Let’s take a concrete example. Say you’re interested in how seabirds forage for small fish, and so you put small GPS recorders on some birds which record their position every second. Examining the tracks, one of them looks like this:

Since I don’t have any real data, I had to make some up. For simplicity’s sake, I used a random walk: at each step, the bird moves a random distance from its current position.^[1] The path starts on the right side near the middle, and then wanders around in a counterclockwise loop, ending up in the top right. This isn’t a terrible approximation of a bird swooping around its foraging grounds, searching for fish. If you use R, you can reproduce the above using these commands.

set.seed(10)
x <- cumsum(rnorm(500, 0, 5))
y <- cumsum(rnorm(500, 0, 5))
plot(x, y, ty='o' , cex=0.2 , xlab='x (meters)' , ylab='y (meters)' )

An important question in foraging ecology is whether the animal is taking in more energy from eating than it is spending from running/swimming/flying around looking for food. To get an estimate of how many calories the bird has burned, you calculate how far it has flown. Figure out how much the x and y change between each time step, and use the pythagorean theorem to get the distance traveled.

distance <- sum(sqrt(diff(x)^2 + diff(y)^2))

This gives a total distance flown of 3114.4 meters.

But suppose that because of budget cuts to the National Science Foundation, your research grant was actually 30% smaller. As a result, you could only afford a cheaper version of the GPS tag, which records the bird's position every 2 seconds instead of every one second. No big deal, right? Calculate the distance, carry on:

subsample <- seq(0, 500, by=2)
x.2 <- x[subsample]
y.2 <- y[subsample]
distance.2 <- sum(sqrt(diff(x.2)^2 + diff(y.2)^2))

Whoa! That gives a total distance of only 2256.0 meters. What happened? The bird flew the exact same path, but when you sample every two seconds instead of one, it appears to have flown 858 meters less. How does that work? Looking at the new path overlaid on the old one helps:

plot(x, y, col='grey' , ty='o' , cex=0.2, xlab='x (meters)' , ylab='y (meters)' )
lines(x.2, y.2, ty='o' , cex=0.2)

Because it skips every other observation, the new path (in black) avoids some of the random squigglyness in the old one (grey). The total measured length is 28% shorter. This is important, because if you were calculating the energy use by the bird, you would get a very different answer with a 1-second and 2-second resolution. Your conclusions would be incomplete at best, and wrong at worst.

But if you're a curious person, you might start wondering what the calculated distance would be at other resolutions. Instead of waiting around passively for the next round of corporate tax cuts before enacting your experiment, you decide to be proactive, and rush to your computer. After dashing off an angry email to your representative, you run a simple simulation, resampling the bird's track at different resolutions, and then plotting the total distance measured as a function of sampling scale.

scales <- 1:200
distances <- rep(0, length(scales))
for (i in 1:length(scales)) {
  subsample <- seq(1, 500, by=scales[i])
  x.sub <- x[subsample]
  y.sub <- y[subsample]
  distances[i] <- sum(sqrt(diff(x.sub)^2 + diff(y.sub)^2))
}
plot(scales, distances)

This graph shows very clearly what we mean when we talk about "scale-dependence." The distance measured depends, in a very real sense, on the scale of measurement. Fortunately, this relationship appears quite regular and predictable. It is, in fact, an example of a power law. Power laws are one of those mathematical relationships that show up all over the place in nature. Ratio of heart rate to body size? Velocity spectrum in a turbulent flow? Distribution of income among the richest Americans? Gravity? All power laws. In our case, the relationship looks like this:

reg.distance <- lm(log(distances) ~ log(scales))
summary(reg.distance)

In this fit, because it is a log-transform of the real relationship, the slope (-0.528) is actually the value of the exponent in the power law, and the y-intercept (8.260) is the natural log of the coefficient a. We transform it back to get the true power law relation.

This relationship can be generalized a bit further as follows:

In the real world, scaling relationships like this won't always appear, and when they do, they will usually only apply over a particular range of scales. Still, if you can deduce a scaling relationship, it's a powerful tool for reasoning about your problem, and may clarify what used to look like inconsistencies. Moral of the (somewhat artificial) story? Sample at multiple scales, or at high enough resolution that you can resample your data at a lower resolution, as I did here. And keep your mind open to the idea of different patterns and processes at different scales—what you see at first is not always the whole story!

[1] In physics, this kind of random walk is known as Brownian motion, and has all kinds of interesting properties.

I just got done reading a new paper [pdf] by Ted Ames, McArthur fellow and founding director of the Penobscot East Resource Center in Stonington, Maine, which speaks directly to this question—namely, how to match the scale of fishing effort to the scale of fish stocks and populations. PERC is an organization I have a personal connection to, having interned there the summer after my Junior year in college. The paper, titled “Multispecies Coastal Shelf Recovery Plan: A Collaborative, Ecosystem-Based Approach,” proposes a new management paradigm for groundfish[1] in the eastern Gulf of Maine aimed at better matching fishery management to both fish and fishermen.

Groundfishing in the Gulf of Maine—in fact, in the entire Northwest Atlantic—has had a troubled past half-century. Stocks collapsed in the late 80s and early 90s, due to a combination of foreign and domestic overfishing, overcapitalization, ineffective management, and, more likely than not, unfavorable ocean conditions. That’s plenty of causes for a stock collapse, but a few more have become apparent since then, too.

For one, we have learned that cod and other groundfish have much more localized stock structure than was previously thought. This is a problem because NMFS sets quotas for large areas, encompassing a number groundfish sub-populations. Even if the overall quota is sustainable, it is still possible for large, mobile boats to fish down local stocks one after another. Stocks in some areas of the Gulf of Maine have seen measured recoveries, but others haven’t. Non-local fishing pressure on local stocks is probably at least partly responsible.

A second wrinkle is the effect particular types of quotas have on the fishing fleet. After several false starts, NMFS settled on a days-at-sea system, with time on the water for fishing allotted based on historical catches. Though it wasn’t the intention, this policy had the effect of shutting out smaller boats. A skipper with too few days-at-sea will find it unprofitable to fish at all, since the majority of his allotted time will be spent steaming to and from the fishing grounds. As a result of the poor recovery of the fish and the days-at-sea restriction, there is very little groundfishing in New England north of Portland, ME. East of Rockland, there is virtually none.

Ames’s proposed management solution borrows a page from another Maine fishery that had a troubled first half of the century: lobster. Maine lobster, Homarus americanus, suffered a collapse in the 1930s, driven by a canning industry that took all sizes of lobster from pre-reproduction juveniles to giant old spawners. In the wake of that collapse, local groups of lobstermen began to self-enforce limits on the number of traps and the size of “keepers.” This informal management regime was gradually folded into state regulations. In 1997, Maine explicitly divided up the coast into seven lobster management zones, yielding considerable management authority in each zone to local lobstermens’ councils. This progression of local management regimes—none of which, by the way, require much data to run—have seen consistently high landings. Just as importantly, they have let fishermen and managers see eye-to-eye when they otherwise might not.

Under the Multi-Species Coastal Shelf Recovery Plan (MSCRP), stocks within 25km of the coast would be limited to small, local boats using hooks or traps, to protect inshore stocks and habitat. An intermediate layer would be open to larger craft with somewhat looser gear restrictions, and beyond 75km, the current federal rules would apply unmodified.

I would be very interested to see a quantitative management-strategy evaluation of this scheme. From my reading and superficial knowledge of the area, it seems as though it could have a good chance at success, at least in allowing near-shore stocks and small-scale inshore fishing a chance to recover. Off the top of my head, I don’t know of any other area where a similar management strategy with nested areas of increasingly local management and effort restrictions has been implemented (if anyone knows of any, please let me know). And in the meantime, I will keep an eye on Maine’s Downeast coast, where a first step towards this idea was implemented this summer—allowing groups of fishermen from several areas to pool the new catch shares, preserving their access to the fishery. (These “sectors” are different from what Ames proposes, as they are attached to groups of fishermen, not to geographical areas.) Optimism is not high, exactly, but it is undeniably present. And when it comes to a fishery that has been more or less dead for as long as yours truly has been alive, any optimism at all is a good thing.

“Groundfish” is a catchall term for fish that live on or near the bottom. In the North Atlantic, these include cod, haddock, hake, halibut, and flounders, among others.

Ames, T. (2010). Multispecies Coastal Shelf Recovery Plan: A Collaborative, Ecosystem-Based Approach Marine and Coastal Fisheries: Dynamics, Management, and Ecosystem Science, 2, 217-231 DOI: 10.1577/C09-052.1

“Grain” refers to the size of the most fundamental sampling unit: a pixel in a satellite image, a net tow on an oceanographic cruise, or a single quadrat in an ecological field study. Values for those examples might be 1 square km of the earth’s surface, 1000 cubic meters of seawater filtered, and a single square meter of a prairie soil, respectively. Any variation within the grain will be averaged out: you can’t tell where the squashed creatures in your net’s cod end came from with any more specificity than that they came from somewhere in the net’s 1000-odd meter track. (There is also the issue of how far apart the sampled units are. In the satellite image, they are contigous, while the net tows might be tens of kilometers apart, and the quadrats separated by tens of meters.)

“Extent” is the total area or distance over which measurements are collected. A satellite image might span a couple thousand kilometers from one side to the other. A long cruise could likewise cover thousands of kilometers, while the prairie grass study would probably only span a few kilometers, at most. Thinking about the grain and extent of your sampling, you can start to imagine the types of patterns and phenomena you would be capable of detecting. To illustrate, look at this clip from 8-Bit Cities, centered above Seattle’s University District—specifically, my office in the UW’s Fishery Science Building.

At this zoom level, you can’t actually see the Fishery Science Building, but if you click the “+” button on the map four times, it will show up. Being zoomed in, the real-world, geographical size of each tile is now smaller that it was zoomed out, and the geographical distance from one side of the map to the other is smaller as well. Both the grain and extent have decreased, and as a result, finer-scale patterns are visible: surface streets and buildings, for example. If you let your eyes go blurry and just look at the patterns as you zoom in and out, you will notice that they look quite different.

You may also notice that as you zoom in, the number of tiles doesn’t actually change. There are always about 26 of them across the map. Though the grain and extent have both changed, their ratio has not. This ratio is known as the “scope,” and it is also an interesting one to ponder. Scope is the extent divided by the grain, and characterizes how densely information is packed inside the boundaries of your measurements. As an example, look at the image below, from Google Maps, showing the same area as the 8-bit map, but with a much smaller grain, and, therefore, a much larger scope (clicking the “+” button four times here will also zoom you in to the same area as above):

View Larger Map

Looks pretty different, huh? We are fairly used to looking at maps, so the loss of relevant information with the 8-bit map is obvious. But for some other space, or measurement, or phenomenon—say, the variance of fish density in space and time in some region—and it is not immediately obvious what the appropriate sampling scales should be to measure the things we’re interested in.

This is just scratching the surface of all the ways you can look at grain, extent, and scope. You can consider them in space, as I did here, or in time, as I tend to do in my thesis work, or in both space and time, or in terms of spectral resolution, or taxonomy…the list goes on. As Camper puts it on 8-Bit Cities:

Maps offer us visual architectures of the world, encouraging us to think about and interact with space in particularly constrained ways. Take some time to think about your surroundings a little differently. Set out on a quest. Be an adventurer.

And as Chuang Chou put it circa 300 B.C., in a truly awesome quote I found at the top of this paper [pdf]:

“This being so,” asked the Earl of the River, “may I take heaven and earth as the standard for what is large, and the tip of a downy hair as the standard for what is small?”

“No,” said the Overlord of the North Sea. “Things are limitless in their capacities, incessant in their occurrences, inconstant in their portions, uncertain in their beginning and ending. For this reason, great knowledge observes things at a relative distance; hence it does not belittle what is small or make much of what is big, knowing that their capacities are limitless.”

Adding machines did exist, and some could even be made to multiply, after a fashion. They were strictly mechanical, however, and the wear and tear on various joints, machine and human, coupled with the high frequency of random inputs, discouraged us from the use of any but the most necessary statistical tools. Perhaps this is why most analytical chemists of this vintage believe that if one does the analyses right, there’s no need for statistics. This is a self-reinforcing fallacy; if you don’t do the statistics, you never discover your limitations or the limitations of your methods and the universe you are sampling.

He also talks about the analysis of water samples for salinity, in the days before CTDs could measure it in-situ with electrical conductivity:

A group of female technicians, the salinity girls, ran the Mohr-Knudsen silver nitrate titrations to a chromate endpoint for eight hours a day. Needless to say, there was a considerable turnover in this group. There was always a shortage of salinity girls, and the sample bottles kept coming in from the ships and stacking up in the hallways.

Those were different times…

The most interesting aspect of the paper is his discussion of the shift in emphasis starting to occur from ship-based sampling to automated sampling with collections of stationary instruments. My thesis research is using a stationary echosounder, so I’ve done a fair amount of thinking about the differences between the traditional ship-based approach and the more recent ocean observatory approach. I appreciated hearing about this shift from the perspective of someone who has seen the entire evolution of modern oceanography firsthand. He doesn’t discuss it in the same vocabulary I might, but he comes to essentially similar conclusions.

Wangersky’s point is that our perception of the ocean and its dynamics is hugely dependent on the tools we have available to study it. It will be a very interesting time in the coming years, as more types of sensors and instruments are deployed long-term at more locations around the world. Just by virtue of observing the ocean at a new spatio-temporal scale, we’re likely to find stuff going on that we missed before. In the old days, oceanographers generally assumed that the ocean was in a steady state. When hydrographic samples were few and far between, and you had to bring them back to shore for the “salinity girls” to analyze before you knew what they meant, this assumption was kind of necessary. These assumptions were also kind of wrong, which is the point that Stommel made in his famous 1963 paper (remember Stommel?).

As instruments, methods, and computing power have increased, these types of oversimplifications have retreated, to the point where we now embrace variability and dynamic changes, and try to understand them in and of themselves, rather than as unwanted noise on top of some supposed equilibrium or steady state that isn’t really there. Every advance in instrumentation and every expansion of the scope of our observations has yielded a new perspective on the oceans, adding on to an understanding that is slowly and gradually becoming more complete.

Peter J. Wangersky (2005). Methods of sampling and analysis and our concepts of ocean dynamics Scientia Marina, 69 (S1), 75-84 : 10.3989/scimar.2005.69s175

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:

Figure 1 from Stommel (1963).

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

The Pacific Ocean, from top to bottom over the Marianas Trench. Click to see full water column.

And the second shows the Atlantic Ocean basin from west to east:

The Atlantic Ocean, from North America to Europe, drawn to scale. Click for full-size image

It’s kind of humbling to look at one and then the other. The first awes you with the immense depth of the ocean at its deepest point, and the second with the relative thinness of the ocean on the surface of the planet. That thin wet film on our ball of rock has so much going on in it: Currents, ecosystems, the lives and deaths as large as whales and as small as photosynthetic archaea. That’s deep, man…but also shallow. You dig?