with p = q is also not de¬ned.

recurrent.™ H0 can be tested with Hotelling T 2

statistic [6.6.10]:

6.10.4 Estimation of the Level of ( p, p)-

n X + nY ’ m ’ 1

recurrence. Zwiers and von Storch [452] con-

T2 = Dˆ. (6.62)

sidered several estimators of the level of ( p, p)- m(n X + n Y ’ 2)

recurrence and found that an estimator originally

Under H0 T 2 has a non-central F distribution (see,

proposed by Okamoto [299, 300] worked well.

e.g., [307]) with m and n X + n Y ’ m ’ 1 degrees

Hense et al. [175] suggested the following modi-

of freedom and non-centrality parameter

¬ed form of this estimator for p:

DS n X nY

p = 1 ’ erf ’

ˆ = D,

(6.59) (6.63)

n X + nY

2

f N (’D S /2) D S (n X + n Y ’ 1)2 ’ 1

+ ’1

where D = 2FN ( p) (6.57).

n X + n Y ’ 2 16 n X nY

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7 Analysis of Atmospheric Circulation

Problems

In the following example the state vector is

7.0.0 Summary. In this chapter we present

only a single variable: the zonal distribution of

examples of hypothesis tests in the contexts of

geopotential height at 500 hPa in the Northern

con¬rming, or validating, Atmospheric General

Hemisphere extratropics. The comparison is often

Circulation Models (AGCMs) (Section 7.1, see

performed with a statistical test of the null

also [1.2.7]) and the analysis of paired sensitivity

hypothesis that the observed and simulated vectors

experiments (Section 7.2, see also [1.2.7]). Similar

have the same distribution.1 Thus, as we noted

applications in the literature include [105, 132,

134, 135, 161, 393]. See also Frankignoul™s review in [6.9.1], given large enough samples we will

of the topic [130], and the recurrence analysis eventually discover that the simulated climate

is ˜signi¬cantly™2 different from that which is

examples presented in Sections 6.9 and 6.10. An

observed because no model is perfect.3

application of the Hotelling test is described in

Section 7.3 and an example of the anthropogenic That is, a fully satisfactory ˜veri¬cation™ or

CO2 signal is discussed in Section 7.4. ˜validation™ is impossible with the hypothesis

testing paradigm. Are there more satisfying ways

to prove the ˜correctness™ of a model? Oreskes

et al. [301] argue that a positive answer can

7.1 Validating a General Circulation be given only if the model describes a closed

Model sub-system of the full system, that is, a sub-system

with completely known ˜external™ forcings. The

atmosphere and the climate system, as a whole,

7.1.1 The Problem. Climate models in gen-

are not closed systems but open to various external

eral, and AGCMs speci¬cally, are mathematical

factors, such as variations in solar radiation,

representations of the climate that are built from

volcanic eruptions, or the Milankovicz cycle. Even

¬rst principles. On short time scales they simulate

if these external factors were known in detail,

the day-to-day variations in the weather, ideally

the part of the climate system represented by an

in such a way that the statistics of the observed

AGCM cannot be viewed as a closed sub-system

climate are reproduced when the model is run

because the atmosphere loses energy and moisture

for a long period of time. A careful strategy is

into other parts of the system.

needed to determine, even partly, whether a model

Sometimes, a possible alternative to the ˜hy-

has achieved this goal. The problem is complex

pothesis testing™ strategy is to use the mod-

because, in principle, we would need to compare

els as forecasting instruments, then assess their

the statistics of a state vector that characterizes

ability to predict atmospheric variations (see

all aspects of the thermo- and hydrodynamics of

Chapter 18) correctly. Unfortunately, this ap-

the atmosphere. The statistics should include time

proach is applicable only in cases when there

averaged ¬elds of various variables at various

levels, and temporal and spatial cross-covariances

1 The test may concentrate on a speci¬c aspect of the

of different variables on different scales.

distributions, such as the means (Section 6.6) or variances

It would be dif¬cult, but not impossible, to (Section 6.7), or it may be concerned with the whole

characterize the simulated climate in this way. On distribution ([5.2.3], [5.3.3] and Sections 6.9 and 6.10)

2 Statistically, not necessarily physically, signi¬cant.

the other hand, it simply cannot be done for the

3 One of the unavoidable errors is due to space-time

observed climate because our observations are far

truncation that determines the modelled sub-space. The part of

from complete. In reality, model validation efforts the phase space that is disregarded by the truncation affects the

must be restricted to an incomplete state vector that real system also in the resolved part of its phase space, but has

represents only a few variables of interest. no impact on the model™s phase space.

129

7: Analysis of Atmospheric Circulation Problems

130

7.1.2 Example: Extratropical Geopotential

Height at 500 hPa. We return to an example

¬rst described in [6.2.6], which dealt with January

mean 500 hPa heights, meridionally averaged

between 30—¦ N and 60—¦ N. In [6.2.6] we asked

whether the individual zonal distributions of

height, X, simulated by a GCM were distributed

similarly to those observed. Here we use the

same model output to test the null hypothesis

that the means and variances of the simulated

zonal distribution are equal to those of the

observations [397].

The permutation test [6.6.12] is used with

statistic (6.36) to test the null hypothesis that the

means of the simulated and observed climates

are equal. The assumptions needed in this case

are (i) the observed and simulated samples

can be represented by iid random vectors

X1 , . . . , Xn X and Y1 , . . . , Yn Y , (ii) the samples

are mutually independent, and (iii) the variances

Figure 7.1: Sample mean (top) and standard

of the simulated meridional means are equal to

deviation (bottom) of the 30—¦ N“60—¦ N meridional

those of the observed means. Assumption (i)

average of 500 hPa height simulated in a GCM

may be violated for the observations since low

(light lines), and derived from observations (heavy

frequency interactions between the ocean and the

lines) (cf. Figure 6.5) [397].

atmosphere, such as the Southern Oscillation, may

result in weak dependence between consecutive

January meridional means. This should not

is predictive skill, as in case of short-term fore- cause major problems with the test procedure.

casts or in case of externally induced anoma- Departures from the third assumption are more

lies (as the injection of volcanic aerosols). Also, obvious, but fortunately Monte Carlo experiments

it is often impractical because we lack inde- have shown that this violation does not lead to

pendent observed data on the time scales of strong biases in the risk of incorrectly rejecting the

interest. null hypothesis.

The result of the test is that the equality of means

Regardless of the validation strategy used,

hypothesis can be rejected at a signi¬cance level of

it is always possible that the model veri¬es