Project 0-4193
Regional Characteristics of Unit
Hydrographs
David Thompson, Texas Tech, RS
Rudy Herrmann, TxDOT, PD
William Asquith, USGS, Co-PI
Xing Fang, Lamar, Co-PI
Ted Cleveland, UH, Co-PI
Unit Hydrograph
• A unit hydrograph is the hydrograph of
runoff that results from a unit pulse of
effective precipitation (runoff) distributed
evenly over a watershed over a specific
duration of time.
• Unit hydrograph theory assumes that
watershed dynamics behave in a linear
fashion.
Objectives
• Is the NRCS Dimensionless unitgraph
representative of Texas hydrology?
• If not, then can an alternative method be
developed?
Tasks
• Literature review (complete)
• Assembly of Database (complete)
• Development of unit hydrographs from
database (in progress)
• Comparison with NRCS unit hydrograph (in
progress)
• Regionalization of computed unit
hydrographs (in progress)
Instantaneous Unit Hydrographs
• The instantaneous unit hydrograph is a transferfunction that relates a unit depth of excess
precipitation applied over a short interval to a
runoff hydrograph at a watershed outlet.
• Its principal advantage is elimination of storm
duration in computing direct runoff hydrographs.
• It can also be used to infer time-to-peak and timeof-concentration (classical concepts).
IUH Concept
Outlet
Unit Depth at Time < 0;
Discharge at Time < 0
Unit Depth at Time = 0;
Discharge at Time = 0
Remaining Depth at Time > 0;
Discharge at Time > 0
Remaining Depth
Cumulative Discharge
Unit Depth
Discharge (Rate)
Time
Instantaneous Unit Hydrographs
• Modeled the conversion of precipitation to runoff
as a simplified hydraulic process that is a
combination of a translation hydrograph, and a
series (cascade) of linear storage elements.
• Resulting IUH function is:
 t p 1 

(t p ) N 1
tp

 exp( 
q(t )  A [ pz0  
)]
N 1 
t _ bar
 t _ bar  ( N  1)!(t _ bar ) 
A=watershed area, zo = input impulse depth : Watershed and precip. data
p=exponent (decay rate), N=res. number (shape,delay), t_bar = timing
parameter (location of peak) : IUH parameters that are estimated
IUH Concept
Outlet
Unit Depth at Time < 0;
Discharge at Time < 0
Unit Depth at Time = 0;
Discharge at Time = 0
Remaining Depth at Time > 0;
Discharge at Time > 0
Remaining Depth
Cumulative Discharge
Unit Depths
Discharge (Rate)
Time
Instantaneous Unit Hydrographs
• Preparation of precipitation and runoff data.
– Separate base flow; remainder is direct runoff
hydrograph (DRH).
– Use a proportional rainfall loss model to ensure
precipitation volume equals direct runoff volume. The
fraction is called the runoff coefficient.
– Recast actual data onto one-minute time intervals
(approximate the impulses with short finite-time
behavior) using linear interpolation and numerical
differencing.
Instantaneous Unit Hydrographs
• Analysis to infer IUH parameters.
– Convolve the one-minute rainfall impulses using the
IUH function, adjust p, N, t_bar to minimize the RMS
error in the model and observed DRH. (Note; A is
fixed by the actual watershed characteristics, zo is the
impulse as approximated by the one-minute derivative
data.
Instantaneous Unit Hydrographs
File : #IUH_1_sta08057320_1973_0603.dat
t he
#SUMERR= .100000+100 #TMEAN=67.5#P_EXP=1.33 #NRES= 5
4.50E-02
Accum. Depth (inches)
4.00E-02
3.50E-02
3.00E-02
2.50E-02
2.00E-02
1.50E-02
1.00E-02
5.00E-03
0.00E+00
0
500
1000
1500
2000
2500
Time (minutes)
#RATE_PRECIP
#RATE_RUNOFF
#RATE_MODEL
• Typical result: The quality of the “fit” varies as measured by the
relative error at the peak from 2% to 40% in some cases, in nearly all
cases the peak rate is underestimated. (Error at largest peak in above
example is 6.5%)
Instantaneous Unit Hydrographs
• Regionalization:
– Type – I Aggregate Models (Ignore Watershed Character).
• Station Data – Use median values of t_bar,p,N for IUH from all
storms at a station.
• Module Data - Use median values of t_bar,p,N for IUH from all
storms in a module.
• All Data – Use median values of t_bar,p,N for IUH from all storms.
– Type –II Regression Models using Watershed Characteristics
• All Data – Use power-law model values of t_bar,p,N for IUH from all
storms.
• Module Data - Use power-law model of t_bar,p,N for IUH from all
storms in a module.
• Station Data – meaningless in this context, nothing to regress.
Instantaneous Unit Hydrographs
File : #IUH_1_sta08057320_1973_0603.dat
t he
#SUMERR= .100000+100 #TMEAN=141.9#P_EXP=1.55 #NRES= 4
4.50E-02
Accum. Depth (inches)
4.00E-02
3.50E-02
3.00E-02
2.50E-02
2.00E-02
1.50E-02
1.00E-02
5.00E-03
0.00E+00
0
500
1000
1500
2000
Time (minutes)
#RATE_PRECIP
#RATE_RUNOFF
#RATE_MODEL
• Example for Type I – Station 08057320
2500
Instantaneous Unit Hydrographs
•
Type-II Power Law Model
yi  wo x1,i 1 x2,i 2 x3,i 3 x4,i 4 x5,i 5 x6,i
w
w
w
x1 
x2 
y1  t _ bar
x3 
y2 
p
x4 
y3  N _ res
x5 
x6 
•
•
w
w
w6
Area
Aspect _ Ratio
Raw _ Slope
Area / Perimeter
ShapeRatio
Stream _ Slope
Weights determined by minimization of RMS error between “observed” IUH
parameters and the power law model.
Predict values of IUH model (t_bar,p,N) from watershed characteristics, then
use resulting IUH.
Instantaneous Unit Hydrographs
File : #IUH_1_sta08057320_1973_0603.dat
t he
#SUMERR= .100000+100 #TMEAN=40#P_EXP=1.53 #NRES= 4
4.50E-02
Accum. Depth (inches)
4.00E-02
3.50E-02
3.00E-02
2.50E-02
2.00E-02
1.50E-02
1.00E-02
5.00E-03
0.00E+00
0
500
1000
1500
2000
Time (minutes)
#RATE_PRECIP
#RATE_RUNOFF
#RATE_MODEL
• Example for Type II – Dallas Module Data
2500
Instantaneous Unit Hydrographs
• Status:
–
–
–
–
Deconvolution of 1600+ storms (essentially complete).
Station Aggregate IUHs (complete).
Watershed properties database (essentially complete)
Regionalization methods to correlate watershed
properties to IUH values in-progress.
– Non-dimensionalization, calculation of traditional Tp,
Tc values (pending).
– Compare to NRCS DUH (started earlier in 2002, idle
until regionalization analysis is firmed)
Unit Hydrograph Derivation by Linear
Programming
• Linear programming is an alternative of deriving
unit hydrograph [U] that minimizes the absolute
value of the error between observed and estimated
DRHs -- [Q] and [Q*], and also ensure all entries
of [U] are nonnegative.
• The general linear programming model is stated in
the form of a linear objective function to be
optimized (maximized or minimized) subject to
linear constraint equations.
Unit Hydrograph Derivation by Linear
Programming
• The constraints can be written (n=1,…,N)
[Q ]  [ n ]  [  n ]  [Q] or,
*
n
PnU1  Pn1U 2  ...  Pn M 1U M   n   n  Qn
Where n is a positive deviation, and n is a
negative deviation of error n.
• Fortran Programs have been developed to
derive UH by linear programming for four
different objective functions and for single or
multiple events. Two-parameter Gamma UH is
fitted to UH derived by linear programming.
Error Analysis to Search Optimum UH
• UH derived are applied to predict direct runoff
hydrograph (DRH) for all events in the
watersheds, and more than 12 error parameters
are developed to evaluate errors between
observed and predicted DRH. The key one
used is to minimize the deviation between the
peak values and the time of peaks for observed
and regenerated hydrographs.
 Q
 Q pc 
po

Z  


Q po


2
2 1 / 2
 Tpo  Tpc 
 
 

 
T
po

 
Gamma Unit Hydrograph
Gamma Unit Hydrograph
Gamma Unit Hydrograph