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Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives A non-Gaussian analysis scheme using rank histograms for ensemble data assimilation Sammy Metref 1 , E.Cosme 1 , C.Snyder 2 and P.Brasseur 1 1 MEOM Team - LGGE, Grenoble, France 2 National Center for Atmospheric Research, Boulder, Colorado 6th WMO SYMPOSIUM October 7th, 2013

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Sequential Ensemble Data Assimilation Analysis step A prior ensemble (information from the model) Observations (information from measurements)

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Sequential Ensemble Data Assimilation Analysis step A prior ensemble (information from the model) Observations (information from measurements) Ensemble Kalman Filter (Formalism of Anderson, 2003) Observed variable : Linear correction z a = z b + K ( z b − z o ) Unobserved variable : Linear regression x a = x b + C ( z a − z b )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives The stakes of non-Gaussian DA Illustration Ensemble of simulations propagated by NEMO-LOBSTER, a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010), on the chlorophyll (CHL) / detritus (DET) plane at the Gulf Stream station (47W/40N).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives The stakes of non-Gaussian DA Illustration Ensemble of simulations propagated by NEMO-LOBSTER, a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010), on the chlorophyll (CHL) / detritus (DET) plane at the Gulf Stream station (47W/40N).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives The stakes of non-Gaussian DA Illustration Ensemble of simulations propagated by NEMO-LOBSTER, a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010), on the chlorophyll (CHL) / detritus (DET) plane at the Gulf Stream station (47W/40N).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives The stakes of non-Gaussian DA Illustration Ensemble of simulations propagated by NEMO-LOBSTER, a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010), on the chlorophyll (CHL) / detritus (DET) plane at the Gulf Stream station (47W/40N). Non-Gaussian Data Assimilation Particle Filter Anamorphosis Rank Histogram Filter

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Rank histogram Filter (Anderson, 2010) EnKF : Observed variable : Linear correction z a = z b + K ( z b − z o ) Unobserved variable : Linear regression x a = x b + C ( z a − z b )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Rank histogram Filter (Anderson, 2010) EnKF : RHF : Observed variable : Linear correction Bayes’ theorem z a = z b + K ( z b − z o ) p ( z | z o ) = p ( z ) p ( z o | z ) Unobserved variable : Linear regression x a = x b + C ( z a − z b )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Rank histogram Filter (Anderson, 2010) Construction of a 1D - p.d.f. from an ensemble by rank histogram :

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Rank histogram Filter (Anderson, 2010) Construction of a 1D - p.d.f. from an ensemble by rank histogram : Bayes’ theorem : p ( z | z o ) ∝ p ( z ) p ( z o | z ) p ( z o | z ) p ( z | z o ) p ( z ) × ∝

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF principle EnKF : RHF : Observed variable : Linear correction Bayes’ theorem z a = z b + K ( z b − z o ) p ( z | z o ) = p ( z ) p ( z o | z ) Unobserved variable : Linear regression x a = x b + C ( z a − z b )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF principle EnKF : RHF : MRHF : Observed variable : Linear correction Bayes’ theorem z a = z b + K ( z b − z o ) p ( z | z o ) = p ( z ) p ( z o | z ) Unobserved variable : Linear regression Joint density decomposition x a = x b + C ( z a − z b ) p ( x , z | z o ) = p ( z | z o ) p ( x | z , z o )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : The assimilation problem

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : Retrieval of the pdf p ( z )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : The observation likelihood

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : Correction on Z : p ( z | z o )

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : Attributing the weights

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : Correction on X

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives MRHF Methodology : Global resampling

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Results on a NEMO-LOBSTER ensemble - CHL/MLD Ensemble of simulations propagated by a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010) represented on chlorophyll (CHL) / mixed layer depth (MLD) plane at the Gulf Stream station (47W/ 40N).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Results on a NEMO-LOBSTER ensemble - CHL/DET Ensemble of simulations propagated by a coupled ocean-biogeochemical model in North-Atlantic (B´ eal et al., 2010) represented on chlorophyll (CHL) / detritus (DET) plane at the Gulf Stream station (47W/ 40N).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Outcomes Experiments on different cases Similar results to Gaussian filters in quasi-Gaussian cases Appropriate ensemble shape in non-Gaussian cases (A submitted article) Perspectives Data Assimilation comparison on MODECOGel, a 1D coupled ocean-biogeochemical model Comparison with other non-Gaussian DA methods

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Bibliography Anderson, Jeffrey L. ; 2003 : A Local Least Squares Framework for Ensemble Filtering. Mon. Wea. Rev., 131, 634–642. Anderson, Jeffrey L. ; 2010 : A Non-Gaussian Ensemble Filter Update for Data Assimilation. Mon. Wea. Rev., 138, 4186–4198. B´ eal D., Brasseur P. , Brankart J.-M., Ourmi` eres Y., Verron J. ; 2010 : Characterization of mixing errors in a coupled physical biogeochemical model of the North Atlantic : implications for nonlinear estimation using Gaussian anamorphosis. Ocean Sci., 6, 247–262. Metref S., Cosme E., Snyder C., Brasseur P. ; submitted : A non-Gaussian analysis scheme using rank histograms for ensemble data assimilation.

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Approximation : one-batch MRHF By Tarantola’s formalism (2005) : p ( x | z ) ∝ p ( x ) L ( x ) , (1) o` u p ( x | z ) posterior density and p ( x ) prior density on x and L ( x ) a likelihood function written : � ρ ( z ) θ ( z | x ) L ( x ) = (2) dz , µ ( z ) ρ ( z ) id the prior information available on z → p ( z | z o ). θ ( z | x ) is the theoretical pdf statistically describing the physical relationship between x and z → p ( z | x ). µ ( z ) is an homogeneous density on z (constant here).

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Approximation : one-batch MRHF Considering the sample of the prior p ( x ), N e � δ ( x − x b p ( x ) = i ) , (3) i =1 The analysis equation on x becomes : � L ( x ) δ ( x − x b p ( x | z ) ∝ i ) , (4) and leads to � p ( z b i | z o ) δ ( x − x b p ( x | z ) ∝ i ) . (5) the weights w i = p ( z b i | z o ) can be either directly used when creating the density or by duplicating the particles accordingly.

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Results on L63 Figure : Scatterplots of 1000 member ensembles in the X − Y plane of Lorenz 63 model phase space : Forecast of the particle-by-particle MRHF experiment at time step 1000. Analyses are performed using the forecast and the same Z observation.

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives Results on L96 Figure : Talagrand Diagram for : EnKF(red), RHF(blue), MRHF(green) and MRHF2(magenta) computed for 5 realisations with the true trajectory as verification on each time step and each grid point for both unobserved (top) and observed (bottom) variables.