# Seasonal copula models for the analysis of glacier discharge at King George Island, Antarctica

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## Abstract

Modelling glacier discharge is an important issue in hydrology and climate research. Glaciers represent a fundamental water resource when melting of ice and snow contributes to runoff. Glaciers are also studied as natural global warming sensors. GLACKMA association has implemented one of their Pilot Experimental Catchment areas at the King George Island in the Antarctica which records values of the liquid discharge from Collins glacier. In this paper, we propose the use of time-varying copula models for analyzing the relationship between air temperature and glacier discharge, which is clearly non constant and non linear through time. A seasonal copula model is defined where both the marginal and copula parameters vary periodically along time following a seasonal dynamic. Full Bayesian inference is performed such that the marginal and copula parameters are estimated in a one single step, in contrast with the usual two-step approach. Bayesian prediction and model selection is also carried out for the proposed model such that Bayesian credible intervals can be obtained for the conditional glacier discharge given a value of the temperature at any given time point. The proposed methodology is illustrated using the GLACKMA real data where there is, in addition, a hydrological year of missing discharge data which were not possible to measure accurately due to problems in the sounding.

## Keywords

Bayesian inference Copulas Glacier discharge Seasonality MCMC Melt modelling## Notes

### Acknowledgments

We would like to thank two anonymous referees for their helpful comments. We are very grateful to the GLACKMA association. The second author acknowledges financial support by UC3M-BS Institute of Financial Big Data at Universidad Carlos III de Madrid. The third author would like to thank the Russian, Argentinean, German, Uruguayan and Chilean Antarctic Programs for their continuous logistic support over the years. The crews of Bellingshausen, Artigas, and Carlini station as well as the Dallmann Laboratory provided a warm and pleasant environment during fieldwork. GLACKMA’s contribution was also partially financed by the European Science Foundation, ESF project IMCOAST (EUI2009-04068) and the Ministerio de Educación y Ciencia (CGL2007-65522-C02-01/ANT).

## References

- Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insurance 44(2):182–198Google Scholar
- Ausin MC, Lopes HF (2010) Time-varying joint distribution through copulas. Comput Stat Data Anal 54:2383–2399CrossRefGoogle Scholar
- Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stati 12:171–178Google Scholar
- Bers AV, Momo F, Schloss IR, Abele D (2013) Analysis of trends and sudden changes in long-term environmental data from King George Island (Antarctica): relationships between global climatic oscillations and local system response. Clim Chang 116:789–803CrossRefGoogle Scholar
- Braun M et al. (2002) Satellite image map of King George Island, Antarctica. Department of Physical Geography, Albert-Ludwigs-Universit Freiburg. doi: 10.1594/PANGAEA.114703
- Brechmann EC, Schepsmeier U (2013) Modeling Dependence with C- and D-Vine Copulas: the R Package CDVine. J Stat Softw 52(3): 1–27. URL:http://www.jstatsoft.org/v52/i03/
- Carnicero JA, Ausín MC, Wiper MP (2013) Non-parametric copulas for circular-linear and circular-circular data: an application to wind directions. Stoch Environ Res Risk Assess 27:1991–2002CrossRefGoogle Scholar
- Cantet P, Arnaud P (2014) Extreme rainfall analysis by a stochastic model: impact of the copula choice on the sub-daily rainfall generation. Stoch Environ Res Risk Assess 28(6):1479–1492CrossRefGoogle Scholar
- Cogley JG, Hock R, Rasmussen LA, Arendt AA, Bauder A, Braithwaite RJ, Jansson P, Kaser G, Mller M, Nicholson L, Zemp M, (2011) Glossary of glacier mass balance and related terms, IHP-VII Technical Documents in Hydrology No. 86, IACS Contribution No. 2, UNESCO-IHP, ParisGoogle Scholar
- Cong R, Brady M (2012) The interdependence between rainfall and temperature: copula analyses. Sci World J. doi: 10.1100/2012/405675 Google Scholar
- Czado C (2010) Pair-copula constructions of multivariate copulas. In: Durante F, Härdle W, Jaworki P, Rychlik T (eds) Workshop on copula theory and its applications. Springer, DortrechGoogle Scholar
- Domínguez MC, Eraso A (2007). Substantial changes happened during the last years in the icecap of King George, Insular Antactica. In: Tyk A, Stefaniak K (eds.): Karst and Cryokarst, Studies of the Faculty of Earth Sciences, University of Silesia vol 45, pp 87–110Google Scholar
- Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, BerlinCrossRefGoogle Scholar
- Embrechts P, Lindskog F, McNeil A (2001) Modellings dependence with copulas and applications to risk management. In: Rachev S (ed) Handbook of heavy tailed distributions in finance. Elsevier, Amsterdam, pp 329–384Google Scholar
- Genest C, Favre AC (2007) Everything you always wanted to know about copula but were afraid to ask. J Hydrol Eng 4:347–368CrossRefGoogle Scholar
- Gray D, Prowse T (1993) Snow and Floating Ice. In: Maidment DR (ed) Handbook of Hydrology, chap 7. McGraw-Hill, Inc., New York, pp. 1–58Google Scholar
- Hock R (2003) Temperature index melt modelling in mountain areas. J Hydrol 282:104–115CrossRefGoogle Scholar
- Hock R (2005) Glacier melt: a review of processes and their modelling. Prog Phys Geogr 29:362–391CrossRefGoogle Scholar
- Hock R, Jansson P, Braun L (2005) Modelling the response of mountain glacier discharge to climate warming. In: Huber UM, Bugmann HKM, Reasoner MA (eds) Global change and mountain regions—A state of knowlegde overview. Springer, DordrechtGoogle Scholar
- Jansson P, Hock R, Schneider T (2003) The concept of glacier storage: a review. J Hydrol 282:116–129CrossRefGoogle Scholar
- Kazianka H, Pilz J (2010) Copula-based geostatistical modeling of continuous and discrete data including covariates. Stoch Environ Re Risk Assess 24:661–673CrossRefGoogle Scholar
- La Frenierre J, Mark BG (2014) A review of methods for estimating the contribution of glacial meltwater to total watershed discharge. Prog Phys Geogr 38(2):173–200CrossRefGoogle Scholar
- Nelsen RB (2006) An introduction to Copulas, 2nd edn. Springer, New YorkGoogle Scholar
- Nazemi A, Elshorbagy A (2012) Application of copula modelling to the performance assessment of reconstructed watersheds. Stoch Environ Res Risk Assess 26(2):189–205CrossRefGoogle Scholar
- Osmanoglu B, Braun M, Hock R, Navarro FJ (2013) Surface velocity and ice discharge of the ice cap on King George Island, Antarctica. Ann Glaciol 54(63):111–119CrossRefGoogle Scholar
- Ohmura A (2001) Physical basis for the temperature-based melt-index method. J Appl Meteorol 40:753–761CrossRefGoogle Scholar
- Patton A (2012) A review of copula models for economic time series. J Multivar Anal 110:4–18CrossRefGoogle Scholar
- Pellicciotti F, Brock B, Strasser U, Burlando P, Funk M, Corripio J (2005) An enhanced temperature-index glacier melt model including the shortwave radiation balance: development and testing for Haut Glacier dArolla. Switzerland. J Glaciol 51:573587CrossRefGoogle Scholar
- R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
- Rückamp M, Braun M, Suckro S, Blindow N (2011) Observed glacial changes on the King George Island ice cap, Antarctica, in the last decade. Glob Planet Chang 79:99–109CrossRefGoogle Scholar
- Saad C, St-Hilaire A, El Adlouni S, Gachon P (2014) A nested multivariate copula approach to hydrometeorological simulations of spring floods: the case of the Richelieu River (Quebec, Canada) record flood. Stoch Environ Resour Risk Assess 29:275–294CrossRefGoogle Scholar
- Scholzel C, Friederichs P (2008) Multivariate non-normally distributed random variables in climate research—introduction to the copula approach. Nonlinear Process Geophys 15:761–772CrossRefGoogle Scholar
- Spiegelhalter DJ, Best NG, Carlin BP, Van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). J R Stat Soc, Ser B 64(4):583–639CrossRefGoogle Scholar
- Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 22(1701):1762Google Scholar
- Toews MW, Whitfield PH, Allen DM (2007) Seasonal statistics: the ‘seas’ package for R. Comput Geosci 33:944–951CrossRefGoogle Scholar
- Venter, G., (2001). Tails of copulas. In Proceedings ASTIN, Washington, USA, pp 68–113Google Scholar
- Warburton J, Fenn CR (1994) Unusual flood events from an Alpine glacier: observations and deductions on generating mechanisms. J Glaciol 40(134):176–186CrossRefGoogle Scholar
- Willis IC, Arnold NS, Brock BW (2002) Effect of snowpack removal on energy balance, melt and runoff in a small supraglacial catchment. Hydrol Process 16:272149CrossRefGoogle Scholar
- Whitfield PH, Bodtker K, Cannon AJ (2002) Recent variations in seasonality of temperature and precipitation in Canada, 197695. Int J Climatol 22:16171644CrossRefGoogle Scholar
- Zhang Q, Xiao M, Singh V, Chen X (2013) Copula-based risk evaluation of hydrological droughts in the East River basin, China. Stoch Environ Res Risk Assess 27(6):1397–1406CrossRefGoogle Scholar