A regional comparative analysis of empirical and theoretical flood peak-volume relationships

Open access


This paper analyses the bivariate relationship between flood peaks and corresponding flood event volumes modelled by empirical and theoretical copulas in a regional context, with a focus on flood generation processes in general, the regional differentiation of these and the effect of the sample size on reliable discrimination among models. A total of 72 catchments in North-West of Austria are analysed for the period 1976–2007. From the hourly runoff data set, 25 697 flood events were isolated and assigned to one of three flood process types: synoptic floods (including long- and short-rain floods), flash floods or snowmelt floods (both rain-on-snow and snowmelt floods). The first step of the analysis examines whether the empirical peak-volume copulas of different flood process types are regionally statistically distinguishable, separately for each catchment and the role of the sample size on the strength of the statements. The results indicate that the empirical copulas of flash floods tend to be different from those of the synoptic and snowmelt floods. The second step examines how similar are the empirical flood peak-volume copulas between catchments for a given flood type across the region. Empirical copulas of synoptic floods are the least similar between the catchments, however with the decrease of the sample size the difference between the performances of the process types becomes small. The third step examines the goodness-of-fit of different commonly used copula types to the data samples that represent the annual maxima of flood peaks and the respective volumes both regardless of flood generating processes (the traditional engineering approach) and also considering the three process-based classes. Extreme value copulas (Galambos, Gumbel and Hüsler-Reiss) show the best performance both for synoptic and flash floods, while the Frank copula shows the best performance for snowmelt floods. It is concluded that there is merit in treating flood types separately when analysing and estimating flood peak-volume dependence copulas; however, even the enlarged dataset gained by the process-based analysis in this study does not give sufficient information for a reliable model choice for multivariate statistical analysis of flood peaks and volumes.

AghaKouchak, A., 2014. Entropy-copula in hydrology and climatology. Journal of Hydrometeorology, 15, 2176–2189. doi:10.1175/JHM-D-13-0207.1.

Bačová Mitková, V., 2012. Vzájomný vzťah objemu a dĺžky trvania povodňových vĺn. [The relationship between volume of the flood wave and the time duration of flood events]. Acta Hydrologica Slovaca, 13, 1, 165–174. (In Slovak.)

Bačová Mitková, V., Halmová, D., 2014. Joint modeling of flood peak discharges, volume and duration: a case study of the Danube River in Bratislava. Journal of Hydrology and Hydromechanics, 62, 3, 186–196. doi:10.2478/johh-2014-0026.

Ben-Aissia, M.-A., Chebana, F., Ouarda, T.B.M.J., Roy, L., Desrochers, G., Chartier, I., Robichaud, É., 2012. Multivariate analysis of flood characteristics in a climate change context of the watershed of the Baskatong reservoir, Province of Québec, Canada. Hydrological Processes, 26, 130–142. doi:10.1002/hyp.8117.

Ben Aissia, M.-A., Chebana, F., Ouarda, T.B.M.J., Bruneau, P., Barbet, M., 2015. Bivariate index-flood model: case study in Québec, Canada. Hydrological Sciences Journal, 60, 2, 247–268. doi:10.1080/02626667.2013.875177.

Bezak, N., Mikoš, M., Šraj, M., 2014. Trivariate frequency analyses of peak discharge, hydrograph volume and suspended sediment concentration data using copulas. Water Resources Management, 28, 8, 2195–2212. doi:10.1007/s11269-014-0606-2.

Blöschl, G., Viglione, A., Merz, R., Parajka, J., Salinas, J.L., Schöner, W., 2011. Auswirkungen des Klimawandels auf Hochwasser und Niederwasser. [Climate impacts on floods and low flows]. Österreichische Wasser- und Abfallwirtschaft, 63, 1, 21–30. doi:10.1007/s00506-010-0269-z. (In German.)

Chapman, T.G., Maxwell, A.I., 1996. Baseflow separation—comparison of numerical methods with tracer experiments. In: 23rd Hydrology and Water Resources Symposium: Water and the Environment, Natl. Conf. Publ., 96/05, Inst. of Eng., Barton, A.C.T., Australia, pp. 539–545.

Chebana, F., Ouarda, T.B.M.J., 2007. Multivariate L-moment homogeneity test. Water Resources Research, 43, W08406. doi:10.1029/2006WR005639.

Chebana, F., Ouarda, T.B.M.J., 2009. Index flood-based multivariate regional frequency analysis. Water Resources Research, 45, W10435. doi 10.1029/2008WR007490.

Chowdhary, H., Escobar, L.A., Singh, V.P., 2011. Identification of suitable copulas for bivariate frequency analysis of flood peak and flood volume data. Hydrology Research, 42, 2–3, 193–216. doi:10.2166/nh.2011.065.

Dalrymple, T., 1960. Flood frequency analysis. U.S. Geological Survey Water-Supply Paper, 1543-A, 80 p.

Favre, A.-C., El Adlouni, S., Perreault, L., Thiémonge, N., Bobée, B., 2004. Multivariate hydrological frequency analysis using copulas. Water Resources Research, 40, W01101. doi:10.1029/2003WR002456.

Gaál, L., Szolgay, J., Kohnová, S., Parajka, J., Merz, R., Viglione, A., Blöschl, G., 2012. Flood timescales: Understanding the interplay of climate and catchment processes through comparative hydrology. Water Resources Research, 48, 4, W04511. doi:10.1029/2011WR011509.

Gaál, L., Kohnová, S., Szolgay, J., 2013. Regional flood frequency analysis in Slovakia: Which pooling approach suits better? In: Klijn, F., Schweckendiek, T. (Eds.): Comprehensive Flood Risk Management: Research for Policy and Practice. London, CRC Press/Balkema, pp. 27–30.

Gaál, L., Szolgay, J., Kohnová, S., Hlavčová, K., Parajka, J., Viglione, A., Merz, R., Blöschl, G., 2014. Dependence between flood peaks and volumes – A case study on climate and hydrological controls. Hydrological Sciences Journal, 60, 6, 968–984. doi:10.1080/02626667.2014.951361.

Ganguli, P., Reddy, M.J., 2013. Probabilistic assessment of flood risks using trivariate copulas. Theoretical and Applied Climatology, 111, 341–360. doi:10.1007/s00704-012-0664-4.

Genest, C., Favre, A.-C., 2007. Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, 4, 47–368. doi: 10.1061/(ASCE)1084-0699(2007)12:4(347).

Genest, C., Rémillard, B., Beaudoin, D., 2009. Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199–213. doi:10.1016/j.insmatheco.2007.10.005.

Gräler, B., van den Berg, M.J., Vandenberghe, S., Petroselli, A., Grimaldi, S., De Baets, B., Verhoest, N.E.C., 2013. Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation. Hydrology and Earth System Sciences, 17, 1281–1296. doi:10.5194/hess-17-1281-2013.

Grimaldi, S., Serinaldi, F., 2006. Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources, 29, 8, 1155–1167. doi: 10.1016/j.advwatres.2005.09.005.

Grimaldi, S., Petroselli, A., Salvadori, G., De Michele, C., 2016. Catchment compatibility via copulas: A nonparametric study of the dependence structures of hydrological responses. Advances in Water Resources, 90, 116–133. doi:10.1016/j.advwatres.2016.02.003.

Hosking, J.R.M., Wallis, J.R., 1997. Regional Frequency Analysis: An Approach Based on L-moments, Cambridge University Press, Cambridge, UK.

IH, 1999. Flood Estimation Handbook. Institute of Hydrology: Wallingford, UK.

Karmakar, S., Simonovic, S.P., 2009. Bivariate flood frequency analysis. Part 2: A copula-based approach with mixed marginal distributions. Journal of Flood Risk Management, 2, 32–44. doi:10.1111/j.1753-318X.2009.01020.x.

Kendall, M.G., 1955. Rank Correlation Methods. Hafner Publishing, New York.

Kohnová, S., Szolgay, J., 1999. Regional estimation of design summer flood discharge in small catchments of northern Slovakia. In: Gottschalk, L., Olivry, C., Reed, D., Rosbjerg, D. (Eds.): Hydrological Extremes: Understanding, Predicting, Mitigating. IAHS publ. 255, IAHS Press, Wallingford, pp. 265–268.

Ljung, G.M., Box, G.E.P., 1978. On a measure of lack of fit in time series models. Biometrika, 65, 297–303. doi:10.1093/biomet/65.2.297.

Merz, R., Blöschl, G., 2003. A process typology of regional floods. Water Resources Research, 39, 12, 1340–1347. doi:10.1029/2002WR001952.

Merz, R., Blöschl, G., 2009. A regional analysis of event runoff coefficients with respect to climate and catchment characteristics in Austria. Water Resources Research, 45, 1, W01415. doi:10.1029/2008WR007163.

Merz, R., Blöschl, G., Parajka, J., 2006. Spatio-temporal variability of event runoff coefficients. Journal of Hydrology, 331, 3–4, 591–604. doi:10.1016/j.jhydrol.2006.06.008.

Nelsen, R.B., 2006. An Introduction to Copulas. 2nd edition. Springer-Verlag, New York.

Parajka, J., Merz, R., Blöschl, G., 2007. Uncertainty and multiple objective calibration in regional water balance modelling – Case study in 320 Austrian catchments. Hydrological Processes, 21, 435–446. doi:10.1002/hyp.6253.

Parajka, J., Kohnová, S., Bálint, G., Barbuc, M., Borga, M., Claps, P., Cheval, S., Dumitrescu, A., Gaume, E., Hlavčová, K., Merz, R., Pfaundler, M., Stancalie, G., Szolgay, J., Blöschl, G., 2010. Seasonal characteristics of flood regimes across the Alpine–Carpathian range. Journal of Hydrology, 394, 1–2, 78–89. doi:10.1016/j.jhydrol.2010.05.015.

Pham, M.T., Vernieuwe, H., Baets, B.D., Willems, B., Verhoest, N.E.C., 2015. Stochastic simulation of precipitation-consistent daily reference evapotranspiration using vine copulas. Stochastic Environmental Research and Risk Assessment, 18 p. doi: 10.1007/s00477-015-1181-7.

Poulin, A., Huard, D., Favre, A.-C., Pugin, S., 2007. Importance of tail dependence in bivariate frequency analysis. Journal of Hydrologic Engineering, 12, 4, 394–403. doi:10.1061/(ASCE)1084-0699(2007)12:4(394).

Reddy, M.J., Ganguli, P., 2012. Bivariate flood frequency analysis of Upper Godavari River flows using Archimedean copulas. Water Resources Management, 26, 14, 3995–4018. doi:10.1007/s11269-012-0124-z.

Remillard, B., Plante, J.-F., 2012. TwoCop: Nonparametric test of equality between two copulas. R package version 1.0 (19-10-2012). http://cran.r-project.org/web/packages/TwoCop.

Remillard, B., Scaillet, O., 2009. Testing for equality between two copulas. Journal of Multivariate Analysis, 100, 377–386. doi:10.1016/j.jmva.2008.05.004.

Requena, A.I., Chebana, F., Mediero, L., 2016. A complete procedure for multivariate index-flood model application. Journal of Hydrology, 535, 559–580. doi:10.1016/j.jhydrol.2016.02.004.

Rosbjerg, D., Blöschl, G., Burn, D.H., Castellarin, A., Croke, B., DiBaldassarre, G., Iacobellis, V., Kjeldsen, T.R., Kuczera, G., Merz, R., Montanari, A., Morris, D., Ouarda, T.B.M.J., Ren, L., Rogger, M., Salinas, J.L., Toth, E., Viglione, A., 2013. Prediction of floods in ungauged basins. Chapter 9. In: Blöschl, G., Sivapalan, M., Wagener, T., Viglione, A., Savenije, H. (Eds.): Runoff Prediction in Ungauged Basins - Synthesis across Processes, Places and Scales. Cambridge University Press, Cambridge, UK, pp. 135–162.

Salinas, J.L., Castellarin, A., Viglione, A., Kohnová, S., Kjeldsen, T.R., 2014. Regional parent flood frequency distributions in Europe – Part 1: Is the GEV model suitable as a pan-European parent? Hydrology and Earth System Sciences, 18, 4381–4389. doi:10.5194/hess-18-4381-2014.

Serinaldi, F., 2013. An uncertain journey around the tails of multivariate hydrological distributions. Water Resources Research, 49, 10, 6527–6547. doi:10.1002/wrcr.20531.

Serinaldi, F., 2015. Can we tell more than we can know? The limits of bivariate drought analysis in the United States. Stochastic Environmental Research and Risk Assessment, 14 p. doi:10.1007/s00477-015-1124-3.

Serinaldi, F., Kilsby, C.G., 2013. The intrinsic dependence structure of peak, volume, duration, and average intensity of hyetographs and hydrographs. Water Resources Research, 49, 3423–3442. doi:10.1002/wrcr.20221.

Shiau, J.-T., Wang, H.-Y., Tsai, C.-T., 2006. Bivariate flood frequency analysis of floods using copulas. Journal of the American Water Resources Association, 42, 6, 1549–1564. doi:10.1111/j.1752-1688.2006.tb06020.x.

Sraj, M., Bezak, N., Brilly, M., 2014. Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River. Hydrological Processes, 29, 2, 225–238. doi:10.1002/hyp.10145.

Szolgay, J., Gaál, L., Bacigál, T., Kohnová, S., Hlavčová, K., Výleta, R., Blöschl, G., 2016. A regional look at the selection of a process-oriented model for flood peak/volume relationships. IAHS publ. 373. IAHS Press, Wallingford, pp. 61–69. doi: 10.5194/piahs-373-1-2016.

Szolgay, J., Gaál, L., Kohnová, S., Hlavčová, K., Výleta, R., Bacigál, T., Blöschl, G., 2015. A process-based analysis of the suitability of copula types for peak-volume flood relationships. Proc. IAHS, 370, 183–188. doi: 10.5194/piahs-370-183-2015.

Vernieuwe, H., Vandenberghe, S., De Baets, B., Verhoest, N.E.C., 2015. A continuous rainfall model based on vine copulas. Hydrology and Earth System Sciences, 19, 6, 2685–2699. doi:10.5194/hess-19-2685-2015.

Werner, P.C., Gerstengarbe, F.-W., 2010. Katalog der Grosswetterlagen Europas (1881–2009) nach Paul Hess und Helmut Brezowsky, 7., verbesserte und ergänzte Auflage. [Catalog of Large Weather Conditions of Europe (1881-2009), after Paul Hess and Helmut Brezowsky, 7th, Improved and Completed Edition]. PIK-Report No. 119, Potsdam Institute for Climate Impact Research, Potsdam, 146 p. (In German.)

Zhang, L., Singh, V.P., 2006. Bivariate flood frequency analysis using the copula method. Journal of Hydrologic Engineering 11, 150–164. doi:10.1061/(asce)1084-0699(2006)11:2(150).

Journal of Hydrology and Hydromechanics

The Journal of Institute of Hydrology SAS Bratislava and Institute of Hydrodynamics CAS Prague

Journal Information

IMPACT FACTOR 2017: 1.714
5-year IMPACT FACTOR: 1.639

CiteScore 2017: 1.91

SCImago Journal Rank (SJR) 2017: 0.599
Source Normalized Impact per Paper (SNIP) 2017: 1.084

Cited By


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 294 294 56
PDF Downloads 204 204 52