and Thus the ¬rst Hilbert EOF describes the dominant

rotational behaviour of the system whereas the

’(X1t + XH )H

1) = 2t

Im(±1 e second Hilbert EOF represents just the ˜residual™

X1t + XH

2t

which is small.

7 This example is easily generalized to the case in which the Note that this interpretation is independent of

1Σ,

noise components are correlated: then Σx x (0) = the frequency interval used to form the integrals

zz

1’r 2

in (16.52) and (16.53).

and the conventional EOFs of Xt coincide with those of Zt .

16.3: Complex and Hilbert EOFs 363

al. [329], Horel [181], Wang and Mooers [413],

16.3.12 Interpretation of Hilbert EOFs. The

Johnson and McPhaden [196], Trenberth and

heuristic argument of the previous subsection is

Shin [373], just to mention a few.

generally used to interpret the outcome of a Hilbert

EOF analysis. Its validity depends crucially on the

validity of (16.60), which is by no means a trivial 16.3.15 Example: Tropical Paci¬c Sea-surface

assumption as demonstrated by the AR examples Temperatures. We now describe a Hilbert EOF

studied in [16.2.3]. analysis of monthly mean SST anomalies in the

tropical Paci¬c Ocean between 20—¦ S and 20—¦ N.

There are no generally applicable techniques

for deciding whether an estimated Hilbert EOF The data used in this example were obtained from

describes a real oscillatory signal. Prudence is COADS (Woodruff et al. [425]) and cover the

clearly advisable. If possible, the data should be period 1951“90.

divided into ˜learning™ and ˜validation™ subsets so The annual cycle was removed by subtracting

that phase relationships identi¬ed in the learning the 40-year mean for each month of the year and

data set can be veri¬ed independently in the variations on time scales shorter than a year were

validation data set. removed by low-pass ¬ltering the anomalies from

the annual cycle. The ¬ltered time series was then

Hilbert transformed (16.19, 16.20). Finally, the

16.3.13 Estimating Hilbert EOFs. The two

covariance matrix of the complexi¬ed process was

different de¬nitions of Hilbert EOFs, either

estimated with equation (16.63). The eigenvectors

directly by means of the covariance matrix of the

of this matrix are the estimated Hilbert EOFs.

complexi¬ed process or by means of the frequency

The dominant Hilbert EOF, which represents

integrated spectral matrix, provide two different

40% of the variance of both the ¬ltered SST

approaches for estimating the Hilbert EOFs.

anomalies and the ¬ltered complexi¬ed process,

The estimation can be done in the time domain,

is shown in Figure 16.6. The real part, shown

in which case the Hilbert transform is ¬rst

in the upper panel, depicts the mature phase of

estimated. This can be done either with the

El Ni˜ o when the corresponding EOF coef¬cient,

n

truncated time domain ¬lter (16.18) or by a Fourier

± 1 (t), is real and positive. It also approximates the

decomposition and phase-shifted reconstruction

mature phase of La Ni˜ a when ± 1 (t) is real and

n

(see [16.2.4]). Then the complex covariance matrix

negative. The imaginary part of the ¬rst Hilbert

is computed in the usual manner by computing

EOF, shown in the lower panel, depicts a transition

n

1 phase between the warm El Ni˜ o and the cool

n

†

x j + i (xt )H x j + i (xt )H (16.63)

n La Ni˜ a.

n

t=1

The time series of complex time coef¬cients of

where x1 , . . . , xn form a sample of size n the ¬rst Hilbert EOF is shown in Figure 16.7. The

and x1 , . . . , xn are deviations from the sample imaginary part, given by the dashed curve, is the

mean. Finally, the eigenvectors of this matrix are Hilbert transform of the real part (recall (16.42)).

determined. By focusing on the 1982/83 El Ni˜ o event, we

n

Estimation can also be done in the frequency can see that the Hilbert transform can indeed be

domain. First the width 2δω and the centre ω0 of interpreted as a crude derivative: the imaginary

the frequency band of interest are selected. Next, part is positive until the warm event peaks in late

an estimate of the spectral matrix with equivalent 1982/early 1983 and then becomes negative as the

bandwidth 2δω is constructed (see Section 12.3 event fades.

for a description of spectral estimation). This Figure 16.8 shows the same time series in polar

estimator is evaluated at frequency ω0 , and coordinate form. The upper panel displays the

eigenvectors are found. amplitude of the EOF coef¬cient as a function

For the estimation in the frequency domain the of time, and the lower panel displays the phase

spectral matrix is estimated for all frequencies ω ≥ in radians. We see that the complex coef¬cient

0 in the frequency band of interest, and then the tends to rotate in a clockwise direction, but not

spectral matrices are summed. at uniform speed. The amplitude varies irregularly

in time. Each ˜sawtooth™ in the lower panel

16.3.14 Applications of Hilbert EOF Analysis. of Figure 16.8 depicts one ENSO-like cycle. It

Hilbert EOF analysis has been pursued extensively begins with the cold version of Figure 16.6a,

in climate research, for instance by Barnett [19] then rotates to the warm version in Figure 16.6b

who pioneered this technique, Wallace and with weak warm anomalies over most of the

Dickinson [408], Brillinger [66], Rasmusson et tropical Paci¬c one-quarter of a period later. This

16: Complex Eigentechniques

364

Figure 16.6: The ¬rst Hilbert EOF of low-pass ¬ltered tropical Paci¬c sea-surface temperatures.

Courtesy E. Zorita. a) Real part (top). b) Imaginary part (bottom).

is followed by the mature warm phase (positive

version of Figure 16.6a) halfway through the cycle 2

and weak negative SST anomalies (Figure 16.6b)

three-quarters of the way through the cycle. The

1

cycle is completed with the mature cold phase

(Figure 16.6a multiplied by ’1).

0

It is clear from Figure 16.8b that there is

-1

signi¬cant variability in the length of an ENSO

cycle. The vertical lines in Figure 16.8 give the

approximate time of warm events (short dashes)

1950 1960 1970 1980 1990

and cold events (long dashes) as identi¬ed by

Kiladis and Diaz [222]. Warm events tend to

occur within one radian of zero phase while cold

events tend to occur 180 —¦ later. The amplitude is Figure 16.7: The time series of complex time

coef¬cients of the ¬rst Hilbert EOF of low-pass

often, but not always, large when a warm or cold

¬ltered tropical Paci¬c SST anomalies. Units: —¦ C.

event is identi¬ed, perhaps because there is large

variability from event to event in the precise spatial

structure of the SST anomalies. 16.3.16 Hilbert POPs. When the POP analysis

In summary, the Hilbert EOF analysis of the (see Chapter 15) is applied to the complexi¬ed

¬ltered SST anomalies captures the essential process, complex patterns are derived. As with

features of ENSO. We have found a pair of patterns the Hilbert EOF, these may be interpreted as

that depict a substantial fraction of the ENSO specifying the ˜state™ and the ˜rate of change™

cycle. The length of the cycle varies from 2 to of the process. B¨ rger [75] has pioneered this

u