Essential Maths for Geoscientists

An Introduction

Paul I. Palmer

• Publisher: Wiley-Blackwell
• Year: 2014
• Print ISBN: 9780470971949, 9780470971932
• eBook ISBN: 9781118527023
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geophysics

Essential Maths for Geoscientists

An Introduction

Essential Maths for Geoscientists: An Introduction is an accessible, student-friendly introduction to the mathematics required by those students taking degree courses within the geosciences. Clearly structured throughout, this book carefully guides students step by step through the first mathematics they will encounter and provides numerous applied examples throughout to enhance students' understanding and to place each technique in context.

Opening with a chapter explaining the need for studying mathematics within geosciences, this book then moves on to cover algebra, solving equations, logarithms and exponentials, uncertainties, errors and statistics, trigonometry, vectors and basic calculus. The final chapter helps to bring the subject all together and provides detailed applied questions to test students' knowledge.

Worked applied examples are included in each chapter along with applied problem questions which are a mix of straightforward maths questions, word questions and more involved questions that involve the manipulation and interpretation of real and synthetic data.

The emphasis in the book is on the application of relatively rudimentary mathematics to real-life scientific problems within the geosciences, enabling students to make use of current-day research problems and real datasets.

1
How Do You Know that Global Warming Is Not a Hoax?

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:

1. popular media (internet, TV, radio, newspapers);
2. rigorous scientific reasoning and/or debate;
3. (blind) faith in scientists; or
4. other.

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.)

The Earth system: how do we know what we know?

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:

• First, we have a basic physical understanding of the Earth. We know, for example, about the heat-trapping properties of gases in the atmosphere, based on work first started in the nineteenth century. Another example is continental drift, a theory describing how Earth’s continents move relative to each other, which has been known since the twentieth century. These are well-established science theories that have stood up to decades/centuries of scientific scrutiny.
• Second, we have circumstantial evidence. We make qualitative connections between observations of disparate quantities and results from computer models² of the Earth system, for example, warming of oceans, lands, and the lower atmosphere, cooling of the middle atmosphere, and increases in water vapour.
• Third, we have palaeoclimate evidence. We can reconstruct past climate using a variety of data, for example, ice core, lake sediment core, coral reefs, pollen. This places contemporary warming trends in the longer-term context. Although there is debate about whether the past is any guide to the future, they do provide us a history of how Earth has behaved in the past.
• Finally, we have so-called ‘fingerprint’ evidence. The underlying philosophy is that individual (natural and human-driven) processes will leave their own unique signature (or fingerprint) on measurements of the Earth. By comparing these data that naturally include these signatures with computer models of climate with/without descriptions of the processes responsible for these signatures we can understand the importance of individual processes. This can also potentially identify the need for additional processes that are currently not present in the model.

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:

• 1995: The balance of evidence suggests a discernable human influence on global climate.
• 2001: Most of the observed warming over the last 50 years is likely to have been due to the increase in greenhouse gas concentrations.
• 2007: Most of the observed increase in globally averaged temperatures since the mid-twentieth century is very likely due to the observed increase in anthropogenic greenhouse gas concentrations.

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.

Notes

¹ A report prepared by a ...

Table of contents

• Cover
• Titlepage
• Copyright
• Dedication
• Contents
• Preface
• 1 How Do You Know that Global Warming Is Not a Hoax?
• 2 Preamble:  
• 3 Algebra
• 4 Solving Equations
• 5 Logarithms and Exponentials
• 6 Uncertainties, Errors, and Statistics
• 7 Trigonometry
• 8 Vectors
• 9 Calculus 1: Differentiation:  
• 10 Calculus 2: Integration
• 11 Bringing It All Together
• A Answers to Problems
• B A Brief Note on Excel:  
• C Further Reading
• Index