than the North American trough. A secondary

analysis (Chapter 13).

trough can be identi¬ed over eastern Europe and

two minor ridges are located over the northeast Third, a climatological mean should be under-

Paci¬c and the east Atlantic. stood to be a moving target. Today™s climate is

different from that which prevailed during the

4 The geopotential height ¬eld is a parameter that is

Holocene (6000 years before present) or even

frequently used to describe the dynamical state of the

during the Little Ice Age a few hundred years ago.

atmosphere. It is the height of the surface of constant pressure

at, e.g., 300 hPa and, being a length, is measured in metres. We

5 A zonal wavenumber 2 pattern contains two ridges and two

will often simply refer to ˜height™ when we mean ˜geopotential

height™. troughs in the zonal, or east“west, direction.

1: Introduction

4

We therefore need a clear understanding of To demonstrate the point, consider the following

our interpretation of the ˜true™ mean state before two procedures for estimating the long-term mean

interpreting an estimate computed from a set of January air pressure in Hamburg (Germany). Two

observations. data sets, consisting of 104 observations each, are

available. The ¬rst data set is taken at one minute

To accomplish this, it is necessary to think of

intervals, the second is taken at weekly intervals,

the ˜January 300 hPa height ¬eld™ as a random

and a mean is computed from each. Both means

¬eld, and we need to determine whether the

are estimates of the long-term mean air pressure in

observed height ¬elds in our 15-year sample are

Hamburg, and each tells us something about our

representative of the ˜true™ mean state we have in

parameter.

mind (presumably that of the ˜current™ climate).

The reliability of the ¬rst estimate is question-

From a statistical perspective, the answer is a

able because air pressure varies on time scales

conditional ˜yes,™ provided that:

considerably longer than the 104 minutes spanned

1 the time series of January mean 300 hPa by the data set. Nonetheless, the estimate does

height ¬elds is stationary (i.e., their statistical contain information useful to someone who has

no prior information about the climate of locations

properties do not drift with time), and

near sea level: it indicates that the mean air

2 the memory of this time series is short relative pressure in Hamburg is neither 2000 mb nor 20 hPa

to the length of the 15-year sample. but somewhere near 1000 mb.

The second data set provides us with a

Under these conditions, the mean state is much more reliable estimate of long-term mean

representative of the random sample, in the sense air pressure because it contains 104 almost

that it lies in the ˜centre™ of the scatter of the independent observations of air pressure spanning

individual points in the state space. As we noted two annual cycles. The ¬rst estimate is not

above, however, it is not representative in many ˜wrong,™ but it is not very informative; the second

other ways. is not ˜right,™ but it is adequate for many purposes.

The characteristics of the 15-year sample may

not be representative of the properties of January

mean 300 hPa height ¬elds on longer time scales 1.2.2 Correlation. In the statistical lexicon,

when assumption 1 is not satis¬ed. The uncertainty the word correlation is used to describe a

of the 15-year mean height ¬eld as an estimator linear statistical relationship between two random

of the long-term mean will be almost as great variables. The phrase ˜linear statistical™ indicates

as the interannual variability of the individual that the mean of one of the random variables is

January means when assumption 2 is not satis¬ed. linearly dependent upon the random component

We can have con¬dence in the 15-year mean of the other (see Section 8.2). The stronger the

as an estimator of the long-term mean January linear relationship, the stronger the correlation.

300 hPa height ¬eld when assumptions 1 and 2 A correlation coef¬cient of +1 (’1) indicates a

hold in the following sense: the law of large pair of variables that vary together precisely, one

numbers dictates that a multi-year mean becomes variable being related to the other by means of a

an increasingly better estimator of the long-term positive (negative) scaling factor.

mean as the number of years in the sample While this concept seems to be intuitively

increases. However, there is still a considerable simple, it does warrant scrutiny. For example,

amount of uncertainty in an estimate based on a consider a satellite instrument that makes radiance

15-year sample. observations in two different frequency bands.

Statements to the effect that a certain estimate Suppose that these radiometers have been designed

of the mean is ˜wrong™ or ˜right™ are often made in such a way that instrumental error in one

in discussions of data sets and climatologies. Such channel is independent of that in the other. This

an assessment indicates that the speakers do not means that knowledge of the noise in one channel

really understand the art of estimation. An estimate provides no information about that in the other.

is by de¬nition an approximation, or guess, based However, suppose also that the radiometers drift

on the available data. It is almost certain that the (go out of calibration) together as they age because

exact value will never be determined. Therefore both share the same physical environment, share

estimates are never ˜wrong™ or ˜right;™ rather, some the same power supply and are exposed to the same

estimates will be closer to the truth than others on physical abuse. Reasonable models for the total

average. error as a function of time in the two radiometer

1.2: Some Typical Problems and Concepts 5

Figure 1.2: The monthly mean Southern Oscillation Index, computed as the difference between Darwin

(Australia) and Papeete (Tahiti) monthly mean sea-level pressure (˜Jahr™ is German for ˜year™).

Figure 1.3: Auto-correlation function of the index shown in Figure 1.2. Units: %.

Correlations manifest themselves in several dif-

channels might be:

ferent ways in observed and simulated climates.

e1t = ±1 (t ’ t0 ) + 1t , Several adjectives are used to describe corre-

e2t = ±2 (t ’ t0 ) + 2t , lations depending upon whether they describe

relationships in time (serial correlation, lagged

where t0 is the launch time of the satellite and correlation), space (spatial correlation, telecon-

±1 and ±2 are ¬xed constants describing the rates nection), or between different climate variables

of drift of the two radiometers. The instrumental (cross-correlation).

errors, 1t and 2t , are statistically independent of A good example of serial correlation is the

each other, implying that the correlation between monthly Southern Oscillation Index (SOI),6 which

the two, ρ( 1t , 2t ), is zero. Consequently the

total errors, e1t and e2t , are also statistically 6 The Southern Oscillation is the major mode of natural

independent even though they share a common climate variability on the interannual time scale. It is frequently

systematic component. However, simple estimates used as an example in this book.

of correlation between e1t and e2t that do not It has been known Walker, the end that the last pressure

since of century

(Hildebrandson [177]; 1909“21) sea-level

account for the deterministic drift will suggest that (SLP) in the Indonesian region is negatively correlated with that

these two quantities are correlated. over the southeast tropical Paci¬c. A positive SLP anomaly

1: Introduction

6

is de¬ned as the anomalous monthly mean spatially correlated. The Southern Oscillation In-

pressure difference between Darwin (Australia) dex (Figure 1.2) is a manifestation of the negative

and Papeete (Tahiti) (Figure 1.2). correlation between surface pressure at Papeete

The time series is basically stationary, although and that at Darwin. Variables such as pressure,

variability during the ¬rst 30 years seems to be height, wind, temperature, and speci¬c humidity

somewhat weaker than that of late. Despite the vary smoothly in the free atmosphere and con-

noisy nature of the time series, there is a distinct sequently exhibit strong spatial interdependence.

tendency for the SOI to remain positive or negative This correlation is present in each weather map

for extended periods, some of which are indicated (Figure 1.5, left). Indeed, without this feature,

in Figure 1.2. This persistence in the sign of the routine weather forecasts would be all but impos-

index re¬‚ects the serial correlation of the SOI. sible given the sparseness of the global observing

A quantitative measure of the serial correlation network as it exists even today. Variables derived

is the auto-correlation function, ρ S O I (t, t + ), from moisture, such as cloud cover, rainfall and

shown in Figure 1.3, which measures the similarity snow amounts, and variables associated with land

of the SOI at any time difference . The auto- surface processes tend to have much smaller spa-

correlation is greater than 0.2 for lags up to tial scales (Figure 1.5, right), and also tend not to

about six months and varies smoothly around zero have normal distributions (Sections 3.1 and 3.2).

with typical magnitudes between 0.05 and 0.1 While mean sea-level pressure (Figure 1.5, left)

for lags greater than about a year. This tendency will be more or less constant on spatial scales of

of estimated auto-correlation functions not to tens of kilometres, we may often travel in and out

converge to zero at large lags, even though the of localized rain showers in just a few kilometres.

real auto-correlation is zero at long lags, is a This dichotomy is illustrated in Figure 1.5, where

natural consequence of the uncertainty due to ¬nite we see a cold front over Ontario (Canada). The

samples (see Section 11.1). left panel, which displays mean sea-level pressure,

A good example of a cross-correlation is the shows the front as a smooth curve. The right panel

relationship that exists between the SOI and displays a radar image of precipitation occurring

various alternative indices of the Southern Os- in southern Ontario as the front passes through the

cillation [426]. The characteristic low-frequency region.

variations in Figure 1.2 are also present in area-

averaged Central Paci¬c sea-surface temperature