El Niño, the Trend, and SST Variability

orIsolating El Niño

Cécile Penland and Ludmila MatrosovaNOAA-CIRES/Climate Diagnostics Center

Review of Linear Inverse Modeling

Assume linear dynamics: dx/dt = Bx +

Diagnose Green function from data: G() = exp(B)

= <x(t+)xT>< x(t)xT>-1 .

Eigenvectors of G() are the normal modes {ui}.

Most probable prediction: x’(t+) = G() x(t)

Optimal initial structure for growth over lead time :

Right singular vector of G() (eigenvector of GTG() )

Growth factor over lead time : Eigenvalue of GTG().

SST Data used:

• COADS (1950-2000) SSTs in the tropical strip 30N – 30S.

• Subjected to 3-month running mean.• Projected onto 20 EOFs (eigenvectors of <xxT>)

containing 66% of the variance.• x, then, represents the vector of SST anomalies,

each component representing a location, or else it represents the vector of Principal Components.

• This is what we call “unfiltered” data.

This optimal initial pattern…

…evolves into this one 6 to 9 months later.

Cor. = 0.65

T3.

4(t)

Pat. Cor. (SST,O.S.)(t – 8mo)

Decay mode, = 31 months

0

0.5

1

1.5

0 5 10 15 20 25Mode number

momo

momo

momo

decay timeT = Period

Projection of adjoints onto O.S. and modal timescales.

-15

-10

-5

0

5

10

1950 1960 1970 1980 1990 2000 2010Date

EOF 1 of Residual

u1 of un-filtered data The pattern correlation

between the longest-lived mode of the unfiltered data and the leading EOF of the residual data is 0.81.

Location of indices: N3.4, IND, NTA, EA, and STA.

-3

-2

-1

0

1

2

3

1950 1960 1970 1980 1990 2000Date

-3

-2

-1

0

1

2

3

1950 1960 1970 1980 1990 2000Date

-2

-1

0

1

2

1950 1960 1970 1980 1990 2000Date

El Niño

El Niño + Trend

Background

Niño 3.4 Time Series

10-4

10-3

10-2

10-1

100

101

1 10 100 1000Period (months)

Red: Spectrum of unfiltered Niño 3.4 SSTA

Blue: Spectrum of residual Niño 3.4 SSTA

-5

0

5

10

15

20

25

1 10 100 1000Period (months)

66.4 mo39.9 mo18.1 mo

15.3 mo

Spectral difference: (Spectrum of unfiltered data – spectrum of residual) / Spectrum of residual.

-505

101520253035

100 101 102 103 104

43.9 mo

16.5 mo

5.2 mo

Period (weeks)

Weekly SST data with its own climatology removed, then projected onto COADS EOFs.

0

0.5

1

1.5

0 5 10 15 20 25Mode number

momo

momo

momo

decay timeT = Period

Projection of adjoints onto O.S. and modal timescales.

Trend mode = 31mo

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.36

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.44

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.45

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino) = 0.61

EA

STA

IND

NTA

R = 0.36 R = 0.45

R = 0.44 R = 0.61

Indices. Black: Unfiltered data. Red: El Niño signal.

-0.8

-0.6

-0.4

-0.2

0

0.2

-100 -50 0 50 100Lead (months)

8 months

STA leads PC1 leads

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-100 -50 0 50 100Lead (months)

IND leads PC1 leads

-0.6

-0.4

-0.2

0

0.2

0.4

-100 -50 0 50 100Lead (months)

EA leads PC1 leads

9 months

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-100 -50 0 50 100Lead (months)

NTA leads PC1 leads

Lagged correlation between El Niño indices and PC 1.

STA leads PC1 leads PC1 leads

PC1 leads PC1 leads

EA leads

IND leads NTA leads

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino +Trend) = 0.75

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino+Trend) = 0.79

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino+Trend) = 0.77

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino + Trend) = 0.62

EA S

STA

(C)

STA

SST

A (C

)IN

D S

STA

(C)

NTA

SST

A (C

)

R = 0.75 R = 0.77

R = 0.79 R = 0.62

Indices. Black: Unfiltered data. Green: El Niño signal + Trend.

…evolves into this one 6 to 9 months later.

Cor. = 0.65

T3.

4 (t)

Pat. Cor. (SST,O.S.)(t-8mo)

This optimal initial condition…

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25Lead (months)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25Lead (months)

Black: “Unfiltered”

Red: El Niño

Green: El Niño + Trend

Blue: El Niño + Parabolic Trend

MA Curve

Lagged correlation C(): O.S., Niño 3.4

Eige

nval

ue o

f G

T G(

) and

ex

pect

ed e

rror

.

Error variance normalized to climatology

Niñ

o 3.

4 (A

R1

Erro

r Var

ianc

e)N

iño3

.4 (E

xpec

ted

Erro

r Var

ianc

e)N

iño3

.4 (O

bser

ved

Erro

r Var

ianc

e)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

Error variance normalized to climatology

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

IND

(AR

1 Er

ror

Var

ianc

e)IN

D (E

xpec

ted

Erro

r Var

ianc

e)IN

D (O

bser

ved

Erro

r Var

ianc

e)

NTA

(AR

1 Error V

ariance)N

TA (Expected

Error Variance)

NTA

(Observed

Error Variance)

Error variance normalized to climatology

EA (E

xpec

ted

Erro

r Var

ianc

e)EA

(Obs

erve

d Er

ror V

aria

nce)

STA (A

R1

Error Variance)

STA (Expected

Error Variance)

STA (O

bserved Error V

ariance)0

0.5

1

1.5

0 5 10 15 20 25Lead (months)0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

0

0.5

1

1.5

0 5 10 15 20 25Lead (months)

EA (A

R1

Erro

r V

aria

nce)

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R=0.36

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R=0.30

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R=0.36

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R=0.48

Black: “Unfiltered” data. Blue: Background (No Niño, no Trend)

R = 0.36 R = 0.36

R = 0.30 R = 0.48

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R = -0.37; Trend = u1

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R = -0.49; Trend = PC1 of residual

BLUE: NTA

No Niño, No Trend

RED: STA

No Niño, No Trend

Conclusions• Two different ways of identifying the trend lead to

qualitatively similar results.• The pattern-based filter can be applied to data of

any temporal resolution.• The El Niño signals in the tropical Indian and

North tropical Atlantic are highly correlated (R = 0.84).

• El Niño signals in EA and STA precede that in Niño 3.4 by about 8 months. This won’t help the predictions, though.

Conclusions (cont.)

• El Niño plus the trend appear to dominate SSTA variability in IND, EA and STA.

• The trend seems to cause overestimation of nonmodal growth of El Niño.

• Isolating the signals with this filter seems to be more valuable for diagnosis than prediction, except in IND.

• The tropical Atlantic dipole is significant in the background SSTA field.