Abstract
The characterisation and quantification of long-term sea-level variability is of considerable interest in a climate change context. Long time series from coastal tide gauges are particularly appropriate for this purpose. Long-term variability in tide gauge records is usually expressed through the linear slope resulting from the fit of a linear model to the time series, thus assuming that the generating process is deterministic with a short memory component. However, this assumption needs to be tested, since trend features can also be due to non-deterministic processes such as random walk or long range dependent processes, or even be driven by a combination of deterministic and stochastic processes. Specific methodology is therefore required to distinguish between a deterministic trend and stochastically-driven trend-like features in a time series. In this chapter, long-term sea-level variability is characterised through the application of (i) parametric statistical tests for stationarity, (ii) wavelet analysis for assessing scaling features, and (iii) generalised least squares for estimating deterministic trends. The results presented here for long tide gauge records in the North Atlantic show, despite some local coherency, profound differences in terms of the low frequency structure of these sea-level time series. These differences suggest that the long-term variations are reflecting mainly local/regional phenomena. © 2008 Springer-Verlag Berlin Heidelberg.

Abstract
This work addresses the autoregressive modelling of sea level time series from TOPEX/Poseidon satellite altimetry mission. Datasets from remote sensing applications are typically very large and correlated both in time and space. Multivariate analysis methods are useful tools to summarise and extract information from such large space-time datasets. Multivariate autoregressive analysis is a generalisation of Principal Oscillation Pattern (POP) analysis, widely used in the geosciences for the extraction of dynamical modes by eigen-decomposition of a first order autoregressive model fitted to the multivariate dataset of observations. The extension of the POP methodology to autoregressions of higher order, although increasing the difficulties in estimation, allows one to model a larger class of complex systems. Here, sea level variability in the North Atlantic is modelled by a third order multivariate autoreerressive model estimated by stepwise least squares. Eigen-decomposition of the fitted model yields physically-interpretable seasonal modes. The leading autoregressive mode is an annual oscillation and exhibits a very homogeneous spatial structure in terms of amplitude reflecting the large scale coherent behaviour of the annual pattern in the Northern hemisphere. The phase structure reflects the seesaw pattern between the western and eastern regions in the tropical North Atlantic associated with the trade winds regime. The second mode is close to a semi-annual oscillation. Multivariate autoregressive models provide a useful framework for the description of time-varying fields while enclosing a predictive potential.

Abstract
The North Atlantic Oscillation (NAO) is one of the most important climatic patterns in the Northern Hemisphere. Indices based on the normalised pressure difference between Iceland and a Southern station, such as Lisbon or Gibraltar, have been defined in order to describe NAO temporal evolution. Although exhibiting interannual and decadal variability, the signals are statistically rather featureless and therefore it is difficult to discriminate between different types of stochastic models. In this study, Lisbon and Gibraltar NAO winter indices are analysed using the discrete wavelet transform discrete wavelet transform(DWT). A multi-resolution analysis (MRA) is carried out for a scale-based description of the indices and the wavelet spectrum is used to identify and estimate long-range dependence. The degree of association of the two NAO indices is assessed by estimating the wavelet covariance for the two signals. The scale-based approach inherent to the discrete wavelet methodology allows a scale-by-scale comparison of the signals and shows that although the short-term temporal pattern is very similar for both indices, the long-term temporal structure is distinct. Furthermore, the degree of persistence or 'memory' is also distinct: the Lisbon index is best described by a long-range dependent (LRD) process, while the Gibraltar index is adequately described by a short-range process. Therefore, while trend features in the Lisbon NAO index may be explainable by long-range dependence alone, with no need to invoke external factors, for the Gibraltar index such features cannot be interpreted as resulting only from internal variability through long-range dependence. Copyright (C) 2006 Royal Meteorological Society.

Abstract
Sea level is an important parameter in climate and oceanographic applications. In this work the scaling behavior of sea level is analyzed from time series of sea level observations. The wavelet domain is particularly attractive for the identification of scaling behavior in an observed time series. The wavelet spectrum from a scale-by-scale wavelet analysis of variance reproduces in the wavelet domain the power laws underlying a scaling process, allowing the estimation of the scaling exponent from the slope of the wavelet spectrum. Here the scaling exponent is estimated in the wavelet domain for time series of sea level observations in the North Atlantic: at coastal sites from tide gauges, covering 50 years of monthly measurements, and in the open ocean from satellite altimetry, covering 12 years of satellite measurements at 10 days intervals. Both tide gauge and altimetry time series exhibit scaling behavior. Furthermore, the degree of stochastic persistence is spatially coherent and distinct at the coast and in the open ocean. Near the coast, the stochastic structure of the sea level observations is characterized by long-range dependence with a moderate degree of persistence. Larger values of the scaling exponent, consistent with weaker persistence, are concentrated in the northern Atlantic. At mid-latitudes the stochastic dependence of sea level observations is characterized by strong persistence in the form of strong long-range and 1/f dependence.

Abstract
A comparative study is carried out for sea level observations in the North Atlantic from tide gauges and satellite altimetry. Monthly tide gauge records from 12 stations in both sides of the North Atlantic from January 1993 to December 2003 and monthly time series of sea level anomalies derived from TOPEX measurements are considered. The degree of association between tide gauge and altimetry observations is analysed for different scales by computing the correlation between the sea level components resulting from a multiresolution analysis based on the maximal overlap discrete wavelet transform. A similar correlation analysis is carried out to assess the relationship between the sea level observations and climate variables: sea surface temperature, precipitation rate and wind speed. The results show that altimetry and tide gauge observations are strongly correlated, as expected, but also that the relation is scale dependent, with covariability driven by the seasonal signal for most stations. For all variables the obtained correlation patterns exhibit significant spatial variability reflecting the diversity of local conditions affecting coastal sea level.

Abstract
Sea level is a key variable in the context of global climate change. Climate-induced variability is expected to affect not only the mean sea level but also the amplitude and phase of its seasonal cycle. This study addresses the changes in the amplitude and phase of the annual cycle of coastal sea level in the extra-tropical North Atlantic. The physical causes of these variations are explored by analysing the association between fluctuations in the annual amplitude of sea level and in ancillary parameters [atmospheric pressure, sea-surface temperature and North Atlantic Oscillation (NAO) winter index]. The annual cycle is extracted through autoregressive decomposition, in order to be able to separate variations in seasonality from long-term interannual variations in the mean. The changes detected in the annual sea level cycle are regionally coherent, and related to changes in the analysed forcing parameters. At the northern sites, fluctuations in the annual amplitude of sea level are associated with concurrent changes in temperature, while atmospheric pressure is the dominant influence for most of the sites on the western boundary. The state of the NAO influences the annual variability in the Southern Bight, possibly through NAO-related changes in wind stress and ocean circulation.