This book concerns the application of mathematics to problems in the physical sciences, and particularly to problems which arise in the study of the environment. Much of the environment consists of fluid—the atmosphere, the ocean—and even those parts which are solid may deform in a fluid-like way—ice sheets, glaciers, the Earth’s mantle; as a consequence, one way into the study of the environment is through the study of fluid dynamics, although we shall not follow that approach here. Rather, we shall approach the study of environmental problems as applied mathematicians, where the emphasis is on building a suitable mathematical model and solving it, and in this introductory chapter, we set out the stall of techniques and attitudes on which the subsequent chapters are based <...>

Mathematical modelling and computer simulations are an essential part of the analytical skills for earth scientists. Nowadays, computer simulations based on mathematical models are routinely used to study various geophysical, environmental and geological processes, from geophysics to petroleum engineering, from hydrology to environmental fluid dynamics. The topics in earth sciences are very diverse and the syllabus itself is evolving. From a mathematical modelling point of view, therefore, this is a decision to select topics and limit the number of chapters so that the book remains concise and yet comprehensive enough to include important and interesting topics and popular algorithms. Furthermore, we use a ‘theorem-free’ approach in this book with a balance of formality and practicality. We will increase dozens of worked examples so as to tackle each problem in a step-by-step manner, thus the style will be especially suitable for non-mathematicians, though there are enough topics, such as the calculus of variation and pattern formation, that even mathematicians may find them interesting. <...>

This chapter develops mathematical concepts and tools underlying techniques for estimation of material structures from observations of reflected waves. These techniques belong in some sense to the theory of inverse problems for partial dierential equations, but many of them have their origins outside of modern mathematics. Most of the ideas developed in this chapter arose rst in the seismology, particularly in that branch devoted to exploration for petroleum. Therefore I shall take viewpoint and terminology from exploration seismology. A principal goal of this chapter is to provide a natural \Courant|Hilbert" setting for these ideas and terms, which play a very important role in the following chapters.<...>

Th e book MATLAB Recipes for Earth Sciences is designed to help undergraduate and postgraduate students, doctoral students, post-doctoral researchers, and professionals fi nd quick solutions for common data analysis problems in earth sciences. It provides a minimal amount of theoretical background and demonstrates the application of all described methods through the use of examples.