no assumption is made about the structure of the analysis (see Chapter 16). When Z = (z , z )T ,

t 1t 2t

connection between the index Zt and the analysed equation (17.24) takes the form

vector variable Vt . The ˜associated correlation

˜ ˜

pattern™ approach, on the other hand, is based on Vt = z1t q 1 + z2t q 2 + noise. (17.30)

17.3: Composites and Associated Correlation Patterns 381

This model states that the conditional mean of Vt

is given by

˜

E Vt |˜ 1 = ±, z2 = β = ± q 1 + β q 2 .

z (17.31)

The interpretation is that Vt = q 1 , on average,

˜ ˜

when z1 = 1 and z2 = 0, and that Vt = q 2 ,

˜ ˜

on average, when z1 = 0 and z2 = 1. However,

individual realizations vt may differ substantially

.

from these long-term mean states.

The associated correlation patterns q 1 and q 2

are derived by minimizing the expected mean Figure 17.8: Map of correlations between annual

mean sea-level pressure at a point (˜Darwin™) over

squared error,

north Australia with annual mean SLP everywhere

1’z q 2 2 ,

˜ ˜ 2t

= E Vt ’ z1t q (17.32) else on the globe. The data are taken from a 47-

year GCM experiment with prescribed observed

1 and q 2 are the solutions of

that is, q sea-surface temperatures.

‚ ‚

= = 0.

‚q 1 ‚q 2 17.3.5 Examples. We now consider two exam-

ples: one with a univariate Southern Oscillation

By differentiating and taking expectations we ¬nd

index and the other with a bivariate MJO index.

σ1 σ12 (q 1 )T (σ1v )T In the ¬rst example, Zt is the annual mean

2

= (17.33)

(σ2v )T

(q 2 )T

σ12 σ2 sea-level pressure near Darwin, Australia, in a

2

47-year GCM simulation in which SST and sea-ice

˜ ˜

where σ1 = Var z1 , σ2 = Var z2 , σ12 =

2 2 observations are prescribed from observations.

˜˜ ˜

Cov z1 , z2 , σ1v = Cov z1 , Vt , and σ2v = The Darwin pressure index is a widely used

ENSO index that carries information similar to the

˜

Cov z2 , Vt . Equation (17.33) has solution

standard SOI [426]. The ¬eld Vt in this example is

σ2 σ1v ’ σ12 σ2v

2 the corresponding annual mean sea-level pressure.

q1 = (17.34) The associated correlation pattern that is obtained

σ1 σ2 ’ σ12

22 2

is shown in Figure 17.8. As expected, sea-

σ1 σ2v ’ σ12 σ1v

2

level pressure variations occur coherently over

q2 = . (17.35)

σ1 σ2 ’ σ12

22 2 broad regions, and variations in the eastern

tropical Paci¬c are opposite in sign to those

The relative importance of associated correla- occurring over the western tropical Paci¬c. The

tion patterns can be measured by the ˜proportion of present diagram compares favourably with similar

variance™ they represent, either locally or in total. diagrams computed from observations. See, for

The proportion of the total variance represented by example, Peixoto and Oort [311, p. 492], or

the patterns is given by Trenberth and Shea [372].14

Our second example uses the same bivariate

T

E(Vt Vt ) ’

rv|z = MJO index (Figure 10.3) employed in the

2

T

E(Vt Vt ) composite analysis of [17.3.3] (Figure 17.7).

The estimated associated correlation patterns are

where is given by (17.27) or (17.32). Locally, the shown in Figure 17.9 [389]. As explained in

proportion is given by Section 15.1, the two POP coef¬cients (z1 , z2 )

t t

tend to have quasi-oscillatory variations of the type

E(V2 ) ’ j

jt

= ,

2

rv|z, j

· · · ’ (0, 1) ’ (1, 0) ’ (0, ’1) ’

E(V2 )

jt

(’1, 0) ’ (0, 1) ’ · · · .

where V jt is the jth element of Vt and j is

the local version of .13 The last representation is At the same time, the V ¬eld tends to evolve as

useful because it can be displayed as a function of

· · · ’ q 2 ’ q 1 ’ ’q 2 ’ ’q 1 ’ q 2 ’ · · · .

location.

13 Note that both (17.27) and (17.32) can be easily re- 14 See also Berlage [46], who published a similar diagram in

=

expressed as j. 1957.

j

17: Speci¬c Statistical Concepts

382

q

2

q

1

Figure 17.9: The associated correlation patterns

V K j of tropical velocity potential anomalies

derived from MJO indices z. Compare with the

composites shown in Figure 17.7. From [389].

Thus the patterns in Figure 17.9 provide the same

information as the composites in Figure 17.7.

The signal has a zonal wavenumber 1 structure

Figure 17.10: Top: The correlation between

that propagates eastward around the world. The

DJF monthly mean 500 hPa geopotential height

oscillation is most energetic when the ˜wave

simulated by a GCM at (50—¦ N, 90—¦ W) and all

crest™ (or ˜valley™) is positioned over the Maritime

other points in the model™s grid.

Continent.

Bottom: As top, except the reference point is

A signi¬cant conclusion from the discussions

located at (2—¦ N, 90—¦ W).

here and in [17.3.3] is that both techniques provide

useful information. The associated correlation

pattern technique is superior in the MJO case

The upper panel in Figure 17.10 shows

since fewer parameters must be estimated from the

teleconnections for a ¬xed point located over Lake

available data (speci¬cally, two patterns instead of

Superior. The main feature is an arched wave