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Library of Congress Cataloging-in-Publication Data
Palmer, Paul I.
Essential maths for geoscientists : an introduction / Paul I. Palmer.
pages cm
Includes bibliographical references and index.
ISBN 978-0-470-97193-2 (cloth) – ISBN 978-0-470-97194-9 (pbk.) 1. Geology–Mathematics. 2. Mathematics–Study and teaching. 3. Ecology–Mathematical models. 4. Environmental protection–Mathematical models. I. Title. II. Title: Essential math for geoscientists.
QE33.2.M3P35 2014
510.24′55–dc23
2013044549
A catalogue record for this book is available from the British Library.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
To A.L.M., L.M.P., and J.S.P.
This book includes the lectures and problem sets from the one-semester course ‘Earth Modelling and Prediction’ that I teach at the University of Edinburgh. The course is aimed at first-year geoscience undergraduates who want to understand the Earth and its evolving climate but do not have the necessary quantitative skills to move beyond qualitative studies. My primary and most ambitious objective for this course was to help students overcome the psychological barrier of applying mathematics to problems associated with the Earth. It is this barrier that artificially limits students’ ability to gain a deeper understanding of the underlying science. My second objective was to show that the relatively simple mathematics covered in this course could be applied to learn something relevant to current areas of scientific research.
The focus of the book is the application of mathematics to scientifically relevant problems. Rather than being comprehensive, the material should be seen as providing a background for more advanced geoscience courses, which practise the application of mathematics and introduce the students to additional mathematics. I support the use of real data in teaching and so in recent years I have included progressively more exercises that involve the analysis of real measurements, many of which form the backdrop to a major news story in that year, for example, increased/decreased tropical deforestation rates or the reduction in the spatial extent of Arctic sea ice. I hope to include in future editions more varied data analysis problems that reflect the breadth of geoscience research.
I thank Patience Cowie, Roger Scrutton, and Roger Hipkin for recognizing the need for this course and for helping me to establish it at Edinburgh. For helping to teach topics within the course over the years I thank Patience Cowie, Godfrey Fitton, Gabriele Hegerl, Roger Hipkin, Ian Main, Chris Merchant, Mark Parrington, Simon Tett, and Thorvaldur Thordarson. I thank all the tutors who helped to make the course work well: Amber Annett, Louise Barron, Dave Bell, Anthony Bloom, Matthew Brolly, Iain Cameron, Ruth Carley, Craig Duguid, Leon Kapetas, Simon King, Jack Lonsdale, Malcolm McMillan, Simone Morak, Heather Nicolson, Katie Noak, Luke Ridley, Robert Shore, Luke Smallman, Lorna Street, Oliver Sus, Sarah Touati, Matthew Unterman, Lucia Viegas, and Adam Wilson. Finally, I thank Martin Wooster (King’s College London) for proofreading and providing useful comments on an earlier draft of the manuscript.
Paul I. Palmer
University of Edinburgh
May 2013
The title of this introductory chapter is the question I pose at the start of my course in Edinburgh. It seems like a ridiculous question to ask a bunch of bright young students, especially ones who have chosen to study the Earth system. But up until walking through the doors of the university many students have not had the resources, inclination, and/or ability to question what they are told; the key to being an effective scientist is to ask the right questions, ones that probe at the very heart of the problem being studied. I provide the student with four possible choices to answer the question and ask for a show of hands:
Typically, choice 1 represents the vast majority of hands. Why? Because we are bombarded with scientific and political coverage of climate change. Why is this dangerous? Because companies need to sell newspapers and to get people to watch TV, and politicians are invariably biased in their opinions. Much of the coverage is accurate but some programmes are biased, loosely based on fact, with a damaging effect on the science education of the general public. Sensationalism about Earth’s climate (particularly looking to the future) is rife, but some aspects of Earth’s climate are genuinely remarkable and awe-inspiring. So how do you know what to believe?
Choice 2 often represents the second highest show of hands, but a much smaller proportion than choice 1. This is fine up to a point. Scientists are some of the biggest sceptics around and are generally very careful about what they say. For instance, we see later in this chapter that the wording used in the Intergovernmental Panel on Climate Change (IPCC) report1 has very strict statistical interpretation that is difficult to misinterpret. But you only learn from the scientists what they tell you. How did they reach their conclusions? Could they have approached the problem from a different perspective and reached a different conclusion? With the renewed call for transparency in science, particularly related to climate, most data used to draw conclusions about Earth’s climate are online and freely available to download. Often the only barrier to pursuing option 2, given that data are now freely available, is the confidence to understand and interrogate quantitative data. The aim of this book is to increase that confidence.
This mix of responses is reasonably similar to the general public response to the question ‘How well do you feel you understand the issue of global warming?’ that has been asked frequently by Gallup (www.gallup.com) for the past quarter century (Figure 1.1). For this admittedly crude comparison I have equated ‘Great deal’ with ‘Rigorous scientific reasoning’, ‘Fair amount’ with ‘Popular media’, and ‘Only a little’ with ‘(Blind) faith in scientists’.
Figure 1.1 Results from a Gallup poll question ‘How well do you feel you understand the issue of global warming?’ that has been asked since 1989.
How can mathematics help? In simple terms, mathematics (at this level) is a tool that allows us to move far beyond what we can learn from descriptive analysis. How much has sea ice changed? If we use the current rate of change, how long will it be before the Arctic is free of ice? These are simple example questions that cannot be answered without mathematics.
Figure 1.2 A schematic describing the broad-scale subcomponents of the Earth system. Graphics reproduced with permission from the UK/NERC National Centre for Earth Observation. (Image courtesy of NASA.)
I define the Earth system as the land, ocean, and atmosphere, all the physical, chemical, biological, and social processes and their interactions (Figure 1.2). This is a big unwieldy interconnected system that is coupled on a wide spectrum of spatial and temporal scales. To minimize the risk of discussing current science results that might be superseded by new data, I have decided to focus on how scientists generally know what they know about the Earth system and the recent role of human activity and not what they know:
It is important to acknowledge that several independent lines of inquiry are used to investigate phenomena and provide evidence to test a hypothesis. The IPCC is testing the overarching hypothesis that human activity has determined recent changes in climate. As we will see in the next chapter, the hypothesis is right at the crux of the scientific method. In successive IPCC reports the headline result has been stronger and stronger:
In the IPCC nomenclature the term ‘likely’ refers to a probability greater than 66% and ‘very likely’ to a probability greater than 90%. In 2001 the IPCC was more than 66% certain that climate change was caused by human activity. By 2007 it was more than 90% certain that recent climate change is due to anthropogenic greenhouse gas concentrations. And most recently, in 2013, the IPCC increased this confidence to 95%. It is possible that climate change is due to other causes, but the IPCC regards this as unlikely. It is unfortunate that this level of scientific ‘honesty’ also represents an inroad to climate scepticism.