## 1. Introduction

Since the pioneering work of Barnett and Preisendorfer (1987) and Barnston (1994), statistical seasonal forecasting primarily has been based on the relationships between sea surface temperatures (SSTs) and subsequent seasonal time averages of predictands such as near-surface temperatures. The physical basis for the predictive relationships captured by these statistical forecasting approaches is that the effects of slowly varying boundary conditions such as SSTs are subsequently felt through their influences on the large-scale atmospheric circulation (e.g., Palmer and Anderson 1994; Goddard et al. 2001).

One of the issues that must be confronted when computing seasonal temperature forecasts is the nonstationarity of the climate and, in particular, the clear warming trend that has been evident, especially over the past 40 or so years. Neglecting the effect of this nonstationarity on seasonal forecasts has led to pronounced cold biases, for both dynamical (Barnston and Mason 2011; Peng et al. 2012; Wilks and Godfrey 2002) and statistical (Wilks 2008; 2014a,b) methods. Reduction of these biases can be extremely important for some users of seasonal temperature forecasts (Wilks and Livezey 2013).

Except during strong ENSO episodes, incorporation of the warming trend into the forecast formulation appears to provide the dominant source of current seasonal predictive skill for midlatitude temperatures, improving skill scores by reducing the forecast cold bias (Livezey and Timofeyeva 2008; Peng et al. 2012). The importance of representing trend in seasonal forecasts is manifested by the weak decreases in forecast skill with increasing lead time (Livezey and Timofeyeva 2008; Wilks 2008, 2014a), because the trend estimators can be computed nearly 1 year ahead. A straightforward method for incorporating the trend information into seasonal forecasts is to add an estimate of the current, evolving climate mean to forecasts of climate anomalies that are expressed relative to a more traditional 30-yr average (Higgins et al. 2004; Livezey et al. 2007; Wilks 2014a). Wilks (2013) and Wilks and Livezey (2013) compared a variety of estimators for computing current and near-future climate means for this purpose and concluded that the most recent 15-yr averages are presently the most successful for representing the nonstationarity of U.S. seasonal temperatures.

Although SST is the boundary condition that most often provides the predictor data for seasonal forecasts, other persistent surface conditions may also contribute skill. In particular, autumn Eurasian snow cover has been found to provide skill in statistical forecasts of North American winter temperatures (Foster et al. 1983; Cohen and Fletcher 2007; Lin and Wu 2011; Brands 2013). The physical basis for this predictive skill appears to be the connection between Eurasian snow cover and the Arctic Oscillation (Cohen and Entekhabi 1999; Cohen et al. 2007; Gong et al. 2007; Cohen and Jones 2011; Smith et al. 2011), and possibly also the Pacific–North America pattern (Clark and Serreze 2000; Wu et al. 2011). The spatial pattern of this predictability may complement that of SSTs for North American seasonal temperatures (Cohen and Fletcher 2007). The utility of antecedent snow cover as a predictor for seasonal forecasts may be limited by its short data record, which begins in November of 1966. Experience with seasonal forecasting on the basis of SSTs has been that using training data (i.e., the data used to fit statistical forecasting equations, which must be distinct from the data being forecast so as to avoid spurious artificial skill) from 1895 onward substantially improves forecast skill over use of the conventional 1950-onward training period, even though the older reconstructed SST data are of lesser quality (Wilks 2008, 2014a). Statistical forecasting equations using snow-cover predictors can be trained using data from 1966 onward only, and it is not clear whether these will be sufficiently skillful to compensate for the inability to use simultaneously SST training data prior to 1966 also.

The purpose of this paper is to investigate the use of SSTs, a representation of nonstationarity in the mean, and antecedent Northern Hemisphere snow cover, alone and in combination, for statistical forecasting of North American seasonal temperatures. The data sources are described in section 2. Section 3 documents the forecasting procedures. Section 4 presents the results, and section 5 concludes the paper.

## 2. Data

The SST predictor data have been derived from the extended reconstructed ERSST.v3b dataset (Smith et al. 2008; http://www.ncdc.noaa.gov/ersst/#grid). As in Wilks (2014a), the original 2° × 2° data have been averaged onto a 4° × 4° grid to reduce the computational burden, yielding 708 SST grid boxes over the Indo-Pacific basin (13°S–55°N, 51°E–81°W). These data have been expressed as anomalies relative to the 1951–80 period. Leading principal components of the seasonal averages (running 3-month means) from January–March (JFM) 1894 through December–February 2013/14 were computed as described in section 3. The reconstructed SST data are serially complete.

Monthly Eurasian and North American areal snow-extent data (Dewey and Heim 1982; Robinson and Frei 2000) were downloaded from the Rutgers University Global Snow Laboratory (http://climate.rutgers.edu/snowcover). Again, seasonal (running 3 month) averages have been computed for November–January (NDJ) 1966/67 through NDJ 2013/14. These predictor data have been expressed as anomalies relative to the 1966–80 reference period. The nine missing months early in the record were set to zero anomaly before computing the seasonal averages.

*T*

_{g,t}is the seasonally averaged temperature for year

*t*in grid box

*g*. Predictand anomalies for the training years 1895–1925 were centered using the 1895–1924 averages.

## 3. Forecast procedures

### a. Maximum covariance analysis

The forecasts here have been computed using maximum covariance analysis (MCA), which is a linear multivariate statistical procedure that maximizes covariances between linear combinations of predictor and predictand vectors (e.g., Bretherton et al. 1992; Wilks 2011). This method is similar to the more familiar canonical correlation analysis (CCA) and the related method of redundancy analysis. Although the three methods often yield similar results, MCA has been chosen here because it has yielded more accurate forecasts for shorter training-data series (Wilks 2014a), since use of the snow-extent predictors strongly constrains the training-data sample sizes. The appendix contains the mathematical details underlying construction of probabilistic MCA forecasts.

### b. Data processing and predictor selection

Out-of-sample seasonal temperature hindcasts were computed for, and verification results were aggregated over, the period JFM 1991–FMA 2014, at lead times of 0 (the end of the initial-time predictor season occurs immediately before the beginning of the target season) through 9 months (the end of the predictor season occurs 9 months before the beginning of the target season). The construction of the hindcasts avoids any use of the target-year data or subsequent data, and therefore the protocol emulates results that could have been achieved operationally. The 1991–2014 forecast period is shorter than the 1981–2013 period used in Wilks (2014a) because the snow-extent predictor records begin only in November of 1966.

The SST data enter the predictor vector **x** through the leading principal components for the appropriate initial-time season, and these have been recomputed for each forecast year using the additional data that would have become available most recently; thus, the training period is extended by 1 year for each successively later target year. These analyses were based on covariance matrices for the entire training-data records beginning in 1894, even when used in conjunction with the shorter snow-extent series, with the underlying SST anomalies being weighted using the square roots of latitude cosines so that the variances being analyzed are scaled according to the cosines of latitudes (North et al. 1982; Wilks 2011).

The snow-extent predictors enter the predictor vector **x** as the three raw monthly anomalies within the predictor season, separately for Eurasian and North American snow cover, so that linear combinations of these anomalies in appendix Eq. (A2a) can represent both seasonal averages and also rates of change of snow cover within the season that would correlate positively with within-month trends that are based on daily data (Cohen and Jones 2011). Unlike the SST predictors, the snow-extent data are not represented as principal components relating to spatial patterns, and therefore no attempt has been made to identify possible effects of subcontinental snow-cover variations, as was done by Mote and Kutney (2012).

As noted in section 2, the predictand vector **y** consists of the leading PCs of the seasonal gridbox temperature averages, computed using anomalies centered with the most recent 30-yr averages not including the target year,

Separately for each forecast year, season, lead time, and predictor variable combination, the numbers and identities of predictor variables have been chosen using a leave-one-out cross-validation procedure over each available training period, optimizing the continuous ranked probability score (CRPS; Matheson and Winkler 1976; Wilks 2011), again as in Wilks (2014a). For forecasts that are based on SST only, up to *I* = 10 leading SST PCs and at least *J* = 3 predictand PCs were made available for the MCA calculations, enforcing the constraint 3 ≤ *J* ≤ *I* ≤ 10. For forecasts that include snow-extent predictors, the most recent, the two most recent, or all three months of the initial-time season, individually for Eurasian and North American snow extents, were made available in the cross validation, again allowing at least three predictand PCs in **y** but no more than the number *I* of predictor variables in **x**. For forecasts that involve both snow and SST predictors, the two leading SST PCs were allowed as predictors in addition to the (up to) six monthly snow-extent anomalies. That combination of numbers and identities of predictor variables and number of predictand PCs, minimizing the CRPS averaged over training years and grid boxes, was selected to be used in a given forecast. Gaussian predictive distributions were assumed, on the basis of appendix Eq. (A4) for the gridbox means and the diagonal elements of Eq. (A7) for the corresponding variances, forecasting the observed anomalies centered using

### c. Formulation and verification of probability forecasts

*t*for each grid box

*g*:andwhere Φ

^{−1}( ) denotes the quantile (inverse cumulative distribution) function for the standard Gaussian distribution, and the climatological standard deviations are also based on the most recent 30 years, not including the target year:

The forecasts are evaluated using the ranked probability skill score (RPSS), which compares the ranked probability score (RPS; Epstein 1969; Wilks 2011) in the conventional way with the RPS for the (preceding 30-yr) climatological forecast, for which

## 4. Results

The black bars in Fig. 1 show RPSS for the three-category probability forecasts constructed without adjusting for the recent climate nonstationarity [Eq. (5)], with the whiskers indicating 90% bootstrap confidence intervals. Within each of the four seasonal stratifications, each group of four bars aggregates results over the three lead-time stratifications: 0–1, 2–5, and 6–9 months. The four bars within each lead-time stratification show results for (from left to right): snow-extent predictors only, SST predictors only (with the MCA trained using data from 1966 onward so as to compare fairly with the snow-extent-based forecasts), both snow extent and SST allowed as predictors (training data from 1966 onward), and SST predictors only (training data from 1895 onward).

The RPSS results in Fig. 1 are modest overall, particularly for the winter and spring seasons, but the snow + SST forecasts at 0–1 months in spring and 2–5 months in autumn and the (1895 onward) SST-based forecasts in summer and autumn exhibit skill levels that are reasonably large in relation to the sampling variability indicated by the bootstrap confidence limits. The gray bars and 90% bootstrap confidence intervals in Fig. 1 show the corresponding forecast biases (forecast minus observed). These biases are large and uniformly negative (cold), reflecting the fact that defining the predictive distributions using the conventional 30-yr climatological means fails to capture much of the recent climate nonstationarity.

Figure 2 shows that the somewhat naive forecast formulation in Eq. (7) is a strong competitor, exhibiting RPSS that is exceeded by the MCA-based forecasts only in the 12 cases indicated by the star symbols. Note that these include the snow-only forecasts for winter and spring and the snow + SST forecasts for spring, at the 0–1-month lead times. These snow-extent-based forecasts also exhibit larger skill than do the 1895-onward SST forecasts, indicating that, in these cases at least, there is sufficient predictive information in the snow-extent data to justify their use even though the available long SST training series cannot be used with them. Other instances outperforming the CPC-15-only [Eq. (7)] RPSS include the SST-based MCA forecasts trained from 1895 onward for all lead times in summer and autumn and the snow + SST forecasts at the two longer lead times for autumn.

Spatial distributions of nonnegative RPSS for the 0–1-month lead times during the winter and spring seasons, for which the snow-extent predictors contribute skill to the forecasts, are shown in Figs. 3 and 4, respectively. Figure 3a shows generally modest but widespread positive RPSS for winter MCA forecasts that are based on the snow-extent predictors only, in a spatial pattern that is reminiscent of the correlations with snow extent and the AO index in Cohen and Fletcher (2007) and Cohen and Jones (2011), and of areas of positive skill for heating degree-days in Brands (2013). Figure 3b shows that the skill of winter MCA forecasts that are based only on the Indo-Pacific SST predictors (trained from 1966 onward) is weak and concentrated along the northwestern portion of the domain but is generally geographically complementary to the nonnegative skill distribution for the snow-only forecasts in Fig. 3a. The winter CPC-15-only forecasts [from Eq. (7)] in Fig. 3c show relatively strong skill throughout the domain. Figure 3d shows the result when the snow-extent forecasts are combined with the CPC-15 nonstationarity adjustment using Eq. (6), which yielded the best winter forecasts at these lead times (Fig. 2a) of all methods considered.

Figure 4a shows grid boxes having positive RPSS when the spring MCA forecasts are based on the snow-extent data only, which are displaced westward and exhibit generally weaker skill relative to their winter counterparts in Fig. 3a. Figure 4b shows that positive RPSS for spring forecasts that are based only on the (1966 onward) Indo-Pacific SST predictors are concentrated along the northern Pacific and Gulf coasts, again complementing the spatial pattern in Fig. 4a. In Fig. 4c the spring CPC-15-only forecasts show relatively strong skill except in the north-central and northeastern portions of the domain. Figure 4d shows the result when all of the predictors are combined, which appears to draw strength from each of the three underlying predictor sources and which yielded the best forecasts overall for spring at these lead times (Fig. 2b).

Both Eurasian and North American snow-extent data are chosen by the cross-validation predictor-selection procedure for nearly all forecasts involving the snow-extent predictors. For the overwhelming majority of cases in which appendix Eq. (A4) does not yield a zero anomaly vector, snow extents in all three of the antecedent months for both Eurasian and North American snow cover are selected as predictors, regardless of whether the SST predictors are being used also (results not shown). Figure 5 shows spatial distributions of nonnegative RPSS for forecasts using only one or the other of Eurasian or North American monthly snow extents as MCA predictors, again for the 0–1-month lead times in winter and spring. Skill for the winter forecasts using only Eurasian snow extent (Fig. 5a) is stronger than for North American (Fig. 5c) snow extent alone, with the pattern of positive skill in Fig. 5a again resembling the spatial extent of positive correlations in Cohen and Fletcher (2007) and Cohen and Jones (2011). Both are individually weak, however, with neither being as skillful as the winter forecasts using snow extents from both continents jointly (Fig. 3a). In a similar way, neither the Eurasian (Fig. 5b) nor North American (Fig. 5d) snow-extent predictors alone perform as well for spring as both do together (Fig. 4a). The Eurasian snow-extent results for spring in Fig. 5c are weaker overall than those for winter in Fig. 5a, whereas there are relatively strong results in the mountainous southwestern United States in Fig. 5d using the North American snow predictors, the physical mechanism behind which might derive from a positive correlation between late-winter snow extent and subsequent spring soil moisture (Walsh et al. 1982; Bao et al. 2011).

## 5. Conclusions

Eurasian and North American snow extents, a simple representation of recent climate nonstationarity, and the more conventional antecedent SST conditions, individually and in combination, have been investigated as statistical predictors for seasonal North American temperatures. Because of the clear warming trend during recent decades, the strongest contribution to forecast skill was derived from adding the CPC-15 statistic [Eq. (2)], which is simply the average of the previous 15 years’ predictand values, to the forecast anomalies computed using MCA. It is evident that the increased RPS skill contributed by this more responsive representation of the current climate mean derives from its strongly reducing the cold biases that are typically exhibited by seasonal forecasts and that were exhibited here by the MCA forecasts computed using the more conventional 30-yr means. Indeed, in many instances MCA forecasts involving the SST and/or snow-extent predictors did not outperform the simple mean extrapolations computed using CPC-15 together with climatological standard deviations.

That recent trends often provide the dominant source of forecast skill is consistent with the results of Livezey and Timofeyeva (2008), who noted seasonal forecast skill diminishing only slightly or not at all with increasing lead time. Using CPC-15 to represent the recent climate warming is more effective than extrapolating a simple linear trend over the period of record because the warming trajectory has accelerated beginning in the mid-1970s (Livezey et al. 2007; Wilks and Livezey 2013). More-sophisticated nonstationarity adjustments may become preferable as the current warming trend continues to unfold, and therefore this issue should be revisited periodically in the coming years (Wilks 2013; Wilks and Livezey 2013).

The data history for snow extent is relatively short but nevertheless was seen to contribute positive predictive skill for North American temperatures in winter and spring at lead times of 0–1 months. Both Eurasian and North American snow cover, and for all three months of the initial-time seasons, appeared to contribute positively to the overall skill. Best results for winter and spring at 0–1-month lead time were achieved using the snow-extent predictors, even though their inclusion precluded simultaneous use of the long (1894 onward) training history of SSTs that prior work has found to improve seasonal forecast skill relative to using only the higher-quality recent (1950 onward) SST training data (Wilks 2008, 2014a). For the summer and autumn seasons, the best forecast skill was achieved using only the longer SST training data together with the CPC-15 nonstationarity adjustment. The snow-extent predictors appeared to provide similar overall skill levels at the 2–5- and 6–9- month lead times for autumn when used together with (1966 onward) SSTs and CPC-15, however. This result should be interpreted cautiously since it is not clear what physical mechanism might provide the basis for this skill, but it is consistent with the correlation results reported in Fig. 1a of Saito and Cohen (2003).

A clear limitation of using existing snow predictors for statistical seasonal forecasts is that they extend only from November of 1966, because they are based on satellite observations. Experience with extending the training-data sample size for SST predictors (Wilks 2008, 2014a) has shown that this straightforward change to the forecast protocol can greatly increase the skill, even though the older training data are known to be of lesser quality. Since snow-extent predictors add to seasonal forecast skill in some circumstances and in ways that appear to be complementary to the more usual SST predictors, extending the hemispheric snow-cover data record farther back in time using land-based measurements (Brown and Robinson 2011) might similarly improve seasonal-forecast skill. This result would be expected both because of the additional training data for the snow extents and also because it would allow fuller simultaneous use of longer SST training series. Poor availability of instrumental Eurasian snow-cover data (e.g., Brown 2000) appears to limit the prospects for such a data extension, however.

## Acknowledgments

I thank both anonymous reviewers for comments that led to improved clarity of the presentation. This research was supported by the National Science Foundation under Grant AGS-1112200.

## APPENDIX

### Probabilistic MCA Forecasts

*I*× 1) vector of predictor variables

**x**and the (

*J*× 1) vector of predictand variables

**y**, MCA is computed using the singular-value decomposition of the (

*I*×

*J*) matrix

_{xy}of covariances among all combinations of the elements of

**x**and

**y**,with

*J*≤

*I*, where the superscript T denotes matrix or vector transpose. Here,

*I*×

*J*) matrix of left singular vectors that is used to transform the predictor variables

**x**, and

*J*×

*J*) matrix of right singular vectors used to transform the predictand variables

**y**, according toand the (

*J*×

*J*) diagonal matrix

**Ω**of singular values is not used in the forecasting procedure. Given a predictor vector

**x**

_{0}, an MCA forecast for the transformed predictand vector is computed using the

*J*simultaneous regressions:where the

*J*columns of the square regression matrix

**x**and predictands

**y**will have been centered (have zero mean), there are no “intercept” terms in the regressions of Eq. (A3). Unlike in CCA, the pairs of transformed predictor and predictand variables in the corresponding positions of the vectors

**v**and

**w**are not independent, and therefore the

*j*th column of

*j*th element of

**w**may contain nonzero elements for any of the predictor elements in

**v**. As was also done in Wilks (2014a), the regression coefficients in each column of

*J*elements of

**v**as predictors and successively removing the weakest until all are nominally significant at the 1% level. The backward-elimination procedure ensures that one of the candidate regression equations includes all available predictors.

**y**will consist of the leading

*J*principal components of North American seasonal temperatures for the target season, which are computed as the projection of the 58 gridbox temperature anomalies onto the corresponding

*J*eigenvectors in the columns of the (58 ×

*J*) matrix

_{y}. Dimensional forecast anomalies can then be recovered by reversing Eq. (A2b) and then computing the truncated PCA synthesis:Requiring

*J*≤

*I*ensures that

*J*MCA regressions are nominally significant, all elements of

*n*is the length of the training-data series, the columns of the (

*J*×

*n*) matrix

*J*training-data predictor vectors

**v**,

**v**

_{0}=

^{T}

**x**

_{0}is the specific predictor vector on which the current forecast is based, and the matrix

**of training-data errors isThe columns of the (**

*J*×

*n*) matrix

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