Error (E) Covariance Diagnostics. in Ensemble Data Assimilation. Kayo Ide, Daryl Kleist, Adam Kubaryk University of Maryland

Transcription

1 Error (E) Covariance Diagnostics in Ensemble Data Assimilation Kayo Ide, Daryl Kleist, Adam Kubaryk University of Maryland 7 th EnKF Data Assimilation Workshop May 2016

2 Outline Part I. Background: E-diagnostics E-Vectors E-Dimension Part II. Application of E-diagnostics to GFS hybrid ensemble Ensembles of operational EnKF Case study Ensemble diagnostics references (not limited to) Patil et al, 2001 (low) dimensionality and instability Ockzowski et al, 2005 Bishop and Hodyss 2011 Enomoto et al 2015: instability vectors Yang et al 2015:

3 Error Characteristics by P The special ellipsoid (x x) T P 1 (x x) = 1 can be characterized by p G (x)=constant u : unit orthogonal vector along n-th principle axis for n=1,.., N σ : corresponding std more uncertain σ (1) σ (2) σ (N) [ 0] less uncertain 3 2 u (2) 1 u (1 ) using SVD on P= U S 2 U T U=(u (1),, u (N) ) S=diag {(σ (1),, σ (N) )} x σ (2) σ (1) x x 1 Shape of P determines the uncertainty à The lower the mode (n small), the larger the uncertainty (σ ) in the corresponding direction (u )

4 E-Diagnostics: E-Vectors on X en X en / (M-1) provides the same (and more) information as P en : unit orthogonal vector along n-th principle axis for n=1,.., M-1 σ en : corresponding std more uncertain σ en(1) σ en(2) σ en(n) [ 0] less uncertain using SVD on X en / (M-1) = U en S en F T en U en =((1),, (M) ) S en =diag {(σ en(1),, σ en(m) )} F en =(f en(1),, f en(m) ) x 2 In the space covered by U en (2) σ en(2) σ en(1) (1) x 1 Same concept as the EOF on X en =(x 1, x 2,, x M ) with ensemble corresponding to time series [ No need to compute P ] F en : projection on X en onto U en

5 Error Characteristics by P en P en = X en X ent / (M-1) can be characterized by (for M N+1) : unit orthogonal vector along n-th principle axis for n=1,.., N σ en : corresponding std more uncertain σ en(1) σ en(2) σ en(n) [ 0] less uncertain x 2 Ex) 2D ensemble 3 perturbation 2 (2) 1 (1) σ en(2) 0 E-vectors using SVD on P en = U en S en2 U en T U en =((1),, (M) ) S en =diag {(σ en(1),, σ en(m) )} σ en(1) x 1 Shape of P determines the uncertainty à The lower the mode (n small), the larger the uncertainty (σ ) in the corresponding direction (u )

6 E-Diagnostics: E-Dimension E-diagnostics based on P en = U en S en U ent from X en = U en S en V en T Characteristics of P en - Shape/sizeof uncertainty» : E-vector» D en : E-dim D en = ( n M = 1 1 σ /σ (1) ) 2 n M = 1 1 (σ /σ (1) ) 2 - Size of uncertainty M» trace P en = (σ ) 2 = σ 2 n=1 = M 1 if σ =σ (1) for all n >1 1 if σ = 0 for all n > Dim 2: blobby P en 3 Dim 1: skinny P en with larger uncertainty tr P en 2 1 x 2 0 x (σ (1) en, σ (2) en ) =(0.72,0.36) x (D 1 en, tr P en, det P en ) =(1.8,0.64,0.3) Same u en but different σ en -3 à D en is defined by σ en (σ (1) en, σ (2) en ) =(1.44,0.18) x (D 1 en, tr P en, det P en ) = (1.24,2.1,0.3)

7 Data Assimilation Process Characteristics of practical DA process:à Lives in the space of U b [=U enb ] Analysis increment: Δx a = K d d= y o h(x b ) Δx a N = α a b K = P b H T (R b +R o ) -1 u a α n=1 = (σ b ) 2 (v b ) T (R b +R o ) 1 d with R b = HP b H T v = H u Analysis Error Perturbation: X a = X b W a : W a = (I + ( V b ) T (R o ) -1 V b ) -1/2 U a (S a ) 2 (U a ) T = U b (T a ) 2 (U b ) T : (T a ) 2 =S b (F b ) T (W a ) 2 F b S b 3 3 partial obs x P b x b space of instability by U b R y o o x xa P a Δx a for non-observed variable by correlaion -2 Δx a & P a limited in the space of U b x x 1

8 Data Assimilation Process Characteristics of practical DA process à Tends to suppress errors of the day Analysis increment: Δx a = K d d= y o h(x b ) Δx a N = α a b K = P b H T (R b +R o ) -1 u with R b = HP b H T v = H u Analysis Error Perturbation: X a = X b W a : W a = (I + ( V b ) T (R o ) -1 V b ) -1/2 partial obs U a (S a ) 2 (U a ) T = U b (T a ) 2 (U b ) T : (T a ) 2 =S b (F b ) T (W a ) 2 F b S b - Tends to knock down lower order e-vectors by analysis 3 - Den a < Den b x a α P b n=1 = (σ b x b space of instability by U b ) 2 (v b ) T (R b +R o ) 1 d R y o o x xa P a Instability is suppressed Δx a for non-observed variable by correlaion x x 1

9 Effect of Instability on P en Low dimensionality ß dynamic instability in model forecast (one way) blobby P en skinny P en with larger uncertainty tr P en x b en,m x a en,m y o

10 Local E-Diagnostics E-Diagnostics: X en = U en S en V en T For a complex system with spatio-temporal chaos, e-diagnostics should be performed locally. General E-dim based on S en = diag {(σ en (1),, σ en (N*) =0) - Smaller dimension may be associated with dynamic instability and ensemble perturbation growth (reversely related to ensemble spread E-vector: U en = ( (1),, (N*) ) in N-dim - The lower the mode, the more power it has - When E-dim is small, the lower mode may represent the direction of instability Projection of X en onto U en : V en = (v en (1),, v en (N*) ) - Can be used to study distribution of ensemble members v en in M-dim

11 Application to Ensemble from GFS/GDAS Hybrid Hybrid 3DEnVar system (just replaced by Hybrid 4DEnVar) T574 deterministic T member ensemble Dates 12/09-14/13 [Selected for space weather Interests] Model state for diagnostics X en : (u,v,t) E-diagnostics Horizontal local box - 19x19 grid - 39x39 grid

12 Local E-Dimension and Synoptic Conditions z: T at 500 mb X : en N x M dimensional - N=1083=19*19*3 (u,v,t) - M=80 ensemble Local E-dim based on X en Locally lower E-dim? Background - Ensemble mean - Ensemble spread Low-dim dynamic instability? x mean and var(x en ) Locally large spread? (K) Large uncertainty

13 Local E-dim Difference (LETKF)- (EnSRF) EnSRF vs LETKF: Background Local spread std Difference Very similar (K)

14 EnSRF: Local E-dim 34.3 EnSRF vs LETKF: Background (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

15 LETKF: Local E-dim 39.4 EnSRF vs LETKF: Background (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en ) Very similar - sign can flip

16 Local E-vector correlation EnSRF vs LETKF: Background LETKF E-vector mode Occasional flips EnSRF E-vector mode EnSRF and LETKF have very similar ensemble perturbation structures, from P-view point

17 Difference: Background-Analysis E-dim EnSRF Analysis Spread Analysis E-vector mode Background E-vector mode Knocking down errors of the day

18 EnSRF Analysis Difference: Analysis pre- & post- inflation Spread Post-Inflation Analysis E-vector mode E-dim Analysis E-vector mode RTPS putting back errors of the day???

19 Localization Size z: T at 500 mb X : en N x M dimensional - N=1083=19*19*3 (u,v,t) - M=80 ensemble Local E-dim based on X en Locally lower E-dim? Background - Ensemble mean - Ensemble spread Low-dim dynamic instability? x mean and var(x en ) Locally large spread? (K) Large uncertainty

20 Localization Size Smaller scale modes 39x39 box 19x19 box

21 Local E-Dimension and Synoptic Conditions z: T at 500 mb X : en N x M dimensional - N=1083=19*19*3 (u,v,t) - M=80 ensemble Local E-dim based on X en Locally lower E-dim? Background - Ensemble mean - Ensemble spread Low-dim dynamic instability? x mean and var(x en ) Locally large spread? (K) Large uncertainty

22 Case Study: Evolution 500mb T local E-diagnostics : E-dim=35.3 (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

23 Case Study: Evolution 500mb T local E-diagnostics : E-dim=36.8 (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

24 : E-dim=36.1 Evolution 500mb T local E-diagnostics (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

25 : E-dim=36.8 Evolution 500mb T local E-diagnostics (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

26 : 850mb T E-dim 26.9 Vertical Variation (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en )

27 Vertical Variation : 825mb T E-dim 33.5 (σ en ) 2 (1) (2) (3) E-dim (4) (5) (6) x mn (7) (8) (9) std(x en ) Eastward tilt between 825 and 850mb

28 Summary Local E-diagnostics: One way to diagnose Local instability represented by ensemble spread Application to GSI Based on P, EnSRF and LETKF are extremely similar - Due to the same inflation (RTPS)? Background vs Analysis - DA supresses the erros of the day. - Impact of stochastic perturbation in model forecast?

29 Thank You. Questions?