6
1: Introduction
is defined as the anomalous monthly mean
pressure difference between Darwin (Australia)
and Papeete (Tahiti) (Figure 1.2).
The time series is basically stationary, although
variability during the first 30 years seems to be
somewhat weaker than that of late. Despite the
noisy nature of the time series, there is a distinct
tendency for the SOI to remain positive or negative
for extended periods, some of which are indicated
in Figure 1.2. This persistence in the sign of the
index reflects the serial correlation of the SOI.
A quantitative measure of the serial correlation
is the auto-correlation function, ρ
SOI
(t, t + 1),
shown in Figure 1.3, which measures the similarity
of the SOI at any time difference 1. The auto-
correlation is greater than 0.2 for lags up to
about six months and varies smoothly around zero
with typical magnitudes between 0.05 and 0.1
for lags greater than about a year. This tendency
of estimated auto-correlation functions not to
converge to zero at large lags, even though the
real auto-correlation is zero at long lags, is a
natural consequence of the uncertainty due to finite
samples (see Section 11.1).
A good example of a cross-correlation is the
relationship that exists between the SOI and
various alternative indices of the Southern Os-
cillation [426]. The characteristic low-frequency
variations in Figure 1.2 are also present in area-
averaged Central Pacific sea-surface temperature
(Figure 1.4).
7
The correlation between the two
time series displayed in Figure 1.4 is 0.67.
Pattern analysis techniques, such as Empiri-
cal Orthogonal Function analysis (Chapter 13),
Canonical Correlation Analysis (Chapter 14) and
Principal Oscillation Patterns (Chapter 15), rely
upon the assumption that the fields under study are
(i.e., a deviation from the long-term mean) over, say, Darwin
(Northern Australia) tends to be associated with a negative
SLP anomaly over Papeete (Tahiti). This seesaw is called
the Southern Oscillation (SO). The SO is associated with
large-scale and persistent anomalies of sea-surface temperature
in the central and eastern tropical Pacific (El Ni
˜
no and
La Ni
˜
na). Hence the phenomenon is often referred to as
the ‘El Ni
˜
no/Southern Oscillation’ (ENSO). Large zonal
displacements of the centres of precipitation are also associated
with ENSO. They reflect anomalies in the location and intensity
of the meridionally (i.e., north–south) oriented Hadley cell and
of the zonally oriented Walker cell.
The state of the Southern Oscillation may be monitored with the
monthly SLP difference between observations taken at surface
stations in Darwin, Australia and Papeete, Tahiti. It has become
common practice to call this difference the Southern Oscillation
Index (SOI) although there are also many other ways to define
equivalent indices [426].
7
Other definitions, such as West Pacific rainfall, sea-level
pressure at Darwin alone or the surface zonal wind in the central
Pacific, also yield indices that are highly correlated with the
usual SOI. See Wright [427].
spatially correlated. The Southern Oscillation In-
dex (Figure 1.2) is a manifestation of the negative
correlation between surface pressure at Papeete
and that at Darwin. Variables such as pressure,
height, wind, temperature, and specific humidity
vary smoothly in the free atmosphere and con-
sequently exhibit strong spatial interdependence.
This correlation is present in each weather map
(Figure 1.5, left). Indeed, without this feature,
routine weather forecasts would be all but impos-
sible given the sparseness of the global observing
network as it exists even today. Variables derived
from moisture, such as cloud cover, rainfall and
snow amounts, and variables associated with land
surface processes tend to have much smaller spa-
tial scales (Figure 1.5, right), and also tend not to
have normal distributions (Sections 3.1 and 3.2).
While mean sea-level pressure (Figure 1.5, left)
will be more or less constant on spatial scales of
tens of kilometres, we may often travel in and out
of localized rain showers in just a few kilometres.
This dichotomy is illustrated in Figure 1.5, where
we see a cold front over Ontario (Canada). The
left panel, which displays mean sea-level pressure,
shows the front as a smooth curve. The right panel
displays a radar image of precipitation occurring
in southern Ontario as the front passes through the
region.
1.2.3 Stationarity, Cyclo-stationarity, and Non-
stationarity. An important concept in statistical
analysis is stationarity. A random variable, or a
random process, is said to be stationary if all
of its statistical parameters are independent of
time. Most statistical techniques assume that the
observed process is stationary.
However, most climate parameters that are
sampled more frequently than one per year are
not stationary but cyclo-stationary, simply because
of the seasonal forcing of the climate system.
Long-term averages of monthly mean sea-level
pressure exhibit a marked annual cycle, which is
almost sinusoidal (with one maximum and one
minimum) in most locations. However, there are
locations (Figure 1.6) where the annual cycle is
dominated by a semiannual variation (with two
maxima and minima). In most applications the
mean annual cycle is simply subtracted from the
data before the remaining anomalies are analysed.
The process is cyclo-stationary in the mean if it is
stationary after the annual cycle has been removed.
Other statistical parameters (e.g., the percentiles
of rainfall) may also exhibit cyclo-stationary
behaviour. Figure 1.7 shows the annual cycles
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