A wavelet cross-spectral analysis of solar/ENSO
connections with Indian monsoon rainfall
Subarna Bhattacharyya, Roddam Narasimha
Jawaharlal Nehru Centre For Advanced
Scientific Research (JNC), Bangalore India
Acknowledgement: Dr. Vanessa George, LASP, Univ. of Colorado
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Outline
Data Analysed
ENSO-Solar-Rainfall Connections Using
™Time Domain Analysis
™Wavelet Power Spectral Analysis
™Wavelet Cross-Spectral Analysis With Monte Carlo Methods
Conclusions
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Data analysed (ENSO)
™global SST ENSO index (henceforth referred to as ENSO)
http://jisao.washington.edu/data/globalsstenso/. uses the eastern
equatorial Pacific SST index as an ENSO index over (20N--20S)
minus (20N--90N,20S--90S) over the longitude grid 0degE-358degE.
The anomalies are in hundredths of a degree Celsius and are
measured with respect to the period 1950-79.
™ Nino 3 SST anomalies: The Nino 3 Region is bounded by
90W-150W and 5S-5N. Monthly means were computed for each
time series for the period 1950-1979.
http://www.cgd.ucar.edu/cas/catalog/climind/Nino\_3\_3.4\_indices
.html. Also at NOAA Climate Prediction Center
http://www.cpc.ncep.noaa.gov/data/
™Nino3.4 SST anomalies: 120W-170W and 5S-5N
ftp://ftp.cgd.ucar.edu/pub/CAS/TNI\_N34.
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Data analysed (contd)
™ SST anomaly tendency affects the AISM rainfall to a larger
extent than the SSTs or the SST anomalies themselves (e.g.
Rajeevan et al. 2005).
™Nino 3.4 Tendencies: The anomalies in SST over the months
December, January and February are subtracted from those
over the succeeding months March, April and May to get the
SST anomaly relevant to the rainfall of the monsoon season
beginning in May--June.
™A positive SST Nino 3.4 anomaly tendency represents a
strong El Nino over the Pacific, which in turn results in
weakening of the monsoon circulation over Asia. Similarly, a
negative implies weakening of the El Nino over the Pacific and
strengthening of the monsoon over Asia.
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Data Analysed (Rainfall)
HIM
™HIM region covers the central
and north western parts of India
amounting to 55 % of the total
land area of the country.
™HIM is considered to be the
most characteristic index of the
component of Indian rainfall
dominated by the dynamics of the
south west monsoon.
™All India Summer Monsoon
(AISM) Rainfall is also studied
Homogeneous rainfall region map
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Data analysed (Solar Activity)
Solar cycle processes:
™ Sunspot number
™ Group sunspot number
™ Solar Irradiance
Source: NOAA
ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA
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Solar-rainfall connections: previous results
SB & RN, Geophys. Res. Lett. 2005
™ Mean rainfall is higher in test-periods of higher solar
activity at confidence levels varying from 75% to 99%
™ Strong regional variations: AISMR misleading indicator;
NEI behaves radically differently than rest of
homogeneous regions
™Over the two test-periods of high and low solar activity,
the average cross power for rainfall and solar activity
index exceeds that between noise and solar activity index
at confidence levels of 99.99% or higher at the 8-16 y, 913 y and 10-12 y bands using the z-test. SB, PhD Thesis, JNC, 2005
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ENSO – AISM – Solar Irradiance
Over TP1 and TP2 solar activity exhibits
Test Period 1 (TP1):
maximum contrast.
1878-1913
Test Period 2 (TP2):
1933-1964
™Rainfall, Solar
process in phase
™ENSO, Rainfall out of
phase
™ENSO, Solar
processes out of phase
™% confidence
levels (using z-test)
shows, for significant
difference between
means for the two 8test
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periods
Nino 3, Nino 3.4 SST
Nino 3, Nino 3.4
SST anomalies are
out of phase with
rainfall and solar
processes over the
two test periods,
although the
confidence levels
for the differences
in these cases are
low.
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Continuous Wavelet Transform
Let xn= x(nt), n=0..N-1.
Continuous Wavelet Transform of xn is
2
Wn (s) ´
P
0
3
N¡ 1
? 64 (n ¡ n)±t 75
0
0
x
,
ª
n
s
n =0
(1)
where for a Morlet wavelet
µ
¶
ª (´ ) = ¼¡ 1=4 exp (i ! 0´ ) exp ¡ ´ 2=2 ,
µ
¶
2¼s 1=2 ^
Ã0(s! k );
±t
ª ? = Complex conjugate of ª .
^ k) =
Ã(s!
R1
0
San Juan Islands,
j Ã^0(! ) j d!SORCE=2006,
1:
Washington
9/29/2006
¡ 1
0
2
(2)
(3)
(4) 10
Wavelet Power Spectrum
Local wavelet power spectrum:
Wn (s)
(5)
?
¤ Wn (s)
Wavelet power:
j Wn (s)
?
¤ Wn (s)
(6)
j.
Global wavelet power:
¹ (s) =
W
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1
N
P
N¡ 1
n= 0
j Wn (s)
?
¤ Wn (s)
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j
(7)
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ENSO – AISM – Solar Irradiance
Wavelet cross power spectra using Morlet continuous
wavelets. Contours in green and blue mark regions where the
cross spectrum exceeds the reference spectrum at 90% and
95% confidence levels respectively.
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ENSO – AISM – Solar Irradiance
Torrence & compo (1998) Test:
Global wavelet cross
power spectrum
between
(a) ENSO index &
solar irradiance,
(b) AISM rainfall &
solar irradiance,
(c) ENSO index &
sunspot number,
(d) AISM rainfall &
sunspot
(e) ENSO & AISM.
ENSO-AISM cross power shows lowest confidence levels (70%),
ENSO-IRRAD shows highest confidence levels (97.5%).
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ENSO – Monsoon – Sunspot: Spatial Variation
Note here SN denotes Sunspot Numbers
High cross power in band 4 for Nino 3.4 and SN
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Wavelet cross
power averaged
over different
period bands as
percentages of
the total wavelet
cross power.
Band (1): 1--2 y,
(2): 2--4 y, (3): 4-8 y, (4): 8--16 y,
(5): 16--32 y, (6):
32--64 y.
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Nino 3.4 T – Monsoon – Sunspot
Nino3.4T &
AISM rainfall
show high cross
power in the 8-16
y band whereas
Nino 3.4 T &
AISM show
scattered cross
power over 2-8 y
band.
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Nino 3.4 T – Monsoon
For all the homogeneous rainfall regions, Nino 3.4 T &
RF shows high cross power scattered over the 2-8 y
band, and even over test period 1, and also over test
period 2 in case of PENSI.
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Consolidated results on ENSO
% Contribution to
wavelet power in
2-7 y and 8-16 y
bands.
NEI in phase with
Nino3, Nino3.4,
Nino3.4 T over
both bands;
global ENSO in
opposite phase
with all of the
above.
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Consolidated results on ENSO
% Contribution to
wavelet cross
power in 2-7 y and
8-16 y bands.
Note high cross
powers for Nino
3.4 Tendency
and HIM in both
2-7 and 8-16 y
bands.
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Consolidated results on ENSO
™ Strongest contributions are in the pairs sunspot-rainfall, Nino
3.4 tendency-rainfall, sunspot-Nino 3.4 and sunspot-Nino 3.4
tendency;
™ Contributions to ENSO-rainfall cross spectra from the 8-16 y
band are not substantially less than from the 2-7 y band.
™On the 2-7 year time scale, an increase in solar activity
increases the ENSO index & decreases the monsoon rainfall
over all regions. (There are regional variations in the magnitude
of such a decrease.)
™On the 8-16 year time scale, an increase in solar activity is
associated with a decrease in the ENSO index and increase in
the rainfall.
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Consolidated results on ENSO
™The net effect of solar activity on rainfall therefore appears
to be the result of counteracting influences on the two bands
i.e. on short and long periods; scale-interactions therefore
appear to be important.
™The effect of positive Nino 3.4 SST tendencies on the
monsoon rainfall is to decrease the rainfall.
™The link between monsoon rainfall and solar activity
emerges as having the strongest evidence; the ENSO-solar
activity connection is stronger than for the ENSO-monsoon
connection.
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Consolidated results on ENSO
The evidence for multi-decadal connections
between Indian rainfall and solar activity, directly
and mediated through ENSO, is thus strong,
as tested by two different procedures:
™
time-domain analysis (which uses
sunspot numbers only to select periods of greatest
contrast in solar activity)
™
wavelet cross spectra (band-averaged
and otherwise, which use quantitative solar activity
data to analyse effects in different period bands).
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Criticism of time domain results
The time domain analysis showed that rainfall was higher in the
more solar--active period. However, apart from a secondary role in
identifying the test periods, no quantitative indicator of solar activity
takes direct part in that analysis. This leaves open the possibility
(however improbable) that some other parameter (such as ENSO, for
example) may be responsible for the differences observed in rainfall
between the two test periods.
To overcome this limitation, we perform wavelet cross power
analysis using synthetic noise signals simulating the spectrum and
pdf of rainfall and ENSO.
WN-white noise,SN-noise with HIM/ENSO spectrum, ANnoise with HIM/ENSO pdf
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Monte Carlo and statistical comparison of band averages
We introduce the following notations for comparing averages statistically:
™Average over a band b : RbSN(t)
™Average over ensemble : ⟨RbSN(t)⟩.
™Average over test-period : [RbSN(t) ]k=1,2.
An average over the 9-13 y band, as well as over the ensemble
for test period 1, will be denoted by [⟨R9-13SN(t)⟩]1 which is just
one number.
Further, we study the cumulative distribution functions of these
averages. We evaluate the confidence levels to which the differences in
the CDFs are significant in each case.
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Monte Carlo Results on ENSO-Rainfall-Sunspot Triad
Typical %
conf. obtd. in
8-16 band
avg. crosspower
N34T-HIM &
N34T-SN is
97.5 % over
TP1, 99.5%
over TP2
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SS-HIM &
SS-SN is
99.99% over
TP1 & TP2
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Conclusions
ENSO-RAINFALL-SUNSPOT TRIAD
Confidence levels at which the hypothesis that the two crossspectra are the same is rejected:
For 8-16 band avg. cross-power
™ N34T-HIM & N34T-SN : 97.5 % over TP1, 99.5% over TP2
™SS-HIM & SS-SN : 99.99% over TP1 & TP2
For 10-12 band avg. cross-power
™SS-N34T & SS-SN : 99.5% over TP1 and TP2
™SS-HIM & SS-SN : 99.99% over TP1 & TP2
For 2-7 y band avg. cross power
N34T-HIM & N34T-SN : less that 50% over TP1, 95% over TP2.
Spatial Variations
MULTIDECADAL CONNECTIONS
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Conclusions
* Rainfall sunspot connections are strong (99.5% confidence level
between sunspot and HIM rainfall)
* ENSO rainfall connections are much weaker (for example 70%
confidence level between ENSO and AISM rainfall)
* ENSO and solar activity connections are stronger than ENSO rainfall
connections (for example 95% confidence level between sunspot
and Nino 3.4 tendency)
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