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

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.

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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).

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Correspondence to Mario Gómez.

Appendix: Algorithm

Appendix: Algorithm

In this appendix, we explain in detail the proposed MCMC algorithm to sample from the posterior distribution of the model parameters, \(\varvec{\vartheta }=(\varvec{\vartheta }_X,\varvec{\vartheta }_Y,\varvec{\vartheta }_C).\) Recall that the loglikelihood is given by:

$$\log l\left( \varvec{ \vartheta }\mid {\mathbf{x,y}}\right)=\sum \limits _{t}\log \left( f_{X}(x_{t}\mid \varvec{\vartheta }_X)\right)$$
(16)
$$+\sum \limits _{t:y_{t}>0}\log \left( f_{Y}(y_{t}\mid \varvec{\vartheta }_Y)\right)$$
(17)
$$+\sum \limits _{t:y_{t}>0}\log \left( c(F_{X}(x_{t}\mid \varvec{\vartheta }_X ),F_{Y}(y_{t} \mid \varvec{\vartheta }_Y)\mid \varvec{\vartheta }_C)\right)$$
(18)
$$+\sum \limits _{t:y_{t}=0}\log \left( C_{t}^{1}(F_{X}(x_{t}\mid \varvec{\vartheta }_X),F_{Y}(y_{\min }\mid \varvec{\vartheta }_Y)\mid \varvec{\vartheta }_C)\right),$$
(19)

We construct a Gibbs sampling scheme where each model parameter is updated separately. Therefore, it is not necessary to compute the whole likelihood for each parameter. In particular, when updating the parameters corresponding to the temperature, \(\varvec{\vartheta }_X\), it is only necessary to consider (16), (18) and (19). When updating the discharge parameters, \(\varvec{\vartheta }_Y\), only (17), (18) and (19) are evaluated. And finally, for updating the copula parameters, \(\varvec{\vartheta }_C\), only (18) and (19) are considered.

The structure of the proposed MCMC method is shown in Algorithm 1. Firstly, it is required to set a vector of initial values for the parameters and the number of MCMC iterations. Then, in each step of the algorithm, each model parameter is updated using a RWMH which is defined in Algorithm 2. Observe that the algorithm is written such that it is not necessary to recalculate the likelihood that was evaluated in previous step for accepted parameters. Finally, Algorithms 3, 4 and 5 separates the computation of the likelihood as the sum of the log-likelihood temperature, discharge and copula, respectively.

These algorithms have been programmed in software R (R Core Team 2013) with the help of the CDVine package (Brechmann and Schepsmeier 2013).

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Gómez, M., Concepción Ausín, M. & Carmen Domínguez, M. Seasonal copula models for the analysis of glacier discharge at King George Island, Antarctica. Stoch Environ Res Risk Assess 31, 1107–1121 (2017). https://doi.org/10.1007/s00477-016-1217-7

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Keywords

  • Bayesian inference
  • Copulas
  • Glacier discharge
  • Seasonality
  • MCMC
  • Melt modelling