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Applications of Scaling to Regional Flood Analysis
Brent M. TroutmanBrent M. Troutman
U.S. Geological SurveyU.S. Geological Survey
Introduction:Flood frequency analysis ObjectiveObjective
Estimate magnitude of flow which is exceeded on Estimate magnitude of flow which is exceeded on the average once every the average once every TT-years at a site (-years at a site (TT-year -year flow) flow)
ProblemProblem Limited flow data!Limited flow data!
ApproachesApproaches Regional flood analysis: Data from nearby sitesRegional flood analysis: Data from nearby sites Rainfall-runoff models: Process knowledgeRainfall-runoff models: Process knowledge Scaling: ConnectionsScaling: Connections
Regional flood analysis methods Regional regression Regional regression
where where qqTT = = TT-year flow; -year flow; B, CB, C = basin, climatic characteristics = basin, climatic characteristics Index-flood methodIndex-flood method
QQ// has same distribution for all sites has same distribution for all sites
wherewhere Q Q = annual peak flow, = annual peak flow, = at-site mean, also often related = at-site mean, also often related to to BB, , CC by regression by regression
eCBqT ...
Scaling invariance
A type of symmetry such that small systems are A type of symmetry such that small systems are “similar” in geometry and/or function to large “similar” in geometry and/or function to large systemssystems
How will scaling help in regional flood analysis?How will scaling help in regional flood analysis? A framework for revealing connections with A framework for revealing connections with
rainfall-runoff processesrainfall-runoff processes Predictions of coefficients in regional Predictions of coefficients in regional
regressionsregressions Indications of appropriate form and Indications of appropriate form and
assumptions for statistical modelsassumptions for statistical models
The role of area Analysis of scaling invariance involves looking at Analysis of scaling invariance involves looking at
changes with respect to a scale parameterchanges with respect to a scale parameter Drainage area Drainage area AA is a logical choice in regional flood is a logical choice in regional flood
analysis: it is often the only or most significant analysis: it is often the only or most significant statistical predictor in regressions:statistical predictor in regressions:
Specific focus of this work: Specific focus of this work: How do scaling ideas help in understanding peak How do scaling ideas help in understanding peak
flow dependence on flow dependence on AA??
A TAq TT
New Mexico scaling exponents
T (years)T (years)
NE NE plainsplains
N N mtns.mtns.
NW NW plateauplateau
SE SE plainsplains
22 0.560.56 0.910.91 0.520.52 0.670.67
55 0.550.55 0.920.92 0.470.47 0.590.59
1010 0.550.55 0.920.92 0.440.44 0.550.55
2525 0.550.55 0.930.93 0.410.41 0.500.50
5050 0.550.55 0.930.93 0.390.39 0.470.47
100100 0.560.56 0.940.94 0.370.37 0.440.44
500500 0.580.58 0.940.94 0.360.36 0.410.41
)( TAq TT
The framework of scaling invariance has been used to look at many of the characteristics known to influence flows …
River basin geometryRiver basin geometry Channel networkChannel network Channel sinuosityChannel sinuosity Downstream Downstream
hydraulic geometryhydraulic geometry Landscape Landscape
roughnessroughness Longitudinal Longitudinal
profilesprofiles
RainfallRainfall Spatial variabilitySpatial variability Temporal Temporal
variabilityvariability IDF curvesIDF curves
SoilsSoils Pore structurePore structure Flow pathwaysFlow pathways
Channel networks:The width function L(x)
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Distance from outlet, x
Wid
th, L
(x)
LL((xx) = number of links at distance ) = number of links at distance xx from the from the outletoutlet
The width function & flow
Under an idealized scenario, flow Under an idealized scenario, flow QQ((tt) at the outlet ) at the outlet has the same shape as the width function has the same shape as the width function LL((xx))
Spatial rainfall pattern: UniformSpatial rainfall pattern: Uniform Temporal rainfall pattern: Instantaneous burst Temporal rainfall pattern: Instantaneous burst
of rain all deposited into networkof rain all deposited into network Channel flow: Translation routing at uniform Channel flow: Translation routing at uniform
velocity velocity vvcc
)()( tvLtQ c
RRT network model: Peano
t0
t1 t2 t3
Generator
Peano network width function
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Distance
Wid
th d
ensi
ty
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Distance
Wid
th d
ensi
ty
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Distance
Wid
th d
ensi
ty
0
10
20
30
0 0.2 0.4 0.6 0.8 1
Distance
Wid
th d
ensi
ty
Peano width function maximum
Straightforward geometric arguments show Straightforward geometric arguments show max width max width LLmaxmax and area and area AA are related by are related by
Implication: Peak flows under “idealized Implication: Peak flows under “idealized scenario” scale as:scenario” scale as:
79.04log
3lognet
netAQ
,maxnetAL
Flint River, GA
Drainage Drainage area:area:
6380 sq km6380 sq km Number of Number of
links:links:
22,95922,959
25 km
Flint River width function
0
50
100
150
200
250
300
0 50 100 150 200 250
Distance (km)
Num
ber o
f lin
ks
Flint R. width function maximum
1
10
100
1000
0.1 1 10 100 1000 10000
Area (sq km)
Max
wid
th
net
0.46
Flint River generators Actual Actual
networks: networks: generators vary generators vary randomlyrandomly
Extract Extract generators and generators and analyze analyze distribution of distribution of no. links per no. links per generator generator
Same for Same for different different replacement replacement levels levels
-4
-3
-2
-1
0
0 5 10 15 20 25
Number Links per Generator
Log(
Freq
uenc
y)
Level 1
Level 2
Level 3
Level 4
Geom (0.40)
Goodwin Creek, MS
Order 1 stream
Order 2 stream
Order 3 streamOrder 4 stream
Basin boundary
Stream gages
1 km
Goodwin Creek width function
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12
Distance from outlet, x (km)
Nu
mb
er o
f lin
ks, L
(x)
Goodwin Cr. width function maximum
1
10
100
0.1 1 10 100
Area, A (sq km)
Wid
th f
un
cti
on
ma
xim
um
net 0.446
Peak flow scaling, Goodwin Cr.
Observed features:Observed features: Average slope Average slope for for
329 events is 0.79329 events is 0.79 Decrease in peak flow Decrease in peak flow
variability as variability as AA increasesincreases
Explanation: hillslope Explanation: hillslope processesprocesses Travel timeTravel time Spatial variability of Spatial variability of
runoff generationrunoff generation
0.01
0.1
1
10
100
0.1 1 10 100
Area (sq km)
Peak
flow
(cm
s)
Travel time: channel and hillslope Width function Width function
LL((xx) = no. links at channel distance ) = no. links at channel distance xx Generalized width functionGeneralized width function
MM((xx,,yy) = no. of pixels at channel distance ) = no. of pixels at channel distance xx and hillslope and hillslope distance distance yy
Assume velocities Assume velocities vvhh and and vvcc such that travel time to outlet such that travel time to outlet isis
Consider flow again with spatially uniform, Consider flow again with spatially uniform, instantaneous rainfallinstantaneous rainfall
hc v
y
v
xt
Peak flow scaling exponent vs. vh/vc
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0001 0.001 0.01 0.1 1 10 100
vh/vc
net = 0.446
This curve is ob-This curve is ob-tained using only tained using only the function the function MM
vvhh/v/vcc large yields large yields
flow proportional flow proportional to width function,to width function, ==netnet
= 0.79 = 0.79 corresponds tocorresponds to
025.0/ ch vv
Spatial variability of runoff generation Assumption: Peak flow is sum of flow Assumption: Peak flow is sum of flow
contributions from a set of links in the contributions from a set of links in the basin, and runoff generation from these basin, and runoff generation from these links (hillslopes) is spatially variablelinks (hillslopes) is spatially variable
Implication: Implication:
Yields a statistical model for log of Yields a statistical model for log of QQ
AQE ~)( 2~)(
AQSD
Statistical model for log of peaks
, , , , storm dependent, storm dependent, ZZ basin effect, basin effect, ee error error
)(loglog 2/ eZAAQ
0.01
0.1
1
10
100
0.1 1 10 100
Area (sq km)
Peak
flow
(cm
s)
Parameter estimates
)(2/ eZA
AQ loglog -4
-2
0
2
4
0.1 1 10 100 1000
Peak at outlet
0
0.4
0.8
1.2
1.6
0.1 1 10 100 1000
Peak at outlet
0
0.5
1
1.5
2
2.5
0.1 1 10 100 1000
Peak at outlet
Pooled variability of residuals, Goodwin Creek
-10
-8
-6
-4
-2
0.1 1 10 100
Area (sq km)
Vari
ance
slope = 1.09 (+/- 0.11)
Conclusions:How will scaling help in regional flood analysis?
Predictions of coefficients in regional Predictions of coefficients in regional regression relations based on network regression relations based on network geometry, basin & storm properties, etc.geometry, basin & storm properties, etc.
Indications of appropriate form and Indications of appropriate form and assumptions for statistical modelsassumptions for statistical models
More generally, a framework for revealing More generally, a framework for revealing connections between regional flood analysis connections between regional flood analysis and rainfall-runoff processesand rainfall-runoff processes