Data Assimilation Using Modulated Ensembles Data Assimilation Using Modulated Ensembles Craig H.
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Data Assimilation Using Modulated Ensembles Data Assimilation Using Modulated Ensembles Craig H. Bishop, Daniel Hodyss Naval Research Laboratory Monterey, CA, USA September 14, 2009 Data Assimilation Using Modulated Ensembles Overview • Motivation for adaptive ensemble covariance localization • Method • Adaptive ensemble covariance localization in ensemble 4D-VAR • Results from preliminary comparison of DA performance with operational covariance model using pseudo-obs • Show that adaptive localization enables ensemble based TLM • Conclusions Data Assimilation Using Modulated Ensembles Motivation Stable flow error correlations x10km Unstable flow error correlations x10km Ensembles give flow dependent, but noisy correlations Data Assimilation Using Modulated Ensembles Motivation Stable flow error correlations Fixed localization Current ensemble DA techniques reduce noise by multiplying ensemble correlation function by fixed localization function (green line). x10km Unstable flow error correlations Resulting correlations (blue line) are too thin when true correlation is broad and Fixed localization too noisy when true correlation is thin. x10km Today’s fixed localization functions limit adaptivity Data Assimilation Using Modulated Ensembles Motivation Stable flow error correlations t=0 • Current ensemble localization functions poorly represent propagating error correlations. x10km Unstable flow error correlations t=0 x10km Today’s fixed localization functions limit ensemble-based 4D DA Data Assimilation Using Modulated Ensembles Motivation Stable flow error correlations • Current ensemble localization functions poorly represent propagating error correlations. t=0 t=1 x10km Unstable flow error correlations t=0 t=1 x10km Today’s fixed localization functions limit ensemble-based 4D DA Data Assimilation Using Modulated Ensembles Stable flow error correlations t=0 t=1 • Green line now gives an example of one of the adaptive localization functions that are the subject of this talk. x10km Unstable flow error correlations t=0 t=1 x10km Want localization to adapt to width and propagation of true correlation Data Assimilation Using Modulated Ensembles Motivation Stable flow error correlations t=0 t=1 Current ensemble localization functions do not adapt to the spatial scale of raw ensemble correlations and they poorly preserve propagating error correlations. km Unstable flow error correlations t=0 t=1 km Data Assimilation Using Modulated Ensembles Method An adaptive ensemble covariance localization technique (Bishop and Hodyss, 2007, QJRMS) _ _ Data Assimilation Using Modulated Ensembles Method Stable flow error correlations • Green line now gives an example of one of the adaptive localization functions that are the subject of this talk. t=0 t=1 km Unstable flow error correlations Key Finding: Moderation functions based on smoothed ensemble correlations provide scale adaptive and propagating localization functions. t=0 t=1 km Data Assimilation Using Modulated Ensembles Modulated ensembles and localization (Bishop and Hodyss, 2009 a and b, Tellus) Data Assimilation Using Modulated Ensembles Modulated ensembles and localization z sj Smooth ensemble member j zk Raw ensemble member k zis Smooth ensemble member i zk z sj zis Modulated ensemble member Data Assimilation Using Modulated Ensembles Modulated ensembles enable global 4DVAR Both incremental and non-incremental AR are possible. Non-incremental weak constraint AR is as follows. Since P f =Z D ZTD where Z D is the very large modulated ensemble, Step 1 is to solve R 1/ 2 HZ D R 1/ 2 HZ D I v R 1/ 2 y H x f T for v using (for example) conjugate gradient. Step 2 is the postmultiply T x a x f Z D R 1/ 2 HZ D v Hybrid mixes of ensemble based TLMs and initial covariances with those from NAVDAS are straightforward. Model space "primal" 4D-VAR form is also straightforward (El Akkraoui et al., 2008, QJRMS) . Data Assimilation Using Modulated Ensembles Application to global NWP model Example of a column of the localization Cs Cs with K 128 06Z Ensemble based localization moves about 1000 km in 12 hrs. This is >=half-width of a typical LETKF observation volume (~900km). Data Assimilation Using Modulated Ensembles Application to global NWP model Example of a column of the localization Cs Cs with K 128 18Z Ensemble based localization moves about 1000 km in 12 hrs. This isNaval >=half-width of a typicalMarine LETKF observation volume (~900km). Research Laboratory Meteorology Division Monterey, California Data Assimilation Using Modulated Ensembles Application to global NWP model Example of a column of PKf Cs Cs with K 128 <vv> Increment 06Z Statistical TLM implied by mobile adaptively localized covariance propagates single observation increment 1000 km in 12 hrs. Data Assimilation Using Modulated Ensembles Application to global NWP model Example of a column of PKf Cs Cs with K 128 <vv> Increment 18Z Statistical TLM implied by mobile adaptively localized covariance propagates single observation increment 1000 km in 12 hrs. Data Assimilation Using Modulated Ensembles Comparison of background covariances Red square NAVDAS f f P P Cs Blue Circle K RMS(Analysis Error)/RMS(Forecast Error) Lower is better Cs Anomaly Correlation Higher is better Anomaly correlation is between analysis correction and the “perfect” correction that would have eliminated all initial condition error. Adaptively localized ensemble covariance produced smaller initial condition errors than covariance model used in operational 3D-PSAS/NAVDAS scheme Data Assimilation Using Modulated Ensembles Adaptive localization enables ensemble TLM Solid – attenuation Dashed – no attenuation 6 hr 12 hr P f PKf Cs Cs Data Assimilation Using Modulated Ensembles Accuracy of ECMWF TLM noise n signal s Data Assimilation Using Modulated Ensembles Comment on TLM/PFM accuracy • It can be argued that the ability of the TLM to represent differences between perturbed and unperturbed trajectories is less important than its ability to accurately describe 4D covariances. • Adaptively localized ensemble covariances have the advantage of incorporating the effects of non-linear dynamics on 4D covariances. • Disadvantage of adaptive localization scheme shown here is that it does not handle linear dispersion as well as typical TLM/PFMs. Data Assimilation Using Modulated Ensembles Conclusions • Adaptive localization should aim to account for propagation and scale variations of error distribution • Proposed adaptive localization given by even powers of correlations of smoothed ensemble • Huge modulated ensembles give square root of localized ensemble covariance matrix • Errors can move over 1000 km in 12 hr window • Modulated ensembles enable 4D-VAR global solve • Adaptively localized covariance beats operational covariance model in idealized experiment with pseudoobs • Adaptive localization enables ensemble based TLMs