Grain, Extent, and 8-bit Cities

A friend sent me a link to this cool website a couple of days ago: a guy named Brett Camper has coded it up to show zoomable maps of several major cities, pixellated à la old video game world maps. It’s visually neat, and it also illustrates how observed patterns change with changing scale of observation. What exactly does “scale of observation” mean? The “scale of observation” is not just one value—it comes in several parts. These parts go by several different names; one of the common frustrations with trying to use scale as an organizing concept is that the terminology is often vague and varied. For the purposes of this blog post, I’ll use the terms “grain” and “extent.”

“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.”

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