modes similar to those found by conventional

signal. These data were low-pass ¬ltered to remove

perturbation analysis, which obtains the matrix

variability on time scales shorter than 15 months.

from ¬rst-principle dynamical reasoning.

The three equatorial data sets, stratospheric

15.2.3 The Southern Oscillation and the Quasi- wind, zonal surface wind, and SST, were subjected

Biennial Oscillation. In this subsection we to a joint POP analysis. The three components

describe how POP analysis was used by Xu [430] were normalized so that they contributed equal

to examine two oscillations in the tropical amounts of variance to the combined data set.

15.2: Examples 343

Figure 15.7: QBO and SO: Two POPs obtained

from a joint POP analysis of zonal 10 m wind, SST,

and stratospheric zonal wind. The real part of each

POP (light curve) is labelled p 2 and the imaginary

part (heavy curve), p 1 . From Xu [430].

Top: The 28-month mode representing the Quasi-

Biennial Oscillation (QBO),

Bottom: The 30-month mode representing the

Southern Oscillation (SO).

Two signi¬cant POP pairs were found, one with

an oscillation period of 28 months, and the other

with a period of 45 months. Cross-spectral analysis

of the POP coef¬cients (not shown) indicates that

the 28-month period is reliably estimated, but that

the period of the ˜45-month™ POP is overestimated.

Figure 15.8: QBO and SO: scatter plots of the

A more realistic estimate of its oscillation period

complex POP coef¬cients associated with the

is approximately 30 months. The two modes are

patterns shown in Figure 15.7. From Xu [430].

shown in Figure 15.7.

Top: The coef¬cients of the QBO mode.

The ¬rst mode (Figure 15.7, upper panel) carries

Bottom: The coef¬cients of the SO mode.

useful information only in the stratosphere where

it represents the downward propagation of a

signal from the upper stratosphere to the lower with the oscillatory intervals. When the Southern

stratosphere over a 14-month period. The POP Oscillation is quiet, the POP coef¬cients are small

coef¬cient time series oscillates regularly (not and noisy. The POP coef¬cients have a unimodal

shown), and occupies a torus-shaped region in distribution in phase space (Figure 15.8, bottom).

phase space (Figure 15.8, top). These modes represent the QBO and the SO,

The second mode, on the other hand, only respectively. They are essentially uncorrelated.

carries useful information at the surface in the SST

and 10 m zonal wind. It describes a 10 m wind

signal that propagates eastward from the Indian 15.2.4 The Madden-and-Julian Oscillation:

Ocean into the Paci¬c, and an almost stationary Sensitivity to Analysis Time-interval and Anal-

feature of SST variability. The POP coef¬cient ysis Area. The Madden-and-Julian Oscillation

time series sometimes oscillate regularly, and the (MJO), also known as the tropical 30“60 day oscil-

occurrence of El Ni˜ o and La Ni˜ a events coincide

n n lation, is particularly well represented in equatorial

15: POP Analysis

344

tropospheric velocity potential. This subsection

describes a POP analysis of ¬ve years NMC8 “

analysed 200 hPa velocity potential from which

the annual cycle was removed. The data cover the

period May 1984 to April 1989.

Six POP analyses were performed in total on

various subsets of the data (see [401]). Two

analyses, ˜A™ and ˜B™, use data along the entire

equator. ˜A™ uses a two-year subset and ˜B™ uses the

whole ¬ve-year data set. Four additional analyses,

labelled ˜C™ to ˜F™, use spatial subsets of the data

that extend over the full ¬ve years. ˜C™ uses data

between 0—¦ and 90—¦ W, ˜D™ uses data from 90—¦ W

to the date line, and so on.

One physically important POP was identi¬ed in

each of the six analyses. The POPs from analyses

˜B™ to ˜F™ were rotated so that their p r patterns

match that obtained from analysis ˜A™ as closely as

possible.9

The POP obtained in the ˜A™-analysis has a

period of 44 days, and an e-folding time of 13

days (about 30% of the period). The squared

coherency of the POP coef¬cients is larger than

68% on time scales between 20 and 50 days

with a maximum value of 96% at 50 days. The

Figure 15.9: MJO: The real (labelled p 1 )

real and imaginary parts of the POP are shown

and imaginary (labelled p 2 ) POPs of equatorial

as solid lines in Figure 15.9a. They are zonal

200 mb velocity potential. From von Storch and Xu

wavenumber 1 type patterns with one minimum

and one maximum. The two patterns are about 90—¦ [401].

a) Analysis of equatorial data from a two-year

out-of-phase, indicating eastward propagation of

subset (analysis ˜A™; solid line) and from the

the signal. The trough and the crest do not move

complete ¬ve-year data set (analysis ˜B™; dashed

at a constant rate.

line).

The pattern in Figure 15.9a is very robust: the

b) The ˜C™ to ˜F™ analyses for 90—¦ -sectors along the

extra three years of data in the ˜B™ analysis (dashed

equator. The real patterns are plotted with a solid

curve) resulted in very little change.

Data in adjacent 90—¦ sectors were considered line, and the imaginary patterns with a dashed

in analyses ˜C™ to ˜F™. The 90—¦ -sector patterns line. Patterns from analysis ˜A™ are shown in dots

resemble the full 360—¦ patterns (Figure 15.9b) for comparison.

closely. The p r patterns appear to match their ˜A™

counterpart somewhat better than the p i patterns

because the rotation was optimized on the former. observation that the 30- to 60-day oscillation is

The e-folding times in the 90—¦ sectors are markedly stronger in the eastern hemisphere.

considerably smaller than in ˜A™ and ˜B™. This

The differences in the periods in the four

difference is reasonable, since the POPs describe a

90—¦ sectors are consistent with the variable

global, travelling feature. Thus the memory in the

longitudinal phase speed of the MJO. The 30“60

system is retained for a longer time in the full 360—¦

day waves travel most slowly in the 90—¦ E to 180—¦

circle than in the 90—¦ sectors. Interestingly, the

sector: the period in this sector was found to be

damping time in the eastern hemisphere (7 days)

62 days. The waves travel most quickly, and the

is about double that in the western hemisphere

period is shortest (33 days), in the 180—¦ to 90—¦ W

(4 days). This ¬nding is consistent with the

sector. The average period for analyses ˜C™ to ˜F™

is 45 days, which is nearly identical to the value

8 National Meteorological Center.

obtained in analyses ˜A™ and ˜B™. Thus ˜C™ to

9 That is, p = p r +i p i was multiplied by eiθ for a suitably

˜F™ further emphasize the robustness of the MJO

chosen θ. This is acceptable since eigenvector p of A can be

uniquely determined only up to a factor eiθ . signal extracted using the POP method.

15.3: POPs as a Predictive Tool 345

15.3 POPs as a Predictive Tool POP forecast. An appropriate POP forecast in this

case is that the system will stay in its ˜quiet phase.™

15.3.1 The POP forecast technique. Fore-

casting is a natural part of the POP ansatz (see, e.g.,

15.3.2 Measures of Skill. The quality of

[432, 429, 401]), because the POP coef¬cient time

the POP forecasts can be determined with the

series evolve similarly to AR(1) processes (i.e.,

correlation skill score ρ„ (18.3) and the root mean

as in (15.11)). Assuming that the forcing noise in

square error S„ (18.1)10

(15.11) is white, the optimal lag-„ forecast of z t+„

from z t is given by Cov z tF„ , z t