Attribution of climatic events using data assimilation

Transcription

1 Attribution of climatic events using data assimilation Alberto Carrassi Nansen Environmental and Remote Sensing Center - Norway University of Maryland College Park 24 th April 2018 A. Carrassi University of Maryland. USA, 24 th April / 33

2 The original goal: the attribution problem Using data assimilation as a tool to diagnose and evidence causal links in weather/climate events. Move toward an operational real-time event attribution system. Work main team: M. Bocquet (CEREA-ENPC, France), M. Ghil (ENS/UCLA, France/USA) and A. Hannart (OURANOS, Canada), J. Ruiz (Un. of Buenos Aires, Argentina), S. Metref (Un. of Buenos Aires, Argentina) Funding partly provided by the ANR project DADA ( ): Data Assimilation for Detection and Attribution of climate change A. Carrassi University of Maryland. USA, 24 th April / 33

3 Outline 1 Data assimilation in a nutshell 2 The attribution problem in climate science 3 Model evidence: a key quantity Definition and interpretation Contextual model evidence Computing model evidence using data assimilation 4 Model evidence: Numerical results Setup Accuracy of the DA-based estimates Low dimensional models - Application for attribution Model evidence with localization Application for model selection Application for the attribution problem (high-dim) 5 Conclusion and future directions A. Carrassi University of Maryland. USA, 24 th April / 33

4 Data assimilation in a nutshell Data assimilation in a nutshell environmental system Filtering observations forecast analysis The problem is in principle solved using a probabilistic framework The quantity of interest are the probability density functions, PDF The PDFs are evolved in time and updated at analysis times using Bayes s rules p(x y) = p(y x)p(x) p(y) model truth From Carrassi et al, t k t k+1 t k+2 time The huge dimensions of typical models and datasets hamper the use of a fully Bayesian approach in the geosciences. The Gaussian approximation is at the core of many successful DA procedures. A. Carrassi University of Maryland. USA, 24 th April / 33

5 Data assimilation in a nutshell Data assimilation in a nutshell A. Carrassi University of Maryland. USA, 24 th April / 33

6 The attribution problem in climate science The attribution problem in climate science A. Carrassi University of Maryland. USA, 24 th April / 33

7 The attribution problem in climate science The attribution problem: trends or events A. Carrassi University of Maryland. USA, 24 th April / 33

8 The attribution problem in climate science The attribution of a climatic event Do we identify a discernible human influence on a single weather or climate-related event? Implications for mitigation and adaptation policy, litigation in courts, insurance, physical understanding of the event. Define the event based on threshold exceedance of an ad-hoc climate index Derive the likelihood of the event in two model ensembles: factual world (e.g. including CO2 emission) = p 1 counterfactual world (e.g. without CO2 emission) = p 0 Derive the fraction of attributable risk (FAR) F AR = 1 p 0 p 1 A. Carrassi University of Maryland. USA, 24 th April / 33

9 The attribution problem in climate science The conventional method August 2003 European heatwave (Stott et al., 2004) They conclude: "It is very likely (> 90%) that CO2 emissions have increased the frequency of occurrence of 2003-like heatwaves by a factor of at least two." Issues with the conventional approach: It requires to run a large ensemble of model simulations, plus extrapolation using EVT for both the factual and counter-factual (very costly). Long delays in producing assessment after an event occurrence. A. Carrassi University of Maryland. USA, 24 th April / 33

10 The attribution problem in climate science The new near real-time event attribution The challenge is to design a system/procedure that allows for near real-time event attribution "The overarching challenge for the community is to move beyond research- mode case studies and to develop systems that can deliver regular, reliable and timely assessments in the aftermath of notable weather and climate- related events, typically in the weeks or months following (and not many years later as is the case with some research-mode)." Stott et al. (2013) Based on data assimilation, our approach addresses this request by relying upon meteo/climate service infrastructures (a routine for prediction and DA already in place). A. Carrassi University of Maryland. USA, 24 th April / 33

11 The attribution problem in climate science The DADA approach: justification We want to compute the FAR in the aftermath of an event of interest: F AR = 1 p0 p 1 The likelihoods p 0 /p 1 can be obtained as a by product of the data assimilation. Recognize that the likelihoods of interest are the normalization factors in the Bayesian updates. DA state update p(x y) = p(y x)p(x) p(y) = p(y) = p(y x)p(x) p(x y) They correspond to the marginal likelihood of the data and are referred to as model evidence in statistical literature. Questions: 1 Can we use DA to estimate them and in the aftermath of a climate extreme? 2 Would this be efficient? 3 How would this compare with the conventional approach? A. Carrassi University of Maryland. USA, 24 th April / 33

12 Model evidence: a key quantity Definition and interpretation Model evidence Definition & Interpretation Model evidence p(y K: M) = p(y K: ) = p(y K, y K 1,..., y ) Likelihood of (the infinite from the far past) observation sequence p(y K: ) = dx p(y K: x)p(x) It is the marginal likelihood of the data. p(x) plays the role of the prior. It depends on the underlying dynamics (the model) but it can be formulated as being dependent on any hypothesis under scrutiny. If the dynamics is ergodic and autonomous = p(x) is the invariant distribution on its attractor. p(y K: ) reflects this invariant distribution at a level of accuracy dependent on that of the observation model. We define p(y K: ) as the climatological model evidence. A. Carrassi University of Maryland. USA, 24 th April / 33

13 Model evidence: a key quantity Definition and interpretation Model evidence Applications General metric for model selection and comparison (see e.g. Metref et al., 2018). It can be used to compare the skill of several candidate models, model settings, or boundary conditions, in representing a given observed phenomenon (see e.g. Carson et al., 2018). Calibrating the state-space model s parameters (see e.g. Pulido et al., 2018). Bayesian inference (see e.g. Elsheikh et al., 2014). Quantifying the evidence supporting several theoretical hypothesis associated to a phenomena. Discriminate sources (i.e. nuclear release - Winiarek et al., 2011) A. Carrassi University of Maryland. USA, 24 th April / 33

14 Model evidence: a key quantity Definition and interpretation Statistical/climatological model evidence - Properties and issues The climatological model evidence embeds global information about the system of interest. Nevertheless: 1 It is very difficult to compute in large dimensional systems (it involves solving an integral over the state vector). 2 Its climatological character is not informative about the system s specific current conditions. 3 If the system is driven a time-dependent forcing (i.e. anthropogenic changes in greenhouse gases) its dynamics is non-autonomous the existence of an invariant p(x) is not guaranteed. A. Carrassi University of Maryland. USA, 24 th April / 33

15 Model evidence: a key quantity Contextual model evidence From climatological to contextual model evidence Proposal: move from a global to a context-dependent formulation of the likelihood p(y K: ) p(y K:1 y 0: ) and p(y K:1 y 0: ) is defined as the contextual model evidence. y 0: is used to specify the context. The informational content from past observations is propagated forward by conditioning on y 0: y K:1 is used for evidence diagnostic. The interval t K t 0 defines the evidencing window. A. Carrassi University of Maryland. USA, 24 th April / 33

16 Model evidence: a key quantity Contextual model evidence Contextual model evidence It narrows the climate perspective to the context at t 0 p(y K:1 y 0: ) = dx 0 p(y K:1 x 0 )p(x 0 y 0: ) We will show that contextual model evidence p(y K:1 y 0: ) can be estimated using DA To this purpose note that: The conditional pdf p(x 0 y 0: ) plays the role of the prior density and substitutes p(x) of the climatological model evidence. p(x 0 y 0: ) is the natural (albeit approximate) outcome of the DA machinery. The contextual model evidence is the product of the individual contextual model evidences p(y K:1 y 0: ) = K p(y k y k 1: ) k=1 A. Carrassi University of Maryland. USA, 24 th April / 33

17 Model evidence: a key quantity Computing model evidence using data assimilation Computing model evidence using data assimilation How it works Schema'c diagram of a forecast- - assimila'on cycle and contextual model evidence (CME) computa'on K- long evidencing window True/Correct model M 1 Incorrect model M 0 observa8ons EnKF- based Trajectory 1 Suppose DA (e.g. EnKF, 4DVar,...) is routinely used in a meteo/climate center using the state-of-the-art environmental model (factual model). 2 If a relevant climate event occurs then use the DA to compute the factual likelihood p 1. This comes without additional cost; it is automatically computable using the innovations and the posterior PDF obtained by the DA procedure. 3 Repeat the DA process over a specified evidencing window that encompasses the event of interest but using the counterfactual model (e.g. without the CO2 forcing). 4 This allows to compute the counterfactual likelihood p 0. 5 Use p 1 and p 0 to compute the fraction of attributable risk. A. Carrassi University of Maryland. USA, 24 th April / 33

18 Model evidence: a key quantity Computing model evidence using data assimilation How to compute contextual model evidence Method 1. Importance sampling Compute the model evidence over the window [t 0, t K ] via Importance Sampling: 1 sample N members, x i o, from p(x 0 y 0: ) and forecast from t 0 to t K 2 weight each member trajectory x i K:0 according to their likelihood p(y K:1 x i K:0 ), so as to obtain N p(y K:1 y 0: ) p(y K:1 x i 0) 1 N Importance sampling (IS): use an ensemble-da method up to t 0 = the assumption N M (i.e. ensemble size much smaller than model dimension) can be used in steps (1) and (2) Evaluation of model evidence using a particle filter proposed by Reich and Cotter, i=1 A. Carrassi University of Maryland. USA, 24 th April / 33

19 Model evidence: a key quantity Computing model evidence using data assimilation Method 2. Filtering: Linear/Gaussian case Kalman filter In the Gaussian linear case (Kalman filter - KF), the likelihood and forecast pdfs are Gaussian The Kalman filter estimate of the contextual model evidence reads p(y K:1 y 0: ) = ( K exp 1 2 y k H k x f k (2π) d R k + H k P f k HT k k=1 2 R k +H k P f k HT k R k, P f k : observation and forecast error covariances H k: observation operator. Notation remark: x 2 A = xt A 1 x. ) A. Carrassi University of Maryland. USA, 24 th April / 33

20 Model evidence: a key quantity Computing model evidence using data assimilation Method 2bis. Filtering: nonlinear/gaussian case ensemble Kalman filter Using an ensemble Kalman filter (EnKF), the Kalman filter formula for model evidence now becomes ( ) K exp 1 2 y k H k x f 2 k R k +Y k Yk p(y K:1 y 0: ) T k=1 (2π) d R k + Y k Yk T Y k = H k X k are the normalized observation anomalies, and X k the forecast anomalies. The inversion of R k + Y k Y T k can be done in ensemble subspace. The estimation is exact when the models are linear, the initial condition and observation errors are Gaussian and when the ensemble perturbation span the full range of uncertainty. A. Carrassi University of Maryland. USA, 24 th April / 33

21 Model evidence: a key quantity Computing model evidence using data assimilation Method 3. Smoothing: nonlinear/gaussian case Use a smoothing DA method to estimate p(y K:1 y 0: ) Compute the evidence with a saddle-point approximation (Laplace method): p(y K:1 y 0: ) = dx 0 p(y K:1 x 0 )p(x 0 y 0: ) ( ) dx 0 exp 1 K y k H k M k:0 (x 0 ) 2 R 2 k + ln p(x 0 y 0: ) k=1 = dx 0 exp ( L(y K:1, x 0 )) exp ( L(y K:1, x 0)) 2 x0 L(x 0 ) How to obtain x 0 characterizes the problem: 4DVar, En-smoother, etc. We shall use En4DVar and IEnKS. A. Carrassi University of Maryland. USA, 24 th April / 33

22 Model evidence: a key quantity Computing model evidence using data assimilation 3. Smoothing En4DVar and IEnKS Ensemble-4DVar (En4DVar) (see nomenclature by Lorenc, 2013) The state is parameterized in the ensemble space: x 0 = x 0 + X 0 w ( exp 1 K 2 k=1 y k H k M k:0 (x 0) 2 R k 1 2 w 2) p(y K:1 y 0: ) (2π) Kd K k=1 R k I N + K k=1 (Y k )T R 1 k Y k (Quasi-static) Iterative Ensemble Kalman Smoother (IEnKS) The initial condition and the ensemble at t 0 are sequentially updated, x 0 = x k 1 + X k 1 w k ) K exp ( 12 y k H k M k:0 (x k ) 2Rk+Y k(y k) T p(y K:1 y 0: ) k=1 (2π) d Rk + Yk (Y k )T A. Carrassi University of Maryland. USA, 24 th April / 33

23 Model evidence: Numerical results Setup Model evidence Numerical experiments Schema'c diagram of a forecast- - assimila'on cycle and contextual model evidence (CME) computa'on K- long evidencing window EnKF- based Trajectory True/Correct model M 1 Incorrect model M 0 observa8ons Forced L63 Model dx dt = σ(y x) + λ i cos θ Factual λ 1 = 0; Counterfactual λ 0 { 8 : 8} ; θ = 140. dy dt = ρx y xz + λ i sin θ L95 Model dx m = x m 1 (x m+1 x m 2 ) x m + F i m = 1,..., M dt Factual F 1 = 8; Counterfactual F 0 {5 : 11}. dz = xy βz dt A. Carrassi University of Maryland. USA, 24 th April / 33

24 Model evidence: Numerical results Accuracy of the DA-based estimates Computation of the model evidence: Accuracy assessment Comparison with two independent highly precise methods: 1 Gauss-Hermite quadrature GHQ with 32 degrees - for L63 2 Massive Monte Carlo MC with N particles (power-law extrapolation) - for L63 and L95 The most sophisticated DA method (IEnKS) provides the most accurate outcome. The En4DVar fails in a highly nonlinear setup represented here by the bi-modal L63 model. The EnKF provides sufficiently accurate estimate at a much reduced cost Suitable operationally. A. Carrassi University of Maryland. USA, 24 th April / 33

25 Model evidence: Numerical results Accuracy of the DA-based estimates Model evidence versus forcing and length of window log(p 0 ) log(p 0 ) (a) L63 (K=10) λ 0 (c) L63 (λ 0 =8) K log(p 0 ) log(p 0 ) (b) L95 (K=10) IS EnKF En-4D-Var IEnKS F (d) L95 (F 0 =11) K A. Carrassi University of Maryland. USA, 24 th April / 33

26 Model evidence: Numerical results Low dimensional models - Application for attribution Low dimensional models - Application for attribution Forced L63 with stochastic noise : dx dt = σ(y x) + λ i cos θ i + v x dy dt = ρx y xz + λ i sin θ i + v y dz = xy βz + vz dt Factual λ 1 = 0; Counter-Factual λ 0 = 0 : 40 GINI index measures the discriminating power, from 0 for random to 1 for perfect discrimination EnKF-based - CONVENTIONAL Gin Index Gin Index λ forcing σ q Model Error Gin Index Gini Index σ q λ forcing σ r Obs Error A. Carrassi University of Maryland. USA, 24 th April / 33

27 Model evidence: Numerical results Model evidence with localization Model evidence with domain localization for high-dimension model In high-dimensional systems EnKF methods require localization We derived the Local CME at each gridpoint s using the local observations y s, K { p(y K:1 s ) (2π) d 2 Σk 1 2 exp 1 } 2 (y k s H k sx f k) T Σ 1 k (y k s H k sx f k) k=1 with Σ k = H k s P f k HT k s + R k s and d the size of y k s, P f k is the prior error covariance matrix, R k s is the local obs. error cov. matrix, H k s is the local linear obs. operator. ρ loc S Based on the local CME we define the Domain localized CME (DL-CME) as an euristic global estimator { } p(y K:1) = exp w(s) ln{p(y s )}, s Γ with w(s), scalar weights inversely proportional to the localization radius. A. Carrassi University of Maryland. USA, 24 th April / 33

28 Model evidence: Numerical results Application for model selection High(er) dimensional models - Application for model selection A global atmospheric model resolving the large scale dynamic (Molteni, 2003 ) The models Res.: O(10 4 ) Vor, Div, T, Q, log(p s) True trajectory: 5 month SPEEDY (01/02-30/06/1983) Two convective relaxation times Correct param.: τ cnv = 6hs Incorrect param.: τ cnv = 9hs The observations Artificial obs.: [u, v, T, Q, p s] Frequ.: 6h Spat. distrib.: 1/4 gridpoints DA setup LETKF 50 members (Miyoshi, 2005, 2007) Hydrostat., σ-coord, spectral-transf. Convect., condens., clouds, radiat. Proba. of selection: number of successfull selection DA using all obs.; CME computed for seperate var. A. Carrassi University of Maryland. USA, 24 th April / 33

29 Model evidence: Numerical results Application for the attribution problem (high-dim) High(er) dimensional models - Application for attribution (ongoing) Factual and counterfactual models Two 5-year intervals of interest: and SPEEDY configurations: Factual Counterfactual Projet CMIP5 CMIP5 Model MPI-ESM-LR MPI-ESM-LR Institute MPI-M MPI-M Experiment historical picontrol Time freq. monthly monthly T (K) T (K) Averaged T Fact. CFact Averaged T differences Anomalies Observations Modeling realm atmos/sea atmos/sea Ensemble r1i1p1 r1i1p1 Factual perturbed: σ u = σ v = 1 m.s 1, σ T = 1 C, σ q = 1 gkg 1, σ ps = 1 hpa. Frequ.: 6h; Spat. distrib.: random 1/4 x grid; DA setup LETKF 50 members; Domain localization (h: 700 km, v: 4 km) A. Carrassi University of Maryland. USA, 24 th April / 33

30 Model evidence: Numerical results Application for the attribution problem (high-dim) Global study: DL-CME differences: p (M 0, M 1 ) = ln{ p(y K: M 1 )} ln{ p(y K: M 0 )} all obs. Periods p u v T q all Percentage of attributed cycles (K = 1) The FAR coefficient: FAR= 1 p(y K: M0) p(y K: M 1) all obs. FAR Periods FAR u 0.18 v 0.13 T 0.40 q 0.07 all 0.59 Monthly FAR averaged over the 5-year intervals A. Carrassi University of Maryland. USA, 24 th April / 33

31 Model evidence: Numerical results Application for the attribution problem (high-dim) Spatial study JJA DJF JJA DJF The CME allows for a geographical assessment of the attribution. A. Carrassi University of Maryland. USA, 24 th April / 33

32 Conclusion and future directions Conclusion We showed analytic formulae to compute model evidence using DA (Gaussian filter and smoother) and proved their validity with numerical experiments. The accuracy of this DA-based estimate scales with the sophistication of the DA method (IS EnKF IEnKS). Applications: 1 Attribution of climate-related event attribution evaluation in the aftermath of the event occurrence using DA. (Hannart et al., 2016; Carrassi et al., 2017) 2 Model selection (Metref et al., 2018) 3 Parameter estimation (Carrassi et al., 2017; Pulido et al., 2018) Open questions and future directions Model error is a central concern. 1 It can mask the factor whose casual attribution is under scrutiny confuse discrimination between factual and counter-factual worlds. 2 A DA-based method has the potential to deal with this issue taking advantage of the knowledge in model error treatment in DA. Applications with real observations is on-going. A. Carrassi University of Maryland. USA, 24 th April / 33

33 Conclusion and future directions Key references Carrassi A., M. Bocquet, A. Hannart and M. Ghil, 2017: Estimating model evidence using Data assimilation. Q. J. R. Meteorol. Soc., 143, , 2017 Carrassi, A., M. Bocquet, L. Bertino and G. Evensen. Data assimilation in the geosciences - An overview on methods, issues and perspectives. Submitted to WIREs Climate Change. Available at: arxiv: v2., Hannart A., A. Carrassi, M. Bocquet, M. Ghil, P. Naveau, M. Pulido, J. Ruiz and P. Tandeo, DADA: Data Assimilation for the Detection and Attribution of Weather- and Climate-related Events. Climatic Change, 136, , Metref S., A. Hannart, J. Ruiz, M. Bocquet, A. Carrassi, and M. Ghil, 2018: Estimating model evidence using ensemble-based data assimilation with localization - The model selection problem. Q. J. R. Meteorol. Soc. - Submitted Pulido, M., P. Tandeo, M. Bocquet, A. Carrassi and M. Lucini, Stochastic parametrization identification using ensemble Kalman filtering combined with expectation-minimization and Newton-Raphson maximum likelihood methods. In press on Tellus. A. Carrassi University of Maryland. USA, 24 th April / 33