Climatic and hydrological variations during the last 117–166 years
in the south of the Iberian Peninsula, from spectral and correlation
analyses and continuous wavelet analyses
B. Andreo
a,*, P. Jime´nez
a, J.J. Dura´n
b, F. Carrasco
a, I. Vadillo
a, A. Mangin
ca
Departamento de Geologı´a, Facultad de Ciencias, Universidad de Ma´laga, E-29071 Ma´laga, Spain b
Direccio´n de Hidrogeologı´a, Instituto Geolo´gico y Minero de Espan˜a, E-28003 Madrid, Spain c
Laboratoire Souterrain de Moulis, 09200 Saint-Girons (Moulis), France
Received 21 April 2004; revised 5 September 2005; accepted 22 September 2005
Abstract
The most complete historical series of instrumental data available, spanning more than a century, on rainfall, temperature and outflow of a karst spring obtained from gauging stations in the south of the Iberian peninsula were analysed by means of spectral and correlation analyses and continuous wavelet analyses. Annual periodicity of the rainfall and temperature distributions was constant over more than 100 years, although weaker (6-month) periodicities have also been observed, as well as rainfall and temperature periodicities of 5 and 2.5 years, which have also been recorded in other areas of Europe. These multiannual scale components can be explained by climatic variations or effects described in the literature in connection with the North Atlantic Oscillation (NAO) and are likely to be the same as the climate variability at decadal to annual scale detected in several proxy data from geological records. No long-term trends in the distribution of precipitation and temperature were detected. q2005 Elsevier B.V. All rights reserved.
Keywords:Climatic change; Correlation and spectral analyses; NAO; Precipitation; South Iberia; Temperature; Continuous wavelet analyses
1. Introduction
Climatic change has been much debated in the scientific world in recent decades. Many investi-gations on the climatic and hydrological variations have been done using carbonate deposits, particularly speleothems because they are less affected by postdepositional processes than superficial sediments.
Speleothems can be dated precisely by means of the U/Th decay series and consequently, they can contribute to our knowledge of the paleoclimatic and paleohydrologic events in continental areas
(Schwarcz, 1986). Geochemical studies can be
performed with a very high time resolution from stable isotopes suchd18O and d13C, which, respect-ively, may reflect paleotemperature and vegetation
(Gascoyne, 1992), and trace elements (Mg, Sr) that
are indicators of paleohydrology (Fairchild et al., 2001). The geochemical and hydrological results have been tested with the actual deposits of speleothems in www.elsevier.com/locate/jhydrol
0022-1694/$ - see front matterq2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.09.010
* Corresponding author. Tel.: C34 95 2132004; fax:C34 95
2132000.
caves (Baker and Smart, 1995; Genty and Deflandre,
1998; Andreo et al., 2002; Tooth and Fairchild, 2003).
Another possibility to study climatic and hydro-logical changes, without taking into account geohydro-logical records, is to examine the period of recorded history (the instrumental period), but it is necessary to possess both time series of adequate length and the mathematical tools that enable us to optimise the results. Thus, the long-term time series are important because they permit to study climatic changes using the real data or to reconstruct longer series using different mathematical tools. The expected modification in rainfall following climate change in the South Iberian Peninsula is a decrease in precipitation in the coming decades or even the whole twenty-first century (Sumner et al., 2003).
In this work, we have analysed the most complete historical series available of real data, spanning more
than a hundred years, obtained from gauging stations in the southern Iberian peninsula (Fig. 1): precipi-tation data obtained at the sprecipi-tations of San Fernando in the province of Ca´diz (127 years) and Gibraltar (166 years), temperature data recorded at San Fernando (117 years) and the outflow of El Tempul spring (133 years). The exceptional length of these time series in the study area, and its geographical location, being influenced both by the Atlantic ocean and the Mediterranean sea, are two key features of our study. Two mathematical tools have been applied: spectral and correlation analyses, and continuous wavelet analyses.
Correlation and spectral analyses were first applied, in surface hydrological systems orientated mainly towards forecasting, completion of data and estimation of parameters for stochastic models (e.g.
Delleur, 1971; Spolia et al., 1980).Mangin (1981a,b,
1984) adapted this methodology to study the
discharge from karstic aquifers. Examples of carbon-ate aquifers studied by this methodology are abundant in the literature, some of them recently published (e.g.
Padilla and Pulido-Bosch,1995; Larocque et al., 1998;
Jime´nez et al., 2002). These studies demonstrate the
application of correlation and spectral analysis to series of both flow data and precipitation in order to determine the flow and characterise aquifer beha-viour. However, correlation and spectral analyses have not often been applied to very long-time series of climatic and hydrological data (Kuhnel et al., 1990) in spite of their advantages in identifying the structures and periodic components (normally average period-icities) in these series.
Wavelet transform techniques have been applied in the fields of hydrology and meteorology to identify coherent convective storm structures and characterise temporal variabilities (Kumar and Foufoula-Geor-giou, 1993; Kumar, 1996; Smith et al., 1998; Szilagyi
et al., 1999), to explain the variability of ocean
temperatures (Meyers and O’Brien, 1994), and the variations in global mean sea level (Breaker et al.,
2001; Chambers et al., 2002). They have also been
used in oceanography (Meyers et al., 1993), and in comparing watersheds of the same region (Gaucherel,
2002).Labat et al. (1999a,b; 2000, 2001)also applied
these techniques in the field of karstic hydrogeology in order to study rainfall rates and outflows of karstic springs located in the Pyrenees and in the Larzac plateau (France). By comparison with correlation and spectral analyses, wavelet tools detect not the average periodicities in time series, but the distribution of the periodic variabilities during the time.
The aims of this paper are to determine whether there are periodicities associated with climatic cycles or oscillations and whether there is any long-term trend that might be related to the climatic changes. In order to validate the results obtained using these series of climatic data (especially rainfall), both method-ologies have also been applied to a time series of the outflow recorded over more than a century at El Tempul spring, which constitutes the main discharge point of the Sierra de las Cabras carbonate aquifer
(Fig. 1). Thus, it is possible to identify the
correspondence between rainfall and hydrological variations.
In a previous paper, the evolution of the annual precipitation recorded at the Gibraltar station from 1791 to 1983 was studied using annual data and it was concluded that a slight decreasing trend exists in the rainfall time series, but without statistical significance
(Moreno and Martı´n, 1986). Similar results were
obtained in other areas of southern Iberia, such as Granada, eastward of the study area, using monthly data for the period 1902–1983 (Benavente et al., 1986) and Huelva, westward of the study area, using daily data for the same period (Romero and Sainz, 1984). These papers used statistical techniques on rainfall data in S Iberia in a preliminary manner, leaving much remaining to be done in terms of detecting climatic changes.
Other previous papers (Rodrigo et al., 2000;
Pozo-Va´zquez et al., 2000) show the results of the
application of different statistical tools to detect changes in meteorological records in S Iberia. These works demonstrate the existence of dry and wet periods into the record, at different timescales, and the contribution of the North Atlantic Oscillation (NAO) to rainfall variability in this region.
2. Materials and methods
The time series data used as the material for this study was the following:
(a) Precipitation and temperature data recorded at the meteorological station of Real Observatorio de San Fernando (Ca´diz). The precipitation data correspond to monthly total values for the period January 1870— December 1997, and the tem-perature data are mean monthly values for the period January 1870— December 1987. The rainfall data available from 1791 to 1870 in Gibraltar station are annual values and they have been not used in the present work.
(b) Series of daily rainfall data from October 1834 to November 2000, obtained at the Gibraltar weather station.
(c) Series of monthly flow rate from El Tempul spring, the main discharge point of Sierra de las Cabras, an aquifer where no pumping exists. Sierra de las Cabras is a diffuse flow carbonate aquifer with a surface of 42 km2 and average
resources of 8 – 9 hm3/year which are used for water supplying to the city of Jerez de la Frontera
(Jime´nez et al., 2001). The data series of spring
flow were calculated using the height of the water surface in the gauging station. Readings were taken on the 20th day of each month and then converted to outflow values by the application of a stage-discharge rating curve. This series is exceptionally long, extending from June 1862 to December 1995.
Therefore, three types of climatic and hydrological data have been used in this work (rainfall, temperature and springflow rate). The location of the stations is in
Fig. 1and the statistical parameters of the time series
are inTable 1.
The first method applied was correlation and spectral analysis, which deals with the whole time series and can be applied in two domains (Mangin, 1984): time (correlation analysis) and frequency (spectral analysis). Normally, the data series are first studied separately by means of a simple analysis to identify the structure and components, and then by cross analysis, considering two series (i.e. precipi-tation and outflow) in order to determine the relationships between the two. Therefore, correlation and spectral analysis can be simple or cross-correlated and, in both cases, in the time (correlation) or frequency (spectral) domains.
The simple correlation analysis (simple correlo-gram or correlocorrelo-gram) of a data series is the auto-correlation function, which shows the linear depen-dency of successive data for increasingly large time
intervals. Thus, the slope of the auto-correlogram flattens quickly to values close to zero if the data have a short-term influence on the time series and, consequently, are of a random nature. Conversely the slope of the correlogram flattens more slowly if the data have a long-term influence on the time series. The simple correlogram,rk, was calculated using the
formula proposed byJenkins and Watts (1968):
rkZ Ck
C0 with CkZn
K1X
nKk
1
ðxiKxÞðx1CkKxÞ
whererkis the value of the correlogram,kis the time
lag varying from 0 to n (cutting point), and xi are
values which have an average of x. The auto-correlogram allows us to quantify the ‘memory effect’, the time in which the correlogram decreases to values of 0.1–0.2.
The simple spectral analysis (spectral density function) is the Fourier transformation of the auto-correlogram, which corresponds to a change from a time mode (time-series space) to a frequency mode. The spectral density function,S(f), is calculated by the formula proposed byJenkins and Watts (1968):
SðfÞZ2 1C2 Xm
1
Dkrkcos 2pFk
" #
wherekis the step andFZj/2mwithjvarying from 1 tom.Dkis a window that is necessary to ensure that
the S(f) estimated values are not biased. The best windows are those of the Tukey filter (Mangin, 1984).
DkZ
1Ccospmk
2
The spectral density function normally shows peaks, which represent periodic phenomena in the time series. When high-density values are found near zero frequencies, this indicates the existence of a long-term trend or a phenomenon in which the periodicity is longer than the time series studied.
The auto-correlogram and the spectral density function have been obtained at two levels: a ‘short-term’ analysis (window 125 months and lag of 1 month) and a ‘long-term’ analysis (window 500 and 475 months and lag of 10 months). Moreover, the long-term analysis has been carried out with different lags in order to verify the results and avoid artificial Table 1
Statistical parameters of the time series of data used in this work
Monthly precipitation Monthly flow rate
Gibraltar station (mm)
San Fernando station (mm)
El Tempul Spring (L/s)
N 1596 1509 1596
Max 528 471 1951
Min 0 0 20
Average 68 49 286
Standard deviation
82 57 280
Variation coefficient
121 117 98
periodicities inherent to the treatment of the data. These analyses enable us to identify and to describe the components (trend, periodicity and randomness) of the time series. Information on the structure of the time series can only be obtained for window values that are between double the lag value and one third of the length of the time series. Thus, the available data series considered in this work permit us to deduce components with a periodicity between 2 months and more than 40 years.
The second method is continuous Morlet wavelet analysis, which allows the completion of time-scale representation of localised and transient phenomena occurring at different time scales. Time-scale dis-crimination is achieved in a more satisfactory way than time-frequency decompositions such as the windowed Fourier method (e.g. Mangin, 1984). Thus, by comparison with the Fourier and correlation analyses, wavelet transforms lead to more precise results especially in the temporal variability of the processes. The continuous wavelet transform provides a time-scale discrimination of the signal into sub-processes. It is defined as the convolution product of a signal x(t) by functions obtained by dilation (or contraction) and temporal translation of a function j(t) called ‘wavelet’, which must satisfy certain admissibility criteria. The algorithm to compute the wavelets and plot the results is from Torrence and
Compo (1998). After discriminating the data with the
wavelets, the amplitude of any variable signal within the data can be determined at various frequencies, as well as the variation of this amplitude with time. For the results plotted here, a dark grey shade is assigned to the lowest value of the wavelet coefficient while a white shade is assigned to the highest value. Since wavelet analysis is dealing with finite-length time series, errors will occur at the beginning and end of the wavelet power spectrum, while the Fourier transform assumes the data is cyclic. The errors are included in the so-called ‘cone of influence’, that is the region of the wavelet spectrum in which edge effects become important.
The continuous Morlet wavelet has been applied because it establishes a clear distinction between random fluctuations and periodic regions. By analogy with Fourier analysis, a global wavelet variance spectrum for continuous wavelet transform can also be defined, which gives a representation of the
variance distribution across time scales (Liu, 1995). A mathematical overview of the wavelet transform (continuous wavelet transform) and its applications to karstic hydrogeology are given byLabat et al. (1999a,
b, 2000, 2001), emphasising also the statistical
interpretation of the wavelet coefficients and introdu-cing the concepts of the wavelet spectrum.
3. Correlation and spectral analyses
3.1. Correlation and spectral analysis of the rainfall time series
We carried out a simple analysis (correlogram and frequency spectrum) of the time series of monthly data recorded at meteorological stations of Gibraltar and San Fernando. The short-term simple correlogram for rainfall data reveals that both weather stations
(Fig. 2A and C) detected the existence of
clearly-defined annual cycles (kZ12 months). The memory effect (value ofkfor rZ0.1–0.2) is several months, although shorter-term analyses previously done demonstrated that the distribution of precipitation is a random phenomenon, with a memory effect of about 2 days (Jime´nez et al., 2001). The spectrum of the variance density for the two time series (Fig. 2B and D) reveals a very clear peak corresponding to the annual periodicity of precipitation (frequencyZ 0.082), together with a less evident 6-monthly periodicity (fZ0.165), which is more evident in the San Fernando data.
Long-term analysis of the time series data from Gibraltar and San Fernando produced simple correlo-grams (Fig. 3A and C) showing a periodicity of about 5 years (kZ60 months). This finding was confirmed by the spectrum of the variance density for the two time series, which clearly shows the approximately quinquennial periodicity of rainfall (fZ0.165 in
Fig. 3B and D). In addition, although less evident,
there was a rainfall periodicity of 2.5 years (fZ0.33), more visible in the San Fernando data. The higher values of the spectra for 1 and 5 years periodicities in
Figs. 2 and 3(B and D) provoke that the peaks of 6
month and 2.5 years periodicities are less marked, but only about 10% of the spectral density variance is greater than these and, therefore, they can be considered significant.
The long-term analysis, autocorrelogram and spectral density, has been also carried out with different lags (7 and 9 months) and the results are the same although only the spectral density functions are represented inFig. 3E and F to avoid repetition. We have done the frequency spectrum with different lags (7 and 9 months) for rainfall data of the San Fernando station and for the flow data of Tempul spring and the results (autocorrelogram and spectral density) are also the same than a lag of 10 months. So, these periodicities are not derived from the math-ematical procedure, as sometime occurs (Burroughs, 1992), but are intrinsic periodicities of the time series. In the long-term frequency spectra obtained as part of the present study, for the Gibraltar and San Fernando stations (Fig. 3B and D), no long-term trends were observed. In order to determine the capacity of
correlation and spectral analysis to establish long-term trends, we generated several series of rainfall data, adding to the original series recorded at the Gibraltar station (the longer series) temporally decreasing tendencies of different rainfall quantities. The corre-lation and spectral analysis has been applied to the new series and we confirm that the correlogram (Fig. 3G) is quite similar to the above one, but the frequency spectra
(Fig. 3H) permits the clear distinction of the tendency
when a rainfall decreasing trend of 50 mm is added over the whole period of the record.
3.2. Correlation and spectral analysis of the temperature time series
In this case, too, we performed a simple analysis (correlogram and frequency spectrum) of the whole
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
100 105 110 115 120 125
k (months)
rk
A
Gibraltar Station lag step 1 month
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
S (f)
B
Gibraltar Station lag step 1 month 1 year
6 months
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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
100 105 110 115 120 125
k (months)
rk
C
San Fernando Station lag step 1 month
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
S (f)
D
San Fernando Station lag step 1 months 1 year
6 months
Fig. 2. Results of short-term correlation analysis (A,C) and spectral analysis (B,D) of monthly precipitation data from the Gibraltar and San Fernando stations.
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0 20 40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
k (months)
rk
A Gibraltar Station lag step 10 months
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 0.0 0 0.0 2 0.0 4 0.0 6 0.0 8 0.1 0 0.1 2 0.1 4 0.1 6 0.1 8 0.2 0 0.2 2 0.2 4 0.2 6 0.2 8 0.3 0 0.3 2 0.3 4 0.3 6 0.3 8 0.4 0 0.4 2 0.4 4 0.4 6 0.4 8 0.5 0 frequency S (f) B Gibraltar Station lag step 10 months
5 years 2.5 years -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 20 40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
k (months)
rk
C San Fernando Station
lag step 10 months
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 0.0 0 0.0 2 0.0 4 0.0 6 0.0 8 0.1 0 0.1 2 0.1 4 0.1 6 0.1 8 0.2 0 0.2 2 0.2 4 0.2 6 0.2 8 0.3 0 0.3 2 0.3 4 0.3 6 0.3 8 0.4 0 0.4 2 0.4 4 0.4 6 0.4 8 0.5 0 frequency S (f) D San Fernando Station
lag step 10 months
5 years 2.5 years 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44
frequency
S (
f)
E Gibraltar Station lag step 7 months
5 years 2.5 years 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.0 0 0.0 2 0.0 4 0.0 6 0.0 8 0.1 0 0.1 2 0.1 4 0.1 6 0.1 8 0.2 0 0.2 2 0.2 4 0.2 6 0.2 8 0.3 0 0.3 2 0.3 4 0.3 6 0.3 8 0.4 0 0.4 2 0.4 4 0.4 6 0.4 8 frequency S (f) F Gibraltar Station lag step 9 months
5 years 2.5 years -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 20 40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
k (months)
rk
G Gibraltar Station lag step 10 months.
tendency 50 mm
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
S (f)
H
Gibraltar Station lag step 10 months, tendency 50 mm
5 years
2.5 years
Fig. 3. Results of long term correlation analysis (A) and spectral analysis (B) of monthly precipitation data, using a lag of 10 months, from the Gibraltar station. Results of long-term correlation analysis (C) and spectral analysis (D) of monthly precipitation data, using a lag of 10 months, from the San Fernando station. Spectral analysis of monthly precipitation data using lag of 7 months (E) and 9 months (F). Correlation (G) and spectral analysis (H) using a lag of 10 months and a decreasing tendency of 50 mm in the rainfall time series data of Gibraltar. To calculate the periodicities (in years) from long-term spectra, the lag should be divided by the frequency and, then, the result should be divided by 12.
monthly series of temperature data recorded at the San Fernando station.
As with the analysis of precipitation data, the short-term simple correlogram (Fig. 4A) clearly revealed the existence of annual cycles. The variance density spectrum (Fig. 4B) indicated a very strong annual temperature periodicity and a somewhat weaker 6-monthly cycle. The long-term analysis of this series of monthly temperature data recorded at the San Fernando station produces a simple correlogram
(Fig. 4C) that shows an average 5-yearly periodicity
of temperature variations. This quinquennial period-icity is corroborated by the variance density spectrum
(Fig. 4D). We also observed a 2.5 year temperature
periodicity, though this was apparently weaker than that obtained for rainfall data at the same station. However is we take into account the different scale for spectral density in Figs. 3D–B and 4B–D and the variance of these (only 10% of the variance is higher) we can deduce that, in both cases, the 6 month and 2.5 years periodicities are significant.
The long-term correlogram of the temperature data
(Fig. 4D) showed no long-term (low frequency) trend
in temperatures, the same as occurs with the analysis of the precipitation data. In addition, in order to validate the capacity of correlation and spectral analysis to establish long-term trends in temperature, we generated series of data, using those originally recorded at the San Fernando station and assuming a tendency to increase between 1 and 38C over the whole period. Correlation and spectral analysis was then applied to this synthetic data (Fig. 4E and F), and it was confirmed that the method was indeed capable of detecting a trend for temperature increases of 18C
(Fig. 4F).
3.3. Correlation and spectral analysis of outflow data series from the El Tempul spring
The short-term correlogram for the series of monthly outflow values from the El Tempul spring
(Fig. 5A) revealed a strong annual periodicity; the
memory effect estimated in a previous work (Jime´nez
et al., 2001) was 95 days, which indicates a high
inertia. In the frequency domain, the spike corre-sponding to the annual periodicity (Fig. 5B) can be clearly observed, which is analogous to that detected for the distribution of precipitation. We also found
weaker periodic components of 4, 2 and 1.4 years, not observed in the rainfall spectrum (Fig. 2B and D), which might be related to the strong inertial effect of the carbonate aquifer of Sierra de las Cabras, that is, the significant natural attenuation for annual vari-ations in precipitation. The simple correlogram of monthly outflow obtained for the long-term analysis and for the precipitation-series analysis reveals a fairly well defined five-year cycle (Fig. 5C). The fact that the quinquennial periodic component for the outflow series is corroborated in the frequency domain
(Fig. 5D), such as occurs with rainfall and temperature
data series, suggests that the origin of the periodicity is climatic, as is the case for the annual periodicity observed in the short-term analysis. Nevertheless, the spectrum of the outflow data (Fig. 5D) presents a long-term (low frequency) trend that is not evident in the spectrum for the precipitation series, despite its greater duration, and consequently cannot be attrib-uted to climatic phenomena.
In addition, cross-correlation in frequency domain has been done with the time series of data from El Tempul spring and Gibraltar station and only the annual effect is deduced in the short term analysis
(Fig. 5E) and the quinquennial one in the long term
analysis (Fig. 5F).
4. Continuous-time Morlet wavelet analysis
4.1. Continuous Morlet wavelet analysis of the rainfall time series
The Morlet wavelet spectrum for rainfall data obtained by the Gibraltar and San Fernando stations are calculated and are depicted inFig. 6A and B. It is a time-scale plot of the signal where the x-axis represents position along the signal (time), the
y-axis represents a periodicity scale, and the shade contour at eachx–ypoint represents the magnitude of the wavelet coefficient at that point. A dark grey shade is assigned to the lowest value of the wavelet coefficient while a white shade is assigned to the highest value.
Over all the period of the record, light shades can be seen in both wavelet spectra with a periodicity of 1 year (Fig. 6A and B), so an annual process is visible but with a higher intensity for the Gibraltar wavelet
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0 5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
k (months)
rk
A
San Fernando Station lag step 1 month
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
S (f)
B
San Fernando Station lag step 1 month
1 year
6 months
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0
20 40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 k (months)
rk
C
San Fernando Station lag step 10 months
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 frequency
S (f)
D
San Fernando Station lag step 10 months
5 years
2.5 years
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0
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480
k (months)
rk
E
San Fernando Station lag step 10 months, tendency 1ºC
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48
frequency
S (f)
F
San Fernando Station lag step 10 months, tendency 1ºC
5 years
2.5 years
Fig. 4. Results of short term (A,B) and long-term (C,D) correlation and spectral analysis of monthly temperature data from the San Fernando station. Correlation (E) and spectral analysis (F) following the addition of a long-term trend of 18C in the temperature time series data of San Fernando.
spectrum (white shade) than for San Fernando (grey and white shades). At large scales, the Gibraltar wavelet spectrum reveals two processes that are more or less localised in time. Areas of white and grey
shades indicate a 2–3 year periodic component during the whole studied period, less visible in the San Fernando wavelet spectrum (only grey coulour appears for this periodicity in Fig. 6B). This first
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
100 105 110 115 120 125
k (months)
rk
A El Tempul spring
lag step 1 month
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
S (f)
B El Tempul spring
lag step 1 month
4 years
1 year
2 years
1.4 years
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 20 40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
k (months)
rk
C El Tempul spring lag step 10 months
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
frequency
frequency frequency
S (f)
D El Tempul spring lag step 10 months 5 years
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
S
(f)
E
El Tempul spring lag step 1 month 1 year
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
S
(f)
F
El Tempul spring lag step 10 months 5 years
Fig. 5. Results of short term (A,B) and long-term (C,D) correlation and spectral analysis for the monthly outflow series from El Tempul spring. Cross-spectral analysis of short-term (E) and long-term (F) time series of the outflow at El Tempul spring and rainfall in Gibraltar station.
component is a multiannual component and it becomes a 4–6 year periodicity during the last six decades: since 1950 appears white shades inFig. 6A (Gibraltar) and grey shades inFig. 6B (San Fernando). Therefore, as for the annual process, this second component shows a higher intensity for the Gibraltar wavelet spectrum. This second component (4–6 years) does not have the same intensity as the first one but reflects a periodicity occurrence in the rainfall large scale distribution.
The Fig. 6A and B also include the cone of
influence which represent the periods with uninter-pretable results due to edge effects of wavelet processing, and consequently they should be ignored in all analysis. These periods are at the beginning and end of the time-series and for periods greater than about 8-10 years. So the apparent periodicity of 8–10 years inFig. 6A and B is not real.
The information provided by the wavelet spectrum can be time-averaged to obtain a global wavelet spectrum without temporal variations (Fig. 6C). The annual process is again marked, whereas the interannual components 2–3 and 4–6 years appear to be much attenuated but visible, which corroborates these periodicities according to the wavelet spectra and correlation and spectral analysis.
4.2. Continuous Morlet wavelet analysis of temperature time series
The Morlet wavelet spectrum for temperature data at San Fernando station is displayed in Fig. 7A. Regarding small scales (scales of less than 1 year), high frequency structures (6–9 months) are visible in several parts of the record. Nevertheless, the annual component is the one that appears with major intensity during overall analyzed period (grey and white shades).
On longer timescales, during the last two decades, a weak 2–3 year component is visible; moreover, in the last decade also there is a 4–6 years component with a similar intensity. Both components had been already made evident from the correlation and spectral analysis, as with the rainfall data. Fig. 6. Continuous wavelet spectra of monthly precipitation data
from the Gibraltar (A) and San Fernando stations (B) and global continuous wavelet spectra of monthly precipitation data for both stations (C). The lighter the grey scales, the higher the value of the
wavelet coefficient. Cross-hatched areas on either end indicate the ‘cone of influence’, where edge effects become important and the results should be ignored.
Nevertheless this last methodology cannot determine when within the series exists a certain frequency component.
The global wavelet spectrum for the whole period
(Fig. 7B) clearly shows the annual component already
mentioned, and this component concentrates practi-cally all value of the total energy of the signal. Hence it is not possible to see other component of the signal. Nevertheless, a detailed analysis of this spectrum allows us to observe peaks of 2–3 and 4–6 years.
4.3. Continuous Morlet wavelet analysis of the outflow data from the El Tempul spring and cross-wavelet analysis of the Gibraltar precipitation—El Tempul outflow
The Morlet wavelet spectrum of El Tempul outflow (Fig. 8A) shows that high frequency or short scale patterns are less visible than in the precipitation spectra (Gibraltar and San Fernando). At this scale of analysis (monthly data) it is only possible to distinguish the classical 1-year process (annual recharge). Nevertheless, these processes are not highly variable and are present during the overall interval of observation as opposed to the precipitation wavelet spectra (Fig. 6A and B). This indicates the strong inertia of the aquifer in filtering the rainfall input signal. The global wavelet spectrum (Fig. 8B)
shows not only the annual component, but this spectrum makes it possible to see other interannual components with the following periodicities (in years): 2–3, 4–6 and 8–10. This last component is not detected in the correlatory and spectral analysis, and it is in the cone of influence, which represent the periods with uninterpretable results.
The precipitation rates and outflow are first studied separately in order to highlight their time-scale characteristics via univariate wavelet transform. Cross-wavelet transforms are then constructed in order to analyse the variability of the precipitatio-n/outflow relationship at the monthly sampling rate. The Morlet wavelet cross-spectrum of the monthly data is displayed in Fig. 8C, which shows again the event of annual periodicity. Moreover, at large scale there are two components (2–3 and 4–6 years), the latter being stronger. This amplification of the signal at large scales must be related to the importance of the internal groundwater reserves, which clearly depends on the long-term behaviour of precipitation rates
(Labat et al., 1999b).
The global Morlet wavelet cross-spectra (Fig. 8D) again shows the importance of the annual component. This component practically concentrates all value of the total energy of the signal; therefore the interannual components (2–3 and 4–6 years) are scarcely visible inside the signal.
Fig. 7. Continuous wavelet spectra of monthly temperature data from the San Fernando station (A) and global continuous wavelet spectrum (B). The lighter the grey scales, the higher the value of the wavelet coefficient. Cross-hatched areas on either end indicate the ‘cone of influence’, where edge effects become important and the results should be ignored.
5. Discussion
The results of correlation and spectral analysis and wavelet continuous analysis applied to long time series of rainfall data recorded in Gibraltar and San Fernando stations (South of the Iberian peninsula) demonstrate that, a short-term, the annual and a lesser periodicity of 6-month exist. Furthermore, long-term analysis reveals a rainfall periodicity of 5 (and 2.5) years. These periodicities have never been detected in
previous work on rainfall data series from the southern Iberian peninsula (Romero and Sainz, 1984; Moreno and Martı´n, 1986; Benavente et al., 1986). However,Rodrigo et al (2000)detected a 2.1-year periodic component in rainfall data series, which could be coherent with the 2.3-year dominant oscillation of the NAO index (Qian et al, 2000).
Analysis of rainfall time series with a recording period similar to those used in this study, mainly obtained at stations located in the Midlands of Fig. 8. Continuous wavelet spectra for the monthly outflow series from the El Tempul spring (A) and global continuous wavelet spectrum (B). Continuous wavelet spectra of monthly precipitation at Gibraltar station and outflows of El Tempul (C) and global continuous cross-wavelet spectrum (D). The lighter the grey scales, the higher the value of the cross-wavelet coefficient. Cross-hatched areas on either end indicate the ‘cone of influence’, where edge effects become important and the results should be ignored.
England and in Wales (Burroughs, 1992), also reveals a periodicity of 5 years in the distribution of precipitation. This fact might be due to a climate presenting more extreme characteristics approxi-mately every 5 years (Burroughs, 1992), which is coherent with the alternation of dry and wet periods of the ‘Joseph effect’ described by Mandelbrot and
Wallis (1968). In addition,Rogers (1984)analysed the
Fourier spectrum of the NAO winter index from 1900 to 1983, using atmospheric pressure data from Iceland and the Azores islands, and found the peak of 5-year period. All the aforementioned data probe the influence of the NAO in the rainfall distribution in South Iberia.
Moreno and Martı´n (1986) detected a slight downward tendency in total annual precipitation values in the south of the Iberian Peninsula, but also remarked that these findings were not statistically significant. Benavente et al. (1986) studied rainfall records from the city of Granada and they concluded that the quantity of precipitation is virtually unchan-ging, although they detect also a general trend towards more extreme variations of climate. In these last works the length of time series data are shorter than the used in the present study. However, taking into account the data used in the present study and the methodologies applied, no long term trends in the precipitation pattern in the south of the Iberian peninsula were detected. If a trend had existed involving a rainfall variation, for exampleR50 mm, it would have been detected by the correlation and spectral analysis.
The analysis of the monthly outflow data series from the El Tempul spring identified two period-icities, of 1 and 5 years, which seem to be of climatic origin. On the other hand, the long-term trends deduced from the outflow data by the present methodology cannot be attributed to an increase in the use of the groundwater, because the aquifer was not exploited by pumping during the study period. In addition, there is no detection of the type of recharge (i.e. diminution of stormy recharge periods). There-fore, the trend detected in the outflow from the El Tempul spring is not of climatic origin, but it is related, with a high modulation capacity, to the long-term distribution of the precipitation.
The continuous wavelet-transform method allowed us to corroborate the precipitation and outflow annual
components during the whole analyzed period. So, as well as annual components there are 2–3 and 4–6 year components, which show a shift from a dominant 2–3 year period in the first part of the record to a 4–6 year period in the later part of the record. These components are more visible for Gibraltar rainfall than for San Fernando, probably because of the Gibraltar data series is larger and with higher variability (Table 1). The transition between the interannual components has already been make evident by others authors in precipitation records in Southern France (Labat et al., 2000), and interannual variations in global mean sea level (Chambers et al., 2002). The multiannual scale components are linked to the alternation of dry and wet periods inherent to the NAO (Rodrigo et al., 2000).
Regarding temperature distribution, the annual periodicity indicative of temperature differences between summer and winter periods has also been constant for over a century. Moreover, we detected a 6-monthly periodicity that is indicative of the similar values of temperature during spring and autumn times in the south of the Iberian Peninsula. Long-term correlation and spectral analysis corroborated the periodicity of approximately 5 (and 2.5) years found for the precipitation series. The approximately 5-year periodicity of temperature distribution has also been detected in other parts of the world, such as Great Britain, Canada and the USA (Burroughs, 1992), where the temperature recording periods used were very similar to those available for the present study. The continuous wavelet transform method allows us to deduce new interannual information, so during the last two decades of the record periodicities of 2–3 and 4–6 years have also been detected in temperature, according to the periodicities deduced from corre-lation and spectral analysis. Nevertheless, no long-term trends were detected for the temperature data time series, if a temperature increase of 18C or more had existed, it would have been detected by the correlation and spectral analysis.
In summary, in spite of the relative brevity of the time series of rainfall, temperature and outflow data, all the periodicities detected in this work can be compared with the climate variability at decadal to annual scale detected in several proxy data from geological records (Alverson and Oldfield, 2000;
6. Conclusions
From the results of the correlation and spectral analysis and wavelet continuous analysis applied to various long time series of climatic and hydrological data recorded in the south of the Iberian Peninsula, the following conclusions are drawn: the annual periodicity of the distribution of precipitation has been constant over more than 150 years, although lesser (6-monthly periods) and weaker periodicities have also been observed. Furthermore, long-term analysis reveals a rainfall periodicity of 5 (and 2.5) years, which has also been recorded in other areas of Europe due to the influence of the NAO. However, no long term trend in the rainfall and temperature distributions has been detected, perhaps because monthly data of rainfall, temperature and flowrate are not the best variables to observe climate change or because the climate change really started a few years ago and the time series do not yet contain these tendencies or perhaps both.
The two methodologies used in this work, correlation and spectral analysis (with different lags) and continu-ous wavelet transform methods, obtain similar results when they are applied to three type of data (rainfall, temperature and flow rate). This means that both methods are useful and complementary to study climatic changes: the correlation and spectral analysis is a powerful tool to detect average periodicities in long term series of climatic and hydrological data, while the wavelet transforms show the distribution of periodic variabilities with time. Therefore climatic and hydro-logical variations deduced are congruent and they can be compared with the geological record.
Acknowledgements
This study is a contribution to the projects IGCP-513 of the UNESCO, PB98-1397 and REN 2002-01797/HID of the DGI and to the Research Group RNM 308 of the Junta de Andalucı´a. We thank the Real Observatorio de San Fernando (Ca´diz) for providing rainfall and temperature data from the San Fernando meteorological station, the Confederacio´n Hidrogra´fica del Sur de Espan˜a and the Gibraltar Meteorological Office for rainfall data from the Gibraltar station and the Aguas de Jerez (AJEMSA) company for outflow data from the El Tempul spring.
We are very grateful to Prof. Ian Fairchild (University of Birmingham) for his suggestions and English corrections on this manuscript, and to the anonymous referees for their contructive criticisms.
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