 So let's talk a little bit about the measurement of an irregular channel cross section. For example here, I have a picture of a dry channel, I haven't put any water in at the moment, but we're going to talk about how we might be able to determine some of the important geometric characteristics of the cross section as the water's flowing through it, knowing that those characteristics will change with the depth of the water. Well, if I can gather enough information about the cross section, I can make very good estimations for the cross sectional characteristics. So we're going to consider this a regular channel here, and I've put a couple stakes into the side of the channel. And the reason for putting those stakes in the side of the channel is because I'm going to start by creating a horizontal datum, a line that I'm going to measure everything from. And I use, tie it to both of the stakes, and I use some sort of level to make sure that it's perfectly horizontal. And I'm going to use it as my measurement point for all the points on the bottom of the channel. I'm also going to need to select an origin, some measurement point along the horizontal axis. Okay, so I'm going to choose this stake over here on the left hand side as being my origin point. Okay, now I've done that, I'm going to need to select an increment, a delta x, okay, an increment here of which I'm going to chop up the channel. And depending on how much work I want to do, more accuracy equals more work will determine how many pieces I want to chop it up into, usually somewhere between 10 and 20. Okay, I'm going to chop it up in this case, I think I have about nine in this case, one, two, three, four, five, six, seven, eight, nine. And I have another data point, which is the point at the end there. Okay, so we chop it up into some increment and it's very useful, it'll make it a lot easier for you to do the calculations if the increments are the same size. They do not have to be, but then you have additional information and it makes it a little bit harder to do the calculations of area and stuff afterwards. So we choose an increment. In this case, my increment is going to be 30 centimeters, not on the page here, but representing 30 centimeters in the reality that I measured this from. Okay, once I've selected that increment, let's go ahead and make a chart over here. Okay, and here's all my values xi measured in centimeters. And I have increments every 30 centimeters or so, 120, 150. So let's see here, zero, 30, 60, 90, 120, 150, 180, 210, 240 and 270. So there's all the measurements that I would like to make. Now, we measure this horizontal datum up here. We're going to have a basis that's actually upside down. We're going to go ahead and create a bunch of measurements down in a direction downward from our datum. Notice this is the only way we really could do it, because if we wanted to measure everything up, we would have to dig down into the ground to measure up to the surface. So realistically, we can only measure down from our datum here. So we're going to go ahead and measure a series of values here. And we'll go ahead and call them, well, their vertical position. So we'll call them y. Okay, we'll label them as y. Let's go ahead and write this as y in this particular direction. And we've already established x as being in this direction. Okay, so let's go ahead and measure a series of values down. Now, if you're out in the field, what you're going to do is you're going to drop something that you know is vertical, maybe using a plum bob or something, and get a measurement for that particular distance, all right, once it hits the surface. So we measure each of those vertical distances. I have a series of vertical distances that are roughly equivalent to the scale that I've drawn here to use as an example. And I'll fill in each of these numbers here, 26, 32. Okay, again, representing each of these distances down from the datum. Now, notice the datum was arbitrary. It doesn't have to do with anything about the stream itself. It was just some value that we picked that was roughly near the top was higher than any portion of the stream we cared about. But it isn't the depth of the water. There's no water necessarily in the stream, or it's probably above any water that is in the stream. So it was kind of arbitrary. Well, we don't want to be completely arbitrary here. So what we're going to do is we're going to change our datum now and do what most people do in this particular case, which is move the datum down and establish the lowest point in our stream bed as being the new datum. That's going to be this point down here. And I should say be clarified here, the lowest measured point in our stream bed. It's possible that there might be a little dip in between where we measured. And somewhere else, we're looking at the lowest measured point. We're going to assume that that's our new datum down here. And that is at a distance y of whatever the highest y max, whatever our maximum value is here in this case. So our y max, our maximum y value was at 98 centimeters. Okay. So now I'm interested instead of having all these measurements down from my datum, I actually want a series of measurements that are going to come up from our new low point, from our new datum. We're going to go ahead and label these as being h. So from our new datum, this is h. And we can establish those heights h. There are actually elevations of each of those points off of our new ground datum. Okay. And they're simply a relationship. We take the maximum y value and subtract the current y value to find any particular height. We should subtract it from the y. So I had my column of y is here. Let me go ahead and convert those all into heights above our lowest point, our datum. And again, this low point is now no longer arbitrary. It's lined up with some point, direct point in the along the bottom of the stream bed. Okay. So if I calculate each of these, I simply subtract 98 minus each of these 98 minus 20 is 78. And if I continue 72, 66, 43, notice one of the points, the lowest point is going to have a height value of 0, 14, 38, 60, and finally 81 for this set of data. Okay. So now we actually have heights, vertical heights here. All right. But any of the things that we want to measure about this particular stream bed, any of the information we want is going to be dependent upon water flowing through it. So there's going to be some, we're going to assume that there's some level of water flowing through it. Let me go ahead and draw a depth of water here. I believe my depth of water runs through these here. So we'll pick a depth that we're going to analyze. We're going to consider what happens when there is water flowing through. And to know that we're going to need a depth. Typically we will find that depth relative again to this low point. Okay. So now I have a depth and that's going to be our maximum depth because we're measuring it up from this lowest point. So we pick a maximum depth. I'm going to go ahead and say that this maximum depth value is going to be 58 centimeters. Okay. Again, in this case it's somewhat arbitrary, but if you're doing an analysis, you might have measured this out in the field. You have a gauge, which is basically a stick stuck into the water somewhere that gives you a level above that deepest point. Okay. And I'm going to go ahead and calculate these depths over here. Let me go ahead and make them in pink. All right. And in this case, what I'm going to do for each of these depths is that depth of each point is going to be equal to that maximum depth minus whatever height we're at. So now I'm flipping things over again to find a depth. Well, let's see here. In this particular case, notice we're going to get negative numbers where we're here. I really don't care about things that are above the surface. So the first points here, I don't really care about it. It's going to be a negative value. This is also going to be a negative value. Okay. Once I get to the point where I'm going 58 minus, let's see here, 58 minus 66 is negative 12. That's still negative. But now I start getting into values. I'm sorry, 58 minus 66 is going to be negative eight. And I start subtracting here. Notice my maximum depth 58 minus zero is at the low point as expected. And I continue to subtract each of these. I am going to go ahead and actually calculate the last two here. Those are the ones that are right at the edge that are located at the sides. Notice the other ones I don't care about, and I might not have drawn this exactly correctly. It looks like we're capturing 1, 2, 3, 4, 5 data points. 1, 2, 3, 4, 5 data points. Perhaps I'm a little high on that version there, but you're getting the kind of idea hopefully. Okay, so now I have a series of depths that represent basically what's happening here. Actually, let me go ahead and draw this just a little bit lower. And we can talk about the depths that were measured here. 1, 2, 3, there's our maximum depth is the fourth one. 1, 2, 3. Well, let's guess it's about the third one. Okay, so if we have a series of depths here, now that we actually have those series of depths, the question sort of becomes, what can we learn from this, or what can we do with these? All right, and we're interested in three of our geometric measurements. If you remember, our geometric measurements are the cross-sectional area, A, the wetted perimeter, Pw, and then the hydraulic radius, which is simply the ratio between those two things, area divided by the wetted perimeter. And notice all of these things are going to be a function of the depth. Okay, they're all going to change as I add more water. We're going to have greater area, we're going to have greater wetted perimeter, and we're going to have a variation in the hydraulic radius. So these are all functions of our depth, D. So how do we go about calculating these things? How do we find the area? How do we find the wetted perimeter? Well, for the area, there's a relatively simple process that we can do here. We can look at each of these chunks and recognize that any particular piece here, any particular slice of my between two of these measurements can be approximated using a trapezoid. We can take the area of a trapezoid, so one of my little pieces of area here, we're going to approximate with the area of a trapezoid, which is one-half base times height one plus height two. Let's see here, one-half of base times height one plus height two is the area for our trapezoid. So in this case, our base, we have one-half, our base is delta x, the distance between, and our heights are our two depths. In this case, we'll take the depths on either side here, di, I'll call this one i, and then di plus one, whatever the next one is, so these two depths here. And that will give us one of these areas. To find the total area, I simply have to add up all of these areas. We can notice that each of the depths is actually going to get included in the calculations for two of our trapezoid, so the one on the left and the one on the right. So if I add all these things up, I'm going to have two of these depths. And in fact, if I take and add up all these pieces, I create something called the trapezoidal rule, which looks a little like this. The area is equal to one-half times that base thickness, and this is why we want to have a consistent base thickness, times a series of depths. Notice the series is the sum of all the depths, but we count every, all the ones in the middle, twice because they're used in each of my trapezoids. The ones in the end, we only count once. So this is a formula we can use, and this is called the trapezoidal rule. It's numerical integration, the idea that you can chop up any shape into pieces like this, and you count all the middle heights twice as much as you count the end heights. In this case, there's some question about what end heights we should actually use. You can make your own choices about how you do this. I actually recommend, because it's basically taking an average, I recommend using these negative values as the end heights, which is going to subtract a little bit, it is going to subtract a little bit from your overall area, but it does make for a reasonably good estimation. If you want to think about different ways that you can actually cut it off, perhaps you can figure out where it intercepts. You can choose to do so, but you're not going to get a lot of additional precision out of the work necessary to do that. So that's how we calculate the area. We use something like the trapezoidal rule. To calculate the wetted perimeter, we recognize that each of these pieces can be considered, if we think about it being a straight line as we did for the trapezoid, we know that the width is our increment, delta x, and we know that this height here is a change in either the two heights h or the two depths. It doesn't really matter. I'll use the depth here, the difference in the two depths. Well, the delta x stays the same for every one, but the change in depth will change with each of our units. But all we do in that case, to find the wetted perimeter, is we're going to use the Pythagorean theorem, a squared plus b squared equals c squared, or in other words, this distance here, let's call this, this little length, we'll call it s. Okay, for a line segment, each segment si is going to be equal to, each segment is going to be equal to the square root of delta x squared plus delta d squared. Okay, well for each of those, we sum them all up, and we will get the wetted perimeter. Now I'm not going to do all those calculations here, but notice I would need to calculate the changes in distance for each of these pieces here. Okay, I have the delta x, that's the 30 centimeters, the distance between each of these here, in order to calculate those distances, and then find these, and then you add them all up, and that gives us the wetted perimeter. Once you have the area, and the wetted perimeter, again, our hydraulic radius is simply the ratio between those two things, that should be a w for wetted perimeter.