Quantile trends in Baltic sea level

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L22704, doi: /2008gl035182, 2008 Quantile trends in Baltic sea level Susana M. Barbosa 1 Received 29 June 2008; revised 7 October 2008; accepted 14 October 2008; published 22 November [1] Quantile regression is applied for characterizing longterm sea-level variability in the Baltic Sea from long tide gauge records. The approach allows to quantify not only variability in the mean but also in extreme heights and thus provides a more complete description of regional sea-level variability. In the Baltic, slopes in minima are similar to the classical mean-based ordinary least squares slope, but maxima exhibit larger trends, particularly at the northernmost stations, in the Gulf of Bothnia, likely associated with changes in north Atlantic atmospheric circulation and particularly regional wind patterns. Citation: Barbosa, S. M. (2008), Quantile trends in Baltic sea level, Geophys. Res. Lett., 35, L22704, doi: / 2008GL Introduction [2] The measurement of sea level has a long tradition in the Baltic Sea region. Tide gauge records of relative sealevel heights have been used for the study of the glacial isostatic adjustment of Fennoscandia [Vermeer et al., 1988; Ekman, 1991, 1996; Vestøl, 2006] and for the study of regional sea level and climatic variability [Ekman and Stigebrandt, 1996; Samuelsson and Stigebrandt, 1996; Ekman, 1999; Plag and Tsimplis, 1999; Ekman, 2003, 2005; Johansson et al., 2001; Chen and Omstedt, 2005; Hünicke and Zorita, 2006, 2008; Ekman, 2007]. [3] The quantification of sea-level long-term variability is a challenging task [e.g., Barbosa et al., 2008]. The analysis of long tide gauge records is often focused on the characterization of long-term variability through linear trends estimated by ordinary least squares. However, the commonly adopted assumption of linearity can be unrealistic [Barbosa et al., 2004; Jevrejeva et al., 2006; Holgate, 2007]. Furthermore, even adopting a linear description for sealevel long-term variability, ordinary least squares only provide information on the mean slope, while variability in other parts of the sea-level distribution can be equally or even more important than trends in mean levels - e.g., changes in extreme high waters can impact considerably coastal locations and populations. [4] Quantile regression [Koenker and Basset, 1978] was developed as an extension to the ordinary linear regression model for estimating rates of change not only in the mean but in all parts of the data distribution. Just as the mean gives an incomplete picture of the data distribution, so the regression curve gives a corresponding incomplete picture [Koenker and Hallock, 2001, p. 154]. By allowing to derive slopes at different quantiles of the data distribution, quantile 1 Departamento de Matematica Aplicada, Faculdade de Ciencias, Universidade do Porto, Porto, Portugal. Copyright 2008 by the American Geophysical Union /08/2008GL regression is able to provide a more complete description of the long-term temporal variability in a time series. [5] In this work, quantile regression is applied to describe sea-level trends in long tide gauge records from the Baltic Sea. The rest of the paper is organized as follows: the quantile regression approach is summarized in section 2. The analyzed sea-level records are described in section 3 and results in the form of quantile trends are presented in section 4. Finally, concluding remarks are given in section Quantile Regression [6] The quantile regression method has been extensively described, mainly in econometrics contexts [Koenker and Basset, 1978; Koenker and D Orey, 1987; Koenker, 2000; Koenker and Hallock, 2001] and in a few ecological applications [Cade and Noon, 2003; Chamaillé-Jammes et al., 2007]. Here, a brief overview of the approach is provided, the reader is referred to the original references for further details. [7] Quantile regression can be viewed as an extension of classical ordinary least squares regression. Within the classical linear regression framework, the response variable (e.g., sea level or GPS heights), denoted by Y, is related linearly with time (t), Y = bt + g with b denoting the linear slope and g the constant intercept, i.e. the variable Y can be written as a function of time t and the unknown parameters b and g, Y = f(b, g, t). Then, the parameters b and g are estimated from the ordinary least squares estimate of the mean of the response variable Y conditional on t, E[Yjt] and therefore obtained by minimizing the sum of squared residuals X ½y i f ðb; g; t i ÞŠ 2 ð1þ i Quantile regression can be understood in a similar manner, by going from conditional mean functions to conditional quantile functions, i.e. replacing the mean of the response variable Y conditional on t, E[Yjt], by the quantile of the response variable Y conditional on t, the conditional quantile function Q[tjt]. Then, for each quantile t, the linear quantile regression model can be written as Y = f 0 (b t, g t, t), and the quantile slope b t and the intercept g t are estimated from the conditional quantile function by minimizing the sum of asymmetrically weighted absolute residuals X r t ½y i f 0 ðb t ; g t ; t i ÞŠ ð2þ i where r t is the tilted absolute value function [Koenker and Hallock, 2001]. L of6

2 Table 1. Analyzed Tide Gauge Records Tide Gauge Period (yrs) Longitude ( E) Latitude ( N) 1 Kungholmsfort Ölands N. Udde Landsort Stockholm Ratan Oulu Helsinki Warnemünde Wismar Gedser København Hornbaek [8] In this work, quantile regression is carried out with R using package quantreg. 3. Data [9] The Baltic Sea has one of the world s densest observational networks and a remarkable number of long and highquality sea-level records. In this study, very long (>100 years) and continuous records of relative sea-level heights from the Baltic Sea area are analyzed (Table 1 and Figure 1). [10] Monthly time series of relative sea-level heights are obtained from the Permanent Service for Mean Sea Level (PSMSL) dataset of Revised Local Reference (RLR) tide gauge measurements [Woodworth and Player, 2003]. As a pre-processing step, the mean seasonal cycle is computed by averaging over the whole period the observations for each month, and is subtracted to each sea-level record. 4. Results [11] For each tide gauge record, quantiles slopes b t are estimated using a modified version of the Barrodale and Roberts algorithm [Koenker and D Orey, 1987, 1994]. As an illustration, slopes at quantiles 0.01, 0.5 and 0.99, corresponding respectively to the lowest 1% (minima), 50% (median) and 99% (maxima) of the sorted sea-level observations are represented in Figure 2 and displayed in Table 2. For comparison, the slope derived using the usual ordinary least squares (OLS) approach is also included in Table 2. Standard errors for the quantile slopes are obtained by computing a Huber sandwich estimate [Huber 1967] using a local estimate of the sparsity [Koenker and Machado, 1999]. Note that since sample variation increases away from the center of the distribution, the standard errors are larger for the more extreme quantiles. Table 2 shows that quantile slopes are considerably different from the ordinary least squares estimate, particularly at some locations (e.g., Oulu). [12] Table 3 displays the differences between the median quantile and extreme quantiles slopes, as well as between the median slope and the classical ordinary least squares slope. A variant of the Wald statistical test [Koenker and Basset, 1982; Gutenbrunner et al., 1993] is applied to test whether the deviations between the quantile regression slopes are due to sampling variation. [13] As the median is a more robust measure of location for a data distribution, so the median (quantile 0.5) slope is more robust than the mean-based ordinary least squares slope. The b 0.5 and b OLS slopes are very similar, suggesting symmetry in the data distribution. However, the classical estimates are larger than the quantile regression estimate for the median quantile, and the difference between the two values, although below statistical confidence thresholds, exhibits spatial coherency, with larger differences (more asymmetric relative sea-level heights) in the Gulf of Bothnia and near the Baltic entrance. [14] The differences between median and lower quantile slopes are not statistically significant at the 95% confidence level at all locations except Ratan and Hornbaek, indicating that at most tide gauges the rate of change of lowest relative heights is similar to the rate of change of mean relative sealevel heights. However, the upper quantile slopes are larger than the median slope, i.e. maximum relative sea-level heights are increasing faster than mean sea-level heights and the difference is statistically significant, at the 95% confidence level, for the northern stations. [15] These features, particularly the increase in the slope of relative sea-level heights for upper quantiles, are confirmed in Figure 3, showing the quantile regression slopes for quantiles 0.1 to 0.9 (in steps of 0.02), along with corresponding standard errors. The horizontal line in each plot depicts the ordinary least squares slope estimated from each record and the corresponding 95% confidence interval (dashed horizontal lines). The increase in the trends for highest relative heights are particularly evident at Stockholm, Ratan, Oulu and Helsinki. 5. Discussion and Conclusions [16] Quantile trends provide a more complete description of regional sea-level long-term variability than classical Figure 1. Study area and tide gauge locations: 1- Kungholmsfort; 2-Ölands Norra Udde; 3-Landsort; 4- Stockholm; 5-Ratan; 6-Oulu; 7-Helsinki; 8-Warnemunde; 9-Wismar; 10-Gedser; 11-København; 12-Hornbaek. 2of6

3 Figure 2. Time series of relative sea-level heights and sea level trends at quantiles 0.01 (dotted line), 0.5 (solid line) and 0.99 (dashed line). 3of6

4 Table 2. Sea-Level Quantile Slopes and Corresponding Standard Errors a Tide Gauge b t=0.01 b t=0.5 b OLS b t=0.99 Kungholmsfort (0.69) (0.11) (0.11) (0.55) Ölands N. Udde (0.27) (0.14) (0.13) (0.67) Landsort (0.58) (0.14) (0.13) (0.60) Stockholm (0.68) (0.15) (0.13) (0.66) Ratan (0.47) (0.16) (0.15) (0.86) Oulu (0.44) (0.17) (0.16) (0.84) Helsinki (0.81) (0.14) (0.15) (0.71) Warnemünde (0.35) (0.084) (0.077) (0.65) Wismar (0.32) (0.077) (0.075) (0.56) Gedser (0.37) (0.086) (0.082) (0.66) København (0.39) (0.094) (0.082) (0.50) Hornbaek (0.33) (0.097) (0.09) (0.54) a Units are given in mm/yr. ordinary least squares trends, allowing to quantify not only sea-level variability in the mean but also in extreme heights. In the Baltic Sea, the results show a larger asymmetry in the distribution of relative sea-level heights for the tide gauges in the Gulf of Bothnia and near the Baltic entrance. Furthermore, although the temporal evolution of lower quantile heights is similar to the temporal evolution of the mean sea level, the increase in maximum relative heights is larger than the rate of change in the mean, in good agreement with Johansson et al. [2001], that found an increase in maximum levels in the Baltic Sea but no trends for the minima. [17] On long time-scales, sea-level variability in the Baltic sea is mainly determined by the exchange of water through the Danish Straits driven by the sea-level difference between the Kattegat and the Baltic proper [Samuelsson and Stigebrandt, 1996], which in turn is influenced by the zonal wind over the Skagerrak and the North Sea [Andersson, 2002]. Thus sea-level long-term variability in the Baltic is mainly governed by the persistent winds over the North Sea and the Baltic entrance driving sea water into or out of the Baltic, depending on the direction of the wind, the effect being larger at the innermost parts of the Baltic [e.g., Ekman, 2007]. Therefore, changes in regional wind patterns, associated to changes in north Atlantic atmospheric circulation patterns, can explain the larger slopes in maximum relative heights observed at the northernmost stations. Sea level in the Baltic is associated with the North Atlantic oscillation index [Johansson et al., 2001; Lehmann et al., 2002; Andersson, 2002; Jevrejeva et al., 2005] and the NAO is expected to impact high-levels, particularly in the northernmost stations, via changes in regional atmospheric pressure and particularly wind climate [Suursaar and Sooäär, 2007; Woodworth et al., 2007]. [18] Quantile regression is a useful approach for identifying distinct rates of change in geophysical time series, and should be included in the arsenal of techniques for quanti- Table 3. Difference Between the Median (Quantile 0.5) Slopes and the OLS, Lower, and Upper Quantile Slopes Tide Gauge b OLS b t=0.5 b 0.01 b t=0.5 b 0.99 b t=0.5 Kungholmsfort Ölands N. Udde Landsort a Stockholm a Ratan a 2.44 a Oulu a Helsinki a Warnemünde Wismar Gedser København Hornbaek a 0.35 a Statistically significant at the 95% confidence level, according to the Wald test. 4of6

5 Figure 3. Quantile slopes (points) and corresponding standard errors (vertical error bars). The horizontal (solid) line indicates the ordinary least squares slope, with the corresponding 95% confidence interval (dashed lines). 5of6

6 fying long-term variability in climatic and oceanographic variables. [19] Acknowledgments. Thanks to Ole B. Andersen, Per Knudsen and M. Eduarda Silva for fruitful scientific discussions. M. Ekman and J. Mäkinen kindly provided access to publications from the Summer Institute for Historical Geophysics, Åland Islands. This work was partially carried out at the Danish National Space Center, Denmark Technical University, with support from FCT (Fundação para a Ciência e Tecnologia, grant SFRH/BPD/23992/2005) and Centro de Investigação em Ciências GeoEspaciais (CICGE), Group 2 (J. Fernandes). References Andersson, H. (2002), Influence of long-term regional and large-scale atmospheric circulation on the Baltic sea level, Tellus, Ser. A, 54, Barbosa, S. M., J. M. Fernandes, and M. E. Silva (2004), Nonlinear sea level trends from European tide gauge records, Ann. Geophys., 22, Barbosa, S. M., M. E. Silva, and M. J. 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