And yes, there are

*at least*four dimensions to talk about here. In much of the work that I've done, and in the hydrologic processes that hold my interest, there are at least four normative dimensions. Then there are also the variables actually being examined, each of which is a "degree of freedom" or a "dimension of variability." I'm really not trying to make this more complicated, so let us go with a concrete example instead of the more abstract conceptual model of thinking. Let us decide to place a precipitation gauge in the backyard in order to keep track of rainfall amounts (which I actually want do right now). In order to contribute the information we collect from this gauge to a larger body of knowledge, such as the CoCoRaHS Network that is now nationwide, we need to know the location of the gauge in space, and then we can provide rainfall amounts over specified time periods (one day, in their case). Space, as you know, is not just latitude and longitude on the surface of the Earth but also elevation, or altitude or depth depending on the measurement taken. However, we'll just work with the standard three-dimensional (

*x*-

*y*-

*z*or longitude-latitude-elevation) space for now. Add the factor of time, and that's four dimensions thus far - these are what I am calling the

*four normative dimensions*. Add any number of variables to be measured, in this case a precipitation amount that changes over time, and that's at least a five-dimensional problem.

Two dimensions, say a flat representation of the Earth, is called a plane. In three dimensions, it becomes a surface. In general, these are called "manifolds." Can you visualize a four-dimensional manifold? Yes, actually, you can--just watch an airplane fly overhead. It is moving in space, over time. You might even be able to trace its path from one side of the horizon to another, especially if it is leaving a visible contrail. Can you visualize a five-dimensional manifold? Hmmm...yes, you can still do that, in a way. That airplane contrail is dispersing over time because of the winds and turbulence in the atmosphere, so the contrail is probably a fine line just behind the airplane but, where the airplane was ten minutes ago, its contrail in the sky is now a wide, cloud-like brush-stroke. Along the path of the airplane, that character differs, with a wiggle here and there and maybe places that the contrail is no longer visible, depending on all kinds of atmospheric variables including wind and temperature and humidity...and so we can see all of this complexity, and our brains can take it in and store it for later so that we can say at another time to our kids "Oh, yeah, that's what happens when an airplane goes overhead sometimes, it's called a contrail, and sometimes you can see lots of them all at once. Cool, huh?" But breaking it down to the fundamentals of what is really happening, physically, to form and then alter that contrail is not something our brains do so well. So this is where we need to begin reducing the complexity of the phenomenon in order to convey information in a way that remains understandable. "Why is the sky blue?" Oh, that's an easy one, but did my daughter ever ask that? Ha! No...I got "How does a tornado happen?" Bring nine or so degrees of freedom (including the four normative dimensions) into the process of system construction, even just conceptually, and maybe you can do that.

Think of it this way: for a single precipitation gauge sitting in a field somewhere, maybe your backyard near the garden, you really just want to know two things, (1) "when did it rain?" and (2) "how much did it rain?" If the last rainstorm occurred a week ago, instead of just last night, then you might need to water those vegetables. If it just drizzled last night, instead of pouring down in buckets, then you might still need to add water to the garden. So you're really only interested in two variables, and you have essentially two dimensions of variability. You already know where the information is located in space, but you really just want time and amount. This is our basic time-series, a simple graph with bars or points that show the amount of rainfall over a known period of time, or a

*p*-

*t*series. Here is an hourly precipitation time series from some of my M.S. work a few years ago:

But this is the data for a single gauge, for which we know its location in space, and we have a few more in our area of interest. So you might have different information, such as the hourly or daily rainfall at several locations around an area of interest, but not right at your garden. Using the given information in a basic form, you have three variables: location (

*x*-

*y*) and rainfall (

*p*). With some analysis, we can visualize these three dimensions as a contour map, similar to a topographic map, but for rainfall:

If you have the time-series of rainfall at several locations, it becomes a four-dimensional problem:

*x*-

*y*-

*p*-

*t*. Working forward from the above 3-D example, it would become an animated series of contour maps over our area of interest. This is, essentially, several individual

*x*-

*y*-

*p*contour maps, each specific to a desired

*t*, in sequence over multiple

*t*. In the following surface weather analyses from Unisys Weather, we see a lot of information at specific times, with data locations repeated over the time spanned by the event I was examining in that work (for your orientation, my area of interest was in eastern Colorado):

Now, we've already introduced the spatial dimensions that make such presentation easier. From there, it's a matter of proper combination of available dimensions and variables that allow for accurate interpretation. Often that required building an analysis by layers, starting with the lowest dimensionality of data combination, 2-D time series and 3-D contour plots, and building our way up to more complex representations using base maps and ancillary information to aid in the analytical process. As it turns out, the base map then becomes our guide - quite often the hierarchy of information is turned on its head, and the more complex representation is the first we see before "drilling down" to the local and specific detail that the 2-D and 3-D variable combinations provide. We'll look at some information systems with that order-of-operations, using examples that are already out there on the web, in part 3 of this series...

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