1 Cse Study: Flood Wtch c 00 Donld Kreider nd Dwight L Animtion: Flood Wtch To get you going on the Cse Study! In this section, we hve lerned tht if we re given the continuous derivtive f of function on the intervl [, b], then one version of the Fundmentl Theorem of Clculus sttes tht f (x) dx = f(b) f() This formul is very useful if we cn find n ntiderivtive for f ; tht is, if we hve n explicit representtion of f. However, in n pplied setting, we my not hve formul for f nd thus cnnot use the fundmentl theorem directly. The derivtive f (x) my be given t only finite number of points of the intervl, nd tht my be ll tht is known. Tht nd the fct tht f (x) is continuous rte of chnge of function f. In these cses, we wnt to keep in mind tht we cn pproximte the integrl using Riemnn sum. This is the procedure tht we will follow in the CSC of this section where the derivtive is non-negtive function on n intervl. We will use the Riemnn sum f (x i ) x i i=1 s n pproximtion to the re under the grph of the continuous function f. As usul, the purpose of CSC is to consider rel ppliction of clculus, with rel dt. In this section, we will work with our erth scientist collegues to study problems relted to river flooding 1. Without being too specific bout our objective t this time, we will stte it in generl terms so tht we hve n ide where we re going. Objective: From dt collected ten yers prt, to determine if the Gorge River hs n incresed or decresed likelihood of flooding. In order to mke the objective more concrete, we re going to hve to lern how erth scientists collect nd ssemble informtion bout rivers. We will ply the role of mthemticins working with geologists to provide the mthemticl nlyses tht regionl nd urbn plnners need for the mngement of resources hving n impct on wtersheds nd the like. We hve to be creful, though, becuse we will find tht mthemticins nd erth scientists sometimes use the sme terminology for different concepts. Becuse this is CSC nd we wnt to get prctice pplying mthemtics to different field, we will use the erthscience terms. But be creful to trnslte them into their mthemticl form so tht you know wht clculus tools to pply. In the next prgrph, we describe ll of the terms we will be using. Often the physicl units (for exmple, volume of wter per unit time) will help with the lnguge conversions from erth sciences to mthemtics. River Flooding: Bckground The flow of wter on the surfce of the erth in rivers nd strems hs n importnt impct on humns. Mny of these wter bodies provide importnt resources s nvigtion routes nd sources of fresh wter for residentil nd industril use. The erosion force of the strems is responsible for the formtion of mny of our lnd forms both by removl s well s deposition of mteril. Perhps their most drmtic impct is flooding, when the discge (volume of wter per unit time) exceeds the norml crrying cpcity of the chnnel, nd the wter spills out onto the djcent flood plin. In n idel world, flood plins would be reserved 1 The CSC is dpted from Skinner, B. J. nd Porter, S. C., 196, The Dynmic Erth, An Introduction to Physicl Geology, nd ed., Wiley, 570 pp.; nd from Leopold, L. B., 1968, Hydrology for Urbn Lnd Plnning, guidebook on the hydrologic effects of urbn lnd use, U.S. Geologicl Survey Circulr 554. We lso owe specil thnks to Professor Dick Birnie, Deprtment of Erth Sciences, Drtmouth College, for collborting on this work.
for rivers nd perhps griculture which rejuventes from fresh silt nd nturl minerl fertilizers left by the river. However, the trnsporttion nd recretionl vlue of rivers hs lured mny humn settlements to river bnks. Over time, mny of these settlements hve grown into mjor metropolitn centers such s St. Louis nd New Orlens on the Mississippi River. When river floods, the humn impct is disstrous. People who live nd work long river should not be surprised by flood. Floods re regulr nd expected events; they hve hppened in the pst nd they will continue to hppen in the future. Erth Scientists mke continuous observtions of the discge (volume of wter per unit time) of strem. These observtions led to n understnding of the flow ccteristics of the river nd its wtershed (re tht drins into the river) nd llow predictions of future behvior. This concept repets itself toughout Erth Science: observtions (mesurements) re mde, n understnding (perhps mthemticl model) is derived by nlyzing (tbulting, grphing, sttisticlly processing) the observtions; nd then predictions re extrpolted from the model. The fundmentl wy to disply the observtions of strem discge is hydrogrph. A hydrogrph hs two components: the mount of rinfll (depth per unit time) nd discge (volume of wter per unit time) (see the exmple below). Prior to rin storm, the strem will be flowing t some bckground level of discge known s bse flow. However, following period of hevy precipittion, the rin flling in the wtershed drins into the strem, nd the discge increses over time. The plot below shows the mount of discge over 1 hour period following n intense rinstorm lsting bout 96 minutes. The discge increses rpidly to pek bout 3 hours fter the storm nd then grdully drops for the next 9 hours. The discge of strem does not rise immeditely with the onset of precipittion, rther it tkes time to flow cross the wtershed nd into the strem. The difference in time between the centroid of the precipittion nd the centroid of the discge (runoff) is known s the lg time nd tht depends on the size nd mkeup of the wtershed. Neglecting bsorption by the ground, the re of the wtershed times the depth of the rinfll equls the volume of discge bove the bseflow. If the totl volume of discge of the strem exceeds the crrying cpcity of the chnnel, the strem overflows its bnks nd floods. Exmple of Hydrogrph A hydrogrph consists of two prts: plot of rinfll, nd plot of discge. Here is set of possible dt giving the discge of specific strem cused by n intense mount of rinfll. First, the rinfll plot in inches per hour: in The rinfll is.1 in in for the first 0.6 hours, then it is 3.7 for the next 0.6 hours, nd finlly it is 1.9 for 0.4 hours. Next, the dt representing the discge D of strem t specific instnts of time beginning t the onset of the intense downpour of rin given bove re s follows, where the discge is mesured in cubic meters per second:
3 Notice tht lthough the discge rises rther quickly, it drops slowly bck to the bse flow level of the strem. To find the lg time, we need to compute the centroids of the plots of rinfll nd discge. The centroid of plne region is the point (x 0, y 0 ) on which the region, thought of s thin plte, would blnce horizontlly. To find the point, we use the formuls for the coordintes. Theorem: Suppose the region of re A is defined s lying between two curves y = f(x) nd y = g(x) where f(x) g(x) nd x b. Then its centroid is locted t the point (x 0, y 0 ) given by x 0 = 1 A y 0 = 1 A x(f(x) g(x)) dx f(x) g(x) dx Becuse the proof is instructive for our discussion of the hydrogrph, we will justify this formul before proceeding further with the rinfll dt. Derivtion of Formuls for the Centroid Suppose two msses m 1 nd m re plced on see-sw, t distnces d 1 nd d, respectively, from the pivot-point. Then how fr do we hve to move one mss reltive to the other so tht the see-sw will blnce? m m 1 d This is n old problem tht hs very simple solution: the see-sw will blnce when m 1 d 1 = m d. For exmple, if m is lrge reltive to m 1, then d must be smll reltive to d 1. We ll hve experienced the d 1
4 effects of this eqution: the lrger person hs to sit closer to the pivot point to blnce the smller person sitting t the opposite end of the see-sw. m m 3 m 4 m 5 m 1 x 3 x x 0 x 4 x 5 x 1 d Suppose now tht we consider n msses m 1, m, m 3,..., m n locted on see-sw t the points x 1, x, x 3,..., x n, nd suppose further tht the pivot point is locted t x 0. Then generlizing from the cse of two msses, it turns out tht blnce is chieved when d 1 m k (x k x 0 ) = 0 Note tht if x k > x 0, then x k x 0 is positive nd is the distnce of m k from the pivot-point, while if x k < x 0, then x k x 0 is negtive nd is the negtive of the distnce of m k from the pivot-point. So, this eqution is indeed generliztion of the one we strted with involving only two msses. If we solve for x 0 in this eqution, then we get tht x 0 n m k = Clling n m k the totl mss, we then hve tht m k x k x 0 = n m kx k totl mss By nlogy with msses, if the rectngles pictured bove re simply two-dimensionl regions, we cn write similr formul for the coordinte of the pivot-point using res: x 0 = n x ka k totl re where A k is the re of the k th rectngle. The product x k A k is clled the moment of the k th rectngle. Suppose now tht we strt with function y = f(x) defined on n intervl [, b], nd s usul we prtition the intervl into n subintervls with the points x 0 < x 1 < x < < x n. Let c k be the midpoint of the k th subintervl. Then c k A k = c k f(c k ) x k is the moment of the k th rectngle with respect to the y-xis.
5 y = f (x) f ( c k ) f ( c k ) c k b We next recognize tht the sum of the moments of the n rectngles is Riemnn sum whose limit is the definite integrl (ssumed to exist) of the function over the intervl [, b]: x k lim n c k f(c k ) x k = xf(x) dx In similr mnner, we cn compute the moment of the k th rectngle with respect to the x-xis. Becuse the y-coordinte of the center of the rectngle is f(c k), the moment with respect to the x-xis is f(c k ) f(c k ) x k, tht is, the y-coordinte of the center times the re of the rectngle. Once gin, we recognize the sum of these moments s Riemnn sum, whose limit is definite integrl: lim n Finlly, the point (x 0, y 0 ) whose coordintes stisfy f(c k ) f(c k ) x k = 1 f(x) dx Ax 0 = Ay 0 = 1 xf(x) dx f(x) dx where A is the re under the grph of f, is clled the centroid of the region. It is the blnce point of the re of the region. Bck to Exmple of Hydrogrph We hd gotten to the point in our discussion of the hydrogrph where we were bout to compute the centroids of the rinfll nd discge dt. The rinfll plot is composed of tee rectngles. We refer to the proof bove of the centroid formul to find the x-coordinte of the centroid. The proof shows tht we cn clculte it s the sum of the midpoint of rectngle times its re (the height of the rectngle times the width of the rectngle), ll divided by the re of the rinfll plot. The re of the rinfll dt is
6 A =.6(.1) +.6(3.7) +.4(1.9) = 4.4 Hence, we find tht the coordintes of the centroid of the rinfll dt re: x 0 = 1 (.3(.1)(.6) +.9(3.7)(.6) + 1.4(1.9)(.4)).8113 A y 0 = 1 ( (.1 )(.6) + (3.7 )(.6) + (1.9 )(.4) ) 1.45094 A Applet: Flood Wtch Try it! To find the centroid of the discge dt, which is given in convenient computtionl form such s n pplet or Mple worksheet, we use Riemnn sum pproximtion to clculte the integrls in question. First, letting y min ( = 150 in this exmple) be the bse level of the flow, the (pproximte) re A under the discge grph is obtined s the sum A = 3600 (f(c k ) y min ) x k 7418 where we hve shown explicitly the multiplier 3600 sec tht is needed to mke consistent the units on the verticl nd horizontl xes. Then, the x nd y coordintes of the centroid of the discge dt due to rinfll re: x 0 = 1 A y 0 = 1 A c k (f(c k ) y min ) x k 4.94 (f(c k ) y min ) x k 691.3 Note tht x 0 is in hours, nd y 0 in m3 sec. The lg time is the difference of the x-vlues of the centroids of the discge nd rinfll dt, or pproximtely 3.487 hours. Moreover, if we wnted mesure of the intensity of the storm, we could compute the difference in the y-vlues of the centroids. To see where the bove formuls for x 0 nd y 0 come from, note tht the re of one rectngle for computing the x coordinte x 0 nd the y coordinte y 0 of the centroid is ( ) f(ck ) y min xk. The distnce of the rectngle s centroid from the y-xis is c k, nd its distnce from the x-xis is 1 ( ) f(ck ) + y min. Thus, ech term in the sum for computing x 0 is 1 A c ( ) k f(ck ) y min xk, nd ech term in the sum for computing y 0 is 1 ( ) ( ) f(ck ) + y A min f(ck ) y min xk = 1 ( f(ck ) y ) A min xk nd both formuls re obtined s bove by summing the respective terms. The CSC: Flood Wtch
7 Now tht we hve considered n exmple of hydrogrph, nd discussed the relevnt mthemtics, we re redy to stte the objective, setup, nd thinking nd exploring issues for the CSC. Objective: We re given two hydrogrphs for the Gorge River tht hve been recorded ten yers prt. The overll objective is to nlyze, compre, nd interpret the two hydrogrphs, thereby reching conclusion bout the incresed or decresed likelihood of the Gorge River flooding. Setup: We will be plotting the rinfll nd discge dt, nd ssuming tht the discge represents smple of mesurements drwn from continuous function. We will be using Riemnn sums to clculte pproximtions to the re under the curve, nd to the centroid of the discge dt. We will keep trck of the vrious units of mesurement to be certin tht we hve consistent set. Thinking nd Exploring: We will be studying ech hydrogrph, nd then compring our findings for the two. The rinfll is mesured in units of inches per hour, the discge in units of cubic meters per second, nd the horizontl time line in hours. Some of the questions we will consider under Thinking nd Exploring re: Wht is the verge bse flow? Wht is the time t which the discge is mximum? Wht is the mximum discge? Wht is the discge due to bse flow over the 1 hour observtion time? Wht is the discge due to the rin storm over the 1 hour observtion time? Wht is the totl discge over the 1 hour observtion time? About how long did the rin storm lst? Wht is the center of mss of the precipittion? Wht is the center of mss of the discge? Wht is the lg time? Wht is the re of the wtershed? Applet: Flood Wtch Try it! As ususl, the lst step in CSC is to write summry of the investigtions. Interprettion nd Summry: After studying the two hydrogrphs of the Gorge River, nd thinking bout the mthemticl tools, it is time to interpret nd summrize the mthemticl results in terms of the originl objective. Pretend tht your synopsis is going to pper in the next issue of mgzine such s Scientific Americn. Include enough detils so tht reder would lern wht the mjor issues of the report re, nd how you went bout ddressing them. Wht will you wnt to tell reders bout your success with regrd to the originl stted objective of the investigtion? Be sure to write in complete sentences using correct rules of stndrd English grmmr. To get you strted, here re two questions tht definitely need to be nswered: 1. Wht chnges in the wtershed might hve occurred in the 10 yers between the hydrogrphs to ccount for the different pttern of the more recent hydrogrph reltive to the erlier one?. How might these chnges influence the likelihood of the Gorge River flooding? Wht dvice do you hve for public policy mkers? Mke the report interesting, compelling. Ask yourself: Would policy mker be ble to understnd it, nd feel compelled to follow its recommendtions? Exercises: Problems Check wht you hve lerned! Videos: Tutoril Solutions See problems worked out!