What math do we need?

Welcome. Before we touch a single seismic trace, we need to build the mathematical toolkit that every processing step quietly assumes you already have.

Seismic processing is, at its heart, applied signal processing on wavefields. Every workflow, from reformatting raw shot gathers to full-waveform inversion, is ultimately convolution, Fourier analysis, sampling, statistics, linear algebra, optimization and wave physics dressed in domain vocabulary. If any of those feel shaky, the rest will feel like memorization. Our goal in Part 0 is that none of them feel shaky.

The map below

Below is an interactive map of the whole book: ten processing steps in book order, from acquisition in Part 1 to time-lapse processing in Part 8, against the nine math topics of Part 0 (Sections 0.2 to 0.10).

Slide through Part 0 as if you were reading it and find out how few steps you could carry out halfway, and which topic to read next to unlock the most. Pick a step and a topic to see the equation where that step uses that math, the sections that teach and use it, and in (d) the topic at work in a small example computed in your browser. The quiz names a symptom from a later chapter and asks which Part 0 section you would reopen. The point of the map is that every bit of math in Part 0 earns its keep; nothing here is academic filler.

The map shows why the math comes first. Every step leans on at least five of the nine topics, and every topic serves at least four steps. Read in book order, the first five topics (through Section 0.6) complete only one step, filtering; the wave equation, the last section, is the last core topic for four steps: acquisition, velocity analysis and NMO, migration and full-waveform inversion all wait for it. Teaching each topic inside the first step that needs it would leave the later steps standing on half a foundation.

How Part 0 is paced

Part 0 assumes no prior signal-processing, linear-algebra, or wave-physics background. Each 0.X section starts from first principles and builds up with its own widget. If you already know the material well, you can skip the section and come back only for the quiz at Section 11.1. If you do not, you will not be left behind: every concept is illustrated with something you can play with.

One practical tip: do not read Part 0 linearly and then close the book. The math lives in your muscle memory only after you have used it. Return to these widgets whenever you are reading a later chapter and something feels opaque. The math and the processing are the same material seen from two angles.

What is next

Section 0.2 opens with complex numbers and phasors, the language underneath every frequency-domain step on the map (the map marks them only where phase is handled directly), and the most consistently under-taught. If you can explain to a colleague what $e^{i\\omega t}$ means by the end of Section 0.2, you are ready for the rest of Part 0.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
• Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge.
• Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.