with K = 3 contains that and the functions

structure varies with the time of year, so the

P2 (φ) cos(») and P2 (φ) sin(») as well, and so on.

1 1

statistical analysis is done separately for each

The hierarchy does not contain an element with

calendar month.

K = 2 guess patterns.

As in [6.1.3], the null hypothesis for each of

¯

The projection of the full signal y ’ x onto a

the 21 months from January 1982 to September

1983 is that the respective monthly anomaly y subset of K guess patterns represents a truncated

is drawn from the random variable X, where signal. The optimal signal is identi¬ed as the

X represents the ˜normal™ monthly mean stream truncated signal that goes with the K for which the

function distribution appropriate to the month evidence against the equality of means hypothesis

in which y is observed. The X sample for a is the strongest. Barnett et al. [22] call this

given month of the year is taken to be the 15 selection rule C.

monthly mean stream function ¬elds observed for The following results were obtained:

that month between 1967 and 1981. The null

hypothesis is tested with Hotelling T 2 [6.6.10], • Results for November 1982 are shown in the

which means that a number of assumptions are bottom panel of Figure 7.6. The statistic that

made implicitly. Speci¬cally, it is assumed that is displayed for each level K of the hierarchy

the monthly mean 500 hPa stream function is is a scaled version of Hotelling T 2 (6.33) that

multivariate normal, that the realizations for a

12 Orthonormal means that the scalar product of any

11 If more than one event is examined, the observations from

two non-identical surface spherical harmonics is zero, and

the events are regarded as samples of another random variable

that of a spherical harmonic with itself is one. In fact,

Y and the null hypothesis is H0 : E(X) = E(Y). If H0 is 2π π m

¯ ¯ 1 n

P j (φ)(cos(m») + i sin(m»))Pk (φ)(cos(n») ’

rejected, the difference y ’ x is understood to be an estimate 2π 2 0 0

i sin(n»)) dφd» = δmn δ jk , where δil is one if i = l and zero

of the mean response of the climate system to the external

events. otherwise.

7.3: Identi¬cation of a Signal in Observed Data 135

MERIDIONAL NODE NUMBER j

6

(16,17) (18,19) (20,21)

5

(12,13) (14,15)

(11)

4

(7,8) (9,10)

3

(4) (5,6)

2

(2,3)

1

(1)

0

0 1 2 3 4 5 6

ZONAL WAVE NUMBER m

99%

1000

Figure 7.5: Two surface spherical harmonics. The 95%

upper panel represents a larger spatial scale than 90%

the lower panel.

Top: P1 (φ) cos(»). Bottom: P2 (φ) cos(»).

1 1

100

is given by

K (n X + n Y ’ 2)

¯

2

D (y ’ x) = 10

n X + nY ’ K ’ 1

’1

1 1

— + T 2,

nX nY

where n X = 15 and n Y = 1. The

critical values are those of the T 2 statistic, 0 2 4 6 8 10 K 12

that is, they are upper tail quantiles of the

F(K , n X + n Y ’ K ’ 1) distribution, also

Figure 7.6: Analysis of extratropical 500 hPa

scaled by the same factor.

height during the 1982/83 El Ni˜ o event [174].

n

The null hypothesis can be rejected at the

Top: Hierarchy in the set of surface spherical

5% signi¬cance level for K = 3, . . . , 8.

harmonics functions, used as guess patterns in

The evidence against H0 is strongest for

Section 7.3.

K = 3. The ¬rst conclusion is that

Bottom: Results for November 1982.

there is a signi¬cant signal in the data.

The second conclusion is that the projec-

tion of the full signal on the three ¬rst optimal signal was often found in the K =

guess patterns, P1 (φ), P2 (φ) cos(»), and

0 1

3 hierarchy. The strongest signals, in terms

P2 (φ) sin(»), yields the optimal model in the

1

of signi¬cance, were found from September

hierarchy. 1982 to June 1983.

• Results for All Months. The hierarchal testing A total of 21 tests were conducted and the

procedure was repeated in each of the 21 null hypothesis was rejected in 15. We would

months from January 1982 to September expect only one or two rejections to occur at

1983. The null hypothesis was rejected at the 5% signi¬cance level if H0 was correct

the 5% signi¬cance level or less for at least throughout the 21-month period, and if the

one member of the hierarchy in every month 21 decisions are statistically independent of

from July 1982 until September 1983. The one another (which they are not). Assuming

7: Analysis of Atmospheric Circulation Problems

136

that we have made the equivalent of seven of institutions, appear to agree broadly with

independent decisions, the probability of observed climate change and also agree broadly

making the reject decision under H0 15 or on the size and distribution of future climate

more times in the 21 tests is well below 1% change. None the less, these simulations are only

[6.8.2]. plausible scenarios for the future since many

aspects of the simulated system, such as the

The major conclusion of this study [174] is that low-frequency variability of the oceans and the

the Northern Hemisphere extratropical circulation role of clouds in regulating climate, are still poorly

during the 1982“83 El Ni˜ o was substantially understood.

n

different from the circulation during the preceding

15 years.

7.4.2 Methodological Considerations. As in

the preceding section, where we dealt with the

˜signal™ excited during an episode with large trop-

7.4 Detecting the ˜CO2 Signal™

ical sea-surface temperature anomalies, the statis-

7.4.1 A Perspective on Global Warming. The tical ˜climate change detection™ problem consists

prospect of man changing the world™s climate of evaluating one event, say the latest record of

by modifying the chemical composition of the the global distribution of near-surface temperature,

atmosphere was ¬rst discussed by Arrhenius [16] in the context of the natural variability of near-

in 1896. He argued that a change in the surface temperature. The problem is to determine

atmospheric concentration of radiatively active whether the recent warming is consistent with the