Optimally weighted ensemble for sub-seasonal forecasting Julien Brajard1∗ François Counillon1,2 Yiguo Wang1 Madlen Kimmritz3 1NERSC, Bergen, Norway 2Geophysical Institute, University of Bergen, Norway 3AWI, Bremerhaven, Germany Introduction 1 Seasonal forecasts are issued as the arithmetic mean (equal weights: EW) of an ensemble of simulations initialized by assimilating observations. 2 During the operational production of the forecast (considering the buﬀer-time needed to ensure timely delivery), new unused observations can be available (typically a week). Objective Deﬁning an optimal weighted average (OW) that can use independent data to enhance the skill of the forecast on a subseasonal time scale with respect to equal weighted averages (EW). Setup of the experiment • Forecasts x: The Norwegian Climate Prediction Model (NorCPM) is providing skillful seasonal forecasts (Wang et al. 2019). A 60 member ensemble reanalysis reanalysis has been recently produced by assimilating monthly sea-surface temperature (SST) and T-S proﬁle data with the ensemble Kalman ﬁlter. Retrospective hindcasts of 60 members - initialised from the reanalysis - have been produced for the period 1985 to 2010 with 4 start dates per year. • Observation y: A weekly NOOA sea-surface , optimally interpolated (Reynolds et al. 2002) after the analysis month is used to determine the weights. Estimation of the Optimal Weights (OW) For a forecast xn of index n (1 ≤n ≤N, N is the size of the ensemble) and a grid point i, the weight is determined by: wn,i ∝exp −1 2(ρi ◦d)TR−1(ρi ◦d) ! , where • d = (y −xn) • ρi is a localization vector whose elements are: ρik = f (d(i, k)/L) , ρik = 0 if d(i, k) > L • R = (λo)2Eobs, is the observation error. Eobs is the diagonal error provided with the NOAA product and λo is an inﬂation factor. Illustration with a Lorenz model The algorithm has been tested using a Lorenz 96 model with a observation at 0.2 time after the analysis time. • The forecast skill (in RMSE) is improved until a time of 1 (> 3 Lyapunov time) • The error of the OW mean at the observation time is comparable to the error of the analysis (initial time) Results The validation is performed on the sea-surface temperature by comparing with future observations. All the conﬁ- dence intervals and signiﬁcativity of the results are computed by bootstrapping with 100 samples. Figure: Global correlation space- and time-wise as a function of the lead month for the optimal weights (OW) and the equal weights (EW) hindcast. ▶The optimal weights hindcast has higher skill up to 2-3 leading months. Figure: Gobal correlation space-wise for a two monhts leading time as a function of the date for the optimal weights (OW) and the equal weights (EW) hindcast. The ﬁlled circles (resp. unﬁlled) correspond to the dates where the diﬀerences are signiﬁcant (resp. not signiﬁcant). ▶The improvement is found for the whole period. Figure: Time-wise correlation 1985-2010 at a two months leading time for the hindcast computed with equal weights (the baseline on the left panel) and with optimal weights (the center panel). The right panel shows the diﬀerence between the weighted hindcast correlation and the equal weights hindcast correlation, positive values indicating that optimal weights have improved the correlation. ▶The improvement is consistent on the whole domain but has diﬀerent magnitude depending on the region. Regional improvement Correlation at a two month leading time globally and for diﬀerent regions Region Correlation OW EW Global 0.50 0.44 ENSO 0.93 0.90 Bar 0.50 0.33 Norw 0.63 0.63 ▶Some regions with low skill using EW have better skill with the optimal weights (OW), e.g. the Barents Sea. Conclusion ▶Seamless update of the forecast between two oper- ational forecast releases with the arrival of fresh data ▶Improvement up to a 3 months lead-time ▶No need to recompute the forecast ▶Little sensitivity to the hyperparameter tuning Sensitivity Two hyperparameters need to be tuned to det