
In this paper, we focus on the decadal prediction of the PDO during winter season (December–January–February, DJF), because many of the physical processes in the North Pacific have undergone a substantial longterm intensification during winter (Mantua et al., 1997; Deser and Phillips, 2006; Yeh et al., 2011; Wang et al., 2012). Fiveyear running mean is used to obtain the decadal variability, and the yearly mark represents the middle year of the 5yr mean period. The decadal PDO index is defined as the leading principal component of the empirical orthogonal function of 5yr running mean DJF SSTAs over the North Pacific (20°–70°N), in which the SSTAs are the anomalies from the climatological annual cycle after removing the global mean SSTs (Mantua et al., 1997).
Four types of monthly mean datasets are involved to predict the decadal PDO, including: sea surface temperature from NOAA Extended Reconstructed SST v3b on a horizontal resolution of 2° x 2° (Smith et al., 2008), sea surface height from SODA (Simple Ocean Data Assimilation) v2.1.6 on a horizontal resolution of 0.5° x 0.5° (Carton and Giese, 2008), sea level pressure from NOAACIRES (Cooperative Institute for Research in Environmental Sciences) 20th Century Reanalysis version 2c on a horizontal resolution of 2° x 2° (Compo et al., 2011), and sea ice concentration on a horizontal resolution of 1° x 1° from the Met Office Hadley Centre (Rayner et al., 2003).
After obtaining the decadal PDO index, the difference of the PDO between the current year and the three years before is calculated as the 3yr decadal increment (DI) of the PDO (namely DI_PDO) [see Eq. (1)]. For example, the DI_PDO in 1906 is the difference between PDO in 1906 and PDO in 1903.
$${\rm{DI}}\_{\rm{PD}}{{\rm{O}}_{{i}}} = {\rm{PD}}{{\rm{O}}_{{i}}}  {\rm{PD}}{{\rm{O}}_{{{i}}  3}},$$ (1) where PDO_{i} and PDO_{i−3} represent the decadal PDO index in current year and three years before the current year, respectively. DI_PDO_{i} indicates the decadal increment of the PDO in current year.
The DI_PDO is predicted firstly through a statistical model with three predictors, which are also in the DI form (DI_Predictors). DI_Predictors are calculated as the 3yr decadal increment of the 5yr running mean for the detrended seasonal mean predictors. The final predicted PDO is derived by adding the predicted DI_PDO to the observed PDO three years ago by using Eq. (2) below,
$${\rm{PD}}{{\rm{O}}_{{i}}} = {\rm{PDO}}_{{{i}}  3}^{{\rm{obs}}} + {\rm{DI}}\_{\rm{PDO}}_{{i}}^{{\rm{s{\text{}}}} {\rm{model}}},$$ (2) where
${\rm{DI}}\_{\rm{PDO}}_{{i}}^{{\rm{s{\text{}}}} {\rm{model}}}$ represents the the statistical model predicted DI_PDO in the target year.${\rm{PDO}}_{{{i}}  3}^{{\rm{obs}}}$ is the decadal PDO index from the observation three years before the target year. PDO_{i} indicates the final predicted PDO in the target year.In order to avoid the artificial builtin skill, we use two modulated validation methods to verify the predictive skill of the statistical model. (1) Crossvalidation with five years left out (Michaelsen, 1987): the predictand in a target year is predicted with the forecast model by using the training samples in all the years except the five years that are centered around the target year. This process is repeated for all the other target years to obtain the crossvalidated reforecast for the whole prediction period. The first/last three years are predicted by leaving the first/last five years out from the prediction period. (2) Independent hindcast: the training period is the forwardrolling 67 yr, which is from 72 (69) yr before the target year to 6 (3) yr before the target year in predictors (predictand) to avoid the prediction model using any information from the prediction period. For example, the DI_PDO in 1975 (1976) is predicted by the DI_Predictors in 1972 (1973) based upon the forecast model built up using the period of 1903–1969/1906–1972 (1904–1970/1907–1973) in DI_Predictors/DI_PDO.
The moving t test (MTT) with a 9yr moving window is used to detect the abrupt change points of the decadal variation of the PDO. The student’s ttest is used to examine the statistical significance, in which the effective sample size N* is computed following Bretherton et al. (1999) and Ding et al. (2012) as follows,
$${N^*} = N\frac{{1  {r_1}{r_2}}}{{1 + {r_1}{r_2}}},$$ where N is the number of available time steps, and r_{1} and r_{2} are the lag1 autocorrelation of the two correlated variables, respectively.

Figure 1a shows the spatial patterns of the decadal DJF global SST associated with the decadal PDO index (the first EOF of decadal DJF SSTAs over the North Pacific). This mode accounts for 27% of North Pacific decadal SST variance. A positive PDO features negative SSTAs over central and western North Pacific and positive SSTAs over eastern North Pacific and eastern tropical Pacific. The decadal DJF PDO index during 1901–2009, as well as the DI_PDO, is shown in Fig. 1b.
Figure 1. (a) Correlation coefficient of the 5yr running DJF global SST with the decadal PDO index during 1901–2009. (b) Decadal PDO index during 1901–2009 (solid line) and the DI_PDO during 1903–2009 (bar).
The DI_PDO is treated as the predictand. The empirical statistical model is built to predict the DI_PDO using DI_Predictors. This empirical statistical model is built through three steps. First, the potential preceding DI_Predictors are derived from the following three main processes driving the PDO as summarized by Newman et al. (2016): a) fluctuations in the Aleutian low; b) teleconnections from the tropics; and c) midlatitude ocean dynamics and coupled variability. Second, the DI_Predictors have to lead the DI_PDO at least three years to avoid the DI_Predictors using any information from the prediction period, since we use 5yr running mean to derive the decadal variability. Third, we use stepwise regression to identify the final DI_Predictors, which should be important and less dependent, to build the empirical statistical model. The fundamental rules in the stepwise regression are to select the DI_Predictor that is most significantly correlated to the predictand, and remove those predictors that are significantly related to this DI_Predictor, thus the DI_Predictors selected this way are independent to each other. We use the 99% confidence level for Fisher’s F test to select the final DI_Predictors.
Based on the above steps, three leading DI_Predictors for three years are selected, which are autumn DI of Aleutian low (DI_AL), winter DI of sea ice over of Greenland Sea (DI_SIC), and spring DI of sea surface height over central Pacific (DI_SSH), respectively [Eq. (4)]. Physically, the SST variability over most of the North Pacific is driven primarily by the atmospheric forcing (Smirnov et al., 2014). Generally, the atmospheric variations lead the SST variations that are resulted from the physical processes of the fluxdriven SSTA pattern or Ekman transports (Davis, 1976; Deser and Timlin, 1997). The preceding Aleutian low (AL) has been suggested to be one of the important atmospheric forcing to drive the PDO variability (Schneider and Cornuelle, 2005; Newman et al., 2016) (Fig. 2a). On the other hand, the SIC anomalies over Greenland Sea is connected to the PDO variability through influencing the Arctic Oscillation anomalies (Fig. 2b) (Lindsay and Zhang, 2005; Sun and Wang, 2006). Additionally, the subArctic frontal zone (SAFZ) over the western Pacific has the large SST variations associated with the PDO (Nakamura and Kazmin, 2003). The preceding thermal capacity over SAFZ can drive the PDO variability through thermodynamic response (Fig. 2c) (Qiu, 2003; Newman et al., 2016).
Figure 2. Correlation coefficients of the DJF DIPDO during 1906–2009 with 3yr leading (a) SON DI of sea level pressure, (b) MAM DI of sea surface height, and (c) DJF DI of sea ice concentration. The area with dots indicate values significant at the 90% confidence level with the student ttest. The boxes indicate the key areas for calculating the DI_Predictors, including DI_AL (36°–44°N, 180°–166°W), DI_SSH (28°–33°N, 155°–140°W), and DI_SIC (70°–75°N, 15°–2°W).
$${\rm{DI}}\_{\rm{PDO}} =  0.31\times {\rm{DI}}\_{\rm{AL}}  0.36\times{\rm{DI}}\_{\rm{SIC}}  0.37\times{\rm{DI}}\_{\rm{SSH}},$$ (3) where DI_AL represents DI of the September–October–November sea level pressure averaged over 36°–44°N, 180°–166°W; DI_SIC indicates DI of the DJF sea ice concentration averaged over Greenland Sea (70°–75°N, 15°–2°W); and DI_SSH is the DI of March–April–May sea surface height averaged over central Pacific (28°–33°N, 155°–140°W). The DI_AL, DI_SIC, and DI_SSH explain 9.6%, 13%, and 13.7% of the DI_PDO variance, respectively.
The correlation coefficients of the DI_PDO during 1906–2009 with DI_AL, DI_SSH, and DI_SIC during 1903–2006 are −0.43, −0.48, and −0.43, respectively. Although three of the DI_Predictors are all significantly correlated with DI_PDO, it is inevitable that these relationships have decadal shifts since the predicted period is more than 100 years (Fig. 3). It is noted that both of the relationships in DI_AL and DI_SSH become weakened around the period of 1960–1980, which may influence the predictive effect in the statistical model during this period (Figs. 3a, b). The moderate explained variances from the three DI_Predictors for the DI_PDO may also hint this decadal shift in the correlations.
Figure 3. The 25yr running correlation of the DI_PDO during 1906–2009 with (a) DI_AL, (b) DI_SSH, and (c) DI_SIC in preceding three years. The thin dashed line denotes the 90% confidence level. The effective sample sizes are the maximum value in the slipping 25 yr, which are 13, 13, and 10 for DI_AL, DI_SSH, and DI_SIC, respectively.
Figure 4a shows the predicted DI_PDO during 1906–2009 in the crossvalidation with 5 years left out. As we can see, the limited skill for the predicted DI_PDO appears over the period around 1960–1980, which may be attributed to the above mentioned weakened relationships in DI_AL and DI_SSH. Excluding the period of 1960–1980, the predicted DI_PDO shows consistency in variability and amplitude with the observed DI_PDO to some extent, especially for the period after 1980. The correlation coefficient (CC) between the observed and predicted DI_PDO is 0.68 during 1906–2009 (significant at the 99.9% confidence level). The statistical model in general displays a reasonable predictive skill for the DI_PDO.
Figure 4. Prediction of the DJF DI_PDO, PDO, and the regime shifts of the PDO detected by moving t test with 9yr moving window in crossvalidation with five years left out for the period of 1906–2009 (left panels) and in independent hindcast for the period of 1975–2009 (right panels). The thick black (red) line with hollow circles (rhombus) represents the result of observation (prediction). The pink shading is the 95% prediction interval. The numbers in the upper right corner is the temporal correlation coefficient between the prediction and observation. The thin black solid lines in panels (e) and (f) indicate the values significant at the 95% and 99% confidence level, respectively.
The predicted DI_PDO is added with the PDO three years ago in the observation to obtain the final predicted PDO (Fig. 4c). The inconsistency between the predicted and observed PDO during 1960–1980 can also be seen in Fig. 4c. In the rest of the prediction period, the final predicted PDO fits the observation very well in amplitude and variability and generally successfully captures the phase change at each time. The CC between the observed and final predicted PDO reaches 0.82 during 1906–2009 (Fig. 4c). Actually, compared to regime shifts in the observation, the errors for each regime shift in prediction is less than or equal to two years (Fig. 4e). Overall, the increment method employed in this study shows effective skill for the decadal prediction of the PDO.
The predictive skill of the statistical model is further investigated by examining the independent hindcast for predicting the DI_PDO during 1975–2009. The magnitude of the predicted DI_PDO during 1975–1980 is lower than 0.5, which is far less than the observation, indicating a limited predictive skill. After 1980, the statistical model displays a high predictive skill for the DI_PDO. The CC between the prediction and observation of the DI_PDO during 1975–2009 can reach 0.78 (significant at the 99% confidence level) (Fig. 4b). The final predicted PDO during 1975–2009 has the CC of 0.81 with the observation (Fig. 4d). There are two regime shifts in observed PDO during 1975–2009, which are 1988/1989 and 1999/2000, respectively. The final predicted PDO has the first regime shift at 1990/1991, which is two years later than the observation. The second regime shift in the final predicted PDO is at 1999/2000, which is precisely consistent with the observation (Fig. 4f). Overall, the statistical model combined with the increment method presents a high skill for the decadal prediction of the PDO, including the prediction for the regime shift.
In order to identify the effect of the increment method, the statistical prediction model in its original form (i.e., without the increment method) is used to directly predict PDO [Eq. (5)]. According to the correlation map of the PDO and the 3yr leading predictors (Fig. 5), the key areas for the AL and SSH move slightly to the highest correlation regions compared to the key areas for DI_AL and DI_SSH. The key areas for the SIC move to the north of Kara Sea, since the region in Greenland Sea is not significant any more.
Figure 5. Correlation coefficients of DJF PDO during 1906–2009 with 3yr leading (a) SON sea level pressure, (b) MAM sea surface height, and (c) DJF sea ice concentration. The area with dots indicate the values significant above the 90% confidence level. The boxes are the key areas for calculating predictors, which are AL (35°–50°N, 175°E–160°W), SSH (30°–35°N, 170°–155°W), and SIC (78°–85°N, 70°–90°E), respectively.
$${\rm{PDO}} =  0.36\times{\rm{AL}}  0.35\times{\rm{SIC}}  0.37\times{\rm{SSH}},$$ (4) where AL indicates the September–October–November (SON) sea level pressure over 35°–50°N, 175°E–160°W; SIC represents DJF sea ice concentration over 78°–85°N, 70°–90°E; SSH is the sea surface height over 30°–35°N, 170°–145°W. AL, SIC, and SSH explain, respectively, 13%, 12.3%, and 13.7% of the PDO variance.
The CCs of the PDO during 1906–2009 with 3yr leading AL, SSH, and SIC are −0.48, −0.55, and −0.56, respectively, which are all above the 90% confidence level. The statistical model in its original form seems to show certain skill in predicting the PDO with significant correlation coefficient between the prediction and the observation (Figs. 6a, c). However, the crossvalidated PDO shows that the extreme negative period of PDO index around 1950 has been missed (Fig. 6a). In the independent hindcast for the PDO during 1975–2009 (Figs. 6b, d), the magnitude of the prediction is much lower than the observation and the predicted PDO keeps in the negative phase after the late 1980s, which is incorrect according to the observation. Generally, the predictive skill of the statistical predictive model without the increment method is lower than that with the increment method.
Figure 6. As in Fig. 4, but for the PDO prediction of the statistical model with 3yr leading AL, SSH, and SIC. (a, c) are the crossvalidated PDO during 1906–2009 and its 9yr MTT results; (b, d) are the predicted PDO during 1975–2009 in the independent hindcast and its 9yr MTT results.

The statistical predictive model combined with the increment method is also employed to predict the decadal variability of the DJF SSTA at each grid point over the North Pacific Ocean (20°–70°N). Equation (6) is the same as Eq. (4), except that the predictand is the DI of DJF SSTA (DI_SSTA) over the North Pacific. The DI_Predictors also lead the DI_SSTA variability by three years. The final predicted SSTA is obtained by adding the predicted DI_SSTA to the observed SSTA three years ago. The predictive skill is tested by the independent hindcast for the DI_SSTA during 1975–2009.
$$ {\rm{D}}{{\rm{I}}_  }{\rm{SSTA}} = {\rm{a}}\times{\rm{D}}{{\rm{I}}_  }{\rm{AL}} + {\rm{b}}\times{\rm{D}}{{\rm{I}}_  }{\rm{SIC}} + {\rm{c}}\times{\rm{D}}{{\rm{I}}_  }{\rm{SSH}}. $$ (5) For the DI_SSTA during 1975–2009, a significantly high predictive skill appears over the Kuroshio–Oyashio Extensions to the central Pacific and the region north of 50°N (Fig. 7a). Although the predictive skill for the final predicted SSTA is not impeccable (Fig. 7b), the final predicted SSTA successfully demonstrates a regime shift at 1999 (Fig. 7c), which is a challenge for current dynamic numerical models (Newman et al., 2016). Moreover, we further investigate the predictive skill of this statistical model combined with the increment method to produce the spatial pattern of the PDO. Figure 7d gives the CC between the final predicted PDO index (Fig. 4d) and the final predicted SSTA over the North Pacific (Fig. 7b) in the independent hindcast. Corresponding to the positive PDO, there are negative SSTAs appearing over the western and central North Pacific and positive SSTAs appearing over the eastern North Pacific, resembling the observation (Fig. 7e). Actually, the pattern correlation coefficients between predictions and observations reaches 0.95. The statistical model combined with the increment method shows a high predictive skill for the regime shift of the SSTA over North Pacific and the spatial pattern of the PDO.
Figure 7. Correlation coefficients of the observation with (a) the predicted DJF DI_SSTA and (b) the final predicted DJF SSTA in the independent hindcast during 1975–2009. (c) Difference of the predicted DJF SSTA in the independent hindcast between 2000–2009 and 1980–1999. (d, e) Correlation coefficient of the PDO index with the SSTA over North Pacific in independent hindcast and observation, respectively. The area with dots indicate the values significant above the 90% (panels a, b) or 95% (panel c) confidence level using the Student’s ttest.
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Huang, Y. Y., and H. J. Wang, 2020: A possible approach for decadal prediction of the PDO. J. Meteor. Res., 34(1), 63–72, doi: 10.1007/s1335102091444. 
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