In this chapter we describe the motivation and requirements for converting analog signals to digital time series so that they can be analyzed using digital computers. The process of digitizing analog geophysical signals, for example continuous voltages from seismometers, must ensure that all information about signal amplitudes (the dynamic range) and range of frequencies (the frequency bandwidth) is retained. Important elements of analog to digital conversion include the number of samples to be taken (the sampling rate) and how to represent samples as numbers in a computer. The chapter also presents the standard statistics used to describe time series and introduces the logarithmic decibel scale as a convenient way to compare the relative magnitudes of signals.

The Leaning Tower of Pisa, used by Galileo to demonstrate the simplicity of science, is also a testament to the complexity of science. Over an 800-year period, multiple attempts were made to fix the errors in the tower’s construction that caused it to lean. Often, the fixes had unanticipated consequences, necessitating additional compensating fixes. Climate models face a similar problem. The models use approximate formulas called parameterizations, with adjustable parameters, to represent processes like clouds that are too fine to be resolved by the model grids. The optimal values of these parameters that minimize simulation errors are determined by a trial-and-error process known as “model tuning.” Tuning minimizes errors in simulating current and past climates, but it cannot guarantee that the predictions of the future will be free of errors. This means that models can be confirmed, but they cannot be proven to be correct.

The Discrete Fourier Transform (DFT) is one of the most important tools in geophysical data processing and in many other fields. The DFT may be understood from a number of viewpoints, but here we emphasize that it is a Fourier series representing a uniformly sampled time series as a sum of sampled complex sinusoids. We refer to the time series as being in the time domain while the set of its complex-valued sinusoidal coefficients computed using the DFT is in the frequency domain. The inverse DFT (IDFT) computes time series values by adding together Fourier frequency sinusoids, each scaled by a frequency domain sinusoidal coefficient. We develop the DFT by converting the ordinary Fourier series to complex form, transitioning to sampled time series, and finally explaining standard normalization and the usual (and often baffling) frequency and time ordering conventions. The DFT came into widespread use only in the 1960s after the development of Fast Fourier Transform (FFT) algorithms. The speed of FFT algorithms has led to many important applications. Those presented here include the interpolation and computation of analytic signals for real-valued time series. In later chapters we show the important roles of the DFT in linear filtering and spectral analysis.

Communicating the strengths and limitations of climate modeling to those outside the field of climate science is a formidable challenge. The nuances of scientific language can be lost in the translation to natural language when climate predictions are presented to a general audience. This loss in translation can lead to misinformation and disinformation that hampers a rational response to the climate crisis. Even simple terms like “model,” “data,” and “prediction” have many different meanings depending on the context. Anytime we talk about the future, we are using a model. In climate science, we might think we are dealing with data from the past, but often this is processed data that is produced by analysis models applied to raw data. The word “prediction” can mean a range of things, from unconditional prophecies to conditional projections.

Global warming became a growing public concern following Jim Hansen’s US Senate testimony in 1988 asserting that the warming was happening. The Intergovernmental Panel on Climate Change (IPCC) was formed in response to this concern. The IPCC issues periodic assessments summarizing recent scientific developments relating to climate change. Climate models were used to attribute global warming to increasing concentrations of carbon dioxide and other greenhouse gases. Certain types of extreme weather can also be probabilistically attributed to these causes. The effect of aerosols and stochastic variability on the past global warming signal is described. The IPCC projects the global warming signal into the future using a range of carbon dioxide emission scenarios, resulting in different degrees of predicted warming. The importance of regional climate change and the difficulty of predicting it are discussed.

The concept of Occam’s Razor, also known as the principle of parsimony, is a motivating force in science. Galileo’s experiment of dropping objects of different weights from atop the Leaning Tower of Pisa to show that they fall at the same rate is an illustration of this principle. But the application of Occam’s Razor to climate modeling is less straightforward. Simple models of the kind used by Manabe provide qualitative insights, but they are not well-suited for quantitative predictions. To understand this, we can make an analogy between the hierarchy of climate models and the hierarchy of biological models, from fruit fly to mouse. Simple models are used to explore “climate tipping points,” where amplifying feedbacks lead to abrupt climate change, but they may not consider all the stabilizing feedbacks. It is therefore important to use a hierarchy of models, with varying degrees of complexity, to study climate phenomena.

The fundamental difference between weather prediction and climate prediction is explained, using a “nature versus nurture” analogy. To predict weather, we start from initial conditions of the atmosphere and run the weather forecast model. To predict climate, the initial conditions matter less, but we need boundary conditions, such as the angle of the sun or the concentration of carbon dioxide in the atmosphere, which control the greenhouse effect. Charles David Keeling began measuring carbon dioxide in the late 1950s, and found that its concentration was steadily increasing. Carbon dioxide concentrations for the past 800,000 years can also be measured using ice cores that contain trapped air. These ice core data show that the rise in carbon dioxide concentrations measured by Keeling was unprecedented. Manabe, and another scientist, Jim Hansen, used climate models to predict that increasing carbon dioxide could cause global warming.

The Aymara people use the metaphor that the future is behind us and the past is in front of us. Imagine that we are driving a car in reverse into our climate future. The past is in front of us; climate models act as the rearview mirror, showing what lies behind us in the future. The view is blurry because there is uncertainty, and the car is moving fast as we continue to emit greenhouse gases. We need to brake quickly – reduce emissions – because we know that the braking distance is very long. The Paris Agreement to reduce worldwide emissions is like a potluck dinner: Each guest decides how much food to bring. If the guests don’t bring enough food for everyone, then some will leave hungry. Similarly, emission reductions pledged in the Paris Agreement are voluntary and may be not be sufficient to strongly mitigate the warming.

The Geophysical Fluid Dynamics Laboratory (GFDL) is a pioneering institution in the field of climate modeling. Its founding director, Joseph Smagorinsky, was a member of the Princeton Meteorology Group. He hired a Japanese scientist, Syukuro Manabe, who formulated a one-dimensional model of climate, known as the radiative–convective model, that was able to calculate the amplifying climate feedback due to water vapor. This model provided one of the first reliable estimates of global warming. Manabe worked with other scientists to build three-dimensional climate models, including the first model that coupled an atmospheric model to an ocean model. The concepts of reductionism and emergentism, which provide the philosophical context for these scientific developments, are introduced.