Regional Characteristics of Unit Hydrographs and Storm

HyetographsTheodore G. Cleveland, Ph.D., P.E.

Instantaneous Unit Hydrograph Approach

• Unit hydrograph is one of several methods examined in this research.

• University of Houston has focused exclusively on this technique.

• Two major components– Analysis (Find IUH from rainfall-runoff data)– Synthesis (Estimate IUH from watershed

character)

Storm Analysis

• Central Texas Database– Analyze all storms using five different IUH

model equations.– Pick a “good” model– Aggregate model parameter values by station.

• Re-run each storm using the aggregated values.• Test these results for acceptability• Interpret results

– Conclusions and Recommendations

Central Texas Database

Different Unit Hydrograph Models

• Five IUH Models– Gamma– Rayleigh– Weibull – NRCS (DUH as an IUH)– Commons

Gamma-family

• Gamma, Rayleigh, and Weibull are all generalized gamma-distributions. The IUH model equation is

• Gamma when p=1; Rayleigh when p=2

)exp()( 1 p

pNp

pNp

p

p

t

t

t

t

t

tp

A

tq

NRCS DUH

• NRCS DUH as an IUH. Using a Gamma-type functional representation is

pt

t

pp

et

t

Aq

tq 88.381.3)(5387.37

)(

NRCS Curve-Fitting Using Gamma function

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

t/Tp

q/qp

NRCS-Tabulation Equation

Commons Model

• Commons’ Hydrograph– Empirically derived for large watersheds

Figure 1. Hydrograph developed by trial to cover a typical flood. from: Commons, G. G., 1942. “Flood hydrographs,” Civil Engineering, 12(10), pp

571-572.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Units of Time

Uni

ts o

f Flo

w

Approximation Original

p

p

p

t

t

p

t

t

p

t

t

p

et

t

et

t

et

t

A

tq

641.5132.0

694.2965.0

707.4176.0

)641.5

()288.0(

88.3

)694.2

()925.0(

58.7

)707.4

()118.0(

001.77)(

Analyze Each Storm

• Supply observed precipitation data to the hydrograph function. – Convolution of sequence of the IUH models to create a

DRH.

– Compare observed runoff with DRH, adjust parameters in IUH to minimize some error function.

Analyze Each Storm

Two different merit functions considered were the sum of squared errors (SSE) and a maximum absolute deviation at peak discharge (QpMAD).  

 

NOBS

iios QQSSE

1

2)(

)()( peakopeaksp tQtQMADQ

Typical Result

0

1

2

3

4

5

6

7

100 600 1100 1600 2100

Time (minutes)

Cu

mu

lati

ve D

epth

(in

ches

)

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

Rat

e (i

nch

es/m

in)

ACC_PRECIP(IN) ACC_RUNOFF(IN)

ACC_MOD_RUNOFF(IN/MIN) RATE_PRECIP(IN/MIN)

RATE_RUNOFF(IN/MIN) RATE_MODEL(IN/MIN)

Figure 7.4 Plot of Observed and Model Runoff, Ash Creek, June 3, 1973 storm using the Weibull IUH model.

Choosing a Model

• Establish acceptance criteria:– Averages

• Bias

• Fractional Bias

• Fractional Variance

• Normalized Mean Square Error

– Peak• Peak Relative Error:• Peak Temporal Bias:

PmPo ttTB

PoPmPo QQQQB /

N

iimiomo QQ

NQQBias

1,,

1

mo

mo

QQ

QQFB 2

qmqo

qmqoFV

22

22

2

mo

mo

QQ

QQNMSE

2

Acceptance Analysis

• All models except Commons’ were similar in performance as measured by the acceptance analysis.– Roughly 60% of the storms could be fit to

within 30 minutes of the peak and within 25% of the peak discharge with the four Gamma-family models.

• Preference is for the Rayleigh model, followed by Gamma – fewer degrees of freedom and they can be expressed in NRCS-type structires (i.e. Qp,Tp)

Synthesis

• Evaluate methods to synthesize hydrographs in absence of data.

• Fundamental assumption: Watershed characteristics (slope, length, etc.) are predictors of hydrologic response and thus are predictors of IUH parameter values, and that there exists a UH.

Synthesis

• Determine watershed characteristics– Area, perimeter, slopes, lengths, etc.

• Relate regression models to IUH parameters to selected watershed characteristics.– Use regression model to determine parameter values by

station.– Run each storm using these values.– Test results for acceptability– Interpret results

• Make Conclusions and Recommendations

Watershed Characteristics

• These are measurements that can be made from a map, air photo, or possibly field visit.– Area, slope, etc.– Manual determination (University of Houston,

checked and corrected by Lamar)– Automated determination (USGS)

Correlation Model

064.0102.0

500..0334.0

206..0109.0

43.2

)(138

137.0

SPN

SP

At

SACr

The resulting correlation equations for QpMAX criterion are: 

[Eqn 3] 

where A is watershed area in square miles, P is perimeter in feet, Sis slope (as a decimal).

 N and Cr are dimensionless, the dimension on the residence time is minutes.

Correlation Model

1

10

100

1000

0.1 1 10 100 1000

Area (square miles)

t_ba

r (m

inut

es)

Model

Observed

Correlation Model

1

10

100

1000

0.001 0.01 0.1

Slope

t_ba

r (m

inut

es)

Model

Observed

Correlation Model

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 100 1000

Area (square miles)

Run

off_

Coe

ffici

ent

Model

Observed

Correlation Model

0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1

Slope

Run

off_

Coe

ffici

ent

Model

Observed

Correlation Model

0

1

2

3

4

5

6

7

8

9

10

0.1 1 10 100 1000

Area (square miles)

Res

ervo

ir N

umbe

r (N

)

Model

Observed

Correlation Model

0

1

2

3

4

5

6

7

8

9

10

10000 100000 1000000

Perimeter (feet)

Res

ervo

ir N

umbe

r (N

)

Model

Observed

Test Case

• Some stations left out of correlation model.– Determine watershed characteristics on these

stations.– Apply correlation equation.– Generate runoff hydrographs and compare to

observed hydrographs.

• Reasonable results

• Terrible results

Remaining Work

• Synthesis– Interpret results in raw form and transform into

conventional Qp,Tp,Tc format. (in-progress).– Test using Bryan storms.– Write research report. (in-progress)

Remaining Work

• Incorporate with NRCS methods to synthesize Unitgraphs

• Directory structure for HEC-HMS is prepared (analysis pending).

• Write research report with methodology and guidelines for use (Report started, quite empty).

Loose Ends

• Rainfall loss model (in-progress). Initial abstraction/constant loss. Saturated K as lower limit of loss rate? Good upper limit?

• Watershed subdivision – part of HEC-HMS study?