1 Flood Days and Good Canoeing Days at Congaree National Park Pick a random day. What is the probability that the water level of Cedar Creek will be within a particular range then? Use USGS stream data to find out. Mark C. Rains 1, David C. Shelley 2, and Len Vacher 1 1 Department of Geology, University of South Florida, Tampa, FL Old-Growth Bottomland Forest Research and Education Center, Congaree National Park, Hopkins, SC University of South Florida Libraries. All rights reserved. SSACgnp.GB661.MCR1.1 Core Quantitative Issue Probability Supporting Quantitative Issues Working with real data Ratios; Percents Variability Core Geoscience Issue Floods and flow frequency 2 Getting started After completing this module you should be able to: Distinguish between classic probability and frequency probability. Distinguish between stage and discharge as measures of flow magnitude. Define flood. Know that flows that inundate natural floodplains are not unusual. Know how to determine the exceedance probability of a given magnitude flow (expressed as a stage height) from a hydrograph. Know how to calculate flow frequency from exceedance probability. You should also know where Congaree National Park is South Carolina Congaree National Park 3 The Setting Congaree National Park Congaree National Park (www.nps.gov/cong) is a bottomland hardwood forest located on a floodplain. It is one of the last-remaining, largest, and best examples of old-growth bottomland forest remaining in North America. The forest has tremendous biodiversity and is home to many state and national champion-sized trees. The Park is a federally designated Wilderness Area, an International Biosphere Reserve, and a globally important bird habitat. The Congaree National Park ecosystem is part of a complex, post-late Pleistocene riverine landscape. Floodplain landforms include terraces and scarps, creek channels, backwater sloughs, oxbow lakes, groundwater rimswamps, Ice Age sand dunes, and more.www.nps.gov/cong 4 Flow Variability: Hydrographs Flow in a river varies over the course of time. We plot these flow variations on hydrographs, such as the one on the right, which you will explore in this module. Hydrographs show time on the x-axis and either discharge (the volume of water passing per unit time) or stage (the height of the water at a point in time) on the y-axis. Useful quantitative descriptors for flows (we will focus on the first three): Magnitude How large is the flow? Frequency How often does this flow occur? Duration How many days a year does this flow occur? Timing At what time of year does this flow occur? Rate of Change How quickly does this flow change to another flow? A flood occurs when the river overtops the channel banks and inundates the adjacent floodplain. 5 The Problem Cedar Creek, a tributary to the Congaree River, runs through the bottomland hardwood forest in the Park. The hydrograph on the preceding slide is from a stream gage along the creek. Park staff have determined that the Park begins to flood when the stage on Cedar Creek exceeds eight feet. At that point, floodwater begins to spread across the floodplain, over the trails, and into some of the most frequently visited parts of the Park. Question 1: What is the probability that, on a randomly selected day, part of the trail system will be flooded? Below flood stage is also relevant to visitors. When the stage is less than two feet, typical Park visitors cannot navigate Cedar Creek by canoe or kayak because the creek is too shallow. When the stage exceeds six feet, typical Park visitors cannot navigate Cedar Creek by canoe or kayak because the creek is moving too swiftly. Question 2: What is the probability that, on a randomly selected day, typical Park visitors will not be able to navigate Cedar Creek by canoe or kayak because either the creek is too shallow or it is too swift? 6 Timeout for Floods, 1: Hazards! Obviously, one of the core geologic subjects of this module is floods. In introductory geology courses, the subject of floods commonly is covered in the section on geologic or environmental hazards. Floods are hazards because human development tends to concentrate around rivers, where land is often level and nutrient-rich, and water is readily available for power-generation, agricultural, municipal, and industrial purposes. But the floods present a potential hazard to human safety and infrastructure. People risk drowning, hypothermia, and disease if they are caught in a flood or become stranded as flood waters inundate trails and roads. Flowing watereven relatively slow flowing waterhas a tremendous capacity to do work, which can include the destruction of property and infrastructure. Knowing about floodsand their probability of occurringis critical to public policy debates about energy and water supply, public health, land use zoning, insurance, and much more. 7 Timeout for Floods, 2: Natural, Frequent, and Vital! Although floods are hazards, they are also natural and frequenta basic element of a functioning river system. Why do you suppose the flat area along a river is called a floodplain? The floodplains have been built by floods (Endnote 1).Endnote 1 The floodplain forest ecosystem at Congaree National Park depends upon periodic flooding from Cedar Creek and the Congaree River. During floods, nutrient-rich sediments are deposited on the floodplain, where they promote primary productivity (i.e., plant growth), through which atmospheric CO 2 is transformed into organic carbon compounds by photosynthesis. These organic carbon compounds form the base of the food web. Fallen leaves and coarse woody debris form deep organic deposits, while trees and coarse woody debris redirect flows, increasing erosion in some locations and deposition in other locations. The resulting floodplain landscape has a complex topography, which create the complex habitats that support diverse assemblages of fish, amphibians, reptiles, birds, and mammals. The big mathematics subject of this module is probabilityas in what is the probability that Cedar Creek has overtopped its banks and flooded the trails in the Park on the day that you visit. Probability, 1: The Classic Approach The classic approach to probability arose from analysis of games of chance in the 18 th century (Endnote 2). To illustrate the basic concept, consider craps, the familiar game where you roll two six-sided dice. Each die is marked 1 to 6, and you sum the numbers on the two face-up sides for the result. For example, if the two face-up sides are 4 and 3, then the outcome of that single roll is 7. Whats the probability of rolling a 7 with two dice?Endnote 2 Before considering two dice, lets consider one die. Suppose you roll that one die once. What are the possible outcomes? How many ways are there to achieve each outcome? What are the chances of achieving each outcome? What numbers should replace the question marks in this table? Outcome Number of ways of achieving outcome Probability of outcome 111/6 21 3?? 4?? 51 6?? Total number of outcomes = 6 Total number of ways of achieving them = 6 Sum of all probabilities = 1 exactly Be sure you understand what an outcome is. In this case, the outcome is the number you get when you roll the die one time. For rolling two dice, the outcome is the sum of the two numbers. For example, rolling a 1 and 6 is the same outcome (7) as rolling a 2 and 5. In fact, there are six ways of rolling a 7 (1,6; 2,5; 3,4; 4,3; 5,2; 6,1). What are the other possible outcomes, and how many ways are there of achieving each of them? What values should replace the question marks in the second column of this table? With 6 chances out of 36 of getting a 7, the probability of getting a 7 is 6/36 (or 16.7%). With 1 chance out of 36 of rolling a 2 (snake eyes), the probability of getting a 2 is 1/36 (2.8%). What are the probabilities for the other outcomes? What values should replace the question marks in the third column? Check that the probabilities add up to one (100%). Why should they? Why should you? Classic definition of probability: the ratio of the number of ways of getting an outcome to the total number of ways of getting all the outcomesexpressed either as a number between 0 and 1, or as a percent between 0 and 100 (Endnote 3).Endnote 3 9 Probability, 1: The Classic Approach, continued Outcome Number of ways of achieving outcome Probability of outcome 211/36 322/36 4?? 5?? 655/36 766/36 8?? 9?? 10?? 11?? 12?? Number of possible outcomes = 11 Total number of ways of getting some outcome = 36 Sum of all probabilities = 1 exactly The frequency approach to probability is a more recent development, originating in the late 19 th to early 20 th centuries. It allows a broader application of the concept of probability. Think of a single roll of the dice as a trial, meaning an experiment. Suppose you conducted 1,000,000,000 (a billion) trials. From what you determined on the previous slide, it is likely that you would roll a 7 on 166,666,667 of those trials and something else on the other 833,333,333 trials. But suppose that you hadnt done the arithmetic of the previous slide. Suppose, instead, that you had simply conducted the billion trials. If you had to say what the probability of rolling a 7 is, you would reasonably say its 166,666,667 divided by a billion (i.e., or 16.7%), or the number of trials in which the outcome 7 occurred divided by the total number of trials. According to this frequency interpretation of probability, all we have to do to determine the probability that flooding conditions occur on Cedar Creek on a randomly selected day is to measure the stage on many days, count the number of days when the stage is 8 ft or higher (i.e., the number of flood days), and divide that number by the total number of days (Endnote 4).Endnote 4 We are assuming that the total number of days is sufficient. Flip a coin 10 or more times (i.e., 10 trials), and record the number of heads and the number of trials. How many trials does it take until the ratio of number of heads to the number of trials equals 0.50? 10 Probability, 2: The Frequency Approach Frequency definition of probability: the ratio of the number of times the particular outcome occurs to the number of times it could occuragain, expressed either as a number between 0 and 1, or as a percent between 0 and 100. Lets get the data, 1 US Geological Survey stream gage records the stage at Cedar Creek. The data are uploaded in real time to the Internet by satellite link. You can check the stage on Cedar Creek at any time (http://waterdata.usgs.gov/nwis/uv? ). You can also retrieve records back to 1982.http://waterdata.usgs.gov/nwis/uv? We will work with daily records for Water Years Water years run from October 1 to September 30 and are named for their ending year. Thus October 1, 1998 through September 30, 1999 is Water Year Retrieve the data on the next slide. 11 If you are connected to the Internet, check the gage now. Is Cedar Creek flooding? On the left is a photograph of the stream gage on Cedar Creek at Congaree National Park. The corrugated standpipe is a stilling well, which is connected to the stream by a subsurface pipe. Therefore, water levels in the standpipe are controlled by water levels in the stream. The wooden housing protects the stilling well head and the stage recording instrumentation. On the right, is a schematic of a typical stream gage. US Geological Survey Lets get the data, 2 Here is what three segments look like: 20 days at the beginning, 20 days at the end, and 20 days in the middle when there is some action. The hydrograph on Slide 4 is a plot of the entire dataset. Here are your data. Click on the icon and save the file. It lists the daily stage in ft at the Cedar Creek gage station from October 1, 1998 through September 30, 2008. Its a long spreadsheet: two columns and 3,654 rows. 12 Explore the data Now here is what we want you to do: determine the highest, lowest, and (!) average stages over the ten-year period of record. You could determine the maximum and minimum by scanning the spreadsheet and picking out the maximum and minimum visually. You could be guided by looking at the hydrograph on Slide 4. Meanwhile, you could compute the average by adding all the Hs and dividing by But we hope you wont do it that way. This is the kind of task that spreadsheets were designed to do! Add the labels in Column D to your spreadsheet. For Cell E3, type =COUNT(B2:B3654), and you should get the value 3653 as shown in this screen shot. A formula must start with =. This one includes a function (COUNT), which must have an argument (B2:B3654). The B2:B3654 refers to the block of cells starting at B2 and ending at B3654. The formula returns the number of cells in the block. For Cell E4, try =MAX(B2:B3654). What do you suppose the formulas are in Cells E5 and E6? Try them. (Use Help, if you get stuck.) 13 Answer Question We want the probability that Cedar Creek will be flooded (i.e., that the stage will be 8 ft or higher on a random day). That means you must count the number of days that the stage was 8 ft or higher over the ten-year period of record. Once again, you should use a function. Specifically, use the COUNTIF function, which is similar to the COUNT function in that it refers to a block of cells specified in an argument. Excel Help says that the format for this function is =COUNTIF(range, criteria). The range, once again, is B2:B3654. The criterion is >=8, meaning greater or equal to 8. Thus, the formula in Cell E5 is =COUNTIF(B2:B3654, >=8). For Cell E6, use the frequency interpretation of probability (divide Cell E5 by Cell E3). For Cell E7, convert the value in E6 to a percent (i.e., multiply by 100). What is the flow duration for H>=8 ft (i.e., the number of days in a year when stage is at 8 ft or higher)? You dont need a spreadsheet for this one. Recreate and save this spreadsheet. You will need it to complete the homework. Answer Question 2 15 Now determine the probability of a good canoe or kayak day (i.e., when the stage will be between 2 and 6 ft on a random day). One way to do this is to determine the probability that the stage is at or above 6 ft (Cells E5 to E7) and the probability that the stage is at or below 2 ft (Cells E9 to E11), and then to subtract the sum of those two probabilities from 1, or 100% (Cells E14 and E15). Why will this work? Again, recreate and save this spreadsheet. You will need it to complete the homework. Wrap Up 16 Stage refers to the water level in the stream. The exceedance probability for a given stage is the probability that that stage will be equaled or exceeded. Probability is expressed as a number between 0 and 1 or, equivalently, as a percent between 0 and 100%. The flow duration for the stage is the number of days per year that stage is equaled or exceeded. Using daily stage from a US Geological Survey gage for the Water Years , and assuming that the 10-year record is long enough, or at least representative enough, to determine frequency probabilities, we have found the exceedance probability and flow duration for three stages of Cedar Creek at Congaree National Park. They are: stage heightexceedance probability (%) flow duration (days) 8 ft ft ft This table suggests it might be instructive to plot stage height against exceedance probability or flow duration. Such a graph is called a flow-duration curve. It is the subject of another module on the Cedar Creek gage at Congaree National Park. Note: the probability of an 8-ft or higher stage at Cedar Creek is a little lower than the probability of throwing a 3 in craps (5.6%). The probability of a 6-ft or higher stage is a little higher than throwing a 4 in craps (8.3%). The probability of a 2-ft or higher stage is about the same as throwing a 3, 4, 5, 6, 7, 8, 9, 10, or 11 (94.4%). 17 End-of-Module Assignment 1.Complete the tables in Slides 8 and 9, and answer the last two questions in Slide 9. 2.Do you have any thoughts about the question in the last green box in Slide 10? Write a sentence using the words limit and frequency probability. 3.Write a couple of sentences describing the water level at Cedar Creek during the past week? (See Slide 11) 4.Your data set consists of daily measurements for 10 years. As shown on Slide 13, there are 3653 daily records in your data set. If there are 365 days per year, there should be 3650 days in 10 years. Why the discrepancy? 5.Use the spreadsheet you developed in Slide 14 to determine the exceedance probability for a stage of 7 ft. 6.To go from probability to flow duration you should multiply by the number of days in the year. You should use for this number. Why? Now answer the question in the green box at the bottom of Slide Use the spreadsheet you developed in Slide 15 to determine the probability for a stage between 3 and 5 ft. 18 End-of-Module Assignment -- Advanced 1.The observation overlook at Weston Lake, which is an oxbow lake, is underwater when the stage reaches 14 ft. Use the spreadsheet you developed in Slide 14 to determine the exceedance probability for a stage this high. 2.The probability of two successive, independent trials is the probability of the first outcome times the probability of the second. For example, the probability of rolling two 7s in a row is (1/6)(1/6) or 2.8%. From your result in Question 1, find a succession of three rolls with about the same probability of the 14-ft stage event. Justify your answer (identify each of three outcomes, state each probability, and state the product of the three). 3.Make a table showing stage height, exceedance probability, and flow duration vs. stage heights for stages 2-14 ft at one-ft increments. 4.Do the data suggest that your calculated exceedance probability of an 8-ft stage is stable? You used data for a 10-year period starting with Water Year What is the exceedance probability of an 8-ft stage using the two-year period starting with Water Year 1999? For the three-year period starting with Water Year 1999? And so on to the 10-year period? Put your results in a table. Do the successive values seem to be approaching a limit? 5.Endnote 1 says that the frequency of bankfull discharge for US streams appears to be equivalent to two times in three years. According to the data in this module, it appears that Cedar Creek floods 15 days a year. What do you make of those two statements together? Endnotes In a well-known study in the mid-20 th century of stream-flow data from all around the country, US Geological Survey hydrogeologists found that bankfull discharge (i.e., any more and the river overtops its banks and floods the floodplain) occurs, on average, with a frequency of about two times in three years (though there is a great deal of variability, largely depending upon climate and geology.) For more information: Dunne, T., and L.B. Leopold, Water in Environmental Planning. W.H. Freeman Co. San Francisco, CA. Return to Slide 7.Return to Slide The origin of probability theory is generally associated with the names of Blaise Pascal ( ) and Pierre de Fermat ( ) and their discussion of the problem of how to divide the stakes fairly if a game of chance is interrupted part way through. Return to Slide 8Return to Slide 8 3. The classic definition of probability is one of the many benchmark contributions of Pierre- Simon Laplace ( ). The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible. (as quoted in Wikipedia from Thorie analytique des probabilits (1812). Return to Slide 9. Return to Slide We need to be careful with what we say here. Notice we are not counting the number of floods, but rather the number of days that flooding conditions occur (flood days). The frequency of floods (floods per year) is not the same as the frequency of flood days (days per year). For example, suppose there were two floods one year, one lasting two days and the other lasting three days; the frequency of flood days that year would be 5/365. Return to Slide 10.