EOT2

EOT1?EOT

Empirical Methods in Short-Term Climate PredictionHuug van den Dool

Price: £49.95 (Hardback)ISBN-10: 0-19-920278-8

ISBN-13: 978-0-19-920278-2Estimated publication date: November 2006

Oxford University Press

Foreword , Professor Edward Lorenz (Massachusetts Institute of Technology) Preface 1. Introduction 2. Background on orthogonal functions and covariance 3. Empirical wave propagation 4. Teleconnections 5. Empirical orthogonal functions 6. Degrees of freedom 7. Analogues 8. Methods in short-term climate prediction 9. The practice of short-term climate prediction 10. Conclusion References Index

Chapter 8. Methods in Short-Term Climate Prediction 1218.1 Climatology 1228.2 Persistence 1248.3 Optimal climate normals 1268.4 Local regression 1298.5 Non-local regression and ENSO 1358.6 Composites 1388.7 Regression on the pattern level 1398.7.1 The time-lagged covariance matrix 1398.7.2 CCA, SVD and EOT2 1418.7.3 LIM, POP and Markov 1458.8 Numerical methods 1468.9 Consolidation 1478.10 Other methods 1498.11 Methods not used 151Appendix 1: Some practical space–time continuity requirements 152Appendix 2: Consolidation by ridge regression 153

EOFBoth Q and Qa Diagonalized

(1) satisfied – Both αm and em orthogonalαm (em) is eigenvector of Qa(Q)

EOT-normalQ is diagonalized

Qa is not diagonalized(1) is satisfied with αm orthogonal

EOT-alternativeQ is not diagonalized

Qa is diagonalized(1) is satisfied with em orthogonal

Rotation

Laudable goal:f(s,t)=∑ αm(t)em(s) (1)

m

iterationRotation

Matrix Q with elements:qij=∑ f(si,t)f(sj,t)

t

Matrix Qa with elements:qij

a=∑f(s,ti)f(s,tj)s

Qa tells about Analogues

Q tells about Teleconnections

Fig.5.6: Summary of EOT/F procedures.

iteration

Discrete Data set f(s,t)1 ≤ t ≤ nt ; 1 ≤ s ≤ ns Arbitrary

state

EOFBoth Q and Qa Diagonalized

(1) satisfied – Both αm and em orthogonalαm (em) is eigenvector of Qa(Q)

EOT-normalQ is diagonalized

Qa is not diagonalized(1) is satisfied with αm orthogonal

EOT-alternativeQ is not diagonalized

Qa is diagonalized(1) is satisfied with em orthogonal

Rotation

Arbitrarystate

Laudable goal:f(s,t)=∑ αm(t)em(s) (1)

m

iterationRotation

Matrix Q with elements:qij=∑ f(si,t)f(sj,t)

t

Matrix Qa with elements:qij

a=∑f(s,ti)f(s,tj)s

Qa tells about Analogues

Q tells about Teleconnections

Fig.5.6: Summary of EOT/F procedures.

Discrete Data set f(s,t)1 ≤ t ≤ nt ; 1 ≤ s ≤ ns

Iteration (as per power method)

iteration

Fig. 4.3 EV(i),the variance explained by single gridpoints in % of the total variance, using equation 4.3. In the upper left for raw data, in the upper right after removal of the first EOT mode, lower left after removal of the first two modes. Contours every 4%. The timeseries shown are the residual height anomaly at the gridpoint that explains the most of the remaining domain integrated variance.

Fig.4.4 Display of four leading EOT for seasonal (JFM) mean 500 mb height. Shown are the regression coefficient between the height at the basepoint and the height at all other gridpoints (maps) and the timeseries of residual 500mb height anomaly (geopotential meters) at the basepoints. In the upper left for raw data, in the upper right after removal of the first EOT mode, lower left after removal of the first two modes. Contours every 0.2, starting contours +/- 0.1. Data source: NCEP Global Reanalysis. Period 1948-2005. Domain 20N-90N

Fig.5.4 Display of four leading alternative EOT for seasonal (JFM) mean 500 mb height.

Shown are the regression coefficient between the basepoint

in time (1989 etc) and all other years (timeseries) and the maps

of 500mb height anomaly (geopotential meters) observed in 1989, 1955 etc . In the upper left for raw data, in the upper right after removal of the first EOT

mode, lower left after removal of the first two modes. A

postprocessing is applied, see Appendix I, such that the

physical units (gpm) are in the time series, and the maps have norm=1. Contours every 0.2, starting contours +/- 0.1. Data

source: NCEP Global Reanalysis. Period 1948-2005. Domain 20N-

90N

0

5

10

15

20

25

EV

0

20

40

60

80

100

EV-cumulative

1 2 3 4 5 6 7 8 910111213141516171819202122232425

mode number

eof

eot

eot-a

EV as a function of modeJFM Z500

Fig 5.7. Explained Variance (EV) as a function of mode (m=1,25) for seasonal mean (JFM) Z500, 20N-90N, 1948-2005. Shown are both EV(m) (scale on the left, triangles) and cumulative EV(m) (scale on the right, squares). Red lines are for

EOF, and blue and green for EOT and alternative EOT respectively.

Laudable goals:f(s,t) = ∑ αm(t)em(s) (1)

m g(s,t +τ) = ∑ βm(t +τ)dm(s) (2)

mconstrained by a connection between

α and β and/or e and d.

Cross Cov Matrix Cfgwith elements:

cij= ∑ f(si,t)g(sj,t+ τ)/nt

t

Fig.x.y: Summary of EOT2 procedures.

Discrete Data set f(s,t)1 ≤ t ≤ nt ; 1 ≤ s ≤ ns

Discrete Data set g(s,t +τ)1 ≤ t ≤ nt ; 1 ≤ s ≤ n’s

Alt Cross Cov Matrix Cafgwith elements:

caij= ∑ f(s,ti)g(s,tj+ τ)/ns

s

EOT2-alternative(1) and (2) satisfied

em and dm orthogonalαm and βm heterogeneously orthogonal Ca

ff (τ=0), Cagg(τ=0) and Ca

fg diagonalized

Two time series, one map.

EOT2(1) and (2) satisfied

αm and βm orthogonal (homo-and-heterogeneous

Cff (τ=0), Cgg(τ=0) and Cfg diagonalized

One time series, two maps.

CCAVery close to EOT2, but

two, maximally correlated, time series.

SVDSomewhat like EOT2a, but

two maps, and (heterogeneously) orthogonal time series.

Laudable goals:f(s,t) = ∑ αm(t)em(s) (1)

m g(s,t +τ) = ∑ βm(t +τ)dm(s) (2)

mConstrained by a connection between

α and β and/or e and d.

Cross Cov Matrix Cfgwith elements:

cij= ∑ f(si,t)g(sj,t+ τ)/nt

t

Fig.x.y: Summary of EOT2 procedures.

Discrete Data set f(s,t)1 ≤ t ≤ nt ; 1 ≤ s ≤ ns

Discrete Data set g(s,t +τ)1 ≤ t ≤ nt ; 1 ≤ s ≤ n’s

Alt Cross Cov Matrix Cafgwith elements:

caij= ∑ f(s,ti)g(s,tj+ τ)/ns

s

EOT2-alternative(1) and (2) satisfied

em and dm orthogonal

Caff (τ=0), Ca

gg(τ=0) and Cafg diagonalized

Two time series, one map.

EOT2(1) and (2) satisfied

αm and βm orthogonal

Cff (τ=0), Cgg(τ=0) and Cfg diagonalized

One time series, two maps.

CCAVery close to EOT2, but

two, maximally correlated, time series.

SVDSomewhat like EOT2a, but

two maps, and (heterogeneously) orthogonal time series.

M = Qf-1 Cfg Qg

-1 CfgT

CCA:

1) Make a square M = Qf-1 Cfg Qg

-1 CfgT

2) E-1 M E =diag ( λ1 , λ2, λ3,… λM)

cor(m)=sqrt (λm)

SVD:

1) UT Cfg V =diag (σ1 , σ2, … , σm)

Explained Squared Covariance = σ2m

Assorted issues:

1) Prefiltering f and g , before calculating Cfg

2) Alternative approach complicated when domains for f and g don’t match

3) Iteration and rotation: CCA > EOT2-normal; SVD > EOT2-alternative ???

Keep in mind

• EV (EOF/EOT) and EOT2

• Squared covariance (SC) in SVD

• SVD singular vectors of C

• CCA eigenvectors of M

• LIM complex eigenvectors of L (close to C)

• MRK no modes are calculated (of L)

En dan nu iets geheel anders…..

                  T-obs T-CA   diff 1991-2006   .52C  .10C   .42C     100.00%  (all gridpoints north of 20N and anomalies relative to 61-90.)

We are warmer by 0.52C, and the circulation explains only 0.10.  So has T850 risen 0.42 ±error due to factor X ???

The plain difference CA minus Obs has a problem, namely that CA (like a regression) 'damps'. (The degree of damping is related to the success of the specification as measured by a correlation). Obs minus CA thus tends to have the sign of the observed anomaly (which is warm these days).  A reasonable explanatory equation reads :    T_obs   =  T_CA *  1/d   + delta, where delta is the true temperature change we are seeking and the 1/d factor undoes the damping of the specification method.

Two unknowns (d(amping) and delta), and I make two equations by doing an averaging separately for gridpoints where T850 was observed above (62.5% of the cases in Jan) and below the mean (37.5%).  (This is a shortcut for a regression thru thousands of points (years times gridpoints))/ For instance, in January we find:                  T-obs T-CA  diff 1991-2006  1.65C   .82C   .83C      62.53%   (at points where T-obs anomaly is positive) 1991-2006 -1.35C -1.09C  -.26C      37.47%        (at points where T-obs anomaly is negative)

Notice that cold anomalies, although covering less area, are also damped. I assume that damping (d; 0<d<1) is the same for cold and warm anomalies. Substituting the T-obs and T-CA twice we get two equations and find (for January):

damp= 0.64 and delta=0.37

I am now building a Table as follows: Month    Damping    Delta 1         0.64         0.37 2         0.67         0.27 3         0.67         0.33 4         0.58         0.15 5         0.57         0.19 6         0.56         0.31 7         0.48         0.33 8         0.50         0.22 9 (sep) 0.51        0.47