## Abstract

The double intertropical convergence zone (ITCZ) bias still affects all the models that participate to CMIP5 (Coupled Model Intercomparison Project, phase 5). As an ensemble, general circulation models have improved little between CMIP3 and CMIP5 as far as the double ITCZ is concerned. The present study proposes a new process-oriented metrics that provides a robust statistical relationship between atmospheric processes and the double ITCZ bias, additionally to the existing relationship between the sea surface temperature (SST) and the double ITCZ bias. The SST contribution is examined using the THR-MLT index (Bellucci et al. in J Clim 5:1127–1145, 2010), which combines biases on the representation of local SSTs and the SST threshold leading to the onset of ascent in the double ITCZ region. As a metrics of a model’s bias in simulating the interaction between circulation and precipitation, we propose to use the Combined Precipitation Circulation Error (CPCE). It is computed as the quadratic error on the contribution of each vertical regime to the total precipitation over the tropical oceans. CPCE is a global measure of the circulation-precipitation coupling that characterizes the model physical parameterizations rather than the regional characteristics of the eastern Pacific. A linear regression analysis shows that most of the double ITCZ spread among CMIP5 coupled ocean–atmosphere models is attributed to SST biases, and that the precipitation large-scale dynamics relationship explains a significant fraction of the bias in these models, as well as in the atmosphere-only models.

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## Notes

The coefficient of determination \(R^{2}\) is the proportion of variability in a data set that is accounted for by the statistical model. It is defined as: \(R^{2}=\frac{\sum _{i}(\hat{SI}_{i}-\bar{SI}_{i})^{2}}{\sum _{i}(SI_{i}-\bar{SI}_{i})^{2}}=1-\frac{\sum _{i}(SI_{i}-\hat{SI}_{i})^{2}}{\sum _{i}(SI_{i}-\bar{SI}_{i})^{2}}\), where \(SI\) is the observed value, \(\hat{SI}\) is the predicted value by the regression model and \(\bar{SI}=\frac{1}{n}\sum _{i}{SI}_{i}\). \(\overline{R^{2}}\) is the proportion of variability in a data set that is accounted for by the statistical model, that accounts for the number of explanatory variables in the model. It is defined as: \(\overline{R^{2}}=1-\frac{n-1}{n-p}\frac{\sum _{i}(SI_{i}-\hat{SI}_{i})^{2}}{\sum _{i}(SI_{i}-\bar{SI}_{i})^{2}}\).

LOESS denotes a method that is also known as locally weighted polynomial regression. At each point in the data set a low-degree polynomial is fitted to a subset of the data, with explanatory variable values near the point whose response is being estimated. The polynomial is fitted using weighted least squares, giving more weight to points near the point whose response is being estimated and less weight to points further away.

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## Acknowledgments

We would like to thank Aurélien Ribes for helpful discussions. We also acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1 of this paper) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.

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## Appendix

### Appendix

### Evaluating the results of a linear regression

To validate the results of a linear regression, it is important to examine the residuals (\(\epsilon\)) from the regression and identify extreme data points (leverage), that can potentially exercise a great influence on the regression line. The residuals are normalized (i.e., divided by the standard deviation of the residuals) in order to make the analysis on a standard scale.

The leverage is based on how the observed values differ from the values predicted by the regression model: \(\hat{SI}=H \ SI\), where \(SI\) is the vector of observed values, \(\hat{SI}\) is the vector of values predicted by the regression model and \(H\) is the hat matrix. The leverage of the i-th value is the i-th diagonal element (\(h_{ii}\)) of the hat matrix \(H\).

Combining both residuals and leverage, we obtain a measure of the actual influence each point has on the slope of the regression line, namely the Cook’s distance. Cook’s distance is a measure of the effect of deleting a given observation on the regression analysis (Cook and Weisberg 1982).

Cook’s distance is calculated as: \(D_{i}= \frac{\sum _{j=1}^{n}(\hat{SI}_{j}-\hat{SI}_{j(i)})^{2}}{p \ MSE}\), where \(\hat{SI}_{j}\) is the prediction from the full regression model for observation *j*, \(\hat{SI}_{j(i)}\) is the prediction for observation *j* from a refitted regression model in which observation *i* has been omitted, \(MSE\) is the mean square error of the regression model and *p* is the number of parameters in the model. Cook’s distance can be expressed as a function of both residuals and leverage: \(D_{i}= \frac{\epsilon _{i}^{2}}{p \ MSE} [\frac{h_{ii}}{(1-h_{ii})^2} ]\), where \(\epsilon _{i}\) is the residual of the regression. Data points with large residuals and/or high leverage may alter the result of the regression.

Smaller Cook’s distances means that removing the observation has little effect on the regression results. Distances larger than 1 are suspicious and suggest the presence of a possible outlier or a poor model.

Figure 17 shows the standardised residuals versus leverage plot of the regression model, described by Eq. (3), performed with AGCMs, with and without INMCM4. The relationship between residuals and leverage is highlighted through a LOESS curve (LOcal regrESSion,^{Footnote 2} Fox 2002). Superimposed on the plot are contour lines for the Cooks distance.

Comparing the two plots, we see that the regression performed without INMCM4 (see Fig. 17b) exhibit smaller residuals and leverage. Indeed, the values of Cook’s distance are inferior to 1. This confirms the robustness of the regression model described by Eq. (3) in AGCMs and validates the exclusion of INMCM4.

Figure 18 shows the same plot of the regression model, described by Eq. (4), performed with OAGCMs, with and without INMCM4. Again, INMCM4 is identified as an outlier (see Fig. 18a). Indeed, after excluding INMCM4, residuals and leverage are smaller and the values of Cook’s distance are inferior to 1 (see Fig. 18b). This validates the regression model described by Eq. (4) in OAGCMs and emphasizes its suitability at explaining the double ITCZ bias through both THR-MLT and CPCE indices.

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Oueslati, B., Bellon, G. The double ITCZ bias in CMIP5 models: interaction between SST, large-scale circulation and precipitation.
*Clim Dyn* **44**, 585–607 (2015). https://doi.org/10.1007/s00382-015-2468-6

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DOI: https://doi.org/10.1007/s00382-015-2468-6

### Keywords

- Double ITCZ
- Atmospheric dynamics
- Coupled ocean–atmosphere feedbacks