Accoun&ng for mul&-scale ver&cal error correla&on - PowerPoint PPT Presentation

Accoun&ng for mul&-scale ver&cal error correla&on within ETKF through eigen-spectral covariance localiza&on Daisuke Hotta 1 and Craig H. Bishop 2 1 MRI/JMA 2 Univ. Melbourne 7 th International Symposium on Data Assimilation 2019/01/21 Kobe, Japan 1

Ba Backg kground: Variou ous covariance loc ocaliza4on 4on schemes used in EnK EnKF (o (or EnV EnVar) Model-space B-localization: R-localiza)on: Obs-space B-localization: • Inflate R for distant obs • Localize B but in obs- • Localize B directly in in place of directly space rather than in model-space: localizing B model-space: B loc = ρ � B K =(ρ 1 � BH T ){ρ 2 � ( H BH T ) + R } -1 • Pros: computationally • Pros: easy to implement • Pros: straighIorward efficient on serial DA treatment of non-local à currently obs mainstream for LETKF • Cons: can be expensive Cons: not evident how to treat non-local obs • à recently overcome by (whose H depends on mulSple grid points) modulated ensemble approach (Bishop, Whitaker and Lei, 2017, MWR ) 2

Model-space B -localiza/on by modulated ensemble + Gain-form ETKF (Bishop, Whitaker and Lei 2017, MWR ) • Given noisy raw ensemble covariance, we can construct a good filtered covariance w/o explicitly storing B • provided that 1) true B has a localized structure (in the sense that correlation between distant grid points are small), and 2) you know how to construct the appropriate localization matrix F (in practice, this is subject to manual tuning that depends the ensemble size K ) Questions: What if • true B is not well-localized, or • appropriate localization matrix From Bishop, Whitaker and Lei (2017) F is not evident? 3

• Goals of this study : To formulate a localization scheme that is • applicable even when the true B is not well- localized • less sensitive to tuning (or does not require it at all) 5

Wha What t is is lo localiz aliza,on? n? What do we exactly y do by y localiza,on? • Common interpretation (as I understand…): 1. Error correlation between distant locations should be weak ( ß which is our a priori knowledge ) 2. à if sampled correlation is large between distant locations, that is a sign of sampling error, so that we should damp it. ( ß which is just one way of imposing our a priori knowledge ) • Extended interpretation: • Impose our a priori knowledge about true B to constrain the structure that a sampled B can take • à Physical location (or distance) is not essential 6

Wha What t is is lo localiz aliza,on? n? What do we exactly y do by y localiza,on? • Common interpreta,on (as I understand…): 1. Error correla,on between distant loca,ons should be weak ( ß which is our a priori knowledge) 2. à if sampled correla,on is large between distant loca,ons, that is a sign of sampling error, so that we should damp it. ( ß which is just one way of imposing our a priori knowledge) • Extended interpreta,on: • Localiza,on imposes our a priori knowledge about true B to constrain the structure that a sampled B can take • à Physical loca,on (or distance) is not essen,al 7

Propo pose sed d metho hod: d: eigen-spe spectral localization Inspired from precondi*oning in the ver*cal direc*on used in VAR schemes Assumed situa+on: • B true stochas*cally changes day-to-day, B true is not well-localized • We have an ensemble X of samples drawn from N (0, B true ) and want to es*mate B true • B true is unknown, but we know its climatological mean B clim . • Each instance of B true may be dissimilar to B clim , but we know that their structures are similar. Key idea: • Eigen-decomposi*on of B true : B true = V true Λ true V trueT • allows us to represent B (or state vector x ) in mode space : B8 true = V trueT B true V true , x̂ = V trueT x • In the mode space, true covariance is diagonal : B8 true = Λ true à Ignoring the off-diagonal elements of B8 ens = V trueT B ens V true should effec*vely reduce sampling noises • But B true and hence V true are unknown, so use B clim = VΛ clim V T in stead to construct the mode space. 8

Propo pose sed d metho hod: d: eigen-spe spectral localiza5on n (con5nued) • Eigenspectral localiza0on: procedure 1. x i � N (0, B true ), i=1,…, K are at our disposal. X =[ x 1 , x 2 ,…, x K ] 2. Project X onto (truncated) mode space : X$ = V T X =[ V T x 1 , V T x 2 ,…, V T x K ] 3. Form the covariance in the mode space, but retain only the diagonal elements: B$ ensdiag =diagm( B$ ens ) = I � B$ ens where B$ ens = X$ X$ T /( K -1) 4. Transform back to physical space by B ensloc = V B$ ensdiag V T ✱ No need to explicitly specify localization matrix à No tuning required for different #ens • In implementa0on … • Retaining all modes of B clim = VΛ clim V T is not feasible (nor necessary) so truncate it by retaining only the leading O(10) modes (20 in the experiments shown later). • Do not hold B$ ensdiag explicitly; instead, represent it with ensemble modula0on: B$ ensdiag =diagm( B$ ens ) = I � B$ ens = ( I I T ) � ( X$ X$ T ) /( K -1)=(I ▵ X$ )(I ▵ X$ ) T 9

Id Idea ealized ed experi erimen ent: Problem em setup • Dimension size n=100. b* L 1 • Shape of B true characterized by two scales L 1 and L 2 . • L 1 and L 2 : fixed in each experiment. • Each instance of B true is generated with b * parameters b* L 1 and b* L 2 L 2 Defined by Eq.(3) of • where b is a stochasCc parameter uniformly Bishop, Whitaker and Lei (2017, MWR; distributed over [0.2, 3.0]. referred to as BWL17 • Model-space and eigen-spectral localizaCon hereafter) schemes both truncated with by retaining only ntrunc=20 leading modes. 10

Diagonal dominance of B/ true = V climT B true V clim • Prerequisite for the success of eigen-spectral localization is that the true covariance expressed in the climatological eigen-mode space V climT B true V clim is close to diagonal. • à Check if this prerequisite is met by plotting V climT B true V clim for different sets of (L 1 ,L 2 ) and for different b 11

Diagonal dominance of B/ true = V T B true V: Well-localized cases V T B true V is Ptrue (L1,L2)=(1.0,=5.0) Pclim b=0.2 1.0 1.0 100 100 0.9 0.9 0.8 0.8 • almost diagonal 80 80 model 0.7 0.7 0.6 0.6 60 60 space 0.5 0.5 • dominated by smaller 0.4 0.4 40 40 0.3 0.3 B clim B true 0.2 0.2 20 20 number of leading modes 0.1 0.1 0 0 20 40 60 80 100 20 40 60 80 100 Ptrue in eig- spc of Pclim Ptrue in eig- spc of Pclim (leading modes enlarged) 0 0 1 1 20 2 2 5 3 3 4 4 mode 40 10 5 5 space 60 6 6 7 15 7 80 8 8 V T B true V 9 9 20 10 10 100 20 40 60 80 100 5 10 15 20 12

Diagonal dominance of B/ true = V T B true V: Well-localized cases V T B true V is Ptrue (L1,L2)=(1.0,=5.0) Pclim b=0.8 1.0 1.0 100 100 0.9 0.9 0.8 0.8 • almost diagonal 80 80 model 0.7 0.7 0.6 0.6 60 60 space 0.5 0.5 • dominated by smaller 0.4 0.4 40 40 0.3 0.3 B clim B true 0.2 0.2 20 20 number of leading modes 0.1 0.1 0 0 20 40 60 80 100 20 40 60 80 100 Ptrue in eig- spc of Pclim Ptrue in eig- spc of Pclim (leading modes enlarged) 0 0 1 1 20 2 2 5 3 3 4 4 mode 40 10 5 5 space 60 6 6 7 15 7 80 8 8 V T B true V 9 9 20 10 10 100 20 40 60 80 100 5 10 15 20 13

Diagonal dominance of B/ true = V T B true V: Well-localized cases V T B true V is Ptrue (L1,L2)=(1.0,=5.0) Pclim b=1.3 1.0 1.0 100 100 0.9 0.9 0.8 0.8 • almost diagonal 80 80 model 0.7 0.7 0.6 0.6 60 60 space 0.5 0.5 • dominated by smaller 0.4 0.4 40 40 0.3 0.3 B clim B true 0.2 0.2 20 20 number of leading modes 0.1 0.1 0 0 20 40 60 80 100 20 40 60 80 100 Ptrue in eig- spc of Pclim Ptrue in eig- spc of Pclim (leading modes enlarged) 0 0 1 1 20 2 2 5 3 3 4 4 mode 40 10 5 5 space 60 6 6 7 15 7 80 8 8 V T B true V 9 9 20 10 10 100 20 40 60 80 100 5 10 15 20 14

Diagonal dominance of B/ true = V T B true V: Well-localized cases V T B true V is Ptrue (L1,L2)=(1.0,=5.0) Pclim b=1.9 1.0 1.0 100 100 0.9 0.9 0.8 0.8 • almost diagonal 80 80 model 0.7 0.7 0.6 0.6 60 60 space 0.5 0.5 • dominated by smaller 0.4 0.4 40 40 0.3 0.3 B clim B true 0.2 0.2 20 20 number of leading modes 0.1 0.1 0 0 20 40 60 80 100 20 40 60 80 100 Ptrue in eig- spc of Pclim Ptrue in eig- spc of Pclim (leading modes enlarged) 0 0 1 1 20 2 2 5 3 3 4 4 mode 40 10 5 5 space 60 6 6 7 15 7 80 8 8 V T B true V 9 9 20 10 10 100 20 40 60 80 100 5 10 15 20 15

Diagonal dominance of B/ true = V T B true V: Well-localized cases V T B true V is Ptrue (L1,L2)=(1.0,=5.0) Pclim b=2.4 1.0 1.0 100 100 0.9 0.9 0.8 0.8 • almost diagonal 80 80 model 0.7 0.7 0.6 0.6 60 60 space 0.5 0.5 • dominated by smaller 0.4 0.4 40 40 0.3 0.3 B clim B true 0.2 0.2 20 20 number of leading modes 0.1 0.1 0 0 20 40 60 80 100 20 40 60 80 100 Ptrue in eig- spc of Pclim Ptrue in eig- spc of Pclim (leading modes enlarged) 0 0 1 1 20 2 2 5 3 3 4 4 mode 40 10 5 5 space 60 6 6 7 15 7 80 8 8 V T B true V 9 9 20 10 10 100 20 40 60 80 100 5 10 15 20 16