The power spectrum describes how the variance of a time series is distributed over frequency. The variance (mean squared signal value per time sample) is a broadband statistic measuring power, while the power spectrum at a given frequency is a statistic measuring the power in a narrow frequency band. Similarly, the coherence spectrum describes how the broadband correlation coefficient between two time series varies with frequency. The DFT is the main tool for estimating both the power and coherence spectra and will be our main focus, but we also compare the DFT (periodogram) results with estimates made using the prediction error filter (PEF) developed in . Using examples we show that PEF estimates tend to be smooth, but the choice of PEF order introduces some variability in these estimates. Periodogram spectrum estimates tend to be erratic but can be tamed at the expense of diminished frequency resolution. We describe standard methods of assigning confidence intervals to periodogram spectrum estimates.

Chapter 20 studies gas, biological, chemical, and nuclear weapons, looking at each of those prohibited weapons in turn. Poisonous gases were first banned in 1925 – but only their use, not their production or sale. There have been numerous uses of poisonous gases, including today in Syria. Biological and toxin weapons were banned in 1972, although their use is unmentioned in the 1972 Convention. A 1993 Chemical Weapons Convention, with a strict verification process, has been more effective. Russia’s continued poisoning of state critics is detailed as an example of state-avoidance of the 1993 Convention. CS (“tear gas”) use is limited by the 1993 convention but still employed as a riot control measure. There is no international law that makes nuclear weapons unlawful. Of the several multinational treaties that bear on nuclear weapons, the most significant is the 2017 Treaty on the Prohibition of Nuclear Weapons, which, so far, has but forty-four State Parties. None of the nine nuclear powers have joined, of course. In short, gas, biological, chemical, and nuclear weapons are abhorred and condemned until they are used; then the international community looks the other way.

There are many systems in nature that are made up of several particles of the same species. These particles all have the same mass, charge and spin, and need to be treated as identical particles. For instance, the electrons in an atom are identical particles. Identical particles cannot be distinguished by measuring their intrinsic properties. While this is also true for classical particles, the laws of classical mechanics allow us to follow the trajectory of each individual particle, i.e. their time evolution in space.

The least squares method is among the most widely used data analysis and parameter estimation tools. Its development is associated with the work of Gauss, the nineteenth century German geophysicist, who introduced many innovations in computation, geophysics, and mathematics, most of which continue to be in wide use today. We will introduce least squares from two viewpoints, one based on probability arguments and considering the data to be contaminated with random errors and the other based on a linear algebra viewpoint and involving the solution of simultaneous linear equations. We will employ these two viewpoints to develop the least squares approach, using, as an example, the fitting of polynomials to a time series. While this might appear to be a departure from linear filters and related topics, we show in subsequent chapters that in fact least squares serves as a powerful digital filter design tool. We will also find that a stochastic viewpoint (in which time series values are considered to be random variables; see ) leads also to the use of least squares in the development of prediction and interpolation filters.

Chapter 3 orients the student to the basics of LOAC as it exists on today’s battlefields. It moves from long-past history to modern, even contemporary, history by relating today’s LOAC, along with its more recent historical foundations and modern law of war incidents. For example, World War I and the ineffective trials of German war criminals by German courts – the Leipzig trials – are shown to be the impetus for World War II’s Nuremberg and Tokyo international military tribunals. The LOAC import of the League of Nations and the Spanish Civil War are detailed, along with the 1929 Geneva Conventions’ (two Conventions) contributions. The bulk of the chapter is an examination of today’s four 1949 Geneva Conventions, including their “common articles,” the varied routes to the charging of war crimes alleged against both combatants and civilians, along with the significance of “grave breach” war crimes. Indicators of war crimes is detailed, as well.

Chapter 17 attempts to pin down a moving target, cyber and its use and utility in armed conflict. The cyber targets chosen by China, Russia, Iran, and North Korea are detailed. The chapter defines cyber “attacks,” differentiating them from cyber “operations” – akin to a felony-misdemeanor distinction. Cyberattacks are found to be a use of armed force, in the sense of UN Charter Article 2(4), raising the right of victim states to respond with armed force in self-defense. Potential cyber conflicts are classified as international or non-international based upon their perpetrators. The difficulties of attribution of cyberattacks are detailed, including sovereignty and military necessity. A counter to cyberattack is belligerent reprisal, which is explored. Cyberattacks on critical national infrastructure (CNI) are a major problem. After defining CNI, the US position on responses is discussed. Recent changes in US cyber policy authorize far greater cyber retaliation and reprisal, as well as providing federal funding to carry them out. Finally, CNI’s weak link, the unwillingness of civilian corporations to fund their own cyber protection, is noted.

Spatial and temporal samples of physical quantities are among the most common data form in the geosciences and many other fields. Such data are called time series (even if they are samples along a profile in space). This chapter presents examples of geophysical time series in order to illustrate the sorts of scientific questions that may be addressed by the methods inthrough . These time series are used in later chapters and exercises, and numerical values can be downloaded from www.cambridge.org/9781108931007. Virtually all the examples are available on the World Wide Web.