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A regional Bayesian hierarchical model for flood frequency analysis

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Abstract

In this study, we propose a regional Bayesian hierarchical model for flood frequency analysis. The Bayesian method is an alternative to the traditional regional flood frequency analysis. Instead of relying on the delineation of implicit homogeneous regions, the Bayesian hierarchical method describes the spatial dependence in its inner structure. Similar to the classical Bayesian hierarchical model, the process layer of our model presents the spatial variability of the parameters by considering different covariates (e.g., drainage area, elevation, precipitation). Beyond the three classical layers (data, process, and prior) of the Bayesian hierarchical model, we add a new layer referred to as the “L-moments layer”. The L-moments layer uses L-moments theory to select the best-fit probability distribution based on the available data. This new layer can overcome the subjective selection of the distribution based on extreme value theory and determine the distribution from the data instead. By adding this layer, we can combine the merits of regional flood frequency and Bayesian methods. A standard process of covariates selection is also proposed in the Bayesian hierarchical model. The performance of the Bayesian model is assessed by a case study over the Willamette River Basin in the Pacific Northwest, U.S. The uncertainty of different flood percentiles can be quantified from the posterior distributions using the Markov Chain Monte Carlo method. Temporal changes for the 100-year flood percentiles are also examined using a 20- and 30-year moving window method. The calculated shifts in flood risk can aid future water resources planning and management.

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References

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723

    Article  Google Scholar 

  • Ballester J, Giorgi F, Rodó X (2010) Changes in European temperature extremes can be predicted from changes in PDF central statistics. Clim Change 98(1–2):277–284

    Article  Google Scholar 

  • Banerjee S, Gelfand AE, Carlin BP (2004) Hierarchical modeling and analysis for spatial data. CRC Press, Boca Raton

    Google Scholar 

  • Berger JO, De Oliveira V, Sansó B (2001) Objective Bayesian analysis of spatially correlated data. J Am Stat Assoc 96(456):1361–1374

    Article  Google Scholar 

  • Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc Ser B 36:192–236

    Google Scholar 

  • Coles S (2001) An introduction to statistical modeling of extreme values. Springer, London

    Book  Google Scholar 

  • Coles S, Pericchi L (2003) Anticipating catastrophes through extreme value modelling. J R Stat Soc: Ser C (Appl Stat) 52(4):405–416

    Article  Google Scholar 

  • Coles SG, Tawn JA (1996) A Bayesian analysis of extreme rainfall data. Appl Stat 45:463–478

    Article  Google Scholar 

  • Cooley D (2009) Extreme value analysis and the study of climate change. Clim Change 97(1–2):77–83

    Article  Google Scholar 

  • Cooley D, Sain SR (2010) Spatial hierarchical modeling of precipitation extremes from a regional climate model. J Agric Biol Environ Stat 15(3):381–402

    Article  Google Scholar 

  • Cooley D, Naveau P, Jomelli V, Rabatel A, Grancher D (2006) A Bayesian hierarchical extreme value model for lichenometry. Environmetrics 17(6):555–574

    Article  Google Scholar 

  • Cooley D, Nychka D, Naveau P (2007) Bayesian spatial modeling of extreme precipitation return levels. J Am Stat Assoc 102(479):824–840

    Article  CAS  Google Scholar 

  • Cooper RM (2005) Estimation of peak discharges for rural, unregulated streams in Western Oregon. US Department of the Interior, US Geological Survey

  • Dalrymple T (1960) Flood-frequence analyses. US Geological Survey

  • Davison AC, Smith RL (1990) Models for exceedances over high thresholds. J R Stat Soc Ser B 52:393–442

    Google Scholar 

  • Dominguez F, Rivera E, Lettenmaier DP, Castro CL (2012) Changes in winter precipitation extremes for the western United States under a warmer climate as simulated by regional climate models. Geophys Res Lett 39(5):L05803

    Article  Google Scholar 

  • Fawcett L, Walshaw D (2006) A hierarchical model for extreme wind speeds. J R Stat Soc: Ser C (Appl Stat) 55(5):631–646

    Article  Google Scholar 

  • Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc 24(2):180–190

    Article  Google Scholar 

  • Fowler HJ, Kilsby CG (2003) A regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000. Int J Climatol 23(11):1313–1334

    Article  Google Scholar 

  • Fowler HJ, Ekström M, Kilsby CG, Jones PD (2005) New estimates of future changes in extreme rainfall across the UK using regional climate model integrations 1. Assessment of control climate. J Hydrol 300(1):212–233

    Article  Google Scholar 

  • Gelfand AE, Smith AF (1990) Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85(410):398–409

    Article  Google Scholar 

  • Gelman A (1996) Inference and monitoring convergence. In Markov chain Monte Carlo in practice. pp 131–143

  • Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian data analysis. CRC, Boca Raton

    Google Scholar 

  • Gnedenko R (1943) Sur la Distribution Limite du Terme Maximum d’une Série Aléatoire. Ann Math 44:423–453

    Article  Google Scholar 

  • Griffis VW, Stedinger JR (2007) Log-Pearson Type 3 distribution and its application in flood frequency analysis. I: distribution characteristics. J Hydrol Eng 12(5):482–491

  • Gupta VK, Mesa OJ, Dawdy DR (1994) Multiscaling theory of flood peaks: regional quantile analysis. Water Resour Res 30(12):3405–3421

    Article  Google Scholar 

  • Haddad K, Rahman A (2011) Selection of the best fit flood frequency distribution and parameter estimation procedure: a case study for Tasmania in Australia. Stoch Environ Res Risk Assess 25(3):415–428

    Article  Google Scholar 

  • Halmstad A, Najafi MR, Moradkhani H (2012) Analysis of precipitation extremes with the assessment of regional climate models over the Willamette River Basin, USA. Hydrol Process 27:2579–2590

    Article  Google Scholar 

  • Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109

    Article  Google Scholar 

  • Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52:105–124

    Google Scholar 

  • Hosking JRM, Wallis JR (1988) The effect of intersite dependence on regional flood frequency analysis. Water Resour Res 24(4):588–600

    Article  Google Scholar 

  • Hosking JRM, Wallis JR (1997) Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hosking JRM, Wallis JR, Wood EF (1985) Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27(3):251–261

    Article  Google Scholar 

  • Interagency Advisory Committee on Water Data (IACWD) (1982) Guidelines for determining flood flow frequency, Bulletin 17B. Hydrology Subcommittee, US Dept. of Interior

  • Jung IW, Chang H (2011) Assessment of future runoff trends under multiple climate change scenarios in the Willamette River Basin, Oregon, USA. Hydrol Process 25(2):258–277

    Article  Google Scholar 

  • Katz RW (2010) Statistics of extremes in climate change. Clim Change 100(1):71–76

    Article  CAS  Google Scholar 

  • Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water Resour 25(8):1287–1304

    Article  Google Scholar 

  • Kroll CN, Vogel RM (2002) Probability distribution of low streamflow series in the United States. J Hydrol Eng 7(2):137–146

    Article  Google Scholar 

  • Kuczera G (1999) Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference. Water Resour Res 35(5):1551–1557

    Article  Google Scholar 

  • Kwon HH, Brown C, Lall U (2008) Climate informed flood frequency analysis and prediction in Montana using hierarchical Bayesian modeling. Geophys Res Lett 35(5):GL032220

    Article  Google Scholar 

  • Laio F, Tamea S (2007) Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrol Earth Syst Sci 11:1267–1277

    Article  Google Scholar 

  • Lettenmaier DP, Wallis JR, Wood EF (1987) Effect of regional heterogeneity on flood frequency estimation. Water Resour Res 23(2):313–323

    Article  Google Scholar 

  • Liang Z, Chang W, Li B (2012) Bayesian flood frequency analysis in the light of model and parameter uncertainties. Stoch Environ Res Risk Assess 26(5):721–730

    Article  Google Scholar 

  • Lima CHR, Lall U (2009) Hierarchical Bayesian modeling of multisite daily rainfall occurrence: rainy season onset, peak, and end. Water Resour Res 45(7):W07422

    Google Scholar 

  • Lima CHR, Lall U (2010) Spatial scaling in a changing climate: a hierarchical Bayesian model for non-stationary multi-site annual maximum and monthly streamflow. J Hydrol 383(3):307–318

    Article  Google Scholar 

  • Madsen H, Rosbjerg D (1997) Generalized least squares and empirical Bayes estimation in regional partial duration series index-flood modeling. Water Resour Res 33(4):771–781

    Article  Google Scholar 

  • Martins ES, Stedinger JR (2000) Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour Res 36(3):737–744

    Article  Google Scholar 

  • Martins ES, Stedinger JR (2001) Generalized Maximum Likelihood Pareto-Poisson estimators for partial duration series. Water Resour Res 37(10):2551–2557

    Article  Google Scholar 

  • Meehl GA, Arblaster JM, Tebaldi C (2007) Contributions of natural and anthropogenic forcing to changes in temperature extremes over the United States. Geophys Res Lett 34(19):L19709

    Article  Google Scholar 

  • Meshgi A, Khalili D (2009) Comprehensive evaluation of regional flood frequency analysis by L-and LH-moments. I. A re-visit to regional homogeneity. Stoch Environ Res Risk Assess 23(1):119–135

    Article  Google Scholar 

  • Milly PCD, Betancourt J, Falkenmark M, Hirsch RM, Kundzewicz ZW, Lettenmaier DP, Stouffer RJ (2008) Stationarity is dead: whither water management? Science 319:573–574

    Article  CAS  Google Scholar 

  • Moradkhani H, Hsu K, Gupta HV, Sorooshian S (2005) Uncertainty assessment of hydrologic model states and parameters: sequential data assimilation using particle filter. Water Resour Res 41:W05012. doi:10.1029/2004WR003604

    Google Scholar 

  • Moradkhani H, DeChant CM, Sorooshian S (2012) Evolution of ensemble data assimilation for uncertainty quantification using the particle filter-Markov chain Monte Carlo method. Water Resour Res 48:W12520. doi:10.1029/2012WR012144

    Google Scholar 

  • Mote PW, Salathé EP Jr (2010) Future climate in the Pacific Northwest. Clim Change 102(1–2):29–50

    Article  Google Scholar 

  • Najafi MR, Moradkhani H (2013a) Analysis of runoff extremes using spatial hierarchical Bayesian modeling. Water Resour Res 49(10):6656–6670

    Article  Google Scholar 

  • Najafi MR, Moradkhani H (2013b) A hierarchical Bayesian approach for the analysis of climate change impact on runoff extremes. Hydrol Process. doi:10.1002/hyp.10113

    Google Scholar 

  • Najafi MR, Moradkhani H, Jung I (2011) Assessing the uncertainties of hydrologic model selection in climate change impact studies. Hydrol Process 25(18):2814–2826

    Article  Google Scholar 

  • Ngongondo C, Li L, Gong L, Xu CY, Alemaw BF (2013) Flood frequency under changing climate in the upper Kafue River basin, southern Africa: a large scale hydrological model application. Stoch Environ Res Risk Assess 27(8):1883–1898

    Article  Google Scholar 

  • Padoan SA, Ribatet M, Sisson SA (2010) Likelihood-based inference for max-stable processes. J Am Stat Assoc 105(489):263–277

    Article  CAS  Google Scholar 

  • Peel MC, Wang QJ, Vogel RM, McMahon TA (2001) The utility of L-moment ratio diagrams for selecting a regional probability distribution. Hydrol Sci J 46(1):147–155

    Article  Google Scholar 

  • Pickands J III (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131

    Article  Google Scholar 

  • Rao AR (2006) Flood frequency relationships for Indiana. Joint Transportation Research Program

  • Rao AR, Hamed KH (1994) Frequency analysis of upper Cauvery flood data by L-moments. Water Resour Manag 8(3):183–201

    Article  Google Scholar 

  • Reis DS Jr, Stedinger JR (2005) Bayesian MCMC flood frequency analysis with historical information. J Hydrol 313(1):97–116

    Article  Google Scholar 

  • Renard B (2011) A Bayesian hierarchical approach to regional frequency analysis. Water Resour Res 47(11):W11513

    Google Scholar 

  • Ribatet M, Sauquet E, Grésillon JM, Ouarda TB (2007) A regional Bayesian POT model for flood frequency analysis. Stoch Environ Res Risk Assess 21(4):327–339

    Article  Google Scholar 

  • Ribatet M, Cooley D, Davison AC (2012) Bayesian inference from composite likelihoods, with an application to spatial extremes. Stat Sin 22:813–845

    Google Scholar 

  • Robinson JS, Sivapalan M (1997) An investigation into the physical causes of scaling and heterogeneity of regional flood frequency. Water Resour Res 33(5):1045–1059

    Article  Google Scholar 

  • Sang H, Gelfand AE (2009) Hierarchical modeling for extreme values observed over space and time. Environ Ecol Stat 16(3):407–426

    Article  Google Scholar 

  • Sang H, Gelfand AE (2010) Continuous spatial process models for spatial extreme values. J Agric Biol Environ Stat 15(1):49–65

    Article  Google Scholar 

  • Sankarasubramanian A, Lall U (2003) Flood quantiles in a changing climate: seasonal forecasts and causal relations. Water Resour Res 39(5):1134

    Google Scholar 

  • Schaefer MG (1990) Regional analyses of precipitation annual maxima in Washington State. Water Resour Res 26(1):119–131

    Article  Google Scholar 

  • Schliep EM, Cooley D, Sain SR, Hoeting JA (2010) A comparison study of extreme precipitation from six different regional climate models via spatial hierarchical modeling. Extremes 13(2):219–239

    Article  Google Scholar 

  • Sivapalan M (2003) Prediction in ungauged basins: a grand challenge for theoretical hydrology. Hydrol Process 17(15):3163–3170

    Article  Google Scholar 

  • Smith EL, Stephenson AG (2009) An extended Gaussian max-stable process model for spatial extremes. J Stat Plan Infer 139(4):1266–1275

    Article  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Ser B (Stat Methodol) 64(4):583–639

    Article  Google Scholar 

  • Stedinger JR (1983) Estimating a regional flood frequency distribution. Water Resour Res 19(2):503–510

    Article  Google Scholar 

  • Stedinger JR, Cohn TA (1986) Flood frequency analysis with historical and paleoflood information. Water Resour Res 22(5):785–793

    Article  Google Scholar 

  • Tobler WR (1970) A computer movie simulating urban growth in the Detroit region. Econ Geogr 46:234–240

    Article  Google Scholar 

  • Viglione A, Merz R, Salinas JL, Blöschl G (2013) Flood frequency hydrology: 3. A Bayesian analysis. Water Resour Res 49(2):675–692

    Article  Google Scholar 

  • Vogel RM, Fennessey NM (1993) L moment diagrams should replace product moment diagrams. Water Resour Res 29(6):1745–1752

    Article  Google Scholar 

  • Vogel RM, Wilson I (1996) Probability distribution of annual maximum, mean, and minimum streamflows in the United States. J Hydrol Eng 1(2):69–76

    Article  Google Scholar 

  • Vogel RM, McMahon TA, Chiew FHS (1993a) Floodflow frequency model selection in Australia. J Hydrol 146(1):421–449

    Article  Google Scholar 

  • Vogel RM, Thomas WO, McMahon TA (1993b) Flood-flow frequency model selection in southwestern United States. J Water Resour Plan Manag 119(3):353–366

    Article  Google Scholar 

  • Yan H (2012) Magnitude and frequency of floods for rural, unregulated streams of Tennessee by L-Moments method. In Masters Abstracts International 50(6)

  • Yan H, Edwards FG (2013) Effects of land use change on hydrologic response at a watershed scale, Arkansas. J Hydrol Eng 18(12):1779–1785

    Article  Google Scholar 

  • Yan H, Moradkhani H (2014) Bayesian model averaging for flood frequency analysis. World Environ Water Resour Congr 2014:1886–1895

    Google Scholar 

  • Yang T, Xu CY, Shao QX, Chen X (2010) Regional flood frequency and spatial patterns analysis in the Pearl River Delta region using L-moments approach. Stoch Environ Res Risk Assess 24(2):165–182

    Article  Google Scholar 

  • Yue S, Wang CY (2004) Possible regional probability distribution type of Canadian annual streamflow by L-moments. Water Resour Manag 18(5):425–438

    Article  Google Scholar 

  • Zwiers FW, Kharin VV (1998) Changes in the extremes of the climate simulated by CCC GCM2 under CO2 doubling. J Clim 11(9):2200–2222

    Article  Google Scholar 

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Acknowledgments

Partial financial support for this project was provided by the National Science Foundation, Water Sustainability and Climate (WSC) program (Grant No. EAR-1038925).

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Correspondence to Hamid Moradkhani.

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Yan, H., Moradkhani, H. A regional Bayesian hierarchical model for flood frequency analysis. Stoch Environ Res Risk Assess 29, 1019–1036 (2015). https://doi.org/10.1007/s00477-014-0975-3

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