The objective of this chapter is to extend the ad hoc least squares method of somewhat arbitrarily selected base functions to a more generic method applicable to a broad range of functions – the Fourier series, which is an expansion of a relatively arbitrary function (with certain smoothness requirement and finite jumps at worst) with a series of sinusoidal functions. An important mathematical reason for using Fourier series is its “completeness” and almost guaranteed convergence. Here “completeness” means that the error goes to zero when the whole Fourier series with infinite base function is used. In other words, the Fourier series formed by the selected sinusoidal functions is sufficient to linearly combine into a function that converges to an arbitrary continuous function. This chapter on Fourier series will lay out a foundation that will lead to Fourier Transform and spectrum analysis. In this sense, this chapter is important as it provides background information and theoretical preparation.

The objective of this chapter is to present some important relations between the Fourier Transform and correlation functions. It turns out that the cross-correlation function and autocorrelation function have some useful relationships to Fourier Transform and power spectrum of the individual functions. As a result, cospectrum and coherence (normalized statistical correlation in frequency domain) can be defined.

The analyses we have discussed in previous chapters include the use of base functions, such as sinusoidal functions with specified frequencies, i.e. harmonic analysis; sinusoidal base functions with a frequency range from 0 to the Nyquist frequency with an interval inversely proportional to the total length of time of the data, i.e. Fourier analysis; and wavelet base functions for wavelet analysis. These base functions, however, are chosen regardless of the nature of the variability of the data themselves. In this chapter, we will discuss a different method, in which the base functions are determined empirically, that is dependent on the nature of the data. In other words, this method will find the base functions from the data and these base functions describe the nature of the data. The method is applicable to many types of data, especially to time series data at multiple locations, e.g. a sequence of weather maps or satellite images. There are several variants of the method, but here we will only provide an introduction for the basics.

This chapter introduces the fast Fourier Transform (FFT) for discrete Fourier Transform, beginning with the discretization of the Fourier Transform to its digital expression with constant time intervals. When the integral in Fourier Transform is replaced by a summation, the continuous Fourier Transform is changed to discrete. The discrete Fourier Transform and its inverse are exact relations. An example of the discrete Fourier Transform is discussed for a simple rectangular window function which results in the sinc function, useful for the interpretation of finite sampling effect. A technique of zero-padding is introduced with the discrete Fourier Transform for better visualization of the spectrum. But the computation of discrete Fourier Transform of a long time series can be quite “labor intensive” or costly in computer time with a direct computation. However, since the base functions are periodic, a direct computation can have many duplications in multiplications of terms. Algorithms can be designed to reduce the duplications so that the speed of computation is increased. The reduction of duplicated computations can be repeatedly done through an FFT algorithm. In MATLAB, this is done by a simple command fft. The efficiency of FFT is discussed.

This chapter introduces MATLAB, aimed at the basic knowledge and skills related to what may be needed in the following chapters for data analysis. This chapter, however, is far from a complete coverage of MATLAB; nor do we need everything provided by MATLAB. For those who are familiar with MATLAB already, this chapter may be either skipped or used as a quick review. The exercises at the end of the chapter may be useful for some data processing, e.g. the selection of a subset of dataset is often needed, and the MATLAB function find is particularly useful for that.

The objective of this chapter is to discuss some background information of tides and the idea, purpose, and method of harmonic analysis of tides. Harmonic analysis is a special application of the least squares method to tidal signals. A list of 37 major tidal frequencies is provided. The basic theory and an example for the analysis is presented. The time origin of expression of tidal time series and longer-term variation of tidal constituents are discussed. A concise equilibrium tidal theory is included at the end of the chapter for reference.

As mentioned earlier, time series data must include time stamps. It may seem trivial, but some attentions are needed to properly use time to avoid mistakes. The objective of this chapter is to review a few concepts of time so that when an analysis of time series data is performed, there is less chance to make mistakes with respect to data consistency, the result of analysis, and interpretation. We will discuss some basic astronomical concepts related to time; different definitions of day; and time measurements, GMT, and UTC. We will learn using MATLAB to construct a time sequence from civil time or time strings, i.e. the year, month, day, hour, minute, and second to a real number of time and vice versa. We will also briefly discuss the Positioning, Navigation, and Timing (PNT) data from the Global Positioning System (GPS).

The objective of this chapter is to review the Taylor series expansion and discuss its usage in error estimation. The unique value of Taylor series expansion is often neglected. The major assumption is that a function must be infinitely differentiable to use the Taylor series expansion. In real applications in oceanography, however, hardly there is a need to worry about a derivative higher than the 3rd order, although one may think of some exceptions. The point is, there is rarely a need in oceanography and other environmental sciences to actually consider calculating a very high order derivative, unless for theoretical investigations or under special situations. So the application of Taylor series expansion usually only involves the first two derivatives. In this chapter, some simple examples are included for a better understanding of the applications.

The objective of this chapter is to introduce rotary spectrum analysis for velocity vector time series. When the two components of a velocity vector have different frequencies, the tip of the displacement vector would draw a figure called a Lissajous Figure. A special case of the Lissajous Figure is when the two components oscillate at the same frequency. Vector time series at a given frequency can only have a few basic patterns or a combination of these patterns: the tip of the vector would draw a line segment back and forth repeatedly, or rotate either clockwise or counterclockwise. This makes it necessary to study the rotary spectra for rotations in both directions. A rectilinear motion is a degenerated version or special case of rotary motion.

The objective of this chapter is to discuss a very important issue of the effect of finite sampling with respect to either the finite length of the record or the finite sampling intervals. A few sampling theorems are discussed.

This chapter discusses some basic concepts quantifying the characteristics of functions with random fluctuations. Deterministic functions are discussed first in order to introduce functions with randomness and ways to quantify them. The concepts of phase space, ensemble mean, ergodic process, moments, covariance functions, and correlation functions are discussed briefly.

Harmonic analysis and Fourier analysis are fundamental tools for oceanographic time series data analysis. Both can be derived from the least squares method. They are different, however, in one major respect: Fourier analysis is based on a complete set of base functions, such that the convergence of relevant Fourier series is guaranteed for continuous functions. In contrast, harmonic analysis almost always has non-zero total error squared unless for pure deterministic functions with tidal frequencies only. This chapter provides additional discussion and some examples aimed at a better understanding of the concepts and techniques. The discussion will involve tidal harmonic analysis and Fourier analysis by contrasting them in concept and through some examples for harmonic analysis.

The objective of this chapter is to discuss digital filters. We start from a review of theory of Fourier Transform for continuous functions. The continuous Fourier Transform is then discretized. The discretized Fourier Transform and inverse Fourier Transform, however, are not approximate equations – they are exact. Using the shifting theorem, a filter can easily be expressed in the frequency domain. A Finite Impulse Response (FIR) filter is then defined. By adding another implicit convolution to the original convolution for FIR filter, the filtered data depends on not only the input (the original time series) but also the output (the filtered data). This is an iterative relation that forms the Infinite Impulse Response (IIR) filter. These filters are examples of so-called linear systems that have an input and output. The gain is defined by the filter, which is the ratio between the input and output in the frequency domain. Several FIR and IIR filter functions in MATLAB are discussed.

This chapter discusses a generic least squares method and a special situation when the base functions are orthogonal to each other, which makes the solution explicit; in addition, we learn that the essence of the least squares method can be viewed as a way to project the target function in a higher dimension onto a lower dimension formed by the base functions. The least squares method ensures that the error vector is “perpendicular” to the projected (or approximate) vector in the base function dimension (a lower dimension) and thus has the shortest “length” or minimized error. Although this chapter does not have much computation involved, it is very important for a good understanding of the meaning of many techniques and methods in the subsequent chapters.

We have learned that, in theory, a single frequency sinusoidal function in a time domain corresponds to an isolated line in the frequency domain. For multiple frequencies with linearly superimposed sinusoidal functions in time, the frequency domain representation is a series of lines (the so-called line spectrum). When finite sampling is performed, the isolated lines will be replaced by sinc functions centered on the locations of the lines in frequency. The lines are “smeared” to become continuous functions; each has a main lobe of finite width (), and side lobes or a series of decaying ringing toward both smaller and larger frequencies. This chapter discusses some techniques to reduce the ringing (side lobes) away from the main lobes. This is the windowing technique. This usually is done at a price: widening of the main lobe, which is sometimes acceptable in order to substantially reduce the side lobes.

The objective of this chapter is to prepare for the use of MATLAB to find solutions for a system of linear equations. The important concepts include the inverse matrix, solution of a system of linear equations, least squares method, and MATLAB’s “left-division,” which is essentially an implementation of the least squares method. MATLAB functions are very useful and efficient in the job. For real problems, the equations are often highly overdetermined, which occurs when there are many more equations (or measurements) than the number of unknowns, and thus it requires the use of the least squares method to solve the solution in a statistical sense. The contents in this chapter lay out a foundation for several later chapters because many theories and methods are related in one way or the other to the solution of a system of linear equations, e.g. Fourier analysis and harmonic analysis.