1. Introduction

Recently, advances have been made in both dynamical and statistical prediction (Goddard et al. 2001). The commonly used statistical tools are canonical correlation analysis (CCA; Barnett and Preisendorfer 1987) and singular value decomposition (SVD; Bretherton et al. 1992). The statistical approach is empirical. The forecasts are made based on the covariability between predictors and predictand during the training period. The CCA method assumes that the relationships between predictors and the predictand are linear. The forecast will be skillful if the relationships determined during the training period remain true during the forecast period.

One of the important predictors for the surface temperature (T_surf) over the United States is the sea surface temperature (SST) field, which represents boundary forcing (Barnston and Smith 1996). In addition to SSTs, atmospheric fields can also be used as predictors. Multiple predictors can enter the CCA together to perform forecasts. However, the forecast skill depends on the weighting function, which has to be given a priori. Barnston (1994) used both SSTs and 700-hPa heights as predictors to predict T_surf over the United States. He discovered that the forecasts have the highest skill when the weighting between SSTs and 700-hPa heights is two to one. In general, it is difficult to find the optimum weighting function among predictors before entering the CCA.

The dynamical forecasts based on global general circulation models (GCMs) do not depend on the past history or weighting function. Because of model errors, statistical forecasts based on the CCA and dynamical model forecasts have comparable skill at the present time. Therefore, model errors need to be removed before the model forecasts can be utilized. In addition to removing the mean model bias, the error correction methods based on CCA or SVD can also be used to improve model forecasts (Feddersen et al. 1999; Smith and Livezey 1999; Ward and Navarra 1997). Model outputs are adjusted based on the common modes between the model forecasts and the observed fields. The CCA method corrects not only the mean field, but also the leading modes of variability. Since the model errors are regime dependent (Mo and Wang 1995; O'Lenic and Livezey 1989), the CCA correction method improves forecast skill.

The nomenclature ensemble has been used to represent a set of forecasts made from different models or forecasts made from the same model but different initial conditions. Lau et al. (2002) developed the ensemble canonical correlation (ECC) prediction method and used the term ensemble to represent forecasts from the CCA based on SSTs from different ocean basins. In this paper, the term ensemble is expanded to cover a set of CCA forecasts based on observations as well as model forecasts. Lau et al. (2002) used the ECC to specify summer precipitation over the United States. The global ocean is separated into different basins. The SSTs from each ocean basin enter the CCA separately to specify summer precipitation. These predicted fields form an ensemble. The ensemble mean is the weighted average of its members. The weighting is determined according to the previous forecast performance. To separate the global ocean into different basins, the weaker and more regional forcing can be recognized.

Later, Mo and Thiaw (2002) adopted the ECC prediction concept to predict summer seasonal mean rainfall over the Sahel. Instead of using SSTs from different ocean basins as predictors, they used different variables, such as SSTs, and 200-hPa streamfunction anomalies over the Tropics. In addition to the CCA forecasts, their ensemble also included model forecasts. Because forecasts from different variables recognize different forcing, they are skillful over different regions of the Sahel. The ensemble mean forecasts have high skill. This method is easy to use. The ensemble can contain as many members as needed. Since each variable enters the CCA prediction separately, the training period for each variable can also vary. This gives the forecast system needed flexibility.

The weighting function among variables can be determined after the CCA. There are many ways to construct the ensemble mean forecasts. The simplest way is to take an equally weighted average of its members (Mo and Thiaw 2002). The superensemble method proposed by Krishnamurti et al. (2000a,b) can also be a candidate. Kharin and Zwiers (2002) called the superensemble forecast the regression-improved forecast to emphasize the regression scheme used. If the forecasts are given in categories, then a Bayesian method developed by Rajagopalan et al. (2002) can also be applied.

For any statistical prediction, the selection of predictors is important. As stated by Goddard et al. (2001), the main sources of predictability on the seasonal and interannual timescales come from the long-term trends in SSTs and El Niño–Southern Oscillation (ENSO). The low-frequency components of the SST forcing contribute to the forecast skill because of its large global impact (Ropelewski and Halpert 1987, 1989; Halpert and Ropelewski 1992; Mason and Goddard 2001; Gershunov and Barnett 1998). In addition to SSTs, the surface temperature field in summer is influenced by soil moisture (Huang et al. 1996; Koster et al. 2000). Atmospheric variables like sea level pressure (SLP; Barnett and Preisendorfer 1987) or 700-hPa heights have also been used as predictors (Barnston 1994). The atmospheric variables are influenced by soil moisture, snow cover, and sea ice in addition to SSTs. Therefore, they may capture variability not associated with the leading SST modes.

The purpose of this paper is to apply the ECC method to forecast T_surf over the United States for summer (July–August–September, JAS) and winter (January–February–March, JFM). These two seasons were selected because the National Centers for Environmental Prediction/Climate Prediction Center (NCEP/CPC) operational seasonal forecasts had the highest skill in JFM and the lowest skill in JAS. Instead of using SSTs from different ocean basins as suggested by Lau et al. (2002), different variables are used as predictors. The data and procedures are outlined in section 2. Section 3 presents the ensemble forecasts for summer T_surf. The model forecasts used in the operational environment are forecasts made with predicted SSTs from a coupled model with a lead time of 1 month. Such forecasts for the current model are available only from September 2000 to the present. Therefore, model simulations from the Atmospheric Model Intercomparison Project (AMIP) were used to represent the model outputs. The current NCEP seasonal forecast model was used to perform the AMIP experiments. They were performed with the observed SSTs. These runs can be considered as the T_surf “forecasts” with perfect SSTs. The actual forecast skill with predicted SSTs will be lower because the errors of SSTs are not included in the AMIP simulations. Since the goal of this paper is to compare the ECC ensemble forecasts with its members, the model simulations serve this purpose. The ECC prediction is applied to the winter T_surf (JFM) forecasts in section 4. Conclusions are given in section 5.

2. Data and procedures

b. Procedure

1) CCA for a single predictor

The CCA procedure for a single predictor is outlined in Barnston (1994) and Barnston and Smith (1996). The predictand is the T_surf over the United States. The predictors are the global SST; SLP over the Northern Hemisphere from 20°–90°N; T_surf, and soil moisture over the United States. In order to compare with the operational seasonal forecasts issued by the NCEP/CPC, the lead time is 1 month (Barnston 1994). For example, forecasting T_surf for JAS based on SST, 1 month lead forecast is performed using SST (March–May; MAM). The reason is that the seasonal forecast for JAS is issued in mid-June. Therefore, the season closest to JAS with data available is MAM.

The forecast skill for each year can be expressed in terms of the anomaly correlation defined as the spacial pattern correlation between the observed and predicted anomalies over the United States for that year. The mean anomaly correlation averaged over all years gives a general indication of the overall skill of the forecasts.

The standard deviations for the predictor and the predictand are averaged over the domain of the datasets. The standardized predictor and predictand separately enter the EOF analysis. The truncated EOF time series then enter the CCA program. The truncation for each variable is determined based on specification using the cross-validation procedure (Barnston and Smith 1996). It will be explained later. For summer, the JAS SSTs are used to specify the JAS T_surf. The mean anomaly correlations (ACs) between predicted and observed T_surf are determined for different modes of truncation. The mean AC increases as the member of modes increases until it reaches a maximum. After that, the AC decreases. The number of modes that gives the highest AC in the specification experiment is used for prediction. The same process is also used to determine the truncation for soil moisture and SLP. The truncation depends on the variable. It varies from 11 modes for SST to 16 modes for soil moisture.

The CCA modes are determined based on the covariability between the predictor and the predictand from data in the training period. The predictor data for the target year T are projected onto the predictor CCA loading patterns, and the predictand values are generated and verified against the observed T_surf for the target year T.

Cross validation is used in all skill evaluations (Barnston 1994; Barnston and Smith 1996). Let N be the data record length. All data for the target year T are withheld. The length of the training period is N − 1. All calculations are repeated for each target year including the means, standard deviations, EOFs, and CCA modes. The predicted field for the target year T should be verified against the observed field for the same target year T. The process then is repeated N times for N different target years. Because of the strong trends in SSTs, the cross validation by taking 1 year out may still inflate skill. The purpose of this paper is to compare the ECC ensemble mean forecasts with the individual members in the ensemble. Therefore, the inflation of skill does not impact conclusions significantly. The CCA procedure produces loading patterns for predictor and predictand as well as the amplitude time series. For diagnostic purposes, the loading patterns and the amplitude time series are determined from the CCA using all data available.

2) Equally weighted ensemble mean (EE)

The ensemble members include the CCA-predicted T_surf fields based on observations and model forecasts from different models or from the same model but different initial conditions. The model outputs are treated the same way as observations. The predictor entering the CCA is the model-forecasted T_surf. The predictand is the corresponding observed T_surf. The CCA gives the model-corrected T_surf (Livezey and Smith 1999).

Let N be the length of the record and M be the number of predictors from observations and from the GCM outputs. For the data period (1950–2000), N = 51. If one uses three predictors from observations (SST, SLP, soil moisture) and the AMIP run, then M = 4.

All observed data are withheld for a given year T. For each single predictor m, the CCA is performed using data for the remaining N − 1 yr using the procedure described in the previous section. There are M predicted T_surf fields R(m, x, T) for the withheld year T, climate division x and predictor m. Here, x = 1, 102 and m = 1, M. The simple ensemble mean E(x, T) for the target year T can be written as

View Expanded

where E(x, T) is the ensemble mean for the climate division x and the target year T. E(x, T) should be verified against the observed field P(x, T) for the withheld year T. In this case, all members in the ensemble are weighted equally.

3) Superensemble mean (SE) or the regression-improved forecast

Another way to determine weights is the superensemble method. As demonstrated by Krishnamurti et al. (2002a,b) and Kharin and Zwiers (2002), the ensemble mean can be written as the linear combination of R(m, x, T) for the withheld year T:

View Expanded

where R(m, x, T) is the T_surf predicted from the observed predictor or model forecast. The linear regression method is used to determine the coefficients b(m, x) for each target year T. To perform regression, one needs to have N − 1 pairs of predicted and observed T_surf. They are obtained the following way:

Within the N − 1 data records, data from another year y are withheld. Now, there are two withheld years: T and y. The CCA is then performed using the remaining N − 2 yr for each predictor m separately. The standard deviations, means, EOFs, and CCA modes are calculated each time for different T and y. Let r(m, x, y) be the predicted field for predictor m, at climate division x for the withheld year y. The ensemble mean e(x, y) can be written the same way as Eq. (2):

View Expanded

where e(x, y) should be verified against the observed field P(x, y) for each year y. This procedure is then repeated N − 1 times for different year y. During this whole process, data from T are not used. There are N − 1 pairs of e(x, y) and the corresponding observed P(x, y).

The coefficients b(m, x) are determined by minimizing the variance V(x) between the ensemble mean e(x, y) and the corresponding observed field P(x, y) for the training period N − 1:

View Expanded

and coefficients b(m, x) are substituted into Eq. (2) to get E(x, T).

In this process, data from the withheld year T are not used to obtain the coefficients b(m, x) or R(x, m, T). This process is then repeated for each year T = 1, N for cross validation. E(x, T) is verified against P(x, T) for the withheld year T. For 51-yr data and 3 predictors, there are 7650 separate CCA calculations performed.

4) Verification

In addition to the mean AC described above, the geographic distribution of the skill is also presented (Barnston and Smith 1996) for verification. The temporal correlation between the predicted and observed pairs of T_surf at each climate division for 51 yr (1950–2000) gives the cross-validated skill for that division. All climate divisions together give the geographic distribution of skill. Assume the T_surf obeys a normal distribution, then the correlation values greater than 0.28 are statistically significant at the 5% level for N = 50.

The NCEP operational forecasts are verified based on the Heidke scores (Barnston et al. 1999). At each climate division, the predicted T_surf is given in three categories: below normal, normal, and above normal. The Heidke score is defined as

SENE(5)

where Hit is the number of correct forecasts, N is the total number of locations verified so here N = 102. The value E is the number of categorically correct matches by chance. For three equally likely categories, E = N/3. The comparison between the Heidke score and other measures of skill were reviewed by Barnston (1992).

The field significance test (Livezey and Chen 1983) is performed by randomly shuffling the forecasts and the corresponding observational fields 1000 times. One counts the number of times that the mean of the correlation in the domain exceeds that of the random statistics. Unless otherwise stated, maps presented below pass the field significance test a the 5% level.

3. Summer surface temperature

The predictand field is the summer mean (JAS) T_surf over United States (25°–49°N, 70°–125°W). The standard deviation for the period 1950–2000 (Fig. 1c) shows large values located over the central United States with a maximum over Oklahoma and Kansas. Over the western United States and Florida, the standard deviations are small. The leading mode of variability can be described by the first rotated EOF. EOF was performed for seasonal mean (JAS) T_surf for the period 1950–2000. Eight EOFs were used for the Varimax rotation. Rotated empirical orthogonal function 1 (REOF 1; Fig. 1d), which explains 27% of the total variance has weak negative loadings over the West Coast and positive loadings over the central United States. The maximum is located over Oklahoma and Kansas. This is the same location of the standard deviation maximum (Fig. 1c).

4. Winter surface temperature

Figure 11 shows the geographical distribution of the cross-validated skill measured by the correlations between T_surf (JFM) forecasts and verifying observations at climate divisions. For the T_surf (JFM) forecasts, the forecast skill decreases as the lead time increases. The mean anomaly correlation for SST (JAS) is only 0.13 and the skillful forecasts are limited to the northern states (not shown). The forecasts based on SST [September–October–November (SON)] have the mean AC = 0.32. The correlations higher than 0.5 are located over the eastern and central part of the United States and the West Coast. Results are consistent with those reported by Barnston and Smith (1996).

The forecasts based on SLP (SON) have the mean AC = 0.15. The forecasts based on the persistence of the T_surf (SON) have AC = 0.1 (Fig. 11c). If data for December are available, then the forecasts based on the October–November–December (OND) mean have the mean AC = 0.15 and skill increases over the northern states (Fig. 11d). The NCEP model simulation is able to capture ENSO responses in general. After the CCA correction, the forecast skill with the mean AC = 0.30 is comparable with forecast skill based on SST (SON). The CCA correction improves the forecast skill.

The sources of skill come from SSTs in the Pacific. Consistent with Barnston and Smith (1996), the leading CCA modes for SST (SON) represent the long-term trends and ENSO (Fig. 12). The first three CCA modes explain 22%, 20%, and 17% of the covariance respectively. (Fig. 12). The degeneracy among the first three CCA modes makes the separation of signal difficult. The first CCA mode represents the long-term trends. The SST loading pattern resembles the ENSO-like decadal mode (Zhang et al. 1997). Positive SST loadings are located over three oceans with a maximum over the central Pacific and negative loadings are found over the North Pacific. The corresponding T_surf pattern shows warming over the west coast of the United States and the northern states. Cooling is found over the Southeast. The second mode shows positive SSTs in the central Pacific and T_surf pattern shows cooling over the southeastern United States and Minnesota. The third mode represents warming over the central Pacific and the west coast of North America. The corresponding T_surf shows cooling over the Southwest and warming over the Northeast.

The leading CCA mode for SLP is influenced by SSTs (Fig. 13). The correlation pattern between the amplitude time series and SST (SON) shows warm SSTs in the central Pacific and negative SSTs over the western Pacific. The T_surf and the SLP loading patterns resemble the composites during the warm ENSO years (Barnston 1994; Barston et al. 1999). The SLP pattern shows large zonal components with positive loadings over high latitudes and negative loadings to the south (Barnston 1994). The T_surf loadings show warming located over the northern states and cooling over the Southeast.

Both the simple ensemble mean EE and super ensemble mean SE forecasts were constructed. The members are the CCA forecasts based on SST, SLP, and T_surf for SON. The skill of the simple EE ensemble forecast depends on its members. The EE (SST, SLP) forecasts have higher skill over the northern states, but lower skill over the southern states in comparison with the forecasts based on SST (SON). Overall, the EE (SST, SLP) forecasts have a slightly higher mean AC = 0.34 (Fig. 14a). The EE (SST, SLP, T_surf) forecasts have lowest skill with mean AC dropping to 0.26 (Fig. 14b). The EE (SST, SLP, model) forecasts have the highest mean AC = 0.37 (Fig. 14c). The improvements are located over the Southeast where the model forecasts are skillful. This indicates that members should be chosen carefully in constructing the EE ensemble forecasts.

The skill of the SE ensemble forecast is not sensitive to its members. The SE (SST, SLP, T_surf), SE (SST, SLP), and SE (SST, SLP, model) forecasts are similar with the mean AC = 0.39–0.40. For example, to add forecasts based on T_surf (SON) to the ensemble does not degrade the SE. The forecasts based on T_surf are consistently poor. The regression assigns lower weights to T_surf (SON), therefore, the SE forecasts are not affected by adding T_surf (SON). The SE mean forecasts (Fig. 14d) are more skillful than forecasts based on any individual member (Fig. 11). The geographic distribution of skill for the SE forecasts (Fig. 14d) shows that correlations greater than 0.5 are located over the southern and the southeastern United States, the Pacific Northwest, and the area over the Great Lakes. Similar to the summer T_surf forecasts, the SE forecasts have high skill because the forecasts for each variable behave consistently.

The NCEP operational forecasts had the highest skill in JFM (Barnston et al. 1999). In comparison with summer, the scores for winter forecasts are much higher (Table 1). During this period, there were two strong warm ENSO events in 1995 and 1998, and one strong cold ENSO event in 1999. The NCEP forecasts were always more skillful during the ENSO winters. The Heidke scores for ENSO seasons vary between 17 and 20. The CCA forecasts were made each year based on variables for SON, and the SE (SST, SLP, T_surf) ensemble mean was constructed. Again, no data after SON year T − 1 were used to predict T_surf (JFM) for year T. Except 2000, the SE gives better or comparable skill than the operational forecast. For winter, the NCEP seasonal forecast model is able to capture the ENSO signals well. That contributes to the successful forecasts.

5. Conclusions

The concept of the ECC prediction method developed by Lau et al. (2002) has been applied to predict summer JAS and winter JFM surface temperature T_surf over the United States. The predictors are the global SST and SLP over the Northern Hemisphere, soil moisture and T_surf persistence over the United States from one to two seasons lead, as well as model AMIP simulations from the NCEP seasonal forecast model. Each variable enters the CCA prediction separately as a single predictor. The predicted fields form an ensemble. The ensemble means are the weighted average of its members. Both the simple ensemble mean and the superensemble mean are tested. The simple ensemble forecast is the equally weighted average of its members. For the superensemble forecast, the weighting function among the members is determined by linear regression (Krishnamurti 2000a,b). Because the linear regression is used, Kharin and Zwiers (2002) called it the regression-improved forecast.

The T_surf winter forecasts have overall higher skill than the summer forecasts. The major sources of skill in winter comes from the decadal SST trends in the central Pacific and ENSO. The T_surf field for summer JAS is influenced by both SST and soil moisture. For JAS, the CCA forecasts based on SSTs recognize the long-term warming trends and ENSO in the tropical Pacific. In addition to the tropical SSTs, SSTs over the North Pacific and the North Atlantic also contribute to the forecast skill because they appear as the second and high CCA modes. One can also test the importance of the extratropical SSTs by performing the CCA forecast based on SST (JFM) in the tropical Pacific (30°S–30°N, 120°E–90°W) alone. The forecasts based on SSTs in the tropical Pacific alone have the mean AC = 0.18. They are less skillful in comparison to forecasts based on the global SSTs. This suggests that SSTs in the North Pacific and the North Atlantic also contribute to the T_surf forecast skill in summer.

For the T_surf JAS forecasts, soil moisture has large influence on T_surf over the central United States. T_surf is also influenced by SLP. The SLP anomalies are associated with SSTs, but they are also influenced by other boundary forcing such as soil moisture, sea ice, snow, and many other factors. The SLP as a predictor may capture forcing that is not represented in the leading modes of SSTs.

Different variables recognize different forcing and give skillful forecasts over different regions of the United States. This is the major reason that the ensemble forecasts have high skill. Both the ensemble EE and SE forecasts are more skillful than forecasts based on a single predictor. The improvements are more impressive for summer (JAS) than in winter (JFM). The skill of the EE forecast depends on its members. The EE members should be chosen in such a way that each members shows forecast skill in different regions of the domain. If the superensemble is constructed, the linear regression will select the best combination of members at each climate division.

The SE forecasts are skillful because forecasts from different variables behave consistently. For example, the summer forecasts for T_surf based on SSTs are skillful over the southern states and over the west coast of the United States. The linear regression scheme will weight forecasts based on SSTs heavily over these areas. On contrary, the forecasts based on soil moisture are more skillful in the central United States, so the regression scheme assigns more weights to forecasts based on soil moisture. The weights do not vary much from one target year to another. This is the major reason that the SE forecasts are skillful.

This does not mean that the SE forecasts are always better than the EE forecasts over the United States everywhere. Therefore, in the operational forecast situation, both EE and SE forecasts or forecasts based on other regression methods (Kharin and Zwiers 2002) should be presented to forecasters together with the geographic distribution of skill determined, based on forecasts from the previous years. The consistency and differences among the ensembles can be used to determine the reliability of the forecasts.

One of the advantages of the ECC method is the flexibility. One can add model outputs to the ensemble. The ensemble simulations of the AMIP runs are used to represent model forecasts. The simulations were performed with observed SSTs. They can be viewed as forecasts made with “perfect SSTs.” In a real forecast situation, the SSTs are predicted from a coupled model or from persistence. Therefore, the skill is much lower. Overall, the CCA correction increases the forecast skill because that the CCA corrects errors in the leading modes of variability.

For cases tested here, to add ensemble model outputs to a SE or EE ensemble only increases the forecast skill slightly. The areas in which the model simulations are skillful are also the regions that the forecasts based on observations are skillful. Therefore, there is little new information. However, this is highly dependent on the region and the model. Mo and Thiaw (2002) applied the same method to prediction of rainfall over the Sahel. In that case, the model simulations and the forecasts based on SSTs are skillful over different parts of the Sahel. Therefore, to add model simulations to an ensemble improves the forecast. The method is flexible and easy to use. It has potential to improve seasonal forecasts.