Up or Out?—Economic-Engineering Theory of Flood Levee Height and Setback
1. 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/
3. 3
Levees
1. 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. 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. 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
6. 6
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 P
D
X X X
0
s s s
EC P B
D
X X X
7. 7
Optimize Analytically 2
If 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…
8. 8
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. 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. 10
Re-Design of Existing Levee
1. 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.
11. 11
Solution 1
1. Compare EV cost for each of the
three alternatives.
2. Some decision rules result.
Height
Setback
Xh0 < c
h0X
c
h0X < 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
12. 12
Solution 2
Decision 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)
LeveeHeight(ft)
Do nothing
Raise to optimal height
at current setback
RebuildRebuild
Optimal
setback
First
Critical
Setback
Second
Critical
Setback
13. 13
Conclusions
1. 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. 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. 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
18. 18
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.
25. 25
Observations
1) Climate changes or urbanization alone can
be accommodated by raising levees
2) Combined effects can raise levees and
increase 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
26. 26
Flood Control Conclusions
1) 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. 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
