P is for Phase

Seismic is about acoustic vibration. The archetypal oscillation, the sine wave, describes the displacement y of a point around a circle. You only need three pieces of information to describe it perfectly: the size of the circle, the speed at which it rotates around the circle, and where it starts from expressed as an angle. These quantities are better known as the amplitude, frequency, and phase respectively. These figures show how varying each of them affects the waveform:

So phase describes the starting point as an angle, but notice that this manifests itself as an apparent lateral shift in the waveform. For seismic data, this means a time shift. More on this later. 

What about seismic?

We know seismic signals are not so simple — they are not repetitive oscillations — so why do the words amplitude, frequency and phase show up so often? Aren't these words horribly inadequate?

Not exactly. Fourier's methods allow us to construct (and deconstruct) more complicated signals by adding up a series of sine waves, as long as we get the amplitude, frequency and phase values right for each one of them. The tricky part, and where much of where the confusion lies, is that even though you can place your finger on any point along a seismic trace and read off a value for amplitude, you can't do that for frequency or phase. The information for those are only unlocked through spectroscopy.

Phase shifts or time shifts?

The Ricker wavelet is popular because it can easily be written analytically, and it is comprised of a considerable number of sinusoids of varying amplitudes and frequencies. We might refer to a '20 Hz Ricker wavelet' but really it contains a range of frequencies. The blue curve shows the wavelet with phase = 0°, the purple curve shows the wavelet with a phase shift of π/3 = 60° (across all frequencies). Notice how the frequency content remains unchanged.

So for a seismic reflection event (below), phase takes on a new meaning. It expresses a time offset between the reflection and the maximum value on the waveform. When the amplitude maximum is centered at the reflecting point, it is equally shaped on either side — we call this zero phase. Notice how variations in the phase of the event alter the relative position of the peak and sidelobes. The maximum amplitude of the event at 90° is only about 80% of the amplitude at zero phase. This is why I like to plot traces along with their envelope (the grey lines). The envelope contains all possible phase rotations. Any event whose maximum value does not align with the maximum on the envelope is not zero phase.

Understanding the role of phase in time series analysis is crucial for both data processors aiming to create reliable data, and interpreters who operate under the assertion that subtle variations in waveform shape can be attributed to underlying geology. Waveform classification is a powerful attribute... but how reliable is it?

In a future post, I will cover the concept of instantaneous phase on maps and sections, and some other practical interpretation tips. If you have any of your own, share them in the comments.

Additional reading
Liner, C (2002). Phase, phase, phase. The Leading Edge 21, p 456–7. Abstract online.