After wavelet decomposition of the time series of annual rainfall values, x , the
n
reconstruction is given by:
1r2
J
d jd t R W s
Ž .
n j
x s ;
1
Ž .
Ý
n 1r2
C c 0 s
Ž .
d j
js0
Ž . symbols have the following values: c 0 removes a scaling introduced during wavelet
y1 r4
w Ž .x
transform; its value is equal to p . C is a constant with value 0.776. R W s
is
d n
j
the real component of the complex wavelet transform. To avoid unnecessary repetition, Ž
. readers are referred to Torrence and Compo 1998 for further details of the Morlet
continuous wavelet transform.
3. Description of decadal and interdecadal fluctuations
3.1. Non-periodic signal analysis of AAÕAR The time series of AAvAR was scrutinized using wavelets to identify non-periodic
components. For each year, the wavelet analysis decomposes the annual rainfall value into contribution from 80 wavelet timescales. Each wavelet timescale makes its own
Ž Ž ..
contribution to the reconstruction of the AAvAR at each year Eq. 1 . The linear correlation coefficient between observed and reconstructed time series of AAvAR is
0.97; they exhibit good agreement; edge effects are small. Result of wavelet time–timescale contribution analysis is shown in Fig. 5. This figure
Ž . depicts local contribution to annual rainfall perturbation, obtained using Eq. 1 , at
Ž . indicated year and from indicated wavelet period. The annual rainfall perturbation r t
ˆ
Ž . Ž .
Ž . is defined by: r t s r t y r, where r t is the AAvAR, t is year, and r is the mean
ˆ
Ž .
Ž . calculated on the period 1905–1983 of r. r t is reconstructed by summing contribu-
ˆ
Ž .
tions along a vertical line that is, summing contributions from all wavelet periods . Fig. 5 shows three main features:
Ž . 1 Contributions to total rainfall perturbation from bands of wavelet timescale larger
than 37 years are small in this region. These bands of wavelet timescales will not be further considered.
Ž . Ž
2 Fig. 5 shows that a chirp-like fluctuation is excited in the mid-1930s a chirp-type .
Ž .
fluctuation has its timescale varying in the band 10 years, 17 years of wavelet timescale; and by late 1950, it expanded to approximately 20-year timescale. Dipole-like
structures form this fluctuation. In Fig. 5, a line joins the center of dipoles of this fluctuation. A center of negative rainfall perturbation harbingers the dipole; followed by
positive rainfall perturbation. This wavelike fluctuation starts at about 1935 with a leading negative rainfall perturbation. Timing between successive dipoles is approxi-
mately coincident with wavelet timescale. The core of negative rainfall perturbation took place in 1935, 1949, and 1969, coincident with severe droughts in the Province of
Cordoba. This fluctuation is amplifying since its excitation in the mid-1930s.
´
Ž . 3 There are large contributions from a band with wavelet timescales between
Ž approximately 2 years to 8 years. The quasi-biennial fluctuation and ENSO El Nino —
˜
. Southern Oscillation are in these wavelet timescales.
Ž .
Fig. 5. Morlet wavelet analysis of the AAvAR. Top: observed time series of AAvAR mm . Bottom: time Ž
elapsed-timescale representation of the time series of rainfall perturbation annual rainfall values minus mean .
Ž .
Ž .
in 1905–1983 . Contours are 0, 5, 10, and 15 solid curves , and y5, y10, y15 dashed curves , in units of millimeters of rainfall at each wavelet timescale. Full line shows the time continuity of the fluctuation excited
in the mid-1930s.
Next, contributions to perturbation of AAvAR were grouped into bands of contribut- ing wavelet timescales. Fig. 6 shows the partial contribution to total rainfall perturbation
from each band of wavelet timescales. Largest partial contributions are supplied by bands of wavelet timescale smaller than 27 years. The band of wavelet timescale smaller
than 10 years encompasses contributions from timescales similar to the quasi-biennial
Ž .
oscillation, and from ENSO El Nino — Southern Oscillation . The following two
˜
bands, with wavelet timescales centered at 13.5 years and 22 years, includes contribution from the fluctuation excited about mid-1930; these contributions show an amplifying
Ž .
wavelike pattern. In particular, the band of wavelet timescales 17 years, 27 years starts to amplify in mid-1930; then about mid-1950 its amplification rate becomes stronger.
Ž .
The contribution from band 10 years, 17 years amplifies in the mid-1930s, reaches a maximum during the early 1940s, and thereafter dampens until the late 1960s when it
starts to amplify again. Contributions to total annual rainfall perturbation can be further synthesized by
grouping them into two bands of wavelet timescale: smaller than 10 years; and larger than 10 years. This is shown in Fig. 7. In this figure, an ascending trend is observed
since the occurrence of the excitation of amplifying fluctuations in the bands of wavelet
Fig. 6. Contribution to perturbation of AAvAR, from indicated bands of wavelet timescales. Y-axes indicate the contribution of band to total annual rainfall perturbation. Bottom panel shows the total perturbation of
Ž .
AAvAR average of three rainfall time series .
Ž .
Ž .
timescales 10 years, 17 years and 17 years, 27 years . In contrast, it is seen in Fig. 6 that contribution from band of wavelet timescales smaller than 10 years does not show
trend. We proceed next to carry out a regression analysis.
Fig. 7. Partial contribution to AAvAR from band of wavelet timescales greater than 10 years.
3.2. Trend in annual rainfall during period of fluctuation amplification Having found an amplifying fluctuation starting in about the mid-1930s in the
decadal and bidecadal bands of contribution to the total perturbation of AAvAR, their effects on the evolution of the mean value of AAvAR is examined next. Because the
amplifying fluctuation started in about the mid-1930s, we want to know if different patterns of evolution in the AAvAR occurred before and after that time. Fig. 8 shows the
time series of 5-year mean of AAvAR, which exhibits two patterns of evolution. The first pattern, before the mid-1930s, lacks a clear trend; the second pattern begins at the
time of excitation of the amplifying interdecadal fluctuations. This second pattern shows a positive trend.
Next, using linear regression we analyze the effect of the fluctuations starting in 1935 on the evolution of the mean of AAvAR. The time series of AAvAR was split into two
different periods. The first period covers the span 1905–1934. The second period is from 1935–1983. Results indicate that along the period 1905–1934, there was not a
statistically significant trend in AAvAR. On the other hand, the period 1935–1983 Ž
. coincident with the amplifying fluctuations at interdecadal timescale shows a rainfall
trend that is statistically significant at the 0.1 level. Graphical representation of regression results is in Fig. 2.
Table 1 shows the linear regression analyses. For the analysis of the linear regression of AAvAR on year, the null hypothesis ‘‘the slope of the regression line is zero during
1905–1934’’ has a p-level equal to 0.14. On the other hand, the p-level of the trend in the period 1935–1983 has a highly significant value of 0.008. During this period, the
slope of the regression line is 5 mmryear.
Fig. 8. Values of the 5-year mean of AAvAR, for 1905–1984. The areal average was calculated using annual rainfall values in Cordoba Observatory, Laboulaye, and Pilar. Dots in the graph are located at the beginning of
´
each 5-year period.
Table 1 Results of regression analyses between annual rainfall and year
Null hypothesis ‘‘Mean annual rainfall is not related to year’’ is rejected if p-level is -0.01. Rain gauges
Slope p-level
r Mean
Standard Number
Ž .
mm error
of data Cordoba Observatory 1873–1934
0.50 0.62
0.06 705.0
144.6 62
´
Cordoba Observatory 1935–1987 5.0
0.0005 0.46
743.6 168.5
53
´
Ž .
AAvAR 1905–1934 4.5
0.14 0.29
707.8 133.9
30 Ž
. AAvAR 1935–1983
5.0 0.0008
0.46 756.1
156.3 49
Ž .
Therefore, during the period with amplifying interdecadal fluctuations 1935–1983 , there is a statistically significant trend in annual rainfall with value 5.0 mmryear. In the
period 1905–1934, there was not a statistically significant trend in annual rainfall. The estimated trend of 4.5 mmryear given by the sample regression for period
1905–1934 is adversely influenced by the length of the time series, which is equal to 30 years. Length of time series has to be larger than twice the length of the 22-year
timescale fluctuation to obtain accurate results. Calculation of the trend before 1935 based on data from Cordoba Observatory provides results that are more accurate. The
´
data set from Cordoba Observatory starts at 1873. The time series of annual rainfall in
´
Ž Cordoba Observatory confirms this absence of trend. A linear regression analysis see
´
. Ž
Table 1 between annual rainfall amounts in Cordoba Observatory and year period
´
. 1873–1934 shows that annual rainfall trend, equal to 0.5 mmryear, is not significant
during this period at the 1 level. Fig. 3 shows the results of regression analysis. The null hypothesis, ‘‘The slope of the regression between annual rainfall amounts and year,
is zero’’, is not rejected at the 1 level. This result supports the previous conclusion on the absence of trend in AAvAR in the period 1905–1934. During the period 1935–1987,
the time series of annual rainfall in Cordoba Observatory shows, as expected, a trend of
´
5 mmryear. This is the same as the trend of the AAvAR. The conclusion is reached that simultaneous to the period during which amplifying
Ž .
fluctuation occurs starting about mid-1930s , there is a trend in annual rainfall. Before the starting of amplifying fluctuation, there is no statistically significant trend.
3.3. Trend produced by decadal and interdecadal fluctuations The question arises whether the trend during the period starting in 1935, is entirely
produced by interdecadal amplifying fluctuations, or also by a contribution from fluctuations with timescale smaller than 10 years. This question is answered using
regression analysis. The band of wavelet timescales between 17 years and 27 years makes an important contribution to reconstruction of annual rainfall amount. For
studying any trend in the contribution from any band of wavelet timescale, it is convenient that the length of time series would be close to or larger than twice the
largest timescale within the band. In turn, this implies that trend from timescale larger than 10 years needs to have data available close to or longer than 40 year. This
requirement is not fulfilled by the time series of AAvAR for the period 1905–1935. For
this reason, to study trend in the band of wavelet timescale larger than 10 years, the time series of annual rainfall at Cordoba Observatory is used. This is adequate because the
´
correlation coefficient is 0.88 between data from common period of the time series of AAvAR and the time series of annual rainfall in Cordoba Observatory. Comparison of
´
Ž .
Ž Fig. 7 for decomposition of AAvAR and Fig. 9 for decomposition of annual rainfall in
. Cordoba Observatory shows the strong similitude between contributions at wavelet
´
timescales larger than 10 years. Table 2 shows the results of regression analyses of contribution from the bands of
timescales smaller than and greater than 10 years on year. Results indicate that fluctuations in rainfall perturbation with timescale greater than 10 years produce the
observed trend in the mean of annual rainfall during 1935–1983. Their contribution is 4.72 mmryear during same period at a significance level of 0.001. Fluctuations with
Ž timescale less than 10 years contribute only with a trend of 0.33 mmryear
p-level of .
test statistics for null hypothesis is 0.78 . Trend of observed annual rainfall is 5.0 mmryear.
Therefore, in central Argentina, an important positive trend in annual rainfall is produced by fluctuations with timescales larger than 10 years. Atmospheric processes
Ž .
related to ENSO El Nino — Southern Oscillation are not contributing significantly to
˜
the generation of this trend in annual rainfall. Contributions from bands of wavelet timescale smaller than and larger than 10 years have a correlation coefficient equal to
0.07. This is not significant even at the 5 level. These results indicate that the rainfall trend of multidecadal extent observed in the
Province of Cordoba starts in conjunction with the excitation of a fluctuation that shows
´
increasing timescale and amplitude. This fluctuation was excited approximately in 1935 with a timescale of about 10 years. By 1981 the fluctuation almost reached a 20-year
timescale. Contributions from the band of fluctuations with timescale larger than 10 years explains almost all the trend.
Fig. 9. Contribution to perturbation of annual rainfall in Cordoba Observatory, from band of wavelet timescale
´
greater than 10 years.
Table 2 Results of regression analyses of annual rainfall for the band of timescales smaller than and greater than 10
years AR: Annual rainfall. AAvAR: Areal Average of Annual Rainfall. Null hypothesis: Slope is zero. Alternative
hypothesis: slope is not zero.
Variable Period
Timescale Slope
p-level AR in Cordoba Observatory
1873–1934 10 years
0.42 0.18
´
AAvAR 1935–1983
10 years 4.72
- 0.0001 AR in Cordoba Observatory
1873–1934 -10 years
0.08 0.93
´
AAvAR 1935–1983
-10 years 0.32
0.79
Time–timescale analysis using the Morlet wavelet indicates that the fluctuation Ž
excited in 1935 does not have precedent in the previous 60 years, since 1873 figure not .
shown .
4. Seasonal variability
