Waves: amplitude, frequency, wavelength

Prerequisites

Learning objectives

Build physical intuition for what a wave is and what moves when a wave moves
Distinguish amplitude, frequency, period, and wavelength
Compute wavelength from velocity and frequency (the key relation $V = f\lambda$ )
Recognize why a short wavelet such as the Ricker, not a pure sine, models a processed seismic pulse

Every seismic image you will ever interpret is built out of waves. Before we talk about rocks, boundaries, or cubes, we need a clear mental picture of what a wave actually is.

What is a wave?

A wave is a disturbance that propagates through a medium while the medium itself stays roughly in place. When you clap your hands, you do not throw air at the listener, you briefly compress the air near your hands, that compression pushes the next bit of air, and the push travels outward at the speed of sound. The air molecules wobble a little and return to where they were. The disturbance moves across the room; the air does not.

A seismic wave is the same idea, in rock. A controlled source (a small explosion, a vibrator truck, or an air gun) briefly compresses the ground. That compression races outward through the earth at the rock's sound speed, from about 1500 m/s in soft, water-saturated sediment to 6000 m/s in hard carbonates and crystalline rock. The rock particles wobble a tiny amount. What we record at the surface is the later arrival of those compressions after they bounce off subsurface boundaries.

The kind of wave that dominates reflection seismology is the P-wave (or compressional wave). The particle motion is along the direction of travel, back and forth like an accordion. P-waves travel fastest through rock, arrive first, and carry the energy we use to image the subsurface. Later in Part 5 we will meet S-waves (shear waves), which move perpendicular to travel and tell us different things. For now, picture P-waves only.

Four numbers describe any simple wave

Amplitude, how big the disturbance is. Louder sound, stronger reflection, bigger number on your seismic trace. Units depend on what you are measuring (pressure, particle velocity, etc.).
Frequency ( $f$ ), how many oscillations per second, in hertz (Hz). A 30 Hz wave completes 30 cycles every second.
Period ( $T$ ), the time for one oscillation. It is just $T = 1/f$ . A 30 Hz wave has period $T = 1/30 \text{ s} \approx 33 \text{ ms}$ .
Wavelength ( $\lambda$ , lambda), the physical distance between two successive peaks, measured in metres.

Amplitude stands on its own, but the other three are tied together. If the wave travels at velocity $V$ , then in one period it moves a distance of exactly one wavelength. Therefore:

$V = f\lambda$

You will use this equation constantly. It lets you go from "a 30 Hz wave in rock that travels at 3000 m/s" to "the wavelength is 100 m" without any further thought. A slower rock at the same frequency gives you shorter wavelengths; a higher-frequency wave in the same rock also gives you shorter wavelengths. Short wavelengths are what let you see small features, more on that when we cover resolution in Section 1.7.

Figure 0.1 lets you check the relation by measurement. Set a frequency and a rock velocity, read the wavelength off the snapshot in (a) and the period off the receiver's trace in (b), and watch which of the two each control moves.

Sines versus wavelets

A pure sine wave oscillates forever. Real sources do not run forever: an air gun or a charge gives a short pulse, and a vibrator's long sweep is compressed into one by correlation. After processing, that pulse often looks like a single central peak flanked by two smaller troughs. The Ricker wavelet is the standard mathematical model for this shape. It is parameterized by one number, its peak frequency, the frequency at which its spectrum peaks, and it looks like a localized burst of a sine wave that quickly damps out. It is a model: it is symmetric about its central peak (zero phase), which no real source pulse is exactly.

Switch the waveform in Figure 0.1 to Ricker, at 30 Hz and 2500 m/s. The spectrum in (c) peaks at 30.0 Hz and stays above half its peak from 14 to 49 Hz: one pulse holds a whole band of frequencies. Its dominant wavelength, $V/f$ with $f$ at the peak of the spectrum, is 83.3 m, but the pulse itself lasts only 26.0 ms from trough to trough in (b), 65 m of rock, about 0.78 of a dominant period. When an interpreter quotes the wavelength of a wavelet, this dominant value is what is meant: a property of its spectrum, not a distance between repeating peaks. Every real seismic trace you ever look at is built from wavelets like this overlapping each other, not from sinusoids.

A note on frequency content. Processed seismic typically contains energy across a band of frequencies, maybe 8 to 80 Hz for a land dataset, or 5 to 150 Hz for a high-resolution marine survey. We speak of the dominant frequency when we need a single number, but any real wavelet has a whole spectrum. Section 1.7 and Part 6 will return to this when we discuss resolution and spectral decomposition.

References

Sheriff, R. E., & Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). Society of Exploration Geophysicists.
Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
Sheriff, R. E. (2002). Encyclopedic Dictionary of Applied Geophysics. Society of Exploration Geophysicists.