Supplemental materials to:

“An empirical model of decadal ENSO variability”

S. Kravtsov[1]

Accepted: June 13, 2012 Climate Dynamics

This document contains Supplemental Materials that essentially duplicate all of the figures shown in the main text of the paper using an alternative ERSSTv.3 data set (Smith et al. 2008). The figures here are labeled Figs. S1–S8 and should be compared with the corresponding main-text Figs. 1–8. Prior to this analysis, the ERSST data were interpolated onto the 5º5º spatial grid used for the Kaplan data set processed in the main text. We argue that while there are quantitative discrepancies between the two data sets used, especially in the 19th century periods, the present results corroborate the essential conclusions of the main text. In particular, there is a strong association between low-frequency modulations of ENSO variance and multidecadal climate modes inconsistent with the null hypothesis of random ENSO events forcing these modes (Fig. S3), while the hindcasts using statistical ENSO model suggest potential predictability of decadal modulations in ENSO variance if the external low-frequency predictors can be forecasted (Fig. S8). Brief discussion of the present figures is given below.

Figs. S1–S2. The DP/CV modes 1 and 3 here correspond to the DP/CV modes 1 and 2. The major difference between leading three DP/CV pairs for the ERSST data is that the CV-2 here (which should be analogous to CV-3 of the main text) varies on a longer time scale than that in the Kaplan data; this naturally results in different DP patterns for this mode. The cause of this discrepancy can be traced to the differences in the SST data sets in the 19th century, which is obviously the period of most uncertainty in these data sets. In particular, the ERSST discriminants computed for the 20th century (not shown) are much more consistent with those of Kaplan’s data. Similarly large differences are seen between the data sets in the 19th century in terms of ENSO index behavior (Fig. S2d here and Fig. 2d in the main text): the ERSST data exhibits a period of abnormally low ENSO variability there. The behavior of ENSO variance during the 20th century is, however, analogous in both data sets, with a large drop of ENSO variance in the middle of the century and higher values prior and after that period.

Fig. S3. The multiple correlation between the observed CVs and ENSO variance falls outside the 95th percentile of surrogate correlations obtained using a multivariate linear stochastic SST model (see main text for details); the fraction of the ENSO STD variance explained by the linear fit ranks above the 93rd percentile of synthetic quantities. Once again, values lower than those in the main text are due to data differences in the beginning of the century, with the ENSO variance values there being essentially an outlier inconsistent with the bulk of the distribution. Restricting the analysis period to the 20th century (not shown) produces higher values and significance levels matching the ones in text.

Figs. S4–S5. Note that the model used here (and in Figs. S6–S7) contains two external predictors — CVs 2 and 3 — and the present plots bear many quantitative similarities with the corresponding figures in the main text. In general, the combination of the restoring structure of the potential and the time-dependent variance of the stochastic forcing determine the time-dependent statistical properties of the model simulations. The results with 3 predictors for ERSST data set produce different contributions of the deterministic/stochastic forcing to the low-frequency variability of ENSO, due, once again, to differences between the data sets in the early years of the record combined with likely overfitting associated with the use of the global-temperature related external predictor (see section 4 of the main text).

Figs. S6–S7. Once again, the results shown here are for the model with two external variables. The “fast” ENSO statistics (Fig. S6) is nearly identical to that of the main-text model, while the multidecadal modulations of the ENSO variance are captured to a lesser extent (Fig. S7a) than for the Kaplan data (Fig. 7a). The present ERSST-based empirical ENSO models clearly suffers from the SST field outliers in the beginning of the record. Furthermore, the dependence of the deterministic operator of the ERSST model plays a relatively larger role in the response of ENSO to the external multidecadal predictors, since Fig. S7b looks more similar to Fig. S7a than the Fig. 7b to Fig. 7a.

Fig. S8. This figure is qualitatively similar to Fig. 8. The ERSST-based model hindcasts of the ENSO variance are less skillful than the Kaplan SST-based model, and the linear CV extrapolation scheme works worse for the present data than for the one in the paper. Hindcasts using fixed CVs (Fig. S8b,d) suggest potential skill in predicting decadal modulations of ENSO variance, underscoring the importance of being able to skillfully forecast these external predictors in future studies of alternative, ideally dynamically based ENSO-variance prediction schemes (see main text for further discussion).


Smith TM, Reynolds RW, Peterson TC, Lawrimore J(2008) Improvements to NOAA’s historical merged land–ocean surface temperature analysis.J Climate21: 2283–2296

Fig. S1. Leading discriminating patterns (DPs) of the observed sea-surface temperature (SST), in ºC, for the period of 1856–2010. The DPs are computed by regressing grid-point SSTs onto the unit-variance time series of canonical variates (CVs; see Fig. S2). The DP/CV pairs describe the SST modes that maximize the ratio of interdecadal-to-subdecadal variability (see main text for details).

Fig. S2. Canonical variates (CVs) and SST indices (1856–2010). (a) CV-1 and area-averaged SST; (b) CV-2 and Pacific Decadal Oscillation (PDO) index; (c) CV-3 and Atlantic Multidecadal Oscillation (AMO) index; (d) ENSO index (blue line, left y-labels) and fractional variations (%) of its 20-yr running standard deviation (STD) (red line, right y-labels), smoothed with 20-yr boxcar running-mean filter. The SST indices in (a–c) were smoothed using 10-yr boxcar running-mean filter. See main text for the detailed definitions of the SST indices.

Fig. S3. Association between CVs and long-term modulation in ENSO STD in 100 surrogate SST realizations produced by a multivariate linear stochastic model (see text for the details of this model). For each realization, the surrogate ENSO STD time series analogous to the one shown in Fig. S2d (red line) was regressed onto 3 leading CVs of the surrogate SSTs, as in Fig. S2a–c (red lines). (a) The histogram of correlations between actual and CV-reconstructed ENSO STD time series; (b) the histogram of fraction of ENSO STD variance (%) explained by the CV reconstruction. Vertical red lines show the corresponding quantities based on the observed SST data.

Fig. S4. The potential [equation (9) of the main text] based on the main level (6a) of the 1-D empirical ENSO-index model [equations (6–8) of the main text] with two decadal predictors given by CVs 2 and 3 (see Fig. S2b,c). Panels (a)–(d) show seasonal dependence of the potential, while the lines defined in the legend depict the potential during positive and negative AMO phases, according to Fig. S2c. Note that despite the potential well is the deepest and not very sensitive to the AMO phase in winter, we will show that ENSO events and their decadal variability are the strongest in winter, consistent with the structure of the potential in the preceding summer and fall seasons.

Fig. S5. The time series of the variance of the main-level residual based on 100 simulations of the 1-D empirical ENSO-index model [equations (6–8) of the main text] with two decadal predictors given by CVs 2 and 3 (see Fig. S2b,c). The variance time series plots are grouped by the season, as indicated in the individual panel captions. Blue lines show raw variance, while red lines represent 10-yr boxcar running means of the variance. Note the low-frequency modulations of the residual-forcing variance, which are most pronounced in the spring and summer seasons and are apparently caused by the dependence of the model’s second-level [equation (6b) of the main text] dynamical operator on the fixed low-frequency CV predictors; see equation (7) of the main text.

Fig. S6. Performance of the empirical 1-D ENSO index model [equations (6–8) of the main text]: autocorrelation function (ACF) — left panel; probability density function (PDF) — middle panel; and ENSO-index variance as a function of the season (1 – winter, 2 – spring, 3 – summer, 4 – fall) — right panel. Observed characteristics are plotted by the solid blue lines, while the 95% spread of 100 synthetic ENSO index simulations using model (6–8) of the main text are plotted by the dashed lines; the solid red line with circles in the right panel shows ensemble-mean ENSO variance as a function of the season. The empirical model uses CVs 2 and 3 (Figs. S2b,c) as fixed predictors and captures the observed characteristics fairly well, except for underestimating positive skewness of the actual ENSO index.

Fig. S7. Observed (heavy red lines) and simulated (lighter blue lines) fractional variations (%) of the ENSO index 20-yr running standard deviation (STD), smoothed with 20-yr boxcar running-mean filter. Light dashed blue lines show interquartile range of the simulated ENSO STDs. (a) Simulated quantities are based on the model (6–8) of the main text driven by the observed CVs 2 and 3 (Fig. S2b–c); (b) Same as in (a), but for the model with suppressed dependence on CVs at the second model level [equation (6b) of the main text]. The full model captures well the decadal variations in ENSO characteristics correlated with (and presumably driven by) the AMO/PDO variability, with the non-overlapping error bars for positive/negative ENSO variance phases indicating statistical significance of these variations.

Fig. S8. Cross-validated hindcasts of the ENSO STD using jack-knifing procedure that trains the ENSO models [equations (6–8) of the main text] on independent data by omitting 15-yr segments from the full observed time series. The models are then validated by running predictions of the observed 20-yr data segments whose starting points are collocated with those of the omitted segments. Panels (a–d) show the observed ENSO STD (heavy red line), as well as lead-time 0–20-yr hindcasts (light colored lines) for: (a) 3-CV model (6–8) and linear CV extrapolation; (b) 3-CV model (6–8) and fixed CV predictors (as in Figs. S2a–c); (c, d) The same as for (a, b), but for the model with two — AMO/PDO related — external predictors, CV-2 and CV-3. (e) Anomaly correlation; and (f) root-mean-square (RMS) error for the hindcasts shown in (a–d), along with persistence and climatology forecasts of ENSO STD. The results suggest potential long-term ENSO STD predictability.


[1] Dept. of Mathematical Sciences, Atmospheric Sciences Group, University of Wisconsin-Milwaukee, P. O. Box 413, Milwaukee, WI 53201. E-mail: