I’ve been thinking a lot lately about issues of scale in ecology, both because I’m taking a fascinating seminar on the topic this quarter, and because my particular research is conducive to thinking about them. “Scale” came to the fore as a topic of interest starting in the late 70’s, and is tied up with other concepts that were on the rise at that time, like chaos theory, fractal geometry, and nonlinear dynamics. An example of one of these ties is Benoit Mandelbrot’s famous question, “how long is the coast of Britain?” The answer, as he shows, depends on how long your ruler is. This turns out to be a recurring conundrum: the pattern detected by an ecological study depends very strongly on the scale of the measurements.
In 1963, fifteen years before most other people started thinking about this stuff, a short paper was published in Science that, in just four pages, managed to lay out how and why we need to consider the scale of our measurements when designing an experiment to measure oceanographic phenomena. It was written by Henry Stommel, which is not at all surprising if you know who he was. For those who don’t, Stommel was one of the original badass physical oceanographers. The kind of guy who could sit down with a pen and paper and demonstrate why the upper ocean circulates the way it is observed to. Or correctly derive the circulation of the deep ocean before it was observed. No big deal. But thanks to his 1963 paper, titled “Varieties of Oceanographic Experience,” he is cited far and wide to this day as one of the first scientists to explicitly consider the importance of scale to experimental design.
To illustrate why scale is important, take one of Stommel’s examples: say we want to measure changes in the height of the sea surface from month to month. At first glance, it would appear that only 12 measurements are necessary: go out once a month for a year, and you’re set. At second glance, however, this is ridiculous on several levels. The ocean has tides, for one. Measuring sea level once a month would give you a near-random sample of different parts of the tidal cycle, and you wouldn’t be able to detect any long-term trends. There are also shorter- and longer-term fluctuations: regular wind-driven waves, and sea-level deviations due to large oceanic eddies. To resolve sea-level changes from month to month, you will actually need to measure it every hour or so. Do it less frequently, and you will get results that are inconclusive or just plain wrong.
Choosing the right scale of measurement for your question of interest is very important. It is also not trivial, especially when the measurement is not as straightforward as the water level on a ruler nailed to the dock. Even today some people are prone to testing hypotheses using data collected at a scale inappropriate to the question. If Stommel had stopped with this message, the paper might still have found a fair number of readers. But the coup de grace was the graphic he came up with to make the message explicit:
It’s a three-dimensional surface, with time along the x-axis and distance along the y. Both are shown on a logarithmic scale, so that each tick mark is a factor of 10 larger than the one before it. Various phenomena that make the sea level go up and down are located on this surface based on their typical size and duration. The height of the surface at each point represents how much the sea level goes up or down—that is, how much energy or variability is concentrated at that space and time scale.
Gravity waves (aka wind waves, the normal ones that crash on the beach and make you seasick on boats) are typically several meters long and perturb the sea surface for a few seconds. They can therefore be placed in the lower left corner. Tides happen every 12-13 hours and affect the entire ocean basin, so they are located near the middle of the time axis, stretching from about a kilometer up to 10,000 km. And every ten thousand years or so, we hit an ice age that lowers the surface of the ocean everywhere about 100 meters, allowing humans to do things like cross the Bering Strait into North America. This figure is an elegant summary of all the different processes that perturb the sea surface, and of the spatio-temporal scales at which they all take place.
This kind of diagram (now known as a Stommel diagram) has found its way over the years into all kinds of different contexts. One direct descendent near and dear to my own heart is the one drawn up by Haury et al. in 1978. It is along similar lines, but shows variability in zooplankton abundance, not sea level. The possibilities for these diagrams are nearly endless, and not limited to the ocean, or even the natural sciences. Any system that has stuff going on over short and long distances and time spans can be clarified by sketching up a Stommel diagram. Drawing a picture like this can help make it clear how to approach your research question, and will hopefully help you avoid screwing it up by choosing the wrong scale of measurements. The hope, as Stommel put it in the last line of the paper, is to “look forward to a time when theory and observation will at last advance together in a more intimately related way.” Amen.
Stommel, H. (1963). Varieties of Oceanographic Experience: The ocean can be investigated as a hydrodynamical phenomenon as well as explored geographically Science, 139 (3555), 572-576 DOI: 10.1126/science.139.3555.572