i

t

where z r = 2Re{z t }, z t = ’2Im{z t }, p r =

i

t

Re{ p }, and p i = Im{ p }. When z 0 = 1, we ¬nd, amplitude pattern A de¬ned by A j = ( p j ) +

2 r2

2

( pij )2 , j = 1, . . . , m, and the local relative phase

by substituting (15.4) into (15.6), that

pattern ψ de¬ned by ψ j = tan’1 ( pij / prj ), j =

p0 = p r

1, . . . , m (Figure 15.2a). The evolution depicted

pt = ξ t cos(·t) p r ’ sin(·t) p i (15.7)

by (15.8) can describe a travelling wave form.

For example, if we re-express p r and p i as

where ξ and · satisfy » = ξ e’i· .

functions of a location vector r , it may turn out

The geometrical and physical interpretation of

that p i is just a translated version of p r (i.e.,

(15.6) and (15.7) is as follows. When » is complex,

that pi (r ) = pr (r ’ r—¦ ) for some displacement

the corresponding eigenvector p is also complex,

and (»— , p — ) is also an eigenvalue/eigenvector r—¦ ). If so, evolution (15.8) describes a wave

train that propagates in the r—¦ direction and has

pair. Thus the sum (15.5) describes two of the

wavelength 4 r—¦ 2 . Amphidromal (rotating) wave

terms in expansion (15.2). Equation (15.6) shows

forms (Figure 15.2b) can also be represented by

that this sum is real and that it describes variations

(15.8).

in a two-dimensional subspace of the full m-

dimensional space that is spanned by the real and

imaginary parts of p . Equation (15.7) shows how

15.1.2 POPs. The only information used so

system (15.1) evolves if its initial state is p r . If

far is the existence of linear equation (15.1) and

ξ = 1, then pattern p r evolves into pattern ’ p i

the assumption that coef¬cient matrix A has no

in π/(2·) time steps, then evolves to pattern ’ p r

repeated eigenvalues. No assumption was made

at time π/·, and eventually returns to pattern p r

about the origins of this matrix. In dynamical

in period T = 2π/·. Schematically, theory, equations such as (15.1) arise from

linearized and discretized differential equations. In

· · · ’ p r ’ ’p i ’ ’p r ’ p i ’ p r ’ · · ·

POP analysis, the state vector X is assumed to

(15.8) satisfy a stochastic difference equation of the form

In the real world, ξ < 1 (otherwise (15.1) would

Xt+1 = AXt + noise (15.9)

describe explosive behaviour; see below). Thus

Multiplying (15.9) on the right hand side by XT

the amplitude of the sequence of patterns decays t

exponentially in time with an e-folding time „ = and taking expectations leads to

’1/ ln(ξ ) so that pt of (15.7) evolves as the spiral

’1

A = E Xt+1 XT E(Xt XT ) .

displayed in Figure 15.1. (15.10)

t t

Note that any eigenvector p is determined up to

a complex scalar ±. To make things unique up to The normalized eigenvectors of (15.10) are

sign, one can choose ± in such a way that p r and called Principal Oscillation Patterns, and the

p i are orthogonal and p r ≥ p i . coef¬cients z are called POP coef¬cients. Their

time evolution is given by (15.3), except that it is

The modes may be represented either by the two

forced by noise:

patterns p r and p i , or by plots of the local wave

z t+1 = »z t + noise.

3 Indices are dropped in the following for convenience. (15.11)

15.1: Principal Oscillation Patterns 337

Figure 15.2: Schematic examples representing a complex-valued POP p = p r + i p i with their

imaginary and real parts parts p i (top) and p r (middle). The corresponding phase (ψ) and amplitude

A are shown in the bottom panel.

a) A linearly propagating wave is shown. If the initial state of the system is P = p i (top), then its state

a quarter of a period later will be P = p r (middle). The wave propagates to the right with a constant

phase speed (bottom), and the amplitude is constant along horizontal lines with maximum values in the

centre.

b) A clockwise rotating wave is displayed. The evolution of the top pattern to the middle pattern takes

one-quarter of a period. The amplitude (bottom) is zero in the centre, and the lines of constant amplitude

form concentric circles around the centre. From [389].

The stationarity of (15.11) requires |»| < 1 (see LXt with an invertible matrix L. The eigenvalues

(10.12)). are unchanged by the transform. The eigenvectors

transform as X, and the adjoints are transformed

by (L’1 )T , as

15.1.3 Transformation of Coordinates. Sup-

p Y = L pX

pose the original time series Xt is transformed

(15.12)

paY = (L’1 )T pa X .

into another time series Yt by means of Yt =

15: POP Analysis

338

• form P = ( p 1 | · · · | p m ), and

The POP coef¬cients are unaffected by the

transformation because

• compute the matrix of adjoint patterns P a =

T ’1 ’1

( p aY ) Y = ( p a X ) L LX = ( p a X ) X.

T T

(P )T .

However, because A is subject to some sampling

15.1.4 Estimating POPs. In practice, when

variability, this approach will produce some POPs

only a ¬nite time series x1 , . . . , xn is available, A

(those with near-zero eigenvalues and poorly

is estimated by ¬rst computing the sample lag-1

organized spatial structure) that re¬‚ect mostly

covariance matrix

noise. These noisy POPs affect all of the adjoint

1 ’1

patterns through the computation of P .

Σ1 = (xt+1 ’ x)(xt ’ x)T

nt The solution to this problem is to add subjective

judgement to the POP ansatz by using experience

and the sample covariance matrix

and physical knowledge to identify the POPs that

1 are related to the dynamics of the system. The

Σ0 = (xt ’ x)(xt ’ x) T

nt coef¬cients and adjoint patterns of these useful

POPs can be estimated by least squares, essentially

and then forming

by assuming that the eigenvalues of the other POPs

’1

(15.13) are zero.

A = Σ1 Σ0 .

Suppose, for simplicity, that there is only one

The eigenvalues of this matrix always satisfy |»| < useful POP. Then the POP coef¬cient can be

1 (i.e., ξ < 1). estimated by minimizing

In many applications the data are ¬rst subjected 2

xt ’ z tr p r ’ z ti p i (15.14)

to a truncated EOF expansion to reduce the number

of spatial degrees of freedom. POP analysis is then

if p is complex, or

applied to the vector of the ¬rst EOF coef¬cients.4

2

A positive byproduct of this procedure is that noisy xt ’ z t p (15.15)

components can be excluded from the analysis.

Also, the sample covariance matrix Σ0 is made if p is real. The solution of (15.14) is

diagonal. ’1

T T T