easterly ¬‚ow across the northern North Sea. The

3 model

50th percentile of the signi¬cant wave heights is

hindcast

increased by 32 cm, while the 90th percentile is

2

reduced by 25 cm. Thus there is a tendency for

1

the wave height distribution to be widened when

pressure anomaly pattern p 2 prevails. The re-

0

versed pattern goes with a narrowed intramonthly

-1

distribution of wave heights. This pair of patterns

accounts for only 5% of the predictable wave

-2

height variance.

-3

The regression model incorporated in the

redundancy analysis was used to estimate the time

-4

1895 1905 1915 1925 1935 1945 1955 1965 1975 1985 1995

series of the percentiles of signi¬cant wave height

in the Brent oil ¬eld from the observed monthly

Figure 14.10: Reconstructed (dashed line) and mean air pressure anomaly ¬elds between 1899

hindcasted (continuous line; 1955“94) anomalies and 1994. The last 40 years may be compared

of the 90th percentile of signi¬cant wave height at with the hindcast data, whereas the ¬rst 50 years

in the Brent oil ¬eld. Units: m. represent our best guess and can not be veri¬ed at

this time. The 90th percentiles of the reconstructed

wave height time series for 1899“94 and the

the three components of a k may be interpreted corresponding hindcasted time series for 1955“94

are shown in Figure 14.10. The link appears to be

as typical anomalies that occur when the pressure

¬eld anomalies are given by √1 p k . strong, as is demonstrated by the correlations and

» k

the proportion of described variance, during the

The pattern a 1 accounts for 94% of the variance

overlapping period:

of Y, and a 2 for 5%. The correlation between the

coef¬cient time series of the ¬rst pair of vectors Wave height percentile

is 0.84 while that between the second pair is only 50% 80% 90%

0.08. Thus, the ¬rst pair establishes a regression Correlation 0.83 0.82 0.77

representing R 2 (Y : B T X) = 94% — 0.842 = Described variance

1 0.70 0.66 0.60

66% of the variance of Y, whereas the second pair

represents only 5% — 0.082 < 0.1% of variance. The amount of percentile variance represented by

Thus the redundancy index for k = 1 (0.66) can the SLP patterns is consistent with the redundancy

index (14.53), which has value 0.66. As with all

not be usefully increased by adding a second vecor.

regression models, the variance of the estimator is

The ¬rst air-pressure pattern is closely related

smaller than the variance of the original variable.

to the North Atlantic Oscillation (see [13.5.5]

This makes sense, since the details of the wave

and Figure 13.6). A weakening of the NAO is

action in a month are not completely determined

associated with a decrease in all three intramonthly

by the monthly mean air-pressure ¬eld. It is also

percentiles of signi¬cant wave height. In effect,

affected by variations in surface wind that occur

this pattern describes a shift of the intramonthly

on shorter time scales.

distribution towards smaller waves.

This Page Intentionally Left Blank

15 POP Analysis

15.0.1 Summary. The Principal Oscillation The POP method is not useful in all applica-

Pattern (POP) analysis is a linear multivariate tions. If the analysed vector time series exhibits

technique used to empirically infer the character- strongly nonlinear behaviour, as in, for example,

istics of the space-time variations of a complex the day-to-day weather variability in the extrat-

system in a high-dimensional space [167, 389]. ropical atmospheric ¬‚ow, a POP analysis will not

The basic approach is to identify and ¬t a linear be useful because a low-dimensional linear sub-

low-order system with a few free parameters. The system does not control a signi¬cant portion of the

space-time characteristics of the ¬tted system are variability. The POP method will be useful if there

then assured to be representative of the full system. are a priori indications that the processes under

This chapter is organized as follows. POPs consideration are linear to ¬rst approximation.

are introduced as normal modes of a discretized

linear system in Section 15.1. Three POP analyses

15.1 Principal Oscillation Patterns

are given in Section 15.2. Since a POP analysis

includes the ¬tting of a time series model to 15.1.1 Normal Modes. The normal modes of a

data, the POP approach has predictive potential discretized real linear system

(Section 15.3). Cyclo-stationary POP analysis is

Xt+1 = AXt (15.1)

explained in Section 15.4. Another generalization,

the Hilbert or ˜complex™ POPs, is introduced are the eigenvectors p of the matrix A. In

brie¬‚y in [16.3.15]. general, A is not symmetric and some or all of

POP models may also be viewed as simpli¬ed its eigenvalues » and eigenvectors p are complex.

state space models. Such models, and in particular However, since A is a real matrix, the complex

the Principal Interaction Pattern (PIP) ansatz1 conjugates »— and p — are also eigenvalues and

(Hasselmann [167]), are a fairly general approach eigenvectors of A.

which allow for a large variety of complex The eigenvectors of A form a linear basis when

scenarios. The merits and limitations of this ansatz all of its eigenvalues are nonzero. Thus any state

are discussed in Section 15.5. X may be uniquely expressed in terms of the

eigenvectors as

15.0.2 Applications of POP Analysis. POP

analysis is a tool [136] that is now routinely used to X = zj p j (15.2)

j

diagnose the space-time variability of the climate

system. Processes that have been analysed with where the pattern coef¬cients z j are given by

POPs include the Madden-and-Julian Oscillation the inner product of X with the normalized

(MJO; also called the 30“60 day oscillation) [388, eigenvectors p aj of AT .2

389, 399, 401], oceanic variability [275, 421], the 2 The eigenvectors of A are linearly independent if

stratospheric Quasi-Biennial Oscillation (QBO) eigenvalues of A are distinct. Making this assumption, allis then

of the

it

[431], the El Ni˜ o/Southern Oscillation (ENSO) easily shown that AT has the same eigenvalues as A, and that

n

phenomenon [20, 50, 75, 242, 243, 429, 430, the eigenvectors p aj of AT are columns of a matrix (P ’1 )T ,

432], and others, tropospheric baroclinic waves where the columns of P are the eigenvectors of A. Then X can

T

[341], and low-frequency variability in the coupled be expanded as X = j z j p j , where z j = p aj X, because

atmosphere“ocean system [431]. T j

zj p j = pa Xp j

1 The word ˜ansatz™ is causing some confusion in the

j j

scienti¬c community. In contrast with meteorologists and T

jT

p j p a X = P (P ’1 )T X = X.

=

statisticians, theoretical physicists and non-statistical applied

mathematicians are generally acquainted with this word. It is j

of German origin and means an ˜educated guess™ that may or

j

may not lead to a successful line of analysis. The eigenvectors p a are called adjoint patterns.

335

15: POP Analysis

336

Inserting (15.2) into (15.1), we ¬nd that

the coupled system (15.1) becomes uncoupled,

yielding m single equations,3

z t+1 p = »z t p (15.3)

where m is the dimension of the process Xt . Thus,

if z 0 = 1,

z t p = »t p . (15.4)

Now let Pt be the vector

Figure 15.1: Schematic diagram of the time

—

zt p —.

Pt = z t p + (15.5) evolution of POP coef¬cients z t with an initial

value z 0 = (z r , z i ) = (0, 1). The rotation time is

Then slightly more than eight time steps. The e-folding

(15.6) time „ is indicated by the large open circle [400].