K is for Wavenumber

Wavenumber, sometimes called the propagation number, is in broad terms a measure of spatial scale. It can be thought of as a spatial analog to the temporal frequency, and is often called spatial frequency. It is often defined as the number of wavelengths per unit distance, or in terms of wavelength, λ:

The units are m^–1, which are nameless in the International System, though cm^–1 are called kaysers in the cgs system. The concept is analogous to frequency f, measured in s^–1 or Hertz, which is the reciprocal of period T; that is, f = 1/T. In a sense, period can be thought of as a temporal 'wavelength'—the length of an oscillation in time.

If you've explored the applications of frequency in geophysics, you'll have noticed that we sometimes don't use ordinary frequency f, in Hertz. Because geophysics deals with oscillating waveforms, ones that vary around a central value (think of a wiggle trace of seismic data), we often use the angular frequency. This way we can also express the close relationship between frequency and phase, which is an angle. So in many geophysical applications, we want the angular wavenumber. It is expressed in radians per metre:

The relationship between angular wavenumber and angular frequency is analogous to that between wavelength and ordinary frequency—they are related by the velocity V:

It's unfortunate that there are two definitions of wavenumber. Some people reserve the term spatial frequency for the ordinary wavenumber, or use ν (that's a Greek nu, not a vee — another potential source of confusion!), or even σ for it. But just as many call it the wavenumber and use k, so the only sure way through the jargon is to specify what you mean by the terms you use. As usual!

Just as for temporal frequency, the portal to wavenumber is the Fourier transform, computed along each spatial axis. Here are two images and their 2D spectra — a photo of some ripples, a binary image of some particles, and their fast Fourier transforms. Notice how the more organized image has a more organized spectrum (as well as some artifacts from post-processing on the image), while the noisy image's spectrum is nearly 'white':

Explore our other posts about scale. 

The particle image is from the sample images in FIJI. The FFTs were produced in FIJI.

Update on Thursday, May 3, 2012 at 1:39 PM by Matt Hall

Following up on Brian's suggesstion in the comments, I added a brief workflow to the SubSurfWiki page on wavenumber. Please feel free to add to it or correct it if I messed anything up. 

Well worth showing off

Have you ever had difficulty displaying a well log in a presentation? Now, instead of cycling through slides, you can fluidly move across a digital, zoomable canvas using Prezi. I think it could be a powerful visual tool and presentation aid for geoscientists. Prezi allows users to to construct intuitive, animated visualizations, using size to denote emphasis or scale, and proximity to convey relevance. You navigate through the content simply by moving the field of view and zooming in and out through scale space. In geoscience, scale isn't just a concept for presentation design, it is a fundamental property that can now be properly tied-in and shown in a dynamic way.

I built this example to illustrate how geoscience images, spread across several orders of magnitude, can be traversed seamlessly for a better presentation. In a matter of seconds, one can navigate a complete petrophysical analysis, a raw FMI log, a segment of core, and thin section microscopy embedded at its true location. Explore heterogeniety and interpret geology with scale in context. How could you use a tool like this in your work?

Clicking on the play button will steer the viewer step by step through a predefined set of animations, but you can break off and roam around freely at any time (click and drag with your mouse, try it!). Prezi could be very handy for workshops, working meetings, or any place where it is appropriate to be transparent and thorough in your visualizations.

You can also try roaming Prezi by clicking on the image of this cheatsheet. Let us know what you think!

Thanks to Burns Cheadle for Prezi enthusiasm, and to Neil Watson for sharing the petrophysical analysis he built from public data in Alberta.

Scales of sea-level change

Click to read about sea level on Wikipedia. Image prepared by Robert Rohde and licensed for public use under CC-BY-SA.Sea level changes. It changes all the time, and always has (right). It's well known, and obvious, that levels of glaciation, especially at the polar ice-caps, are important controls on the rate and magnitude of changes in global sea level. Less intuitively, lots of other effects can play a part: changes in mid-ocean ridge spreading rates, the changing shape of the geoid, and local tectonics.

A recent paper in Science by Petersen et al (2010) showed evidence for mantle plumes driving the cyclicity of sedimentary sequences. This would be a fairly local effect, on the order of tens to hundreds of kilometres. This is important because some geologists believe in the global correlatability of these sequences. A fanciful belief in my view—but that's another story.

The paper reminded me of an attempt I once made to catalog the controls on sea level, from long-term global effects like greenhouse–icehouse periods, to short-term local effects like fault movement. I made the table below. I think most of the data, perhaps all of it, were from Emery and Aubrey (1991). It's hard to admit, because I don't feel that old, but this is a rather dated publication now; I think it's solid enough for the sort of high-level overview I am interested in. 

After last week's doodling, the table inspired me to try another scale-space cartoon. I put amplitude on the y-axis, rate on the x-axis. Effects with global reach are in bold, those that are dominantly local are not. The rather lurid colours represent different domains: magmatic, climatic, isostatic, and (in green) 'other'. The categories and the data correspond to the table.
It is interesting how many processes are competing for that top right-hand corner: rapid, high-amplitude sea level change. Clearly, those are the processes we care about most as sequence stratigraphers, but also as a society struggling with the consequences of our energy addiction.

References
Emery, K & D Aubrey (1991). Sea-levels, land levels and tide gauges. Springer-Verlag, New York, 237p.
Petersen, K, S Nielsen, O Clausen, R Stephenson & T Gerya (2010). Small-scale mantle convection produces stratigraphic sequences in sedimentary basins. Science 329 (5993) p 827–830, August 2010. DOI: 10.1126/science.1190115

The scales of geoscience

Helicopter at Mount St Helens in 2007. Image: USGS.Geoscientists' brains are necessarily helicoptery. They can quickly climb and descend, hover or fly. This ability to zoom in and out, changing scale and range, develops with experience. Thinking and talking about scales, especially those outside your usual realm of thought, are good ways to develop your aptitude and intuition. Intuition especially is bound to the realms of your experience: millimetres to kilometres, seconds to decades. 

Being helicoptery is important because processes can manifest themselves in different ways at different scales. Currents, for example, can result in sorting and rounding of grains, but you can often only see this with a hand-lens (unless the grains are automobiles). The same environment might produce ripples at the centimetre scale, dunes at the decametre scale, channels at the kilometre scale, and an entire fluvial basin at another couple of orders of magnitude beyond that. In moments of true clarity, a geologist might think across 10 or 15 orders of magnitude in one thought, perhaps even more.

A couple of years ago, the brilliant web comic artist xkcd drew a couple of beautiful infographics depicting scale. Entitled height and depth (left), they showed the entire universe in a logarithmic scale space. More recently, a couple of amazing visualizations have offered different visions of the same theme: the wonderful Scale of the Universe, which looks at spatial scale, and the utterly magic ChronoZoom, which does a similar thing with geologic time. Wonderful.

These creations inspired me to try to map geological disciplines onto scale space. You can see how I did below. I do like the idea but I am not very keen on my execution. I think I will add a time dimension and have another go, but I thought I'd share it at this stage. I might even try drawing the next one freehand, but I ain't no Randall Munroe.

I'd be very happy to receive any feedback about improving this, or please post your own attempts!

Update on Tuesday, April 12, 2011 at 11:40 PM by Matt Hall

I have just finished a new version of this graphic, incorporating some comments from readers. I replaced the one in the original post, so please grab it from there if you want to see what I did. 

If you like this post, you might also like the more recent one on the scales of relative sea-level change, and also this older one on the scales of various experiments in oil & gas geoscience. 

Unstable at any scale

Rights reserved, Adrian Park, University of New Brunswick

Studying outcrops can be so valuable for deducing geologic processes in the subsurface. Sometimes there is a disconnect between outcrop work and geophysical work, but a talk I saw a few weeks ago communicated nicely to both.

At the 37th Annual Colloquium of the Atlantic Geological Society, held at the Fredericton Inn, Fredericton, New Brunswick, Canada, February 11-12, 2011, Adrian Park gave a talk entitled: 

Adrian Park, Paul Wilson, and David Keighley: Unstable at any scale: slumps, debris flows, and landslides during deposition of the Albert Formation, Tournaisian, southern New Brunswick.

He has granted me permission to summarize his presentation here, which was one of my favorites talks of the conference.

Click to read more ...