Background on the Superposed Epoch Analysis
The Superposed Epoch Analysis (SEA) technique is a statistical method used to resolve significant signal to noise problems. This is especially desired in cases where the responses to particular events (in our case large radiative forcing from explosive volcanic eruptions) may be obfuscated by noise from other competing influences that operate at similar time scales (internal El Nino variation). Through simple compositing, the SEA method involves sorting data into categories dependent on a ‘key-date’ for synchronization and then comparing the means of those categories. Given sufficient data, a common underlying (causal) response to a forcing event should theoretically emerge in the average (composite) while other noise in the data should cancel. Examples of applications of the SEA method are widespread in various scientific fields of study. For example, the SEA has been employed in studies on the relationship between climate and fire histories (Swetnam and Betancourt, 1990), magnetospheric physics (Lühr et al., 1998), the lunar cycle and rainfall (Brier and Bradley, 1964), and the relationship between volcanic activity and climate variations (Mass and Portman, 1989). Compositing can also lend insight into periodic responses that, because of the relative magnitude of the signal and associated noise, are not detected by spectral analysis.
The SEA method is simple and involves basic arithmetic calculations (averaging). However, the randomization procedure used to determine statistical significance, depending on the number of iterations performed, can be computationally demanding. The method requires a random number generator for the Monte Carlo randomization procedure. We used the commercially available Matlab™ statistical software package to perform our SEA.
As with any statistical method, care must be taken in the application and interpretation of SEA results. For instance, the SEA can be vulnerable to leveraging resulting from the influence of a single large anomaly. This problem typically arises when the ‘key date’ sample size is small. To deal with this explicitly, our approach embraces proxy data to considerably increase the number of climatically-relevant eruptions incorporated into the SEA. We also include a normalization step in our SEA in order to provide a methodological safeguard to leveraging.
Applying the SEA to the ‘Volcano-ENSO Hypothesis’
We employ the SEA method to determine the response (if any) of the El Niño-Southern Oscillation ocean-atmospheric system to negative radiative forcing resulting from volcanic dust veils following large explosive eruptions in the tropics. Such a response to explosive volcanism was proposed from observations by Handler (see references in main text) and was discussed in early model simulations (Hirono, see main text; and MacCrackan and Luther). Because the details of the SEA method can be specifically tailored to each individual application, a detailed description of the basic structure of our statistical approach is included below. This supplements the information provided in the ‘Methods’ section.
First, SEA forcing events are selected based on physical criteria and typically referred to as ‘key dates’. For our application, the forcing component determining the selection of ‘key dates’ are years in which explosive eruptions took place in low-latitude (20 N to 20 S). Data sources included both the Ice-core Volcanic Index (IVI) and/or the Volcanic Explosivity Index (see ‘Methods’ sections for selection criteria and references). Given an array of key dates, we then sampled the time series that we analyze for a possible impact (NINO3 or SOI) using a window of data points centered on each key date. For each key date, the typical window used in our analysis is 21 years with 10 data points on either side of the key date. The extracted windows are then stacked in what we refer to as the ‘eruption matrix’. This symmetrical matrix contains before- and after-event information for all selected eruptions (key dates). Thus, for an 18-eruption key date list, for example, the accompanying eruption matrix would contain 18 rows and 21 columns of data.
Before analyzing the data, an adjustment intended to remove any disproportionate weight any extreme case could have on the overall composite is applied. This involves normalizing the individual data in each key date window by dividing each value in that row by the maximum absolute value of all anomalies in that window. This scales the magnitudes in each row of the eruption matrix so that the chance that a single anomaly may unduly influence the composite analysis is restricted. The overall mean of the eruption matrix is then removed.
As an aside, we additionally employed a ranked approach to account for potential leveraging (results not shown). A ranked SEA is a version of SEA where the ENSO diagnostic anomalies extracted for each eruption are first ranked prior to insertion into the eruption matrix. This is one way of protecting against the undue influence of anomalously large-magnitude events. Results from the ranked SEA method are in good agreement with both normalized and traditional SEA analyses that we performed. Thus we have taken every step possible to protect against this potential bias.
For those eruption matrices that exhibit statistically significant cooling in the post-eruption decade compared to the pre-eruption decade due to large scale volcanic cooling, the mean difference between pre- and post-eruption decades is removed (see ‘Methods’ for more discussion on the removal of the surface cooling expected). The significance of post-eruption cooling is determined using the non-parametric Mann-Whitney difference-of-mean test.
The composite (or averaged) results of the SEA are simply calculated by averaging down each column in the eruption matrix (e.g. the ‘year –10’ column is averaged for all eruptions yielding the composite value for ten years before explosive eruptions). This compositing of the eruption matrix yields what we call the ‘composite matrix’. The composite matrix contains 21 years of averaged NINO3 or SOI anomalies ranging from ten years before to ten years after the selected explosive eruptions (key dates). It is this averaging procedure that allows us to compare the average values of the NINO3 or SOI in any particular year before and after large eruptions. It is the stated goal here to determine if a systematic post-key date response emerges.
In order to compare these composite years and to evaluate the significance of an average anomaly determined in each composite year, several further statistical steps are required. First, the probability of occurrence for each composite value based on chance alone must be determined. To do this, we employ a Monte Carlo randomization procedure that reshuffles blocks of values inside each row of the ‘eruption matrix’ with equal weight given to each eruption. This block resampling procedure generates a randomly generated eruption matrix from the actual eruption matrix that is subsequently averaged to create a new, random composite matrix. Block resampling is based on the fact that serial correlation must be taken into account in estimating null distributions with time series with autocorrelation, or else the resulting confidence limits will be too liberal. The most straightforward way to do this is simply to resample the series in effectively independent blocks, rather than individual values. For a first order autoregressive process, the size of effectively independent blocks is given by the formula we specified (see ‘Methods’).
This is repeated 10,000 times to create 10,000 randomly generated composite matrices. These 10,000 randomized composite values are then sorted and a random composite distribution is created for each column (i.e. year relative to the key year 0). These 21 distributions are used to statistically judge how anomalous the actual composites are. We use these distributions to test the significance of the actual composites at the 90%, 95%, and 99% confidence levels.
Other Applications of the Superposed Epoch Analysis
Brier, G.W., and Donald A. Bradley, 1964: The lunar synodic period and precipitation in the US.
J. Atmos. Sci.21, 386-395.
Lühr, H., M. Rother, T. Iyemori, T. L. Hansen, R. P. Lepping. 1998. Superposed epoch analysis
applied to large-amplitude travelling convection vortices. Annales Geophysicae 16(7):
743-753.
MacCracken, M.C. and Luther, F.M., 1984: Preliminary estimate of the radiative and climatic
effects of the El Cichon eruption. Geofisica Internacional 23(3), 385-401.
Mass, C. F. and D. A. Portman, 1989: Major volcanic eruptions and climate: A critical
evaluation. J. Climate2, 566-593.
Swetnam, T.W. and J.L. Betancourt. 1990. Fire-Southern Oscillation relations in the
southwestern United States. Science 249:1017-1020.
Time Period / Eruption Criteria / pre-event mean (A) / post-event mean (B) / B - A / P-value1649-1979 / IVI Moderate-Large / 0.032 / -0.035 / -0.067 / 0.03
1649-1868 / IVI Moderate-Large / 0.031 / -0.034 / -0.065 / 0.09
1706-1868 / IVI Moderate-Large / 0.030 / -0.033 / -0.063 / 0.09
1706-1977 / IVI Moderate-Large / 0.030 / -0.033 / -0.063 / 0.03
1649-1979 / IVI Large Only / 0.031 / -0.034 / -0.065 / 0.09
1706-1977 / IVI Large Only / 0.042 / -0.046 / -0.088 / 0.03
1649-1979 / VEI >= 4 * / 0.018 / -0.020 / -0.038 / 0.20
1869 - 1979 / VEI >= 4 * / 0.006 / -0.001 / -0.007 / 0.47
1649-1868 / VEI >= 4 * / 0.021 / -0.024 / -0.045 / 0.18
1706-1977 / VEI >= 4 * / 0.028 / -0.030 / -0.058 / 0.17
1706-1868 / VEI >= 4 * / 0.055 / -0.061 / -0.116 / 0.03
1649-1979 / VEI moderate-large / 0.005 / -0.006 / -0.011 / 0.42
1869 - 1979 / Handler List / -0.006 / 0.006 / 0.012 / 0.61
1869 - 1979 / VEI moderate-large / 0.006 / -0.007 / -0.013 / 0.47
1649-1868 / VEI moderate-large / -0.001 / 0.001 / 0.002 / 0.5
1706-1868 / VEI moderate-large / 0.004 / -0.004 / -0.008 / 0.36
1706-1977 / VEI moderate-large / 0.006 / -0.006 / -0.012 / 0.47
1649-1979 / VEI large / 0.029 / -0.032 / -0.061 / 0.09
1706-1977 / VEI large / 0.055 / -0.061 / -0.116 / 0.03
1706-1868 / VEI large / 0.055 / -0.061 / -0.116 / 0.03
Table S1. Results of the non-parametric Mann-Whitney difference-of-mean test for pre- and post- eruption periods for composite NINO3 reconstruction data. Significance values (bold) show composites that exhibit a pre-eruption mean (A) significantly greater than the post-eruption mean (B) at the p<0 .10 level according to a one sided test. The * marks eruption lists whose selection criteria require that no other eruptions of magnitude VEI>=4 occur in the 12 months before or 12 months after the chosen eruption.
Eruption Criteria / Time Period / # of events / -9 / -8 / -7 / -6 / -5 / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9IVI Moderate-Large / 1649-1979 / 25 / N
IVI Moderate-Large / 1649-1868 / 18 / P / P / N / N
IVI Moderate-Large / 1706-1868 / 13 / N
IVI Moderate-Large / 1706-1977 / 19 / P
IVI Large Only / 1649-1979 / 12 / P
IVI Large Only / 1706-1977 / 8 / P*
VEI >= 4, +/- 12m sep. / 1649-1979 / 20 / P / P / P / N / N
VEI >= 4, +/- 12m sep. / 1869 - 1979 / 9
VEI >= 4, +/- 12m sep. / 1649-1868 / 10 / P / P / N* / N
VEI >= 4, +/- 12m sep. / 1706-1977 / 16 / P / P / N / N*
VEI >= 4, +/- 12m sep. / 1706-1868 / 7 / P* / N
VEI moderate-large / 1649-1979 / 31 / P / N / N
Handler List / 1869 - 1979 / 10 / P / P
VEI moderate-large / 1869 - 1979 / 9
VEI moderate-large / 1649-1868 / 22 / P / N
VEI moderate-large / 1706-1868 / 18 / P / N / P / N
VEI moderate-large / 1706-1977 / 26 / P / N / N
VEI large / 1649-1979 / 13 / P / P / N
VEI large / 1706-1977 / 10 / P / P
VEI large / 1706-1868 / 7 / P* / N
Total > positive 95% / 4 / 0 / 0 / 3 / 0 / 0 / 1 / 0 / 0 / 7 / 9 / 1 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 95% / 8 / 17
Total < negative 95% / 0 / 0 / 0 / 0 / 0 / 1 / 0 / 0 / 0 / 0 / 0 / 0 / 1 / 10 / 0 / 2 / 6 / 0 / 0
Total < negative 95% / 1 / 19
Combined total significant @ 95% / 9 / 36
Total > positive 99% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 3 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 99% / 0 / 3
Total < negative 99% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 1 / 0 / 0 / 1 / 0 / 0
Total < negative 99% / 0 / 2
Combined total significant @ 99% / 0 / 5
Table S2. Normalized, Mean-adjusted Annual NINO3 Reconstruction S.E.A Results. Years with positive composite values at the 95 and 99% confidence level are marked by P and P*, respectively; years with negative composite values at the 95 and 99% confidence level are marked by N and N*, respectively.
Eruption Criteria / Time Period / # of events / -8 / -7 / -6 / -5 / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8IVI Moderate-Large / 1706-1868 / 13
IVI Moderate-Large / 1706-1977 / 19 / N
IVI Large Only / 1706-1977 / 8
VEI >= 4, +/- 12m sep. / 1869 - 1979 / 9 / N
VEI >= 4, +/- 12m sep. / 1706-1977 / 16 / N / N
VEI >= 4, +/- 12m sep. / 1706-1868 / 7 / N*
Handler List / Handler List / 10 / P
VEI moderate-large / 1869 - 1979 / 9 / N / P
VEI moderate-large / 1706-1868 / 18 / P / P / N*
VEI moderate-large / 1706-1977 / 26 / N / P / N / N* / P
VEI large / 1706-1977 / 10 / N
VEI large / 1706-1868 / 7 / N*
Total > positive 95% / 1 / 1 / 0 / 0 / 1 / 0 / 0 / 2 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 1 / 0
Total > positive 95% / 5 / 1
Total < negative 95% / 0 / 1 / 0 / 0 / 0 / 4 / 0 / 0 / 1 / 6 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total < negative 95% / 5 / 7
Total > positive 99% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 99% / 0 / 0
Total < negative 99% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 4 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total < negative 99% / 0 / 4
Table S3. Normalized, Annual Winter SOI Reconstruction S.E.A Results. Years with positive composite values at the 95 and 99% confidence level are marked by P and P*, respectively; years with negative composite values at the 95 and 99% confidence level are marked by N and N*, respectively.
Time Period / Eruption Criteria / # of events / -7 / -6 / -5 / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 71856-2001 / Largest IVI/VEI / 6 / p
1856-2001 / VEI >= 4, +/- 12m sep. / 10 / N / n / p / p / n
1856-2001 / VEI Moderate-Large / 11 / N / N / P / n
1856-2001 / IVI Moderate-Large / 9
Total > positive 90% / 0 / 0 / 0 / 0 / 0 / 1 / 0 / 0 / 3 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 90% / 1 / 3
Total < negative 90% / 2 / 2 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 2 / 0
Total < negative 90% / 4 / 2
Total > positive 95% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 1 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 95% / 0 / 1
Total < negative 95% / 2 / 1 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total < negative 95% / 3 / 0
Table S4. Normalized, 3-yr Instrumental (Kaplan) Boreal Cold-season NINO3 S.E.A Results. Years with positive composite values at the 90 and 95% confidence level are marked by p and P, respectively; years with negative composite values at the 90 and 95% confidence level are marked by n and N, respectively.
Eruption Criteria / Time Period / # of events / -6 / -5 / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7Largest IVI/VEI / 1869-2001 / 6 / N
VEI >= 4, +/- 12m sep. / 1869-2001 / 9 / N / n
VEI Moderate-Large / 1869-2001 / 11 / N / N
IVI Moderate-Large / 1869-2001 / 9 / p / n
Total > positive 90% / 0 / 0 / 0 / 0 / 0 / 1 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 90% / 1 / 0
Total < negative 90% / 0 / 0 / 0 / 2 / 0 / 0 / 4 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total < negative 90% / 2 / 4
Total > positive 95% / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total > positive 95% / 0 / 0
Total < negative 95% / 0 / 0 / 0 / 2 / 0 / 0 / 2 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Total < negative 95% / 2 / 2
Table S5. Normalized, 3-yr Instrumental Winter SOI S.E.A Results. Years with positive composite values at the 90 and 95% confidence level are marked by p and P, respectively; years with negative composite values at the 90 and 95% confidence level are marked by n and N, respectively.
Eruption Criteria / IVI M-L / IVI M-L / IVI M-L / IVI M-L / IVI Large / IVI LargeTime Period / 1650-1980 / 1650-1868 / 1706-1868 / 1706-1980 / 1650-1980 / 1706-1980
# of eruptions / 25 / 18 / 13 / 19 / 12 / 8
Eruption Years / 1660 / 1660 / 1721 / 1721 / 1665 / 1747
1665 / 1665 / 1728 / 1728 / 1674 / 1808
1674 / 1674 / 1747 / 1747 / 1694 / 1815
1694 / 1694 / 1753 / 1753 / 1712 / 1831
1712 / 1712 / 1775 / 1775 / 1747 / 1835
1721 / 1721 / 1789 / 1789 / 1808 / 1884
1728 / 1728 / 1795 / 1795 / 1815 / 1903
1747 / 1747 / 1808 / 1808 / 1831 / 1963
1753 / 1753 / 1813 / 1813 / 1835
1775 / 1775 / 1815 / 1815 / 1884
1789 / 1789 / 1824 / 1824 / 1903
1795 / 1795 / 1831 / 1831 / 1963
1808 / 1808 / 1835 / 1835
1813 / 1813 / 1862
1815 / 1815 / 1884
1824 / 1824 / 1890
1831 / 1831 / 1903
1835 / 1835 / 1929
1862 / 1963
1884
1890
1903
1929
1963
1969
Table S6. IVI-based Eruption Lists. M-L denotes lists containing moderate and large explosive eruptions.
Eruption Criteria / Largest IVI/VEI / VEI >= 4* / VEI M-L / IVI M-LTime Period / 1856-2001 / 1856-2001 / 1856-2001 / 1856-2001
# of eruptions / 6 / 10 / 11 / 9
Eruption Years / 1884 / 1869 / 1884 / 1884
1903 / 1884 / 1899 / 1890
1951 / 1899 / 1902 / 1903
1963 / 1902 / 1911 / 1929
1982 / 1911 / 1917 / 1963
1991 / 1917 / 1943 / 1969
1943 / 1951 / 1975
1951 / 1963 / 1982
1963 / 1968 / 1991
1968 / 1982
1991
Table S7. Full Instrumental Period Eruption Lists. M-L denotes lists containing moderate and large explosive eruptions. The * marks lists that require eruptions to be separated from any other VEI>=4 eruptions by at least 12 months.
Eruption Criteria / VEI M-L / Handler / VEI M-L / VEI M-L / VEI M-L / VEI M-L / VEI Large / VEI Large / VEI LargeTime Period / 1650-1980 / 1869 - 1980 / 1869 - 1980 / 1650-1868 / 1706-1868 / 1706-1980 / 1650-1980 / 1706-1980 / 1706-1868
# eruptions / 31 / 10 / 9 / 22 / 18 / 26 / 13 / 10 / 7
Eruption / 1661 / 1869 / 1869 / 1661 / 1720 / 1720 / 1671 / 1745 / 1745
Years / 1665 / 1883 / 1884 / 1665 / 1745 / 1745 / 1673 / 1768 / 1768
1674 / 1899 / 1899 / 1674 / 1756 / 1756 / 1680 / 1772 / 1772
1680 / 1902 / 1902 / 1680 / 1760 / 1760 / 1745 / 1812 / 1812
1720 / 1911 / 1911 / 1720 / 1761 / 1761 / 1768 / 1814 / 1814
1745 / 1917 / 1917 / 1745 / 1764 / 1764 / 1772 / 1815 / 1815
1756 / 1937 / 1943 / 1756 / 1768 / 1768 / 1812 / 1835 / 1835
1760 / 1950 / 1951 / 1760 / 1791 / 1791 / 1814 / 1869
1761 / 1952 / 1963 / 1761 / 1793 / 1793 / 1815 / 1884
1764 / 1963 / 1764 / 1803 / 1803 / 1835 / 1951
1768 / 1768 / 1808 / 1808 / 1869
1791 / 1791 / 1812 / 1812 / 1884
1793 / 1793 / 1813 / 1813 / 1951
1803 / 1803 / 1814 / 1814
1808 / 1808 / 1815 / 1815
1812 / 1812 / 1818 / 1818
1813 / 1813 / 1823 / 1823
1814 / 1814 / 1835 / 1835
1815 / 1815 / 1884
1818 / 1818 / 1899
1823 / 1823 / 1902
1835 / 1835 / 1911
1884 / 1917
1899 / 1943
1902 / 1951
1911 / 1963
1917
1943
1951
1963
1968
Table S8. VEI-based Eruption Lists. M-L denotes lists containing moderate and large explosive eruptions.
Eruption Criteria / VEI >= 4 * / VEI >= 4 * / VEI >= 4 * / VEI >= 4 * / VEI >= 4 *Time Period / 1650-1980 / 1869 - 1980 / 1650-1868 / 1706-1980 / 1706-1868
# of eruptions / 20 / 9 / 10 / 16 / 7
Eruption Years Used / 1671 / 1869 / 1671 / 1745 / 1745
1673 / 1884 / 1673 / 1768 / 1768
1680 / 1899 / 1680 / 1772 / 1772
1745 / 1902 / 1745 / 1812 / 1812
1768 / 1911 / 1768 / 1814 / 1814
1772 / 1917 / 1772 / 1815 / 1815
1812 / 1943 / 1812 / 1835 / 1835
1814 / 1951 / 1814 / 1869
1815 / 1963 / 1815 / 1884
1835 / 1835 / 1899
1869 / 1902
1884 / 1911
1899 / 1917
1902 / 1943
1911 / 1951
1917 / 1963
1943
1951
1963
1968
Table S9. VEI>=4 Eruption Lists. M-L denotes lists containing moderate and large explosive eruptions. The * denotes that the selection criterion requires eruptions to be separated from any other VEI>=4 eruptions by at least 12 months. The criterion for selection used in these lists is the same as that used by Handler and colleagues.