Ensemble Forecast of Analyses With Uncertainty Estimation Vivien - PowerPoint PPT Presentation

Ensemble Forecast of Analyses With Uncertainty Estimation Vivien Mallet 1 , 2 , Gilles Stoltz 3 , 4 , 1 , Sergiy Zhuk 5 , Alexander Nakonechniy 6 1 INRIA 2 CEREA, joint laboratory ENPC - EDF R&D, Université Paris-Est 3 DMA, École normale supérieure, CNRS 4 HEC Paris 5 IBM Research, Dublin 6 Taras Shevchenko National University of Kiev International Conference on Ensemble Methods in Geophysical Sciences Toulouse, November 2012 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 1 / 14

Objective To produce the best forecast of a model state using a data assimilation system, which produces analysis state vectors z t using one or several models, observations and errors description; and a given ensemble of forecasts x ( m ) , possibly provided by the data t assimilation system. Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 2 / 14

Ensemble Forecast of Analyses (EFA) Notation x ( m ) State vector forecast by model/member m at time t t Analysis state vector at time t z t Strategy Forecasting the analysis state vector z t computed by the data assimilation system Rationale: The analyses are the best a posteriori knowledge of the state Aggregated forecast: � M w ( m ) t , i x ( m ) z t , i = � t , i m = 1 z t beat any sequence of forecasts x ( m ) Success if the ensemble forecasts � t Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 3 / 14

Ensemble Forecast of Analyses (EFA) Notation x ( m ) State vector forecast by model/member m at time t t Analysis state vector at time t z t Principle To produce an aggregated forecast � z t as efficient as possible, using the linear combination: M � w ( m ) t , i x ( m ) z t , i = � t , i m = 1 t − 2 t − 1 t + 1 t x ( m ) x ( m ) x ( m ) x ( m ) t − 2 t − 1 t t + 1 w ( m ) w ( m ) w ( m ) w ( m ) t − 2 → � t − 1 → � → � t + 1 → � z t − 2 z t − 1 z t z t + 1 t y t − 2 → z t − 2 y t − 1 → z t − 1 y t → z t y t + 1 → z t + 1 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 4 / 14

Ensemble Forecast of Analyses (EFA) Notation x ( m ) State vector forecast by model/member m at time t t Analysis state vector at time t z t Principle To produce an aggregated forecast � z t as efficient as possible, using the linear combination: M � w ( m ) t , i x ( m ) z t , i = � t , i m = 1 t − 2 t − 1 t + 1 t x ( m ) x ( m ) x ( m ) x ( m ) t − 2 t − 1 t t + 1 w ( m ) w ( m ) w ( m ) w ( m ) t − 2 → � t − 1 → � → � t + 1 → � z t − 2 z t − 1 z t z t + 1 t y t − 2 → z t − 2 y t − 1 → z t − 1 y t → z t y t + 1 → z t + 1 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 4 / 14

EFA Using Machine Learning Computing the Aggregation Weights Ridge regression with discount in time ( λ > 0 and β > 0):  � 2  � � � s < t M � � β v ( m ) x ( m )  λ � v � 2 w t , i = argmin ( t − s ) 2 + 1  ∀ i 2 + z s , i − s , i v ∈ R M s = 1 m = 1 Theoretical Comparison With the Best Linear Combination With Constant Weights � � 2 s ≤ t M � � 1 w ( m ) s , i x ( m ) z s , i − s , i t s = 1 m = 1  � 2  � � ln t � s ≤ t M  1 � � v ( m ) x ( m ) − argmin  � O z s , i − s , i t t v ∈ R M s = 1 m = 1 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 5 / 14

EFA Using Filtering Formulation in Terms of Filtering State equation: w 0 , i = c + e i w t + 1 , i = Aw t , i + ( I − A ) c + e t , i “Observations” (i.e., analyses in our case): z t , i = E t , i w t , i + η t , i � � x ( 1 ) t , i , . . . , x ( m ) E t , i = t , i Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 6 / 14

EFA Using Filtering Kalman Filtering Assignment of variances to initial weight errors, (weight) model errors and analyses errors The filter computes a variance P t , i for the weight error at time t The aggregated forecast has variance E t , i P t E T t , i Minimax Filtering Bounds on errors, described by an ellipsoid T − 1 T � � i Q − 1 e i + t , i Q − 1 A − 1 t η 2 e T e T t , i ≤ 1 t e t , i + t = 0 t = 0 Admissible weights are compatible with weight model, “observations” and errors bounds Weights defined such that, for any direction ℓ , sup ℓ T ( w true sup ℓ T ( w true − � w t ) ≤ − w t ) t t e , e 0 ,..., e t − 1 , η 0 ,..., η t e , e 0 ,..., e t − 1 , η 0 ,..., η t Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 7 / 14

Application to Air Quality Forecast Simulations Description Forecasting ground-level ozone at 15h00 UTC (peak) over Europe Ensemble with 20 members One reference member in the ensemble benefits from data assimilation and actually provides the analyses 180 180 55 55 160 160 140 140 50 50 120 120 100 100 45 45 80 80 60 60 40 40 40 40 35 35 20 20 10 5 0 5 10 15 20 10 5 0 5 10 15 20 � � � � Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 8 / 14

Application to Air Quality Forecast Simulations Description Forecasting ground-level ozone at 15h00 UTC (peak) over Europe Ensemble with 20 members One reference member in the ensemble benefits from data assimilation and actually provides the analyses RMSE ( µ g m − 3 ) , With Respect to Analyses and Observations Analyses Observations Reference model without assimilation 15.8 21.6 Reference model with assimilation 13.5 19.8 EFA with machine learning 11.3 15.6 EFA with filtering 10.9 15.7 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 8 / 14

Ozone Maps ( µ g m − 3 ) Averaged For One Year 55 55 112 50 50 104 45 45 96 40 40 88 35 35 10 5 0 5 10 15 20 10 5 0 5 10 15 20 80 55 55 72 50 50 64 45 45 56 40 40 35 35 48 10 5 0 5 10 15 20 10 5 0 5 10 15 20 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 9 / 14

Uncertainty Map (Standard Deviation in µ g m − 3 ) 19.5 18.0 55 16.5 50 15.0 45 13.5 12.0 40 10.5 35 9.0 10 5 0 5 10 15 20 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 10 / 14

Ozone in One Grid Cell ( µ g m − 3 ) 160 140 120 100 80 60 40 220 240 260 280 300 320 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 11 / 14

Ozone in One Grid Cell ( µ g m − 3 ) 160 140 120 100 80 60 40 220 240 260 280 300 320 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 12 / 14

Conclusions Ensemble Forecast of Analyses With machine learning: guaranteed to beat any linear combination with constant weights With filtering: access to uncertainty quantification Some Perspectives Machine learning with robust uncertainty quantification Some focus on aggregation of fields (with patterns) Ensemble forecast of analyses: Coupling data assimilation and sequential aggregation. Mallet, JGR, 2010. Ozone ensemble forecast with machine learning algorithms. Mallet, Stoltz & Mauricette, JGR, 2009. Air quality simulations with Polyphemus, http://cerea.enpc.fr/polyphemus/ Algorithms from data assimilation library Verdandi, http://verdandi.gforge.inria.fr/ Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 13 / 14

Time Evolution of the Weights Machine Learning and Filtering 1.4 2.0 1.2 1.5 1.0 1.0 0.8 0.5 0.6 0.0 0.4 0.5 0.2 1.0 0.0 1.5 0.2 2.0 n r y l p v a a u e o 0.4 a M J J M S N 0 50 100 150 200 250 300 350 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 14 / 14