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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
QQFB 2
qmqo
qmqoFV
22
22
2
mo
mo
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
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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?