 4/11 Outline • The dispersion of points in the L-moment diagram of the L-moment estimates from the different sites in the region gives an indication of the regional homogeneity. The question is if the observed variability is significant (i.e. the points form a heterogeneous group) or can be explained by sampling uncertainties. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  13. Test of regional homogeneity 5/11 Outline • Regional homogeneity measure: • Comparison of observed regional variability of L-moment statistics with the variability that would be expected in a homogeneous region. L-moment Analysis GLS Regression Observed Simulated Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  14. Test of regional homogeneity 6/11 Outline • Test statistic: • where V is the record-length-weighted standard deviation of the L-CV estimates, and mv and sv are, respectively, the mean and standard deviation of V in a homogeneous region (determined by simulation). • Evaluation of H-statistic: • H < 1: Acceptably homogeneous region • 1 < H < 2: Possibly heterogeneous region • H > 2: Definitely heterogeneous region L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  15. Choice of regional distribution 7/11 Outline • The grouping of points in the L-moment ratio diagram is compared with the theoretical • L-moment relationships for a number of candidate distributions. To discriminate between different 3-parameter distributions the L-skewness/L-kurtosis diagram is used. The • L-Cv/L-skewness diagram is used to discriminate between various 2-parameter distributions. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  16. Choice of regional distribution 8/11 Outline • Goodness-of-fit measure for a 3-parameter distribution: • Comparison of regional average L-kurtosis and the theoretical L-kurtosis for the considered distribution corresponding to the regional average L-skewness. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  17. Choice of regional distribution 9/11 Outline • Test statistic: • where t4R is the regional average L-kurtosis, t4DIST is the L-kurtosis of the fitted regional distribution, and β4 and σ4 are, respectively, the bias and the standard deviation of the regional average L-kurtosis obtained from simulations. • Evaluation of Z-statistic: • The test statistic is evaluated against the quantiles of a standard normal distribution, i.e. |Z|< 1.96: Acceptable fit (corresponding to a 5% significance level). L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  18. Choice of regional distribution 10/11 Outline • Goodness-of-fit measure for a 2-parameter distribution: • Comparison of regional average L-skewness and the theoretical L-skewness for the considered distribution corresponding to the regional average L-CV. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  19. Choice of regional distribution 11/11 Outline • Test statistic: • where t3R is the regional average L-skewness, t3DIST is the L-skewness of the fitted regional distribution, and σ3 is the standard deviation of the regional average L-skewness obtained from simulations. • Evaluation of Z-statistic: • The test statistic is evaluated against the quantiles of a standard normal distribution, i.e. |Z|< 1.96: Acceptable fit (corresponding to a 5% significance level). L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  20. GLS regression 1/6 Outline • Objective: • Generalised least squares (GLS) regression is applied for estimation of regional parameters and testing of regional homogeneity. For parameters that show a significant regional variability the GLS regression procedure is subsequently applied to evaluate the potential of describing the variability from catchment characteristics (Stedinger & Tasker, 1985; Madsen & Rosbjerg, 1997). L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  21. Regional mean model 2/6 Outline • The regional mean model is a special case of the GLS regression model. It is used for estimation of regional parameters of L-moment ratios (L-Cv and L-Cs) and their associated uncertainties. In addition, the regional mean model provides a heterogeneity measure for testing regional homogeneity. • Input: • - L-moment ratio estimates L-Cv and L-Cs. • - Sampling variances of L-Cv and L-Cs estimates. • Output: • - Regional L-moment ratio estimates. • - Variance of regional L-moment ratio estimates. • - Residual model error variance (heterogeneity measure). L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  22. Regression model 3/6 Outline • The GLS regression model is used for estimation of mean drought statistics from catchment characteristics. • The following log-linear model is considered: • where mi is the mean value of the drought characteristic, Aik are the considered catchment characteristics, bk are the regression parameters, ei is a random sampling error, and di is the residual model error. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  23. Regression model 4/6 Outline • The sampling error and the residual model error are assumed to have zero mean and covariance structures • where s2ei is the sampling error variance, reij is the correlation coefficient due to concurrent observations at stations i and j (intersite correlation coefficient), and s2d is the residual model error variance. • Note: Compared to ordinary least squares regression GLS regression accounts explicitly for heteroscedastic (different variances) and cross-correlated sampling errors. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  24. Regression model 5/6 Outline • Input: • - Mean value and associated variance of drought characteristic • - Catchment characteristics • Output: • - Regression parameters and associated covariance • - Residual model error variance • Application: • For given catchment characteristics the mean value and the associated variance are estimated from the GLS regression equation. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  25. Regression model 6/6 Outline • In the case where some at-site data are available the sample mean mAS and the mean value obtained from the regression equation mR can be combined using (Madsen & Rosbjerg, 1997): • where Var{mAS} is the variance of the at-site estimate and Var{mR} is the variance of the regional estimate obtained from the regression model. • The variance of the weighted estimator is: L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  26. Split sample grouping 1/2 Outline • Objective: • Grouping of sites into homogeneous (or fairly homogeneous) regions according to catchment characteristics. • Procedure (Wiltshire, 1995): • 1. The method splits a set of catchments into two groups based on a single partitioning value of one chosen catchment characteristic. Measures of variability of drought characteristics within each group are aggregated into one statistic, and the optimum grouping is achieved at the point where this statistic is minimum. • 2. Step 1 is repeated for all the considered catchment characteristics. The catchment characteristic that provides the minimum variability statistic is chosen as the optimal two-way grouping. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  27. Split sample grouping 2/2 Outline • 3. Steps 1-2 are repeated for a multiple partitioning, i.e. a four-way grouping based on 2 characteristics, an eight-way grouping based on 3 characteristics etc. • 4. After each split sample grouping regional homogeneity is tested using the L-moment H-statistic or the GLS regression statistic. • Variability measure (Pearson, 1991; Madsen et al., 1997): • where ej,i is the deviation of the jth L-moment ratio estimate at site i from its group record-length-weighted average. In this case the L-Cv is weighted ahead of the L-skewness, which in turn is weighted ahead of L-kurtosis, so that homogeneity is primarily influenced by L-Cv and less so by L-kurtosis. L-moment Analysis GLS Regression Split Sample Grouping Introduction Index Method Regional Procedure Application Example References

  28. Karlsruhe Strassburg (France) Stuttgart Freiburg Basel (Switzerland) Application example - data 1/6 Data • The regional frequency analysis procedure was applied to the regional data set with daily streamflow series from Baden-Würtenberg, Germany Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  29. Streamflow drought series 2/6 Data • Streamflow data: • Stations with a record length larger than 20 years were included in the regional analysis, which comprises 46 stations with recording periods ranging between 20 to 37 years with an average of 30.2 years. • Drought series: • For definition of drought events the threshold level approach was applied using the 70% quantile of the daily flow duration curve as the threshold level (Tallaksen et al., 1997). From the drought series annual maximum series of drought duration and deficit volume were extracted. Of the 46 stations, 20 stations experience no zero drought years, whereas the remaining 26 stations have one or more zero drought years. The regional average annual number of drought events is equal to 0.952 years-1. Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  30. Streamflow drought series 3/6 Data • Empirical distributions of AMS of drought duration Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  31. Streamflow drought series 4/6 Data • Empirical distributions of AMS of deficit volume (normalised with the catchment area) Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  32. Catchment characteristics 5/6 Data • Climate: • Mean annual precipitation [mm] • Land use: • Fraction of urbanisation [-] • Fraction of forest [-] • Morphometry: • Catchment area [km2] • Drainage density [km/ km2] • Highest elevation [m a.m.s.l.] • Average elevation [m a.m.s.l.] • Lowest elevation [m a.m.s.l.] • Maximum slope [%] • Average slope [%] • Minimum slope [%] Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  33. Catchment characteristics 6/6 Data • Soil: • Fraction of soils with high infiltration capacity [-] Fraction of soils with medium infiltration capacity [-] Fraction of soils with low infiltration capacity [-] Fraction of soils with very low infiltration capacity [-] Mean hydraulic conductivity of the soils [cm/d] Fraction of soils with low hydraulic conductivity [-] Fraction of soils with high water-holding capacity in the effective root zone [-] Mean water-holding capacity in the effective root zone [mm] • Hydrogeology: • Fraction of rock formations with a very low hydraulic permeability [%] • Weighted mean of hydraulic conductivity [m/s] Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  34. Grouping of sites 1/10 Data • Testing of regional homogeneity of all 46 catchments: • Duration: H = 5.76 • Deficit volume: H = 4.08 • Both for duration and deficit volume the H-statistic indicates that all 46 catchments form a “definitely heterogeneous” group with respect to L-Cv. • Split sample grouping (two-way grouping): • The global minimum value of the variability statistic V as a function of the different catchment characteristics is obtained with the mean annual precipitation (MAP). • “Dry catchments”: MAP < 1000 mm • “Wet catchments”: MAP > 1000 mm Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  35. Grouping of sites 2/10 Data • Variability measure as a function of mean annual precipitation Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  36. Grouping of sites 3/10 Data • Testing of regional homogeneity (two-way grouping): • MAP < 1000 mm: • Duration: H = 1.64 (Possibly heterogeneous) • Deficit volume: H = 0.60 (Acceptably homogeneous) • MAP > 1000 mm: • Duration: H = 1.94 (Possibly heterogeneous) • Deficit volume: H = 1.29 (Possibly heterogeneous) • The grouping of sites with respect to MAP provides more homogeneous groups. Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  37. Grouping of sites 4/10 Data • Split sample grouping (four-way grouping): • MAP < 1000 mm: • For the “dry catchment” group the available catchment characteristics did not provide a well-defined partitioning. • MAP > 1000 mm: • For the “wet” catchments a well-defined partitioning was obtained with the catchment average hydraulic conductivity of the upper hydrogeological unit (HCMEAN) with a minimum variability measure for HCMEAN = 2.510-5 m/s. Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  38. Grouping of sites 5/10 Data • Variability measure as a function of average hydraulic conductivity Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  39. Grouping of sites 6/10 Data • Testing of regional homogeneity (four-way grouping): • Region A (MAP < 1000 mm): • Duration: H = 1.64 (Possibly heterogeneous) • Deficit volume: H = 0.60 (Acceptably homogeneous) • Region B (MAP > 1000 mm, HCMEAN < 2.510-5 m/s): • Duration: H = -2.12 (Acceptably homogeneous) • Deficit volume: H = -0.36 (Acceptably homogeneous) • Region C (MAP > 1000 mm, HCMEAN > 2.510-5 m/s): • Duration: H = 0.24 (Acceptably homogeneous) • Deficit volume: H = -0.64 (Acceptably homogeneous) Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  40. Grouping of sites 7/10 Data • L-Cv and L-Cs estimates of the three regions for drought duration Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  41. Grouping of sites 8/10 Data • L-Cv and L-Cs estimates of the three regions for deficit volume Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  42. Grouping of sites 9/10 Data • L-Cs and L-kurtosis estimates of the three regions for duration Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  43. Grouping of sites 10/10 Data • L-Cs and L-kurtosis estimates of the three regions for deficit volume Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  44. Goodness-of-fit statistics 1/7 Data • Duration (3-parameter distribution): • Duration (2-parameter distribution): Distributions accepted at a 5% significance level marked in red Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  45. Goodness-of-fit statistics 2/7 Data • Deficit volume (3-parameter distribution): • Deficit volume (2-parameter distribution): Distributions accepted at a 5% significance level marked in red Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  46. Goodness-of-fit statistics 3/7 Data • Distributions accepted at a 5% significance level: • 3-parameter distributions (out of 3 regions): • GP: Duration: 2; Volume: 1 • GEV: Duration: 0; Volume: 0 • LN: Duration: 0; Volume: 0 • GAM: Duration: 2; Volume: 2 • WEI: Duration: 2; Volume: 2 • 2-parameter distributions (out of 3 regions): • GP: Duration: 2; Volume: 0 • LN: Duration: 0; Volume: 0 • GAM: Duration: 2; Volume: 3 • WEI: Duration: 1; Volume: 2 •  2-parameter Gamma distribution chosen for both duration and volume in all 3 regions. Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  47. Regional parameters 4/7 Data • Regional parameters (duration): • Regional parameters (deficit volume): Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  48. Normalised regional distributions 5/7 Data • The regional L-Cv estimates and average annual number of drought events (average rates) are used to estimate the normalised regional quantile in the three regions for both drought duration and deficit volume. • Region A catchments have heavier tailed distributions than the Region C catchments, which in turn have heavier tailed distributions than the Region B catchments. In each region the deficit volume have heavier tailed distributions than the duration. Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  49. Normalised regional distributions 6/7 Data • Drought duration Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

  50. Normalised regional distributions 7/7 Data • Deficit volume Grouping of Sites Regional Distribution GLS Regression Quantile Estimation Introduction Index Method Regional Procedure Application Example References

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