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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 Metref1, E.Cosme1, C.Snyder2 and P.Brasseur1

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 za = zb + K(zb − zo) Unobserved variable : Linear regression xa = xb + C(za − zb)

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 za = zb + K(zb − zo) Unobserved variable : Linear regression xa = xb + C(za − zb)

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 za = zb + K(zb − zo) p(z|zo) = p(z)p(zo|z) Unobserved variable : Linear regression xa = xb + C(za − zb)

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|zo) ∝ p(z)p(zo|z) p(z) × p(zo|z) ∝ p(z|zo)

Non-Gaussian Data Assimilation Multivariate Rank Histogram Filter A few results Outcomes and perspectives

MRHF principle

EnKF : RHF : Observed variable : Linear correction Bayes’ theorem za = zb + K(zb − zo) p(z|zo) = p(z)p(zo|z) Unobserved variable : Linear regression xa = xb + C(za − zb)

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 za = zb + K(zb − zo) p(z|zo) = p(z)p(zo|z) Unobserved variable : Linear regression Joint density decomposition xa = xb + C(za − zb) p(x, z|zo) = p(z|zo)p(x|z, zo)

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|zo)

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

• cean-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)

• `

u p(x|z) posterior density and p(x) prior density on x and L(x) a likelihood function written : L(x) = ρ(z)θ(z|x) µ(z) dz, (2) ρ(z) id the prior information available on z → p(z|zo). 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), p(x) =

Ne

• i=1

δ(x − xb

i ),

(3) The analysis equation on x becomes : p(x|z) ∝

• L(x)δ(x − xb

i ),

(4) and leads to p(x|z) ∝

• p(zb

i |zo)δ(x − xb i ).

(5) the weights wi = p(zb

i |zo) 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

• n each time step and each grid point for both unobserved (top) and observed

(bottom) variables.