 So our last piece in this discussion about the thesis in polygons is to talk a little bit about why. Step five, we're going to talk about area weighting, how we use the thesis in polygons. So here, I have a completed region where I've divided using thesis in polygons. And in each of these regions, we have a value. In this case, this value was a rain depth or some other value. In this case, it's a rain depth. Maybe I could label that as d1, d2, d3, and so on and so forth, where we have some number of values that we can index. Perhaps we label those as di, where there are, i goes from 1 to 12. There are 12 different ones in here. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Actually, I should say there are 10 regions here. There were 12 triangles, but there are only 10 regions. d6, d7, d8, d9, and d10. Well now, not only do we have those depths, but each of those depths is associated with a particular area. Now, depending on the problem you're using, you might have different areas that you're concerned with. We might be concerned with the area over the entire map. In other words, we might be concerned with all of the area of the polygon. Or we might just be concerned with an overlapping area, a piece of area that's within the polygon, but is defined by some other portion. For example, we might be interested in that area. Either way, we have some area that is associated with each of these points. We'll label that area with the same index. a1, a2, a3. And again, you have to determine whether the area you're talking about is the overlapping area or whether it is the entire area within the polygon. That depends on the problem you're trying to solve. Here, let's go ahead and consider those areas to be the areas within the map that we're looking at, this map of watersheds that we're looking at. And if you have those values, ai, each of those values is going to be associated with one of the other values, the depth values. One of the common ways to use areas like this is in what we call area weighting. The Thesen polygon method, one of the things we can do is let's say we wanted to find an average depth. Well, one of the ways we can find an average depth here is we can assume that the depth that's associated with very little bit of the area contributes less than the depths that are associated with a large amount of the area. This particular piece here affects only a very little bit of the watershed. And we wouldn't want to count it equally with all the remaining depths. A very simple example would be to just average out all of the d's. But in this case, we could do what's called a weighted average, where we take each value, di, each depth value, we multiply it by the area that it's associated with. And then we sum, add up all those areas. However, this is going to scale our overall value by a substantial amount, because we're multiplying everything by an area. Well, if we then want to reduce that, what we then do is sum up the entire areas, sum up all the areas that are associated. In other words, the entire area here, if we count up all the ai's, it's going to be the area of the gray image. We sum those up and divide this weighted sum by the total average. And when we do so, that's going to give us something, our area units are going to cancel out, and what's going to give us a depth. And that depth is a weighted average or a weighted mean of depths over the area. And that's one of the typical uses of the Thesen-Polygon method, is to establish some areas associated with another value and then use that to determine a weighted mean based on the areas in the Thesen-Polygons.