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Ensemble Kalman Filter basedsnow data assimilation

(just some ideas)

FMI, Sodankylä, 4 August 2011

Jelena Bojarova

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Sequential update problem

Non-linear state space problem

Tangent-linear state space problem

• Optimal interpolation solution• Variaional minimization solution• ExtendedKalman filter

Ensemble Kalman filter (square-root Kalman filter)

global local

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Data assimilation

Data assimilation provides a point estimate of the “true” model state conditional on the observed quantities.

Data assimilation prepares an initial state for NWP models.

A “true” model state is a reflection of the state of atmosphere projected on the discrete space of solution of differential equations which describe phenomena of interest.

Data assimilation provides a statistical combination of the observations and the background model state (usually a short range forecast) using information on their uncertainties.

Filtering away of observation error

Interpolation of the observed information to other model state components

Balancing of model state components (explicit use of cross-dependencies)‏

Background forecast error covariance

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Different approaches for using ensembles in variational data assimilation

– Covariance modelling with parameters of the covariance model determined from an ensemble. Use for example a wavelet-based covariance model (Alex Deckmyn; Loik Berre et al. Meteo-France) ‏

– Use the ensemble-based covariances in a hybrid variational ensemble data assimilation (Barker et al. WRF, UK Met.Office, HIRLAM )

– Ensembles can also be used to determine static background error statistics

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What makes the ensemble based data assimilation scheme attractive

Ensemble of forecast error perturbations contains flow-dependent structures;Ensemble of the forecast error perturbations contains an-

isotropic structures (both large- and small-scales are represented);Ensemble of the forecast error perturbations reflects

relationship between large-scale forcing and meso-scale developments.

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Kalman filter data assimilation scheme is based on very restrictive assumptions.

• The distribution of observation errors, forecast errors and model errors are Gaussian;• The model dynamical propagator and the observation operator are linear;• Observation errors and model errors are zero-mean stochastic variables;• Observation errors are uncorrelated in space and in time;• Model errors are uncorrelated in time;• Observation errors and model errors are mutually uncorrelated

It can be very challenging to meet these requirements for snow data assimilation even approximately....!

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Optimal interpolation

xa = xf + BHT (HBHT + R)- 1(y - Hxf)‏

If observation operator H is linear , the solution of the minimization problem is the BLUE

Hx t ru

eyHxf Hxa

RHBHT

Hxa = Hxf (1 - ) + y──── ────1

1+ ──R

HBHT

1

1+ ──R

HBHT

Var(Hxa) = ────1

─ + ──1

R

1

HBHT

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Efficient (snow) data assimilation scheme requires:careful specification of observation error statistics • biases (non-linear observation operator; complicated retrievals algorithms; impact of surface type, orography and terrain)• representativity errors(model state variables, which represent space and time averaged quantities, versus momentary and descrete observed values; tiling makes problem even more complicated)• quantification of the uncertainty for the data product

Extensive data preprocessing need to be done

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Efficient (snow) data assimilation scheme requires:

adequate sampling of all sources of the forecast error uncertainty:• the uncertainty of the initial model state• the uncertainty of the lateral boundary conditions• the uncertainty of the forcing fields• the uncertainty associated with the model deficiencies (dynamical evolution and physical parameterization) The realistic ensemble spread need to be maintained

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The efficient data assimilation scheme requires

modelling of the realistic cross-dependencies within and between different sources of uncertainty • preservation of realistic/ physically meaningful balances between different model state variable compenents • reproduction of spatial inhomogeneity and unisotropy• accumulated impact of the model error uncertainty on the forecast uncertainty at the analysis time.

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Ensemble based snow data assimilation (Step 1)

• preprocessing of observations (removal of biases, thinning and quality control) • use an appropriate ensemble of upper air perturbations (possibly GLAMEPS product) to sample uncertainty in the forcing and lateral boundaries fields • identify model error components with the strongest impact of the analysis quality• impose the accumulated impact of the model error uncertainty on the forecast error uncertainty• apply ensemble data assimilation scheme to construct the snow analysis (one way interaction: atmospheric forcing -> surface scheme -> snow model)

e

1-D snow model to start with

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Ensemble based snow data assimilation (Step 2)

• Sample uncertinty about the initial uncertainty of snow field. • Project snow analysis uncertainy on the forcing fields uncertainty, preserving important cross-dependencies (two-way interaction scheme: atmospheric forcing -> surface scheme -> snow model -> surface scheme -> atmospheric forcing)• Merge upper-air and surface perturbations • Extend snow data assimilation scheme to the 3-D settings

fi

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+36 h ETKF EPS valid at 24Jan 12 UTC

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3DVAR+ETKF 3DVAR+EnsDA 3DVAR+TEPS

3DVAR

24 Jan 12 UTC

3d-Var analysis versus +36 h forecasts