Environmental Institute of Houston • 2003 Annual Report • 10-15

Management and Monitoring

 

Determination of Instantaneous Unit Hydrographs for
Houston from HCOEM and USGS OFR 96-250 Data

Abstract--An instantaneous unit hydrograph (IUH) is a mathematical function that relates rainfall to runoff. IUH analysis was the subject of intensive research in the 1960s and 1970s, but has been relatively ignored in the last 20 years because of difficulties in relating measurable characteristics of watersheds to IUH functional properties.

A re-examination of the technology in cooperation with the Texas Department of Transportation suggested that the technology has merit and predictive calculations using near-real time updating may allow a reasonable advance in the art of early-warning in a probabilistic sense.

This research reformatted existing USGS Harris County data for IUH analysis, as well as HCOEM’s (some stations are duplicates) historical records of rainfall-runoff at their gaging stations and includes these events into a Harris County database. Still on-going is the analysis to identify IUH parameters and test these against a censored set of real Harris County data to assess the ability of the approach to predict imminent flooding.

A longer term goal of the research is to generate probabilistic inundation forecasts using NWS rainfall forecasts and near-real time updating from the NWS Doppler network. This report examines the first component—initial reformatting of the data.

FLOODING IS A NATURAL DISASTER THAT IS OFTEN ignored as an environmental problem. However, flooding endangers human lives and public health, impacts essential services, and can transport pathogens over a wide area. It is largely impossible to prevent flooding, but it is possible to get out of the way and to estimate the magnitude of flows in the near future from current and past rainfall data.

In principle the physics of the rainfall-runoff process could be encoded for a region and the rainfall field input into a computer program and the near-term runoff and flow depth predicted using methods of computational fluid dynamics, however the computational resources needed to accomplish this task on the scale of Harris county don’t yet exist. An accepted alternative is to use hydrologic methods (essentially highly simplified physics, often linearized) to make such predictions. Exact flow values and depths are not needed for design and early warning, thus these methods are extremely attractive because of computational simplicity. This research is applying methods developed from existing research for small watersheds in central Texas to the Harris County region and evaluate the utility of the approach for imminent flood forecasting from current and past rainfall data.

Background
 A hydrograph is a time-series of either water-surface elevation or instantaneous discharge, taken at a particular point on a stream. The point on the stream represents the outlet from a watershed, or the area topographically above that point from which all drainage passes through that point. The hydrograph represents the integrated response of the watershed to all hydrometeorologic processes extant on the watershed. Hydrologists and engineers use hydrographs to analyze the characteristics of a watershed and for design of engineering projects.

The instantaneous unit hydrograph (IUH) is a special case of a unit hydrograph where the precipitation duration is infinitesimally small (essentially an impulse). The functional form of an IUH is varied, but it has two important properties: the function must integrate to unity over the range zero to infinity and exhibit linearity with respect to the input depth.1 The second requirement allows calculation of the response to a continuous precipitation signal by convolution of the instantaneous responses. Most probability density functions have the requisite properties, and the IUH can be interpreted as a residence time distribution of one unit of excess precipitation on the watershed.

These IUH functions are analogous to impulse-response approaches used in heat flow modeling, well functions, signal processing, potential flow modeling, chemical reactor modeling, and other areas of engineering where linear-systems theory produces usable predictions.1 The attractiveness of an IUH is that the dependence of the hydrograph on the storm duration is decoupled from the analysis. This decoupling lets the analyst apply a storm of any duration over the watershed and calculate the direct runoff hydrograph.

The determination of an IUH for a particular storm event, where historical data exists (analysis) involves the following steps:

  1. Baseflow separation.

  2. Rainfall loss model to extract excess precipitation.

  3. Conversion to consistent time and length units.

  4. Interpolation of precipitation and runoff onto continuous functions.

  5. Selection of candidate IUH functions.

  6. Deconvolution of the observed signals to determine the IUH function parameters.

  7. Aggregation of IUH parameters.

  8. Non-dimensional representation and regionalization.

The determination of an IUH in the absence of rainfall and/or runoff data is called synthesis and is fundamental problem in hydrologic research. Most current approaches are to use correlations and regressions from large sets of observed data and estimate IUH parameters from functional relationships of measurable properties (areas, lengths, slopes, shapes, soil classifications, etc.), the NRCS unit hydrograph methods are an example of this approach as is our current work.

The other approach, which is still in its infancy, is to use a physics-based hydraulic/hydrologic model to generate an equivalent unit hydrograph and then fit this hydrograph to some distribution. Generally this latter approach would represent unnecessary effort (if we can model the physics, why bother with a probability density function?) but these watershed models are extremely complicated and the computation of the response for a real rainfall time series can take hours, thus a scenario approach (run the physics model to generate candidate IUH functions and then use these functions for operational decisions) is still quite attractive. In either case, the synthesis of hydrographs in the absence of data is quite challenging.

The selection of candidate IUH functions that convert the rainfall signal into a runoff signal is a challenge. After considerable literature review effort we examined the classical Nash model, and a related model based on the Weibull probability distribution.2-4 Other approaches based on digital signal processing (recursive filters, cross correlation, empirical orthogonal functions, etc.) were not examined in any detail, although cross correlation analysis appears to be a promising approach for gauged watersheds.

The Nash model can be demonstrated to be a special case of the Weibull distribution, so the results presented here are for the Weibull model.

     Equation 1

(1)

Equation 1 is the IUH discharge function used in this study. The model contains three parameters that are estimated for each watershed, a timing parameter t_bar that locates the center of the distribution, an exponent on the time, p that impacts the rate of decay (drainage) of the hydrograph, and a reservoir number, N, that impacts both the lag time (from end of precipitation to beginning of discharge) and the width of the hydrograph. In the Nash model the value of p is 1. The watershed area is A and the excess precipitation rate is z0 (actually a depth over a short time interval). The output of this function is the discharge from the watershed at any time t after the initial charge of excess precipitation is applied at time t = 0.

De-convolution of the observed signals to determine the parameters of the IUH model that best fit observations is a major and on-going computational effort. The technique to fit the distribution is a simulation-optimization method based on grid search. A set of ordered values (t_bari,pi,Ni) for the undetermined parameters of the IUH is selected from a range of values for each element of the ordered triple as in Eq. 2. The ranges of values are divided into thirty uniform increments, and the exhaustive combination of these increments defines the total set of candidate values.

    

(2)

The actual precipitation signal is convolved through the model to produce a DRH. This DRH is compared with the actual DRH using a square error criterion as in Eq. 3. The subscripts s and o represent the simulated (model) and the observed values, respectively. The index, i, is the particular value at each time in the series.

    

(3)

The value SSE is calculated and saved, then another element of the set is selected and the process is repeated. The set that produces the lowest value of the error criterion is saved as the “non-inferior” set of parameters. The process is repeated several times with smaller and smaller intervals to identify a good non-inferior solution. The grid search for each storm takes less than an hour, but with thousands of storms to analyze the time to search the entire grid (set of sets) takes a couple of days. Despite the lack of elegance, the process is simple to automate, and always produces a result. The final model DRH is plotted against the original DRH data to be sure the results are not absurd.

Plot of a typical de-convolution analysis result.

Figure 1. Plot of a typical de-convolution analysis result. This particular result is the storm-optimal result for station 08057320, for the storm that began on June 3, 1973. To date, 1600 such storms at 90 locations have been analyzed in this fashion.

Figure 1 is a plot of a typical result of the grid search and the visual check of the storm-specific hydrograph results. This figure represents a comparatively good result with a complex (multiple burst storm). Not all the resulting hydrographs are of this quality, but a majority of the results are similar to this result.

These IUH parameters represent a storm-specific unit hydrograph for a given watershed. Each watershed experienced a number of recorded storms, some as few as two, many with at least 30. The parameters for all storms in a watershed are aggregated using the median value, and confidence intervals on this median are computed using order statistics as a way to quantify uncertainty. This watershed aggregate value is called the station median IUH. This quantification of uncertainty is in-progress and the results reported here ignore uncertainty.

EIH Research
 Houston area precipitation and runoff data up to the mid 1990s is contained in Open File Report 96-250. The data is stored electronically but needs significant manual adjustment to fit the structure required for automated IUH analysis. One main task of the EIH supported work is to complete this reformatting.

There is also a significant database of Houston area rainfall-runoff data maintained on the Harris County OEM Web site. Figure 2 illustrates the coverage—most of the gages meet the criterion of “small watershed” as used in an on-going central Texas study and thus the techniques already developed could be applied to the Houston area.

The HCOEM data is stored electronically in a fashion that should be readily transformed into the database structure of the central Texas study. Figure 3 illustrates the HCOEM structure. The second task to the EIH supported work is to download this data and reformat it into the central Texas structure.

Reformatting
 During the project period both datasets mentioned above were reformatted. Currently the data are being assembled into an ASCII database to be publicly served on a University server so anyone can use these data for unit hydrograph (or other work where it is convenient to have a precipitation series and KNOWN discharge from that precipitation series already arranged). As of October 31, 2003 the data are not posted to the server, but this component is to be completed by the end of 2003. The server address is <http://129.7.204.231>.

Analysis of OFR and HCOEM Data
 The IUH model for central Texas is a three-parameter Weibull-type model. It was constructed assuming a cascade of reservoirs. It is essentially identical to the generalized Gamma models of all prior researchers except we have identified a strong correlation between watershed area and aspect ratio and the timing parameter—making the model promising for synthesis of rainfall-runoff in ungauged areas. More importantly, the convolution of the rainfall process can be handled extremely rapidly by use of a finite difference analog to the actual process (a departure from past methods which convolve the actual kernel function). The extremely rapid convolution procedure makes to prospect of near-real time forecasting realistic.

An illustration of the single station above is displayed on Fig. 8. The plot shows the observed results and a model result. The line scheme is the same as in the hydrographs in the introduction. The three-parameter model qualitatively is a reasonable prediction of the observed behavior, with the observation that the model time-to-drain is less than in the observed data, but the peak flow rates and overall shape is well described by this simple hydrograph model. It is especially important to observe that the precipitation signal is quite complex in that there are a series of bursts separated by short intervals of zero precipitation, classical unit graph analysis would likely replace the actual signal with a two or three pulse model to simplify the analysis—here we use the actual input series.

The IUH displayed on the plot is

    

(4)

where the runoff coefficient is 0.334 for the particular watershed (determined in the rainfall loss analysis mentioned in the introduction). Analysis of the remaining storms is in-progress, but the figure illustrates the expected results.

Future Efforts
 The EIH portion of this work has produced the following products:

  1. A Harris County Database in the same structure as the central Texas database. It is to be delivered by a University server by the end of 2003. The database contains both OFR and HCOEM data. Duplicates are not to be presented twice (many HCOEM stations are USGS stations and duplicate data occurs when the two datasets are merged).

  2. An IUH analysis procedure of the Harris County Data with an aim to produce a predictive tool for short-term imminent flood forecasting from current and past rainfall signals.

Future efforts are to:

  1. Perform a complete IUH analysis for the Harris County Dataset.

  2. Test the IUH functions using censored data (remove portions before fitting and test against the censored portions).

  3. Perform a geographic analysis to determine how to project gage data from one location to a downstream receiving location (this task is necessary for effective warning forecasting as well as probabilistic forecasting)

  4. Develop a prototype alert model using these IUH and test both the predictive capability and the forecast failure rates.

References
1J. C. I. Dooge.“Linear Theory of Hydrologic Systems,” U.S. Dept. of Agriculture, Technical Bulletin 1468. 1973.
2J. E. Nash.“The Form of the Instantaneous Unit Hydrograph,” Intl. Assoc. Sci. Hydrology, Pub 42, Cont. Rend. 3 (1958): 114-18.
3J. H. Lienhard and P. L. Meyer. “A Physical Basis for the Generalized Gamma Distribution,” Quarterly of Applied Mathematics 25.3 (1967): 330-34.
4J. H. Leinhard. “Prediction of the Dimensionless Unit Hydrograph,” Nordic Hydrology 3 (1972): 107-09.

Bibliography
Dooge, J. C. I. “A General Theory of the Unit Hydrograph,” J. Geophysical Research 64.2 (1959): 241-56.
Dooge, J. C. I. and M. Bruen. “Unit Hydrograph Stability and Linear Algebra,” J. Hydrology 111 (1989): 377-90.
Nash, J. E. “Systematic Determination of Unit Hydrograph Parameters,” J. Geophysical Research 64.1 (1959): 111-15.
Soil Conservation Service (now NRCS). National Engineering Handbook. Washington D.C.: U.S.D.A., 1972. Chapter 4.

Presentations
 Cleveland, T. G. and M. Smith. “Demonstration of Remote Wireless Access to a Database for Communicating Water Quality Data,” Final Report, Houston Department of Health and Human Services, Environmental Health Division, 2003. <http://cleveland1.cive.uh.edu/Publications/dhhs_report.pdf>
Cleveland, T. G., D. Thompson, and X. Fang. “Instantaneous Unit Hydrographs for Central Texas,” Proceedings, Texas Section Spring Meeting, Dates, 2003, Corpus Christi, TX <http://cleveland1.cive.uh.edu/Publications/asce_tx_2003.pdf>

Funding and proposals
 “Regional Characteristics of Unit Hydrographs.” Texas Department of Transportation, $19,995.
“Regional Characteristics of Storm Hyetographs.” Texas Department of Transportation, $34,758.
“Estimating Time Parameters of Direct Runoff and Unit Hydrographs for Texas Watersheds.” Texas Department of Transportation, $34,198. (EIH shown as 15% credit.)
“Guidance for Design in Areas of Extreme Bed Mobility.” Texas Department of Transportation, $43,950.

Theodore Cleveland, Ph.D., P.E., is a professor of civil
and environmental engineering. He can be reached at
cleveland@uh.edu. Ionan Lazarescu and Xin He are
graduate students in the Department of Civil and
Environmental Engineering, University of Houston.

Copyright © 2004 Environmental Institute of Houston - 2003 Annual Report