 So today, we're going to talk a little bit about rainfall characteristics, and in particular, a set of measurements that we use in measuring rainfall called intensity, duration, and frequency. So we can think about rainfall as being an accumulation of a whole bunch of little teeny, tiny raindrop events. These raindrop events happen, and they happen both in space and they happen in time. They land at different times, and they land in different places. But we can think about a rainfall of event as it being an accumulation of a whole bunch of these teeny, tiny events occurring. And if we want to quantify the amount of rainfall, we have a couple of ways of doing it. One of the ways we can do it is we can assign the rainfall to a point in space. We can have a particular point in space, and typically, what we do is we might create a gauge set of something there. We can think about it as being some sort of catch there, and we allow that catch to fill up. And we measure the amount of rainfall by a depth associated with that one point in space. Or we might associate a larger amount of rainfall with an entire area, some sort of larger space on the ground, in which case we calculate the entire volume of rain as it fills up on that entire area. So volume is associated with area, whereas depth is associated with a single point. And volume, when referred to precipitation, can usually be related to this depth, that we find the volume by multiplying the area, whatever area we're talking about, times the depth. And again, the area is going to be in some sort of units of area. Perhaps they might be in meters cubed, meters squared. And the depth might be in meters, perhaps, in which case square times meters might give us meters cubed, which is a measure of volume. Notice when we're dealing with precipitation, we might get very different units. For example, acre inches, where the depth is measured in inches, but the area is measured in acres. Well, an acre inch is still a two-dimensional thing, an acre, multiplied by a one-dimensional thing, an inch. Even if it is a strange unit, it is still a unit of volume. And in the same way that volume equals AD, the area times the depth, we can sort of reverse and turn that around and find the depth by dividing the volume by the area. In other words, normalizing the volume. And we can basically take whatever this area is here and divide the entire volume measure to figure out sort of an average depth. This process is called normalizing. And it makes it easier for us to compare rainfall at different locations, because there's not a dependency upon the area that's associated with that location. So the depth of the rainfall that is measured at the depth of the rainfall over the Ha River watershed, for example, might be comparable to the depth of the water over the Yadkin watershed, even though the two watersheds have a different area. If you compare the volume, you'd have to know one might be much bigger than the other. But if you simply compare the depth, it doesn't matter what area we're talking about. And sometimes this volume, sometimes you'll see the term volume used or expressed as a depth, volume of two inches when referred to precipitation. You might sometimes hear the term, well, there was a volume of two inches of precipitation. Notice that isn't officially a volume. That's a depth. But sometimes we assume that there's a times A that's in place that you're going to multiply it by whatever area you want when you really want to figure out the actual complete volume of water. So occasionally those terms are used interchangeably. Now, not only does rainfall need to be associated with the location or an area in space, it also must be associated with a time period, because it takes a while for rain to fall. So a hi-etograph is a graphical representation of rainfall as it occurs over time, the distribution of rainfall over time. But note, each part of a hi-etograph must have some time interval associated with it. Let's take a look at a hi-etograph that I've prepared here. Here's an example of a hi-etograph. Notice along the x-axis, we have time measured in hours. And then we have a depth on the vertical axis measured in inches. And if you see here, each of these depths has to be associated with some time interval. We don't accumulate the 0.24 inches of rainfall unless we wait around enough time for that to accumulate. So each of these is measured and represented in this hi-etograph with these columns. And again, this is a hi-etograph. However, if we change the interval, either we use different measurements or we're measuring at a different time, it's very much going to change what the hi-etograph looks like. For example, let's say instead of measuring every hour, we instead measure every two hours. So our interval now is going to be two hours across. We'll notice the same amount of rain is going to fall in two hours. If I consider here, I look and I see that there's 0.24 inches in the first hour and 0.08 inches in the second hour. Well, let me go ahead and plot that. That's going to be a total of 0.32 inches. So in the first two hours, I've accumulated 0.32 inches of rain. And if I similarly add the next pair, that's 0.56 inches of rain, 0.56. And if I consider the next interval, I get 0.22 inches of rain. And finally, my last interval, I get 0.10 inches of rain. Notice this has a very different shape. And it's a little hard to compare these two because we're beginning to sort of stack. Basically, the hi-etograph when we plot depth on that vertical axis gets taller. In fact, I can continue with that process. Let's consider what happens if instead of using a two-hour interval, I break this down into two four-hour intervals. Well, in that case, I get 0.88, somewhere I have to scroll way up to the top here, 0.88 inches in that first four hours, 0.88. And I get something significantly smaller. Looks like 0.32 inches in that second block. And again, my hi-etograph looks very different. And it also gets very difficult to sort of compare the different time periods. We're losing some information. And I don't notice our graph is stacking up here. And finally, we could add both of those values together. I believe this is 0.88. I've written up here in the 0.32 down here. And if we sum those together, we find out that we get 1.2 inches of rainfall. But that 1.2 inches is accumulated over my entire eight-hour time period. And again, that time period is kind of important for knowing we're talking about the depth of that time period. It's important. If we got 1.2 inches over the course of a week, that's a significantly different situation than if we got the 1.2 inches over the course of eight hours, or even more so if we got the 1.2 inches over the course of an hour or 15 minutes. So thinking about this depth, we might actually want to, again, normalize by considering the depth divided by how much time it took. We're going to compare the depth by dividing by the amount of time, eight hours. So for example, 1.2 inches falling in eight hours is going to give us an average of 0.15 inch per hour. Notice that 0.15 inch per hour on our original depth chart, if that was the true for our first hour, it would have been somewhere here in the middle, there's about 0.15 inches happening in an hour. So that would actually be a little more comparable. We can see that first we were above that level, then we were a little below that level, and our storm kind of varied above and below that level over the course of the storm. This value, this 0.15 inch per hour, is called an intensity. And let's consider how the intensity would look if instead of plotting the depth on this side, we instead implot the intensity. So let me erase all these comparisons that we made here when we were summing up the various values and go back and look at my original hyedograph. But now, let's sort of think about instead of expressing depths along the hyedograph, let's express intensities along the hyedograph, intensity. And in this case, our intensity is going to be an inches per hour. Well, notice because my original sort of graph was already broken up into hour increments, this conversion is pretty simple right now. All we have to do is simply say the intensity in inches per hour. And because our interval was only one hour, then each of those values, 0.33 inches falling in that one hour is 0.33 inches per hour. But notice now that we're thinking about intensity, if I instead consider a two hour interval, if I'm considering a two hour interval here, what I can do is add both of those values together, the 0.24 and the 0.08, which I did before gives me 0.32. But that's 0.32 inches in two hours or 0.16. Turns out I've already drawn that right there. So during that time frame, we actually have an intensity of 0.16 inches per hour. And notice that's an average of the intensity in the first hour and the intensity in the second hour. Let's consider the next two hour interval. If I average those two together, add them together, I get 0.56. 0.56 inches average in that two hours is going to be 0.28. And again, is the average of those two values over the two hours. So similarly, I can do that same process here. I get 0.11. And down here, I get 0.05. Notice that shape of the graph, the shape of the graph is the same as the shape of the totals that we had before. But it's a little bit easier and makes a little more sense to compare the intensity of this storm Let's compare the intensity when we've measured things every two hours to the intensity when we measure everything every one hour. Notice the heights are approximately the same. Just the two hours sort of an averaging of the one hour. Let me repeat that process again. Let me go ahead and consider four hour intervals. In that case, we have an average between these two values, 0.16 and 0.28 is 0.44, or an average of 0.22. So if we instead measure every four hours, we'd have an intensity here of 0.22. And then in this other part, we'd have an intensity of 0.08, where we're averaging the 0.11 and the 0.05. And finally, if we look at both those two and average them together, 0.22 plus 0.08 gives us 0.3 inches per hour divided by 2 divided by 2 for the average. We're actually, in this case, we have 0.22 plus 0.08, which are four hour intervals. We're taking that over two of those intervals, and we end up with a value of 0.15. So there's our sort of a little bit below there, 0.15. So our intensity for this entire storm was 0.15 inch per hour. Notice if we multiply that intensity by the length of the storm, it's an eight hour storm, we end up with a value of 1.2 inches, and the hours cancel out. So 1.2 inches is the total rainfall for the storm, which is what we saw before when we were calculating the total depth. So it's, again, typical to normalize depths by dividing by the length of the interval and creating this value that we call intensity. Now, intensity is essentially a flow rate. If we think about our intensity, our intensity, we'll abbreviate this with I, is a flow rate. It's a change in volume over change in time, although, again, we're kind of cheating with our volume because we're assuming that it's a volume divided by time, which is our duration of the storm. I'll talk about that in a second. But often, we'll, again, refer to volume in inches and assume that if you need more information about the actual volume, you'll multiply by whatever area you need here. So often, intensity is given in terms of inches per hour, even though it is essentially, it's often done in inches per hour, but you can immediately turn that into a flow by multiplying it by some area to turn that into a volume per hour. Well, if you have a given intensity, like our 0.15, you can recover the volume, the depth, and then the volume by establishing a duration. And again, using this here, we can say that our volume is equal to the intensity times the duration of the storm or whatever period you're looking at. Again, like we did before, we said that our intensity of the overall storm was 0.15 inch per hour. And we multiply that by the duration of the storm, which gives us of eight hours, which gives us 1.2 total inches. That's the volume. And then we would, in turn, multiply that by whatever area of a watershed or particular roof or whatever sort of surface area we were interested in to find the actual volume of water that fell in our region. So two ideas, this idea of intensity, and then this idea for this measurement of duration. So now we have two quantities that we're looking at, our intensity and the second one, second characteristic quantity that we use for measuring rainfall, which is our duration. Our third quantity that we're interested in looking at recognizes a particular rule that's true about a lot of natural phenomena. And sort of a standard rule of natural phenomena is that large events, and by large events, I mean extremely big hurricanes or larger earthquakes or flood events like rainfall, large events happen less often than smaller events. And I'm sure you've witnessed that in most situations. You don't see major downpours quite as often as you see light sprinkles. You might see three or four small rainfall events occur during a month where you might only see a large, can't go outside, it's pouring too hard, event happen once a month or once every two months. So this is pretty typical, and it's true about things like hurricanes, earthquakes, a number of things where you can think about it as being, if a lot of things need to accumulate to create a large event, sometimes those things don't accumulate quite enough and you have a smaller event. So a measure of how often an event might occur is called the frequency, how often an event occurs. Usually we measure this using some form of statistics. We gather data over a course of periods of time, and then we do some analysis to determine these frequencies. And it can be pretty straightforward. For example, I might say that there are eight storms. We might observe that there are eight rainfall events, which we often call storms, that show us greater than 4.5 inches of rainfall in a given three-hour period. So we recognize that there's some pretty big storm, 4 and 1 half inches of rainfall over one three-hour period. Notice most storms tend to drop off and get less, so we would expect in that larger period, if we got 4.5 inches times 2, if we got 9 inches and 6 hours, that's actually a much bigger event. Usually storms kind of have some sort of peak to them, where they have less rainfall than a lot of rainfall in the middle and then less rainfall at the end. And so we might be considering just the highest rainfall here, the highest three-hour period when we were getting the most rainfall. And if we took the total event and said there's 4.5 inches in that three-hour period, OK? Well, we look at all the storms that are greater than that amount, and we count them. And we measure all the storms for a period of time. Let's say, for example, that we did this for 10 years. So A, we find that eight storms occur in a 10-year period. So that's less than one a year, on average, which is an important concept. So we're going to call this our frequency. So we have, if we relate those two, that's eight storms. The ratio between the two, eight storms in 10 years. Well, that gives you a value. If we do the math, it's 0.8 storms per year. Well, what does 0.8 storms per year mean? You can't have a fraction of a storm. It's not like you walk out and say, oh, that was 8 tenths of a storm that we had this year. Let's wait for our next 8 tenths of a storm next year. I mean, I suppose you could make the storm smaller, but that's not really what we're trying to do. We're trying to say, what are the chances of having one of these bigger storms? So 8 tenths of having a big storm can be interpreted in a couple ways. One way is something called the exceedance probability. So what we do is we take this number eight, and we express it as a percentage, 0.8, 80% chance. And we say that there is an 80% chance of having an event occur in a given time. And usually we'll talk about that as being a year. So in this case, we have an 80% chance of 0.8 storms per year as 80% storms per year. But this is 80% chance of having one storm in one year. So in other words, if I say, I'm wondering what will happen in the year 2017, well, if we've observed eight storms in the past 10 years, then I can make an estimate that there's an 80% chance that I will see a similar sized storm sometime in 2017, assuming that the same sort of natural patterns occur. So that's one where we express it. 80% of chance of seeing a storm in a given year. We can also flip that over. I'm going to take that ratio that we did, eight storms in 10 years, and flip it over. Let's go ahead and make the exceedance probability equal to a letter P for probability. And we're going to create something, T equals 1 over P, which is going to give us a value of 10 years over eight storms, or in other words, 1.25 years per storm. This is what we call a return period. It's a period of time. And what it represents is the average time between the events we're looking at, the average time between these large storm events of 4.5 inches in a three-hour period. Now, you have to be a little careful here. This 1.5 years per storm does not mean that we're going to go, here's a storm, and then we go 1.25 years in another storm, and then we go to 2.5 years, and there's another storm. It does not mean that they are going to be perfectly equal like that. What it does mean is that they are going to average out to have that distance. For example, if you start your clock at the beginning of 10 years, you might have a storm here, then you might have a storm right after that, maybe two in the same year, it's possible. And then maybe there's a long period of time before you have the next storm. And if they're sort of randomly distributed, which is what we assume, what we will see as we go further and further into time is as we get up to 10 years, we will expect to see eight storm events. One, two, three, four, five, six, seven, eight. We'd expect to see that. Now we may not, if there's still probability involved, we might see seven in one year, we might see eight, I mean seven in one 10 year period, we might see nine or 10 in the next 10 year period. But we use this as an estimate to sort of determine how often we're going to see storms of a certain size. So you will sometimes hear the terminology of something like a 100 year storm. What is a 100 year storm? Okay, what that says, that's a storm that happens on average once in every 100 years. Or the other way we can think about it, it's a storm that has a 1%, that's the one over 100, 1% chance of having in a given year. In other words, it's not very frequent. You're likely to see only one in your lifetime, assuming again that there are not major changes, natural changes that are happening in your area. Okay, again in note, just because a 100 year event just happened, for example, Hurricane Matthew here in North Carolina just happened and they classified this as being a 100 year storm. But just because you just had a 100 year storm does not mean that you could not have a 100 year storm in a much shorter time period, next year, or within two years. It just means that the average observation that we've seen is this storm only comes around on average every 100 years. Think about this in terms of a die. Might be easier to think about it. If you had a six-sided die and you have your, let's see here, I guess the four might be on this side of the die. Okay, if you have a six-sided die and you roll the die and let's say that you wanna know the chance of having rolling a number six, for example. Well, the number six is one side of the die, which is the six, out of six total sides of the die. So your chances of rolling a number six, using probability, are one out of six, or 0.167, or 16.7%. So that's your chance of a six in one roll. So if you take one roll, you have a 16.7% chance of rolling a six. If we flip that over, we basically have six rolls to get a six on average, that you will score a six on average every six rolls. Well, if I take my die here, I have a die. Let me roll it a couple times. I roll a five, I roll a two, I roll a one, I roll a one, I keep rolling, I roll a four, I roll a two. Notice I rolled six times and I have not yet rolled a six. Okay, but then I roll a six. Now, the question is, since I rolled a six and it only happens every six rolls, does that mean I'm not gonna roll another six again for six rolls? Well, no, not necessarily. You notice we rolled a whole bunch of times before we have sixes, but over length of time, if I keep rolling again, here's a two, here's a three, and there's my other six. Well, that only occurred three rolls away, but total, one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, right now, two out of 13, which is almost one out of six. Two out of 13 in my rolls have been sixes. So hopefully that gives you a little bit of a picture of the idea behind what we're talking about when we talk about a 100 year storm. A 100 year storm can happen once every 100 years or we can think about it as being a 1% chance of happening any one year. So that's our frequency, the term we use here. And why do we use frequency? Well, there's a couple of reasons for it. Generally, we use it for design purposes. If we're trying to figure out how much rain might occur in a certain area, so we can calculate how much volume to expect, we can use the longer history, and then we make a selection. We say, okay, I want to be able to deal with this flood, for example, once every 100 years. I can't stop something that is too big. I mean, I could plan to stop any sort of rainfall that would ever, ever happen, but if it's not going to happen for 1,000 years, it seems unlikely that I should have to worry about it, or if I do worry about it, well, it would probably easier not to spend the money to design for something like that, but instead to clean up after the fact. So when you're trying to design a system, you try to pick a length of time that you'll design for. And how do you go about relating these things? Well, again, we use history and we have an entire government entity, NOAA, the National Oceanic and Atmospheric Association, that gathers all of our precipitation data and creates for us something called an IDF curve. They gather the data, and you can use an IDF curve to relate intensity, duration, and frequency. Notice we've already established that the volume of a storm is equal to the intensity times the duration, and we've already established this idea that the bigger something is, in other words, more volume of rain, the less frequent it is. In other words, the period, I mean the percentage chance of it occurring is lower, or the time period in between when it happens is higher. In other words, it becomes less frequent as it becomes bigger. Let's go take a look at the NOAA website. Here's an example of the, there's a page on the Hydro Meteorological Design Studies Center. It's a website with free accessibility operated by NOAA. And if I look here, I can choose my data type, precipitation intensity, and I can pick a place on the map. In this case, I've picked a place close to the airport in Durham, North Carolina. And down below here, there is some data. This is the data for our curves here. You'll notice there is duration on the side here, five minute, 10 minute, 15 minute, 30 minute, and so on and so forth. And across the horizontal axis, we have an average recurrence interval in years. So for example, I could look here and I could say, if I was looking for a hundred year storm, a storm that comes out every a hundred years, I might expect to see 8.86 inches per hour of rainfall, an intensity in inches per hour of 8.86 inches per hour to occur in a five minute interval. Now notice as the interval gets longer, the intensity gets less and less because it's very difficult for it to continue to rain at a rate of 8.86 inches, but you'll probably have noticed in any large storm that you experience that there are as one or two periods where it's very intense, usually in the middle of the storm and that a large portion of the rainfall falls in a very short period of time. That's true about a lot of major storms and that's what that's demonstrating here. Let's see the graphical version here. If we click, we can create some graphs and notice this is effectively a three dimensional graph. What we have is we have intensity on one of our axes and notice this intensity goes from a thousandth to a tenth, to a, I'm sorry, a thousandth to a hundredth to a tenth to a singular unit to tens of units and these are spaced out equally, so this is what we call a logarithmic curve that it's counting by multiples instead of counting by direct intervals. Along the bottom here, we have a similar thing, five minutes, 10 minutes, 15 minutes, 30 minutes for a duration and then we have lines of different colors that represent the recurrence interval. The green line would be a storm that would happen every year. The yellow line every two years and all the way up to the light blue line which is every 500 years and the gray line which is every a thousand years and notice they're pretty close together. There's not that much variation in these things. Okay, but notice they sort of slope down that your duration, the longer thing is not likely to keep up the intensity. The longer the storm goes, the lower the intensity over time. Okay, let's see if we can consider a problem here. Let's say for example that we know that 2.7 inches, okay, we go and we look up some rainfall for a recent storm at the airport and we find out that 2.7 inches of rain fell but obviously that's not meaningful unless we know the time period over which it fell in 24 hours at Raleigh International Airport, R-D-U. Well, that's a depth and we're talking about all these measured in intensity so I'm gonna go ahead and calculate the intensity. Intensity of 2.7 inches divided by my 24 hours gives me 0.113 inches in an hour. And we've already established our duration. Our duration is 24 hours. That's the duration of the storm that we're interested in. Well, now I have those two values, an intensity and a duration. Let me go look at this curve. It's gonna be a little bit easier for me to read the curve down here at the bottom. I'm going to find a duration. The duration is 24 hours, which is this sky blue curve here in the middle. And then I'm going to move to an intensity of 0.11. Well, let's see here, here's 0.1 and 0.11's probably a little bit above that. Maybe that's a little higher, maybe 0.12. It's kind of hard, again, to read on a logarithmic scale, but that's a sum 0.11 is right about in that area there. Okay, and if we read down the vertical axis, down to the horizontal axis here, we see that that's a recurrence interval of about two years. Okay, similar, I could try to read it from the other one. I go up on the 24 hour axis until I hit around 1.1, which looks like it's the yellow curve is the closest, which again, the yellow curve is the two year recurrence interval curve. So I'm establishing that my frequency, okay, my return period is about two years. Again, I'm not real accurate with that, but it's about two years. This means I would see a storm of this size is a pretty normal size storm. I'd see it once every two years or so. Okay, following those curves. Note, with this same data, if you prefer not to do it, you can not to use the intensity, you can plot this data using precipitation depth curves, in which case you can have depths that sort of go along with each of these things, but notice you have to know which depth, well, the depth is gonna be obviously associated with a particular interval. So your depth has to be associated with whatever interval you did. So in this case, our depth was 2.7 inches. I can move up to 2.7 inches, which is right about there, read across to the 24 hour and see if we can do a better job there. Now notice there that depth of 2.7 inches in 24 hours, looks like it gives us something a little closer to maybe the green line. Maybe that's a one year recurrence interval. It might be a little bit easier to read it if you use the depth. The other thing you can do is you can always compare it to the table. If I know, for example, that I have a duration of 24 hours, I can move across the table until I get to something that's close to 2.7 inches. Hey, here's the 2.9, which is actually a little bit higher, and so we're able to make a little bit better estimate by looking not at the graphic, but instead looking at the tabular numbers and we could see that that's definitely something that's closer to a one year interval, probably even less than one year, maybe something like 11 months, 10 or 11 months, so something we see multiple times per year. So, intensity, duration, and frequency, often IDF if you're talking about the curves. Those three are characteristics that we often use for measuring precipitation.