a. GEOS modeling system

The analyses in this paper focus largely on a 45-member ensemble of AMIP-style simulations with the National Aeronautics and Space Administration (NASA) Goddard Earth Observing System (GEOS) atmospheric general circulation model (AGCM; Molod et al. 2015). The version of the AGCM used here is essentially the same as that underlying the NASA Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2), reanalysis (Gelaro et al. 2017); it is run on a cubed-sphere grid and includes state-of-the-art representations of a host of atmospheric and land surface processes. Two important differences, however, should be noted between the AGCM used here and that underlying MERRA-2. First, we run the AGCM here at a coarser resolution (1° × 1°, rather than the ∼1/2° × 1/2° resolution underlying MERRA-2). Second, we use a tendency bias correction (TBC) approach to mitigate some of the model’s long-term biases (Chang et al. 2019). We apply diurnally varying and seasonally varying TBC corrections derived from the increments (forecast minus analysis) computed during the long-term MERRA-2 reanalysis itself; because the corrections applied are climatological, containing no prescribed interannual variability, the TBC-corrected AGCM remains a fully free-running global atmospheric model.

Each simulation in the 45-member ensemble covers the period 1981–2014, a period chosen for consistency with the other datasets used in our analysis. The simulations follow AMIP protocols, being forced by prescribed, observations-based fields of sea surface temperature and sea ice that vary daily and annually. For the present analysis, we process the simulated near-surface air temperature (T2M) values into monthly averages, our aim being to understand the sources of monthly scale T2M variability over continental areas.

Detrending the data was found to be necessary to isolate the signals borne of interannual SST variability from those associated with long-term SST trends. Trends were removed on a monthly basis; at each grid cell for a given month, the ensemble mean monthly T2M was computed for each year in 1981–2014, and the resulting time series was regressed against the year index. Then, for a given year, the T2M value computed with the regressed relationship was subtracted from each ensemble member’s monthly T2M.

b. Observed monthly temperature data

Gridded daily temperature (T2M) data at 0.5° × 0.5° resolution are available from the Climate Prediction Center (CPC; https://www.esrl.noaa.gov/psd/data/gridded/data.cpc.globaltemp.html). The raw source of the data is station-based measurements of minimum daily temperature T_min and maximum daily temperature T_max at the 2-m height; these point measurements were spatially interpolated by CPC onto the 0.5° × 0.5° grid. We converted the gridded T_min and T_max values into daily T2M using T2M = (T_min + T_max)/2, and we then spatially and temporally averaged the daily temperatures into monthly averages on the atmospheric model’s 1° × 1° grid. Finally, the T2M values for the period 1981–2014 were detrended on a monthly basis using the approach outlined above for the GEOS data, though computing the trend, of course, from the single time series of T2M values for a given month rather than from an ensemble mean.

We should note that using the mean of the minimum and maximum observed temperatures to approximate the daily T2M, while a common practice, can result in T2M values that are biased high (Bernhardt et al. 2018). This is an unavoidable limitation of the available observational data. Nevertheless, for our analysis framework, the relevant information in the observed T2M values lies not in their absolute magnitudes but in their time variability (specifically, their time correlation with other fields at the monthly scale), which arguably is minimally affected by this bias.

c. Supplemental AMIP experiments using independent models

For our analysis, we require estimates of T2M variations from a number of additional atmospheric general circulation models. For this, we utilize AMIP-style simulations produced by five other AGCMs as part of their contribution to the CMIP6 project (Eyring et al. 2016). See Table 1 for a list of these models and suitable references. While some models produced data over longer time periods, each model produced monthly T2M data for the 1981–2014 period with at least 10 ensemble members. To simplify the interpretation of our results, we use only 10 ensemble members from each AGCM.

Table 1.

Modeling systems providing monthly T2M data. Each model performed AMIP-style simulations as part of their contribution to the CMIP6 project; for each model, data from 10 ensemble members covering the period 1981–2014 were extracted for this study.

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For the present work, for each of the five models, we regridded the T2M data for a given month to the GEOS model’s 1° × 1° grid using a nearest-neighbor approach. The 1° × 1° data for each model were then detrended independently on a monthly basis using the approach outlined above for the GEOS model’s data. One difference between these CMIP6 simulations and the GEOS simulations is the use in the former of time-varying constituent forcing—a forcing that, like SSTs, could impart a signal to the generated monthly T2M fields. We presume this impact to be small, even negligible, given that the applied detrending would largely remove the impacts of trends in the imposed constituents. Still, the lack of such time variations in the GEOS simulations should be interpreted as an additional source of error for the GEOS time series.

a. Relative contributions of boundary forcing and unpredictable variability

The quantity ${\rho }_o^2$ indeed describes a fundamental property of the real world’s predictability: stated another way, it is the relative degree to which an observed quantity such as monthly T2M is controlled by SST variability rather than by chaotic noise. Unfortunately, while fundamental, ${\rho }_o^2$ cannot be directly measured. Past studies have approximated ${\rho }_o^2$ using model-based ${\rho }_m^2$ estimates (e.g., Koenigk and Mikolajewicz 2009), i.e., by assuming that climate models have accurate signal-to-total variance ratios. We make a nod to this approach ourselves below, though we will be emphasizing throughout that ${\rho }_m^2$ can be quite different from ${\rho }_o^2$ and does not even provide its upper limit—in some cases, the signal-to-noise ratio is too low in models, leading to an underestimate of the real world’s ${\rho }_o^2$.

The new estimates of ${\rho }_o^2$ derived with (14) are independent of those determined through the more common approach of simply equating ${\rho }_o^2$ and ${\rho }_m^2$, for in the conceptual framework above, B and ρ are independent of each other. In other words, a high consistency in teleconnections, as characterized by Corr²(B_o, B_m) being close to one, does not imply a high consistency between the model’s and Nature’s signal-to-total variance ratios (${\rho }_m^2$ and ${\rho }_o^2$), and vice versa. The key distinction in the two approaches is that values of the fundamental real-world predictability metric ${\rho }_o^2$ obtained with (14) do not rely on an assumption of accurate signal-to-total variance ratios in the model; they instead rely on our ability to estimate the model’s independent success in simulating ocean–land teleconnections—i.e., on our ability to estimate Corr²(B_o, B_m).

We must emphasize here that in this analysis, we will not be evaluating the absolute magnitudes of the boundary-forced variance or of the variance induced by internal variability. An excessive ρ² for a model, for example, could result from either an underestimate of the internal variability or an overestimate of the impact of the signal. Here, we are interested only in whether the model captures the correct fraction of the total variance induced by boundary forcing as opposed to noise, a critical quantity in itself.

This overall framework—particularly the part associated with estimating model values of ${\rho }_m^2\hspace{0.17em}$—is essentially consistent with those offered already by Eade et al. (2014), Siegert et al. (2016), and Scaife and Smith (2018). Murphy (1990) also provides similar but alternative forms of some of the above equations; indeed, (2) of that paper can be shown to be mathematically identical to (3) above. The framework as expressed above, however—particularly the focuses on Corr²(B_o, B_m) and Corr²(B_m, B_q)—will facilitate our forthcoming analysis.

b. Details regarding ensemble sampling and correlation calculations

In some of our analyses, we will be examining correlations as a function of ensemble size. For a given ensemble size K in the determination of Corr²(Y_o, Y_m), we randomly select K members (without replacement) from the available members (45 available members for GEOS, 10 for the other models) and call this our reduced ensemble; after computing the desired correlation with this reduced set, we repeat the process 99 times. The correlation value for ensemble size K is the average over the 100 computed correlations.

For the model-based Corr²(Y_mn, Y_m) calculations, we consider five individual ensemble members in turn as Y_mn in (3). Note that Y_m in the equation accordingly differs each time, since each choice of Y_mn defines a different set of remaining ensemble members. The five resulting correlations for a given ensemble size, each already representing an average over 100 computed values, are then averaged into the final result for that size. The use of five ensemble members (rather than 45 or 10) as Y_mn here was considered a reasonable and tractable compromise given the need to perform at least some averaging in the face of the tremendous computational burden associated with each calculation.

Each individual correlation calculation is based on 102 sample pairs: 3 monthly averages per year for 34 years (1981–2014). As a consequence of the detrending described earlier, climatological monthly averages are removed from each value before computing the correlations so as not to capture trivial correlations associated with seasonality.

a. Quantification of unpredictable variability within the GEOS model

Our analysis focuses first on the GEOS model rather than on the CMIP models. Given its much larger ensemble size, we expect the characterization of noise for GEOS to be the most robust.

Figure 1 shows, with blue dots, the degree to which the GEOS model’s ensemble mean captures the monthly T2M time series produced by a single ensemble member [i.e., Corr²(Y_mn, Y_m)] as a function of ensemble size for a grid cell in North America. We use (3) to produce the blue curve through the dots, with ${\rho }_m^2$ determined via a fitting procedure—we find the single value of ${\rho }_m^2$ that produces the lowest root-mean-square error (RMSE) between the dots and the curve. The ${\rho }_m^2$ value so found indeed serves as the asymptote for the relationship, shown in the figure as a dashed blue line. This asymptote is interpreted as the model’s underlying strength of connection between the boundary forcing and a single simulation’s monthly T2M time series. Stated another way, 1 minus this asymptote is interpreted as the fraction of monthly T2M variability in a simulation that stems from chaotic noise.

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Representative examples of how key model relationships vary with ensemble size, focusing on the simulation of monthly mean air temperature, T2M. Blue dots: Variation of Corr²(Y_mn, Y_m) with ensemble size, where Corr²(Y_mn, Y_m) characterizes the ability of the ensemble mean to capture the variability produced by a single ensemble member. Red dots: Variation of Corr²(Y_o, Y_m) with ensemble size, where Corr²(Y_o, Y_m) characterizes the ability of the ensemble mean to capture the variability seen in the observations. The lines through the dots are determined through a fitting procedure, which provides as a matter of course the indicated asymptotes, shown as dashed lines (see text). Results shown are for JJA.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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It is important to emphasize that our procedure does not simply equate ${\rho }_m^2$ to the correlation achieved with all 45 ensemble members. By fitting the curve and determining the asymptote, we are effectively equating ${\rho }_m^2$ to the correlation that would be achieved with an infinite number of ensemble members.

Note that the results in Fig. 1 are representative; all of the grid cells we examined show similarly strong (often even better) fits through their corresponding blue dots, though we caution that this success may be specific to the particular variable, monthly T2M, that we are examining here. Such success in the fitting provides confidence that the analysis framework represented by (1)–(4) properly characterizes the relative contributions of signal and noise in the modeling system. In Fig. 2, we show global maps of ${\rho }_m^2$ so obtained for GEOS as a function of season. The salient spatial pattern is a strong signal-to-total variance ratio in the tropics and much smaller values for the ratio as one moves poleward, a feature that is well documented in previous studies (e.g., Koster et al. 2000) and that leads to enhanced predictability in tropical seasonal forecasts (Scaife et al. 2017; Kumar et al. 2013). Remoteness from the oceans also appears to be a factor, with the lowest potential values of ${\rho }_m^2$ often appearing deep in the heart of Eurasia. The ${\rho }_m^2$ maps show some interesting seasonal variations, with, for example, North America showing lower ${\rho }_m^2$ values in SON, Australia showing somewhat lower values in JJA, and the Sahel showing low values in MAM but higher values in JJA.

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Derived global distribution of ${\rho }_m^2$ (the fractional contribution of SST boundary forcing to monthly T2M variability in the GEOS atmospheric model) for (a) DJF, (b) MAM, (c) JJA, and (d) SON. The shading contour at 0.029 represents the 99% confidence level that the null hypothesis (i.e., no underlying signal) is invalid, as determined by Monte Carlo analysis. Similarly, values above 0.037 indicate that the null hypothesis is invalid at the 99.9% confidence level.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Note that in these maps, any colored shading indicates at least a 99% confidence that the null hypothesis of no underlying signal is invalid. The value of 0.025 delimiting the lowest colored shading was determined from a Monte Carlo analysis in which appropriately sized ensembles of randomly generated time series, i.e., time series of normally distributed numbers produced with a random number generator, were processed in precisely the same manner as the AGCM data to quantify the statistics associated with that single null hypothesis. That same Monte Carlo analysis, by the way, indicates a 99.9% confidence that the null hypothesis is invalid for any value above 0.037.

b. Comparison of the GEOS model’s noise and teleconnection characteristics with those in other models

Now that the equations and fitting approach have been demonstrated for the GEOS model, we can take a look at the CMIP AGCMs listed in Table 1 to shed some light on how different models compare in regard to boundary forcing and noise levels. Because only 10 ensemble members are used for each of the CMIP models, the estimates obtained for them are presumably more uncertain than those for GEOS; still, enough data are available for some useful joint analyses. Indeed, as a check, we estimated ${\rho }_m^2$ using a 10-member subset of the 45-member GEOS ensemble and found substantial agreement with the values found using all 45 members (see section S3 in the supplemental material), supporting the idea that an ensemble of 10 is adequate, at least to first order, for fitting the curve and determining the asymptote ${\rho }_m^2$. Results below are for boreal spring (MAM). Results for other seasons are provided in the supplemental material (see section S4).

For each of the five CMIP models, curve fitting along the lines of that shown in Fig. 1 (blue curve) was performed to determine spatial distributions of ${\rho }_m^2$, the fraction of the T2M variance explained by the boundary-forced signal. Results are shown in Fig. 3, with the GEOS results for MAM repeated in Fig. 3f. The different models agree on many features of the distributions, showing, for example, higher values in the tropics, negligible values in central Asia, and a dipole of higher values along the western coast of North America. The overall similarity is encouraging given that the six models use distinct structures and formulations (with the exception of NorCPM1 and CESM2, which use similar atmospheric models). The overall similarity supports the idea that the models are capturing, to first order, the same elements of the spatial distributions of the relative strength of the boundary-forced signal in Nature.

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Spatial distribution of ${\rho }_m^2$, the fraction of the simulated T2M variance explained by the boundary-forced signal, for six different AGCMs. (a) MIROC6. (b) CESM2. (c) NorCPM1. (d) ACCESS. (e) IPSL. (f) GEOS. The GEOS results are based on a 45-member ensemble, and those for the other AGCMs are based on 10-member ensembles (though in all cases, the values effectively represent what would be obtained with an infinite number of ensemble members). The shading contour at 0.052 represents, for the CMIP models, the 99% confidence level that the null hypothesis (i.e., no underlying signal) is invalid, as determined by Monte Carlo analysis. The corresponding value for the 99.9% confidence level is 0.067.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Corr²(B_m, B_q), the degree to which two different models m and q agree with each other on the boundary-forced signal at a given location once the impact of noise is removed, is computed with (10). Figures 4a–e shows Corr²(B_m, B_q) between the GEOS model and each of the five CMIP models. We focus here on the GEOS model’s correlations with the CMIP models rather than on correlations among the different CMIP models because the size of the GEOS ensemble would presumably provide some extra robustness to the estimates. Also note that theoretically, while a large agreement in the boundary-forced signal is possible in the presence of substantial model noise, the presence of noise can make quantifying the agreement in this quantity difficult. Indeed, with ${\rho }_m^2$ or ${\rho }_q^2$ close to zero, the calculation of a reliable Corr²(B_m, B_q) value using (10) is essentially impossible. To account for this, we assign undefined values to the Corr²(B_m, B_q) calculations from (10) if either ${\rho }_m^2$ or ${\rho }_q^2$ falls below an arbitrary threshold of 0.05. The undefined values are whited out in the figure.

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Spatial distribution of Corr²(B_m, B_q), the degree to which the different CMIP models agree with the GEOS model on the temporal variations in the boundary-forced signal. (a) Corr²(B_GEOS, B_MIROC6). (b) Corr²(B_GEOS, B_CESM2). (c) Corr²(B_GEOS, B_NorCPM1). (d) Corr²(B_GEOS, B_ACCESS). (e) Corr²(B_GEOS, B_IPSL). (f) Arithmetic mean of the Corr²(B_m, B_q) values. Values are considered undefined (and shown as white) if the internal variability of either model involved in a calculation is overwhelmingly high (i.e., if ${\rho }_m^2$ or ${\rho }_q^2$ falls below a threshold of 0.05). Undefined values are not utilized in the calculation of the arithmetic mean.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Figures 4a–e indicates that the teleconnections inherent in the GEOS system generally agree with those of the CMIP models in the Americas and along the northern coast of Australia, with squared correlations often exceeding 0.7. Agreement is weaker in Africa (though still in the neighborhood of 0.5) and is particularly weak in southern Asia. Excessive internal variability prevents a determination of the agreement in Europe and most of the rest of Asia. Figure 4f encapsulates these results by providing the arithmetic mean of the five Corr²(B_m, B_q) fields, with undefined values excluded from the mean calculations. To our knowledge, Fig. 4 represents a unique, first-of-its-kind comparison of teleconnection behavior between models. The indicated degree of disagreement underlines some uncertainty in this quantity; it indeed suggests that models could be improved in this regard, which could lead to substantial gains in long-range prediction.

Again, results for other seasons are provided in the supplemental material (section S4). For these other seasons, we find that, as with MAM, the ${\rho }_m^2$ distributions are very similar across the models, both in terms of pattern and magnitude. We also find that, as with MAM, Corr²(B_m, B_q) for DJF is strong in the Americas and Australia and weaker in Africa and South Asia. The Corr²(B_m, B_q), however, is somewhat weaker in the Americas during JJA and SON, with values typically closer to 0.5, again indicating substantial room for the improved modeling of teleconnections.

c. Quantification of maximum realizable skill with the GEOS model

The red dots in Fig. 1 show the square of the correlation, as a function of ensemble size, between the ensemble mean T2M from GEOS and the observed monthly value (section 2b) at the sample grid cell. Here, having already obtained the model’s ${\rho }_m^2$ from the calculations in section 4a, we need only fit the value of the product ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ in such a way that the red curve, computed with (12), best captures the relationship indicated by the red dots as determined by minimizing the RMSE between the two. The product ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ serves as the asymptote for the relationship, shown as the red dashed line in Fig. 1. Again, the strong fit of the red curve through the dots is representative of the fits obtained across the globe.

Figure 5 shows, as a function of season, the global distribution of the ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ values derived in this way {equivalent to ${[{\rho }_o^2{\text{Corr}}^2(B_o,B_m)]}_{\text{est}}$ in (14)}—the degree to which the model can capture, with an infinite number of ensemble members, the observed time variations of T2M. Monthly temperature variations in tropical South America are seen to be well reproducible by the model in DJF and MAM, sometimes with almost half the variance explained. Moderate levels of skill are possible during DJF in eastern Australia and much of Europe and during JJA in the Sahel and northwest Asia.

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Derived global distribution of ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ (the fraction of the monthly T2M variance that the GEOS model can hope to reproduce, assuming an infinite number of ensemble members) for (a) DJF, (b) MAM, (c) JJA, and (d) SON. The shading contour at 0.059 represents the 99% confidence level that the null hypothesis (i.e., no underlying signal) is invalid, as determined by Monte Carlo analysis. The corresponding value for the 99.9% confidence level is 0.083.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Of course, an infinite number of ensemble members is hardly realizable, limiting the practical usefulness of Fig. 5. Note, however, that having quantified both ${\rho }_m^2$ and ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$, we can use (12) to determine how well a given ensemble size might perform. Figure 6a shows, for JJA, the square of the correlation between observed monthly T2M and the model ensemble mean value when the ensemble consists of five members. Figures 6b–d then show, respectively, the increase in this skill metric when the ensemble size is increased from 5 to 20 members, from 20 to 50 members, and from 50 to 100 members.

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(a) Square of the correlation [as determined with (12)] between the observed and ensemble mean monthly T2M for JJA, assuming a five-member ensemble. (b) Increase in this skill metric when the ensemble size is increased from 5 to 20 members. (c) Increase in the skill metric when the ensemble size is increased from 20 to 50 members. (d) Increase in the skill metric when the ensemble size is increased from 50 to 100 members.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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As expected, increasing the ensemble size has little impact in areas like northeastern Asia, which has little maximum realizable skill in the first place (Fig. 5c). Increasing the ensemble size also has little impact in the tropics, since most of the skill is already achieved with the small ensemble size of 5 (cf. Figs. 5c and 6a, noting the differences in the color bars). In contrast, going to 20 ensemble members has a significant impact on skill in many other areas, including the Sahel and much of Canada. Based on the more limited skill increases in going from 20 to 50 members (Fig. 6c), one might conclude that 20 members is adequate for capturing the SST-forced signal in monthly T2M. A section of northern Eurasia (just east of Scandinavia), however, continues to show a benefit in skill as the ensemble size is increased, even in going from 50 to 100 members (Fig. 6d). This appears to be consistent with the large ensemble size needed to provide skillful predictions of the winter NAO (Scaife et al. 2014, their Fig. 3). Of course, the skill obtained there with even 1000 members would still be suboptimal due to the imperfect reproduction of teleconnections in the model.

d. Implications for quantifying the signal-to-total ratio present in nature

The quantity ${\rho }_o^2$, as used in (12) and (13), is a fundamental characterization of predictability in Nature, one that exists independently of models. Again, ${\rho }_o^2$ describes the fraction of the real world’s T2M variance that is controlled by the time variation of SST fields as opposed to chaotic atmospheric noise. We emphasize again that ${\rho }_o^2$ cannot be directly measured or otherwise quantified with observations alone. Past estimates of the quantity have relied on the determination of model-based predictability, assuming the two are the same (e.g., Koenigk and Mikolajewicz 2009). Such model-based estimates of ${\rho }_o^2$ are indeed effectively provided in Fig. 3, which shows ${\rho }_m^2$, which is the model climate’s version of ${\rho }_o^2$, for six different models. In a sense, the average ${\rho }_m^2$ from the six models, presented in Fig. 7, could be interpreted as a fully model-based estimate of this unmeasurable property. However, one must bear in mind that such model-based predictability can be less than the true predictability; the model-based estimates do not serve as upper bounds for Nature’s values (Kumar et al. 2013; Scaife and Smith 2018).

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Spatial distribution of ${\rho }_m^2$, the fraction of the simulated T2M variance explained by the boundary-forced signal, averaged over the individual values obtained for six different AGCMs. (a) DJF. (b) MAM. (c) JJA. (d) SON. Note that in contrast to Figs. 2, 3, and 5, colored shading here is not tied specifically to a 99% confidence level.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Of course, one of the reasons using ${\rho }_m^2$ to estimate ${\rho }_o^2$ is unsatisfying is that no observational data whatsoever are used to produce the ${\rho }_m^2$ estimates—we are fully at the mercy of the models’ abilities to reproduce the true predictable signals and the behavior of chaotic dynamics in Nature. For this reason, we now examine our alternative and independent second approach to estimating ${\rho }_o^2$, one that does use observations. Analyzing the time series of observational T2M data in conjunction with model data allows the estimation of the product ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$—the maximum skill (relative to observations) a given model might attain with an unlimited number of ensemble members. The distribution of this product was provided for the GEOS model in Fig. 5. All that would be needed to obtain ${\rho }_o^2$ from such fields is an estimate of Corr²(B_o, B_m)—an estimate for how the teleconnections in the real world, in the absence of noise, agree with those in the model. As noted in section 3a, we obtain this estimate by pairing GEOS with each of the five CMIP6 models in turn and then averaging the five resulting Corr²(B_m, B_q) estimates obtained with (10). We thus make the admittedly major assumption, supported by the analysis in section S2 of the supplemental material, that this average is a reasonable proxy for Corr²(B_o, B_m).

We reemphasize here an important point: the quantities Corr²(B_o, B_m) and ${\rho }_m^2$ are conceptually independent of each other— they represent distinctly different facets of model behavior. Accordingly, dividing ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ by our model-based estimate of Corr²(B_o, B_m) provides a second estimate of ${\rho }_o^2$, one that is fully independent of the more common approach of simply utilizing ${\rho }_m^2$. It is this second approach that represents a novel application of the framework in section 3a to the estimation of real-world values of ${\rho }_o^2$.

We present in Fig. 8 estimates of ${\rho }_o^2$ obtained by dividing the ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ in Fig. 5 by the arithmetic mean of the Corr²(B_m, B_q) values derived with (10). Again, we focus on the ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ estimates from GEOS given the larger ensemble size available for that model. Values are not available everywhere; as before, if internal variability is so large that it overwhelms our ability to determine Corr²(B_m, B_q)—that is, if the derived ${\rho }_m^2$ or ${\rho }_q^2$ lies below the imposed 0.05 threshold—we assume the associated ${\rho }_o^2$ values are undefined.

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Estimates of the degree to which SST boundary forcing controls the time variability of T2M in the real world (as measured with ${\rho }_o^2$) obtained by dividing the estimates of ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ in Fig. 5 by the arithmetic mean of the model-based Corr²(B_m, B_q) estimates, as shown for MAM in Fig. 4f. (a) DJF. (b) MAM. (c) JJA. (d) SON. Values deemed undefined are masked in white. Note that in contrast to Figs. 2, 3, and 5, colored shading here is not tied specifically to a 99% confidence level.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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Where the ${\rho }_o^2$ values are defined, we see in Fig. 8 some of the patterns that were determined for ${\rho }_m^2$ in Fig. 7, particularly the high relative control of boundary forcing on T2M variability in the tropics and the lower control toward midlatitudes. Some more specific features seen in the average ${\rho }_m^2$ fields of Fig. 7 are also seen in the new ${\rho }_o^2$ estimates in Fig. 8, such as higher ${\rho }_o^2$ values in the Amazon during DJF than in JJA or SON and a dipole of high values along western North America in MAM. The “splotchiness” of the patterns in Fig. 8 presumably stems from a similar quality of the estimated ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ fields, which in turn must relate to the character of the T2M observations being used—the observations have their own sources of error (with a limited density of measurement stations in many parts of the world; see section 5) and represent, in any case only a single 36-yr time series with concomitant sampling error. Presumably, the true underlying patterns are smoother. The point of Fig. 8 is to provide first-order patterns of ${\rho }_o^2$ based on this new approach.

Naturally all of these ${\rho }_o^2$ estimates, whether from Fig. 7 or Fig. 8, are uncertain, and because ${\rho }_o^2$ is an unmeasurable property of Nature, definitive uncertainty estimates for the quantity are themselves unattainable. To the extent that, as in past studies, ${\rho }_m^2$ can be said to approximate ${\rho }_o^2$, the intermodel ${\rho }_m^2$ variations seen in Fig. 3 and Figs. S9–S11 provide a flavor for the uncertainty. (It is worth noting in this context that the different models are indeed generally consistent in the spatial structures, and even in the magnitudes, of their ${\rho }_m^2$ fields.) Considering the newer approach presented in Fig. 8, we can illustrate the uncertainty qualitatively by dividing, at each grid cell, the value of ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ by both the maximum and minimum value of Corr²(B_o, B_m) obtained from the five models rather than by their arithmetic mean. The resulting “lower bounds” and “upper bounds” for ${\rho }_o^2$ are provided for MAM in Fig. 9 and for the other seasons in the supplemental material (section S5). The blackened areas in Fig. 9b indicate an unrealistic perfect predictability, a reflection of the limitations of the analysis approach. Even with such limitations, the minimum and maximum fields do show the same spatial structures, and they do significantly constrain the potential ranges of the ${\rho }_o^2$ values in most areas.

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Qualitative indication of the uncertainty associated with estimating ${\rho }_o^2$ from ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$. (a) Estimates of the lower bound of ${\rho }_o^2$ obtained by dividing ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ by the maximum value of Corr²(B_o, B_m) found at the grid cell among the five model combinations. (b) As in (a), but obtained by dividing ${\rho }_o^2{\text{Corr}}^2(B_o,B_m)$ by the minimum value of Corr²(B_m, B_q) found at the grid cell. Values deemed undefined are masked in white.

Citation: Journal of Climate 38, 6; 10.1175/JCLI-D-23-0740.1

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