Abstract

Decadal climate variability of sea surface temperature (SST) over the Pacific Ocean can be characterized by interdecadal Pacific oscillation (IPO) or Pacific decadal oscillation (PDO) based on empirical orthogonal function (EOF) analysis. Although the procedures to derive the IPO and PDO indices differ in their regional focuses and filtering methods to remove interannual variability, the IPO and PDO are highly correlated in time and are often used interchangeably. Studies have shown that the IPO and PDO conjointly (IPO/PDO for conciseness) play a vital role in modulating the pace of global warming. It is less clear, however, how externally forced global warming may, in turn, affect the IPO/PDO. One obstacle to revealing this effect is that the conventional definitions of the IPO/PDO fail to account for the spatial heterogeneity of the background warming trend, which causes the IPO/PDO to be conflated with the warming trend, especially for the twenty-first-century simulation when the forced change is likely to be more dominant. Using a large-ensemble simulation in the Community Earth System Model, version 1 (CESM1), it is shown here that a better practice of detrending prior to EOF analysis is to remove the local and nonlinear trend, defined as the ensemble-mean time series at each grid box (or simply as the quadratic fit of the local time series if such an ensemble is not available). The revised IPO/PDO index is purely indicative of internal decadal variability. In the twenty-first-century warmer climate, the IPO/PDO has a weaker amplitude in space, a higher frequency in time, and a muted impact on global and North American temperature and rainfall.

1. Introduction

The interdecadal Pacific oscillation (IPO) is an internal climate mode characterizing the variability in the Pacific Ocean sea surface temperature (SST) at decadal and multidecadal time scales [i.e., beyond the interannual variability, such as El Niño–Southern Oscillation (ENSO)] (Folland et al. 1999; Power et al. 1999). It has been recognized recently as a primary source of global-mean temperature variability, and it plays a vital role in modulating the actual pace of global warming that is largely externally forced by greenhouse gas (GHG) increase (e.g., Meehl et al. 2011; Kosaka and Xie 2016; England et al. 2014; Dai et al. 2015; Meehl et al. 2016b). Another index related to decadal variability over the Pacific Ocean is the Pacific decadal oscillation (PDO) (Mantua et al. 1997; Zhang et al. 1997). Both the IPO and PDO are based on empirical orthogonal function (EOF) analysis of SST, but with distinct regional focus: the former is for the entire Pacific Ocean, while the latter for the North Pacific only. Nevertheless, these two indices have been shown to be highly correlated in time (Deser et al. 2004; Newman et al. 2016) and are often used interchangeably in the literature on the regional climate impact of Pacific variability (e.g., Dai 2013; Dong and Dai 2015). These two indices were first brought up in the analysis of the recent twentieth-century observational dataset (Zhang et al. 1997). More recently, the usage of the IPO and PDO (IPO/PDO for conciseness) has been extended to a much wider time domain, such as paleoclimate records (Henley et al. 2011) and climate model projection into the twenty-first century and beyond (Meehl et al. 2013), where the temperature changes over a much greater range than observed during the twentieth century.

Ambiguity exists regarding the definitions of the IPO/PDO in previous literature, however, particularly concerning the detrending (to remove background long-term trend) and smoothing procedures (to remove higher-frequency variability such as ENSO). As we will demonstrate in this study, the ambiguity in the analysis procedure prevents the full utilization of the IPO/PDO as an indicator of pure natural variability over a wide range of climate state.

The first part of the paper aims to identify how these different detrending procedures prior to EOF analysis will affect the obtained IPO/PDO as the principal component time series. We then present a revised procedure that accounts for 1) the regional pattern of SST change at the ocean basin scale and 2) the nonlinearity of the warming pace. The revised procedure thus allows for a unified IPO/PDO definition that can be applied to historical observations, preindustrial climate model simulations, and twentieth- and twenty-first-century climate model simulations. Since the IPO/PDO time scale is approximately 20–40 yr, it is possible that climate change induced by the anthropogenic forcing on centennial time scale may alter the frequency and amplitude of the IPO/PDO. Extensive studies have looked at how long-term climate change might affect short-term climate variability such as ENSO (Cai et al. 2015; Collins et al. 2010) and monsoons (Zhu et al. 2012). Recently Pascale et al. (2017) reported a simulated reduction of monsoonal rainfall in the U.S. Southwest as a response to twenty-first-century global warming. But similar investigations concerning potential changes in IPO/PDO are sparse. The refined procedure proposed here enables us to examine consequences of climate change (as a background mean state) on Pacific decadal variability.

The second component of this study deals with the connection between the IPO/PDO and regional climates. Although there is currently no consensus on understanding the origins and the underlying mechanisms of the IPO/PDO (Han et al. 2014; Newman et al. 2016), many regional climate phenomena over North America and East Asia have been empirically shown to strongly correlate with fluctuations of the IPO/PDO (e.g., Meehl and Hu 2006; Dai 2013; Dong and Dai 2015; Meehl et al. 2016a,b; Si and Ding 2016; Si and Hu 2017). It remains unclear how the impact of the IPO/PDO on regional climate will change in a future warmer climate.

This paper is organized as follows. In section 2, we describe the modeling dataset and present a brief description of the EOF-based procedure to obtain the IPO/PDO, including the conventional approach and the revised approach, and the regression or composite method to quantify regional climate impact as a result of the IPO/PDO. In section 3, we review those procedures step by step across a wide range of model simulations and thoroughly assess the advantages and drawbacks of each method. We thus propose the optimal method for long-term climate change studies. Section 4 shows the IPO/PDO influences on the global and North American climates, as well as the changes in the IPO/PDO and its regional influences resulting from future warming. Further discussion and a summary of findings are presented in section 5.

2. Methods

a. The coupled global climate model

The climate model used here is the Community Earth System Model, version 1 (CESM1; Hurrell et al. 2013), which is a fully coupled global climate model. The atmospheric component is the Community Atmospheric Model, version 5 (CAM5), with the nominal 1° horizontal resolution and 30 levels vertically. The ocean component is the Parallel Ocean Program, version 2 (POP2), at a 1° horizontal resolution with enhanced meridional resolution in the equatorial tropics (⅓°) and 60 levels vertically. The sea ice component is the Community Ice Code, version 4 (CICE4), and the land component is the Community Land Model, version 4 (CLM4). CICE4 had the same horizontal resolution as POP2, and CLM4 had the same horizontal resolution as CAM5.

b. Model simulations

The simulations used here were 1) the 1800-yr preindustrial control run with all forcings kept at the 1850 level and 2) 35-member ensemble transient climate simulations from 1920 to 2080. The twentieth-century ensemble simulations (1920–2005) include time-evolving observed external forcings, such as CO2, CH4, N2O, and anthropogenic aerosols. The only difference among ensemble members was a small round-off level perturbation in the atmospheric temperature field on 1 January 1920. Note that although the CESM1 twentieth-century simulation was initiated from 1850 (preindustrial as in many other models), CO2 and other forcing is fairly weak until after 1920. That is why only post-1920 runs are repeated to form an ensemble.

From 1 January 2006 onward, the ensemble simulations followed the emission scenarios of representative concentration pathway 8.5 up to 2080. The ending year for the twentieth-century analysis is 2005 following CMIP5 protocol. Changing that from 2005 to 2015 does not impact the main results of this study.

This large-ensemble simulation was described in greater detail by Kay et al. (2015). An overall evaluation of CESM1 twentieth- and twenty-first-century simulations has been documented by Meehl et al. (2013). Specifically related to our application, the simulation of the twentieth-century IPO/PDO spatial pattern by CESM1 was found to be one of the best among all major CMIP5 models [see Fig. 1 of Newman et al. (2016)].

The original dataset (temperature and precipitation) analyzed here was monthly mean output from the model. We noted that sea surface temperature (TS in CESM1 subsampled over ocean surface) was used for the EOF analysis to obtain the IPO/PDO indices. Precipitation was the total precipitation including both convective rainfall and large-scale rainfall (PRECC and PRECL in CESM1, respectively). All model data used here were publicly available through the Earth System Grid website.

c. Revised procedures for data smoothing and detrending

Many different methods have been used to define the IPO/PDO in the literature and will be thoroughly assessed in section 3. In this subsection, we first briefly describe the major steps in the conventional methods of deriving the IPO/PDO and our revision to the conventional approach. The proposed revision is motivated by the following reason: The global climate will likely undergo a significant warming with a heterogeneous regional pattern in the twenty-first century, and the conventional methods to obtain the IPO/PDO could be seriously affected by this extensive warming.

A three-step process is used in the conventional method to obtain the IPO/PDO: 1) removing the seasonality, 2) filtering the high-frequency variability, and 3) removing the long-term trend. As demonstrated in section 3, the choices on how to do the first two steps do not affect the EOF patterns and the principal components much. The removal of the long-term trend from the data, however, is more complicated and is a major focus of this study. Without doing so prior to EOF analysis, the first EOF mode is usually assumed to be a trend pattern induced by external forcings, and the second EOF is an IPO/PDO-like pattern (e.g., Meehl et al. 2009; Dai 2013; Dong and Dai 2015). As we will demonstrate in section 3, this implicit trend removal usually works fine with weak GHG forcing, such as in the twentieth century. With an expected much larger rate of GHG increase in the twenty-first century, this will often lead to a contaminated EOF2 pattern with residue warming trend since the rate of warming is not uniform globally. Explicit detrending methods commonly used in previous studies include 1) removal of a linear fit of global-mean temperature change in time dimension (Knight et al. 2005) and 2) removal of global-mean temperature time series from each grid point (Trenberth and Shea 2006). As explained in detail in section 3c, these detrending methods fail to address the nonlinearity or the spatially heterogeneous feature of the background warming for the twenty-first century. Motivated by the deficiency of existing approach, in section 3c we propose a detrending procedure that removes ensemble-average temperature time series at each grid point.

d. Calculation of the IPO/PDO and the triple Pacific index

The area-weighted EOF (centered and correlation EOF) was used to derive the IPO and PDO. We follow the conventional definition that the PDO is obtained by taking EOFs over the North Pacific SST (20°–70°N, 110°E–100°W), whereas the IPO is obtained by taking EOFs over the entire Pacific SST (60°S–70°N, 110°E–70°W). Despite these differences, IPO and PDO time evolutions in observations are highly correlated (Han et al. 2014; Newman et al. 2016). The correlation matrix, as opposed to covariance matrix, was used to lessen the influence of polar regions on the calculation, although the results of those two approaches are quite similar. We showed only the first two leading modes of EOFs, which explained more than 75% of the total variance (with the EOF3 explaining the remaining part). The obtained principal component (PC) time series were always standardized in this analysis (with the mean removed and then divided by the standard deviation of the time series) before calculating global SST correlation to PC time series. Thus, the amplitude of the IPO/PDO can be easily assessed from EOF spatial patterns shown in this study. Using nonstandardized PC time series to evaluate the variance of the IPO/PDO is difficult as the total variation of temperature field is embedded in both PC time series and the EOF pattern.

In addition to the EOF-based IPO/PDO, we also test the importance of proper preprocessing of data in deriving a third index called the tripole Pacific index (TPI). This index is simply the difference between SST over the central eastern Pacific (10°S–10°N, 170°E–90°W) and the average of the northwest (25°–45°N, 140°E–145°W) and southwest Pacific (50°–15°S, 150°E–160°W). This index is easy to compute and highly correlated with the EOF-based IPO (Henley et al. 2015).

3. A revised procedure to derive the IPO/PDO in the context of long-term warming

In this section, the conventional methodology to derive the IPO/PDO is thoroughly reviewed, and after identifying the main pitfalls, we propose a revised method for deriving the IPO/PDO for both twentieth- and twenty-first-century climates. Some technical aspects previously documented in section 2c are reiterated here for clarity and reasoning. To better illustrate the potential problems associated with the conventional methodology, we primarily analyze the twenty-first-century simulations in this section.

a. Removal of the climatological seasonal cycle

To quantify decadal variability, a common practice is to remove the seasonal cycle first from the monthly mean temperature time series. This seasonality can be removed either by eliminating the climatological annual cycle or by simply calculating the annual average of the monthly data. The resulting EOF patterns and the PC time series do not significantly depend on the method used to remove this seasonal cycle. However, without this step, the resultant first EOF pattern (EOF1, following the IPO procedure) represents the antisymmetric temperature changes between the Northern and Southern Hemispheres and the first principal component (PC1) time series clearly represent the high-frequency annual cycle (figure not shown). The global warming appears in EOF2 with a faster warming in the Northern Hemisphere and a monotonic upward trend in PC2 time series representing the twenty-first-century warming trend, agreeing with previous twentieth-century observational analyses (Xu and Ramanathan 2012; Friedman et al. 2013).

b. Removal of interannual variability such as ENSO

In previous IPO-related work, a low-pass filter was applied before the EOF analysis in order to focus on the decadal variability (e.g., Power et al. 1999; Meehl and Hu 2006; Phillips et al. 2014). Without applying the low-pass filter, the PC time series was considerably noisier and was dominated by interannual variability such as ENSO (Fig. 1b).

Fig. 1.

Comparing the effects of applying low-pass filtering. (a),(b) EOF2 and PC2 time series from ensemble member 33, when the seasonal cycle was removed by taking the annual average, but no low-pass filter was applied. (c),(d) As in (a),(b), but a low-pass filter was applied before the EOF analysis. Note that the PC2 time series gets smoother than (b). (e),(f) As in (a),(b), but a low-pass filter was applied directly to the PC2 time series after the EOF analysis, instead of to the SST data before the EOF analysis. The EOF analysis was performed over the entire Pacific SST (i.e., the IPO approach). No detrending, as discussed in later figures, was applied here.

Fig. 1.

Comparing the effects of applying low-pass filtering. (a),(b) EOF2 and PC2 time series from ensemble member 33, when the seasonal cycle was removed by taking the annual average, but no low-pass filter was applied. (c),(d) As in (a),(b), but a low-pass filter was applied before the EOF analysis. Note that the PC2 time series gets smoother than (b). (e),(f) As in (a),(b), but a low-pass filter was applied directly to the PC2 time series after the EOF analysis, instead of to the SST data before the EOF analysis. The EOF analysis was performed over the entire Pacific SST (i.e., the IPO approach). No detrending, as discussed in later figures, was applied here.

The low-pass filter can be applied in two ways: 1) directly to the annual mean SST before the EOF analysis (Figs. 1c,d) or 2) to the PC time series after the EOF analysis (Figs. 1e,f). The resultant patterns obtained from an individual run of the model simulation were nearly identical for these two methods (Fig. 1c vs Fig. 1e), both indicative of the IPO pattern, which showed a broader and weaker warming anomaly in the tropics than did the ENSO pattern in Fig. 1a. This is consistent with previous findings that EOF patterns of low-pass- and high-pass-filtered SSTs in the Pacific differ noticeably (Zhang et al. 1997; Meehl and Hu 2006), and thus the ENSO and IPO modes are separable in their spatial patterns (Dong and Dai 2015). The PC time series, however, show some differences between these two different filtering methods (Fig. 1d vs Fig. 1f). The second method of filtering the PC time series (as in Figs. 1e,f) could cause some ambiguity as to whether one should use the filtered SST dataset for regression analysis or the unfiltered original dataset. For consistency, we recommend the first method: applying any low-pass filtering (or smoothing) to the SST field prior to EOF analysis [as in Dong and Dai (2015)].

c. Detrending to remove the influence of the external forcing

In general, studies show that the IPO/PDO is primarily an internal climate mode independent from external forcings (Si and Hu 2017). In other words, we assume for now that externally forced responses and internally generated variability are linearly separable. Toward the end of this paper (section 4c), however, we will reexamine this assumption and demonstrate that despite the effort to remove the long-term externally forced trend, a discernable impact of the external forcings on the internal variability remains, and thus the centennial trend and decadal variability are entangled in a nonlinear fashion.

1) Why is detrending necessary?

One may argue that we can simply use the EOF analysis to separate the long-term trend from IPO variability, without first detrending the data. This is especially helpful because we do not need to assume the linearity of the trend. We test this approach first below.

If the underlying temperature trend was not removed, such as in the previous PDO study of Phillips et al. (2014) or the IPO studies of Power et al. (1999), Meehl et al. (2009), Dai (2013), and Dong and Dai (2015), EOF1 and PC1 were, not surprisingly, indicative of the global warming signal (Figs. 2a,b). The IPO/PDO mode is expected to appear as EOF2. (Note that all panels in Figs. 1 and 2 are EOF2, regardless what procedures were taken to remove seasonality and/or ENSO, which only explained a small percentage of the total variance at 2%–10%).

Fig. 2.

The EOF patterns and PC time series obtained if no detrending was applied prior to EOF analysis. (a),(b) Ensemble average of EOF1 and PC1 showing the expected background warming trend. (c),(d) EOF2 from ensemble member 32, whose PC2 time series was a residual warming trend instead of expected decadal variability. (e),(f) As in (c),(d), but for the ensemble average of EOF2 and PC2, which still had a nonnegligible residual trend. The EOF analysis was performed over the entire Pacific SST (i.e., the IPO approach). EOF was based on the annual dataset to which a low-pass filter was applied.

Fig. 2.

The EOF patterns and PC time series obtained if no detrending was applied prior to EOF analysis. (a),(b) Ensemble average of EOF1 and PC1 showing the expected background warming trend. (c),(d) EOF2 from ensemble member 32, whose PC2 time series was a residual warming trend instead of expected decadal variability. (e),(f) As in (c),(d), but for the ensemble average of EOF2 and PC2, which still had a nonnegligible residual trend. The EOF analysis was performed over the entire Pacific SST (i.e., the IPO approach). EOF was based on the annual dataset to which a low-pass filter was applied.

To our surprise, we found that for many ensemble members (e.g., ensemble member 32), there was still a residual warming trend even in the PC2 time series (Fig. 2d), and the associated EOF2 pattern was not the “tripole” pattern expected for the IPO/PDO (Fig. 2c). In fact, about 70% of the 35 members of the CESM1 twenty-first-century ensemble showed a residual warming trend in the PC2 time series, instead of the decadal fluctuation of the IPO/PDO. The ensemble mean of the PC time series should be close to zero if each PC time series reflects only unforced variability with different phases when the ensemble size is large enough. Figure 2f, however, shows that a residual trend was still visible in the ensemble average of PC time series (weaker than in Fig. 2d, however). The accompanying EOF2 pattern in Fig. 2e is virtually positive everywhere.

The bottom line is that if no explicit detrending is applied before the EOF analysis, EOF2 fails to clearly reveal the IPO/PDO pattern, which was particularly problematic for the twenty-first-century large warming case. Another problem with using EOF1 as the effective detrending is that the derived IPO/PDO may be mixed with the warming trend if the study period is relatively short. Therefore, a recent study (Dai et al. 2015) chose not to use the EOF decomposition to derive the forced signal. The explicit detrending, rather than using EOF1 as the trend, is also preferable from a mathematical perspective. Most of the statistical methods require time series to be stationary and ergodic, and thus removing trends should be performed prior to further statistical analysis.

For a time period during which the underlying trend is less significant (such as preindustrial simulation or early twentieth century), the detrending might not be necessary. But for time periods longer than a few decades, removal of the long-term trend becomes critical. For the purpose of consistency, we recommend performing a detrending procedure prior to any EOF analysis, and thus the resulting EOF1 pattern will be the desired internal variability component. We suggest avoiding using EOF1 as an effective detrending method and EOF2 as the variability component (as in many previous studies; e.g., Parker et al. 2007; Meehl et al. 2009; Hu and Bates 2017, manuscript submitted to Nat. Commun.), in particular when the underlying trend is large.

2) Why does “global detrending” fail?

As argued by Trenberth and Shea (2006), simply removing a linear fit of global-mean temperature in earlier studies such as Enfield et al. (2001) as the forced response to external forcings is not a good choice, since the actual warming trend might not be linear. Thus, removal of the global-mean temperature time series (nonlinear in nature) from all SST grid boxes (so-called global detrending) is a better alternative. However, when the global-mean temperature was removed from the twenty-first-century simulation for IPO, the resulting EOF1 pattern still showed an asymmetrical change between the North and South Pacific, with a larger value in the former (Fig. 3a). The PC1 time series in Fig. 3b depicted an upward trend, suggesting that EOF1 was merely a pattern of the differentiated warming trend between the North and South Pacific. The ensemble-averaged EOF2 was indeed the expected IPO pattern (Fig. 3c) with the ensemble-averaged PC2 being close to zero (Fig. 3d) because of the random phase of IPO present in each ensemble members.

Fig. 3.

The pitfalls of performing “global” detrending. (a),(b) The ensemble average of EOF1 and PC1 when the global-mean temperature of the ensemble average was removed from all grid boxes at each time step before the EOF analysis. It was expected that the EOF1 would provide an IPO pattern since the background warming was removed. On the contrary, the residual global warming pattern persisted in the PC time series. (c),(d) As in (a),(b), but for EOF2 that more closely resembled the IPO and PC2 that was closer to zero because of phase cancellation of individual IPO time series. (e),(f) As in (a),(c), but for EOF1 and EOF2 based on the North Pacific SST (i.e., PDO procedure). (g),(h) As in (e),(f), but for the average of TPI in group 1 (15 runs) and group 2 (20 runs). All results here are the ensemble average, based on the annual dataset to which a low-pass filter was applied.

Fig. 3.

The pitfalls of performing “global” detrending. (a),(b) The ensemble average of EOF1 and PC1 when the global-mean temperature of the ensemble average was removed from all grid boxes at each time step before the EOF analysis. It was expected that the EOF1 would provide an IPO pattern since the background warming was removed. On the contrary, the residual global warming pattern persisted in the PC time series. (c),(d) As in (a),(b), but for EOF2 that more closely resembled the IPO and PC2 that was closer to zero because of phase cancellation of individual IPO time series. (e),(f) As in (a),(c), but for EOF1 and EOF2 based on the North Pacific SST (i.e., PDO procedure). (g),(h) As in (e),(f), but for the average of TPI in group 1 (15 runs) and group 2 (20 runs). All results here are the ensemble average, based on the annual dataset to which a low-pass filter was applied.

This differentiated warming also appears in EOF1 if we follow the PDO procedure (using North Pacific SST to calculate PC), and EOF2 better resembles the expected PDO pattern that has a warming center located more toward the North Pacific (Fig. 3f). Since the PDO calculation is based on the North Pacific SST (Fig. 3e), rather than the entire Pacific as in PDO (Fig. 3a), the differential warming problem is less severe.

In addition, we tested if the global detrending is also problematic if the index is not based on EOF procedure. A simple metric (TPI; see section 2) is calculated for each member of the twenty-first-century simulation. We found that 15 out of the 35 ensemble members suffer the problem with the averaged TPI pattern skewed toward the NH (group 1 in Fig. 3g) and the averaged PC showing a monotonic increasing trend similar to averaged IPO PC in Fig. 3b. The other 20 of the 35 ensemble members (group 2 in Fig. 3h) appear to be less affected by the regional warming problem with the average TPI pattern showing the tripole as expected, and the average PC closed to zero (similar to Fig. 3d).

The analysis above shown in Fig. 3 suggests that a proper removal of background warming is essential to both EOF-based and non-EOF-based indices. Obviously, the global detrending procedure as used in many studies failed to eliminate the regional warming trend; as a result, decadal variability does not consistently emerge as the EOF1 pattern. The persistent residual regional warming pattern suggests that global detrending is unsuitable to remove the externally induced warming trend (Bonfils and Santer 2011; Dai et al. 2015). This issue of regionally differentiated warming rate is not as crucial for the twentieth-century observation/simulations, which is likely why the problem of global detrending did not receive much attention in recent well-cited studies (e.g., Phillips et al. 2014) and studies on the Atlantic multidecadal oscillation (AMO) index by removing the global-mean time series after obtaining the index (e.g., Trenberth and Shea 2006).

In the presence of multiple ensemble opportunities, it would be better to use the ensemble-mean global-mean time series, as it likely represents the climate response to the external forcing better than does the time series from individual members, especially at a shorter time scale (Deser et al. 2014; Dai et al. 2015). But a sensitivity test suggested that the issue of regional warming becomes less severe, but only marginally, if the global detrending used ensemble average (not shown) as opposed to individual ensemble members (as in Fig. 3).

3) how to do “local detrending”

The problem of differential warming will be mitigated by conducting a so-called local detrending, that is, removing a different trend at each grid box. A few choices exist concerning how to define the local trend.

(i) Linear local detrending

Removing the linear fit of the temperature trend at each grid box appeared to be the easiest choice (Meehl and Hu 2006). We found that this also led to an expected problem: the linear fit overfitted warming in the early and late twenty-first century while underfitting the warming in the midcentury, as seen in PC1 (Fig. 4b). Note that the PC time series in Fig. 4b is the ensemble average and would be close to zero if the obtained results were indeed the IPO because of the random phase cancellation. A close examination of each ensemble member showed that more than two-thirds of them had the over- or underfitting problem (e.g., see ensemble member 32 in Figs. 4c,d), which was clearly a result of the warming trend being highly nonlinear in the twenty-first century. If we were to analyze the twentieth-century dataset, where the warming is smaller, instead of the twenty-first century in Fig. 4, the nonlinearity would be a much smaller problem.

Fig. 4.

Comparing different methods of performing “local” detrending. (a),(b) Ensemble average of EOF1 and PC1 when a local linear detrending was applied. The ensemble average of PC1 depicted a fake variability because of the overfitting at the beginning and the end of the century. (c),(d) As in (a),(b), but for ensemble member 32 for which the over- or underfitting problem was more evident. (e),(f) As in (a),(b), but the ensemble-average time series was removed locally (i.e., nonlinear local detrending). This is the proposed revision to the IPO procedure. Note that the PC time series of the first 10 individual members are shown in thin gray lines in (f) to illustrate the random phase. (g),(h) As in (a),(b), but the local detrending was based on a quadratic fit rather than a linear fit. All results here are based on the annual dataset to which a low-pass filter was applied.

Fig. 4.

Comparing different methods of performing “local” detrending. (a),(b) Ensemble average of EOF1 and PC1 when a local linear detrending was applied. The ensemble average of PC1 depicted a fake variability because of the overfitting at the beginning and the end of the century. (c),(d) As in (a),(b), but for ensemble member 32 for which the over- or underfitting problem was more evident. (e),(f) As in (a),(b), but the ensemble-average time series was removed locally (i.e., nonlinear local detrending). This is the proposed revision to the IPO procedure. Note that the PC time series of the first 10 individual members are shown in thin gray lines in (f) to illustrate the random phase. (g),(h) As in (a),(b), but the local detrending was based on a quadratic fit rather than a linear fit. All results here are based on the annual dataset to which a low-pass filter was applied.

(ii) Nonlinear local detrending: Using the ensemble average

To cope with the nonlinearity problem of the local temperature trend, the ensemble-average time series at each grid box can be considered as the forced trend. After the removal of the local ensemble-average time series, the IPO pattern clearly stood out as the EOF1 (Fig. 4e). The clear separation of the IPO from the forced trend was true not only for the ensemble mean of EOF patterns (Fig. 4e), but also for all individual members (not shown). And because of the phase cancellation, the ensemble average of PC1 was close to zero (Fig. 4f, first 10 individual PCs shown in thin gray lines), providing a confirmation that the nonlinear local detrending method worked.

(iii) Nonlinear local detrending: Alternative methods

Note that local detrending conducted by removing the ensemble average requires (large) ensemble simulations, and therefore it is not directly applicable to observations or single-member simulations. As a compromise, a quadratic fit, instead of a linear fit, to the local temperature time series can be applied to account for the nonlinear warming trend. This was found to be a reasonably good approximation, as the calculated EOF1 (Fig. 4g) is highly similar to the EOF1 from the ensemble-based local detrending method (Fig. 4e).

Another alternative approach provided by Dai et al. (2015) is to first calculate the multimodel mean global-mean temperature time series as the forced response, and then scale this global-mean temperature time series at different locations to account for the spatial heterogeneity of forced response. The scaling factors for different locations were derived by linearly correlating local temperature time series with the multimodel mean global temperature time series. The method in Dai et al. (2015) can be viewed as an intermediate step between our proposed method (i.e., the nonlinear local detrending using the ensemble average) and the problematic linear local detrending in section 3c(3)(i), because the local trend defined in this method is “nonlinear” in nature but the “nonlinearity” shares a common shape globally (scaled to the same global-mean time series), unlike our method in section 3c(3)(ii). The difference between these two approaches is expected to be small, as we have tested for the quadratic fit approach (Fig. 4g).

d. A summary of the revised practice for filtering and detrending

1) On the filtering (smoothing) to remove high-frequency fluctuation

The conventional way in the literature to derive the PDO is by conducting EOF analysis on monthly data (with seasonal cycle removed but monthly time series retained), and therefore the PC time series contains subseasonal variability. For the IPO, a low-pass filtering is conventionally applied to the annual mean data, so that both the seasonal and interannual variability is removed (Fig. 1). The latter procedure is the one we recommend for both IPO and PDO analysis.

2) On the detrending to remove the forced response

The absence of a detrending procedure leads to the following issue: the internal decadal variability could not be consistently identified as EOF2 because it is contaminated by the warming trend in EOF1 (Fig. 2).

Removing only global-mean temperature (global detrending) can also lead to biases because the regional trend has a different pace than the global-mean trend (Fig. 3). The global detrending procedure as commonly used in many studies, therefore, fails to account for the regionality of the warming pattern and is not sufficient for capturing the decadal variability component in EOF1, especially when warming becomes more significant than that in the twentieth century.

We hereby recommend using a local detrending method. Specifically, the procedure requires removing the ensemble average of the temperature time series at each grid box to account for the nonlinearity of the warming (Figs. 4e,f), as opposed to removing a linear fit of time series (Figs. 4a,b). The latter approach will still be negatively affected by the fact that the GHG changes and the associated warming are nonlinear and accelerating during the twenty-first century. The benefit of combining these two approaches together, regardless of whether local detrending or global detrending is performed first, is marginal because once the local trend is removed, there is no longer much of a global trend.

In the literature, some have taken an approach using the annual SST (therefore the seasonality is removed) for the EOF analysis, but low-pass filtering or detrending was only applied to the PC time series after the EOF analysis. This method is somewhat inconsistent because, in the regression/correlation step for obtaining global EOF pattern, the PC time series is low-pass filtered but the original SST field is not. In contrast, our approach is self-consistent because the high-frequency variability and background trends are always removed before the EOF analysis.

In summary, for the preprocessing before EOF analysis, we recommend 1) removing the seasonal cycle by taking the annual average, 2) smoothing out ENSO by applying a low-pass filtering to the SST field, and 3) applying a local detrending by removing the ensemble average of temperature time series at each grid box. After the proper preprocessing, we use a centered correlation EOF method over the entire Pacific basin to calculate PC1 as the IPO index for the following analysis on regional influence.

4. Regional influences of IPO over North America: Seasonality and future changes

a. Quantifying the regional influence of the IPO/PDO on temperature and precipitation

After the IPO index was calculated following the procedure described in section 3, we used three methods to quantify the connection between IPO and regional climate:

  1. Regression coefficient: A commonly used method is to calculate the regression coefficient (regCoef) of the spatially varying and time-evolving temperature and precipitation on the derived index. The resulting two-dimensional regression coefficient field provides a measure of the connection between the regional climate and the IPO index. For consistency, the same detrending procedure as shown in section 3 is also applied to precipitation before the calculation of regression.

  2. Composite analysis used to obtain the anomalies. Alternatively, we conducted composite analysis by sampling the years in which the IPO index exceeded above (or fell below) the one standard deviation. The results were highly similar to those obtained in method 1, as expected, because statistically the regression coefficient is mainly determined by the most anomalous values in the time series.

  3. As in method 1, but also calculating the correlation coefficient (ecscore; ranging from −1 to 1) between SST field and the index. The downside of using the correlation coefficient is that it is unitless and thus does not provide an absolute measure of temperature or rainfall variation.

b. The regional influences of IPO in the twentieth century

The global temperature variation between the composite positive (larger than 1σ of the IPO index) and negative (lower than −1σ of the IPO index) IPO in the twentieth century was approximately 0.12°C (Fig. 5a). Regionally, the SST’s response to positive IPO in the Pacific is the typical IPO pattern. In other regions, there is a strong warming the Arctic, eastern Siberia, and Alaska, especially during the winter season (Fig. 5e). A warming signal also shows in Africa, the Middle East, and tropical South America, but a cooling in Europe and North America. Regarding precipitation, the tropical ocean tended to be wetter, featuring two midlatitude branches of enhancement extended into the United States and South America (Figs. 5b,d). The northern branch was more profound during DJF (Fig. 5f), and the Southern Pacific branch was more visible in Southern Hemisphere wintertime [June–August (JJA)]. The boreal summertime response mainly featured tropical enhancement of the rainfall, with a global enhancement resulting from IPO fluctuation being approximately 0.01 mm day−1. These results are consistent with previous analyses such as Dong and Dai (2015).

Fig. 5.

The influence of IPO on twentieth-century hydroclimate. The ensemble mean (a) temperature and (b) precipitation anomaly by contrasting the years when standardized IPO index is larger than 1 vs the years when standardized IPO index is smaller than −1. (c),(d) As in (a),(b), but for the correlation coefficient field (unitless). (e),(f) As in (a),(b), but for DJF. The analysis was based on the twentieth-century simulation (1920–2005).

Fig. 5.

The influence of IPO on twentieth-century hydroclimate. The ensemble mean (a) temperature and (b) precipitation anomaly by contrasting the years when standardized IPO index is larger than 1 vs the years when standardized IPO index is smaller than −1. (c),(d) As in (a),(b), but for the correlation coefficient field (unitless). (e),(f) As in (a),(b), but for DJF. The analysis was based on the twentieth-century simulation (1920–2005).

The impact of the IPO on temperature and precipitation in the continental United States is shown in Fig. 6 in the form of anomalies corresponding to a one standard deviation change in IPO. The main feature of the surface temperature in response to the IPO is a negative correlation with the IPO in the United States (Fig. 6a), especially over the eastern part. This is consistent with recent studies indicating that the U.S. “warming hole”—a smaller than global mean warming in the central-eastern United States from 1970 to 2000—is mainly due to natural variability associated with the IPO (Meehl et al. 2015), whereas local air pollution sources (Leibensperger et al. 2012) may have only played a secondary role. The rainfall in the southern part of the United States increases by up to approximately 1.0 mm day−1 corresponding to a positive one standard deviation change of the IPO, and this signal is stronger during DJF (Fig. 6d), especially for California, consistent with many previous studies (e.g., Meehl and Hu 2006; Dai 2013).

Fig. 6.

(a)–(d) As in Figs. 5a–d, but for the U.S. hydroclimate. (e),(f) As in (c),(d), but using observed temperature from GISS and observed precipitation from GPCC.

Fig. 6.

(a)–(d) As in Figs. 5a–d, but for the U.S. hydroclimate. (e),(f) As in (c),(d), but using observed temperature from GISS and observed precipitation from GPCC.

The model simulated anomaly as a result of the IPO is also consistent with analysis based on observed temperature (2° GISS temperature record; Hansen et al. 2010) and land precipitation (1° GPCC precipitation record; Schneider et al. 2017) for the same period (1920–2005) and following the same revised procedure. The eastern U.S. cooling and southern U.S. wetting as a result of the positive IPO appears even stronger in the observation (Figs. 6e,f) than in the model. The anomaly associated with the IPO as in Fig. 6 suggests that corresponding to a negative phase of the IPO such as observed during the last 15 years, one can expect warmer and drier conditions in many parts of North America, especially the western and southern regions. For example, California has experienced a persistent drought condition for the last few years possibly in association with the negative phase of IPO. These results are consistent with recent analysis of Chylek et al. (2017) based on twentieth-century observations, indicating that western U.S. precipitation is more subject to the influence of the IPO/PDO.

c. The change of the IPO in a warmer climate

Now we address the question posed in the title of this paper: How will IPO variability and its connection with North American rainfall change in the future?

Previously, in the procedure deriving the IPO (section 3) we only used twenty-first-century ensemble simulations (2006–80; Figs. 14). In section 4b, the influence of the IPO on the global and regional surface temperature and precipitation were discussed using twentieth-century ensemble simulations (1920–2005; Figs. 57). We repeated the same analysis for the preindustrial (PI) control simulation. To derive the IPO from the 1800-yr PI simulation, we used the local linear detrending method but it is unnecessary as there is hardly any noticeable trend in the simulated surface climate (Kay et al. 2015).

Fig. 7.

Understanding how IPO changes with warming. (a) The changes in the correlation coefficient of global temperature and IPO between the twenty-first century and PI. The dotted areas depict regions where the changes are significant at 90% confidence level considering the ensemble variability. (b) As in (a), but for the changes between the twentieth century and PI. The anomalous pattern that was opposite to the original IPO pattern [as in Fig. 4e for the twenty-first century and in Fig. 5c for the twentieth century] indicated a weaker IPO in a warmer climate. (c),(d) As in (a),(b), but for regression coefficient (°C). (e) As in (a), but using TPI instead of IPO. (f) EOF2 of the detrended dataset (EOF1 is IPO shown in Fig. 4e).

Fig. 7.

Understanding how IPO changes with warming. (a) The changes in the correlation coefficient of global temperature and IPO between the twenty-first century and PI. The dotted areas depict regions where the changes are significant at 90% confidence level considering the ensemble variability. (b) As in (a), but for the changes between the twentieth century and PI. The anomalous pattern that was opposite to the original IPO pattern [as in Fig. 4e for the twenty-first century and in Fig. 5c for the twentieth century] indicated a weaker IPO in a warmer climate. (c),(d) As in (a),(b), but for regression coefficient (°C). (e) As in (a), but using TPI instead of IPO. (f) EOF2 of the detrended dataset (EOF1 is IPO shown in Fig. 4e).

To assess whether the IPO signal and its impact on surface temperature and precipitation change over time, we compare the differences of the IPO pattern between the PI control run and the twentieth or the twenty-first century simulations. The ensemble-average correlation of surface temperature with the IPO index in the twentieth and twenty-first-century simulations is contrasted with that in the PI control simulation (Figs. 7a,b). The anomalous correlation pattern bears similarity to a negative IPO, especially for the twenty-first century minus PI case (Fig. 7a), suggesting that a warmer climate would dampen the climatological IPO amplitude. But note there is a subtle difference between the change of IPO and IPO itself even over Pacific: a negative–positive–negative–positive–negative pattern instead of the negative–positive–negative pattern associated with IPO. Moreover, the global-mean temperature change associated with the IPO decreases from 0.2°C for the PI simulation to 0.12°C for the twentieth-century simulations, and further down to 0.1°C for the twenty-first-century simulations. The weakening of the IPO can also be seen from regression field of surface temperature and the IPO (Figs. 7c,d).

To further test whether the change of decadal variability with warming is an artifact of EOF analysis procedure, the non-EOF-based TPI index is also used. Figure 7e shows that the temperature correlation with TPI weakens too in the twenty-first century. The similarity between the TPI-based and EOF-based anomalous patterns suggests that the influence of global warming on decadal variability is not an artifact of EOF analysis procedure.

Interestingly, the change of IPO pattern at a warmer climate (Figs. 7a,b) was strikingly similar to the EOF2 from the smoothed and detrended dataset (Fig. 7f; EOF1 being the IPO signal itself as in Fig. 6e). The similarity suggests that a proper EOF analysis on the twenty-first-century dataset itself might also reveal the influence of external forcing on internal variability. We consider EOF2 as the nonlinear interaction term of trend (removed) and variability (EOF1), but the physical mechanism of this pattern needs to be further investigated.

When the IPO phase becomes less persistent in time, the influence of the IPO on regional and global surface temperature could become less significant. For example, previous studies showed that in a lower CO2 emission scenario the IPO is capable of producing a warming hiatus, but in a high CO2 emission scenario the IPO cannot induce any hiatus warming events (Meehl et al. 2011, 2013). Consistently, we find the impact of the IPO on global-mean precipitation [measured by the rainfall anomaly resulting from a one standard deviation (1σ) fluctuation of the IPO index] would be weakened by approximately 50% in a warmer climate, from 0.01 mm day−1 in the PI run to 0.005 mm day−1 in the twenty-first-century simulation.

Regionally, the weaker magnitude of the IPO in a warmer climate also clearly reflected on the precipitation impact over the United States (Fig. 8). The wetting tendency as a result of a positive IPO in the southwest and southern United States (shown as red color in Fig. 6) was muted in a warmer climate (shown as blue color over the same regions in Fig. 8), especially during the twenty-first century. Note that global warming, in general, will increase the precipitation over southern United States and decrease the precipitation over the U.S. Southwest (at least in this model; see Meehl et al. 2013; Kay et al. 2015), so one inference we can make here is that the worsening of southwestern U.S. drying as a result of the negative IPO as argued recently (and by the same token the worsening of the southeastern U.S. flooding as a result of the positive IPO) will be less likely to occur in the future.

Fig. 8.

Understanding how IPO regional impact changes with warming. (a) As in Fig. 7a, but for the changes in the correlation coefficient of U.S. precipitation and IPO between the twenty-first century and PI. The dotted areas depict regions where the changes are significant at 90% confidence level considering the ensemble variability. (b) As in (a), but for the changes between the twentieth century and PI. The change of rainfall anomaly was opposite to the original anomaly pattern (as in Fig. 6b for the twentieth century) indicated a weaker IPO influence for U.S. rainfall. (c),(d) As in (a),(b) but for the rainfall anomaly (mm day−1). The regression results are similar to the anomaly.

Fig. 8.

Understanding how IPO regional impact changes with warming. (a) As in Fig. 7a, but for the changes in the correlation coefficient of U.S. precipitation and IPO between the twenty-first century and PI. The dotted areas depict regions where the changes are significant at 90% confidence level considering the ensemble variability. (b) As in (a), but for the changes between the twentieth century and PI. The change of rainfall anomaly was opposite to the original anomaly pattern (as in Fig. 6b for the twentieth century) indicated a weaker IPO influence for U.S. rainfall. (c),(d) As in (a),(b) but for the rainfall anomaly (mm day−1). The regression results are similar to the anomaly.

Since the IPO pattern and amplitude change with background climate, the IPO temporal characteristics may also change. This is examined in Fig. 9 by showing the power spectra of the IPO index (PC1 time series) for different periods in two ways. Figure 9a shows the ensemble mean of power spectra of IPO index for which the power spectra of each individual member are derived before the ensemble mean is taken. One cannot derive the power spectra of ensemble-averaged IPO index as ensemble-averaged IPO index is close to zero (Fig. 4f). To compare the long PI IPO index with the twentieth-century and twenty-first-century ensemble on an equal footing, we divide the 1800-yr PI IPO index into 20 chunks of 90-yr-long (similar to twentieth- and twenty-first-century runs) time series. The smoothed PI power spectrum is generated by averaging this constructed ensemble. Doing so also allows us to obtain an estimate of ensemble variance shown as the two thin blue lines bracketing the “ensemble” averaged PI power spectrum in Fig. 9a. The ensemble variances from the 35-member twentieth-century and twenty-first-century runs are of comparable size and are thus not shown. The shift of the power spectrum curve toward the high-frequency domain is evident, especially for the twenty-first-century runs (red line in Fig. 9a). The shift of the IPO power spectrum toward the high-frequency domain in the twentieth century (orange line) is less clear and barely falls out of the ensemble uncertainty of PI.

Fig. 9.

The power spectra of IPO time series as a function of frequency (cycles per century) for three time periods [PI (blue), twentieth century (orange), and twenty-first century (red)]. (a) The ensemble-averaged power spectrum. Note that the ensemble for PI is constructed by breaking the 1800-yr simulation into 20 time series to be of similar length with the twentieth- and twenty-first-century simulation. The standard deviation of power from the constructed ensemble is shown in thin blue lines as a measure of uncertainty of ensemble-averaged power. (b) The power spectrum calculated from the time series of IPO index directly (see section 4c). The uncertainty range of PI power as shown in the thin blue lines is from (a). The upper 99% confident internal of the red noise spectrum is shown in thin dashed lines. The IPO variance is not separable from red noise power at lower frequencies (smaller than 4 cycles per century) and thus is not shown.

Fig. 9.

The power spectra of IPO time series as a function of frequency (cycles per century) for three time periods [PI (blue), twentieth century (orange), and twenty-first century (red)]. (a) The ensemble-averaged power spectrum. Note that the ensemble for PI is constructed by breaking the 1800-yr simulation into 20 time series to be of similar length with the twentieth- and twenty-first-century simulation. The standard deviation of power from the constructed ensemble is shown in thin blue lines as a measure of uncertainty of ensemble-averaged power. (b) The power spectrum calculated from the time series of IPO index directly (see section 4c). The uncertainty range of PI power as shown in the thin blue lines is from (a). The upper 99% confident internal of the red noise spectrum is shown in thin dashed lines. The IPO variance is not separable from red noise power at lower frequencies (smaller than 4 cycles per century) and thus is not shown.

Since the power spectrum curve in Fig. 9a is ensemble averaged, the peak of the curve is hard to identify. We thus also calculated the power spectrum in a different way by not taking ensemble average. Instead, Fig. 9b shows the power spectrum calculated from the time series directly (1800 yr for PI, 35 multiplied by 85 yr for the twentieth century, and 35 multiplied by 75 yr for the twenty-first century). Note that the very low-frequency signal (less than 4 cycles per century) is not shown as they are not larger than the red noise variance (thin dashed lines in Fig. 9b). In comparison to the PI control simulation, the IPO power spectrum appeared to shift from the lower-frequency band with a corresponding period of 28 yr (16–33 yr) in PI control to a higher-frequency band with a period of 19 yr (12–25 yr) in the twenty-first-century simulations. The higher frequency (shorter period) of IPO, combined with a muted amplitude, suggests that its modulating effects on global climate will be smaller in the future.

5. Discussion and conclusions

There have been numerous studies on the influence of the IPO on regional temperature and precipitation (i.e., Dai 2013; Deser et al. 2014; Dong and Dai 2015). The general results are consistent with our findings (Figs. 5 and 7). However, to our best knowledge, the changes of the IPO (and its influence) under a warmer climate have not been carefully examined before, which is probably due to the difficulty of clearly separate the forced responses from the unforced variability when the underlying trend is large. In this study, we comprehensively reviewed and evaluated various approaches to obtaining the IPO/PDO spatial pattern and time series, and made a recommendation of which is better suited for long-term climate change studies (especially when the background climate change is rapid).

The first key issue is the removal of the interannual variability, such as the ENSO signal, before EOF analysis, which is important for isolating the decadal variability component. The removal of ENSO is particularly critical for the IPO index, which is more subject to influences from tropical variability.

The second key issue that is often overlooked is the proper detrending method. We recommend applying a local ensemble-average detrending to account for both the spatial heterogeneity and temporal nonlinearity of the warming trend. This revised method is essential to reveal how future warming affects the IPO/PDO and its connection to regional rainfall. The detrending, again, is more critical for the IPO index because the North Pacific warms more slowly than the tropics. Not surprisingly, a proper selection of the detrending method is less important for the analysis of the observations and model simulations in the twentieth century than those in the twenty-first century because of weaker warming.

We note that although this study focused on Pacific variability, the “local” nonlinear detrending approach also has implications for variabilities in other ocean basins, such as the AMO in the Atlantic. Model simulations show that the North Atlantic warming is muted compared to the projected global warming resulting from the weakening of the Atlantic meridional overturning circulation (AMOC) (e.g., Hu et al. 2013). Moreover, the benefit of adopting a local nonlinear detrending procedure is that it allows for a better quantification of how the long-term climate change might affect the decade variability (i.e., interaction of forced response and internal variability). This nonlinear interaction between background climate change and decadal variability has not received much attention in the past because to the first order one can assume these two terms are linearly additive, and decadal variability can be superimposed on the background forced trend and only plays a modulating role in the overall trend (Meehl et al. 2016a). A future warming of 3°–5°C toward the end of twenty-first century, however, could shift the magnitude or frequency of the natural variability modes, as clearly demonstrated in many ENSO or monsoon studies (e.g., Cai et al. 2015), and therefore a nonlinear interaction between centennial trends and decadal variability cannot be ignored.

By applying the recommended procedure to derive the IPO/PDO index, we further quantified the anomalous regional temperature and precipitation as a result of the IPO phase swing, using both correlation and regression methods. Comparing the IPO patterns in the PI control run to those in the twentieth-century transient simulations and the twenty-first-century projections, we found that IPO would have a weaker amplitude and higher frequency as the climate warms. The weakening of the IPO amplitude also results in a weaker response of the regional hydroclimate over the United States to the IPO changes.

Last, we note that the changes of the IPO pattern in a warmer climate were similar to the EOF2 pattern of unforced simulation or the detrended transient simulation (with EOF1 being the IPO signal itself). The mechanism behind the IPO weakening requires further investigation, as the underlying physical process for past IPO/PDO is still poorly understood (Han et al. 2014; Newman et al. 2016). Brown et al. (2017) recently also found a muted natural variability in future warming conditions and attributed that to reduced snow-albedo effect. Our speculation here is that the enhanced ocean stratification and a shallower mixed layer predicted in a warmer climate [also seen in recent analysis of Fasullo et al. (2017)] would tend to lessen the exchanges of heat between upper and deeper oceans. These oceanic governing factors might also provide clues to explain 1) the member-to-member difference of IPO magnitude and impact on regional and global climate and 2) the model-to-model differences if such an analysis were repeated for the multimodel dataset.

Acknowledgments

We acknowledge support from the U.S. Department of Energy’s Office of Science (BER; DE-FC02-97ER62402). The National Center for Atmospheric Research is supported by the U.S. National Science Foundation. Y.X. thanks the National Science Foundation.

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