1. Introduction
a. The need for statistical seasonal forecasts
In the last decades, numerical weather prediction (NWP) models have been extended to seasonal forecasting by adapting atmospheric convection scales and data assimilation schemes (Wu et al. 2021; Xuan et al. 2022). The current long-range forecasting system [ECMWF’s Seasonal Forecast System 5 (SEAS5)] of the European Center for Medium-Range Weather Forecasts (ECMWF) has substantially reduced uncertainty in the ocean and sea ice states thanks to initial conditions provided by an operational ocean analysis system (OCEAN5) that combines historical Ocean Reanalysis System 5 (ORAS5) and real-time (daily) ocean analysis (ORTA5) (Zuo et al. 2017a, 2019). Uncertainty analysis for the ocean states is performed in ORAS5 using a five-member ensemble analysis of perturbations to the forcing fields and the assimilated observations, both at the surface and at the depth (Zuo et al. 2017b).
The role of the ocean as a driver of subseasonal and seasonal forecasts is widely recognized because of its coupling with the atmosphere (surface forcing fields) and its internal circulation, which is more energetic and slower than that of the atmosphere, as well as its increasing heat budget due to climate change (Wild et al. 2015; Wang et al. 2021; Marti et al. 2022). However, dynamical models still present some initialization problems due to the poor availability of deep ocean data, which are required for the model assimilation process (Sandery et al. 2020; Zaron and Elipot 2021). This limitation can be partially mitigated using statistical approaches, especially ones based on the coupled ocean–atmosphere variability and quasi-periodic schemes (see section 1b).
An opportunity to complement dynamical model outputs is granted by the statistical analysis of remote oceanic–atmospheric connections, known as teleconnection patterns, and represented by indices that are defined from spatial and temporal patterns of oceanic or atmospheric variables (Redolat et al. 2020; Liang et al. 2021). An example of the seasonal role of that coupling is given by the connection between the Madden–Julian oscillation (MJO) index and tropical anomalies in the Atlantic, African, and European regions (Hu and Guan 2018; Wills et al. 2019; Wu et al. 2021). Other findings are related to the delayed effects of the oceans, especially represented by tropical and extratropical currents. A lesser effect of the sea ice anomaly on the Atlantic multidecadal oscillation (AMO) influences the winter North Atlantic Oscillation (NAO) (Peings and Magnusdottir 2016), while the Greenland current impacts the Atlantic meridional overturning circulation (AMOC) anomalies, which are transferred in turn to the NAO index itself (Gastineau et al. 2018; Wills et al. 2019). In general terms, the large variability modes of some atmospheric patterns are affected by oceanic anomalies. This is also the case for the quasi-periodic phenomena of the Western Mediterranean Oscillation (WeMO) and the Atlantic jet stream latitude (AJSL) patterns (Redolat et al. 2019). In fact, the AMOC carries up to 25% of the global northward atmosphere–ocean heat transport in the Northern Hemisphere (Bryden and Imawaki 2001), in particular influencing large variability modes of the major atmospheric patterns. Oceanic teleconnections have also been proven as good indicators of changes in thermohaline circulation and the vertical fluxes between surface and deep ocean fluxes (Su et al. 2019; Diansky and Bagatinsky 2019).
Analysis of the self-predictability of teleconnection indices could improve dynamical models as an added value in the model output statistics. Under this framework, the variability of subtropical regions is still a challenge due to their high interannual and seasonal variability, as is the case in the Mediterranean region, which is susceptible to a high recurrence of long dry periods followed by short spells of high precipitation (Monjo et al. 2020).
b. Statistical analysis of quasi-periodic oscillations
Since the climate system is highly nonlinear and sensitive to initial conditions (i.e., a chaotic system), seasonal anomalies can be divided into a predictable part (e.g., inertial trends or cycles) and an unpredictable part (i.e., likened to noise or other uncertainty sources) that increases with the time horizon. This conceptual model is known as quasi-periodic oscillation or quasi-oscillation (Sun et al. 2015; He et al. 2021; Wang and Sobel 2022).
To model the predictable part of quasi-oscillations, several statistical approaches have been tested in meteorology and climatology. For instance, Redolat et al. (2019, 2020) applied the technique of filtering main oscillation modes from fast Fourier transform (FFT) and wavelet analysis (WA) to check the predictability of temperature and precipitation on seasonal and decadal time scales. Anshuka et al. (2019) used wavelet-based artificial neural networks (ANNs) to predict seasonal droughts in the Middle East and Asia. The wavelet method has proven to be effective for analyzing series in which there is seasonality with a large amount of noise. In this way, it improves the existing overfitting of various methods by decomposing the different frequencies forming the series. The result shows the different frequencies and whether they are statistically significant for each teleconnection index analyzed (Joo and Kim 2015).
Furthermore, the characteristics of this function allow a global view of the set of oscillations across a scalar field. Its division of regions with similar patterns of dynamic variability allows the comparison of large regions of the planet, as well as the identification of regions with temporally equivalent patterns.
For operational uses, the discrete wavelet transform (DWT) is commonly chosen for fast computation of the wavelet transform. Dyadic decomposition can be implemented by using a univariate time series {X(t)} and selecting a discrete subset of the theoretical frequency band, which consists of points proportional to 2−j, where j is an integer number. With this, both translation and scale parameters are also proportional to 2−j, and the discrete wavelet function can be expressed using a set of functions (known as child wavelets) that generates an orthonormal basis. Therefore, 2−j-signal X(t) representing finite energy can be reconstructed by the expansion in terms of these functions, similarly to the reconstruction obtained using the Fourier series of an analytic function.
Modifications of the DWT method are commonly used for time series analysis and forecasting (Box and Jenkins 2008; Percival and Walden 2013; Gordu and Nachabe 2021). For instance, Paul and Birthal (2016) applied maximal overlap DWT (MODWT) for forecasting rainfall monsoon trends in India, while Gordu and Nachabe (2021) used this approach to predict monthly groundwater recharges at multidecadal time scales. MODWT is a highly redundant, nonorthogonal transform in which the maximum overlap ensures no loss of information during the decomposition of a function and also preserves the sampled values at each level of decomposition, unlike the DWT process. In fact, MODWT is well defined by any sample size and therefore is very suitable for application in time series analysis (Paul 2015).
The resulting components of the decomposed series of MODWT are used as a basis for modeling the provided variable X(t). The model can be combined with other techniques, such as the clustering–regression hybrid approach and the autoregressive integrated moving average (ARIMA) (Castán-Lascorz et al. 2022). As a commonly used method, ARIMA is able to predict a value in response to the time series by linear combination of its own past values. It is able to seasonally accommodate and combine the following processes: 1) the autoregressive process, 2) differencing to eliminate integration, and 3) a moving average process. Thanks to the persistence or inertial prediction of the parameters fitted to the signal, ARIMA can be combined with the wavelet technique by using MODWT, and this combination is known as the wavelet–ARIMA approach (Nury et al. 2017). Therefore, the problem of deep ocean data scarcity and its effects on dynamical models can be partially relieved by statistical forecasting of teleconnection patterns.
Taking advantage of wavelet–ARIMA techniques, this paper aims to exploit the quasi-periodic anomalies represented by teleconnection indices as complementary information that tries to compensate for the lack of data on the atmosphere–ocean coupling and, in particular, the delayed effects of deep ocean currents. Thus, the combination of teleconnection pattern representatives with classical statistical methods (e.g., ARIMA and wavelet) allows incorporating 1) additional data related to the atmosphere–ocean coupling as well as to the surface and deep ocean that are poorly assimilated in dynamical models and 2) historical data from ground stations that help to reduce the bias of the model.
c. Novel ideas of this study
To support subseasonal-to-seasonal forecasting of coupled meteorological variables and teleconnection indices, this paper establishes a novel combination of 1) delayed correlations of teleconnection indices with respect to the predicted variables and 2) self-predictability of time series with wavelet–ARIMA techniques. This teleconnection + wavelet–ARIMA combination aims to improve the detection of possible ocean–atmosphere anomalies that can affect the often-limited results provided by dynamical models at large-range time scales.
The key idea is based on the fact that the climate is a nonlinear dynamical system whose evolution has two contributions: 1) high-amplitude variations like turbulent eddies far from equilibrium (playing the role of unpredictable noise) and 2) low-amplitude perturbations of variables close to equilibrium (playing the role of predictable signal). Thus, the periodic part of the large-scale atmospheric and oceanic patterns provides operational forecast windows for each teleconnection index. However, chaotic contributions can be found in several time periods and scales, determined by the action of disruptive factors of global phenomena, such as internal natural variability, tropical cyclones, sudden stratospheric warming, oceanic extreme phases linked to El Niño–Southern Oscillation (ENSO), AMO patterns, or dust storms. In light of this fact, these variables are widely recognized as an important limitation in predictability at the subseasonal scale (Kolstad et al. 2020). Thus, because of introducing the delayed but noisy contributions of these disruptive factors (represented by extreme phases of teleconnection indices), the quasi-periodic oscillations will complement the model outputs in detecting subseasonal signals (from daily assimilated variations of teleconnection patterns).
2. Study area and data
a. Climate of reference observatories
The study area is represented by 11 reference stations covering the Euro-Mediterranean region, including North Africa and central and northern Europe (Fig. 1). These observatories, collected from the Global Surface Summary of the Day (GSOD; NCEI 2022), have been selected on the basis of quality and geographical criteria and hold precipitation and 2-m temperature data for the period 1982–2021. For the quality analysis, outliers and inhomogeneity filters were based on the Kolmogorov–Smirnov (KS) test (Monjo et al. 2013). The KS filters revealed no remarkable problems since none of the significant inhomogeneities detected lie within the reference period considered, 1993–2021 (minor inhomogeneities were detected and corrected in temperature time series for the Salzburg observatory and in precipitation for Mikra, Mersa Matruh, and Constantine).
Location of selected rainfall and temperature observatories (red circles).
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
The geographic criteria were designed to ensure representation of the high climatic variability of Europe and North Africa; thus, according to the Köppen climate classification, hot desert (BWh), hot or cold semiarid (BSh/BSk), hot- (Csa) or warm-summer Mediterranean (Csb), oceanic (Cfb), continental (Dsa/Dsb), and hemiboreal (Dfb) climates are covered. This variety is a characteristic of the study area because of the great geographic extent of the sea, the main large-scale synoptic patterns, and the irregular orography, which generate a wide range of nuances in temperature and rainfall regimes.
Most observatories in North Africa and southern Europe have Mediterranean climates; for example, Rabat, Rome, and Larnaca have hot-summer Mediterranean climates with low annual temperature oscillation, distinctive of coastal areas. However, the duration of the dry season is longer in Rabat and Larnaca compared to Rome. Madrid, Constantine, and Thessaloniki are examples of the transition to steppe climates because of their proximity to desert zones or their location on the leeward side of a mountain range, although summer is drier in Madrid and Constantine than in Thessaloniki. Marsa Matruh, on the other hand, has a hot desert climate but is relatively wetter than almost all of Egypt. Moving north, we find observatories in Benson and Paris that have oceanic climates with a low annual temperature range (lower in Benson). This oceanic characteristic results from the zonal flow of westerlies that dominate these latitudes in Europe and bring humidity from subtropical regions of the Atlantic Ocean. This mild climate becomes colder as we move to inner Europe. In the example of Salzburg, due to its remoteness from the temperate effect of the ocean, it has a humid continental climate with four contrasted seasons. Helsinki is a similar case, but it is farther north, at the border of the subarctic climate, which dominates the majority of the Scandinavian Peninsula; because of this, it has colder winters than other cities at similar latitudes, such as Stockholm, Oslo, and Edinburgh.
b. Predictor selection
Initially, as a basis for our statistical forecasting, a total of 16 teleconnection indices were collected, all of which were chosen as potential predictor variables to represent the climate variability modes of the Euro-Mediterranean region. However, after statistical tests [Akaike’s information criterion (AIC)-based backward stepwise regression], no cases were sensitive to the Pacific decadal oscillation (PDO) index, so only the actual set of 15 indices used is shown in Table 1. Indices’ data were provided by several sources, such as the National Oceanic and Atmospheric Administration (NOAA) for AMO index (AMOi), Arctic Oscillation index (AOi), east Atlantic pattern index (EAi), east Atlantic/western Russia index (EAWRi), ENSO index (ENSOi), NAO index (NAOi), QBO index (QBOi), and Scandinavia pattern index (SCANDi), the University of Colorado Boulder for Northern Hemisphere ice extent index (NHIEi), the Climate Research Unit of East Anglia University for Mediterranean Oscillation index (MOi), and the University of Barcelona for WeMO index (WeMOi); the indices of AJSL index (AJSLi), global jet stream latitude-N index (GJSLi), stratospheric sudden warming index (SSWi), and ULMO index (ULMOi) were calculated using data from the NOAA (2022).
Teleconnection indices (i) selected for this study. Variables: SST, sea level pressure (SLP), geopotential height at 500-hPa level (Z500), surface wind speed (WS), and ice. Physical mechanisms over the Euro-Mediterranean region are indicated.
The choice of indices was based primarily on the region features and the linked physical mechanisms, that is, their impact on seasonal variability in the Euro-Mediterranean region according to previous works and how most of them represent the characteristic atmospheric patterns of Europe and North Africa (Redolat et al. 2019; Mathbout et al. 2020; Lü et al. 2020; Benassi et al. 2022). Secondarily, the criteria were also intended to represent a wide range of index types whose amplitude, frequency, and duration vary greatly from one another, providing a sufficiently broad base from which to examine the existence of optimal prediction windows. Finally, the ease of replicating the indices on an operational time scale was also considered.
To compare with the statistical method, the NWP model of reference in Europe was considered. Particularly, ECMWF’s SEAS5 outputs provided by Copernicus (2022) were collected for the period 1993–2021 (i.e., 1993–2016 hindcast and 2017–21 archive forecast). SEAS5 forecasts have a 6-month time horizon with a monthly resolution, and they run once a month. This work considered the forecast product of anomalies (with respect to 1993–2021) for the first 3 months of prediction. To avoid the bias effects of the SEAS5 product, all the data were unbiased for each lead time separately by using the fitting time period (training period), common to the reference observations. Finally, to analyze the lagged correlation between oceanic and atmospheric anomalies (coupling), the Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) was considered at a monthly scale in the period of 1982–2021.
3. Methodology
a. Hypotheses
Our study investigates the self-predictability of quasi-periodic oscillations found in remote connections among climate anomalies of atmospheric and oceanic patterns, which should present delayed responses in a continuous time scale. Our methodology of subseasonal-to-seasonal forecasting establishes three main hypotheses:
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On the physical causality: There exists a physical link (predictor–predictand) with delayed causality between teleconnection patterns (predictors) and surface variables, such as temperature or precipitation (predictands). The delayed connection between oceanic and atmospheric variables implies some kind of energy propagation (e.g., atmospheric “bridges”).
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On the signal/noise components: The dynamical behaviors of the remotely connected modes of climate variability (i.e., teleconnection patterns represented by indices) can be statistically modeled by quasi-periodic oscillations (quasi-oscillations); that is, their predictability ratio is statistically significant despite the level of chaoticity (low signal/noise ratio).
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On the continuous time-scale predictability: The predictability of a quasi-oscillation at a given time scale depends on the time granularity considered for smoothing or filtering noises, always with a similar magnitude to detect signal persistence or inertia. For instance, at seasonal scales, let the effective granularity (step size) be about 1 month. In this example, the “inertial signal” hypothesis is that the predictable time horizon is about 3 months ahead (three steps), predicted in accordance with the behavior in the previous 6 months (six steps), which are chosen as a training period to detect the inertial trends or oscillation phases. The effective granularity can be therefore defined by some smoothing or filtering of the real granularity (e.g., daily) to the time scale most adequate for the future time horizon of interest.
According to these assumptions, our statistical seasonal prediction model should have at least three main components: 1) the ocean–atmosphere coupling represented by correlations between surface variables with delayed teleconnections and 2) the self-predictability of the residual anomalies by trends or 3) cycles (quasi-oscillations). These requirements are detailed below.
b. TeWA: Teleconnection and self-predictability combination
To represent the predictable relation between subseasonal variability and lead-time horizon, the time scale considered here is linearly set by the accumulated days from the first day to the horizon itself. In this way, the real granularity is daily, but a moving average is considered between the first horizon day and the final lead-time horizon (target period) to obtain an effective granularity that coincides with the final horizon (e.g., first month forecast). The predictability analysis in these target periods aims to reduce high-frequency noises, and therefore, it results in smoothed subseasonal anomalies, which are easier to analyze.
This work uses a three-stage approach with three predictor components (delayed, linear, and cyclic). The first two stages consist of separate predictions, the delayed, linear, and cyclic components, while the third stage is a combination of both predictions (Fig. 2): the teleconnection-based approach (Redolat et al. 2019, 2020) and self-predictability of residuals by using wavelet–ARIMA models (Conejo et al. 2005; Joo and Kim 2015). Therefore, the total method can be considered a teleconnection + wavelet + ARIMA (TeWA) approach, which represents the three signal components (delayed, cyclic, and linear).
Scheme of the TeWA method: (a) input dataset composed by time series of teleconnection indices and the target variable, (b) teleconnection-based approach for predicting time series via the statistically significant lag-correlated indices with at least 90 days with respect to the target variable, (c) wavelet–ARIMA-based approach with a wavelet analysis of the residual signal plus an ARIMA prediction, and (d) combination of both contributions to obtain the final prediction in the target period, which coincides with the validation time period.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
In more detail, the first stage analyzes the delayed cross correlation (XCOR) between the target variable (predictand) and the teleconnection indices (predictors) with a lag between 90 days (3 months) and 200 days (about 6 months) prior to the start of the forecast target period. For each index, between 0 and 2 lagged correlation peaks of statistically significant correlation were considered (i.e., the list of candidate predictors consists of 2 × 15 members). After adding the restrictions of minimum/maximum lag allowed, the number of potential predictors is typically reduced to m = 5 ± 3 (one standard deviation) (Fig. 3).
Example of the XCOR between different lags and indices for each observatory. Only the four best XCOR values are displayed. Green dots represent positive XCOR, and red dots represent negative XCOR. The size represents how high the XCOR is.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
These m predictors were then fitted to the previous observations (i.e., training period) with a minimum length of 200 days. The selection of the final predictors was performed on the basis of AIC over the linear regression in the training period with no other performance measure on the target/validation period, thereby avoiding possible data dredging.
For instance, to forecast or hindcast N = 90 days of a surface variable using self-prediction, we identify all the peaks with at least N days of delay from the time-lagged XCOR applied for the past (the entire time series prior to the targeted N days to be predicted). From this, we select a list of m time series consisting of the best-delayed predictors and therefore the N “future days” for each of these m indices (some of them can be repeated if two lagged correlation peaks are significant). Using this set of m time series, a prediction of the surface variables is performed up to N days.
The residuals of the multilinear model outputs (fitted in the training period) are then used to fit the wavelet–ARIMA functions. These functions are built from the MODWT decomposition, in which the redundancy of the coefficients decreases the variance of wavelet-based estimates by increasing the effective degrees of freedom at each scale (Paul 2015).
The combined TeWA model was run with 12 different fitting periods (moving windows with different sizes) equally distributed between 700 and 14 000 days. This 12-member ensemble was weighted according to the Pearson correlation of the fitting simulation, resulting in a final (weighted) average ensemble that reduces the fitting noise of the prediction (Redolat et al. 2019). The predictability analysis was performed by cross validation, distinguishing between the training (700–14 000 days before) and validation (1–90 days ahead) periods. All the statistical processes were performed with R language code (Redolat and Monjo 2022). Particularly, wavelet filtering was implemented using the packages WaveletArima and TSPred with the ARIMA model (Salles et al. 2017; Paul and Samanta 2018; Salles and Ogasawara 2021).
c. Validation metrics
Normalized anomalies of daily temperature and precipitation were considered on a daily-year basis to avoid biases in the validation process. That is, for each variable V = {υij}ij, let aij be the anomaly of the day i ∈ {1, 2, … 365} of the year j ∈ {1, 2, … n}, obtained by subtracting the i-average of all set Vi = {υij}j of the n values recorded in day i of all the years, and then, normalized anomaly
These statistics were estimated for the TeWA, wavelet–ARIMA, ARIMA, and “inertial forecast” approaches at a daily scale up to H = 90 days of time horizon and compared to the reference given by the climatic forecast. Since the SEAS5 forecast has a monthly time scale, the same statistics were also estimated at a monthly time resolution for a horizon of up to H = 3 months. To compare the results, the seasonal forecast of SEAS5 was used as a reference for NWP models commonly used in Europe. It should be noted that the TeWA method uses a daily time scale, but comparisons with the SEAS5 seasonal model itself are monthly due to its intrinsic time limitation.
4. Results and discussion
a. Predictability analysis of precipitation and temperature
Based on the normalized historical analysis of the SEAS5, SMAEs were obtained with respect to the observed time series (Fig. 4 and appendix). For the first-month forward prediction of precipitation, poor values (above 1.0) are found for all seasons of the year, especially summer (JJA) and fall (SON), with SMAEs of about 1.3. On the other hand, the TeWA-based prediction shows significant SMAE results of around 0.3, oscillating between 0.28 in summer and 0.34 in winter (DJF) and spring (MAM), an improvement of up to 70% over the forecast of SEAS5 for all lead-time horizons. For the second and third forecast months, TeWA produces an SMAE very close to 1, but this is still significantly lower than that of SEAS5 (which still oscillates between approximately 1.3 and 1.6, an improvement of 30%–40%).
Comparative SMAE between SEAS5 and TeWA monthly forecasts (H = 30 days) for precipitation in the pilot observatories for each season. See the appendix for detailed values.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
For temperature as well, the results of the seasonal forecast substantially improve (50%–60%) when the TeWA method is used (Fig. 5 and appendix). This enhancement is greater during summer; SEAS5 presents its minimum error in winter, but this error is still slightly greater than that of TeWA. The skill of the second month ahead is not statistically significant compared to the climate averages. Specifically, for the second forecast month, the TeWA temperature prediction presents a SMAE only slightly lower than 0.9, and for the third forecast month, its SMAE is 1.0; meanwhile, SEAS5 always presents errors greater than 1.3.
As in Fig. 4, but for temperature.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
Spatial distribution of the error presented only small differences, with no general statistical significance, among regions and seasons. This is found under both approaches and for both variables analyzed, despite the geographical diversity of the climates. For instance, the Mediterranean coast experiences very concentrated precipitation in a few days for all the seasons, with long dry spells, in contrast to northern Europe (Cortesi et al. 2012; Monjo et al. 2020). However, only spring shows some noticeable differences in the Mediterranean coast, especially for temperature predicted by SEAS5.
Comparison with inertial and climate-based reference forecasts leads to similar results for first-month predictability. Concerning the forecast of precipitation, a curve with substantially better SMAE values than climatology is observed up to a lead time of 20–26 days (Fig. 6). Seasonally, a better prediction behavior is specially found for the spring months (MAM), more so than for the summer months (JJA).
Validation of the TeWA method for predicting subseasonal anomalies of (left) precipitation and (right) temperature according to SMAE for all the observatories in the first row and for the Pearson correlation in the second row. Boxplot horizontal line: median; box: 25th and 75th percentiles; vertical whiskers: 3 times the interquartile range (25th and 75th percentiles); and circles: outliers. Smoothed average at a daily resolution is used for the analysis.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
In terms of lead-time horizons, the results of the TeWA method show better predictability for temperature than for precipitation. An assumable prediction horizon for precipitation is the 20th–24th days, when the median SMAE starts to be clearly higher than 1.0 for the first forecast month. In the case of temperature, this does not occur until the 60th–70th days of prediction, with a smoother decay of the skill when compared to precipitation. However, the prediction window with an SMAE of less than 0.9 is about 30 days for temperature and 16 days for precipitation.
These optimum horizons of TeWA imply an improvement of seasonal prediction relative to the simple wavelet–ARIMA approach, which presents an average prediction window of 15–17 days for temperature and 10–12 for precipitation with an SMAE of less than 0.9 on average.
The higher predictability of temperature (in windows between the 15 and 30 days) is also testified by the statistically significant (p value < 0.01) Pearson correlation between observations and TeWA predictions (R = 0.34 for the 1st–15th-day averaged values and R = 0.22 for the 1st–30th-day averaged values), which is between 30% and 50% higher than for the wavelet–ARIMA forecast (R = 0.21 for the 15-day prediction window and R = 0.12 for the 30-day prediction window). Regarding total precipitation, this improvement as a result of TeWA is observed in the second forecast week (R = 0.60 for 7-day accumulation and R = 0.21 for 14-day accumulation), raising the correlation between 30% and 40% relative to the wavelet–ARIMA forecast (R = 0.48 and R = 0.13, respectively). For the second and third months of lead-time horizons, the TeWA approach’s forecast skill is only slightly better than the climatology-based prediction skill, with an improvement of up to 15% in some observatories for both temperature and precipitation.
b. Interpretation of the predictability ranges
The third starting hypothesis (on the continuous time scale predictability) is only partially proved or validated. Despite using a moving prediction window (target period) to smooth climate anomalies and therefore reduce high-frequency noise, our method cannot be directly used for any time horizon. The predictability of both precipitation and temperature with the TeWA approach is optimum for the lead-time horizons slightly shorter than the first-month forecast (particularly about 20 days), and it rapidly decays for the second-month forecast. This can be explained by the fast variability of the natural variability modes at a subseasonal time scale, which is represented by interrelated atmospheric and oceanic patterns.
Teleconnection patterns are linked to one another, especially by “atmospheric bridges” that connect remote oceanic anomalies (Alexander et al. 2002; Trinh et al. 2021; Zhang and Liang 2022). While some variability modes are considered more robust (higher correlation with statistical significance) with zero lag (i.e., in the same month), there are other patterns that propagate their effects over several months. Under this conceptual scheme, remote currents or sea surface temperature (SST) anomalies can help to explain some subseasonal-to-seasonal anomalies in the Mediterranean region; however, their effective physical–statistical link is susceptible to change with changes in the decadal cycles themselves. For instance, atmosphere–ocean coupling is seasonally modulated by changes in ocean currents generated by SST anomalies in the North Atlantic or ENSO in the Pacific. Nevertheless, oceanic connections with atmospheric variables (e.g., temperature and precipitation) at small time scales of 0–2 months are useful for seasonal forecasting. Moreover, as this study has shown, in combination with other statistical methods (wavelet, ARIMA, etc.), they reinforce the predictive skill of SEAS5 over 2 additional weeks on average. In general, any SST predominant signal in the North Atlantic is reflected in the AMO index, but it could be due to other mechanisms different from the AMO pattern. Although the main role of the AMO is to modulate the multidecadal scales, its inclusion was considered to analyze if the small internal oscillations of the AMO affect seasonal predictability.
Another remarkable coupling between ocean and atmosphere is demonstrated by the ULMO variability modes. The ULMOi is selected as a representative atmospheric predictor throughout the Mediterranean basin because it shows the highest predictive ability (statistical variance explained) without lags for annual precipitation and, in combination with the AMOi, also for the annual temperature for the whole basin (Redolat et al. 2019). Examined on a monthly scale, the connection between the AMO and the ULMO is noticeable during most months of the year (Fig. 7). The Gulf Stream SST influences the Mediterranean region quickly (within 0–2 months). Moreover, sea surface temperature anomalies of the North Atlantic show significant effects (correlation between SST and ULMO) in winter and spring. It is worth mentioning that some areas do not necessarily follow the same causality direction (from the Atlantic Ocean to the Mediterranean basin), especially the areas with 0 months of delay in response (e.g., to the west of Portugal).
The Pearson correlation between SST and ULMOi in the same month in areas with statistical significance (p value < 0.05).
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
SST correlations among teleconnection-related anomalies are consistent with previous studies. For instance, the nonstationary relationship between ENSO and NAO also influences ULMO (Redolat et al. 2019). This connection is delayed by several months, modulated by AMO (Zhang et al. 2019). Another example is the synchronization between the Gulf Stream and the Kuroshio found in February–June, as noted by Kohyama et al. (2021).
Contrary to the case of midlatitude ocean anomalies (usually inactive in air–sea interactions), tropical anomalies produce relative differences in local convection that lead to extratropical atmospheric anomalies and further ocean anomalies (e.g., ocean overturning) via the “atmospheric bridge” mechanism (Alexander et al. 2002; Zhang et al. 2019). On the other hand, AMOC is strengthened by warm subpolar temperatures, thus causing anomalous local heat fluxes from the Atlantic Ocean into the atmosphere, with a subsequent communicated warming in tropical areas (Wills et al. 2019). In this way, subpolar and tropical oceanic areas are connected not only by surface-deep currents in the ocean but also by coupled atmospheric dynamics. This insight is also confirmed by the lagged correlation of ULMOi with several areas of the North Atlantic (Fig. 8), aligned to the multidecadal (40–55 year) coherence between the Mediterranean and Subpolar Gyre SSTs (Marullo et al. 2011). However, the influence of the Northern Pacific Subpolar Gyre on the ULMOi (Fig. 6) is clearer during the cold season of the Mediterranean region (January–February), consistent with Hay et al. (2022). At the same time, SST anomalies in the Mediterranean contribute to anomalous precipitation patterns in other remote regions, such as the Sahel (Mohino et al. 2011).
Maximum lag month of oceanic areas with statistically significant (p value) correlation between retarded SST anomalies and ULMOi.
Citation: Weather and Forecasting 39, 6; 10.1175/WAF-D-23-0061.1
Despite the lower memory of atmospheric anomalies compared to the oceanic anomalies, statistically significant signals are found on large variability modes of some teleconnection patterns, such as that of the WeMO and the AJSL (Redolat et al. 2019). These energy transfers are not constant and also not (linearly) significant in most of the regions, but where and when they appear, they can be exploited to detect possible seasonal trends or patterns that can be resolved from teleconnection indices in combination with seasonal dynamical models.
In counterpoint, this study used the AIC-based stepwise algorithm to select candidate predictors because it is very easy to implement in operational scales and it is already available in the R package MASS (Venables and Ripley 2002). To reduce the loss of signal from “true predictors,” we considered the ensemble strategy as a consensus criterion. In future work, we will consider additional statistical criteria, such as maximum-likelihood estimation and bootstrap processes (Harrell 2022). Moreover, our technique will be combined with other methodologies, such as the deep learning of neural networks (Zheng et al. 2022) and ensemble oscillation correction (EnOC; Bach et al. 2021). These approaches will enable the correction of dynamical outputs and the deepening of the applications for ocean–atmosphere coupling, the oscillatory modes, and their statistical prediction.
5. Closing remarks and conclusions
This study on subseasonal forecasting in the Mediterranean region was based on three main hypotheses: 1) There exists a delayed physical link between teleconnection patterns (predictors) and weather variables (predictands); 2) teleconnection patterns can be modeled by quasi-periodic oscillations; and 3) their predictability depends on the (continuous) time granularity used.
The results proved the first and second hypotheses and partially proved the third hypothesis since the predictability of the analyzed time series rapidly decays from the second month to all longer time horizons. Consistent with the first point, the study has found significant delayed links (with a physical basis) between these results and teleconnection patterns. The atmospheric dynamics related to the ULMO modes are coupled to oceans by atmospheric bridges that remotely connect tropical and subpolar anomalous fluxes, in addition to the surface-deep oceanic streams. Concerning the second hypothesis, teleconnection indices were satisfactorily modeled by self-predictability and wavelet–ARIMA approaches, improving the results provided by reference predictions. In particular, the TeWA method showed predictability of monthly temperature and precipitation anomalies that were 70% better than the forecast of SEAS5. On a subseasonal time scale, the prediction window of accumulated anomalies (from the first day) is up to 30 days for temperature and 16 days for precipitation but is limited for longer time horizons, which partially validates the third hypothesis on the time-resolution dependency of the method.
Therefore, the combination of the teleconnection indices and wavelet–ARIMA approaches (TeWA) improves the predictability of subseasonal anomalies (at least for 15–30 days) in the Mediterranean region by more than 30% with respect to the reference forecasts used. Nevertheless, their effective physical–statistical link is not necessarily stationary for long-term subseasonal forecasting (>30 days), causing possible changes in the statistical significance or even the sign of the correlation between some teleconnection indices and the predictand variables (temperature or precipitation). To detect possible breakpoints (e.g., in correlation signs) that impact our forecasting performance, several activation/modulation mechanisms could be analyzed in further studies. For instance, activation functions can be implemented in ANN approaches and combined with the TeWA approach. Moreover, additional statistical criteria should be considered to enhance the selection of candidate predictors in the multilinear regression stage.
The main physical interpretation of the remarkable TeWA results is that the seasonal and subseasonal anomalies are dominated by the heat–humidity fluxes, which are mainly transferred from the oceans to the atmosphere and then returned to the sea surface via the atmospheric bridges. The different sensitivity signals associated with the delay and time scale (or impact duration) of each indicator are determined by 1) the extension of the oceanic anomalies in area and depth, 2) the activation or coupling mechanism that causes the propagation of these anomalies, and 3) the path usually followed by the atmospheric currents that transfer the anomalies to the Euro-Mediterranean region.
Although with limited prediction range, the improvement of TeWA over the NWP models in the Mediterranean region suggests the potential of continued efforts in developing and implementing this method, especially in regions of the globe susceptible to regular teleconnection patterns, such as tropical regions and areas of the Pacific Ocean. As NWP models improve their assimilation of initial oceanic data, the combination of statistical and dynamical approaches will provide a thrilling window of opportunity for subseasonal-to-seasonal forecasting.
Acknowledgments.
This work is partially supported by the Improving ClimAte Resilience of critical Assets (ICARIA) project, funded by the European Commission through the Horizon Europe Programme (Grant 101093806). The work is also supported by the Ministry for the Ecological Transition and the Demographic Challenge (MITECO) of the Spanish Government under the two phases of the IMpacts of climate change on wetlands Affected by GroUndwAter (IMAGUA II) project. Finally, we truly appreciate all the constructive comments and suggestions from the reviewers.
Data availability statement.
An R code about the TeWA approach and an example of test data are available on GitHub at https://github.com/robertmonjo/subseasonal. Data on the Upper-Level Mediterranean Oscillation index (ULMOi), Atlantic Jet Stream Latitude (AJSL), and Global Jet Stream Latitude-N (GJSL) are available on the server of the Climate Research Foundation (FIC) at https://www.ficlima.org/ulmo/. The other indices are available in their respective sources detailed in Table 1 and detailed in the reference section.
APPENDIX
Reference Observatories
Table A1 shows the list of reference observatories used in this study.
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