324

Atlantic [14.3.1,2]. In the second example, one of

the vector times series is again North Atlantic sea-

level pressure but the second ˜partner™ in the CCA

is a regional scale variable, namely, precipitation

on the Iberian Peninsula [14.3.3,4]. This example

is used to demonstrate statistical downscaling of

GCM output. The last example [14.3.5] illustrates

the Principal Prediction Patterns introduced in

[14.1.7].

The literature also contains many other exam-

ples of applications of CCA. Bretherton et al. [64]

cite several studies, including classic papers by

Barnett and Preisendorfer [21], Nicholls [293] and

Barnston and colleagues [26, 28, 30, 346].

14.3.1 North Atlantic SLP and SST: Data and

Results. CCA is used to analyse the relationship

between X = monthly mean sea-level pressure

(SLP) and Y = sea-surface temperature (SST) over

the North Atlantic in northern winter (DJF) (see

Zorita et al. [438] for details). The data are time

series of monthly means of SLP and SST on a grid

over the North Atlantic north of about 20—¦ N, for

DJF of 1950 to 1986. Anomalies were obtained

at each grid point by subtracting the long-term

Figure 14.3: The second pair of canonical patterns

monthly mean from the original values.

for monthly mean SLP and SST over the North

The coef¬cients of the ¬rst ¬ve EOFs of both

Atlantic in DJF. The dark shading on each pattern

¬elds were retained for the subsequent CCA. They

identi¬es the main positive feature of the opposing

represent 87% and 62% of the total variance

pattern.

or SLP and SST respectively. To check the

Top: SLP, contour interval: 1 hPa,

sensitivity of the results to EOF truncation, the

Bottom: SST, contour interval: 0.1 K.

same calculations were performed using ¬ve SLP

From Zorita et al. [438].

EOFs and either 10 or 15 SST EOFs (77% and

84%, respectively) and essentially the same results

were obtained.

ocean surface is warmer than normal when the

The CCA yields two pairs of patterns that

westerly wind is reduced. West of the cyclone, just

describe the coherent variations of the SST and

downstream from the cold American continent, the

SLP ¬elds. The two patterns are dominant in

ocean is substantially cooled. The SST anomalies

describing SLP and SST variance.

off the African coast are a local response to

The ¬rst pair of patterns, FS1L P and FSST , 1

anomalous winds; coastal upwelling is reduced

which corresponds to a canonical correlation of

when there are weaker than normal northerly

0.56, represents 21% of the variance of monthly

winds. In contrast, when the circulation produces

mean SLP and 19% of the variance of monthly

enhanced westerlies and anomalous anticyclonic

mean SST (Figure 1.13).9 The two patterns are

¬‚ow in the southern part of the area, opposite SST

consistent with the hypothesis ¬rst suggested by

anomalies are expected. The canonical correlation

Bjerknes that atmospheric anomalies cause SST

coef¬cient time series also support the Bjerknes

anomalies. The main features of the SLP pattern

are a decrease of the westerly wind at about 50—¦ N, hypothesis: the one month lag correlation is 0.65

when SLP leads SST but it is only 0.09 if SLP lags.

and an anomalous cyclonic circulation centred at

40—¦ W and 30—¦ N.10 North of the cyclone, the The coef¬cients of the second pair of pat-

terns, FS2L P and FSST , have correlation 0.47

2

9 We have dropped the ˜ · ™ notation for now, but be aware

(Figure 14.3). The SLP pattern represents 31%

that the patterns are parameter estimates. The same applies to

of the total variance and is similar to the ¬rst

canonical coordinate time series when they are discussed.

SLP EOF (Figure 13.8), which is related to the

10 We use the geostrophic wind relationship for the derivation

North Atlantic Oscillation (see also [13.5.5] and

of approximate wind anomalies from pressure anomalies.

14.3: Examples 325

from a number of rain gauges on the Iberian

Peninsula is related to the air-pressure ¬eld over

the North Atlantic (see [403] for details). CCA

was used to obtain a pair of canonical correlation

pattern estimates FS1L P and Fpr e (Figure 14.4), and

1

pr e

corresponding time series β1 L P (t) and β1 (t)

S

of canonical variate estimates. These strongly

correlated modes of variation (the estimated

canonical correlation is 0.75) represent about

65% and 40% of the total variability of seasonal

mean SLP and Iberian Peninsula precipitation

respectively. The two patterns represent a simple

physical mechanism: when FS1L P has a strong

positive coef¬cient, enhanced cyclonic circulation

advects more maritime air onto the Iberian

Peninsula so that precipitation in the mountainous

1

northwest region ( Fpr e ) is increased.

Since the canonical correlation is large, the

results of the CCA can be used to forecast or

specify winter mean precipitation on the Iberian

peninsula from North Atlantic SLP. The ¬rst

pr e

step is to connect β1 (t) and β1 L P (t) with a

S

pr e

simple linear model β1 (t) = aβ1 L P (t) + .

S

pr e

Since β1 (t) and β1 L P (t) are normalized to unit

S

variance, the least squares estimate of coef¬cient a

Figure 14.4: First pair of canonical correlation

is the canonical correlation ρ1 . Given a realization

patterns of the North Atlantic winter mean sea-

of β1 L P (t), the canonical variate for precipitation

S

level pressure Y and a vector X of seasonal means pr e

can be forecast as β 1 (t) = ρ1 β1 L P (t), and thus

S

of precipitation at a number of Iberian locations

the precipitation ¬eld is forecast as

[403].

pr e

Figure 13.6). The structure of this pair of patterns R = β 1 (t) Fpr e = ρ1 β1 (t) Fpr e .

SL P

1 1 (14.27)

is also consistent with the Bjerknes hypothesis.

The one month lag correlation is 0.48 when SLP Similarly, if several useful canonical correlation

leads and 0.03 when SLP lags. patterns had been found, Iberian winter mean

precipitation could be forecast or speci¬ed as

14.3.2 North Atlantic SLP and SST: Discussion.

We described conventional and rotated EOF k

analysis of the same data in [13.5.6,7]. The CCA of R = ρi βiS L P (t) Fpr e .

i

SLP and SST suggests why rotation had a marked i=1

effect on the SST EOFs but not on the SLP EOFs.