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\begin{center}
\noindent {\Large Use of the Rational Method and Modified Rational Methods for TxDOT Hydraulic Design: Development of Guidance for Runoff Coefficient Selection and Examination of Temporal and Rainfall Depth Sensitivity}

\vspace*{0.25in}
Theodore G.\ Cleveland, University of Houston, Houston, Texas

William H.\ Asquith, U.S.~Geological Survey, Austin, Texas

Xing Fang, Lamar University, Beaumont, Texas

David B.\ Thompson, R.O.~Anderson Engineering, Inc., Minden Nevada

\vspace*{0.5in}
Original 4 February 2007
\end{center}
\vspace*{0.5in}

\begin{abstract}
The rational methods is a tool hydraulic engineers use to estimate design discharge for sizing a variety of drainage structures. The method are relatively simple: The peak discharge $Q_p$ is equal to the product of the drainage area $A$, the rainfall intensity $I$, and a runoff coefficient $c$. The last two terms, $I$ and $c$, are dependent on analyst estimates of time-of-concentration and watershed conditions. Furthermore there is evidence that the runoff coefficient, $c$, is dependent on rainfall depth, thus the two terms are correlated and the simple model becomes quite non-linear.

The modified rational method is an extension of the rational method used to generate a runoff hydrograph for applications where the peak discharge is not sufficient to execute a design. The modified rational method uses the peak discharge produced by application of the rational method. A hydrograph is created by using the time of concentration for the time to peak discharge and using twice the time of concentration for the time base of runoff.

The purpose of this project is to evaluate appropriate conditions for use of the rational and modified-rational methods for design on small watersheds, evaluate and refine, if necessary, current tabulated values of the runoff coefficient, and construct guidelines for TxDOT analysts for the selection of appropriate parameter values for Texas conditions.
\end{abstract}

\section{Background and Significance of Work\label{sec:Background}}
Texas Department of Transportation (TxDOT) analysts design roadways and proximal infrastructure. Each year, billions of dollars are spent on new construction\footnote{As described by George ``Rudy'' Herrmann, TxDOT, on March 10, 2006, approximately 40~percent of construction dollars are spent on drainage-related facilities.}. The infrastructure must accomodate storm-water drainage and conveyance.  

The rational method \citep{kuichling89} is a tool for estimating peak (maximum) discharge from relatively small drainage areas. General TxDOT guidance is that the rational method should be applied to watersheds with drainage areas of 200~acres or smaller\footnote{Hydraulic Design Manual, pg. 5--29, TxDOT 3/2004, located electronically at \url{http://manuals.dot.state.tx.us/dynaweb/colbridg/hyd} at the time of this writing.}. The rational method relates peak discharge to contributing drainage area, average rainfall intensity for a duration equal to a watershed response time (typically the time of concentration), and a coefficient  that represents hydrologic abstractions and hydrograph attenuation. The coefficient is generally termed the \emph{runoff coefficient\/} and has a range from 0 (no runoff) to 1 (complete runoff). The modified rational method extends the idea to parameterize simple runoff hydrographs (typically triangular), where the peak of the hydrograph is the peak discharge estimated by application of the rational method, and the time base of the hydrograph (start-to-finish) is twice the time of concentration. Equation~\ref{eqn:classical_rational_method} is representative of the classical rational equation.

\begin{equation}
Q_p = ciA
\label{eqn:classical_rational_method}
\end{equation}

In equation~\ref{eqn:classical_rational_method}, $c$ is the runoff coefficient (dimensionless if all other terms are in consistent length and time units) that relates the ratio of input volume rate $i A$ to the output volume rate $Q_p$. The various assumptions of the method are listed in the TxDOT Design Manual; of particular note is that the runoff coefficient can vary with rainfall intensity $i$. The rainfall intensity $i$ is, by definition, the rainfall rate when the rainfall duration equals the time of concentration. 

Because the rational method remains widespread in TxDOT, as well as outside the agency, TxDOT research supervisory personnel issued a problem statement, \emph{Use of the rational and modified rational methods for TxDOT hydraulic design}, in the spring of 2007. The project objectives (from the problem statement) are:
\begin{quote}
$\ldots$to evaluate the appropriateness of using both rational and modified rational methods for small watershed design, evaluate the tabulated values of the runoff coefficient, and construct guidelines for TxDOT analysts for selection of appropriate parameter values for Texas watershed conditions. 
\end{quote}

Use of the rational method for estimating peak discharges for design of drainage structures and the modified rational method to estimate inflow hydrographs for design of best management practices require judgment to determine appropriate parameter values. From the problem statement,
\begin{quote}
Use of either method depends on the analyst�s estimate of the time of concentration and the runoff coefficient. Both of these estimates can vary substantially depending on watershed conditions. Therefore, research to document appropriate values is needed.
\end{quote}
In addition to estimates of both time of concentration and runoff coefficient, design-rainfall intensity directly impacts resulting estimates. An initial investigation of the rational method, in the context of watershed scale, was reported by \citet{thompson07}. That work was done using data from about 20 Texas watersheds, mostly rural (undeveloped). \citet{thompson07} reported
\begin{quote}
For simple watersheds, the rational method may be applied when only an estimate of the peak discharge from the runoff hydrograph is required. Watershed drainage area does not seem to be an important consideration. However, this observation must be tempered with a caveat that only 20 watersheds were examined. Furthermore, watershed complexity was not examined as part of this research. Because the rational method is a simple procedure, application of the method to a complex watershed would be an error of judgment and may result in substantial errors in estimated design discharge. Further study through expansion of the study database also is in order.
\end{quote}
So, \citet{thompson07} recommends expansion of the relatively small database used for his study. The research team assembled for this proposal has access to a larger database.

\subsection{Database}

A database is presented in \citet{asquith04b}, comprising about 100 watersheds ranging in drainage area from about 0.5~square miles to 150~square miles. These stations are listed in table~\ref{tbl:asquith}. 

\footnotesize
\begin{longtable}{cp{42ex}ccc}
\caption{Stations included in paired rainfall and runoff database [adapted from \citet{asquith05b}] \\
\footnotesize
[sub., subwatershed; U, undeveloped watershed; D, developed watershed; ---, not applicable] \\
} \label{tbl:asquith} \\
\hline
Number & Station name & Latitude & Longitude & Landuse \\
\hline
\endfirsthead
\caption[]{\normalsize Stations included in paired rainfall and runoff database --- Continued} \\
\hline
Number & Station name & Latitude & Longitude & Landuse \\
\hline
\endhead
\hline
\multicolumn{5}{r}{\emph{Continued on next page}}
\endfoot
\hline
\endlastfoot

08042650 & North Creek sub. 28A near Jermyn, Tex. & 33$^\circ$14'52" & 98$^\circ$19'19" & U \\
08042700 & North Creek near Jacksboro, Tex. & 33$^\circ$16'57" & 98$^\circ$17'53" & U\\
08048520 & Sycamore Creek at IH 35W, Fort Worth, Tex. & 32$^\circ$39'55" & 97$^\circ$19'16" & D\\
08048530 & Sycamore Creek tributary above Seminary South Shopping Center, Fort Worth, Tex. & 32$^\circ$41'08" & 97$^\circ$19'44" & D\\
08048540 & Sycamore Creek tributary at IH 35W, Fort Worth, Tex. & 32$^\circ$41'18" & 97$^\circ$19'11" & D\\
08048550 & Dry Branch at Blandin Street, Fort Worth, Tex. & 32$^\circ$47'19" & 97$^\circ$18'22" & D\\
08048600 & Dry Branch at Fain Street, Fort Worth, Tex. & 32$^\circ$46'34" & 97$^\circ$17'18" & D\\
08048820 & Little Fossil Creek at IH 820, Fort Worth, Tex. & 32$^\circ$50'22" & 97$^\circ$19'20" & D\\
08048850 & Little Fossil Creek at Mesquite Street, Fort Worth, Tex. & 32$^\circ$48'33" & 97$^\circ$17'28" & D\\
08050200 & Elm Fork Trinity River sub. 6 near Muenster, Tex. & 33$^\circ$37'13" & 97$^\circ$24'15" & U\\
08052630 & Little Elm Creek sub. 10 near Gunter, Tex. & 33$^\circ$24'33" & 96$^\circ$48'41" & U\\
08052700 & Little Elm Creek near Aubrey, Tex. & 33$^\circ$17'00" & 96$^\circ$53'33" & U\\
08055580 & Joes Creek at Royal Lane, Dallas, Tex. & 32$^\circ$53'43" & 96$^\circ$41'36" & D\\
08055600 & Joes Creek at Dallas, Tex. & 32$^\circ$51'33" & 96$^\circ$53'00" & D\\
08055700 & Bachman Branch at Dallas, Tex. & 32$^\circ$51'37" & 96$^\circ$51'13" & D\\
08056500 & Turtle Creek at Dallas, Tex. & 32$^\circ$48'26" & 96$^\circ$48'08" & D\\
08057020 & Coombs Creek at Sylvan Ave, Dallas, Tex. & 32$^\circ$46'01" & 96$^\circ$50'07" & D\\
08057050 & Cedar Creek at Bonnieview Road, Dallas, Tex. & 32$^\circ$44'50" & 96$^\circ$47'44" & D\\
08057120 & McKamey Creek at Preston Road, Dallas, Tex. & 32$^\circ$57'58" & 96$^\circ$48'11" & U\\
08057130 & Rush Branch at Arapaho Road, Dallas, Tex. & 32$^\circ$57'45" & 96$^\circ$47'44" & D\\
08057140 & Cottonwood Creek at Forest Lane, Dallas, Tex. & 32$^\circ$54'33" & 96$^\circ$45'54" & D\\
08057160 & Floyd Branch at Forest Lane, Dallas, Tex. & 32$^\circ$54'33" & 96$^\circ$45'34" & D\\
08057320 & Ash Creek at Highland Road, Dallas, Tex. & 32$^\circ$48'18" & 96$^\circ$43'04" & D\\
08057415 & Elam Creek at Seco Boulevard, Dallas, Tex. & 32$^\circ$44'14" & 96$^\circ$41'36" & D\\
08057418 & Fivemile Creek at Kiest Boulevard, Dallas, Tex. & 32$^\circ$42'19" & 96$^\circ$51'32" & D\\
08057420 & Fivemile Creek at US Highway 77W, Dallas, Tex. & 32$^\circ$41'15" & 96$^\circ$49'22" & D\\
08057425 & Woody Branch at IH 625, Dallas, Tex. & 32$^\circ$40'58" & 96$^\circ$49'22" & D\\
08057435 & Newton Creek at IH 635, Dallas, Tex. & 32$^\circ$39'19" & 96$^\circ$44'41" & D\\
08057440 & Whites Branch at IH 625, Dallas, Tex. & 32$^\circ$39'26" & 96$^\circ$44'25" & D\\
08057445 & Prarie Creek at US Highway 175, Dallas, Tex. & 32$^\circ$42'17" & 96$^\circ$40'11" & D\\
08057500 & Honey Creek sub. 11 near McKinney, Tex. & 33$^\circ$18'12" & 96$^\circ$41'22" & U\\
08058000 & Honey Creek sub.12 near McKinney, Tex. & 33$^\circ$18'20" & 96$^\circ$40'12" & U\\
08061620 & Duck Creek at Buckingham Road, Garland, Tex. & 32$^\circ$55'53" & 96$^\circ$39'55" & D\\
08061920 & South Mesquite Creek at SH 352, Mesquite, Tex. & 32$^\circ$46'09" & 96$^\circ$37'18" & D\\
08061950 & South Mesquite Creek at Mercury Road, Mesquite, Tex. & 32$^\circ$43'32" & 96$^\circ$34'12" & D\\
08063200 & Pin Oak Creek near Hubbard, Tex. & 31$^\circ$48'01" & 96$^\circ$43'02" & U\\
08094000 & Green Creek sub. 1 near Dublin, Tex. & 32$^\circ$09'57" & 98$^\circ$20'28" & U\\
08096800 & Cow Bayou sub. 4 near Bruceville, Tex. & 31$^\circ$19'59" & 97$^\circ$16'02" & U\\
08098300 & Little Pond Creek near Burlington, Tex. & 31$^\circ$01'35" & 96$^\circ$59'17" & U\\
08108200 & North Elm Creek near Cameron, Tex. & 30$^\circ$55'52" & 97$^\circ$01'13" & U\\
08111025 & Burton Creek at Villa Maria Road, Bryan, Tex. & 30$^\circ$38'48'' & 96$^\circ$20'57'' & D\\
08111050 & Hudson Creek near Bryan, Tex. & 30$^\circ$39'38'' & 96$^\circ$17'59'' & U\\
08136900 & Mukewater Creek sub. 10A near Trickham, Tex. & 31$^\circ$39'01" & 99$^\circ$13'30" & U\\
08137000 & Mukewater Creek sub. 9 near Trickham, Tex. & 31$^\circ$41'40" & 99$^\circ$12'18" & U\\
08137500 & Mukewater Creek at Trickham, Tex. & 31$^\circ$35'24" & 99$^\circ$13'36" & U\\
08139000 & Deep Creek sub. 3 near Placid,Tex. & 31$^\circ$17'25" & 99$^\circ$09'22" & U\\
08140000 & Deep Creek sub. 8 near Mercury, Tex. & 31$^\circ$24'08" & 99$^\circ$07'17" & U\\
08154700 & Bull Creek at Loop 360, Austin, Tex. & 30$^\circ$22'19" & 97$^\circ$47'04" & U\\
08155200 & Barton Creek at SH 71, Oak Hill, Tex. & 30$^\circ$17'46" & 97$^\circ$55'31" & U\\
08155300 & Barton Creek at Loop 360, Austin, Tex. & 30$^\circ$14'40" & 97$^\circ$48'07" & U\\
08155550 & West Bouldin Creek at Riverside Drive, Austin, Tex. & 30$^\circ$15'49" & 97$^\circ$45'17" & D\\
08156650 & Shoal Creek at Steck Avenue, Austin, Tex. & 30$^\circ$21'55" & 97$^\circ$44'11" & D\\
08156700 & Shoal Creek at Northwest Park, Austin, Tex. & 30$^\circ$20'50" & 97$^\circ$44'41" & D\\
08156750 & Shoal Creek at White Rock Drive, Austin, Tex. & 30$^\circ$20'21" & 97$^\circ$44'50" & D\\
08156800 & Shoal Creek at 12th Street, Austin, Tex. & 30$^\circ$16'35" & 97$^\circ$45'00" & D\\
08157000 & Waller Creek at 38th Street, Austin, Tex. & 30$^\circ$17'49" & 97$^\circ$43'36" & D\\
08157500 & Waller Creek at 23rd Street, Austin, Tex. & 30$^\circ$17'08" & 97$^\circ$44'01" & D\\
08158050 & Boggy Creek at US 183, Austin, Tex. & 30$^\circ$15'47" & 97$^\circ$40'20" & D\\
08158100 & Walnut Creek at FM 1325, Austin, Tex. & 30$^\circ$24'35" & 97$^\circ$42'41" & U\\
08158200 & Walnut Creek at Dessau Road, Austin, Tex. & 30$^\circ$22'30" & 97$^\circ$39'37" & U\\
08158380 & Little Walnut Creek at Georgian Drive Austin, Tex. & 30$^\circ$21'15" & 97$^\circ$41'52" & D\\
08158400 & Little Walnut Creek at IH 35, Austin, Tex. & 30$^\circ$20'57" & 97$^\circ$41'34" & D\\
08158500 & Little Walnut Creek at Manor Road, Austin, Tex. & 30$^\circ$18'34" & 97$^\circ$40'04" & D\\
08158600 & Walnut Creek at Webberville Road, Austin, Tex. & 30$^\circ$16'59" & 97$^\circ$39'17" & D\\
08158700 & Onion Creek near Driftwood, Tex. & 30$^\circ$04'59" & 98$^\circ$00'29" & U\\
08158800 & Onion Creek at Buda, Tex. & 30$^\circ$05'09" & 97$^\circ$50'52" & U\\
08158810 & Bear Creek below FM 1826, Driftwood, Tex. & 30$^\circ$09'19" & 97$^\circ$56'23" & U\\
08158820 & Bear Creek at FM 1626, Manchaca, Tex. & 30$^\circ$08'25" & 97$^\circ$50'50" & U\\
08158825 & Little Bear Creek at FM 1626, Manchaca, Tex. & 30$^\circ$07'31" & 97$^\circ$51'43" & U\\
08158840 & Slaughter Creek at FM 1826, Austin, Tex. & 30$^\circ$12'32" & 97$^\circ$54'11" & U\\
08158860 & Slaughter Creek at FM 2304, Austin, Tex. & 30$^\circ$09'43" & 97$^\circ$49'55" & U\\
08158880 & Boggy Creek (south) at Circle S Road, Austin, Tex. & 30$^\circ$10'50" & 97$^\circ$46'55" & U\\
08158920 & Williamson Creek at Oak Hill, Tex. & 30$^\circ$14'06" & 97$^\circ$51'36" & D\\
08158930 & Williamson Creek at Manchaca Road, Austin, Tex. & 30$^\circ$13'16" & 97$^\circ$47'36" & D\\
08158970 & Williamson Creek at Jimmy Clay Road, Austin, Tex. & 30$^\circ$11'21" & 97$^\circ$43'56" & D\\
08159150 & Wilbarger Creek near Pflugerville, Tex. & 30$^\circ$27'16" & 97$^\circ$36'02" & U\\
08177600 & Olmos Creek tributary at FM 1535, Shavano Park, Tex. & 29$^\circ$34'35" & 98$^\circ$32'45" & D\\
08177700 & Olmos Creek at Dresden Drive, San Antonio, Tex. & 29$^\circ$29'56" & 98$^\circ$30'36" & D\\
08178300 & Alazan Creek at St. Cloud Street, San Antonio, Tex. & 29$^\circ$27'29" & 98$^\circ$32'59" & D\\
08178555 & Harlendale Creek at West Harding Street, San Antonio, Tex. & 29$^\circ$21'05" & 98$^\circ$29'32" & D\\
08178600 & Panther Springs Creek at FM 2696 near San Antonio, Tex. & 29$^\circ$37'31" & 98$^\circ$31'06" & U\\
08178620 & Lorence Creek at Thousand Oaks Boulevard, San Antonio, Tex. & 29$^\circ$35'24" & 98$^\circ$27'47" & D\\
08178640 & West Elm Creek at San Antonio, Tex. & 29$^\circ$37'23" & 98$^\circ$26'29" & U\\
08178645 & East Elm Creek at San Antonio, Tex. & 29$^\circ$37'04" & 98$^\circ$25'41" & U\\
08178690 & Salado Creek tributary at Bitters Road, San Antonio, Tex. & 29$^\circ$31'36" & 98$^\circ$26'25" & D\\
08178736 & Salado Creek tributary at Bee Street, San Antonio, Tex. & 29$^\circ$26'38" & 98$^\circ$27'13" & D\\
08181000 & Leon Creek tributary at FM 1604, San Antonio, Tex. & 29$^\circ$35'14" & 98$^\circ$37'40" & U\\
08181400 & Helotes Creek at Helotes, Tex. & 29$^\circ$34'42" & 98$^\circ$41'29" & U\\
08181450 & Leon Creek tributary at Kelly Air Force Base, Tex. & 29$^\circ$23'12" & 98$^\circ$36'00" & D\\
08182400 & Calaveras Creek sub. 6 near Elmendorf, Tex. & 29$^\circ$22'49" & 98$^\circ$17'33" & U\\
08187000 & Escondido Creek sub. 1 near Kenedy, Tex. & 28$^\circ$46'41" & 97$^\circ$53'41" & U\\
08187900 & Escondido Creek sub. 11 near Kenedy, Tex. & 28$^\circ$51'39" & 97$^\circ$50'39" & U\\
SSSC & Seminary South Shopping Center drainage, Fort Worth, Tex. & --- & --- & D\\
\end{longtable}%
\normalsize

Since publication of \citet{asquith04b}, additional watersheds were added to the database to bring the total number of watersheds available for analysis to about 130. Table~\ref{tbl:cleveland} lists these additional stations and some relevant properties\footnote{Table \ref{tbl:cleveland} is in SI Units, converted from U.S. Customary for ASCE publication}. Figure~\ref{fig:station_map} is a map locating the stations in tables~\ref{tbl:asquith}, \ref{tbl:cleveland}, and \ref{tbl:watersheds07}. Figure~\ref{fig:station_map} presents the current spatial coverage status of the collective database(s) in use by the research team. 

\footnotesize
%\begin{longtable}{cp{42ex}ccc}
\begin{longtable}{lcccccccc}
\caption{Locations, Selected Physical Characteristics, and Unit Hydrograph Values for Study Watersheds in Houston, Texas. adapted from (Cleveland and others 2007) \\
 \\
\footnotesize
[ TDA, Total drainage area in $km^2$; MCL, Main channel length in $km$; SLOPE, Dimensionless main channel slope; DEVF, Basin development factor (0=undeveloped, 1=developed); $Q_p$, Peak rate factor in $\frac{m^3/s-hr}{mm-km^2}$; $T_p$, Time to peak in hours]} 
\label{tbl:cleveland} \\
\hline
Station No. & Latitude & Longitude & TDA  & MCL  & SLOPE & DEVF & Qp  & Tp\\
\hline
\endfirsthead
\caption[]{\normalsize Locations, Selected Physical Characteristics, and Unit Hydrograph Values for Study Watersheds in Houston, Texas. --- Continued} \\
\hline
Station No. & Latitude & Longitude & TDA  & MCL  & SLOPE & DEVF & Qp  & Tp \\
\hline
\endhead
\hline
\multicolumn{5}{r}{\emph{Continued on next page}}
\endfoot
\hline
\endlastfoot

 \\
8068438 & 30$^\circ$08'38" & 95$^\circ$28'09" & 1.4 & 1.2 & 0.0077 & 0 & 0.341 & 0.41 \\
8068440 & 30$^\circ$08'24" & 95$^\circ$28'33" & 1.8 & 2.1 & 0.0062 & 0 & 0.133 & 1.10 \\
8077100 & 29$^\circ$36'09" & 95$^\circ$16'41" & 3.4 & 3.0 & 0.0015 & 0 & 0.051 & 3.27 \\
8075780 & 29$^\circ$56'56" & 95$^\circ$31'10" & 20.9 & 6.9 & 0.0015 & 0 & 0.022 & 6.66 \\
8074780 & 29$^\circ$39'55" & 95$^\circ$35'42" & 22.4 & 9.4 & 0.0019 & 0 & 0.046 & 5.76 \\
8068400 & 30$^\circ$11'34" & 95$^\circ$29'09" & 67.7 & 14.3 & 0.0024 & 0 & 0.020 & 8.81 \\
8068450 & 30$^\circ$08'04" & 95$^\circ$28'38" & 89.4 & 22.9 & 0.0018 & 0 & 0.017 & 10.77 \\
8073630 & 29$^\circ$46'32" & 95$^\circ$32'23" & 3.6 & 1.2 & 0.0041 & 1 & 0.086 & 2.31 \\
8074145 & 29$^\circ$51'31" & 95$^\circ$29'09" & 0.5 & 1.1 & 0.0011 & 1 & 0.182 & 2.77 \\
8075300 & 29$^\circ$37'33" & 95$^\circ$29'56" & 9.9 & 5.1 & 0.0028 & 1 & 0.046 & 3.95 \\
8075600 & 29$^\circ$39'00" & 95$^\circ$14'18" & 4.1 & 3.6 & 0.0017 & 1 & 0.027 & 4.57 \\
8075750 & 29$^\circ$48'00" & 95$^\circ$20'02" & 3.1 & 2.9 & 0.0013 & 1 & 0.052 & 2.42 \\
8178555 & 29$^\circ$21'05" & 98$^\circ$29'32" & 4.9 & 6.5 & 0.0024 & 1 & 0.093 & 2.69 \\
8075550 & 29$^\circ$38'32" & 95$^\circ$13'22" & 6.6 & 5.4 & 0.0018 & 1 & 0.056 & 2.23 \\
8074910 & 29$^\circ$39'44" & 95$^\circ$29'11" & 0.8 & 2.3 & 0.0008 & 1 & 0.115 & 1.26 \\
8074100 & 29$^\circ$51'24" & 95$^\circ$30'55" & 18.3 & 8.1 & 0.0023 & 1 & 0.030 & 6.42 \\
8075760 & 29$^\circ$48'22" & 95$^\circ$19'50" & 6.7 & 6.5 & 0.0016 & 1 & 0.041 & 5.40 \\
8075700 & 29$^\circ$21'05" & 95$^\circ$15'11" & 12.6 & 6.8 & 0.0016 & 1 & 0.046 & 3.23 \\
8075650 & 29$^\circ$40'35" & 95$^\circ$14'37" & 27.7 & 7.9 & 0.0016 & 1 & 0.023 & 6.69 \\
8074750 & 29$^\circ$43'11" & 95$^\circ$39'37" & 2.3 & 2.8 & 0.0005 & 1 & 0.026 & 2.46 \\
8075400 & 29$^\circ$37'07" & 95$^\circ$26'45" & 52.4 & 10.6 & 0.0017 & 1 & 0.029 & 8.74 \\ 
8074150 & 29$^\circ$51'04" & 95$^\circ$29'16" & 19.4 & 10.7 & 0.0013 & 1 & 0.024 & 6.10 \\
8074540 & 29$^\circ$47'33" & 95$^\circ$22'06" & 46.7 & 14.5 & 0.0016 & 1 & 0.038 & 7.68 \\
8075730 & 29$^\circ$41'40" & 95$^\circ$12'58" & 21.4 & 8.4 & 0.0009 & 1 & 0.033 & 6.15 \\
8075770 & 29$^\circ$47'35" & 95$^\circ$16'04" & 41.7 & 12.5 & 0.0012 & 1 & 0.044 & 3.80 \\
8076200 & 29$^\circ$54'07" & 95$^\circ$25'21" & 23.3 & 12.4 & 0.0010 & 1 & 0.016 & 11.09 \\
8074900 & 29$^\circ$39'01" & 95$^\circ$29'11" & 29.0 & 12.2 & 0.0009 & 1 & 0.012 & 13.00 \\
8074760 & 29$^\circ$42'39" & 95$^\circ$35'13" & 36.5 & 13.8 & 0.0008 & 1 & 0.019 & 6.91 \\
8074850 & 29$^\circ$41'16" & 95$^\circ$30'20" & 11.4 & 10.9 & 0.0005 & 1 & 0.021 & 7.44 \\
8076500 & 29$^\circ$51'42" & 95$^\circ$20'05" & 74.4 & 24.0 & 0.0010 & 1 & 0.010 & 14.46 \\
8074500 & 29$^\circ$46'30" & 95$^\circ$23'49" & 223.7 & 33.1 & 0.0011 & 1 & 0.022 & 14.79 \\
8075500 & 29$^\circ$40'27" & 95$^\circ$17'21" & 163.3 & 30.3 & 0.0010 & 1 & 0.010 & 11.52 \\
8074810 & 29$^\circ$40'21" & 95$^\circ$31'41" & 137.9 & 22.6 & 0.0007 & 1 & 0.016 & 12.44 \\
8075000 & 29$^\circ$41'49" & 95$^\circ$24'43" & 246.0 & 33.8 & 0.0007 & 1 & 0.008 & 15.50 \\
\end{longtable}%
\normalsize

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4in]{station_map.pdf} 
   \caption{Station locations of stations with paired rainfall-runoff data}
   \label{fig:station_map}
\end{figure}

\subsection{Runoff Coefficient\label{sec:runoff}}
The rational-method runoff coefficient is generally taken from tables of values published in any standard hydrology text\footnote{See, for example, \citet{viessman03}. Also, a table of runoff coefficients is published in the TxDOT on-line Hydraulic Design Guidelines, located electronically at \url{http://manuals.dot.state.tx.us/dynaweb/colbridg/hyd} at the time of this writing.}. The genesis of those standard tables remains unknown. However, given contemporaneous measurements of rainfall and runoff, the accepted method for estimating the runoff coefficient is to take the ratio of total runoff to total storm depth,
\begin{equation}
  C_v = \frac{\int Q(t)dt}{A\int P(t)dt},
\end{equation}
where $C_v$ is the volumetric runoff coefficient, $Q$ is the instantaneous runoff rate (discharge), $A$ is the watershed drainage area, and $P$ is the instantaneous rainfall rate. 

\citet{thompson07} used measurements of rainfall and runoff from the watersheds listed in table~\ref{tbl:watersheds07} to estimate values of the volumetric runoff coefficient. The watersheds of table~\ref{tbl:watersheds07} constitute a subset of those listed in table~\ref{tbl:asquith}. The method used to estimate observed runoff coefficients\footnote{Two estimates of runoff coefficient were produced. One was termed the \emph{observed\/} runoff coefficient because it was extracted from site-specific contemporaneous measurements of rainfall and runoff. The second was termed the \emph{predicted\/} runoff coefficient because it was taken from tabular estimates.} was that presented by \citet{schaake67}. Estimates of both observed and predicted (table) runoff coefficient are also presented on table~\ref{tbl:watersheds07}.

\begin{table}[h!]
\caption{Watersheds, watershed characteristics, and estimates of observed and predicted (table) runoff coefficients from \citet{thompson07}.\label{tbl:watersheds07}}\vspace*{12pt}
\begin{center}
{\footnotesize 
\begin{tabular}{ccccccc}
\hline
USGS Gage &	USGS Quadrangle & Drainage & Channel & Channel & Observed & Table\\
Station ID & Location & Area (mi$^2$) & Length (ft) & Slope (ft/ft) & $C$ & $C$\\ \hline
08088100 & True 		& 11.8 & 24,874 & 0.003 & 0.45 & 0.40 \\
08093400 & Abbot 		& 12.4 & 52,689 & 0.004 & 0.44 & 0.37 \\
08160800 & Frelsburg 	& 17.3 & 42,407 & 0.005 & 0.27 & 0.41 \\
08167600 & Fischer 		& 10.9 & 29,393 & 0.017 & 0.26 & 0.68 \\
08156800 & Austin 		& 12.3 & 55,144 & 0.008 & 0.70 & 0.68 \\
08158700 & Austin 		& 124 & 175,221 & 0.004 & 0.30 & 0.40 \\
08158840 & Austin 		& 8.24 & 25,690 & 0.010 & 0.40 & 0.46 \\
08178640 & San Antonio & 2.45 & 15,786 & 0.020 & 0.35 & 0.46 \\
08181400 & San Antonio & 15.0 & 51,257 & 0.011 & 0.38 & 0.48 \\
08098300 & Cameron 		& 23.0 & 72,642 & 0.003 & 0.67 & 0.38 \\
08096800 & Moody 		& 5.47 & 34,715 & 0.006 & 0.34 & 0.36 \\
08137000 & Bangs 		& 4.02 & 22,238 & 0.004 & 0.35 & 0.36 \\
08182400 & Martinez 	& 7.01 & 25,586 & 0.006 & 0.28 & 0.34 \\
08187000 & Lenz 		& 3.29 & 14,477 & 0.010 & 0.16 & 0.38 \\
08187900 & Lenz 		& 8.43 & 25,011 & 0.005 & 0.41 & 0.39 \\
08050200 & Muenster 	& 0.77 & 13,497 & 0.011 & 0.58 & 0.41 \\
08058000 & Weston 		& 1.26 & 10,283 & 0.009 & 0.65 & 0.36 \\
08052700 & Marilee 		& 75.5 & 122,054 & 0.002& 0.54 & 0.41 \\
08042700 & Senate 		& 21.6 & 60,538 & 0.006 & 0.32 & 0.44 \\
08063200 & Coolidge 	& 17.6 & 45,847 & 0.004 & 0.53 & 0.36 \\ \hline
\end{tabular}}
\end{center}
\end{table}

However, the rational method produces an estimate of the discharge, a rate, using an estimate of the rainfall rate. Therefore, it follows that the runoff coefficient should be based on a relation between the rate of runoff (discharge) and the rate of rainfall. That is,
\begin{equation}
  C=\frac{Q_p}{i}, \label{eqn:rate-c}
\end{equation}
where $Q_p$ is the target runoff rate (the peak) and $i$ is a measure of rainfall intensity.

The importance of these two views of how to determine the runoff coefficient is briefly illustrated in the next two sections, where the sensitivity to the time base is examined and evidence of dependence on storm depth is presented.

\subsection{Sensitivity to Time of Concentration}

The rainfall intensity is sensitive to changes in the time base. To illustrate consider the 2\%-chance depths (50-year frequency) for Harris County as shown in table~\ref{tab:intensity}. The values in the time column and depth column are estimates from the Texas depth-duration-frequency atlas \citep{asquith04a}, the intensity is calculated as the ratio of the mapped depth and associated duration \footnote{Table~\ref{tab:intensity} is constructed from the maps in \citet{asquith04a}; not from the provided L-moments}.  

\begin{table}[h!]
   \centering
   \caption{Example Rainfall Intensities}
   \begin{tabular}{c c c} 
   Time (min) & Depth (inches) & Intensity (in/min) \\
   \hline
   15 & 1.9 & 0.126 \\
   30 & 2.6 & 0.086 \\
   60 & 3.8 & 0.063 \\
    & & \\
   \end{tabular}
   \label{tab:intensity}
\end{table}

Figure~\ref{fig:figure_3} is a plot of the relation between rainfall intensity and rainfall duration (storm duration) using values listed in table~\ref{tab:intensity}. The curve on the plot is a hyperbolic interpolating function used to illustrate the sensitivity of rainfall intensity to small changes in the time base. 

For the curve presented in figure~\ref{fig:figure_3}, a ten-percent change in time at a duration of 30 minutes causes about a five-percent change in indicated rainfall intensity. That is $i_{30} = 0.084$ while $i_{33}=0.079$ for values on the curve. At smaller values of time, hence for small watersheds, the change in time greatly impacts the estimated intensity (notice the magnitude of the slope increases at smaller time). At larger values, the effect is reduced --- however, if the time of concentration values are large (exceeding, say, about 300~minutes), then either the watershed under consideration is too large for application of the rational method or the watershed of interest is in a low-slope region and other methods should be considered\footnote{One proposal author suggests that dimensionless slopes in the range 0.002 to 0.0002 define the boundary where kinematic-wave theory begins to fail for hydrologic modeling}.

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4in]{figure_3.pdf} 
   \caption{Intensity-duration relationship for 2\%-chance storm, Harris County, TX. Markers are values estimated from \citet{asquith04a}; curve is a hyperbolic interpolating function passed through the markers}
   \label{fig:figure_3}
\end{figure}

The true nature of sensitivity to time depends on the shape of the curve and the slope (first derivative) of the curve near the time of interest. Small variations at large time are probably irrelevant, small variations at small time will have considerable impact on intensity. Time estimates by several different methods as examined in TxDOT project 0-4696 vary about one-half log cycle of each other so the effect of the time estimate methodology alone is expected to exert considerable influence on rational method peak discharge values on smaller watersheds.

\subsection{Storm Depth Dependence}
The runoff coefficient is known to be dependent on storm depth as well as watershed properties, and indeed this dependence is implicitly acknowledged in the TxDOT design manual in the coefficient adjustment factors for the rational method. Further illustration of the dependence is offered in the following discussion.

Figure \ref{fig:q over p vs p} is a plot of runoff depth as a function of storm depth for for about 2,600 events in Texas on 130 watersheds. The data for the plot were derived from data reported in \citet{asquith04b}\footnote{Watersheds in table~\ref{tbl:asquith}} supplemented by Houston area watersheds\footnote{Watersheds in table~\ref{tbl:cleveland}}. These data were analyzed in support of TxDOT projects 0--4193, 0--4194, 0--4405, and 0--4696. The line on the plot is an equal value line that illustrates, except for a few percent of the storm pairs, that storms produced runoff depths less than the storm depth --- an anticipated response.  Also in the upper panel, the slope of a reasonable trend line plotted through the marker cloud would differ from, and probably be steeper than, the equal value line (a slope of unity). The difference in slopes suggests that the conversion of rainfall depth to runoff depth depends on the rainfall depth, perhaps in a non-linear fashion (the slope of a trend line is conceptually a runoff coefficient).

%The lower panel of figure~\ref{fig:q over p vs p} is a plot of the relation of a volumetric runoff coefficient, (Essentially $C_v$) and storm depth, $D_{\mbox{\it\tiny storm}}$. The horizontal line is equivalent to the equal value line $D_{\mbox{\it\tiny runoff}}=D_{\mbox{\it\tiny storm}}$. The other curve is a power-law trend line that was fit through the marker cloud using ordinary least-squares. The slope of the line is positive, and more importantly, statistically different from zero, thus providing some evidence of a relationship between the volumetric runoff coefficient and the storm depth. 

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4in]{figure_1.pdf} 
   \caption{Relationship of $C_v = (\int Q(t)dt / \int P(t)dt)$ to $\int P(t)dt$ for 2600 Texas storms}
   \label{fig:q over p vs p}
\end{figure}

Figure~\ref{fig:rc_vs_peakrain} is a set of figures from the same underlying dataset. Figure~\ref{fig:rc_vs_peakrain} differs from figure~\ref{fig:q over p vs p} in that peak rainfall intensity and peak discharge for a one minute interval from each of the 2,600 storms were extracted and selected for plotting. An implicit assumption in figure~\ref{fig:rc_vs_peakrain} is that peak rainfall rate causes peak discharge --- similar in concept to the rational method. The curves on each plot are power-law fits ($y=ax^b$) through the marker clouds --- in these cases the explanatory variable is the peak one-minute rainfall intensity.

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4in]{figure_2.pdf} 
   \caption{Relationship of $Q_p$ to $I$ (upper panel), and $c = Q_p/I$ (lower panel) for 2600 Texas storms}
   \label{fig:rc_vs_peakrain}
\end{figure}

In contrast to volumetric-based runoff coefficients these plots displayed on figure~\ref{fig:rc_vs_peakrain} exhibit the opposite behavior, but the interpretation is the same --- similar to depth-based runoff coefficients the rate-based runoff coefficients appear to be intensity dependent.

The very simplicity of the rational method makes it attractive for use in many applications. Care is recommended in the TxDOT manual. However, guidance is needed to consistently manage issues of temporal sensitivity, intensity dependence, and estimation of reasonable values for runoff coefficient when computing estimates of peak discharge from Texas watersheds.


%This proposal continues with further background on existing regression regression equations in Texas in Section~\ref{sec:History}. Mathematical background is provided in Section~\ref{sec:Stats}. A focus on theoretrical L-moments of a distribution and L-moments of a sample is provided in Section~\ref{sec:Lmoments}. The favored candidate distributions by the authors for site-specific peak-streamflow frequency are presented in Section~\ref{sec:Dist}. An example computation of site-specific peak-streamflow frequency that demonstrates several caveats and technical considerations is provided in Section~\ref{sec:Example}. Study objectives are summarized in Section~\ref{sec:Objectives}, and the technical approach is outlined in Section~\ref{sec:Approach}. Implementation of project results in the context of TxDOT requirements is described in Section~\ref{sec:Implementation}.

\subsection{History\label{sec:History}}

TxDOT uses only a handful of methods for estimating discharges for hydraulic designs. Among these methods are the rational method (and modified rational method when a hydrograph is required), the unit hydrograph method, regional regression equations, and streamgage analysis. Although the Hydraulic Design Guidelines offer guidance on method selection, the final choice is left to the analyst.

Over the last decade, TxDOT funded research focused on improved understanding of their suite of hydrologic methods for estimating design discharges. In addition, their research objective was to develop their suite of methods such that increased confidence in design discharge estimates would result. One benefit of improved estimates is reduced reliance on simple conservatism\footnote{One definition of conservatism is the artificial inflating of method parameters to increase discharge estimates with the hope that such practices reduces the risk of hydraulic failure without regard to the impact on project cost.}. Much of the research undertaken by this research team focused on a dataset assembled from USGS records encompassing about 100 watersheds and about 1,600 storm events. This dataset includes watersheds from central and eastern Texas and was recently supplemented with additional watersheds and storm events for the Houston area.

An exploratory research project was completed \citep{thompson07} to examine the relation between hydrologic scale\footnote{Hydrologic scale is often measured in terms of watershed drainage area. One of the conclusions of \citet{thompson07} is that drainage area may not be the best parameter for assessing hydrologic scale. More work is needed on this topic.} and appropriate method for estimating design discharge. While substantial work on hydrologic scale remains to be done, TxDOT Research Project 0--4405 \citep{thompson07} resulted in an alternative approach to the \citet{asquith97} regression equations, published in \citet{asquith05b}. The direct result of \citet{asquith05b} was another TxDOT research project (TxDOT Project 0--5521). In addition, \citet{thompson07} suggested that drainage area alone may not be a good discriminator for choosing a hydrologic method for estimating a design discharge.

Another result of \citet{thompson07} dealt with the rational method and estimates of the runoff coefficient. Two conclusions from \citet{thompson07} are pertinent:
\begin{quote}
\begin{enumerate}
	\item{Estimates of $n$-year discharge using the rational method are based on best estimates for the runoff coefficient taken from a comparison of measured rainfall and runoff events. Selection of proper estimates of the runoff coefficient is critical for correct estimation of $n$-year discharges. Refinement of this process beyond simple selection of values from a table is appropriate. 
  
  This observation does not constitute a recommendation for a probabilistic adjustment of runoff coefficient (as recommended in current hydraulic design guidelines). Rather, the observation that runoff coefficient varies with runoff depth (as demonstrated by the figures presented in this report) is important and suggests that a single value for runoff coefficient may not be appropriate. Additional work is necessary to further investigate runoff coefficients for use with the rational method.}

  \item{For simple watersheds, the rational method may be applied when only an estimate of the peak discharge from the runoff hydrograph is required. Watershed drainage area does not seem to be an important consideration. However, this observation must be tempered with a caveat that only 20 watersheds were examined. Furthermore, watershed complexity was not examined as part of this research. Because the rational method is a simple procedure, application of the method to a complex watershed would be an error of judgment and may result in substantial errors in estimated design discharge. Further study through expansion of the study database also is in order.}
\end{enumerate}
\end{quote}
Therefore, the research proposed herein represents an important development of, and extension to, research already completed under TxDOT auspices.

\section{Project Objectives \label{sec:Objectives}}

The objectives, as stated in the problem statement, are:
\begin{quote}
$\ldots$to evaluate the appropriateness of using both rational and modified rational methods for small watershed design, evaluate the tabulated values of the runoff coefficient, and construct guidelines for TxDOT analysts for selection of appropriate parameter values for Texas watershed conditions. 
\end{quote}
Therefore, using the database developed by \citet{asquith04b}, supplemented with data added after publication of \citet{asquith04b} for the Houston, Texas, region, an evaluation of the rational method and the modified rational method will be undertaken. The technical approach suggested by the analyses presented in section~\ref{sec:runoff} will be used, and is developed below.

\section{Implementation \label{sec:Implementation}}

The results of the proposed project are anticipated to include a number of implementable results. These include:
\begin{itemize}
	\item Review of the range of applicability of the rational method and modified rational method
	\item Calibration and/or validation of rational method runoff coefficients for selected Texas watersheds
	\item An assessment of the utility of both the rational method and modified rational method for use in design of both drainage structures and best management practices
\end{itemize}
The above results are considered likely --- additional results may come from the research. These results should be directly implementable by TxDOT and useful for providing guidance to TxDOT analysts involved in making estimates of hydrologic response for design activities. 

\section{Technical Approach and Work Plan\label{sec:Approach}}

The technical approach and work plan for the proposed project is summarized in this section. The tasks approximately parallel project objectives described in Section~\ref{sec:Objectives}. The major facets are

\begin{itemize}
   \item Literature and database review
   \item Technical investigation of the rational method
   \item Assessment of results
   \item Reporting
\end{itemize}

\subsection*{Task 1: Literature and Database Review\label{task:1}}
As part of a previous TxDOT research project (Project 0--4405), a literature review encompassing the rational method was completed \citep{thompson04}. Because a few years passed since publication of that work, a review of the professional literature will be undertaken to evaluate new publications pertinent to the project or to determine that no new literature contributes knowledge to the problem. A brief literature review is anticipated.

Much of the existing database compiled for Texas is documented by \citet{asquith04b}. However, subsequent to publication of \citet{asquith04b}, additional data from 34 watersheds located in and around Houston, Texas, were added to the database. These events comprise 1,094 rainfall-runoff events and significantly extend the database. Therefore, a review of the current database is considered appropriate to ensure data for recently-added watersheds are correct and complete.

A secondary objective of Task~1 is to locate additional rainfall-runoff measurements from very small watersheds (less than 1 square mile). Although the geographic location of additional watersheds is not considered critical\footnote{Watersheds located in Arizona, New Mexico, Colorado, Kansas, Oklahoma, Arkansas, and Louisiana could be useful, if they prove hydrologically similar to watersheds located in Texas.}, location in Texas is considered ideal. The Agricultural Research Service (ARS) maintains a number of experimental watersheds across the United States. While their mandate differs from other researchers (ARS research is primarily concerned with agriculture), ARS scientists use measurements of rainfall-runoff responses from small watersheds. Therefore, contacts\footnote{A few watersheds have already been located and a few hydrographs are available, however these data have not been integrated into the larger database. This will be an sub-objective of Task~1.} will be made in an effort to supplement the existing database, particularly for very small watersheds (drainage areas less than one square mile). However, one of project researchers (Asquith) has been working closely (August 2006--present) with Daren Harmel (ARS, Blacklands Experiment Station, Texas) to prepare their voluminous rainfall-runoff database for digital processing. The ARS database apparently has over 1,000 station years of data. The database suffers from some limited, actually surprizingly few, transcription errors and bad time stamps. Once this database is ``clean,'' the research team will unleash computational tools for investigation of the rational method.

The research team has considerable experience with Texas hydrology and an existing collection of literature on relevant studies. In addition, members of the research team authored several papers on rainfall-runoff studies in Texas over the last decade. The literature review in this study will extend and complement the team's already sizeable literature collection.
 
A technical memorandum documenting results of Task~1 will be prepared and submitted on completion of this task.

\subsection*{Task 2: Technical Review of the Rational Method\label{task:2}}

Groundwork for studying the rational method (and modified rational method) was developed by \citet{thompson07} based on work of earlier researchers. In particular, the work of \citet{schaake67,hawkins85,hawkins90,hawkins93,hjelmfelt80} are important contributions to hydrologic methods. While not all of these papers deal directly with the rational method, the knowledge presented in them is useful to interpretation of runoff responses of watersheds and pertinent to the proposed research.

Two principal threads of research are appropriate: The first approach is more  traditional for estimating runoff coefficients based on runoff volume,
\begin{enumerate}
	\item Extract estimates of the observed runoff coefficient (volumetric) from rainfall-runoff events in the database\footnote{It is suggested that the rank-ordering approach documented by \citet{schaake67} for the rational method and used by \citet{hawkins90} for curve numbers be applied. By ordering rainfall and runoff, separately, from greatest to least, events are coupled by probability of occurrence and not temporally.}.
	\item Compare table estimates of runoff coefficient with observed values of runoff coefficient.
	\item Examine the relation between runoff coefficient and storm depth, if any. 
\end{enumerate}

The second approach differs radically from the first. Because the rational method is a rate-based method, it should be possible to extract estimates of a rate-based runoff coefficient from the project database using equation~\ref{eqn:rate-c}. While the procedure remains to be refined, an approach is suggested that includes,
\begin{enumerate}
	\item Extract peak discharges from the database; adjust from units of $L^3/T$ to units of $L/T$ to be consistent with precipitation rate units.
	\item Extract peak rainfall rates for the watershed time of concentration.
	\item Estimate the rate-based runoff coefficient by dividing the discharge rate by the rainfall rate using equation~\ref{eqn:rate-c}.
	\item Compare observed rate-based runoff coefficients with table (or predicted) values of the runoff coefficient.
\end{enumerate}

A technical memorandum documenting results of Task~2 will be prepared and submitted on completion of this task.

\subsection*{Task 3: Assessment of Results \label{task:3}}

Substantial work on Texas rainfall depths \citep{asquith04a} as well as methods for estimating time of concentration \citep{roussel05} is complete. Given the database developed for Texas hydrologic research, it is possible to compare results derived from analysis of the database to produce estimates of the runoff coefficient with site-specific flood-frequency curves. The protocol for such comparisons could follow \citet{thompson07}, in which site-specific flood frequency curves (FFC) were developed from study watersheds and used as a baseline for assessment of study methods.

Possible comparisons include:
\begin{itemize}
	\item Compute watershed time of concentration with methods presented in \citet{roussel05}.
	\item Use table runoff coefficients with existing approximations of the IDF relation to compute peak discharge. Compare with site-specific FFC.
	\item Use observed runoff coefficient with existing IDF curves to compute peak discharge. Compare with site-specific FFC
	\item Repeat the process with IDF curves derived from \citet{asquith04a}.
\end{itemize}
Other combinations may come from interactions of the research team with TxDOT project oversight personnel. The objective is to measure method performance in quantifiable terms. The approach is best developed in cooperation with research team and the project oversight team. This process has been fruitful in other completed projects.

A technical memorandum documenting results of Task~3 will be prepared and submitted on completion of this task.

\subsection*{Task 4: Reporting \label{task:4}}

Project reports are the principal deliverable for this project. Methods and materials used to execute the research and research results will be described by project research personnel and presented in a pair of reports. The final project report is intended for dissemination to TxDOT and other interested technical personnel; the project summary report is for wide distribution for publicizing TxDOT-supported research results.

Project researchers, not graduate students, will undertake report-writing tasks. Further, the research team desires ancillary publications concerning the results of this research. The research team has a consistent history of textual production.

% End of workplan

\section*{Information Technology \label{sec:IT}}
Computer programs may be developed and used during the conduct of the research. However, no computer program will be part of the deliverables for this project. 

\section{Assistance by TxDOT Personnel \label{sec:Assistance}}

No direct involvement of TxDOT personnel is expected or required to complete the proposed research. However, technical review and further input on project direction will be sought from the technical leaders within the TxDOT hydrologic community. It is understood that the broad mandate of the project requires continual discussions with TxDOT personnel to ensure that project results are readily used or adapted by design engineers of all experience levels.

\section{Budget Summary \label{sec:Budget}}
A budget summary for the participating institutions is presented below.\footnote{UH portion includes professional services subcontract to R.O. Anderson for Dr.~Thompson's involvement}
$$
\begin{tabular}{cD{.}{.}{0}D{.}{.}{0}D{.}{.}{0}}

Institution 	& \multicolumn{1}{c}{FY 2008} & 
				  \multicolumn{1}{c}{FY 2009} & 
				  \multicolumn{1}{c}{FY 2010} \\ \hline
UH				& \$60,000		& \$60,000	& \$25,000 \\
USGS			& \$90,000		& \$90,000	& \$20,000 \\
Lamar			& \$40,000		& \$40,000	& \$10,000 \\ \hline
TxDOT Cost		& \$190,000		& \$190,000	& \$55,000 \\ \hline
Total TxDOT Cost &				&			& \$435,000 
\end{tabular}
$$

\newpage

\section{Project Deliverables \label{sec:Deliverables}}

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=7.0in]{DelTable_Cut.pdf} 
   \caption{Deliverable Table for 0-6070}
   \label{fig:DelTable_Cut}
\end{figure}

\newpage

\section{Schedule of Research Activities \label{sec:Schedule}}

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=7.0in]{Schedule_6070_Cut.pdf} 
   \caption{Project Schedule for  0-6070}
   \label{fig:Schedule_6070_Cut}
\end{figure}

\newpage
\bibliographystyle{chicago}
\bibliography{hydrology,books,thesis}

\end{document}

% Holding area

\begin{thebibliography}{9}

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Aragon Long, S.C., Reece, B.D., Eames, D.R., 2005. Water resources data Texas water year 2004. U.S. Geological Survey Water-Data Report TX--04--4.

\bibitem[Asquith and Slade(1997)]{AS1997}
Asquith, W.H., Slade, R.M., 1997. Regional equations for estimation of peak-streamflow frequency for natural basins in Texas. U.S. Geological Survey Water-Resources Investigations Report 96--4307, 68 p.

\bibitem[Asquith(2001)]{Asquith2001}
Asquith, W.H., 2001. Effects of regulation on L-moments of annual peak streamflow in Texas: U.S. Geological Survey Water-Resources Investigations Report 01--4243, 66 p.

\bibitem[Asquith and Roussel(2004)]{AR2004}
Asquith, W.H., and Roussel, M.C., 2001. Atlas of depth-duration frequency of precipitation annual maxima in Texas: U.S. Geological Survey Scientific-Investigations Report 2004--5041, 106 p.

\bibitem[Asquith and Thompson(2005)]{AT2005}
Asquith, W.H., Thompson, D.B., 2005. Alternative regression equations for estimation of annual peak-streamlfow frequency for undeveloped watersheds in Texas using PRESS minimization. Texas Department of Transportation Research Report 0--4405--2, Texas Tech University, Center for Multidisciplinary Research in Transportation, 27 p.

\bibitem[David(2003)]{David03}
David, H.A., 2003. Order statistics. Wiley, New York.

\bibitem[Elamir and Seheult(2003)]{Elam03}
Elamir, E.A.H., Seheult, A.H., 2003. Trimmed L-moments. Computational Statistics and Data Analysis, 43, 299--314.

\bibitem[Heitmuller and others(200X)]{Heit200X}
Heitmuller, F.T., Asquith, W.H., Cleveland, T.G., Fang, Xing, and Thompson, D.B., 200X, A rapid method to estimate main channel length from basin length for selected watersheds in Central Texas: Journal of Hydrologic Engineering, 
submitted on July 7, 2005.

\bibitem[Helsel and Hirsch(1992)]{HH92}
Helsel, D.R., Hirsch, R.M., 1992. Statistical methods in water resources---Studies in environmental science 49, Elsevier, New York.

\bibitem[Hosking(1990)]{Hosk90}
Hosking, J.R.M., 1990. L-moments: Analysis and estimation of distributions using linear combination of order statistics. Journal Royal Statistical Society Series B, 52(1) 105--124.

\bibitem[Hosking(1992)]{Hosk92}
Hosking, J.R.M., 1992. Moments or L-moments? An example comparing two measures of distributional shape. American Statistician, 46(3) 186--189.

\bibitem[Hosking(1994)]{Hosk94}
Hosking, J.R.M., 1994. The four-parameter kappa distribution. 
IBM Journal of Research and Development, 38(3) 251--258.

\bibitem[Hosking(1996)]{Hosk96}
Hosking, J.R.M., 1996. FORTRAN routines for use with the method of L-moments, Version 3. IBM Research Report RC12822, IBM Research Division, Yorktown Heights, New York.

\bibitem[Hosking and Wallis(1997)]{HW97}
Hosking, J.R.M., Wallis, J.R., 1993. Regional frequency analysis---An approach based on L-moments. Cambridge University Press, Cambridge.

\bibitem[Karian and Dudewicz(2000)]{KD00}
Karian, Z.A., Dudewicz, E.J., 2000. Fitting statistical distributions---The generalized lambda distribution and generalized bootstrap methods. CRC Press, Boca Raton, Florida.

\bibitem[Stedinger et al.(1992)]{Sted92}
Stedinger, J.R., Vogel, R.M., Foufoula-Georgiou, E., 1992. Frequency analysis of extreme events, In: D.A. Maidment (Ed.), Handbook of Hydrology, McGraw-Hill, New York, 18.1--18.66.

\end{thebibliography}

% eof
