DETECTING SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL USING MULTICHANNEL SINGULAR SPECTRAL ANALYSIS

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. of Climatol. 26: (26) Published online 3 May 26 in Wiley InterScience ( DOI: 1.12/joc.1358 DETECTING SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL USING MULTICHANNEL SINGULAR SPECTRAL ANALYSIS SÍLVIA ANTUNES, a OLIVEIRA PIRES b and ALFREDO ROCHA c, * a Departamento de Acompanhamento do Clima e das Alterações Climáticas, Instituto de Meteorologia, Rua C do Aeroporto, Lisboa, Portugal b Macrofísica, Consultoria Lda, Portugal c CESAM, Universidade de Aveiro, Campus de Santiago, Aveiro, Portugal Received 4 May 25 Revised 23 March 26 Accepted 5 April 26 ABSTRACT A spatio-temporal variability study of annual precipitation in Portugal has been performed using the multichannel singular spectral analysis (MSSA). The spatio-temporal variability patterns, which can represent progressive signals, reveal significant periodic characteristics that have not been unambiguously detected in previous studies using other methods. In this study, MSSA detects periodic behaviour in annual precipitation of about 3 and 9 years. Similar periodic signals of about 4 and 8 years are detected when singular spectral analysis (SSA) is applied to a series of a North Atlantic oscillation (NAO) index. The strong correlation between the NAO and precipitation variability in Portugal reinforces the importance of detection of these signals with similar periodic characteristics. Copyright 26 Royal Meteorological Society. KEY WORDS: precipitation; significant oscillations; singular spectral analysis; multichannel singular spectral analysis; spatio-temporal patterns 1. INTRODUCTION The influence of the North Atlantic atmospheric pressure field in the weather and climate of southwestern Europe (Portugal and Spain) has been recognized by weather forecasters for a long time. One of the main factors of this influence is the position and strength of the Azores High. Since a large fraction of the surface atmospheric pressure field variance in the North Atlantic can be explained by the North Atlantic oscillation (NAO) (Hurrel, 1996), it is not surprising that very significant correlations between the series of NAO indexes and climatic data in Portugal can be found, i.e. for precipitation amount climatic series. As should be expected, these correlations are different, and sometimes of opposite sign to what is found over northwestern Europe. In fact, during a stronger positive NAO the zonal flux tends to increase in the northern regions, with an increase in the frequency of occurrence and strength of the weather systems. However, in these situations the weather systems moving at higher latitudes tend to affect less the southern regions. The NAO is a natural mode, internally generated by the atmosphere, which represents a quasi-meridional mass oscillation (Hurrel, 1996; Hurrel and van Loon, 1997). The NAO has been widely documented as a known source of inter-annual and decadal climate variability in the Euro Atlantic region, particularly over the Iberian Peninsula (Zorita et al., 1992; Rodriguez-Puebla et al., 1998; Trigo et al., 22; Munõz-Díaz and Rodrigo, 23; Nieto et al., 24) and Portugal (Ulbrich et al., 1999). * Correspondence to: Alfredo Rocha, Departamento de Física, Universidade de Aveiro, Aveiro, Portugal; arocha@fis.ua.pt Copyright 26 Royal Meteorological Society

2 22 S. ANTUNES, O. PIRES AND A. ROCHA Time series analyses of the NAO index performed by other authors have shown that the NAO has a significantly different behaviour from what should be expected from white- or red-noise realizations. Stephenson et al. (2) detected 2 5 year variations, decadal trends and a long-range memory. Pozo- Vázquez et al. (21) identified some periodic behaviour at inter-annual and decadal timescales, particularly for periods of about 8 years. However, the same authors point out that this period is highly non-stationary, and showed that this oscillation has not always been present in the past. A similar oscillation has also been identified in other atmospheric variables in the region (Zveryaev, 1999). Recently, Mills (24) has identified a stable periodicity of 7.5 years in the wintertime NAO index. In this study is also presented an inter-annual analysis of an NAO index using singular spectral analysis (SSA). Although a significant correlation can be found between the NAO index series and precipitation amount series in Portugal, it has been, till this study, difficult to detect in these precipitation series traces of the periodic behaviour of the NAO. In a previous study, the longest records of annual and seasonal precipitation time series of Portuguese stations analysed by maximum entropy methods (MEM) did not show any characteristics significantly different from a white-noise process (Antunes, 1998). SSA was also unable to detect any significant characteristics in the annual precipitation variability (Antunes et al., 2). Serrano et al. (23) studied the spatial variability of annual precipitation in Portugal using principal component analysis (PCA). In that study, the first mode represented 75% of the total variance and was associated with precipitation anomalies of the same sign over the whole region. The second mode explained 9% of the total variance and reflected a seesaw of simultaneous precipitation anomalies between the north and the south, while the third mode represented anomalies of a different sign between the coastal and interior regions. Temporal principal components (T-PCs) associated with these patterns do not reveal, by the application of Monte Carlo tests, significant signals for the white- and red-noise null-hypothesis (Antunes, personal communication). Despite not being statistically significant, spectral analysis applied to the first pair of components extracted from the first T-PC revealed periodic behaviour of about 2.7 and 9 years. Corte-Real et al. (1998) have also identified an 8-year period in the winter monthly precipitation at Évora (Portugal). In this study, multichannel singular spectral analysis (MSSA) is applied to the annual precipitation time series recorded at 11 locations in Portugal. The main objective is to detect space and temporal oscillatory modes in the data, to evaluate their statistical significance and to assess the eventual existence of relations with the NAO. 2. DATA The analysis of multivariate time series precipitation has been performed using 11 stations located in mainland Portugal from 1949 to 2 (Figure 1). The data consist of annual totals recorded from January to December. These data have been previously centred around the mean and standardized by the standard deviation. Table I shows the station metadata and a climatologic summary of the annual precipitation recorded at these stations over this period. The data represent the most highly quality-controlled series recorded in Portugal. These data have no missing values and have been subjected to a previous homogeneity analysis. Besides these series, the normalized precipitation long-term annual time series recorded in Lisbon since 1871 has also been used in the cross correlation analysis with the NAO index. The NAO index (1871/1997) was constructed using the Stykkisholmur mean sea-level pressure, provided by David Stephenson in his NAO thematic page ( and the corrected Lisbon mean sea-level pressure (Ferreira and Antunes, 2). The index is calculated as the normalized difference between the normalized annual mean sea-level pressures at each of these stations. Annual means are calculated for the January to December period. Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

3 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL 221 Figure 1. Location of stations used in precipitation variability analysis 3.1. Multichannel singular spectral analysis 3. METHODS PCA is frequently used to compress variability into a reduced number of modes that explain a considerable part of the field variance. However, this method allows only the evaluation of the mode and how the respective spatial pattern evolves in time. The analysis of spatio-temporal variability, i.e. the temporal analysis of modes evolving in space and time, may provide more information about the underlying physical system (Vautard, 1995). This can be accomplished through the use of MSSA, which is an extension of SSA, used in time series analysis and described by Vautard et al. (1992). The method computes the lagged cross covariances, and a multichannel trajectory matrix is created. This is done first by generating each channel with M lagged copies of itself, M being the number of temporal lags, and then forming the full augmented trajectory matrix (Allen and Robertson, 1996). Each block of the lag-covariance matrix contains the lag covariances of pairs of channels. Singular value decomposition (SVD) was the method applied to this matrix to obtain two orthogonal bases, the right- and the left-singular vectors (e.g. Robertson, 1996). The first of these, the eigenvectors of the matrix, is a sequence of spatial modes referred to as spatio-temporal empirical orthogonal functions (ST-EOFs) and the others, the associated principal components representing the way these patterns Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

4 222 S. ANTUNES, O. PIRES AND A. ROCHA Table I. Metadata and annual precipitation climatologic summary of stations used in study. Station name, latitude (Lat.), longitude (Lon.) and station altitude (H) are presented as station metadata. Annual mean and standard deviation (SD), minimum and maximum annual amounts and the corresponding years of occurrence are also presented as a climatic summary of these stations Station metadata Annual precipitation Name Lat. ( ) Lon. ( ) H (m) Mean (mm) SD (mm) Min. (mm) Year Max. (mm) Year Bragança.8 N 6.73 W Braga.55 N 8. W Porto.13 N 8.6 W P. Douradas. N 7.55 W Castelo Branco.83 N 7.47 W Cabo Carvoeiro.35 N 9. W Portalegre.28 N 7. W Lisboa.72 N 9.15 W Évora 3 N 7.9 W Beja.2 N 7.87 W Alvalade.95 N 8. W evolve in time, are referred to as spatio-temporal principal components (ST-PCs). The diagonal of the matrix corresponds to the associated explained variances. More details of MSSA can be found in Plaut and Vautard (1994) Statistical significance testing As opposed to the classical spectral analysis, where the basis functions are the prescribed sines and cosines, SSA produces data-adaptive filters capable of isolating oscillations spells, which makes the method more flexible and better suited for the analysis of non-linear, anharmonic oscillations (Vautard et al., 1992). Like in the case of SSA, if the variance of the series is dominated by an oscillation, MSSA generates a pair of EOFs nearly periodic, with the same period, and in quadrature (Plaut and Vautard, 1994). However, these pairs can also be randomly generated realizations of processes that do not have an oscillatory nature, and consequently, once detected, must be tested to determine whether they correspond to real oscillations or could have arisen as an artefact of the method. After testing and rejecting the null-hypothesis that precipitation data are pure realizations of whitenoise (not shown here), we used a more stringent null-hypothesis as formulated by Allen and Robertson (1996). This null-hypothesis is that the data have been generated by L first-order, autoregressive (AR(1)) independent processes, L being the number of channels. The null-hypothesis can be rejected if the spectrum of the eigenvalues associated with the modes detected by MSSA is higher than that expected in the data generated by red-noise processes. The confidence intervals for a given significance level are estimated using Monte Carlo simulations. A large ensemble of normally distributed (Gaussian) surrogate noise time series is generated with the null-hypothesis characteristics, in the case of autoregressive first-order processes, with the same length, variance and temporal lag-one autocorrelation as the series that form the centred input channels. Allen and Robertson (1996) suggest two different significance tests based on the same null-hypothesis: the first using the ST-PCs and ST-EOFs of the data, and the second using the ST-PCs and ST-EOFs of the null-hypothesis. In the first test, the lag-covariance matrix is computed from the data, whereas in the second, the lag-covariance matrix is computed from data generated by Monte Carlo methods. The subsequent procedure of projecting data and noise surrogates onto the vector basis is similar to both tests. The second test is more robust, since the former implicitly assumes the existence of a signal before any signal has been identified (Allen and Smith, 1996). Besides that, the second test does not present the artificial Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

5 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL 223 variance compression problem that boosts the significance of the first ST-PCs in detriment of higher-order modes. 4. ANNUAL NAO INDEX VARIABILITY ANALYSIS The annual NAO index analysed by SSA reveals a significant trend component (Antunes et al., 21). After removing this trend from the series, a new SSA has been performed. Results still show a significant pair of components in relation to the red-noise null-hypothesis at the.5 significance level. The methodology used is based on the WMO Technical Note No. 79 prepared by Mitchell (1966). A spectral peak is considered significant at the.5 level if it goes above the upper limit of the 9% confidence interval with limits 5% and 95%. As we are interested only in spectral values above a specified value, this is a unilateral or one-tailed test. Figure 2 shows the eigenvalues, the error bars associated with the estimation of these eigenvalues and obtained following Ghil and Mo (1991) equation, and the confidence intervals estimated by Monte Carlo methods to the red-noise null-hypothesis at the.5 significance level. Spectral analysis of this pair of components reveals a periodicity of about 8 years (Figure 3). Although not significant, the second paired component spectra with a periodicity of about 4 years are also included in this figure. The condition to form an oscillation is also verified, since the band of frequencies associated with these components is less than.75/2m, M being the used window width (Vautard et al., 1992). The periodic significant signal of about 8 years in the NAO index is consistent with the results presented by Hurrell et al. (23), who used a winter NAO index (December March) and detected power in the 8 1 years band, although not statistically significant. 5. SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL The first two annual precipitation ST-PCs estimated by MSSA are nearly periodic and in quadrature (Figure 4), satisfying the conditions to represent an oscillation according to the Plaut and Vautard (1994) criteria. They error bars.5 significance level 1 Spectral Density Eigenvalue Order Figure 2. Eigenvalue spectrum of the NAO index after removing the trend component estimated by SSA. Error bars and the.5 significance level (one-tailed test) for the red-noise null-hypothesis are also shown Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

6 224 S. ANTUNES, O. PIRES AND A. ROCHA 7. Spectral Density ST T-PC 2ND T-PC 3RD T-PC 4TH T-PC Frequency (1/year) Figure 3. Paired component variance spectra of the annual NAO index estimated by MEM. Periodicities of these T-PCs are also presented (in years) 3 2 1ST PRECIPITATION ST-PC 2ND PRECIPITATION ST-PC 1 Amplitude Time step (year) Figure 4. Temporal evolution of the first two annual precipitation ST-PCs in Portugal also satisfy a third more stringent condition proposed by the same authors which establishes the existence of the oscillatory pair: in order to guarantee that at least one cycle of the oscillation is coherent, a periodic behaviour in the ST-PCs cross correlation function is required, with the absolute values of correlation higher than.5 for the two successive extremes on each side of lag zero (Figure 5). Following the methodology presented in Section 3.2, the statistical significance of this oscillation was tested by Monte Carlo methods in relation to the null-hypothesis of red noise at the.5 significance level (onetailed test). Results using a base derived from data and a base derived from the null-hypothesis are presented in Figure 6(a) and (b), respectively. In the first case, two pairs of significant eigenvalues are detected. These Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

7 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL Correlation Lag (year) Figure 5. Lagged cross correlation function between the first two annual precipitation ST-PCs oscillatory modes reveal periods of about 3 and 9 years, respectively. The second test confirms the occurrence of a significant signal with a period of about 9 years. The choice of the window length M is simpler when looking for an oscillatory behaviour in a certain band. Taking into account the experiments performed by Plaut and Vautard (1994), which assume that the use of a window of length M typically allows the distinction of oscillations of periods in the range (M/5, M), and that MSSA does not distinguish between different oscillations of period longer than M, we selected a window of length M = 25. In order to evaluate the dependence of the results on the window length, other values of M have been tested. The stability of the results obtained for M = 17 and M = 2 and projected onto the null-hypothesis basis at the.5 significance level is presented in Figures 7 and 8, respectively. The consistency in identifying a significant periodic signal with a similar period of about 9 years confirms the rejection of the null-hypothesis that annual precipitation series are pure realizations of red-noise processes. This periodic characteristic is also consistent with that estimated as not being significant in the T-PCs spectral analysis performed elsewhere. Figure 9 shows the space time evolution of one of the ST-EOFs that represents the principal mode of precipitation variability in Portugal, for one-year steps for lags up to 16 years. This mode explains 21% of the total spatio-temporal variance. The most frequent pattern is, as in the study using PCA of Serrano et al. (23), the occurrence of precipitation anomalies of the same sign in the whole region. The spatial evolution is towards the occurrence of precipitation below the mean during a period of about 3 to 4 years following a period of the same duration with precipitation above the mean, plus a transition period of about 1 to 2 years to complete the 8 9 year cycle. A complete period is identified by years (t + 2 ) to (t + 5 ) with precipitation below the mean, following precipitation above the mean from year (t + 6 ) to (t + 9 ) and a transition year represented by the (t + 1 ) lag. In transition years, the most frequent pattern represents an opposite behaviour between the northwestern and the rest of the region. The dry and wet parts of the cycle begin first in the northwest region, showing a progressive behaviour from NW to SE. The second mode of about 3-year periodicity is captured by the third and fourth ST-PCs, which are presented in Figure 1. Figure 11 shows the space time evolution of one of the ST-EOFs that represents this second mode for one-year steps for lags up to 12 years. This mode, explaining 13% of the total spatio-temporal variability, reveals, like the first mode, the occurrence of precipitation anomalies of the same sign in the whole region as being the most frequent pattern. However, the patterns representing an opposite sign over Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

8 226 S. ANTUNES, O. PIRES AND A. ROCHA 3.5E+ 3.E+ 2.5E Spectral Density 2.E+ 1.5E+ 1.E E+1 (a).e Frequency (1/year) 2.5E+ 2.E Spectral Density 1.5E+ 1.E+ 5.E+1 (b).e Frequency (1/year) Figure 6. (a) Monte Carlo significance tests for annual precipitation oscillations using a window length M = 25. Diamonds show the data eigenvalues projected onto the data-adaptive basis. The vertical bars indicate the.5 significance level (one-tailed test) for the red-noise null-hypothesis. Periodicities of significant eigenvalues for this hypothesis are also shown. (b) Monte Carlo significance tests for annual precipitation oscillations using a window length M = 25. Diamonds show the data eigenvalues projected onto the null-hypothesis basis. The vertical bars indicate the.5 significance level (one-tailed test) for the red-noise null-hypothesis. Periodicities of significant eigenvalues for this hypothesis are also shown the country are more frequent than in the first mode. The most frequent seems to be the opposite behaviour between the northern and southern regions. 6. CORRELATION BETWEEN THE NAO INDEX AND PRECIPITATION VARIABILITY IN PORTUGAL The significant trend detected in the annual NAO index is not present in the precipitation time series. In spite of this, the long-term time series of annual precipitation observed in Lisbon (1871/1997) presents a highly Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

9 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL E+ Spectral Density 3.E+ 2.E E+.E Frequency (1/year) Figure 7. Monte Carlo significance tests for annual precipitation oscillations using a window length M = 17. Diamonds show the data eigenvalues projected onto the null-hypothesis basis. The vertical bars indicate the.5 significance level (one-tailed test) for the red-noise null-hypothesis. Periodicities of significant eigenvalues for this hypothesis are also shown 4.E+ 3.E+ Spectral Density 2.E E+.E Frequency (1/year) Figure 8. Monte Carlo significance tests for annual precipitation oscillations using a window length M = 2. Diamonds show the data eigenvalues projected onto the null-hypothesis basis. The vertical bars indicate the.5 significance level (one-tailed test) for the red-noise null-hypothesis. Periodicities of significant eigenvalues for this hypothesis are also shown significant, lag, correlation of.6 with the annual NAO index (Figure 12). The analysis was performed using the de-trended NAO index, since the usual definition of correlation is valid only for stochastic processes that are stationary to second order, and so have a constant mean value. This negative correlation means that Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

10 228 S. ANTUNES, O. PIRES AND A. ROCHA t t+4 year t+8 year t+12 year t+1 year t+5 year t+9 year t+13 year t+2 year t+6 year t+1 year t+14 year t+3 year t+7 year t+11 year t+15 year Figure 9. Space time evolution of the first annual precipitation variability mode (mm) for one-year lags during 16 years. This figure is available in colour online at Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

11 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL RD PRECIPITATION ST-PC 4TH PRECIPITATION ST-PC 1 Amplitude Time step (year) Figure 1. Temporal evolution of the third and fourth annual precipitation ST-PCs in Portugal positive NAO phases tend to occur simultaneously with precipitation below the mean and negative NAO phases tend to occur with precipitation above the mean. The cross correlation function presented in the same figure reveals some evidence of the 8 9 year periodic behaviour just as detected in the NAO index and in the precipitation over Portugal. In order to verify further the link between the near-9-year periodicity detected in precipitation using MSSA and the near-8-year periodicity detected in the NAO index using SSA, some additional correlation tests have been performed. The annual NAO index reduced to 1949/1997 has been lag cross-correlated to groups of reconstructed components (RCs) obtained from the ST-PCs and ST-EOFs estimated by MSSA from the precipitation data in Portugal. The results, shown in Figure 13, reveal a significant correlation for lag zero, even when a few components corresponding to the leading modes of precipitation are used. This correlation becomes stronger as more components are added, meaning that other signals, besides the 8 9 year detected in the leading components, are common to both series. This can be related to the detection of similar shorter periodic characteristics in the NAO index and in precipitation of about 4 and 3 years, respectively. The analysis of the figure leads to the conclusion that considering more than ten RCs does not improve the correlation value for zero lag. As can be observed in the same figure, the lag-one cross correlation is still significant when only the first two precipitation modes are included in the analysis. This result shows that the NAO can be considered as forcing the precipitation in Portugal. That means that the precipitation in Portugal may allow some predictability using NAO indexes as predictors. 7. CONCLUSIONS In this study, MSSA has been applied to annual precipitation at 11 Portuguese stations for the period. Results reveal the presence of periodicities of about 9 years in precipitation variability over Portugal, detected also by other studies that have used different methods and data. Unlike in those studies, these periodicities are statistically significant, revealing MSSA as an efficient tool in the detection of non-purely random characteristics in series with a weak signal/noise relation. Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

12 221 S. ANTUNES, O. PIRES AND A. ROCHA t t+3 year t+6 year t+9 year t+1 year t+4 year t+7 year t+1 year t+2 year t+5 year t+8 year t+11 year Figure 11. Space time evolution of the second annual precipitation variability mode (mm) for one-year lags during 12 years. This figure is available in colour online at The spatial evolution of this principal mode is towards the occurrence of precipitation below the mean during a period of about 3 to 4 years, following a period of the same duration with precipitation above the mean, plus a transition period of about 1 to 2 years to complete the 8 9 year cycle. These transition years show a progressive behaviour from NW to SE. This study also identifies statistically significant periodic signals of about 8 years in the annual NAO index, similar to those found for precipitation. Besides this similar periodicity, the results suggest the existence of spectral peaks for higher frequencies corresponding to periods of nearly 3 4 years both in the NAO and in the precipitation time series. Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

13 SPATIO-TEMPORAL PRECIPITATION VARIABILITY IN PORTUGAL Correlation Lag (year) Figure 12. Cross correlation function between annual NAO index and annual precipitation in Lisbon (1871/1997) for positive and negative lags up to 12 years. Also presented are the confidence intervals at the.5 significance level. Positive lags correspond to delays in the precipitation series.4.2 Correlation RC1 RC1:4 RC1:1 RC1:2 -.8 Lag (year) Figure 13. Cross correlation functions between annual NAO index and groups of RCs estimated from the annual precipitation data in Portugal for positive and negative lags up to 12 years. Also presented are the confidence intervals at the.5 significance level. Positive lags correspond to delays in the precipitation series The significant correlations obtained between the annual NAO and precipitation over Portugal reinforce the importance of this well-documented relationship. REFERENCES Allen M, Robertson A Distinguishing modulated oscillations from coloured noise in multivariate datasets. Climate Dynamics 12: Copyright 26 Royal Meteorological Society Int. J. of Climatol. 26: (26) DOI: 1.12/joc

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