INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 30: 1423–1430 (2010) Published online 7 July 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/joc.1981 Short Communication Hierarchical Bayesian modelling of wind and sea surface temperature from the Portuguese coast Ricardo T. Lemos, a,b * Bruno Sans´ o c and F. D. Santos d a Instituto de Oceanografia, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b Maretec - Instituto Superior T´ ecnico, Universidade T´ ecnica de Lisboa, Sec¸ c˜ ao de Ambiente e Energia, Dpt. Mecˆ anica, Av. Rovisco Pais, 1049-001 Lisboa, Portugal c Department of Applied Mathematics and Statistics, University of California, 1156 High St. MS:SOE2, Santa Cruz, CA-95064, USA d SIM, Faculdade de Ciˆ encias da Universidade de Lisboa, Campo Grande, Edifcio C1, Piso 4, 1749-016 Lisboa, Portugal ABSTRACT: In this work, we revisit a recent analysis that pointed to an overall relaxation of the Portuguese coastal upwelling system, between 1941 and 2000, and apply more elaborate statistical techniques to assess that evidence. Our goal is to fit a model for environmental variables that accommodate seasonal cycles, long-term trends, short-term fluctuations with some degree of autocorrelation, and cross-correlations between measuring sites and variables. Reference cell coding is used to investigate similarities in behaviour among sites. Parameter estimation is performed in a single modelling step, thereby producing more reliable credibility intervals than previous studies. This is of special importance in the assessment of trend significance. We employ a Bayesian approach with a purposely developed Markov chain Monte Carlo method to explore the posterior distribution of the parameters. Our results substantiate most previous findings and provide new insight on the relationship between wind and sea surface temperature off the Portuguese coast. Copyright  2009 Royal Meteorological Society KEY WORDS upwelling; wind; SST; space-time models; Bayesian modelling Received 6 December 2008; Revised 10 June 2009; Accepted 11 June 2009 1. Introduction The central task of climate change detection studies is to determine whether the observed changes or trends in environmental time series are ‘significant’, that is, highly unusual relative to the background of natural variability, and unlikely to have occurred by chance alone (Santer et al., 1996). Because most statistical models hinge on the assumption that the resulting residuals are inde- pendent and identically distributed (viz. Gaussian white noise), they are required to incorporate the most impor- tant sources of variability in the observed data; otherwise, this assumption is not verified and inference about the sig- nificance of trends is compromised. Commonly, environ- mental processes operate on various spatial and temporal scales, producing time series that are impregnated with a number of features that make statistical modelling a chal- lenging task. These include cycles, long-term linear and non-linear trends, short-term memory, spatial covariance and crossed covariance. When starting from a simple statistical model that aims to detect long-term trends (viz. a model with intercepts, * Correspondence to: Ricardo T. Lemos, Maretec - Instituto Superior T´ ecnico, Universidade T´ ecnica de Lisboa, Sec¸ c˜ ao de Ambiente e Energia, Dpt. Mecˆ anica, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: rtl@net.sapo.pt trends and uncorrelated Gaussian errors), red noise resid- uals are bound to result. In face of that, one of the three approaches may be followed: (1) modify the model, usually by making it more complex; (2) modify the input, by thinning the data set (Szunyogh et al., 2008) or pre- whitening (Rodionov, 2006); (3) modify the output, by correcting estimates based on the effective sample size (von Storch and Zwiers, 1999). At some point in (1), parsimony becomes an issue, which may occur before the residuals conform to white noise. If the residual struc- ture is small when compared to the modelled counterpart, ‘redness’ is regarded as a nuisance property of residuals, and the modeller attempts to eliminate it by means of (2) or (3). In cases where residuals still contain relevant information, a different approach is needed to include it in the model. Such was, for example, the conclu- sion of Lemos and Pires (2004, henceforth, LP04), who used ordinary least squares regression to analyse wind and sea surface temperature (SST) data in the west Por- tuguese coast (37–42 ° N, 9–15 ° W) and found significant and readily interpretable amounts of information in the residuals. This paper describes a more comprehensive method, based on hierarchical Bayesian modelling, which accom- modates spatial, temporal and cross-covariance structures, Copyright  2009 Royal Meteorological Society