## 1. Introduction

The 500-hPa geopotential height is one of the most fundamental and widely used meteorological variables for characterizing the general atmospheric circulation. Skilful seasonal forecasts of hemispheric 500-hPa heights have obvious potential benefits for understanding large-scale atmospheric circulations and seasonal predictions of other meteorological variables. A lot of effort has focused on developing seasonal forecast schemes based on coupled general circulation models (CGCMs). However, the predictive skill achieved by CGCMs is still not very satisfactory, especially outside of the Tropics (Francis et al. 2003; Francis and Renwick 2004).

In the extratropics, it is known that a substantial component of interannual variability of seasonal mean fields arises from variability within the season (see, e.g., Madden 1981; Zheng et al. 2004). This component mainly arises from day-to-day weather variability with a time scale longer than the deterministic prediction period (about 10 days), and therefore it is essentially unpredictable on seasonal, or longer, time scales. For this reason, this component has been referred to as the “intraseasonal component” (Zheng and Frederiksen 2004) or the “weather noise component” (Madden 1976; Madden 1981; Zheng and Frederiksen 1999). After removing this component from seasonal mean fields, the residual component is more likely to be associated with slowly varying external forcings (e.g., SSTs) on the atmospheric climate system and from low-frequency (interannual and longer; slower than intraseasonal time scale) internal atmospheric variability, and therefore, it is more potentially predictable at the long range (Madden 1976). As a result, this component of seasonal mean fields has been referred to as the “slow component” (Zheng and Frederiksen 2004) or the “potentially predictable component” (Madden 1976; Zheng and Frederiksen 1999).

Zheng and Frederiksen (2004) recently developed a methodology for estimating, from monthly mean data, spatial patterns of the slow and intraseasonal components. This methodology provides a way to better identify and understand the sources of predictive skill as well as the sources of uncertainty in climate variability. By applying this methodology to reanalysis datasets, they successfully identified the potentially predictable and unpredictable patterns of the 500-hPa geopotential height field for the Northern Hemisphere (Frederiksen and Zheng 2004) and the Southern Hemisphere (SH; Frederiksen and Zheng 2007). Also, they were able recently to improve the predictive skill of New Zealand rainfall forecasts using a statistical prediction scheme based on the prediction of the principal component time series of the slow components of rainfall variability (Zheng and Frederiksen 2006). In this paper, we will utilize this methodology to construct a statistical prediction scheme for austral summer [December–February (DJF; hereafter 3-month periods are denoted by the first letter of each respective month)] and winter (JJA) SH reanalysis 500-hPa geopotential height anomalies. Out of interest, we compare the skill of our statistical forecasts with those from a dynamical seasonal forecast scheme.

The paper is arranged as follows. The data and methodology are described in sections 2 and 3, respectively. The proposed prediction scheme is studied in section 4 and compared with forecasts from a dynamical scheme. Some discussions and our conclusions are presented in section 5.

## 2. Datasets

### a. Reanalysis data

The monthly mean 500-hPa geopotential height reanalysis datasets used in this paper are taken from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalyses (1948–2004; Kalnay et al. 1996) and the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis dataset (ERA-40; 1957–2002; Uppala et al. 2004). Our main aim in this paper is to show that our proposed methodology is able to produce skilful statistical forecast schemes based on observed datasets that can be used to train and verify the schemes. For this purpose, either of the two datasets could be used to illustrate the procedure. In this paper, we choose to use the NCEP–NCAR dataset to set up and verify the proposed scheme. The ERA-40 dataset is used only as a verification dataset to provide a fairer comparison with the dynamical forecast scheme.

The NCEP–NCAR data for the period December 1952 through August 1992 is used as the training period for the statistical scheme. The verification period is taken as 1993–2002 for forecasts of NCEP–NCAR and ERA-40 height. Both datasets have been subsampled on a 5° × 5° latitude–longitude grid. As in Frederiksen and Zheng (2004, 2007), a variable resolution in longitude has been used in order to avoid biasing the analysis toward high latitudes. In the Southern Hemisphere, the longitude spacing is taken as 5° between 0° and 45°S, 10° between 50° and 65°S, 15° at 70°S, 20° at 75°S, 30° at 80°S, 60° at 85°S, and a single point at the Pole.

### b. Sea surface temperature

To help identify important interannual SST forcing of the SH atmospheric patterns of variability, we have used the SSTs from the 1° × 1° (latitude–longitude grid) Hadley Centre Sea Ice and SST dataset (HADISST1.1; Rayner et al. 2003) from September 1952 to August 2004. The area used to produce the one-point correlation maps of section 4, between the principal component time series of the slow modes of height variation and the SST, is global in longitude but restricted between 60°N and 60°S.

### c. Model-predicted data

It is of interest to compare the skill of our statistical forecasts of SH 500-hPa geopotential heights with that of a dynamical seasonal forecast model. In this paper, we analyze the hindcast skill of such a model, namely, the Coupled Oasis Commonwealth Scientific and Research Organisation (CSIRO) Ocean–Atmosphere Model version 2 (COCA2). This model is a global CGCM and is used routinely to provide seasonal to interannual forecasts.

The COCA2 dynamical prediction system (see Smith et al. 2005 for details) is based on the Australian CSIRO’s Mark 3 (Mk3) global CGCM (see Gordon et al. 2002 for details). The atmospheric model is spectral T63 (1.875° latitude × 1.875° longitude per grid box) with 18 vertical levels on a hybrid sigma-pressure vertical coordinate. The ocean model has a horizontal resolution matching that of the atmospheric model’s grid in the east–west direction, and twice that in the north–south direction (i.e., 0.937° latitude × 1.875° longitude). There are 31 levels in the vertical, with the spacing of the levels increasing with depth, from 10 m at the surface to 400 m in the deep ocean. Hindcasts for the period 1980–2004 were used to determine the skill of this model (see Smith et al. 2005 for details of hindcast runs). To set up the initial state of the coupled system, COCA2 uses a process called coupled-surface nudging (Smith et al. 2005) in which surface wind stress and SST are anomaly coupled at each time step in the period leading up to the forecast period. In brief, during the initialization phase, the COCA2 seasonal forecast model uses a “blending” of observational and model anomalies added to the model climatology to couple the surface wind stresses and SST between the atmosphere and ocean. This results in a “nudging” of the model toward the observations during the initialization process.

In this study, we shall compare the hindcasts for austral summer and winter SH 500-hPa geopotential height anomalies from the dynamical forecast model, relative to their climatology, and our statistical scheme. The forecasts are initialized in the previous month (i.e., November or May for summer and winter, respectively). For ease of comparison, the model forecasts have been interpolated onto the same horizontal grid as the reanalysis data used in the statistical analysis (see section 2a above).

## 3. Methodology

For predicting the height field, we will use the methodology proposed by Zheng and Frederiksen (2004) to construct potentially predictable patterns of the height field, and then we will predict the time series associated with the potentially predictable patterns. For the reader’s convenience in sections 3a*–*d, we reproduce the details of Zheng and Frederiksen (2004). Then in sections 3e*–*f, we will outline the methodology for predicting the principal components associated with the potentially predictable patterns, and how to use the predicted principal components to construct the predicted height field. The prediction skill scores are introduced in section 3g.

### a. Model for monthly mean fields

*x*

_{ym}be a climate variable (e.g., pressure), where

*y*(=1, . . . ,

*Y*) represents a particular year, and

*m*(=1, 2, 3) is a month in a specified season. After the annual cycle is removed from the data,

*x*

_{ym}is assumed to be a sample from a random population with a seasonal “population” mean

*μ*

_{y}, that is,where

*ε*

_{ym}is a residual monthly departure of

*x*

_{ym}from

*μ*

_{y}. Moreover, the vector

*ε*

_{y1},

*ε*

_{y2},

*ε*

_{y3}is assumed to comprise a stationary and independently distributed random vector with respect to year. The linear form Eq. (1) implies that month-to-month fluctuations, or intraseasonal variability, arise entirely from

*ε*

_{y1},

*ε*

_{y2},

*ε*

_{y3}and

*x*

_{y1}−

*x*

_{y2}=

*ε*

_{y1}−

*ε*

_{y2}.

*m*or

*y*) by replacing that variable subscript with “o.” For example,

*x*

_{yo}indicates the seasonal average of

*x*

_{ym}in year

*y,*and

*x*

_{oo}, the average of

*x*

_{ym}taken over all months and years. With this notation, a sample seasonal mean can be expressed aswhere

*ε*

_{yo}is associated with intraseasonal variability and the seasonal population mean

*μ*

_{y}is more likely associated with the interannual variability of external forcings (such as SST) and low-frequency variability (interannual or longer) of internal dynamics. For convenience, we shall refer to these as the intraseasonal and “slow” component, respectively, of the seasonal mean.

*x*

_{ym}and

*x*′

_{ym}that satisfy Eq. (1). Then,

*x*

_{ym}and

*x*′

_{ym}can represent either the same or different physical quantity at different locations or the same location. The following assumptions about monthly means, of the climate variables, are necessary for estimating the covariance of the intraseasonal component between two climate variables [denoted here by

*V*(

*ε*

_{yo},

*ε*′

_{yo})]. Since the daily time series of a climate variable, within a season is, in general, assumed to be stationary, so are the monthly statistics. In particular, the covariance between two valuables is assumed to be independent of months; that is,The same is also assumed to be true for the intermonthly covariance, that is,Strictly speaking, these assumptions may not apply well in transition seasons, but they are probably reasonable for winter or summer seasons. Recognizing that day-to-day weather events are unpredictable beyond a week or two, we further assume that the intraseasonal components are uncorrelated if they are a month or more apart, that is,

### b. Covariance arising from intraseasonal variability

*E*denotes the expectation value based on all years,

Since, from Eq. (1), *ε*_{y1} − *ε*_{y2} = *x*_{y1} − *x*_{y2} and *ε*_{y2} − *ε*_{y3} = *x*_{y2} − *x*_{y3} are known, the moment estimation can be applied to derive

*σ*

^{2}and

*ϕ*in Eq. (6) with estimates

*σ̂*

^{2}and

*ϕ̂*, we can derive an estimate for the covariance of the intraseasonal components. In particular,Since we have assumed that

*V*(

*ε*

_{y1},

*ε*′

_{y3}) = 0 and

*V*(

*ε*′

_{y1},

*ε*

_{y3}) = 0, then

To reduce the estimation error, we constrain the estimated *ϕ̂* to lie within the interval [0, 0.1]. The lower bound can be justified as follows. Owing to persistence, daily climate variables, such as pressure and temperature, are positively autocorrelated, and so are their monthly means. Thus, *V*(*ε*_{y1}, *ε*′_{y1}), *V*(*ε*_{y1}, *ε*′_{y2}), and *V*(*ε*_{y2}, *ε*′_{y1}) are likely to have the same sign. This leads to the lower bound [see Eqs. (7) and (8)]. The upper bound is based on the assumption that the daily meteorological field is a multivariate first-order autoregressive process. A detailed proof appears in the appendix of Zheng and Frederiksen (2004). A simulation study (Zheng and Frederiksen 2004) demonstrated that the truncation is necessary to reduce the estimation error.

### c. Decomposition of the covariance between two seasonal means

*V*(

*ε*

_{yo},

*ε*′

_{yo}) can be estimated using Eq. (16), the covariance of two seasonal means can be decomposed aswhere the first term on the left-hand side can be rewritten asand will be referred to as the “residual” covariance after removing the intraseasonal variability. It is worth emphasizing that this covariance, in general, consists of not only the covariance between

*μ*

_{y}and

*μ*′

_{y}(i.e., the slow components of the climate variables), but also their interaction terms with

*ε*

_{yo}and

*ε*′

_{yo}. In the case where the intraseasonal and slow components are independent, the residual covariance reduces to the covariance of the slow component. When this is not the case,

*V*(

*x*

_{yo},

*x*′

_{yo}) −

*V*(

*ε*

_{yo},

*ε*′

_{yo}) may still be better related to the variability in the slow component than is

*V*(

*x*

_{yo},

*x*′

_{yo}). However, another independent test would need to be performed to show this.

### d. Decomposition of the covariance matrix of the seasonal mean field

The covariance matrix of an *R*-dimensional seasonal mean field arising from intraseasonal variability, could be constructed by estimating the covariance between every pair of seasonal means arising from intraseasonal variability using Eq. (16). However, some care needs to be taken so that the rank of the covariance matrix is not overestimated. Estimation errors can result in a matrix with rank greater than *Y* − 1. This is unreasonable, since there are only *Y* independent annual fields, so the rank of the covariance matrix should not be more than *Y* − 1.

This can be overcome by adopting an approach similar to a truncated EOF analysis.

We propose the following procedure to estimate the covariance matrix of the intraseasonal component of the seasonal mean.

- (i) An EOF analysis is applied to the sample covariance matrix of a seasonal mean field to derive the eigenvalues and EOFs. Since there are only
*Y*years of data available, the number of positive eigenvalues is less than*Y*. - (ii) The monthly fields are projected onto the dominant
*Y*-1 EOFs to form the*Y*-1 truncated monthly principal component (PC) series. - (iii) For every pair of monthly truncated PCs, estimate the covariance arising from intraseasonal variability using the method documented in section 3b. In this way, a symmetric matrix with dimension (
*Y*− 1) × (*Y*− 1) is formed. - (iv) Multiply the matrix of EOFs [with dimension
*R*× (*Y*− 1)] and its transpose on the left-hand side and the right-hand side, respectively, of the symmetric matrix to form a matrix with dimension*R*×*R*. This is the estimated component of the covariance matrix arising from intraseasonal variability. - (v) Subtract the estimated component arising from intraseasonal variability from the sample covariance matrix of the seasonal mean field to get the residual covariance matrix.

This procedure ensures the correct rank of the intraseasonal covariance matrix and residual covariance matrix, which for convenience we shall refer to as the slow convariance matrix.

### e. Principal component

*x*

_{ym}(

*r*) represent sample monthly values, within a season, of the 500-hPa geopotential height anomaly field at geographical location

*r*(

*r*= 1, . . . ,

*R*), in month

*m*(

*m*= 1, 2, 3) of the winter or summer season in year

*y*(

*y*= 1, . . . ,

*Y*). For each EOF of the slow covariance matrix, or intraseasonal, covariance of the field

*x*

_{ym}(

*r*) (see section

*3*d), we define an associated PC time series to be the dot product between the monthly time series

*x*

_{ym}(

*r*) and the EOF. Equations (16) and (17) can then be used to determine what fraction of the interannual variance of this time series is due to intraseasonal variability. For example, if

*p*

_{ym}represents the PC time series and eof(

*r*),

*r*= 1, . . . ,

*R*a particular EOF then, using Eq. (1),Here, ϵ̃

_{ym}and

*μ̃*

_{y}represent the intraseasonal and slow components, respectively, of the PC time series. Thus, Eqs. (16) and (17) can be used to estimate the fraction of interannual variance due to the intraseasonal component [i.e., V(ϵ̃

_{yo})/V(

*p*

_{yo})]. The fraction [1 − V(ϵ̃

_{yo})/V(

*p*

_{yo})] represents the fraction remaining after the removal of the intraseasonal component from the PC time series. We shall refer to this as the residual variance fraction or potential predictability of the PC.

### f. Prediction scheme

**x**

_{ym}= [

*x*

_{ym}(1), . . . ,

*x*

_{ym}(

*R*)]

^{T}denote a hemispheric 500-hPa geopotential height field in month

*m*and in year

*y*and 𝗩 denote the matrix with columns that are the EOFs derived from a singular value decomposition (SVD) analysis of the residual covariance matrix (section 3d) of the height field in all

*Y*years. Let

**p**

_{y}denote the projection of the height field onto 𝗩 in year

*y*, that is,The row vector

**p**

_{y}consists of the amplitudes of the PC time series in year

*y.*Because 𝗩 is orthonormal, the 500-hPa geopotential height field can be written asWe will use Eq. (22) to predict the height field. The proposed statistical prediction comprises the following steps.

- (i) Select the first
*M*years as the training period, and the last*Y*−*M*years as the verification period. - (ii) Estimate 𝗩 by an SVD analysis on the residual covariance matrix of the height fields in the training period (see section 3d).
- (iii) Construct the projection
**p**_{y}in the training period by Eq. (21). - (iv) For each projection time series (i.e., a row vector of
**p**_{y}), select candidate predictors based on the corresponding EOFs and the correlation map between the projection time series and one season lagged SST field (see section 4a for the details). Only the top few significant EOFs with potential predictability larger than 50% need to be considered. The others are dropped from the analysis. - (v) Apply stepwise linear regression to establish the relationship between each projection time series and its candidate predictors in the training period. Use the trained regression equation to predict projection time series in the verification period.
- (vi) Use the predicted projection time series and Eq. (22) to construct the predicted height field.

*μ*_{y}= [

*μ*

_{y}(1), . . . ,

*μ*

_{y}(

*R*)]

^{T}. As for the total field [Eq. (22)], the slow height field can be represented bywhere

*μ̃*_{y}represents the slow principal component time series, andHowever, while it is possible to separate the variance, or covariance, of the total field into slow and intraseasonal components, it is not possible to separate the total field itself. Thus,

*μ*_{y}in the training period is unknown, and therefore

*μ̃*_{y}is unknown. Consequently, we have to use

**p**

_{y}instead of

*μ̃*_{y}. The main predictors of

**p**

_{y}, that is, tropical SST indexes and the annual linear trend, have a small intraseasonal component, and, therefore, so does the predicted

**p**

_{y}. Consequently, from Eq. (22), the predicted height field consists predominantly of

*μ*_{y}and only a small component of intraseasonal variability.

### g. Prediction skill

Let *o*_{y}(*r*) and *f*_{y}(*r*) be the anomaly of the observed and forecasted seasonal mean of 500-hPa heights in year *y* and at grid *r* (=1, . . . , *R*), respectively, and *b*(*r*) be the baseline climatology at grid *r* in the training period (*y* = 1, . . . , *M*). To prevent the skill scores from being dominated by regions of high interannual variability (e.g., the polar region), it is important to normalize the anomalies by *s*(*r*), the observed standard deviation of the seasonal means at grid *r* estimated for the training period, that is, we use *o*′_{t}(*r*) = *o*_{y}(*r*)/*s*(*r*), *f* ′_{t}(*r*) = *f*_{y}(*r*)/*s*(*r*), and *b*′(*r*) = *b*(*r*)/*s*(*r*).

*V*

_{expl}to be negative, especially when

*f*

_{t}(

*r*) is significantly biased.

An important application of the geopotential height field is to depict the general circulation, and, therefore, the similarity of the spatiotemporal variability between the observed and predicted is also a major concern. In this paper, we introduce a skill score called the spatiotemporal correlation (STC) as a measure of similarity. It is defined as the correlation between [ *f* ′_{t}, *t* = 1, . . . , *R* × (*Y* − *M* + 1)]^{T} and [*o*′_{t}, *t* = 1, . . . , *R* × (*Y* − *M* + 1)]^{T}, where *f* ′_{t} = *f* ′_{y}(*r*) and *o*′_{t} = *o*′_{y}(*r*) for *t* = (*y* − *M* − 1) × *R* + *r*. Large STC values indicate that the spatiotemporal variability between predicted and observed is similar. An STC value of one indicates that the observation and prediction are equal up to a linear transformation. For this reason, the STC skill score is relatively insensitive to the prediction bias compared with the percent of explained variance skill score.

## 4. Results

In this section, our aim is to construct a statistical prediction scheme that will be skilful in forecasting the NCEP–NCAR SH 500-hPa geopotential heights following the procedure documented in section 3f. For the point of the exercise, we will take the NCEP–NCAR dataset as representing true observations of the geopotential height field.

### a. Slow patterns

Here, we follow steps (i)–(ii) of the proposed statistical prediction scheme documented in section 3f. First, we choose December 1952–August 1992 as the training period, and December 1993–August 2002 as the verification period of our predictive scheme. Before performing the analysis, the seasonal cycles of the reanalysis data in the 40-yr training period are estimated and are removed from the 50-yr NCEP–NCAR SH 500-hPa geopotential heights. This ensures that the anomaly data in the training period are completely independent of that in the verification period.

Figures 1 and 2 show the four most dominant EOFs of the slow component of NCEP–NCAR SH 500-hPa geopotential height field for summer and winter calculated using the training period. In summer, they explain 38%, 17%, 8%, and 5%, respectively, of the variance in the slow component. The potential predictability associated with these EOFs (i.e., the fraction of variance of slow component over the variance of the total value for the PC) is 0.74, 0.85, 0.79, and 0.61, respectively. In winter, the corresponding values are 29%, 17%, 10%, and 7%, respectively, of the variance, and 0.72, 0.75, 0.62, and 0.73, respectively, for the potential predictability. Also shown, in Figs. 3 and 4, are the one-season lead correlations between SST and the PCs for summer and winter, respectively, to give some indication of possible SST forcing of these slow patterns. Correlations above about 0.4, 0.3, and 0.25 are significant at the 1%, 5%, and 10% levels, respectively. It is worth noting also that the slow patterns and intraseasonal patterns (not shown) derived from this 40-yr dataset are very similar to those of Frederiksen and Zheng (2007) derived from a 52-yr dataset (see their Figs. 1 and 2).

As discussed by Frederiksen and Zheng (2007), there is some similarity between the patterns in both seasons, with the EOFs falling into four categories reflecting 1) high-latitude variability associated with the Southern Annular Mode (SAM; the first slow pattern for both summer and winter), 2) the cold event ENSO pattern [see, e.g., Fig. 8 of Frederiksen and Zheng (2007), for composite cold events for DJF and JJA] seen in the second and fourth slow pattern for summer and winter, respectively, 3) the warm event ENSO pattern (again, refer to Fig. 8 of Frederiksen and Zheng 2007) seen in the third and second slow pattern for summer and winter, respectively, and 4) variability associated with a South Pacific wave train extending from Australia into the Atlantic Ocean (the fourth summer slow pattern and the third winter slow pattern).

Frederiksen and Zheng (2007) found that the patterns for the cold and warm event composites during austral summer were very similar, suggesting a much more linear response to ENSO for this season compared to the austral winter. However, ENSO-related variability in the height field, in general, requires a combination of EOF2 and EOF3 for summer, and EOF2 and EOF4 for winter.

### b. Statistical prediction experiment

Here, we follow steps (iii)–(iv) of the proposed statistical prediction scheme documented in section 3f. The slow height patterns for the training period are those estimated in section 4a. Because the heights are known in the training period, the corresponding PCs are also known. For each PC time series, select candidate predictors based on the corresponding EOFs and the correlation map between the projection time series and one-season lagged SST field (Figs. 3 –4). The aim then is to estimate the PC time series in terms of a set of relevant predictors and to use this functional relationship to estimate the seasonal height anomalies in the verification period. Here, we estimate the relationship between the PCs and the candidate predictors using multivariate linear regression. To add a little more flexibility, we will consider both one-season lead and one-month lead values of our circulation and SST predictors. Stepwise regression is used to select those predictors that are most appropriate to train the regression coefficients. The results of the final regression formulas are listed in Table 1, and the observations and the predictions of the PCs in the verification period are shown in Figs. 5 –6.

As discussed in section 4a, the SAM is clearly related to the first slow pattern of DJF heights. Although the correlation between the first slow PC and SSTs is very significant in the tropical Indian Ocean (see Fig. 3a), this is mainly due to the positive annual linear trends in both the SAM and the SSTs (Frederiksen and Zheng 2007). Therefore, possible predictors for DJF PC1 might be the Southern Annular Mode index (SAMI; adopted from the Web site of the University of Washington; Thompson and Wallace 2000), at one-month (November) or one-season lead (SON), and the annual linear trend calculated over the training period. In this case, a stepwise regression analysis shows that the November SAMI and the linear trend are the best predictors. Since the SAMI will also have a trend, it is possible that its contribution is mainly due to the linear trend and not the tropospheric circulation over the Southern Ocean (characterized by the detrended SAMI). To check this point, the linear trend and the detrended SAMI were also tested as possible predictors for regression. Our analysis shows that the detrended SAMI is also a very statistically significant (at the 0.01% level) predictor. This is further confirmation that, in addition to the linear trend, the persistence of the tropospheric circulation over the Southern Ocean also contributes to the predictability of PC1. The predicted PC time series seems to be well correlated with that observed, but a negative bias is evident (top-left panel of Fig. 5). The bias indicates that the trend in the first slow PC estimated in the training period is smaller than the trend present during the verification period. This coincides with the results presented in Marshall (2003) that suggest that there is a steeper increase in SAM after 1990.

The correlation map between the second slow PC and one-season lead SST field (Fig. 3b) indicates the close association between PC2 and ENSO SST variability. Both one-month and one-season lead ENSO SST indices (Niño-3, Niño-4, etc.) were trialed as possible predictors. Our analysis shows that the SON Niño-3 SST (5°N–5°S, 150°–90°W) is the best predictor. The top-right panel of Fig. 5 shows how well the observed and predicted PC2 time series correspond. This is possible, because the chaotic component of the second slow PC2 is only 15%.

Figure 3c shows that the third slow DJF PC time series is highly correlated with SST variability in both the region of the Coral Sea (negatively) and the central Pacific Ocean (positively) near the date line. Again one-month and one-season SST indices over these regions were trialed. Our regression analysis showed that a Coral Sea SST index [defined as the average SST over the region 15°S–30°S, 150°E–180°, following Mullan (1998)] for November is the best predictor.

As mentioned above in section 4a, the first slow pattern of JJA heights is also associated with interannual variability of the SAM. As for DJF, we have considered the annual linear trend, the one-season lead (MAM) and one-month lead (May) SAMI as possible predictors for JJA PC1. In this case, stepwise regression selected the MAM SAMI as the best predictor rather than May.

The second slow pattern of JJA height is clearly associated with ENSO SST variability (Fig. 4b), with high positive correlations in the central Pacific and central Indian Oceans, and negative correlations in the region of the South Pacific convergence zone. A stepwise regression of one-month and one-season SST indices in these regions indicates that the May Niño-4 index (defined as the average SST over 5°N–5°S, 160°E–150°W) is the best predictor of JJA PC2. This is consistent with the fact that El Niño often changes phase in April, and therefore May Niño-4 SST might be expected to be a better predictor of the phase of ENSO in JJA.

The fourth slow JJA pattern, at one-season lag, has largest correlation with central Indian Ocean SSTs. Using the central Indian Ocean SST of Mullan (1998; defined as the average SST over the region 0°–15°S, 60°–85°E), at leads of one month and one season, our analysis shows that the MAM index is the best predictor.

Using the prediction formulas in Table 1 and Eq. (22), predictions of the SH 500-hPa geopotential height field were made for each season and year during the verification period. Tables 2 and 3 show the spatiotemporal correlation skill score for the predicted height field with the NCEP–NCAR reanalysis heights in DJF and JJA, respectively, indicating the skill derived from using 1, 2, or 3 of the predicted PC time series. For the entire SH, and using all three predicted PCs, the spatiotemporal correlation between the prediction and the observation is 0.66 and 0.56 for DJF and JJA, respectively. During DJF, using only the predicted PC1, PC2, and PC3 produces a skill of 0.38, 0.43, and 0.08, respectively, indicating that PC1 and PC2 are the dominant predictors, and PC3 adds only a small additional level of skill. This contrasts with JJA where, PC1, PC2, and PC4 provide a skill of 0.33, 0.33, and 0.44, respectively.

An interesting fact is that the winter slow EOF4 explains less variance than EOF1, but provides a higher skill score. The reason may be as follows. Slow EOFs are derived from the nonscaled height field. So the explained variance of an EOF is largely determined by the loading in the polar region and extratropics where the amplitude is larger. Since EOF1 has higher loading in higher latitudes than EOF4 does, it should explain higher variance than EOF4 does. On the other hand, the predictive skill used in this study is constructed using scaled data. So the predictive skill of a slow EOF largely depends on its loadings in the lower latitudes (where the area is larger than at higher latitudes). Since EOF4 represents the cold ENSO phase and predicts large variability in lower latitudes, it might be expected to have a higher skill score.

Marshall (2003) suggested that the SAM trend in the reanalyses is greater and more in error, relative to station observations, in JJA than in DJF. However, even when we drop the JJA SAM from the predictor set in this study, the STC skill score is still 0.52 (Table 3).

The percent of explained variance skill scores Eq. (23) are shown in Table 4. Their magnitudes and spatial patterns are similar to the STC skill scores in Tables 2 –3.

The observed and predicted 500-hPa geopotential height anomalies, for each season in the period December 1993–August 2004, are shown in Figs. 7 –10. The year-to-year pattern correlations between the predicted and the NCEP–NCAR reanalysis heights are shown in Figs. 11 –12. Pattern correlations are generally higher for DJF and during periods of strong ENSO, for example, 1997/98.

### c. Predictability of a CGCM model

In this section, we compare the skill of our statistical scheme with that from the COCA2 forecast scheme. To make the comparison a little fairer, we compare the skills in forecasting both the NCEP–NCAR and ERA-40 reanalyses. We do this because it might be argued that our scheme has the advantage in predicting the NCEP–NCAR verification data because it has been trained by the earlier period data, whereas the model has been tuned to reproduce observed variability in the atmosphere–ocean system.

To reduce the difference between the two reanalysis anomaly data, the seasonal cycles of ERA-40 data are also estimated using the ERA-40 data in the training period, that is, December 1957–August 1992, and is removed from the 45-yr ERA-40 SH 500-hPa geopotential heights. While the NCEP–NCAR data have assimilated observations, they are also highly model dependent, as are the ERA-40 data. In fact, the nonscaled spatiotemporal correlation between the 45-yr ERA-40 and NCEP–NCAR reanalysis height field is 0.85 for summer and 0.78 for winter, indicating model differences. Comparing both schemes with ERA-40 data should give neither the advantage.

Owing to the relatively small sample size, the seasonal cycle of COCA2 prediction is estimated using all the data available. Even then, the estimated COCA2 prediction anomalies are still subject to bias, but the small sample size prevents the bias being accurately corrected. Since the STC skill score is less sensitive to the bias than is the percent explained variance skill score (section 3g), we only use the STC skill score in evaluating the COCA2 prediction.

The spatiotemporal correlations between the 45-yr ERA-40 and statistical predicted height fields are listed in Table 5. Comparing the skill scores (Tables 2 –3) for predictions of the NCEP–NCAR SH reanalysis heights, it is clear that the statistical scheme is significantly more skilful. In particular, the skill score for the COCA2 predictions are 0.40 and 0.34 for DJF and JJA, respectively, compared with 0.66 and 0.56, respectively, for our scheme. For the ERA-40 data (Table 5), our scheme, while showing some reduction in skill, still displays significantly more skill (0.62 and 0.45 for DJF and JJA, respectively) compared with the model (0.44 and 0.35, respectively).

It is important to clarify here that the higher skill score of our proposed model only indicates that the statistical model predicts the two reanalysis height fields better than the CGCM. We cannot conclude that it predicts the real heights better. This is because, as mentioned above, any reanalysis data, while they have assimilated observations, are highly model dependent. The larger the differences between the CGCM used for prediction and the CGCM used in the reanalysis are, the lower the prediction skill score is likely to be. Since the statistical model is trained by NCEP–NCAR reanalysis data, we have to bear in mind that the NCEP–NCAR CGCM also pays an important role in our proposed prediction. Therefore, it is not surprising that our scheme predicts the NCEP–NCAR reanalysis heights better than the ERA-40.

Also shown in Figs. 11 and 12 are the year-to-year pattern correlations for the model and statistical forecasts of the NCEP–NCAR and ERA-40 datasets. In general, our scheme has higher correlation than the model except during the 1997/98 ENSO.

## 5. Discussion and conclusions

It is of interest to investigate the performance of our proposed prediction scheme in the Tropics (0°–22.5°S), the subtropics (17.5°–32.5°S), the extratropics (32.5°–62.5°S), and the south polar region (62.5°–90°S). We would also expect the dynamical forecast scheme to perform better at lower latitudes. For our own interest, we also investigate the performance of our forecast scheme in the Australia (0°–60°S, 100°E–180°) and New Zealand regions (25°–60°S, 140°E–160°W). The spatiotemporal correlation scores with the NCEP–NCAR dataset for each region are shown in Tables 2 –3 for DJF and JJA, respectively.

For DJF, the SAM is clearly the dominant source of predictive skill in the extratropical and polar region, while ENSO (Niño-3 SST) makes the dominant contribution to the predictability in the Tropics and subtropics, including over the Australian region and to a lesser extent over New Zealand. Coral Sea SSTs only contribute a small additional amount of predictive skill over that provided by PC1 and PC2, and are limited to the subtropics and the extratropics.

For JJA, variability related to the warm and cold ENSO phases contributes about the same to the predictability in the Tropics and subtropics. The SAM only contributes marginally to the predictability in the subtropics and extratropics. Interestingly, the largest skill is over the Australian region.

The dynamical forecast scheme has its largest skill in the Tropics and subtropics. During DJF, the skill of the model in the Tropics and the subtropics is slightly higher than our statistical scheme, and the model appears to capture the variability dominated by ENSO. In the extratropics and polar region, the model does not perform as well as our proposed statistical approach. During JJA, the model again shows highest predictive skill in the Tropics, but reduced skill when compared with DJF. The skill of our statistical scheme again is slightly higher than the model in all regions. COCA2 has reasonable skill in all regions. Generally, these conclusions also hold true for the ERA-40 heights. Generally speaking, the skill of our statistical scheme is higher than those of the model for predicting the two reanalysis heights.

Overall, our proposed methodology has resulted in a statistical forecast scheme for the reanalysis data that has a reasonably high predictive skill generally over the SH. In comparison with a single prediction of a fairly sophisticated dynamical forecast scheme, it generally has higher skill, except for the Tropics in summer where it has equivalent or slightly lower skill. Outside the Tropics, it tends to have higher predictive skill.

A major reason why the statistical scheme may be more skillful than a single dynamical prediction, is the fact that the statistically predicted height field is mainly the slow component of the observed height field, whereas the model predicted height field generally comprises a significant intraseasonal component. Since the intraseasonal component of the observed height field is largely unpredictable at the long range, this component in the model would tend to add noise to the model prediction. A conventional way to remove the intraseasonal component from dynamical prediction is ensemble forecasting.

Another feature of the statistical scheme is the longer sample used to train the relationship between the height field and its predictors. Also, statistical predictors, such as tropical SST indices, tend to combine interannual to interdecadal variability, so that statistical schemes have the possibility of capturing this variability. Any linear trends, as we found in the DJF SAM, can also be incorporated. In addition, statistical predictions tend to have less bias than the dynamical predictions. In fact, the spatial averaged bias is less than 5 m for the statistical prediction, but greater than 50 m for the single COCA2 prediction.

We fully acknowledge the importance of dynamical models in seasonal prediction. Our statistical prediction scheme is trained by the NCEP–NCAR reanalysis height field. Without dynamical models, the reanalysis height field could not have been derived. Through the use of ensembles, dynamical models can also explore the intraseasonal variability of a given year and attempt to separate the predictable signal from the chaos. This approach leads naturally to probability forecasts. In contrast, with a statistical method, one has only one case per season at one’s disposal, and therefore estimates of the noise can be derived only from the day-to-day variability. Finally, models can generally produce output at a much higher resolution and for multiple fields.

It might be expected that with improvements in models, better initialization and data assimilation, the use of ensemble forecasts, as done operationally, and changes in ozone and greenhouse gas concentration over the forecast period that the predictive skill of the dynamical models would be expected to be higher. This study does demonstrate that the variance decomposition approach we have developed is a useful tool in studying climate variability and predictability, and for constructing simple but skillful statistical forecast schemes. In future work, we plan to assess the relative merits of our statistical methodology and ensemble dynamical seasonal forecast models.

This work was supported by the New Zealand Foundation for Research, Science and Technology (Contract C01X0202). We wish to thank BMRC for funding several visiting fellowships for XZ. We also wish to thank Dr. Steve Wilson for giving us access to the model results and Drs. David Straus, Brett Mullan, James Renwick, and the two anonymous reviewers for useful comments, which have helped improve this paper.

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Prediction formulas for the slow height PCs for the training period.

Spatiotemporal correlation of the DJF heights for the verification period. The results used in the prediction experiment are bold.

Same as in Table 2, but for JJA.

Percent explained variance skill score of the proposed prediction.

Spatiotemporal correlation between the predicted heights and ERA-40 heights for 1993–2002. “Proposed” indicates the statistical prediction model proposed in this paper.