It is often best to time-¬lter the data prior to the = xt

z ti T T T

pr pi pi pi pi

POP analysis if there is prior information that the

expected signal is located in a certain frequency

(15.16)

band. A somewhat milder way to focus on selected

time scales is to derive the EOFs from time-¬ltered and that of (15.15) is

data, but to project the un¬ltered data onto these T

EOFs. Note that the resulting sample covariance z t = p xt . (15.17)

p Tp

matrix is no longer diagonal.

Criteria for distinguishing between POPs that Note that (15.16) can be written in terms of

contain useful information and those that re¬‚ect estimated adjoint patterns as

primarily sample effects are given in [389]. The

most important rule-of-thumb is related to the z t = p a xt T

cross-spectrum of the POP coef¬cients z r and z i :

The coef¬cient time series should vary coherently where p a = p a + i p a ,

r i

and be 90—¦ out-of-phase in the neighbourhood of rT T T T

p i p i ’p r p i pr

pa

=κ

the POP frequency ·. iT T T T

’p r p i pr pr pi

pa

15.1.5 Estimating POP Coef¬cients. Two and κ = ( p r T p r )( p i T p i ) ’ ( p r T p i )2 ’1 .

approaches can be used to estimate the POP Equation (15.17) can also be interpreted as a

coef¬cients z t . The straightforward approach is to projection onto an estimated adjoint pattern. When

there are two or more useful POPs, the coef¬cients

• compute the eigenvectors p of A, (15.1),

are estimated simultaneously by minimizing

4 Note, however, that there is a small cost. The results of such

2

xt ’ z jt p j

a POP analysis will generally change if the data are transformed j

to another coordinate system since EOFs are invariant only

where the sum is taken over the useful POPs.

under orthonormal transformations.

15.2: Examples 339

15.1.6 Associated Correlation Patterns. The Then the power spectrum of z t has a single

maximum at frequency ω = · that is different

POP coef¬cients can often be regarded as an index

from zero when » is complex. The width of the

of some process, such as the MJO or ENSO. It is

spectral peak is determined by ·. As ξ becomes

then often desirable to be able to relate the index to

other ¬elds. This can be achieved by means of the smaller, the spectrum becomes broader (in the

limit as ξ ’ 0, the spectrum becomes white).

associated correlation patterns [389] discussed in

[17.3.4]. The MJO is presented as an example in Thus, the POP analysis yields a multivariate

[17.3.5]. AR spectral analysis of a vector time series [167].

A ¬rst attempt to simultaneously derive several

15.1.7 POPs and Hilbert EOFs. The POP signals with different spectra from a high-

method is an approach for identifying modal dimensional data set was made by Xu [431]. For

structures in a vector time series that has been a more complete discussion of the POP technique

demonstrated to work well in real applications. as a type of multivariate spectral analysis, refer to

There are certainly other techniques that can be J. von Storch [405].

used successfully for similar purposes. An alter-

native is Hilbert Empirical Orthogonal Function

analysis [19, 408].5 The Hilbert EOFs of a ¬eld Xt 15.2 Examples

are EOFs of the complex vector ¬eld that has Xt as

its real part and the Hilbert transform of Xt as its 15.2.1 Overview. Three examples of POP

imaginary part.6 analysis are presented in this section. The purpose

The main differences between Hilbert EOFs and of the ¬rst example, which is of the tropospheric

POPs are that Hilbert EOFs are orthogonal and baroclinic waves [341], is to demonstrate the

they maximize explained variance. The proportion normal mode interpretation of the POPs. The best

of variance represented by the POP is not optimal, de¬ned POP coincides, to good approximation,

and it must be diagnosed from the POP coef¬cients with the most unstable modes obtained from a

after the POP analysis has been completed. conventional stability analysis of the linearized

Another difference is that the period and e-folding dynamical equations. The other two examples

time (i.e., damping rate) are not an immediate show that the POP analysis can detect signals

result of the Hilbert EOF analysis; they must in different situations. A joint POP analysis of

be derived empirically from the Hilbert EOF tropospheric and stratospheric data [430] identi¬es

coef¬cient time series. The POPs, on the other two independent modes with similar time scales,

hand, are constructed to satisfy a dynamical the Southern Oscillation (SO) and the Quasi-

equation, and the characteristic times are an output Biennial Oscillation (QBO). A POP analysis of

of the analysis. A third difference is that the the Madden-and-Julian Oscillation (MJO; [401]),

POP coef¬cients z t are not pairwise orthogonal. shows that its signal has a well-de¬ned signature

This makes the mathematics less elegant, but it is all along the equator. We will see that this is a very

not a physical drawback because there is usually robust signal. It is possible to detect the signal in

—¦

no reason to assume that different geophysical data that are restricted to 90 subsectors on the

processes are stochastically independent of each equator, and in two-year sub-samples of the full

¬ve-year data set.

other.

15.1.8 POPs as Multivariate Spectral Analysis.

15.2.2 Tropospheric Rossby Waves, from POP

The power spectrum of the POP coef¬cients,

and Stability Analyses. POPs can be seen as

zz (ω), is determined by the eigenvalue » and the

empirical estimates of the normal modes of a

power spectrum nn (ω) of the noise:

linear approximation to a dynamical system. The

nn (ω) estimated normal modes are the eigenvectors of a

zz (ω) = . (15.18) matrix A (15.13). An alternative to estimating A is

|eiω ’ »|2

to derive it by linearizing the dynamical equation

’i· and that the noise is

Assume that » = ξ e that governs the system. The eigenmodes of the

approximately white, that is, nn (ω) ≈ constant. linearized system can then be computed directly.

Schnur et al. [341] compared these two

5 Hilbert EOFs are frequently referred to as Complex EOFs

approaches in the context of the tropospheric

in the climate literature. However, the term ˜complex EOFs™ is

baroclinic waves that are responsible for much

a misnomer (see [16.1.1]).

6 For details, see Section 16.2. of the high-frequency atmospheric variability

15: POP Analysis

340

hPa hPa

Phase Phase

Imag

Real

North Latitude North Latitude

Amplitude

hPa hPa

Amplitude

Real Imag

North Latitude North Latitude

Figure 15.3: Baroclinic waves: The Northern Hemisphere zonal wavenumber 8 POP. This mode

represents 54% of the total zonal wavenumber 8 variance in the 3“25 day time scale. The oscillation

period is 4 days and the e-folding time is 8.6 days. The amplitude Ar and Ai (bottom) and phase r

and i (top) of the real and imaginary parts of the POP are shown (see text). From Schnur et al. [341].

Figure 15.4: Baroclinic waves: The coef¬cient time series z r (dashed) and z i (solid) of the POP shown

in Figure 15.3. From Schnur et al. [341].