1

Up or Out?Levee setback vs. levee height

Tingju Zhu

International Food Policy Research Institute

Jay R. Lund

University of California, Davis

http://cee.engr.ucdavis.edu/faculty/lund/CALVIN/

2

Overview

1. Levees

2. Height vs. setback

3. A mathematical formulation

4. Some solution results

5. Implications

3

Levees1. Levees are common for land protection

2. Imperfect but sometimes optimal protection

3. Failure by: overtopping, slope failure, seepage, false sense of security

4. Economic controversies over costs, benefits, and risks

5. Environmental controversies over aquatic and terrestrial stream corridors and interactions

4

Height vs. Setback

1. Every levee design involves a trade-off of levee height against levee setback

2. Greater height costs more, but allows less setback to protect more land

3. Less setback (greater height) driven by difference in land value between protected and unprotected floodplain land

4. Can be seen as a benefit-cost tradeoff

5. Might be expanded to include environmental benefits of flood-prone land

5

A Mathematical Formulation

EC(.)= expected annualized total cost

Xs = designed levee setback

Xh = designed levee height

P(.)= failure probability for levee height & setback

D = damageable property value (potential loss in a flood disaster)

C(.) = annualized cost to build a levee of height

B(.) = annual value of floodplain land (both leveed and unleveed)

, , ,s h s h h s hMin EC X X P X X D C X B X X

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Optimize Analytically 1

C(.) = annualized cost to build a levee of height

B(.) = annual value of floodplain land (both leveed and unleveed)

0

h h h

C BEC PD

X X X

0

s s s

EC P BD

X X X

7

Optimize Analytically 2If levee fails only by overtopping, and

given a levee overtopping flow Q(Xs, Xh) …

hh X

Q

Q

P

X

P

ss X

Q

Q

P

X

P

By the chain rule and some algebra…

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Solution

Optimal height vs. set-back trade-off

LHS = marginal economic value of less setback/marginal channel capacity with increased setback

RHS = marginal economic value of greater height/ marginal channel capacity with increased height

s s h h

C BB Q Q

X X X X

9

Implications

Optimal height vs. set-back trade-off is only a function of land and construction economics and relative hydraulic effectiveness.

Not affected directly by flood frequency or flood damage.

Optimal channel capacity is separate.

s s h h

C BB Q Q

X X X X

10

Re-Design of Existing Levee1. Previous formulation was for a new levee.

2. What if a levee exists – with a setback and height designed long ago.

3. Three choices:

a) Keep levee as is.

b) Raise levee to an optimal height.

c) Build new levee at optimal location.

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Solution 1

1. Compare EV cost for each of the three alternatives.

2. Some decision rules result.

Height

Setback Xh0 <

ch0X

ch0X < Xh0 <

*h0X ELSE

Xs0 <1c

sX Move to ),( **hs XX

Raise current levee to *0hX Do nothing if

Xh0 > *

h0X

1c

sX < Xs0 <

2c

sX Move levee inward and resize to ),( **

hs XX Do nothing if

Xh0 > 2c

h0X

Xs0 >2c

sX Move levee inward and resize to ),(**hs XX

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Solution 2Decision rules for an example.

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400 1600

Levee Setback (ft)

Levee H

eig

ht

(ft)

Do nothing

Raise to optimal height

at current setbackRebuildRebuild

Optimal

setback

First

Critical

Setback

Second

Critical

Setback

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Conclusions1. Levee height vs. setback trade-off

2. Optimal trade-off determined by construction costs vs. relative land value, with hydraulic efficiencies.

3. Damage and flood frequency do not affect optimal substitution of height for setback.

4. Levee re-design also can be analyzed with some intuitive decision rule results.

5. As conditions change, so sometimes should levees.

14

Other Big Changes

1. Population growth

2. Land use

3. Social values

4. Economic well-being

5. Crop prices, yields,

etc.

6. Others?

John Landis, UCB, estimates 2002

15

Adaptation Studies for Climate Change

Planning studies more than “impact” studies

Allow and explore substantial adaptation,

preferably with multiple options

Use future population, land use, and

economic conditions

For complex systems, some optimization will

be required

Interpretation and limitations

16

Flooding on the Lower American River

Climate Change and

Urbanization

17

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

1900 1950 2000 2050 2100 2150

Time (yr)

Mea

n A

nn

ua

l F

loo

d P

eak

(m

3/s

) HCM2 Regression

Historical Trend

Stationary History

=

Three-Day Peak Inflows at Folsom Lake

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Method Optimize levee heights & setbacks over time

Minimize average total cost of:

• flood damage and frequency

• levee construction

• lost urban and floodplain land value

Considers changing flood probabilities

Changing urban land and flood damage

values – 150 year time frame.

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0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Time (yr)

Le

ve

e H

eig

ht

(m)

0

30

60

90

120

150

180

210

240

270

300

Le

ve

e S

etb

ack

(m

)

HCM2

Historical Trend

Stationary Historical

Climate Change Alone Without Urban Growth

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0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Time (yr)

Le

ve

e H

eig

ht

(m)

0

30

60

90

120

150

180

210

240

270

300

Le

ve

e S

etb

ack

(m

)

0% 2% 4%Series6 Series7 Series8

Urbanization rate:

Urban Growth Alone

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0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Time (yr)

Le

ve

e H

eig

ht

(m)

0

30

60

90

120

150

180

210

240

270

300

Le

ve

e S

etb

ack

(m

)

0% 2% 4%

Urbanization rate:

Combined Effects with Historical Trend in Floods

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0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Time (yr)

Le

ve

e H

eig

ht

(m)

0

30

60

90

120

150

180

210

240

270

300

Le

ve

e S

etb

ack

(m

)

0% 2% 4%

Urbanization rate:

Combined Effects with HCM2 Scenario

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Costs: 2% Urbanization & HCM2 Climate

0

200

400

600

800

1,000

1 21 41 61 81 101 121 141

Time (yr)

Co

st

($M

illio

n/y

r)

Average Flood Damage

Forgone Land Value

Construction Cost

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2% Urbanization & HCM2 Hydrology

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

0 20 40 60 80 100 120 140Time (yr)

Flo

od

ing

Fre

qu

en

cy

(y

ea

rs)

0

5

10

15

20

25

30

Ch

an

ne

l C

ap

acit

y (

10

3 m

3/s

)Flood Recurrence Period

Channel Capacity

“100-year” flood

“500-year” flood

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Observations1) Climate changes or urbanization alone can

be accommodated by raising levees

2) Combined effects can raise levees andincrease levee setbacks

3) Adding loss of life accelerates levee raising and floodway widening

4) Adding climate change uncertainty could slow or speed adaptation

5) Non-levee adaptations are also likely

6) Raising American River levees & perhaps widening floodway might be desirable

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Flood Control Conclusions1) People and societies adapt all the time.

2) Combined effects of climate change and other factors are important for adaptations

3) Increasing Central Valley flooding problems

– Continued urbanization

– Wet climate warming & apparent flood trends

– Other tributaries have similar problems

– Limits of levees and levee heights alone

4) “100-year” flood planning is a bad wager.

27

Adaptation Studies for Climate Change

Planning studies more than “impact” studies

Allow and explore substantial adaptation,

with multiple options

Use future population, land use, and

economic conditions

For complex systems, some optimization will

be required

Interpretation and limitations