variation of the system.10

whereas Shawinigan has maintained more of its

rural character.

Top: The raw records. The abrupt drop of several 1.2.4 Quality of Forecasts. The Old Farmer™s

degrees in the Sherbrooke series in 1963 re¬‚ects Almanac publishes regular outlooks for the climate

the move of the instrument from downtown Sher- for the coming year. The method used to prepare

brooke to its suburban airport. The reason for these outlooks is kept secret, and scientists

the downward dip before 1915 in the Shawinigan question the existence of skill in the predictions.

To determine whether these skeptics are right or

record is unknown.

Bottom: Corrected time series for Sherbrooke wrong, measures of the skill of the forecasting

and Shawinigan. The Sherbrooke data from 1963 scheme are needed. These skill scores can be used

onward are increased by 3.2 —¦ C. The straight lines to compare forecasting schemes objectively.

are trend lines ¬tted to the corrected Sherbrooke The Almanac makes categorical forecasts of

data and the 1915“90 Shawinigan record. future temperature and precipitation amount in

Courtesy L. Vincent, AES Canada. two categories, ˜above™ or ˜below™ normal. A

suitable skill score in this case is the number of

correct forecasts. Trivial forecasting schemes such

called inhomogeneities. An example is contained as persistence (no change), climatology, or pure

in the temperature records of Sherbrooke and chance can be used as reference forecasts if no

Shawinigan (Quebec) shown in the upper panel other forecasting scheme is available. Once we

of Figure 1.9. The Sherbrooke observing site have counted the number of correct forecasts made

was moved from a downtown location to a with both the tested and the reference schemes, we

suburban airport in 1963”and the recorded can estimate the improvement (or degradation) of

temperature abruptly dropped by more than 3 —¦ C. forecast skill by computing the difference in the

The Shawinigan record may also be contaminated counts. Relatively simple probabilistic methods

can be used to make a judgement about the

by observational errors made before 1915.

Geophysical time series often exhibit a trend. 10 This is an example of the importance of time scales

in climate research, an illustration that our interpretation of

Such trends can originate from various sources.

a given process depends on the time scales considered. A

One source is urbanization, that is, the increasing

short-term trend may be just another swing in a slowly varying

density and height of buildings around an obser- system. An example is the Madden-and-Julian Oscillation

vation location and the corresponding changes in (MJO, [264]), which is the strongest intra-seasonal mode in the

tropical troposphere. It consists of a wavenumber 1 pattern that

the properties of the land surface. The temper-

travels eastward round the globe. The MJO has a mean period

ature at Sherbrooke, a location heavily affected of 45 days and has signi¬cant memory on time scales of weeks;

by development, exhibits a marked upward trend on time scales of months and years, however, the MJO has no

after correction for the systematic change in 1963 temporal correlation.

1: Introduction

10

(e.g., skill may be high during the dry season, and

low during the wet season). The skilfulness of a

forecast also often depends on the low-frequency

state of the atmospheric ¬‚ow (e.g., blocking

or westerly regime). Thus, in most forecasting

problems there are physical considerations (state

dependence and the memory of the system) that

must be accounted for when using statistical tools

to analyse forecast skill. This is done either

by conducting a statistical analysis of skill that

incorporates the effects of state dependence and

serial correlation, or by using physical intuition

to temper the precise interpretation of a simpler

analysis that compromises the assumptions of

stationarity and non-correlation.

There are various pitfalls in the art of forecast

Figure 1.10: Correlation skill scores for three evaluation. An excellent overview is given by

forecasts of the low-frequency variations within Livezey [255], who presents various examples in

the Southern Oscillation Index (Figure 1.2). A which forecast skill is overestimated. Chapter 18

score of 1 indicates a perfect forecast, while a zero is devoted to the art of forecast evaluation.

indicates a forecast unrelated to the predictand

1.2.5 Characteristic Times and Characteristic

[432].

Spatial Patterns. What are the temporal char-

acteristics of the Southern Oscillation Index illus-

signi¬cance of the change. We will return to the trated in Figure 1.2? Visual inspection suggests

Old Farmer™s Almanac in Section 18.1. that the time series is dominated by at least two

Now consider another forecasting scheme time scales: a high frequency mode that describes

in which quantitative rather than categorical month-to-month variations, and a low-frequency

statements are made. For example, a forecast mode associated with year-to-year variations. How

might consist of a statement such as: ˜the SOI can one objectively quantify these characteristic

will be x standard deviations above normal next times and the amount of variance attributed to

winter.™ One way to evaluate such forecasts is to these time scales? The appropriate tool is referred

use a measure called the correlation skill score to as time series analysis (Chapters 10 and 11).

ρ (Chapter 18). A score of ρ = 1 corresponds Indices, such as the SOI, are commonly used

with a perfect forecasting scheme in the sense that in climate research to monitor the temporal

forecast changes exactly mirror SOI changes even development of a process. They can be thought

though the dynamic range of the forecast may be of as ¬lters that extract physical signals from a

different from that of the SOI. In other words, multivariate environment. In this environment the

the correlation skill score is one when there is signal is masked by both spatial and temporal

an exact linear relationship between forecasts and variability unrelated to the signal, that is, by spatial

reality. Forecasts that are (linearly) unrelated to the and temporal noise.

predictand yield zero correlation. The conventional approach used to identify

The correlation skill score for several methods indices is largely subjective. The characteristic pat-

of forecasting the SOI are displayed in Figure 1.10. terns of variation of the process are identi¬ed and

Speci¬cally, persistence forecasts (Chapter 18), associated with regions or points. Corresponding

POP forecasts (Chapter 15), and forecasts made areal averages or point values are then used to

with a univariate linear time series model indicate the state of the process.

(Chapters 11 and 12). Forecasts based on Another approach is to extract characteristic

persistence and the univariate time series model patterns from the data by means of analytical

are superior at one and two month lead times. The techniques, and subsequently use the coef¬cients

POP forecast becomes more skilful beyond that of these patterns as indices. The advantages

time scale. of this approach are that it is based on

Regretfully, forecasting schemes generally do an objective algorithm and that it yields the

not have the same skill under all circumstances. characteristic patterns explicitly. Eigentechniques

The skill often exhibits a marked annual cycle such as Empirical Orthogonal Function (EOF)

1.2: Some Typical Problems and Concepts 11

Figure 1.11: Empirical Orthogonal Functions

(EOFs; Chapter 13) of monthly mean wind stress

over the tropical Paci¬c [394].

a,b) The ¬rst two EOFs. The two patterns are

spatially orthogonal. Figure 1.12: A schematic representation of the

c) Low-frequency ¬ltered coef¬cient time series spatial distributions of simultaneous SST and SLP

of the two EOFs shown in a,b). The solid curve anomalies at Northern Hemisphere midlatitudes in

corresponds to the ¬rst EOF, which is displayed in winter, when the SLP anomaly induces the SST

panel a). The two curves are orthogonal. anomaly (top), and when the SST anomaly excites

the SLP anomaly (bottom).

The large arrows represent the mean atmospheric

analysis and Principal Oscillation Pattern (POP) ¬‚ow. The ˜L™ is an atmospheric low-pressure

analysis are tools that can be used to de¬ne system connected with geostrophic ¬‚ow indicated

patterns and indices objectively (Chapters 13 and by the circular arrow. The hatching represents

15). warm (W) and cool (C) SST anomalies [438].

An example is the EOF analysis of monthly

mean wind stress over the tropical Paci¬c [394].

The ¬rst two EOFs, shown in Figure 1.11a in fact, may be associated with the Southern

and Figure 1.11b, are primarily con¬ned to the Oscillation.

equator. The two ¬elds are (by construction)

orthogonal to each other. Figure 1.11c shows the 1.2.6 Pairs of Characteristic Patterns. Almost

time coef¬cients of the two ¬elds. An analysis of all climate components are interrelated. When one

the coef¬cient time series, using the techniques component exhibits anomalous conditions, there

of cross-spectral analysis (Section 11.4), shows will likely be characteristic anomalies in other

that they vary coherently on a time scale T ≈ components at the same time. The relative shapes

2 to 3 years. One curve leads the other by a time of the patterns in related climate components are

lag of approximately T /4 years. The temporal lag- often indicative of the processes that dominate the

relationship of the time coef¬cients together with coupling of the components.

the spatial quadrature leads to the interpretation To illustrate this idea we consider large-scale

that the two patterns and their time coef¬cients air“sea interactions on seasonal time scales at

describe an eastward propagating signal that, midlatitudes in winter [438] [312]. Figure 1.12

1: Introduction

12

illustrates the two mechanisms that might be anomalies off the North American coast. Peng and

involved in air“sea interactions in the North Fyfe [312] refer to this as the ˜atmosphere driving

Atlantic. The lower panel illustrates how a sea- the ocean™ mode. See also Luksch [261].

surface temperature (SST) anomaly pattern might Canonical Correlation Analysis is explained in

induce a simultaneous sea-level pressure (SLP) detail in Chapter 14 and we return to this example

anomaly pattern. The argument is linear so we in [14.3.1“2].

may assume that the SST anomaly is positive. This

positive SST anomaly enhances the sensible and

1.2.7 Atmospheric General Circulation Model