1. 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

2. 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.

3. Objectives
Is the NRCS Dimensionless unitgraph representative of Texas hydrology?
If not, then can an alternative method be developed?

4. 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)

5. Instantaneous Unit Hydrographs
The instantaneous unit hydrograph is a transfer-function 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 time-of-concentration (classical concepts).

6. IUH Concept Outlet Unit Depth at Time < 0; 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
Unit Depth
Cumulative Discharge
Time
Discharge (Rate)
Remaining Depth
Graphics illustrate the IUH concept.
The upper left figure depicts a watershed prior to the uniform addition of one unit of excess precipitation depth. The watershed area is “A”.
(2) At Time zero, the IUH theory assumes that a unit depth of excess precipitation is uniformly distributed over the watershed as depicted in the center figure. The total volume of accumulated precipitation is the product of the unit depth and watershed area. This volume is depicted in the hyetograph/hydrograph chart as the light orange “unit depth” and the solid orange horizontal line.
(3) As time proceeds, the watershed drains and the remaining cumulative depth decreases, while the cumulative discharge (runoff) increases. This relationship is depicted on the hydrograph as the blue arrow and red arrow, showing cumulative remaining precipitation (orange) and cumulative discharge (red). The rightmost “watershed” depicts the remaining depth and the discharge using the same color scheme.
(4) The hydrograph chart shows the cumulative discharge as the solid red line, and the rate (volume/time) as the dashed line. The rate is the derivative of the cumulative line.
(5) The IUH is the function that produces the dashed red line in response to the single unit depth input over an infinitesmal unit of time.
(6) The goal of this component of the research is to find suitable functional forms for the IUH function(s) and the values of the parameters of those functions.

7. 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:
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
Orignially we used a cascade of linear reservoirs as a conceptual model. It produced the Nash model that is used in a lot of earlier work. One problem was the need to include a delay-time that could not be easily explained. More research, and the observation that a combination of a pure translation model (discharge is the effective precip translated by elapsed time) and linear reservioirs (discharge proportional to residual accumulated precip) produced a weibull distribution, which is the equation in the figure.
Currently we are still using integral values for N, but prior workers have demonstrated that it need not be integer. Departure from integer is harder to explain in terms of a series of linked reservoirs, but can be developed from statistical-mechanical methods. We think the additional complexity is probably not necessary.
This discharge function is the time series of discharge over the watershed of area A, from a unit depth zo applied instantly at time 0. The DRH is produced by breaking the rainfall time series into a sequence of zo’s each with different value, each at different starting time, use the equation for each impulse, and sum the results at a particular elapsed time from the beginning of the precip event.

8. 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.
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
Before we analyzed the data, we took the raw data and prepared it for analysis.
First baseflow separation using the constant baseflow method was separated from runoff series where there was baseflow present in the data.
The precipitation was multiplied by a fractional value so that the total depth of precip and the total depth of runoff (total runoff/watershed area) were equal. The precip remaining is the effective precip. This method is called the constant fraction rainfall loss model.
Lastly the prepared data were linearly interpolated onto one-minute time intervals, then finite-differencing to convert cumulative values of precipitation into instantaneous values (depth/minute) and convert cumulative runoff (where supplied) into instantaneous values. The choice of one-minute time steps was arbritrary, although it is the smallest time increment that the original database could represent It greatly simplifies differencing calculations as the denominator is a constant.
Once the data are prepared, values of t_bar, p and N are guessed, and the precip is convolved using the IUH and the response compared to the observed response. The values are changed systematically to make the root mean squared error between the convolved and observed responses as small as possible.
A grid searching technique was used because it is east to program and control. The compromise is that a large number of trials must be endured to get meaningful results. Because the goal was IUH values for further analysis, it was decided that a sophisticated optimization tool for parameter estimation was not justified.

9. Instantaneous Unit Hydrographs
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%)
This figure is a graphical display of typical results of AUTOMATED analysis of storms.
The yellow curve is the precipitation input prepared as just described.
The red curve is the runoff prepared as just described.
The black line is the IUH model for direct runoff for this station and this storm.
Ideally the solid black and solid red should exactly overlay one another, but because of noise in the data, the assumption of linearity, etc, this is not the case in any of the storms. The result displayed here is one of the better cases.
For clarity the cumulative curves are not displayed, but would be a series of “S” shaped curves with plateaus between the peaks on the figure.
The IUH parameters are displayed at the top of the graph with the following results:
N=5 (fair amount of storage-delay)
P=1.33 (relatively fast decay after precip ends)
T_bar = (This unit is minutes when p=1)

10. 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.
The previous result is relatively useless unless we can regionalize the analysis for watersheds where data are not available, unless the IUH is to be used for other purposes (reservoir operations, etc. – not the intent of the present work).
As first steps to regionalization have examined two approaches. The first, is to simply construct aggregate models by some averaging technique. Called Type-I in this presentation. No relation to statistical error.
The promise of an aggregate IUH is that we might find a single equation that characterizes all behavior for a station (plausible)(nice example presented), module (maybe), entire database (unlikely).
The station specific values would produce 80+ such equations, each specific for a gaging station; Module specific is five, and entire database is a single equation.
Hypothesis testing determined that the IUH parameters were statistically different between modules, that suggests a single aggregate IUH is unlikely for Texas.
Within a module station to station differences were significant, but the p-value at rejection was relatively small (nearly “accept” the null hypothesis that values are not different). Within a station, hypothesis testing did not make sense.
The second approach is called Type –II where we used regression analysis to relate watershed characteristics to IUH values in an attempt to develop a unified approach that could be used based on data easily determined from maps or air photos. In this case we will have a regression equation that predicts values of the IUH parameters based on watershed area, slope, shape, perimeter, etc. This approach is useful because it can also study the influence of these characteristics on the parameter. For example, slope does not have much effect on the reservoir number N, and thus its regression weight is quite small.
We used a power law model, so the weights are really exponents. Values close to zero mean that the particular predictor variable does not contribute much to the predicted variable.

11. Instantaneous Unit Hydrographs
Example for Type I – Station
This figure is an example of a station specific aggregate unit hydrograph applied to the rainfall sequence presented as an example of typical results. In this case, and many others in the database, the station specific IUH does a nice job of predicting the DRH. The IUH is derived from all storms at the station, these values are then used as the IUH for any storm at the station.
This result is promising for GAGED watersheds, and might have application in near real-time flood forecasting (which is not the scope of the current research), but it is not especially useful for design in ungaged areas, unless they are arguably near the gaged watershed, and have similar characteristics (the idea behind Type II regression).

12. Instantaneous Unit Hydrographs
Type-II Power Law Model
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.
The next exploration is to incorporate watershed features to infer IUH parameters from measured watershed properties. We selected what we considered relatively simple measurements:
Area
Aspect ratio – defined in this study as the rectangle that encloses the watershed. The ratio is the NS distance over the EW distance.
Raw Slope is the straight-line distance from the outlet to the most remote point of the watershed (determined using calipers) divided by the change in elevation in the watershed (higest elevation minus the outlet elevation)
Area/Perimeter –Thought to convey some shape information; Result is a “characteristic length” that might be useful in slope type relations.
Shape ratio – same as Wu’s 1963 paper. It is a ratio of measured perimeters to perimeter of a circle with same area as the watershed.
Stream slope is the ratio of stream length (as opposed to straight-line distance) to elevation change along the stream.
In the initial results presented here, some of the exponents are forced to zero, so the particular varaible does not contribute to the estimate of the IUH parameter.

13. Instantaneous Unit Hydrographs
Example for Type II – Dallas Module Data
This figure is the result of regression analysis to construct a power law model that produces values for t_bar, N, p based on watershed area, slopes, etc.
These values are then used in an IUH to model response to a specific storm at the station. Timing is early, but matching of peaks magnitues is better than expected. This analysis is still-in progress.

14. Instantaneous Unit Hydrographs
Status:
Deconvolution of 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)
I forgot to note that the IUH are a dimensional approach – If we find a way to get the IUH values independently of rainfall-runoff analysis then the analyst can run any rainfall sequence through the watershed and get an idea of the peak discharge magnitude and timing without having to estimate Tp, Tc, and Qp in the NRCS fashion.
The concept here is fundamentally the same as NRCS, just the information is buried into different parameters.

15. 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.

16. Unit Hydrograph Derivation by Linear Programming
The constraints can be written (n=1,…,N)
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.

17. 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.

19. Gamma Unit Hydrograph

21. Gamma Unit Hydrograph

22. Gamma Unit Hydrograph