Rank Histograms – measuring the reliability of an ensemble forecast You cannot verify an ensemble forecast with a single observation. The more data you have for verification, (as is true in general for other statistical measures) the more certain you are. Rare events (low probability) require more data to verify => as do systems with many ensemble members. From Barb Brown

From Tom Hamill

Troubled Rank Histograms Slide from Matt Pocernic Ensemble # Ensemble # Counts Counts

From Tom Hamill

Example of Quantile Regression (QR) Our application Fitting T quantiles using QR conditioned on: 1)Ranked forecast ens 2)ensemble mean 3)ensemble median 4) ensemble stdev 5) Persistence R package: quantreg

T [K] Time forecastsobserved Regressor set: 1. reforecast ens 2. ens mean 3. ens stdev 4. persistence 5. LR quantile (not shown) Probability/°K Temperature [K] climatological PDF Step I: Determine climatological quantiles Step 2: For each quan, use “forward step-wise cross-validation” to iteratively select best subset Selection requirements: a) QR cost function minimum, b) Satisfy binomial distribution at 95% confidence If requirements not met, retain climatological “prior” Step 3: segregate forecasts into differing ranges of ensemble dispersion and refit models (Step 2) uniquely for each range Time forecasts T [K] I.II.III.II.I. Probability/°K Temperature [K] Forecast PDF prior posterior Final result: “sharper” posterior PDF represented by interpolated quans

Rank Probability Score for multi-categorical or continuous variables

Scatter-plot and Contingency Table Does the forecast detect correctly temperatures above 18 degrees ? Slide from Barbara Casati Brier Score y = forecasted event occurence o = observed occurrence (0 or 1) i = sample # of total n samples => Note similarity to MSE

Other post-processing approaches … 1) Bayesian Model Averaging (BMA) – Raftery et al (1997) 2) Analogue approaches – Hopson and Webster, J. Hydromet (2010) 3) Kalman Filter with analogues – Delle Monache et al (2010) 4) Quantile regression – Hopson and Hacker, MWR (under review) 5) quantile-to-quantile (quantile matching) approach – Hopson and Webster J. Hydromet (2010) … many others

Quantile Matching : another approach when matched forecasts-observation pairs are not available => useful for climate change studies 2004 Brahmaputra Catchment-averaged Forecasts -black line satellite observations -colored lines ensemble forecasts -Basic structure of catchment rainfall similar for both forecasts and observations -But large relative over-bias in forecasts ECMWF 51-member Ensemble Precipitation Forecasts compared To observations

Pmax 25th50th75th100th Pfcst Precipitation Quantile Pmax 25th50th75th100th Padj Quantile Forecast Bias Adjustment -done independently for each forecast grid (bias-correct the whole PDF, not just the median) Model Climatology CDF“Observed” Climatology CDF In practical terms … Precipitation 01m ranked forecasts Precipitation 01m ranked observations Hopson and Webster (2010)

Brahmaputra Corrected Forecasts Original Forecast Corrected Forecast => Now observed precipitation within the “ensemble bundle” Bias-corrected Precipitation Forecasts