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**Calibration of EPS’s Renate Hagedorn**

European Centre for Medium-Range Weather Forecasts

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Outline Motivation Methods Training data sets Results

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Motivation EPS forecasts are subject to forecast bias and dispersion errors, i.e. uncalibrated The goal of calibration is to correct for such known model deficiencies, i.e. to construct predictions with statistical properties similar to the observations A number of statistical methods exist for post-processing ensembles Calibration needs a record of prediction-observation pairs Calibration is particularly successful at station locations with long historical data record (-> downscaling)

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**Calibration methods for EPS’s**

Bias correction Multiple implementation of deterministic MOS Ensemble dressing Bayesian model averaging Non-homogenous Gaussian regression Logistic regression Analog method

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Bias correction As a simple first order calibration a bias correction can be applied: This correction factor is applied to each ensemble member, i.e. spread is not affected Particularly useful/successful at locations with features not resolved by model and causing significant bias with: ei = ensemble mean of the ith forecast oi = value of ith observation N = number of observation-forecast pairs

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Bias correction OBS DET EPS

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**Multiple implementation of det. MOS**

A possible approach for calibrating ensemble predictions is to simply correct each individual ensemble member according to its deterministic model output statistic (MOS) BUT: this approach is conceptually inappropriate since for longer lead-times the MOS tends to correct towards climatology all ensemble members tend towards climatology with longer lead-times decreased spread with longer lead-times in contradiction to increasing uncertainty with increasing lead-times Experimental product at but no objective verification yet…

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Ensemble dressing Define a probability distribution around each ensemble member (“dressing”) A number of methods exist to find appropriate dressing kernel (“best- member” dressing, “error” dressing, “second moment constraint” dressing, etc.) Average the resulting nens distributions to obtain final pdf

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Ensemble Dressing (Gaussian) ensemble dressing calculates the forecast probability for the quantiles q as: key parameter is the standard deviation of the Gaussian dressing kernel with: Φ = CDF of standard Gaussian distribution xi = bias-corrected ensemble-member ~ error variance of the ensemble-mean FC average of the ensemble variances over the training data

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**Bayesian Model Averaging**

BMA closely linked to ensemble dressing Differences: dressing kernels do not need to be the same for all ensemble members different estimation method for kernels Useful for giving different ensemble members (models) different weights: Estimation of weights and kernels simultaneously via maximum likelihood, i.e. maximizing the log-likelihood function: with: w1 + we (nens - 1) = 1 g1, ge = Gaussian PDF’s

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**BMA: example OBS 90% prediction interval of BMA single model**

ensemble members OBS Ref: Raftery et al., 2005, MWR

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**BMA: recovered ensemble members**

100 equally likely values drawn from BMA PDF OBS single model ensemble members Ref: Raftery et al., 2005, MWR

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**Non-homogenous Gaussian regression**

In order to account for existing spread-skill relationships we model the variance of the error term as a function of the ensemble spread sens: • The parameter a,b,c,d are fit iteratively by minimizing the CRPS of the training data set • Interpretation of parameters: bias & general performance of ens-mean are reflected in a and b large spread-skill relationship: c ≈ 0.0, d ≈ 1.0 small spread-skill relationship: d ≈ 0.0 • Calibration provides mean and spread of Gaussian distribution (called non-homogenous since variances of regression errors not the same for all values of the predictor, i.e. non-homogenous)

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Logistic regression Logistic regression is a statistical regression model for Bernoulli- distributed dependent variables • P is bound by 0,1 and produces an s-shaped prediction curve steepness of curve (β1) increases with decreasing spread, leading to sharper forecasts (more frequent use of extreme probabilities) parameter β0 corrects for bias, i.e. shifts the s-shaped curve

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**How does logistic regression work?**

GP: 51N, 9E, Date: , Lead: 96h + training data 100 cases (EM) height of obs y/n + test data (51 members) height of raw prob calibrated prob event observed yes/no (0/1) event threshold

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**LR-Probability worse! + training data + test data**

GP: 51N, 9E, Date: , Lead: 168h + training data 100 cases (EM) height of obs y/n + test data (51 members) height of raw prob calibrated prob event observed yes/no (0/1) event threshold

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**LR-Probability better!**

GP: 15.5S, 149.5W, Date: , Lead: 168h + training data 100 cases (EM) height of obs y/n + test data (51 members) height of raw prob calibrated prob event observed yes/no (0/1) event threshold

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Analog method Full analog theory assumes a nearly infinite training sample Justified under simplifying assumptions: Search only for local analogs Match the ensemble-mean fields Consider only one model forecast variable in selecting analogs General procedure: Take the ensemble mean of the forecast to be calibrated and find the nens closest forecasts to this in the training dataset Take the corresponding observations to these nens re-forecasts and form a new calibrated ensemble Construct probability forecasts from this analog ensemble

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**Analog method Forecast to be calibrated Closest re-forecasts**

Corresponding obs Probabilities of analog-ens Verifying observation Ref: Hamill & Whitaker, 2006, MWR

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Training datasets All calibration methods need a training dataset, containing a number of forecast-observation pairs from the past The more training cases the better The model version used to produce the training dataset should be as close as possible to the operational model version For research applications often only one dataset is used to develop and test the calibration method. In this case cross-validation has to be applied. For operational applications one can use: Operational available forecasts from e.g. past days Data from a re-forecast dataset covering a larger number of past forecast dates / years

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**“Perfect” Reforecast Data Set**

2009 Feb March Apr 27 28 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 30 31 1971 1972 1973 1974 1975 . 1996 1997 1998 1999 2000

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**Early motivating results from Hamill et al., 2004**

Raw ensemble Bias corrected with refc data Achieved with “perfect” reforecast system! Bias corrected with 45-d data LR-calibrated ensemble

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**The 32-day unified VAREPS**

Unified VarEPS/Monthly system enables the production of unified reforecast data set, to be used by: EFI model climate 10-15 day EPS calibration Monthly forecasts anomalies and verification Efficient use of resources (computational and operational) “Perfect” reforecast system would produce for every forecast a substantial number of years of reforecast “Realistic” reforecast system has to be an optimal compromise between affordability and needs of all three applications

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**Unified VarEPS/Monthly Reforecasts**

2009 Feb March Apr 27 28 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 30 31 1991 1992 1993 1994 1995 . 2004 2005 2006 2007 2008

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**Unified VarEPS/Monthly Reforecasts**

2009 Feb March Apr 27 28 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 30 31 1991 1992 1993 1994 1995 . 2004 2005 2006 2007 2008

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**Calibration of medium-range forecasts**

A limited set of reforecasts has been produced for a preliminary assessment of the value of reforecasts for calibrating the medium-range EPS Test Reforecast data set consists of: 14 refc cases (01/09/2005 – 01/12/2005) 20 refc years (1982 – 2001) 15 refc ensemble members (1 ctrl pert.) Model cycle 29r2, T255, ERA-40 initial conditions Used to calibrate the period Sep-Nov 2005 (91 cases) Calibrating upper air model fields vs. analysis demonstrated less scope for calibration Greater impact for surface variables, in particular at station locations

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Main messages ECMWF forecasts, though better than GFS forecasts, can be improved through calibration Main improvement through bias correction (60-80%), but when using advanced methods (e.g. NGR) calibration of spread adds to general improvements, in particular at early lead times Improvements occur mainly at locations with low skill Operational training data can be used for short-lead forecasts and/or light precipitation events, however, reforecasts beneficial at long leads and/or extremer precipitation events Usually, near-surface multi-model forecasts are better than single model forecasts, however, reforecast calibrated ECMWF forecasts are competitive for short lead times and even better than the MM for longer lead times

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I. ECMWF vs. GFS Hagedorn et al., 2008 Results from cross-validated reforecast data: 14 weekly start dates, Sep-Dec (280 cases)

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Main messages ECMWF forecasts, though better than GFS forecasts, can be improved through calibration Main improvement through bias correction (60-80%), but when using advanced methods (e.g. NGR) calibration of spread adds to general improvements, in particular at early lead times Improvements occur mainly at locations with low skill Operational training data can be used for short-lead forecasts and/or light precipitation events, however, reforecasts beneficial at long leads and/or extremer precipitation events Usually, near-surface multi-model forecasts are better than single model forecasts, however, reforecast calibrated ECMWF forecasts are competitive for short lead times and even better than the MM for longer lead times

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**II. Bias-correction vs. NGR-calibration**

2m temperature forecasts (1 Sep – 30 Nov 2005), 250 European stations REFC-data: 15 members, 20 years, 5 weeks CRPSS BC-spread = DMO-spread NGR-spread RMSE & spread NGR-calibration Bias-correction DMO

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Main messages ECMWF forecasts, though better than GFS forecasts, can be improved through calibration Main improvement through bias correction (60-80%), but when using advanced methods (e.g. NGR) calibration of spread adds to general improvements, in particular at early lead times Improvements occur mainly at locations with low skill Operational training data can be used for short-lead forecasts and/or light precipitation events, however, reforecasts beneficial at long leads and/or extremer precipitation events Usually, near-surface multi-model forecasts are better than single model forecasts, however, reforecast calibrated ECMWF forecasts are competitive for short lead times and even better than the MM for longer lead times

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**III. Individual locations**

CRPSS: 2m-Temperature, Sep-Nov 2005, Lead: 48h NGR - DMO DMO NGR

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Main messages ECMWF forecasts, though better than GFS forecasts, can be improved through calibration Main improvement through bias correction (60-80%), but when using advanced methods (e.g. NGR) calibration of spread adds to general improvements, in particular at early lead times Improvements occur mainly at locations with low skill Operational training data can be used for short-lead forecasts and/or light precipitation events, however, reforecasts beneficial at long leads and/or extremer precipitation events Usually, near-surface multi-model forecasts are better than single model forecasts, however, reforecast calibrated ECMWF forecasts are competitive for short lead times and even better than the MM for longer lead times

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**IV. Operational training data vs REFC data**

ECMWF GFS Hagedorn et al., 2008 Operational training data can give similar benefit for short lead times REFC data much more beneficial for longer lead times

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**IV. Operational training data vs REFC data**

NGR calibrated Bias correction only 20y REFC training data 45-day operational data DMO 20y REFC training data 45-day operational data DMO NGR calibration method particularly sensitive to available training data

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**IV. REFC beneficial for extreme precip**

Precipitation > 1mm Precipitation > 10mm Hamill et al., 2008 REFC data much more beneficial for extreme precipitation events Daily reforecast data not more beneficial than weekly REFC

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Main messages ECMWF forecasts, though better than GFS forecasts, can be improved through calibration Main improvement through bias correction (60-80%), but when using advanced methods (e.g. NGR) calibration of spread adds to general improvements, in particular at early lead times Improvements occur mainly at locations with low skill Operational training data can be used for short-lead forecasts and/or light precipitation events, however, reforecasts beneficial at long leads and/or extremer precipitation events Usually, near-surface multi-model forecasts are better than single model forecasts, however, reforecast calibrated ECMWF forecasts are competitive for short lead times and even better than the MM for longer lead times

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**V. TIGGE multi-model Multi-Model ECMWF Met Office NCEP Solid: no BC**

T-2m, 250 European stations – (60 cases) Multi-Model ECMWF Met Office NCEP Solid: no BC

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**V. TIGGE multi-model Multi-Model ECMWF Met Office NCEP Dotted: 30d-BC**

T-2m, 250 European stations – (60 cases) Multi-Model ECMWF Met Office NCEP Dotted: 30d-BC Solid: no BC

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**V. TIGGE multi-model Multi-Model ECMWF Met Office NCEP**

T-2m, 250 European stations – (60 cases) Multi-Model ECMWF Met Office NCEP Dashed: REFC-NGR Dotted: 30d-BC Solid: no BC

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Summary The goal of calibration is to correct for known model deficiencies A number of statistical methods exist to post-process ensembles Every methods has its own strengths and weaknesses Analog methods seems to be useful when large training dataset available Logistic regression can be helpful for extreme events not seen so far in training dataset NGR method useful when strong spread-skill relationship exists, but relative expensive in computational time Greatest improvements can be achieved on local station level Bias correction constitutes a large contribution for all calibration methods ECMWF reforecasts very valuable training dataset for calibration

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**References and further reading**

Gneiting, T. et al, 2005: Calibrated Probabilistic Forecasting Using Ensemble Model Output Statistics and Minimum CRPS Estimation. Monthly Weather Review, 133, Hagedorn, R, T. M. Hamill, and J. S. Whitaker, 2008: Probabilistic forecast calibration using ECMWF and GFS ensemble forecasts. Part I: 2-meter temperature. Monthly Weather Review, 136, Hamill T.M. et al., 2004: Ensemble Reforcasting: Improving Medium-Range Forecast Skill Using Retrospective Forecasts. Monthly Weather Review, 132, Hamill, T.M. and J.S. Whitaker, 2006: Probabilistic Quantitative Precipitation Forecasts Based on Reforecast Analogs: Theory and Application. Monthly Weather Review, 134, Hamill, T. M., R. Hagedorn, and J. S. Whitaker, 2008: Probabilistic forecast calibration using ECMWF and GFS ensemble forecasts. Part II: precipitation. Monthly Weather Review, 136, Raftery, A.E. et al., 2005: Using Bayesian Model Averaging to Calibrate Forecast Ensembles. Monthly Weather Review, 133, Wilks, D. S., 2006: Comparison of Ensemble-MOS Methods in the Lorenz ’96 Setting. Meteorological Applications, 13,

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