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