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

Ensemble Forecast of Analyses With Uncertainty Estimation

Vivien Mallet1,2, Gilles Stoltz3,4,1, Sergiy Zhuk5, Alexander Nakonechniy6

1INRIA 2CEREA, joint laboratory ENPC - EDF R&D, Université Paris-Est 3DMA, École normale supérieure, CNRS 4HEC Paris 5IBM Research, Dublin 6Taras 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 zt using one or several models, observations and errors description; and a given ensemble of forecasts x(m)

t

, possibly provided by the data assimilation system.

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 2 / 14

Ensemble Forecast of Analyses (EFA)

Notation

x(m)

t

State vector forecast by model/member m at time t zt Analysis state vector at time t

Strategy

Forecasting the analysis state vector zt computed by the data assimilation system

Rationale: The analyses are the best a posteriori knowledge of the state Aggregated forecast:

• zt,i =

M

• m=1

w (m)

t,i x(m) t,i

Success if the ensemble forecasts zt beat any sequence of forecasts x(m)

t

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 3 / 14

Ensemble Forecast of Analyses (EFA)

Notation

x(m)

t

State vector forecast by model/member m at time t zt Analysis state vector at time t

Principle

To produce an aggregated forecast zt as efficient as possible, using the linear combination:

• zt,i =

M

• m=1

w(m)

t,i x(m) t,i

t − 2 t − 1 t t + 1 x(m)

t−2

x(m)

t−1

x(m)

t

x(m)

t+1

w(m)

t−2 →

zt−2 w(m)

t−1 →

zt−1 w(m)

t

→ zt w(m)

t+1 →

zt+1 yt−2 → zt−2 yt−1 → zt−1 yt → zt yt+1 → zt+1

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 4 / 14

Ensemble Forecast of Analyses (EFA)

Notation

x(m)

t

State vector forecast by model/member m at time t zt Analysis state vector at time t

Principle

To produce an aggregated forecast zt as efficient as possible, using the linear combination:

• zt,i =

M

• m=1

w(m)

t,i x(m) t,i

t − 2 t − 1 t t + 1 x(m)

t−2

x(m)

t−1

x(m)

t

x(m)

t+1

w(m)

t−2 →

zt−2 w(m)

t−1 →

zt−1 w(m)

t

→ zt w(m)

t+1 →

zt+1 yt−2 → zt−2 yt−1 → zt−1 yt → zt yt+1 → zt+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): ∀i wt,i = argmin

v∈RM

 λv2

2 + s<t

• s=1
• β

(t − s)2 + 1 zs,i −

M

• m=1

v(m)x(m)

s,i

2 

Theoretical Comparison With the Best Linear Combination With Constant Weights

1 t

s≤t

• s=1
• zs,i −

M

• m=1

w(m)

s,i x(m) s,i

2

− argmin

v∈RM

 1

t

s≤t

• s=1
• zs,i −

M

• m=1

v(m)x(m)

s,i

2  O ln t

t

• Mallet, Stoltz, Zhuk, Nakonechniy

Ensemble Forecast of Analyses November 2012 5 / 14

EFA Using Filtering

Formulation in Terms of Filtering

State equation: w0,i = c + ei wt+1,i = Awt,i + (I − A)c + et,i “Observations” (i.e., analyses in our case): zt,i = Et,iwt,i + ηt,i Et,i =

• x(1)

t,i , . . . , x(m) 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 Pt,i for the weight error at time t The aggregated forecast has variance Et,iPtET

t,i

Minimax Filtering

Bounds on errors, described by an ellipsoid eT

i Q−1ei + T−1

• t=0

eT

t,iQ−1 t et,i + T

• t=0

A−1

t η2 t,i ≤ 1

Admissible weights are compatible with weight model, “observations” and errors bounds Weights defined such that, for any direction ℓ, sup

e,e0,...,et−1,η0,...,ηt

ℓT(wtrue

t

− wt) ≤ sup

e,e0,...,et−1,η0,...,ηt

ℓT(wtrue

t

− wt)

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

• 10
• 5

5 10 15 20 35 40 45 50 55 20 40 60 80 100 120 140 160 180

• 10
• 5

5 10 15 20 35 40 45 50 55 20 40 60 80 100 120 140 160 180

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 (µg 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 (µg m−3 ) Averaged For One Year

10 5 5 10 15 20 35 40 45 50 55 10 5 5 10 15 20 35 40 45 50 55 10 5 5 10 15 20 35 40 45 50 55 10 5 5 10 15 20 35 40 45 50 55 48 56 64 72 80 88 96 104 112

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 9 / 14

Uncertainty Map (Standard Deviation in µg m−3 )

10 5 5 10 15 20 35 40 45 50 55 9.0 10.5 12.0 13.5 15.0 16.5 18.0 19.5

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 10 / 14

Ozone in One Grid Cell (µg m−3 )

220 240 260 280 300 320 40 60 80 100 120 140 160

Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 11 / 14

Ozone in One Grid Cell (µg m−3 )

220 240 260 280 300 320 40 60 80 100 120 140 160

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

J a n M a r M a y J u l S e p N

• v

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

50 100 150 200 250 300 350 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Mallet, Stoltz, Zhuk, Nakonechniy Ensemble Forecast of Analyses November 2012 14 / 14