frequentist approach to statistics we use in

that the methodology described here is a current

this book. Bayesians, as they are known,

re¬‚ection of the art of applied statistics as it is

incorporate a priori beliefs into a statistical

conducted by statisticians. Statistics as it is applied

analysis of a sample in a rational manner (see

in climatology is far removed from the cutting

Epstein [114], Casella [77], or Gelman et al.

edge of methodological development. This is

[139]).

partly because statistical research has not come yet

to grips with many of the problems encountered • Geostatistics, which is widely used in geol-

by climatologists and partly because climatologists ogy and related ¬elds. This approach deals

have not yet made very deep excursions into the with the analysis of spatial ¬elds sampled at

world of mathematical statistics. Instead, this book a relatively small number of locations. The

presents a subjectively chosen discourse on the most prominent technique is called kriging

tools we have found useful in our own research on (see Journel and Huijbregts [207], Journel

climate diagnostics. [206], or Wackernagel [406]), which is re-

We will discuss a variety of statistical concepts lated to the data assimilation techniques used

and tools which are useful for solving problems in in atmospheric and oceanic science (see, e.g.,

climatological research, including the following. Daley [98] and Lorenc [258]).

• The concept of a sample. A collection of applications of many statistical

techniques has been compiled by von Storch and

• The notions of exploratory and con¬rmatory Navarra [395]; we recommend this collection as

complementary reading to this book and refer to

statistics.

ix

x

• Thanks for discussion, review, advice and

its contributions throughout. This collection does

useful comments: Gerd B¨ rger, Bill Burrows,

u

not cover the ¬eld systematically; instead it offers

Ulrich Callies, Susan Chen, Christian Eckert,

examples of the exploitation of statistical methods

Claude Frankignoul, Marco Giorgetta, Sil-

in the analysis of climatic data and numerical

vio Gualdi, Stefan G¨ ß, Klaus Hasselmann,

u

experiments.

Gabi Hegerl, Patrick Heimbach, Andreas

Cookbook recipes for a variety of standard

Hense, Hauke Heyen, Martina Junge, Thomas

statistical situations are not offered by this book

Kaminski, Frank Kauker, Dennis Letten-

because they are dangerous for anyone who does

maier, Bob Livezey, Ute Luksch, Katrin

not understand the basic concepts of statistics.

Maak, Rol Madden, Ernst Maier-Reimer, Pe-

Therefore, we offer a course in the concepts

ter M¨ ller, D¨ rthe M¨ ller-Navarra, Matthias

u o u

and discuss cases we have encountered in our

M¨ nnich, Allan Murphy, Antonio Navarra,

u

work. Some of these examples refer to standard

Peter Rayner, Mark Saunders, Reiner Schnur,

situations, and others to more exotic cases. Only

Dennis Shea, Achim St¨ ssel, Sylvia Venegas,

o

the understanding of the principles and concepts

Stefan Venzke, Koos Verbeeck, Jin-Song von

prevents the scientist from falling into the many

Storch, Hans Wackernagel, Xiaolan Wang,

pitfalls speci¬c to our ¬eld, such as multiplicity

Chris Wickle, Arne Winguth, Eduardo Zorita.

in statistical tests, the serial dependence within

samples, or the enormous size of the climate™s

phase space. If these dangers are not understood, • Thanks for making diagrams available to

then the use of simple recipes will often lead to us: Howard Barker, Anthony Barnston,

erroneous conclusions. Literature describes many Grant Branstator, Gerd B¨ rger, Bill Burrows,

u

cases, both famous and infamous, in which this has Klaus Fraedrich, Claude Frankignoul, Euge-

occurred. nia Kalnay, Viacheslaw Kharin, Kees Ko-

We have tried to use a consistent notation revaar, Steve Lambert, Dennis Lettenmaier,

throughout the book, a summary of which is Bob Livezey, Katrin Maak, Allan Murphy,

offered in Appendix A. Some elements of linear Hisashi Nakamura, Reiner Schnur, Lucy Vin-

algebra are available in Appendix B, and some cent, Jin-Song von Storch, Mike Wallace,

aspects of Fourier analysis and transform are listed Peter Wright, Eduardo Zorita.

in Appendix C. Proofs of statements, which we do

not consider essential for the overall understand-

• Thanks for preparing diagrams: Marion

ing, are in Appendix M.

Grunert, Doris Lewandowski, Katrin Maak,

Norbert Noreiks, and Hinrich Reichardt, who

Thanks helped also to create some of the tables in the

Appendices. For help with the L TEX-system:

A

We are deeply indebted to a very large number J¨ rg Wegner. For help with the Hamburg

o

of people for their generous assistance with this computer network: Dierk Schriever. For help

project. We have tried to acknowledge all who con- with the Canadian Centre for Climate Mod-

tributed, but we will inevitably have overlooked elling and Analysis computer network in Vic-

some. We apologize sincerely for these oversights. toria: Mike Berkley. For scanning diagrams:

• Thanks for her excellent editorial assistance: Mike Berkley, Jutta Bernl¨ hr, and Marion

o

Grunert.

Robin Taylor.

1 Introduction

its enormously large phase space.1 Thus it is not

1.1 The Statistical Description and

possible to map the state of the atmosphere, the

Understanding of Climate

ocean, and the other components of the climate

system in full detail. Also, the models are not

Climatology was originally a sub-discipline of deterministic in a practical sense: an insigni¬cant

geography, and was therefore mainly descriptive change in a single digit in the model™s initial

(see, e.g., Br¨ ckner [70], Hann [155], or Hann

u conditions causes the model™s trajectory through

and Knoch [156]). Description of the climate phase space to diverge quickly from the original

consisted primarily of estimates of its mean state trajectory (this is Lorenz™s [260] famous discovery,

and estimates of its variability about that state, which leads to the concept of chaotic systems).

such as its standard deviations and other simple Therefore, in a strict sense, we have a

measures of variability. Much of climatology is ˜deterministic™ system, but we do not have

still focused on these concerns today. The main the ability to analyse and describe it with

purpose of this description is to de¬ne ˜normals™ ˜deterministic™ tools, as in thermodynamics.

and ˜normal deviations,™ which are eventually Instead, we use probabilistic ideas and statistics to

displayed as maps. These maps are then used describe the ˜climate™ system.

for regionalization (in the sense of identifying Four factors ensure that the climate system is

homogeneous geographical units) and planning. amenable to statistical thinking.

The paradigm of climate research evolved from

• The climate is controlled by innumerable

the purely descriptive approach towards an

factors. Only a small proportion of these

understanding of the dynamics of climate with the

factors can be considered, while the rest

advent of computers and the ability to simulate the

are necessarily interpreted as background

climatic state and its variability. Statistics plays an

noise. The details of the generation of this

important role in this new paradigm.

˜noise™ are not important, but it is important

The climate is a dynamical system in¬‚uenced to understand that this noise is an internal

not only by immense external factors, such as solar source of variation in the climate system

radiation or the topography of the surface of the (see also the discussion of ˜stochastic climate

solid Earth, but also by seemingly insigni¬cant models™ in Section 10.4).

phenomena, such as butter¬‚ies ¬‚apping their

• The dynamics of climate are nonlinear.

wings. Its evolution is controlled by more or

Nonlinear components of the hydrodynamic

less well-known physical principles, such as the

part include important advective terms, such

conservation of angular momentum. If we knew

‚u

as u ‚ x . The thermodynamic part contains