Recent studies using general circulation models to project the response of ENSO to greenhouse warming find, despite a shift in the mean state toward a warmer climate, no significant change in the amplitude of ENSO. Here, an intermediate coupled model of tropical Pacific is shown to capture such insensitivity of ENSO to a change in mean climate. This is accomplished by systematically searching in the model parameter space for the configurations that enable the model to reproduce the observed variance, skewness, kurtosis, and autocorrelation of the Niño 3 index of the past 150 years. Three model configurations are identified to give similar approximation of Niño 3 index, which represent the realization of ENSO in different climate regimes.
 The El Niño–Southern Oscillation (ENSO) is the strongest mode of interannual variability of the coupled atmosphere-ocean system in the tropical Pacific with global impacts because of its ability to force changes in the global planetary wave structure and precipitation patterns. There is much interest in understanding the evolution of ENSO behavior upon a background of changing climate.
 Despite the recent improvements in the ENSO simulations in coupled ocean-atmosphere models [AchutaRao and Sperber, 2006], the projections of ENSO's response to global warming are generally diverse and highly uncertain. Zelle et al. , using the Community Climate System Model (CCSM, version 1.4), found no significant change in the period, amplitude and spatial pattern of ENSO as greenhouse gas concentration increases. Toniazzo [2006a], using the Third Hadley Centre Coupled Model (HadCM3), found that ENSO in the 4 × CO2 integration exhibits a westward shift in the sea surface temperature (SST) pattern and an increase in the frequency. Meehl et al.  analyzed the stabilized greenhouse gases integrations of the Parallel Climate Model (PCM) and the CCSM version 3, found reduction in the amplitude of ENSO, but no appreciable change in the frequency. Survey of the multiple General Circulation Models (GCMs) experiments for the Fourth Assessment Report of the Intergovernmental Panel on Climate Change shows that the projections of future ENSO are largely model-dependent. Under doubling CO2 scenarios, models dominated by local SST-wind feedback show decrease in the amplitude of ENSO, whereas models dominated by remote thermocline feedback show an amplitude increase [Merryfield, 2006]. The largest ENSO amplitude change occurs in those models that exhibit an increase in the strength of thermocline feedback as CO2 increases [Guilyardi, 2006]. For the models with the most realistic simulations of present day ENSO, there is a reasonable balance of feedback mechanisms, and the feedbacks achieve a new equilibrium as climate moves toward a warming state [Philip and van Oldenborgh, 2006], where no statistically significant amplitude changes are observed in the projections of future ENSO [van Oldenborgh et al., 2005].
 The dependence of ENSO properties on the local SST feedback and the remote thermocline feedback has been proposed by Fedorov and Philander . In their linear stability analysis of the tropical ocean-atmosphere system, the ENSO mode is uniquely controlled by mean background climate parameters that include the intensity of the trade winds, the mean depth of the thermocline, the temperature difference across the thermocline. The mean climate change affects the strength of the ocean-atmosphere feedbacks, hence the period and growth rate of the ENSO mode. However, there is an obvious gap in trying to relate the easily observed ENSO statistics (variance, skewness, persistence) from observation and GCM simulations to the more understandable ENSO stability from linear theory [Toniazzo, 2006b]. For example, the GCMs (at least some) exhibits an insensitivity of ENSO to mean climate change, i.e., different equilibrium climates corresponding to similar ENSO variability [van Oldenborgh et al., 2005; Philip and van Oldenborgh, 2006], while Fedorov and Philander  suggest a high sensitivity of ENSO to changes in the background climate state.
 In an attempt to bridge the gap, here we apply the Bayesian stochastic inversion to an intermediate coupled model (ICM) under the constraint of 150-year observation of present day ENSO. First, six parameters of the ICM are selected to describe the mean climate states, then the ICM simulation under a mean climate state is compared to the observed ENSO in terms of Niño 3 index. By exploring the parameter space, we try to determine the climate regimes that produce interannual variability similar to ENSO observation. Furthermore, we discuss the implications of these climate regimes for interpreting the GCM projections of the response of ENSO to greenhouse warming.
2. Model and Parameters
 The model used in this study is an intermediate coupled model. It consists of an empirical atmosphere and a 1.5 layer reduced gravity ocean embedded with a constant-depth mixed surface layer. The model domain covers the tropical Pacific Ocean (30°S ∼ 30°N, 120°E ∼ 80°W) with a resolution of 1° in latitude and 2° in longitude. It was developed to study the physical processes controlling the seasonal cycle and the interannual variability in the tropical Pacific [Chang, 1994].
2.1. Model Parameters to Perturb
 In the ICM, the wind stress is represented by a linear stochastic model in vector form
where is the mean wind stress, T′ is the SST anomaly, η is the atmospheric white noise forcing. The linear feedback AT′ describes the dynamic response of the atmosphere to the SST anomalies. α and β are two nondimensional parameters, controlling the strength of the mean winds and the noise forcing.
 The oceanic component is a derivate of the Lamont model [Zebiak and Cane, 1987] with a parameterization of the temperature of the entrained water
where THmix and ∂zTHmix are the observed mean temperature and its vertical derivative at the base of mixed layer, γ and δ are two adjustable parameters controlling the dependence of Te on the sea surface temperature T and the thermocline perturbations h′, a and b are predetermined coefficients. In the SST equation, the effectiveness of the vertical upwelling term W(T − Te)/Hmix is determined by the contributions of the local SST-wind interaction γ and the remote SST-thermocline feedback δ.
 Within the ICM, six parameters were identified as controlling key characteristics of the model simulation: the coefficient of Rayleigh friction in the Ekman layer rs, the sensitivity of entrained temperature to SST γ, the sensitivity of entrained temperature to thermocline anomaly δ, the mean thermocline depth H, the strength of mean wind stress α, and the strength of noise forcing β. To explore the full range of model possibilities, we allow these parameters to vary freely within subjectively chosen uncertainty ranges.
2.2. Optimal Parameter Estimation
 For each combination of parameter values rs, γ, δ, H, α, β (hereafter referred to as a model configuration), we run the model forward for 300 years and calculate the Niño 3 index as the SST anomalies averaged over the eastern equatorial Pacific (5°S–5°N, 150°W–90°W). Then, we evaluate the model performance by comparing the 300-year Niño 3 simulation with the 150-year Niño 3 observation of Kaplan Extended SST [Kaplan et al., 1998]. The model-data comparison is quantitatively measured by a cost function, which is defined as a weighted average of the mean square error of multiple characteristics of ENSO statistics,
 Here var, skw, kur, and cor are variance, skewness, kurtosis, and autocorrelation (persistence) of Niño 3 time series; σs are the data uncertainties; and τ = 3,6,…,36 months is the time lag. We choose these statistics because they are easy to calculate and they are directly related to properties of the system's dynamics. For example, variance depends on stability [An et al., 2004]; skewness and kurtosis measure nonlinearity [Burgers and Stephenson, 1999; Hannachi et al., 2003]; autocorrelation is connected to power spectrum (via Fourier transform). As a sensitivity test, we also try different cost functions by adding more (less) weight to the individual terms of equation (3). As long as the general statistics of Niño 3 index is concerned, our results are insensitive to the changing cost.
 The data uncertainty σ consists of measurement error (which we ignore) as well as the natural fluctuations. Here, we first divide the 150-year observation of Niño 3 index into 15 ten-year segments, then calculate variance, skewness, kurtosis, and persistence for these ten-year time series and get 15 (independent) estimates. σs are the variances of these statistical quantities. It can be viewed as a measure of the decadal changes in ENSO statistics. Large σ (mainly the skewness and autocorrelation at long lag time) indicates large data uncertainty, thus a great tolerance in the model-data mismatch and a small contribution to the total cost. Since data uncertainty does not vary with model parameters, we can compare the model results for different parameter values by using the cost function (equation (3)). Smaller costs mean smaller errors, hence better ENSO simulations.
 In order to find the optimal parameters, i.e., the parameter values corresponding to the minimum cost, one needs to search through the six-dimensional parameter space spanned by (rs, γ, δ, H, α, β). In this study we apply an effective and efficient technique to sample the parameter space—Bayesian stochastic inversion with fast annealing [see Jackson et al., 2004; Sen and Stoffa, 1996]. The solution is cast in terms of Bayesian statistics and the result is an estimate of the joint probability density function (pdf) for the model parameters under the constraint of Niño 3 observation. From this joint pdf, we can determine the likelihood of parameter settings in reproducing the properties of observed ENSO. The optimal parameter setting is then inferred through the maximum likelihood estimation.
3. Parameter Regimes Consistent With Present ENSO
3.1. Probability Distribution in Parameter Space
 In our experiment, totally 5000 model configurations were explored, which represents 5000 (nonuniform) sampling points in the parameter space. The likelihood of a model configuration is the probability of the ICM's simulation close to the observation at that particular parameter value. By integrating the six-dimensional pdf over four parameters, we obtain the marginal probability of two parameters as well as the correlation between them. Figure 1 shows the marginal probability as a function of the mean thermocline depth H and the mean wind strength α. Obviously, probability tends to distribute within three distinct regions of parameter space. Taking H = 120 m, α = 1 as threshold for mean thermocline and wind, we shall refer to these high-probability regions (shaded areas) as weak wind-shallow thermocline (WWSH), weak wind-deep thermocline (WWDH), and strong wind-deep thermocline (SWDH). Given the intermediate model and the observational data, these are the parameter regimes where present-day ENSO is most likely to reside, with 56% chance in WWSH, 15% in WWDH, and 5% in SWDH.
 To test whether these high probability features truly reflect the six-dimensional distribution, we select the 100 best simulations (smallest cost) and mark them on various two-dimensional parameter maps of marginal probability. If the identified regime is really a simply connected domain in six-dimensional parameter space, then these best solutions should fall into the high-probability areas of two-dimensional maps. Our results show that most of the best solutions indeed concentrate in the three identified regimes, with some scattering between WWDH and SWDH.
 The congregation of best solutions in parameter space also reflects the inter-dependence between model parameters, which is well illustrated by the correlation among parameters (Table 1). As we can see, subsurface sensitivity (γ and δ), mean thermocline depth (H), and strength of noise forcing (β) are all significantly correlated. Here, high correlation means that in the model the effect of change in one parameter could be counteracted by change in the other parameter. For example, an increase in H is stabilizing while an increase in δ is destabilizing [Fedorov and Philander 2001]. In order for the ICM to reproduce observation, the effect of deepening H should be balanced by the effect of strengthening δ, which implies a positive correlation between H and δ. Such parameter correlation poses a challenge for parameter tuning in the development of climate models. As model parameters are not independent, simply tuning them one by one will not achieve the best performance [Stainforth et al., 2005].
It is calculated by integrating the probability density functions over parameters. Bold numbers indicate high correlations which are further discussed in the text.
3.2. Optimal Solutions
 The three parameter regimes are represented by their optimal solutions, whose parameter values are listed in Table 2. It is interesting to notice that these three optimals differ not only in mean thermocline depth and mean wind strength, but also in other parameters. The only exception is the Rayleigh friction coefficient, which is characterized by a damping timescale of 3 ∼ 4 days, indicating little parameter uncertainty and thus being ignored from the rest of discussion. Another observation is the coherent behavior of H and δ, confirming their high correlation shown in Table 1.
Table 2. Optimal Parameter Values for the Three Climate Regimesa
Figure 2 shows the simulated Niño 3 indices from optimal solutions along with the 150-year observation. Observation indicates that ENSO is an interannual variability with variance of 0.65 K2, skewness of 0.72, and kurtosis of 4.0 [Burgers and Stephenson, 1999]. The three model simulations all produce some sort of interannual variations, resemble the observation in terms of variance, skewness, kurtosis, and persistence. Considering the intermediate complexity of the model, we might conclude that the characteristics of observed ENSO are well reproduced within these three parameter regimes.
 Nevertheless, there are also some differences among the optimal solutions. For example, WWSH and SWDH produce more skewness (skw = 0.92) while WWDH produces less (skw = 0.2). An intuitive explanation is that saturation of SST is easier to occur in the cases of WWSH and SWDH than WWDH, which limits the full development of cold events. Spectral analysis of the simulated Niño 3 time series reveals a 2 ∼ 3 year power-band for SWDH, 3 ∼ 4 year band for WWDH, and dual peaks (3 year and 5 year) for WWSH. By switching on/off the noise forcing (set β = 0), we further investigate the free oscillations in these three regimes. Our study show that the ENSO-like variability in the WWSH regime is a self-sustaining oscillation disturbed by stochastic forcing (β = 1.3); WWDH is a highly damped oscillation forced by strong stochastic noise (β = 3.9); SWDH is also a damped oscillation but with moderate damping and modest stochastic forcing (β = 1.9). An in-depth explanation of these differences requires a dynamic understanding of the coupled tropical ocean-atmosphere system, which will be reported elsewhere.
4. Conclusion and Discussion
 With a novel inversion technique, we thoroughly sample the parameter space of an intermediate coupled model of the tropical Pacific, find three climate regimes, weak wind-shallow thermocline, weak wind-deep thermocline, strong wind-deep thermocline, that are consistent with the variance, skewness, kurtosis and autocorrelation of the Niño 3 index in the present day ENSO observation. This result has important implications for the interpretation of GMC results concerning the future state of ENSO under greenhouse warming. In those GCM experiments, the models first are validated by comparing with present day ENSO observation, then are used to make projection about ENSO under CO2 doubling or quadrupling. Only results from models with realistic ENSO simulations are considered reliable [Collins et al., 2005; van Oldenborgh et al., 2005; Merryfield, 2006]. Those models, however, differ in their configuration and parameterization. The simulated ENSOs then may represent a realization in different dynamic regimes, just like the ICM simulations in section 3. The diversity in model dynamics is definitely going to affect the projection of ENSO's sensitivity to mean climate change caused by greenhouse warming [Zelle et al., 2005; Merryfield, 2006; Toniazzo, 2006a; Yeh and Kirtman, 2007].
 We further note that the deep thermocline regimes are dominated by the remote SST-thermocline feedback, while the shallow thermocline regime is dominated by the local SST-wind feedback, implying a shift in the coupling loop during regime transition. Such shift has been demonstrated in several GCM experiments concerning the impact of global warming on ENSO. Namely, as the mean climate of equatorial Pacific warms, the trade winds becomes weaker, and the thermocline becomes shallower [Philip and van Oldenborgh, 2006], the coupling processes shift accordingly between the local wind feedback and the remote thermocline feedback [Guilyardi, 2006; Toniazzo, 2006a], however no statistical significant changes in the amplitude and period of ENSO variability are detected [van Oldenborgh et al., 2005].
 This paper primarily focuses on changes in the statistics of Niño 3 index in response to changes in the mean climate. Other aspects of ENSO, including pattern, phase, and propagation of the oscillation are not explored in our analysis. Therefore, the similarity of Niño 3 indices in three climate regimes does not necessarily suggest that ENSO also behaves similarly in all the other aspects. In addition, we assume that the combination of parameter values is self-consistent. In other words, every model configuration identified is physical realizable by assumption. In reality, wind, current, thermocline, and SST form a coupled system of mean climate states which involve feedbacks just like ENSO [Latif et al., 2001]. In addition, many important feedbacks associated with increasing greenhouse gases (e.g., the cloud forcing) are not contained in our intermediate model, which make it difficult to draw a firm conclusion on the projection of ENSO. It would require a full GCM to investigate all these feedbacks and determine the future state of the mean climate as well as ENSO.
 We thank Charles Jackson for initiating this research and providing his BSI code for the calculation. He is also greatly thanked for the stimulating discussions and constructive suggestions during the course of the work. Ping Chang, Matthew Collins, and an anonymous reviewer are acknowledged for valuable comments. This work has been supported by the G. Unger Vetlesen Foundation and the Chinese NSF grant 40576016. F. Wang is supported by the Hundred Talents Program, Chinese Academy of Sciences.