Up or Out? Economic-Engineering Theory of Flood Levee Height and Setback

Up or Out? Economic-Engineering Theory of Flood Levee Height and Setback Tingju Zhu 1 and Jay R. Lund 2 Abtract: Levee etback location and height are important iue in flood levee ytem deign and modification. Thi paper derive an economic-engineering theory of the optimal trade-off of levee etback for height both for original and redeigned flood levee, demontrating the interconnection of levee etback, height, cot and rik, and economically optimal deign. Thee analye aume tationary flood hydrology and tatic ratio among damageable property value, unit contruction cot, and land price. The economic trade-off of levee etback for height depend on economic cot and benefit and hydraulic parameter, and only indirectly on flood frequency and economic damage parameter. The redeign rule derived in thi paper indicate condition where exiting levee hould be raied or moved in repone to change in condition. Numerical example illutrate the reult. Thi paper demontrate everal idea and theory for economic flood levee ytem planning and policy rather than providing guideline for direct deign practice. DOI: 10.1061/ASCE0733-94962009135:290 CE Databae ubject heading: Economic factor; Flood; Levee; Model. Introduction Levee ytem have long been built for flood protection. Early flood levee uually were deigned with cant quantitative analyi, relying primarily on a few oberved flood tage and human judgment. The achievement in experimental and theoretical hydraulic ince the 18th century Roue and Ince 1957, rational etimation of torm dicharge in the mid 19th century Biwa 1970, and early economic-engineering analyi Humphrey 1861 made poible the modern deign of flood levee. In recent decade, many tudie have addreed economic apect of flood levee deign, uually with benefit-cot analyi and optimization technique Davi et al. 1972; Wurb 1983; Goldman 1997; Olen et al. 1998; Jaffe and Sander 2001; Lund 2002; USACE 2002, 2006. Levee contruction or rehabilitation often follow flood diater. There i ometime coniderable controvery over whether flood channel capacity hould be obtained more from levee height or from greater levee etback. Thi iue ha received increaed public attention due to concern for riparian recreation and environmental ue of unprotected floodplain land. Some tudie have examined the optimal levee height and etback under tatic or dynamic hydrologic and economic condition Tung and May 1981; Zhu et al. 2007 but they only provided the optimal levee deign without more general theoretical inight into the trade-off of levee etback for height or flood levee redeign. 1 Senior Scientit, International Food Policy Reearch Intitute, 2033 K St., N.W., Wahington, D.C. 20006 correponding author. E-mail: t.zhu@cgiar.org 2 Ray B. Krone Profeor of Environmental Engineering, Dept. of Civil and Environmental Engineering, Univ. of California, Davi, CA 95616. E-mail: jrlund@ucdavi.edu Note. Dicuion open until Augut 1, 2009. Separate dicuion mut be ubmitted for individual paper. The manucript for thi paper wa ubmitted for review and poible publication on January 4, 2007; approved on July 9, 2008. Thi paper i part of the Journal of Water Reource Planning and Management, Vol. 135, No. 2, March 1, 2009. ASCE, ISSN 0733-9496/2009/2-90 95/$25.00. Economic deign of a levee ytem for flood protection involve balancing cot of levee height, loe of land value acrificed by floodway expanion etback, and flood damage from inadequate channel capacity. The mot common economic objective for floodplain management i minimization of expected annual damage and flood management expene Lund 2002; Olen et al. 2000; USACE 2006. Under tatic condition, annual peak flood flow uually fit an independent and identical probability ditribution, and economic factor, uch a the value of damageable property, contruction cot, and floodplain land value, are contant. Optimality condition can be applied to examine flood levee deign under uch tatic condition. The approach developed here applie to relatively common cae where the expected net cot function i convex with levee etback and levee height. Flood levee contruction involve large irreverible invetment, o rigorou examination i deirable before implementation deciion are made. Thi paper examine the optimal levee height and etback deciion from a theoretical and analytical perpective. The approach taken here i to examine deign baed on overall economic efficiency, conidering both flood control action cot and flood damage. Flood warning ytem are aumed to allow u to neglect, for now, loe of life in thee evaluation. Hypothetical river and levee type are explored to provide conceptual inight, rather than practical levee deign. Optimal Static Trade-Off of New Levee Setback for Height A tatic model i formulated to minimize the annualized net economic cot, which equal the expected flood damage plu the net annual flood management cot. The net annual flood management cot include annualized levee contruction cot and annualized economic benefit negative cot of levee-protected land and unleveed land by the river. The land value uually decreae with levee etback, ince a maller land area i protected. Thi imple model allow preliminary quantitative examination of the optimal 90 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / MARCH/APRIL 2009

economic trade-off between etback and height in deigning a new levee. The objective function i + Min EC,X h fqdqdq + C B,X h =,X h 1 where EC.=expected annual net cot, a a function of levee etback X and height X h ; the firt term on the right hand ide =expected annual flood damage, in which f. repreent the flood frequency ditribution; q = flood flow larger than the levee overtopping flow Q., and D. denote the damageable property value lo when the levee fail a a function of flood flow q; C. refer to the annualized cot to build a levee with height X h ; and B.=annual value of floodplain land both leveed and unleveed floodplain. Eq. 1 eentially aume the levee doe not fail with flow le than overtopping capacity. In reality, levee failure and flood damage can alo be caued by geotechnical failure mechanim uch a lumping, eroion, or water eepage undermining levee foundation. For example, the catatrophic flood damage in New Orlean during Hurricane Katrina were brought by both levee overtopping and tructural failure at approximately 50 location in the city hurricane protection ytem ASCE-ERP 2007. Levee overtopping may not alway lead to levee failure, but the rik i very high ince overtopping erode the levee back a water pill over. For implicity, in thi paper levee breach and flood damage are aumed to occur only when river tage overtop the levee. Geotechnical failure are neglected. Thi aumption i reflected in the firt term of Eq. 1 that only take into account damage caued by flood flow exceeding the overtopping capacity. Thi integral term can be approximated with dicrete flood frequencie of a number of flood flow interval and the flood damage aociated with thoe flow. A contructed flood levee i alway expected to function properly without the need for major modification for a period of decade, depending on local economic development, magnitude of flood, and other river characteritic. Thi i due to the large irreverible invetment involved in levee contruction and modification. To make the analyi valid for a period of time of interet to floodplain planner, two additional aumption are made implicitly in Eq. 1. Firt, the flood hydrology i aumed to be tationary. Second, ratio among the economic term involved in Eq. 1, damageable property value, unit contruction cot, and land price, are aumed to be contant over the planning horizon. Thee aumption can often be jutified in the real world. For example, the econd aumption hold true when damageable property value, contruction cot, and land price grow at ufficiently cloe rate within the planning period. Statitical tationarity i a widely adopted aumption in floodplain planning practice although it i increaingly being challenged by the preence of climate change Zhu et al. 2007. The land value function B depend on levee etback, and alo levee height becaue the bottom width of the levee cro ection commonly increae with levee height. B include the annual value of land protected from flood by the levee typically dominated by urban and agricultural ue value and the value of unprotected land on the tream ide of the levee which can have ubtantial recreational and environmental value. The firt-order condition for minimizing the expected total cot of flood control require the firt partial derivative of EC,X h with repect to X and X h equal zero EC = fd C B + =0 2 EC = fd B =0 In the above equation, / 0, / 0, C/ 0, B/ 0, and B/ 0. Auming uniform flow in the river channel, the overtopping capacity Q i determined olely by river cro-ection geometry, that i, X and X h in thi cae, beide energy lope and channel roughne Sturm 2001, which are not uppoed to be affected by levee modification. Eq. 2 and 3 lead to = C B B Eq. 4 hold for the optimal levee height X h * and etback X *. The optimal levee height X h * and etback X * can be found by numerically olving combined Eq. 2 and 3 and verifying that a global minimum i attained. In Eq. 4, the left hand ide i in economic parlance the marginal ubtitution rate MSR of levee etback for height for the optimal overtopping flow Q *,X h *. The right hand ide i the ratio of marginal contruction and land value cot of levee height X h to marginal land value lo due to etback X, that i, the MSR of levee etback for height for the optimal net cot. Note that B repreent land benefit, a negative cot, therefore a minu ign appear on the right hand ide. Eq. 4 implie that optimal economic efficiency occur where the floodway capacity MSR of levee etback for height equal the cot MSR of levee etback for height. In Eq. 4, neither flood hydrology nor value of potential flood damage affect the optimal trade-off between levee height and etback, although thee factor do affect the optimal flood channel capacity. Thu, change in flood hydrology from climate change, human activity impact on waterhed, or tructure meaure in uptream and/or damageable property value due to floodplain zoning, flood warning, or urbanization do not affect the economic optimal ratio of ubtitution between levee height and etback, depite change in optimal flood channel capacity. Rearranging Eq. 4 lead to B = C B If we define B/ and C B/ a cot efficiencie of etback and height, and / and / a hydraulic efficiencie of etback and height, Eq. 5 mean the ratio of cot efficiency to hydraulic efficiency of levee etback hould equal that ame ratio for levee height. There i an optimal trade-off of levee height for etback and it i affected by the relative economic value of protected veru unprotected land, contruction cot, and hydraulic characteritic, but i not directly affected by flood frequency or damage potential. Where decreaed etback increae damage potential becaue the larger protected area contain more damageable property, D Dq,X,X h. Auming damageable property value i in proportion to the width of levee-protected floodplain land, we have Dq,X,X h = w X ax h D q 6 in which w = total width of floodplain land behind a levee; a=levee ide lope; and D q=average damageable property 3 4 5 JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / MARCH/APRIL 2009 / 91

Table 1. Flood Levee Redeign Rule under Static Condition Setback X 0 X* X * 0 c1 c X h0 0 Move levee outward to *,X h * Move levee inward to *,X h * Height X c h0 0 X* h0 ELSE Raie current levee to X* h0 Do nothing if X h0 X* h0 X c1 0 Move levee inward and reize to *,X h * Do nothing if X h0 0 X 0 Move levee inward and reize to *,X h * value per unit width of floodplain, for flood flow q. Therefore, Eq. 5 now become Eq. 7 and then Eq. 8 B + X D = C B + ax hd 7 where D + = QX,X h fqd qdq, the expected flood damage for a unit width of floodplain behind the levee B + D = ax h f,x h D,X h + C B ad Eq. 8 reflect the ituation that decreaed etback increae damage potential or vice vera. Note that D =function of overtopping flow Q. For thi cae, greater damage potential encourage: 1 greater flood control capacity; and 2 greater etback, and the probability of flooding and etback effect on damage potential enter into the optimal trade-off of levee etback for levee height Eq. 8. Levee Redeign under Static Condition Thi ection derive theoretical optimal rule for when levee height and etback hould be changed. The previou analyi aume that no levee currently exit for flood protection. But commonly, levee already exit and one of three deciion mut be made: retain the exiting levee, raie the exiting levee to an optimal height at the current etback, or build a new levee with an optimal height and etback. We till neglect geotechnical failure. Let G 1 0,X h0, G 2 0,X h0, and G 3 X * denote the annualized expected cot of flood control under the three deciion, repectively, a given below G 1 0,X h0 =EC0,X h0 C0 9 8 G 2 0,X h0 =EC0 0 0 C0 10 G 3 *,X h * =EC *,X h * 11 where EC.=function of annualized expected cot of flood control; and C.=levee contruction cot, a previouly defined. Here, levee contruction cot i aumed to be linear with levee volume quadratic with levee height for trapezoidal levee. Value of function Eq. 9 and 10 depend on the etback and height of an exiting levee, X 0 and X h0, and Eq. 11 i a function of the optimal levee height X * h and etback X *.InEq.9, the optimal levee height X * h0 at the current etback X 0 without a pre-exiting levee can be found from Eq. 2. Thee particular formulation neglect any fixed cot to prepare and permit for any levee-raiing contruction; uch fixed cot could be added without lo of generality, and would tend to increae the range of condition under which no change in height i optimal. Thee three cot function allow u to develop rule for raiing and relocating levee under fairly general circumtance. The deciion criterion here i to minimize expected net economic cot. Thee rule are ummarized after their derivation in Table 1. If the exiting levee height i le than the optimal height at the current location 0 X * h0, the optimal levee height at the exiting etback lead to the relationhip EC0,X h0 EC0 0 0 0, and conequently, G 1 0,X h0 G 2 0,X h0 0. Therefore, if an exiting levee height i le than the optimal height at the current etback, the levee hould be raied to the optimal height unle it hould be relocated. If the current levee height exceed the optimal height for the current etback 0 X * h0, the levee hould not be further raied at the exiting etback, given the convex nature of the flood control cot function. However, it might be economical to relocate the levee. If G 1 0,X h0 G 3 X * 0, the exiting levee etback X 0 and height X h0 are preferable to relocating the levee; otherwie, moving the levee to the optimal etback X * and height X * h i preferable to the exiting levee. Where G 2 0,X h0 G 3 X * =0, the annualized overall cot to raie the exiting levee equal that of a new relocated levee. For a fixed levee location X 0, the levee height where moving and raiing the levee are economically equivalent i a critical cae. Thi critical levee height X c1 h0 for the current levee etback i a partitioning point for levee redeign option. If the exiting levee height i le than thi critical levee height for thi location, then the levee hould be moved to the optimal etback X* and height X * h. If the exiting levee height exceed thi critical height, but i le than the optimal height, then it hould be raied to X * h0 at the ame location. However, a critical height le than the optimal height for the current etback might not exit when the current etback i too large. For very large etback, lo of land value can be o high that no exiting levee at optimal height or greater can compenate the lo of land value and it i alway optimal to move the levee cloer to the tream. At the optimal etback X * the critical height X c1 h0 =0. The more 92 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / MARCH/APRIL 2009

thee equation become more complex. In Eq. 12, let =k/n S 1/2 and the partial derivative with repect to X and X h are Fig. 1. Rectangular channel and levee cro ection. = 5 3 X h 2/3 13 a etback deviate from the optimal etback, the more it correponding critical height approache it optimal height until they cro. Beyond a firt critical etback X c1, where the levee optimal height coincide with it critical height, X * h0 =X c1 h0, it i never optimal to raie an exiting levee. Beyond uch a etback, it i either optimal to retain the levee or move the levee to reduce the etback. Such a critical etback X c1 i a contant for given hydrologic, hydraulic, and economic condition, independent of current levee location or height. Beyond thi etback, an exiting etback would be too large owing to the lo of economic value from land on the tream ide of the levee. For uch a cae, the height of an exiting levee mut exceed the optimal height for current location to avoid being relocated. The deciion criterion become: If current levee height exceed a econd critical height, X h0, where G 1 0,X h0 G 3 X * =0, no action hould be taken to the exiting levee; otherwie, the levee hould be relocated to the optimal etback and height, X * and X * h. A econd critical etback X alo exit, beyond the firt critical etback, beyond which no pre-exiting levee height i preferable to relocating the levee; no X h0 exit where G 1 X,X h0 G 3 X * =0. When a levee etback exceed thi critical etback, the levee alway hould be relocated, a no additional height of exiting levee can compenate for the land value gained from moving the levee toward the tream. The econd critical etback occur when G 1 0,X h0 G 3 X * 0, even for the larget X h0. Theoretically, the value of the econd critical etback hould roughly equal the annualized expected economic value of the optimal deign ECX * divided by the product of annualized unit value of protected land B/ and levee length. Table 1 ummarize thee theoretical deciion-making rule for redeign of a levee. For generality, we aume there are two critical etback, the mall one X c1 and the larger one X. Numerical example of thee cae are explored later in the paper. Illutrative Example The theory developed i now applied to a common pecial cae. Optimal Trade Off of Flood Wall Setback for Height for Broad Flat Floodplain Conider a new levee for an ideal primatic wide hallow rectangular channel Fig. 1 with width X and levee height X h on one ide of the channel the other ide being high and fixed, with the overtopping flow given by Manning equation Sturm 2001 Q = k n S1/2 X X h 5/3 12 5/3 = X 14 If levee width doe not change with levee height for example, a flood wall, land value depend on etback only, leading to linear cot function B = LX and C = 0 Lw 0 X h, where =unit land protection value per year the difference between the annual value of protected veru unprotected floodplain land, per unit area; 0 =annualized unit contruction cot; L=length of levee reach; and w 0 =width of levee. Embedding thee cot function and Eq. 13 and 14 into Eq. 5 lead to X* X* = 3 0w 0 15 5 h Eq. 15 how that the optimal ratio of levee etback to height i a contant for an idealized wide hallow rectangular channel and rectangular levee cro ection under tatic condition. Thi ideal ratio of levee etback to height i the ratio of marginal hydraulic effectivene of levee etback to height in producing channel capacity 3/5 in thi cae multiplied by the ratio of the relative economic cot of levee etback and height 0 w 0 /. The optimal ratio of X to X h i unaffected by hydrology or the damage potential of protected propertie which do affect optimal channel capacity. In addition, Manning roughne and channel longitudinal lope alo do not influence the optimal ratio of X to X h but would affect optimal channel capacity. The trade-off between levee height and etback for levee planning and deign i thu primarily driven by economic factor, related to the unit cot of levee contruction and the increaed economic value of protected land compared with unprotected floodplain land. A the ociety come to value recreational and environmental benefit from unprotected floodplain, decreae and the optimal etback increae. However, a urban land value increae, increae and the optimal etback decreae. Urbanization, increaing damage potential and protected land value imultaneouly, will increae optimal channel capacity and preference for levee height over etback a a ratio. However, growth of optimal channel capacity ometime drive increae in abolute value of optimal etback. Levee Redeign for Broad Flat Floodplain The following preent an analyi of the levee redeign deciion for the wide hallow rectangular channel and rectangular croection levee hown in Fig. 1. Auming the flood have mean and tandard deviation, and fit a lognormal ditribution, the mean and tandard deviation of thi lognormal ditribution hould be where n = Manning roughne; S = energy lope, equaling longitudinal channel bed lope for uniform flow cae; and k=1.49 for Englih unit and 1 for metric unit. Where the oppoite bank become uceptible to flood damage with greater flow tage, M = 1 2 ln 2 2 +1 16 JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / MARCH/APRIL 2009 / 93

Table 2. Hydrologic, Hydraulic, and Economic Parameter, and Reultant Optimal and Critical Value Parameter Reult L 0 w 0 D X* X* h c1 X X ft 3 / ft 3 / ft $/ft 3 year $/acre year ft n $10 9 ft ft ft ft 20,000 20,000 52,800 0.05 10,000 150 0.035 1 519 27 1,440 1,520 S =ln 2 +1 17 For lognormal ditribution, the probability ditribution function i 1 lnq M 2 fq = e /2S 2 18 qs 2 Embedding Eq. 13 and 18, and the land value function B and contruction cot function C into Eq. 2, we obtain e ln X 5/3 h M 2 /2S 2 3 0Lw 0 =0 19 X h S 2 5D The parameter in Eq. 19 are the ame a previouly defined. Combining Eq. 19 and 15, the optimal levee etback X* and height X * h can be found for given hydrologic, hydraulic and economic condition. With an exiting levee etback X 0, the optimal levee height X * h0 for thi exiting etback can be olved from Eq. 19. With X * h0, X *, and X * h, for the exiting etback X 0, the c critical levee height X h0 can be olved by olving G 1.=G 2.. Table 2 give the parameter to calculate optimal and critical levee height and critical levee etback. Reult are hown in Fig. 2. Fig. 2 preent the optimal levee redeign deciion with critical etback and the optimal and critical levee height for variou etback for a river reach characterized by parameter in Table 2. The olid quare repreent the optimal levee height and etback for building a new levee for the river reach. The optimal levee height decreae a exiting etback increae. The critical height i convex with levee etback with a zero minimum at the optimal etback of the river reach, about 520 ft. If an exiting levee i located at it optimal etback it i not economical to move the levee and the levee hould be raied to it optimal height. If levee etback deviate from the optimal etback, whether more or le, an exiting levee greater than the critical height i needed to avoid Exiting Levee Height (ft) 70 60 50 40 30 X* h0 Do nothing 20 X c h0 Raie to optimal height at 10 Rebuild current etback - outward Rebuild - inward 0 0 200 400 600 800 1000 1200 1400 1600 Optimal Firt Second etback Critical Critical Exiting Levee Setback (ft) Setback Setback Fig. 2. Redeign rule with optimal and critical levee height and etback X c h0 Alway rebuild - inward relocating the levee. The firt and econd critical levee etback are alo illutrated, howing where ditant levee etback require more than optimal exiting height to jutify their retention between the firt and econd critical etback and the etback beyond which even the tallet exiting levee cannot jutify the land value gained from moving the levee toward the tream beyond the econd critical etback. The optimal levee redeign deciion are indicated in each zone defined by the optimal levee height curve upper curve before critical etback and critical levee height curve. Concluion Dicuion of levee deign and levee etback have increaed in recent year, particularly in the context of riing public value for recreation and environmental ue of floodplain and riing economic land value and damage potential in leveed floodplain. Thi paper develop ome theoretical apect of implified flood levee planning problem, with particular attention paid to levee etback and height, under tatic hydrologic and economic condition. While the reult developed here are largely theoretical, they illutrate everal point of common importance for floodplain planning and policy. There i an optimal trade-off of levee height for etback, which can be examined analytically. Under tatic condition, the optimal trade-off of levee etback for height i determined by levee contruction cot and floodplain land value in protected veru nonprotected part of the floodplain, beide hydraulic factor. Flood frequency and flood damage potential do not influence the optimal ubtitution between levee etback and height rate except for ome effect when etback affect damage potential, though they affect the optimal overall channel capacity. Levee redeign deciion rule are developed for tatic condition where a levee already exit. The critical height for a given etback and critical etback for making deciion to raie or relocate levee are derived and analyzed for a river reach. The critical etback are contant for a given river reach, depending only on hydrologic, hydraulic, and economic factor. Levee redeign deciion rule are derived baed on thee optimal and critical value of a given problem. A condition of flood frequency, channel hydraulic, and floodplain economic change, it i likely that levee hould be modified in height and/or etback. Thi too can be examined analytically. A the economic value of unprotected floodplain rie for recreational and environmental purpoe, the relative difference between protected and unprotected land value decreae and optimal etback hould increae imultaneouly decreaing optimal levee height. However, in growing metropolitan area, increae in urban land value can outweigh riing value for unprotected floodplain, often tending to decreae the optimal floodway width. While the reult here are unlikely to provide directly ueful quantification for actual levee problem, uch theoretical formulation and reult hould help provide qualitative and conceptual inight for levee and floodplain management problem. 94 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / MARCH/APRIL 2009

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