Ensemble Forecas Sensiiviy o Observaions EFSO) and Flow-Following Localisaion in 4DEnVar Andrew C Lorenc 07/20/14 Me Oce andrew.lorenc@meoce.gov.uk Prepared for DAOS meeing, Augus 2014, Monreal.
Inroducion The global FSO algorihm, developed wih he help of DAOS, has become a useful ool for OS managers. How can we mainain his in fuure EnVar mehods, wihou an adjoin forecas model? Ensemble covariance esimaes are noisy! Filering usually localisaion) is essenial for accurae resuls. EFSO has o ler 4D covariances for longer windows han DA. ETKF can deec he large signals from major arge areas wihou localisaion.) This is work in progress: 1. Flow-following localisaion in 4DEnVar 2. EFSO using 4DEnVar
The Advanages of 4DEnVar DA Can iniialise higher resoluion NWP han an ensemble Model-space ensemble covariance faciliiaes many forms of lering as well as simple spaial localisaion: - Hybridisaion, balance, ime smoohing, - spaial smoohing via specral localisaion Buehner and Charron 2007)), - muli-scale localisaion Does no need repeaed runs of linear and adjoin model
The need for ow following localisaion in 4DEnVar Top: 6 hour ime-evoluion of incremen from localised ensemble in 4DEnVar. Boom: Evoluion of he same iniial incremen by he PF model in ensemble-4dvar Lorenc e al., 2014).
Flow-following localisaion in 4DEnVar Could do his saisically Bishop and Hodyss, 2009a,b; Gasperoni and Wang, 2014). Insead we choose o apply he dynamical idea of advecing he α-eld using an esimae of he group velociy e.g. 0.75 of he smoohed wind eld). Each α i is consruced such ha α i α T i = L, he 4D localisaion marix. Basic 4DEnVar Lorenc e al., 2014) uses a persisence forecas α i = Iα i so ha L = ILI T. Our plan is o use a simple semi-lagrangian advecion α i = U α α i o give a fully 4D L wih localisaion following he ow.
The Me Oce FSO sysem Lorenc and Marrio, 2014). )) δx 0 = K y o H x b 0 Variaional daa assimilaion: Dierence, due o obs: The dierence beween scores: is he Impac: J = Linearise abou nie dierence: Aribue o each ob: J = δx fa y o H x fa = M x a 0) ), = M x fb δx = x fa x b 0 x fb δx fa = x fa x, δx fb = x fb x ) T C 1 δx fa J = δx T C δx fa δx fb + δx fb ) T C 1 δx fb ) T x0)) b K T M T C δx fa + δx fb ) K and K T are solved implicily using minimisaions. M T is an adjoin model.
ETKF FSO sysems Liu and Kalnay, 2008; Li e al., 2010). )) ETKF wrien as: δx 0 = X b K y o H x b 0 K isn' marix-mul.: A = K )) y o H x b evaluaed each 0 grid-poin o give rows of δx = A i α i x i )))) whose columns are he α i : δx 0 = X b K y o H x b 0 1 K )) T Aribue o each ob J = y o H x b heir noaion 0 KT X f T 6) C δx fa + δx fb ) marix noaion: ) T J = 1 T K y o H x0)) b KT X f ) ) T 6 C δx fa + δx fb ) K and K T come from LETKF. There is no ow-following localisaion unless he ETKF is amended for KT. Kunii e al. 2012) evaluaed EFSO in a 12 hr WRF forecas of a yphoon very plausible resuls, validaing well agains OSE.
Relies on EnKF mehods producing an analysis ensemble: Alernaive form for he Kalman gain: Aribue o each ob unlocalised) Normal EnKF localisaion: EnKF FSO sysems Kalnay e al., 2012). P a e = X a X a ) T K = P a H T R 1 T J = y o H x0)) b R 1 HX a X a ) T C δx fa + δx fb ) T J = y o H x0)) b R 1 L om Y )) a X a ) T C δx fa + δx fb ) Simple mehod which works in any EnKF sysem. Localisaion L om can be made ow-dependen. Several sudies on is bes form: Kalnay e al. 2012) in a oy 40 variable model ried displacing hem wih a mean group velociy clearly beer han xed localisaion. Oa e al. 2013) in he NCEP GFS, found i bes o advec he cenre of he localisaion funcion by 0.75 imes mean wind a each level.
4DEnVar FSO 4DEnVar sysems do no usually generae ensemble, so we rever o adjoin mehod, bu wihou using M T. Exend ensemble & 4DEnVar a lease in principle) o give incremen a ime : Aribue o each ob J = δx = K ex 4DEnV ar y o H y o H )) x b 0 T x0)) b K T ex 4DEnV ar C δx fa Schur produc is inside K T ex 4DEnV ar his is like LETKF FSO, excep for ow-dependen α advecion. + δx fb ) K T ex 4DEnV ar is solved implicily by minimisaion algorihm. As in oher ensemble mehods, no need for adjoin model M T.
Long erm prospecs Ensemble mehods can work, bu ow-following localisaion is he key. Bu here are heriage ETKF mehods???) Ensemble mehods are likely o be more accurae for more nonlinear and longer windows. Convecive-scale mehods will need a new deniion of impac J since accurae vericaion analyses are unavailable. Me Oce is using J o observaion penaly from VAR. Unlikely o have single impac measure his won' please OS managers. Can in principle measure impac of larger-scales, via boundary condiions. Bu is i worh he eor?
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