[10.5.5]). We assume that this integrated noise is

white on the inter-cycle time scale.

A conventional POP analysis can be applied

to each of the n models described by (15.25).

This results in n collections of eigenvectors p „

and eigenvalues »„ that are obtained from the n

eigenproblems

B „ p „ = »„ p „ . (15.27)

As usual, all eigenvectors are normalized to

unit length. Note that the eigenvalues »„ are

independent of „ , because

B „ p „ = »„ p „

” A„ +n B „ p „ = »„ A„ +n p „

” B „ +1 A„ p „ = »„ A„ p „ .

The last step is a consequence of (15.26) and the

periodicity of A„ .

Thus we now have a recursive relationship that

can generate eigenvectors for all n eigenproblems

by solving only the ¬rst problem. That is, A„ p „ is

Figure 15.11: MJO: Skill scores of POP (solid) and an eigenvector of B „ +1 when p „ is an eigenvector

persistence (dashed) forecasts of the MJO. From of B „ . These eigenvectors are unique up to

multiplication by a complex constant.13 If we now

von Storch and Xu [401].

a) Correlation skill (15.22), ρ„ . normalize p 1 to unit length and set

b) Root mean square error (15.23), S„ .

’1

p „ +1 = r„ eiφ A„ p „ (15.28)

where r„ = A„ p „ , φ = ·/n, and · satis¬es

15.4.1 De¬nition. Assume that time is given

»„ = » = ξ e’1· , then the resulting eigenvectors

by a pair of integers (t, „ ), where t counts the

cycles (e.g., annual cycle), and „ indicates the will be unique up to multiplication by a factor eiθ ,

and will be periodic (i.e., p „ +n = p „ ). Thus the

˜seasonal date™ (e.g., months), or time steps within

cyclo-stationary POP is damped by the factor ξ

a cycle. Assume that a cycle has n time steps so

and rotated by an angle ’· in one cycle.

that „ = 1, . . . , n. Note that (t, n + 1) = (t + 1, 1)

or, generally, (t, „ + n) = (t + 1, „ ). As with The cyclo-stationary POP coef¬cients evolve in

ordinary POP analysis, we then assume that the time as a cyclo-stationary auto-regression that is

cyclo-stationary process can be approximated by similar to the auto-regression (15.10) that applies

to ordinary POP coef¬cients. Speci¬cally,

Xt,„ +1 = A„ Xt,„ + noise (15.24)

z t,„ +1 = r„ e’iφ z t,„ + noise. (15.29)

where Xt,„ +n = Xt+1,„ and A„ +n = A„ . 13 We assume throughout that B , „ = 1, . . . , n, (and hence

„

A„ ) are non-singular and that all eigenvalues of B„ are distinct.

Substituting (15.24) into itself n consecutive times,

15: POP Analysis

348

15.4.2 Example: The Southern Oscillation.

Time series of surface wind and SST along the

equator between 50—¦ E and 80—¦ W (described in

[15.2.2]) are good candidates for a cyclo-stationary

POP analysis because the Southern Oscillation

is known to be phase-locked to the annual

cycle [330]. Monthly anomalies are analysed so

that n = 12. The data are time-¬ltered to suppress

the month-to-month variability. A conventional

POP analysis was performed for comparison.

Both analyses identi¬ed a single dominant

POP with comparable periods (31 months for

the cyclo-stationary analysis, 34 months for the

conventional analysis). The mode identi¬ed in the

conventional analysis is similar to the ENSO mode

described in [15.2.3] (see Figure 15.7, bottom, and

Figure 15.8b).

The amplitude, r„ , exhibits a marked annual

cycle (Figure 15.12) which is strongly non-

sinusoidal. Ampli¬cation takes place from April

to September, with a maximum in June. The

process is damped from October to March, with

a minimum in February. Note that the amplitude

Figure 15.12: ENSO: Amplitudes obtained in the

increases from minimum to maximum in only four

conventional and cyclo-stationary POP analyses

months, but then it takes eight months to return

of equatorial 10 m wind and sea-surface tempera-

to minimum. The annually averaged amplitude is

ture. Bars labelled ˜J™, ˜F™, etc., indicate the ampli-

almost identical to the amplitude obtained in the

tudes obtained from the cyclo-stationary analysis

conventional analysis.

in January, February, etc. The bar labelled ˜J“D™

The zonal wind patterns (Figure 15.13, left

is the amplitude obtained from the conventional

column) show eastward progression of the main

analysis.

centre of action with the annual cycle. The

imaginary component is strongest during the ¬rst

Substituting (15.29) into itself n times, we obtain half of the year whereas the real component is

a conventional auto-regression strongest during the second half.

n The imaginary part of the SST patterns

r„ +s+1 e’i· z t,„ + noise

z t+1,„ = (Figure 15.13, top right) has substantial amplitude

s=1 in the Indian Ocean and East Paci¬c in northern

= »z t,„ + noise winter, but not at other times of year. In contrast,

the real component (Figure 15.13, bottom right)

for POP coef¬cients at one cycle increments that is

has large amplitude (at least 0.2) throughout the

consistent with model (15.25).

year in the East Paci¬c that coincides with large

The time coef¬cients at a given time t may be

amplitudes of opposite sign in the West Paci¬c.

obtained by projecting the full ¬eld Xt,„ onto the

„ The signal in the real pattern is strongest in the East

respective adjoint p a or by using a least square

Paci¬c in northern fall.

approximation similar to (15.14) and (15.15). The

adjoint patterns p a and p a +1 are related to each

„ „ The average of these cyclo-stationary modes

is similar to the pattern obtained from the

other through a simple formula similar to (15.28):

conventional POP analysis described in [15.2.3]

’1

p a = r„ eiφ AT p a +1 .

„ „ (15.30)

„ (Figure 15.8, bottom).

The cyclo-stationary system matrices A„ can be Note that the wind data were normalized

estimated with (15.6) for each „ = 1, . . . , n as into unit variance before the POP analysis. To

’1 transform the patterns to meaningful physical

A„ = Σ„,1 Σ„,0 ,

units, the wind patterns (Figure 15.13, left)

where Σ„,1 is the estimated lag-1 cross-covariance must be multiplied by 0.45 times the standard

matrix between Xt,„ and Xt,„ +1 , and Σ„,0 is the deviations of the POP coef¬cients (Figure 15.14).

estimated covariance matrix of Xt,„ . Similarly, typical SST amplitudes are obtained

15.4: Cyclo-stationary POP Analysis 349

Figure 15.13: ENSO: Cyclo-stationary POPs analysed from a combined normalized zonal wind/SST

data set. The horizontal axis represents the longitude along the equator, and the vertical axis the annual

cycle of the patterns.

Top row: Imaginary part. Bottom row: Real part.

variance extremes are delayed relative to those of

the amplitudes.

As with conventional POPs, it is possible to

build scenarios that describe the ˜typical™ evolution

of the ¬eld from a given initial state. Suppose that

a ¬eld X is well represented by a cyclo-stationary