Hydrologic Engineering Center

US Army Corps of Engineers

Hydrologic Engineering Center

Corps of Engineers' Experience with Automatic Calibration of a Precipitation-Runoff Model

May 1980

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May 1980

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Technical Paper

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Corps of Engineers' Experience with Automatic Calibration of a Precipitation-Runoff Model

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David T. Ford, Edward C. Morris, Arlen D. Feldman

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US Army Corps of Engineers Institute for Water Resources

Hydrologic Engineering Center (HEC) 609 Second Street

Davis, CA 95616-4687

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13. SUPPLEMENTARY NOTES

Presented at International Federation of Automatic Control Symposium on Water and Related Land Resource Systems, Cleveland, Ohio, 28-31 May 1980.

14. ABSTRACT

Computer program HEC-1, a precipitation-runoff model widely used throughout the United States, includes the capability to estimate automatically any of twelve parameters necessary to model the precipitation-runoff process and the channel routing process. The parameter estimation scheme employs Newton's method to minimize a weighted sum of squares of differences between observed and computed hydrograph values. Applications of this parameter estimation procedure are presented, and typical steps of the procedure for determining optimal parameter estimates are outlined. Recent efforts to improve the estimation algorithm and recent use of the calibration capability to update sequentially parameter estimates in a flood forecasting application are discussed.

15. SUBJECT TERMS

water resources, hydrology, rainfall-runoff modeling, parameter estimation, computer applications

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Corps of Engineers' Experience with Automatic Calibration of a Precipitation-Runoff Model

May 1980

US Army Corps of Engineers Institute for Water Resources Hydrologic Engineering Center 609 Second Street

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Papers in this series have resulted from technical activities of the Hydrologic Engineering Center. Versions of some of these have been published in

technical journals or in conference proceedings. The purpose of this series is to make the information available for use in the Center's training program and for distribution with the Corps of Engineers.

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CORPS OF ENGINEERS1 EXPERIENCE WITH AUTOMATIC CALIBRATION OF A PRECIPITATION-RUNOFF

MODEL.^

David T. Ford

Hydraulic Engineer, T r a i n i n g & Methods Branch, The Hydrologic Engineering Center, Davis, C a l i f o r n i a

Edward C. M o r r i s

Research H y d r a u l i c Engineer, Research Branch, The Hydrologic Engineering Center, Davis, C a l i f o r n i a

A r l e n D. Feldman

Chief, Research Branch, The Hydrologic Engineering Center, Davis, C a l i f o r n i a

Abstract. Computer program HEC-1, a p r e c i p i t a t i o n - r u n o f f model w i d e l y used

--

throughout t h e U n l t e d States, includes t h e c a p a b i l i t y t o estimate automati- c a l l y any o f twelve parameters necessary t o model t h e p r e c i p i t a t i o n - - r u n o f f process and the channel r o u t i n g process. The parameter e s t i m a t i o n scheme employs Newton's method t o minimize a weighted sum o f squares o f differences between observed and computed hydrograph values. A p p l i c a t i o n s o f t h i s pa- rameter e s t i m a t i o n procedure a r e presented, and t y p i c a l steps o f t h e procedure f o r determining optimal parameter estimates a r e o u t l i n e d . Recent e f f o r t s t o improve t h e e s t i m a t i o n a l g o r i t h m and r e c e n t use o f t h e c a l i b r a t i o n c a p a b i l i t y t o update s e q u e n t i a l l y parameter estimates i n a f l o o d f o r e c a s t i n g a p p l i c a t i o n a r e discussed.

Keywords. Water Resources, Hydrology, R a i n f a l l - R u n o f f Modelling, Parameter Estimation, Computer A p p l i c a t i o n s .

INTRODUCTION Laboratory computer.

Computer program HEC-1, which simulates t h e h y d r o l o g i c response o f urban and r u r a l water- sheds, was developed as a t o o l t o a s s i s t t h e s t a f f o f t h e U.S. Army Corps o f Engineers i n meeting t h e i r water management r e s p o n s i b i l i - t i e s . The basic concepts embodied i n t h e pro- gram were conceived i n 1966 when Beard and o t h e r members o f The Hydrologic Engineering Center (HEC) developed a s e t o f small programs t h a t could be used independently t o s o l v e t h e i n d i v i d u a l tasks t y p i c a l l y r e q u i r e d i n a hydro- l o g i c study. Included i n t h e s e t o f programs was one which employed a u n i v a r i a t e v e r s i o n o f Newton's technique t o c a l i b r a t e a u t o m a t i c a l l y u n i t hydrograph and l o s s r a t e parameters (HEC, 1967). I n 1967, when t h e s e t o f programs was combined i n t o t h e s i n g l e computer program e n t i t l e d HEC-1 Flood Hydrograph Package, t h i s technique was adopted and has been r e t a i n e d through subsequent r e v i s i o n s , i n c l u d i n g t h e l a t e s t 1973 version. Over 400 copies o f t h e l a t e s t v e r s i o n have been d i s t r i b u t e d t o p r i - vate c o n s u l t i n g f i r m s , u n i v e r s i t i e s , and governmental agencies i n both t h e U n i t e d States and o t h e r c o u n t r i e s . The program has been executed n e a r l y 4000 times annually by Corps personnel using the Lawrence Berkeley

The automatic parameter e s t i m a t i o n o p t i o n o f t h e program has been e x t e n s i v e l y used f o r sev- e r a l reasons. F i r s t , t h e Hydrograph Analysis Package i s composed o f a s e t o f simple con- ceptual procedures employing lumped parameters t h a t are intended t o model t h e general behav- i o r o f h y d r o l o g i c phenomena. The parameters o f the conceptual procedures are i n f e r r e d from p r e c i p i t a t i o n and streamflow records r a t h e r than by d i r e c t measurement o f physical water- shed c h a r a c t e r i s t i c s . As a r e s u l t o f i n a c - curacies i n the modelling process and i n the measurement o f i n p u t data, even experienced users have d i f f i c u l t y i n determining p r e c i s e parameter values. Second, many users a r e i n - experienced w i t h t h e model, and t h e automatic c a l i b r a t i o n f e a t u r e e l i m i n a t e s t h e f r u s t r a t - i n g t r i a l and e r r o r approach t o e s t i m a t i n g acceptable parameter values.

This paper describes t h e major t e c h n i c a l com- ponents o f t h e program and d e t a i l s the basic s t r u c t u r e o f i t s c a l i b r a t i o n technique. The s t r a t e g y f o r employing t h e automatic c a l i b r a - z i o n f e a t u r e s o f t h e program i n r e g i o n a l stud- i e s i s discussed, and several t y p i c a l a p p l i c a - t i o n s a r e described. A f l o o d f o r e c a s t i n g

'presented a t I n t e r n a t i o n a l Federation o f Automatic Control Symposium on Water and Related Land Resource Systems, 28-31 May 1980, Cleveland, Ohio.

application is summarized that involves sequentially updating the model 's parameters as successive forecasts are calculated during a storm event. In addition, application of an alternative optimization scheme, involving the random search method of Box (1965), is presented.

DESCRIPTION OF THE HEC-1 FLOOD HYDROGRAPH PACKAGE

The HEC-1 computer program consists of three major hydrologic components that determine the average precipitation and snowmelt and the amount of effective precipitation contributing to direct runoff from a subbasin, compute the subbasin runoff hydrograph from

the effective precipitation, and route and combine the subbasin runoff hydrographs. A1 1 components in HEC-1 employ lumped parameters for each subbasin or routine reach. This means that the model's input, parameters, and output are considered to be average values over the entire subbasin of interest. When subbasin averages are not appropriate, smaller subbasins can be defined to obtain a better spatial definition.

The first computational step in the program determines the average subbasin precipitation from either historical gaged data or hypo- thetical storms. This is followed by an accounting of interception and accumulation of soil moisture by a loss rate function to compute the subarea rainfall or snowmelt excess. The moisture excess is then distri- buted in time by a unit hydrograph function, added to a base flow function, and recessed by a logarithmic decay function once a spec- ified recession flow rate is reached on the falling limb of the hydrograph. This yields the total runoff hydrograph at the subbasin's outlet. Next, the runoff hydrograph from the subbasin is conveyed downstream using a stream- flow routing function. Tributary streamflows are added at the confluences as the simulation proceeds downstream. Snowfall and snowmel t are simulated in each subarea according to temperatures in various elevation bands.

Either the degree-day or energy budget method may be used to compute the snowmelt. Complex stream systems can be simulated if the unit graph, loss rate and routing parameters are specified along with either observed precipi- tation or a specified precipitation depth-area relationship. In addition, expected annual damages can be calculated at any point in the stream system where f 1 ow-frequency and flow- damage relationships are provided. The model 's automatic calibration features can be used to select unit hydrograph and loss rate parameters in a single subbasin or choose the routing parameters in an individual river reach. In both cases the optimization scheme is based on comparisons with the observed and simulated hydrographs .

Loss Rate Functions

Precipitation losses to interception, depres- sion storage and infiltration may be simulated by one of three loss rate functions: initial loss followed by a constant loss rate, the SCS curve number technique (Soil Conservation Service, 1972), or the HEC exponential loss rate function. The latter computes precip- itation losses as a function of antecedent soil moisture, precipitation intensity, and an infiltration rate that is a non-linear function of accumulated losses, as shown in Fig. 1. For a snow-free basin the equations for the HEC-1 exponential loss rate are as fol 1 ows

:

AI.OSS

=

(AK

+

DL.TK) (RAIN) ERAIN

_(,)

for (CUML./DLT'KR) <1

;

otherwise DLTK

= 0.

(3) where ALOSS is the loss rate in mm/hr; AK is the basic loss coefficient, DLTK is the incre- mental loss coefficient; RAIN is the rainfall intensity in mm/hr; ERAIN is an exponent that reflects the influence of precipitation inten- sity on the basin average loss characteris- tics; STRKR is the starting value of the basic loss index on the exponential recession curve in mrn/hr; RTIOL is the ratio of the loss rate coefficient AK to that after 254 mm more of accumulated loss occurs; CUML is the accumu- lated loss in mm; and DLTKR is an incremental loss index.

The program contains a separate set of loss rate equations that are employed when the snowmelt capabilities of the program are des

i

red

:

AL.0SS

=

AK (RAIN + ^{SNWMT) ERAIN} (4) AK

=

(ST'RKS)/ (RTIOK) 0.1 CUML

(5) where SNWMT is the snowmelt in mm/hr, STRKS is the basic loss coefficient for snowmelt in mm/hr, and RTIOK is similar to RTIOR for snowmelt conditions. Equations 4 and 5 are used in lieu of equations 1 and 2 whenever snowmelt occurs. The amount of snowmelt is calculated separately in each elevation zone based on the air temperature which is cal- culated from a base temperature at the lowest elevation and a user supplied adiabatic lapse rate. The degree-day method for computing snowmelt is:

SNWMT

=

COEF (TMPR - ^FRZTP) (6)

where TMPR is the air temperature in

OC

lapsed

to the mid~oint of the elevation zone, FRZTP

is the temperature in

OC

at which snow melts,

and COEF is the melt coefficient in mm per

degree-day (OC). Energy-budget equations f o r m e l t d u r i n g r a i n o r m e l t d u r i n g r a i n f r e e periods can a l s o be employed. The losses are subtracted from r a i n f a l l and snowmel t i n each zone, and t h e excesses a r e summed t o y i e l d t h e excess p r e c i p i t a t i o n from t h e subbasin.

I n t h e l o s s r a t e and snowmel t equation, t h e f o l 1 owing parameters must be determined by c a l i b r a t i o n : STRKR, RTIOL, DLTKR, ERAIN, COEF, STRKS, RTIOK, and FRZTP.

U n i t Hydrograph Functions

-

The p r e c i p i t a t i o n excess-to-runoff t r a n s f o r - mation i s accomplished by t h e use o f t h e u n i t hydrograph. Sherman (1932) d e f i n e d t h e u n i t hydrograph as f o l l o w s :

If a given one-day r a i n f a l l produces a one-inch depth o f r u n o f f over t h e given drainage area, the hydrograph showing t h e r a t e s a t which the r u n o f f occurred can be considered a u n i t I

graph f o r t h e watershed.

A p p l i c a t i o n o f t h i s technique t o p r e c i p i t a t i o n excess amounts o t h e r than one i n c h i s accom- p l i s h e d by m u l t i p l y i n g p r e c i p i t a t i o n excess amounts by t h e u n i t hydrograph ordinates because the r u n o f f ordinates f o r a given dura- t i o n are assumed t o be d i r e c t l y p r o p o r t i o n a l t o t h e r a i n f a l l excess. U n i t hydrograph o r d i n - ates can be supplied d i r e c t l y t o t h e program o r t h e u n i t hydrograph can be c a l c u l a t e d using techniques proposed by Clark (1945), Snyder (1938) o r t h e SCS (1972).

The Clark method uses two parameters and a time-area r e l a t i o n s h i p t o d e f i n e an i n s t a n - taneous u n i t hydrograph. Experience by t h e HEC has i n d i c a t e d t h a t t h e use o f a d e t a i l e d time area r e l a t i o n s h i p i s u s u a l l y n o t warran- t e d and t h a t one based on a generalized water- shed shape (contained w i t h i n t h e program) i s s a t i s f a c t o r y i n most instances. When used i n t h i s fashion, C l a r k ' s technqiue r e q u i r e s o n l y TC, t h e time o f concentration, and R, a storage constant, t o d e f i n e t h e o r d i n a t e s o f t h e u n i t hydrograph. This f u n c t i o n i s a t t r a c - t i v e because i t avoids the d i f f i c u l t i e s assoc- i a t e d w i t h the c a l i b r a t i o n o f many i n d i v i d u a l u n i t graph ordinates. The general shape o f t h e hydrograph i s f i x e d and problems assqc- i a t e d w i t h negative ordinates and i n f e a s i b l e f l u c t u a t i o n s o f t h e u n i t hydrograph are e l i m - inated.

I n a d d i t i o n t o t h e Clark parameters, Snyder's c o e f f i c i e n t s TP and CP, which d e f i n e t h e peak o f the u n i t hydrograph, can be provided as i n - put. The program i n t e r n a l l y uses the C l a r k procedure by i n t e r a t i v e l y varying TC and R u n t i l t h e peak o f the u n i t hydrograph corres- ponds t o t h e one described by t h e s p e c i f i e d Snyder's parameters. The SCS dimensionless u n i t hydrograph technique, which uses a s i n g l e l a g parameter TLAG t o d e f i n e t h e shape o f a t r i a n g u l a r u n i t hydrograph, can a l s o be used.

The u n i t hydrograph parameters r e q u i r e d f o r c a l i b r a t i o n are TC and R, o r TP and CP, o r T LAG.

Streamflow R o u t i n 9

----

Subbasin r u n o f f i s r o u t e d downstream u s i n g one o f t h e f o l l o w i n g ' h y d r o l o g i c 1 r o u t i n g tech- niques: Muskingum, working R&D, s t r a d d l e - stagger, Tatum, m o d i f i e d Puls o r mu1 t i p l e storage. Parameters f o r t h e f i r s t f o u r meth- ods can be a u t o m a t i c a l l y c a l i b r a t e d . These parameters i n c l u d e the f o l l o w i n g :

1. Number o f r o u t i n g steps t o be used f o r r o u t i n g by t h e Tatum method, Muskingum method, o r modified Puls method (NSTPS).

2. Number of o r d i n a t e s t o be averaged i n the straddle-stagger r o u t i n g (NSTDL.)

.

3. Number o f i n t e r v a l s the hydrograph i s t o be lagged i n t h e straddle-stagger r o u t i n g ( L.AG )

.

4. C o e f f i c i e n t s o f t h e Muskingum r o u t i n g f u n c t i o n (AMSKK and

X).

5. Time-of-storage c o e f f i c i e n t f o r t h e mu1 t i p l e storage r o u t i n g (TSK).

The non-linear storage o u t f l o w r e l a t i o n s h i p r e q u i r e d f o r t h e m o d i f i e d Puls and working R&D methods can be obtained using steady- s t a t e water surface p r o f i l e computations o r using a separate o p t i m i z a t i o n program such as the one suggested by Slocum and Dandekar (1 975).

PARAMETER ESTIMATION TECHNIQUES I f HEC-1 were a p e r f e c t model o f watershed hydrology, and i f t o t a l p r e c i p i t a t i o n and t o t a l d i r e c t r u n o f f could be measured accurate1 y, the parameters o f t h e p r e c i p i t a - t i o n - r u n o f f t r a n s f o r m a t i o n f u n c t i o n s f o r a p a r t i c u l a r storm event could be determined d i r e c t l y by i n v e r s e s o l u t i o n o f t h e t r a n s f o r - mation equations. However, these c o n d i t i o n s are n o t s a t i s f i e d i n r e a l i t y , and i n v e r s e s o l u t i o n o f t h e equations i s d i f f i c u l t , so t h e parameters are found i n s t e a d by selec- t i o n o f those values t h a t y i e l d the " b e s t "

reproduction o f a measured r u n o f f event w i t h the a v a i l a b l e measured p r e c i p i t a t i o n data and t h e a v a i l a b l e model. This parameter selec- t i o n has o f t e n been accomplished by a system- a t i c t r i a l - a n d - e r r o r procedure: parameter values a r e selected, t h e model i s exercised w i t h these values, and, the r e s u l t i n g r u n o f f hydrograph i s compared w i t h t h e observed hydrograph. I f t h e " f i t " i s l e s s than s a t i s - f a c t o r y , d i f f e r e n t parameter values are se-.

l e c t e d , t h e model i s exercised w i t h these values, and, t h e r e s u l t i n g r u n o f f hydrograph i s compared w i t h t h e observed hydrograph. I f t h e " f i t " i s l e s s than s a t i s f a c t o r y , d i f f e r e n t parameter values a r e selected, and the e n t i r e process i s repeated.

An alternative to the trial-and-error approach to parameter selection i s an automatic cal- ibration approach in which the necessary tasks f o r calibration are automated. Automatic cal- ibration requires selection of an e x p l i c i t index of the acceptability of a l t e r n a t i v e par- ameter estimates, definition of the range of f e a s i b l e values of the parameters, and develop- ment of some techniques for correction of the parameter estimates until the "best" estimates a r e determined. Thus the parameter estimation problem can be c l a s s i f i e d as an optimization problem: there i s an objective function f o r which an optimal value i s sought, subject t o certain constraints on the decision variables ( t h e parameters). The HEC-1 program includes the capability to solve t h i s optimization pro- blem, thereby automatically determining op- timal parameter estimates.

Objective Function

The objective function of the parameter e s t i - mation optimization problem must define the difference between the computed runoff hydro- graph (with any parameter estimates) and the recorded runoff hydrograph. Presumably, t h i s difference will be a t a minimum f o r the o p t i - mal parameter estimates. In HEC-1, the f o l - lowing objective function i s employed as an index of the errors:

where STDER

=

the e r r o r index; QOBS.

=

observed runoff hydrograph ordinate'for time period i ;

QCOMP. =

the runoff hydrograph ordinate f o r time period i , computed by HEC-1 with the current parameter estimates;

N =

total number of hydrograph ordinates;

WTi =

a weight f o r the hydrograph ordinate. The weight, WTi, i s defined as follows:

WTi =

(QOsSi

+ QAVE) /

(2 *

^{QAVE,) ( 8 )}

where

QAVE =

the average computed discharge.

This weighting function emphasizes accurate reproduction of peak flows rather than low flows by biasing the objective function. Any e r r o r s f o r discharge ordinates t h a t exceed the average discharge will be weighted more heavily, and hence the optimization scheme should focus on reduction of these errors.

Constraints

--

The range of f e a s i b l e values of the parame- t e r s i s bounded because of physical limita- tions on the values t h a t the various u n i t hydrograph, loss r a t e , and snowmelt parame- t e r s may have, and also because of numerical limitations imposed by the mathematical func- tions employed to model watershed behavior.

In addition to bounds on the maximum and min- imum values of certain parameters, the i n t e r - action of some parameters i s also r e s t r i c t e d because of physical or numerical limitations.

These constraints a r e summarized in Table 1.

The constraints shown here are limited t o those imposed e x p l i c i t y in the program.

Additional constraints may be appropriate in certain circumstances; however, these must be imposed externally t o the program when the user must decide whether t o accept, t o modify, or t o r e j e c t a given parameter s e t , based on engineering judgment.

(TC

+

R ) >

1.03/(1. - [R/(TC

+ R ) ] ) R/

(TC + RT

<

0.521

(TC +. R )

R / ( T C

+ R) 5 1.0 - (1.03/(TC

+

R)) ERAIN

<

1.0

RTIOL 1. 1.0 RTIOK , ^1.0

FRZTP 2 -1

. l l

^{O C} F'RZTP 5 3.33

"C

OPTIMIZATION TECHNIQUE

The constrained optimization scheme employed in HEC-1 i s a univariate search technique t h a t uses Newton's method. Application of such a technique permits use of the sirnula-, tion c a p a b i l i t i e s of

HEC-1

in a t r a d i t i o n a l manner and does not require development of analytical derivatives. Steps in applica- tion of t h i s technique, as implemented in

HEC-1,

are as follows:

1 . I n i t i a l values are assigned f o r a l l par- ameters. These values may be assigned by the program user, or program-assigned default values may be used. The default

~ a r a m e t e r values a r e shown in Tables 2 and 3.

2. The response of the watershed i s simula- ted with the i n i t i a l parameter estimates, and the value of the objective function i s computed by comparison of the ordin- ates of the computed and observed runoff hydrographs.

3.

In the order shown in Tables

2

and

3,

each parameter t o be estimated i s de- creased by one percent and then by two percent, the system response i s eval- uated, and the objective function cal- culated f o r each change, respective1 y.

This gives three separate system evalu- ations a t equal1 y-spaced values of the parameter with a l l other parameters held constant. The "best" value of the par- ameter i s then estimated using Newton's method.

4.

Step

3

i s repeated, using the "bestN estimates of the parameters.

5. Step

3

i s repeated f o r the parameter t h a t

most improved the value of the objective

function in i t s l a s t change until no s i n -

gle change in any parameter yields a

r e d u c t i o n o f t h e o b j e c t i v e f u n c t i o n o f more than one percent.

6. One more complete search o f a l l param- e t e r s i s made.

7. Step 5 i s repeated, and t h e f i n a l param- e t e r estimates a r e i d e n t i f i e d as optimal.

For the integer-valued parameters, t h e estimate i s increased o r decreased f o r each i n t u r n u n t i l a minimum i s determined.

The scheme employed f o r e s t i m a t i n g t h e "best"

value o f each parameter i n Step 3 i s based on t h e concept t h a t t h e optimum o f the objec- t i v e f u n c t i o n occurs a t a r o o t o f the f i r s t p a r t i a l d e r i v a t i v e o f t h e f u n c t i o n w i t h respect t o each o f the parameters. These d e r i v a t i v e s cannot e a s i l y be evaluated a n a l y t i c a l 1 y because t h e o b j e c t i v e f u n c t i o n i n d i r e c t 1 y includes a l l t h e f u n c t i o n s and equations contained i n t h e HEC-1 watershed response. Therefore, numerical approxima- t i o n s of t h e d e r i v a t i v e s are used.

TABLE 2 Program HEC-1 D e f a u l t I n i t i a l U n i t Hydrograph and Loss

-

Rate Parameter Estimates

Parameter Parameter I n i t i a l

Number Number

--

Value

3 COEF 0.07

4 STRKR 0.2

5 STRKS 0.2

6 RTIOK 2.0

7 ERAIN 0.5

8 FRZTP 0.0

9 DLTKR 0.5

10 RTIOL 2.0

TAREA = Drainage area, i n square miles TRHR = Computation i n t e r v a l , i n hours

TABLE 3 Program HEC-1 D e f a u l t I n i t i a l Routing Parameter Estimates

-

Parameter I n i t i a l

Name

---

Value

NSTPS NSTDL LAG AMSKK

X T'S K

1 1 1 TRHR .2 3*(TRHR) APPLICATION OF THE CALIBRATION CAPABILITY

Due t o t h e v a r v i n g q u a n t i t y and form o f data a v a i l a b l e f o r p r e c i p i t a t i o n - r u n o f f analysis, the exact sequence o f steps i n a p p l i c a t i o n o f t h e automatic c a l i b r a t i o n c a p a b i l i t y o f HEC-1 v a r i e s from study t o study. An o f t e n - used s t r a t e g y employs t h e f o l l o w i n g steps when using t h e complete exponential l o s s

r a t 1.

2.

3.

4.

5.

6 .

7 .

8.

:e equation:

Determine f o r each storm selected f o r use i n c a l i b r a t i o n t h e base f l o w and reces- s i o n parameters t h a t a r e event dependent and a r e n o t i n c l u d e d i n the s e t o f param- e t e r s t h a t can be estimated a u t o m a t i c a l l y . These parameters are t h e recession f l o w f o r antecedent r u n o f f (STRTQ), t h e d i s - charge a t which recession f l o w begins (QRCSN)

,

and t h e recession c o e f f i c i e n t t h a t i s t h e r a t i o o f f l o w a t some time t o t h e f l o w t e n time periods l a t e r (RTIOR)

.

These parameters are i 11 us- t r a t e d i n Fig. 2 . The HEC-1 Users Manual (HEC, 1973) suggests techniques f o r e s t i m a t i n g these parameters.

For each storm a t each gage, determine the optimal estimates o f a l l unknown u n i t hydrograph and l o s s r a t e parameters using t h e automatic c a l i b r a t i o n f e a t u r e o f HEC-1.

I f ERAIN i s t o be estimated, s e l e c t a r e g i o n a l value o f ERAIN, based on anal- y s i s o f t h e r e s u l t s o f Step 2 f o r a l l storms f o r the r e p r e s e n t a t i v e gages.

Using the o p t i m i z a t i o n scheme, estimate t h e unknown parameters w i t h ERAIN now f i x e d a t the selected value. S e l e c t an a p p r o p r i a t e r e g i o n a l value o f RTIOL i f RTIOL i s unknown. I f t h e temporal and s p a t i a l d i s t r i b u t i o n o f p r e c i p i t a t i o n i s n o t w e l l defined, an i n i t i a l loss, f o l l o w e d by a u n i f o r m l o s s r a t e may be appropriate. I n t h i s case, ERAIN = 0 and RTIOL = 1. I f these values a r e used, as they o f t e n a r e i n s t u d i e s accomplished a t HEC, Steps 2, 3, and 4 are omitted.

With ERAIN and RTIOL f i x e d , estimate the remaining unknown parameters using t h e o p t i m i z a t i o n scheme. S e l e c t a value of STRKR f o r each storm being used f o r c a l i b r a t i o n . I f parameter values f o r adjacent basins have been determined, check t h e selected value f o r r e g i o n a l consistency.

With ERAIN, RTIOL, and STRKR f i x e d , use the parameter e s t i m a t i o n a l g o r i t h m t o compute a l l remaining unknown parameters.

DLTKR can be generalized and f i x e d i f d e s i r e d a t t h i s p o i n t , although t h i s parameter i s considered t o be r e l a t i v e l y event-dependent

.

Using t h e c a l i b r a t i o n c a p a b i l i t y o f HEC-1, determine values o f TR+R and R/ (TCtR)

.

S e l e c t a p p r o p r i a t e values o f TC+R f o r each gage. I n order t o determine TC and R, ^an average value o f R/(TC+R) i s t y p i c a l l y selected f o r t h e region.

Once a l l parameters have been selected, t h e values should be v e r i f i e d by sim- u l a t i n g t h e response o f t h e gaged basins t o o t h e r events f o r which p r e c i p i t a t i o n and r u n o f f records are a v a i l a b l e .

APPL,ICATIONS

OF

HEC-1

AT

THE

HEC

The HEC-1 optimization scheme has been used by the HEC in numerous studies f o r nearly 10 years. The applications have focused on developing frequency curves f o r ungaged locations and on modelling the iapact of basin modifications, of channel improvements, or of additional control measures a t selected locations.

Many of the recent applications of HEC-1 accomplished a t the Hydrologic Engineering Center have employed the automatic calibra- tion scheme in development of data f o r ungaged areas. Typically in these studies data a r e available from stream and precipitation measurement s t a t i o n s in the proximity of a location f o r which detailed stage and dis- charge data a r e unavailable

b u t

a r e desired.

The automatic calibration technique i s used to estimate unit hydrograph, loss r a t e , and routing parameters f o r the gaged locations, and t h i s data i s "transferred" to the ungaged locations using regression techniques. The particular strategy f o r estimating parameters and the methods f o r transferring the param- e t e r s to ungaged locations i s a function of the basin c h a r a c t e r i s t i c s , the available data, the parameters t h a t a r e found via c a l i b r a t i o n , and the time and money available f o r the study.

While the sequence of steps f o r estimation of a1 1 parameters of t h i s r a i n f a l l -runoff model has been employed in a t l e a s t one major study, the f l e x i b i l i t y gained by use of four param- e t e r s in the exponential loss r a t e equation i s not always necessary. Often ERAIN i s s e t to zero, RTIOL i s s e t t o one, and calibration proceeds with Step 5 t o t h i s sequence. This approach has been employed in studies of the She1 lpot and Naaman Creeks

(HEC,

1976), the Schuyl k i l l River (HEC, 1976), the Maurice River (HEC, 1976), and the Lehigh River (HEC, 1978). In a hydrologic study on the Oconee River Basin, ERAIN and DLTKR were s e t to zero, yielding a simple exponential decay loss function (1976).

The Soil Conservation Service's dimensionless unit graph and rainfall-runoff relationship based on the s o i l c l a s s i f i c a t i o n curve numbers were recent1 y added to HEC-1. An application was made on the Pennypack Creek as part of an expanded flood plain information report (HEC, 1978). The 145

km2

study area was broken into 65 subareas, each requiring unit graph, loss r a t e , and channel storage param- e t e r s . The curve number

( C N )

and lag param- e t e r s , which a r e the only variables necessary t o define the SCS unit graph and loss r a t e , were estimated via optimization f o r the gaged basins. These were used as a guide in estab- lishing

CN

and LAG values f o r the ungaged sub- areas.

FORECASTING APPLICATION

In addition t o the previously described applications of

HEC-1,

the model has recently been applied t o develop reservoir inflow forecasts f o r

W .

Kerr Scott Reservoir on the Yadkin River of North Carolina. For t h i s application, the basic model was modified so t h a t the calibration technique could be used to update sequentially the model parameters. The parameters a r e then used to calculate forecasted streamflows.

In the

W.

Kerr Scott system, 20 gages, nine of them recording, were available t o deter-2 mine mean areal precipitation in the 900

km

basin. Seven storm events were model led, using the optimization procedure t o estimate

TC,

R, ERAIN, RTIOL, and STRKR f o r the basin.

Conceptual 1 y, these parameters wi 11 remain constant from storm t o storm on the same basin, with DLTKR alone indicating the ante- cedent basin wetness. In r e a l i t y , a l l the parameters vary due t o storm centering, t o inaccuracies in precipitation data, t o non- homogeneity of the basin, and t o the approx- imate nature of the hydrologic model. Three combinations of estimated and fixed parameters were investigated. In the f i r s t case, basin average values of

TC

and R were used, and the four loss r a t e parameters were established by calibration. In the second case, ERAIN was s e t t o zero and RTIOL was s e t t o one, while the remaining parameters were estimated by calibration. Finally, a l l s i x parameters were estimated via calibration. When tested f o r seven storm events, the l a t t e r approach provided more accurate predictions when the time of forecast occurred before the peak of the hydrograph. In each case, the param- e t e r s were estimated using the computed re- servoir inflows and observed mean areal precipitation t h a t were available a t the time of forecast. The purpose of the calibration was t o adjust loss r a t e s t o r e f l e c t the ante- cedent conditions and f i n e tune the unit graph parameters t o best f i t each storm.

Forecast Procedure

In order t o t e s t the procedure in a r e a l i s t i c forecasting application, data from four storms were used in a s i t u a t i o n t h a t i s similar t o t h a t encountered by Corps ' f i e l d off ices.

Whenever the calculated reservoir inflow exceeded 50 cms (twice the long term average inflow f o r Scott Reservoir), a forecast was calculated using the data available a t that time. Subsequent forecasts were calculated every two hours, using additional data obser- ved between forecasts, until the inflow hydrograph s t a r t e d to recede. In addition, forecasts were issued whenever subsequent r i s e s occurred during the receding limb.

Thus forecasts were made f o r a l l the peaks in a complex hydrograph.

Two s t r a t e g i e s were investigated f o r estab-

lishing the i n i t i a l values f o r calibration

of the unit graph and loss r a t e parameters.

The i n i t i a l values f o r TC, R, ERAIN, RTIOL f o r method 1 were based on average values determined from t h e c a l i b r a t i o n o f seven storms w h i l e program d e f a u l t values were used f o r STRKR and DLTKR. These s t a r t i n g values were used f o r every f o r e c a s t d u r i n g t h e event, and d i d n o t consider t h e r e s u l t s o f the pre- vious forecast. Method 2 employed t h e same s t a r t i n g values f o r t h e f i r s t f o r e c a s t o f each storm, b u t i n i t i a l values f o r subsequent f o r e c a s t s were s e t equal t o optimal parameter values from the previous forecast.

Analysis o f Forecasting Results,

-

The performance o f a f l o o d f o r e c a s t model can be f u l l y evaluated o n l y a f t e r t h e event has occurred and t h e complete observed i n f l o w hydrograph i s a v a i l a b l e . For r e s e r v o i r operation, both t h e shape and the volume of t h e f l o o d hydrograph a r e important. The f o l l o w i n g s t a t i s t i c a l measures were d e f i n e d t o measure discrepancies between t h e f o r e - casted and observed i n f l o w s , beginning when t h e f o r e c a s t i s issued and ending when t h e observed f l o w recedes t o 20% o f the peak flow:

Volume E r r o r =

Average Discharge E r r o r =

where QO. = observed discharge a t o r d i n a t e i;

QSi = simulated discharge a t o r d i n a t e i; N = number o f h y d r o g r a p h ~ r d i n a t e s .

The o v e r a l l accurancy o f the f o r e c a s t s i s ' a f u n c t i o n o f storm type and i n t e n s i t y , repre- sentativeness o f t h e computed basin average p r e c i p i t a t i o n , and t h e degree o f storm develop- ment. Despite t h e d i f f i c u l t i e s i n p r o v i d i n g h i g h q u a l i t y data f o r the c a l i b r a t i o n , t h e forecasted volumes f o r t h e f o u r storms are reasonable; t h e p r e d i c t e d i n f l o w volumes a r e w i t h i n 25 percent o f t h e observed values 8 t o 10 hours p r i o r t o t h e peak. As expected, f o r e c a s t s made b e f o r e t h e t o t a l p r e c i p i t a t i o n had f a 1 1 en g e n e r a l l y underestimated t h e t o t a l r u n o f f volume.

I n order t o improve t h e performance o f t h e f o r e c a s t i n g scheme, t h e sources o f d i f f e r e n c e s between t h e observed and forecasted discharge must be i s o l a t e d . T y p i c a l l y these sources i n c l u d e e r r o r s i n h e r e n t w i t h i n t h e f o r e c a s t - i n g procedure, e r r o r s i n t h e basin h y d r o l o g i c model, and e r r o r s i n t h e model c a l i b r a t i o n . E r r o r s i n h e r e n t w i t h i n any f o r e c a s t i n g pro- cedure i n c l u d e u n c e r t a i n t y i n measurement o f p r e c i p i t a t i o n and streamflow data, and under-

e s t i m a t i o n o f t h e hydrograph before t h e t o t a l p r e c i p i t a t i o n occurs. I n t h e Yadkin appl i c a - t i o n , i n f l o w i n t o the W. Kerr S c o t t Reser- v o i r was c a l c u l a t e d using the change i n eleva- t i o n of t h e pool l e v e l , elevation-storage tab1 es

,

and t h e r e s e r v o i r re1 eases determined by t h e gate-discharge r a t i n g t a b l e s . F l u c t u - a t i o n s i n t h e r e s e r v o i r pool due t o wind s e t - up and seiching, combined w i t h an unknown amount o f gate slippage caused t h e c a l c u l a t e d i n f l o w s t o o s c i l l a t e considerably. The adopted i n f l o w hydrograph used f o r c a l i b r a t i n g and f o r e c a s t i n g was based on a smoothed curve, so considerable e r r o r s c o u l d occur i n t h e slope o f t h e r i s i n g 1 imb. Because t h i s e a r l y por- t i o n o f the observed hydrograph i s used t o v e r i f y and a d j u s t t h e simulated hydrograph, t h e c a l i b r a t i o n scheme could have f o r c e d a f i t w i t h erroneous data and thus could have degraded the forecasted hydrograph.

The basic s t r u c t u r e o f t h e h y d r o l o g i c model and t h e manner i t i s a p p l i e d can a l s o l e a d t o e r r o r s . Because o f the appreciable l a g time between t h e measurement o f r a i n f a l l on t h e basin and i t s occurrence a t the streamgage, when a f o r e c a s t i s issued e a r l y i n t h e f l o o d event, o n l y a small p o r t i o n o f t h e recorded p r e c i p i t a t i o n may c o n t r i b u t e t o t h e simulated discharge p r i o r t o t h e time o f f o r e c a s t . When t h i s occurs i t i s p o s s i b l e t h a t t h e optim-, i z a t i o n scheme may cause s u b s t a n t i a l changes i n t h e model's parameters w i t h o u t a c o r r e - esponding change i n t h e f i t between t h e sim- u l a t e d and observed discharge. This e f f e c t was e v i d e n t when a l t e r n a t i v e i n i t i a l param- e t e r values were employed. I n 17 o f 25 forecasts, method 2 e x h i b i t e d a lower c a l - i b r a t i o n e r r o r ( e r r o r p r i o r t o f o r e c a s t ) , w h i l e i n a l l b u t two cases method 1 e x h i b i t e d a lower t o t a l e r r o r f o r t h e event. I n seven o f t h e cases f o r which method 2 had t h e lower c a l i b r a t i o n e r r o r , the t o t a l event e r r o r was two t o f i v e times g r e a t e r than t h a t w i t h nethod 1. This demonstrates t h a t an e x c e l - l e n t f i t between t h e observed and simulated discharges does n o t n e c e s s a r i l y l e a d t o a correspondingly good f o r e c a s t .

Several c h a r a c t e r i s t i c s o f t h e parameter e s t i m a t i o n scheme can i n f l u e n c e t h e amount o f f o r e c a s t e r r o r . F i r s t , i f the u n i t hydro- graph and l o s s r a t e parameters are constrained t o o l o o s e l y , as i n t h e HEC-1 scheme,

t h e adopted parameters may be unreasonable.

Method 2, which c o n s i s t e n t l y y i e l d e d b e t t e r r e s u l t s d u r i n g c a l i b r a t i o n , o f t e n c a l c u l a t e d u n r e a l i s t i c values o f TR, R and Rl'IOL.. A second f a c t o r i s t h e allowable e r r o r between t h e observed and simulated values i n t h e c a l i b r a t i o n step. To reduce t h e c a l i b r a t i o n e r r o r , t h e parameter e s t i m a t i o n a l g o r i t h m may produce s i g n i f i c a n t f l u c t u a t i o n s i n par- ameters when o n l y a small p o r t i o n o f precip- i t a t i o n and discharge data are a v a i l a b l e . However, i f t h e a1 lowable t o l e r a n c e i s i n - creased s u f f i c i e n t l y e a r l y i n the storm event and i s g r a d u a l l y reduced, t h e number and mag- n i t u d e o f parameter m o d i f i c a t i o n s would be reduced. This concept was adopted f o r a f l o o d f o r e c a s t i n g model developed by t h e National Weather Service (1 979).

I n summary, t h e performance o f the c a l i b r a - t i o n scheme was o n l y one c o n t r i b u t i n g f a c t o r t o t h e accuracy o f t h e f l o o d f o r e c a s t . While t h e a d d i t i o n o f a v a r i a b l e tolerance o r more l i m i t i n g c o n s t r a i n t s t o t h e present method may be b e n e f i c i a l , t h e r e s u l t s o f t h e HEC-1 model were q u i t e s a t i s f a c t o r y .

ALTERNATIVE OPTIMIZATION ALGORITHMS FOR CALIBRAT ION

I n 1978 t h e Hydrologic Engineering Center began a study o f a l t e r n a t i v e techniques f o r estima- t i o n o f parameters f o r HEC-1. This study was motivated by i n c r e a s i n g use by Corps f i e l d o f f i c e s o f t h e automatic c a l i b r a t i o n f e a t u r e o f t h e model. I n these many a p p l i c a t i o n s o f the model, any s i g n i f i c a n t improvement i n t h e parameter e s t i m a t i o n technique i s desirable, i n terms o f a r e d u c t i o n i n t h e c o s t o f c a l - i b r a t i o n o r i n terms o f a r e d u c t i o n i n t h e e r r o r o f the parameter estimates.

Analysis o f t h e e x i s t i n g u n i v a r i a t e search technique employed f o r parameter e s t i m a t i o n i n HEC-1 i n d i c a t e d t h a t the technique o f t e n does n o t e f f e c t i v e l y handle c o n s t r a i n t s on the parameters, so a n o n l i n e a r programming a l g o r i t h m t h a t does was sought. An addi- t i o n a l r e s t r i c t i o n on s e l e c t i o n o f an op- t i m i z a t i o n scheme i s imposed by the number of parameters t h a t must be estimated. I f a l l parameters o f t h e p r e c i p i t a t i o n - r u n o f f process a r e t o be estimated, t h e o p t i m i z a t i o n algo- r i t h m must be capable o f s o l v i n g e f f i c i e n t l y a problem o f as many as t e n variables.

The a l t e r n a t i v e o p t i m i z a t i o n technique s e l - ected f o r i n i t i a l t r i a l a p p l i c a t i o n was a version o f the random search method o f Box (1965), as suggested by Johnston and P i l g r i m (1 976), Chu and Bowers (1 978), and

Sorooshian and Dracup (1 978). This technique i s an extension o f the polyhedron search o f Nelder and Mead (1965), and as programmed a t HEC, uses t h e p r e c i p i t a t i o n - r u n o f f model d i r e c t l y i n e v a l u a t i o n of t h e o b j e c t i v e func- t i o n . Implementation o f t h e technique re-.

q u i r e s d e f i n i t i o n o f e x p l i c i t upper and lower bounds on the variables; t h i s proved t o be a formidable task because t h e parameters o f t h e r a i n f a l l - r u n o f f model a r e n o t r e l a t e d c l e a r l y t o physical a t t r i b u t e s o f t h e drainage basin.

C u r r e n t l y the c o n s t r a i n t s a r e defined on t h e basis o f knowledge o f t h e l i m i t a t i o n s o f t h e mathematical f u n c t i o n s combined w i t h exper- ience i n manual c a l i b r a t i o n o f t h e model f o r Corps' s t u d i e s nation-wide.

Although t h e i n v e s t i g a t i o n o f a l t e r n a t i v e o p t i m i z a t i o n techniques o r c a l i b r a t i o n schemes i s n o t y e t complete, p r e l i m i n a r y analyses i n d i c a t e t h e f o l l o w i n g : ( 1 ) t h e Box tech- nique does n o t s e l e c t an optimal s e t o f param-

e t e r values i n s i g n i f i c a n t l y l e s s time o r w i t h fewer f u n c t i o n evaluations than does the u n i v a r i a t e g r a d i e n t a l g o r i t h m c u r r e n t l y employed; and ( 2 ) t h e Box technique does n o t s e l e c t parameter values t h a t y i e l d a s i g - n i f ic a n t l y b e t t e r r e c o n s t i t u t i o n o f runoff

events, as measured by t h e least-.squares o b j e c t i v e f u n c t i o n . As a r e s u l t o f t h e research, t h e f o l l o w i n g conclusions were reached regarding t h e parameter e s t i m a t i o n procedure: ( 1 ) f o r e f f i c i e n c y , an automatic c a l i b r a t i o n technique should a l l o w s p e c i f i - c a t i o n o f and should e x p l o i t t h e a v a i l a b i l i t y o f a p p r o p r i a t e i n i t i a l values o f t h e param- eters; and ( 2 ) t h e least-squares o b j e c t i v e f u n c t i o n i s r e l a t i v e l y i n s e n s i t i v e t o v a r i - a t i o n s o f t h e parameters o f HEC-1.

The Box technique i s executed by i n i t i a l l y e v a l u a t i n g t h e o b j e c t i v e f u n c t i o n f o r sets o f parameter values s c a t t e r e d randomly throughout t h e f e a s i b l e r e g i o n (a "complex").

Box recommends t h a t 2n sets o f values be e s t a b l i s h e d f o r the i n i t i a l complex where n equals t h e number o f unknown parameters. For t e n parameters, t h i s r e q u i r e s 20 e v a l u a t i o n s o f t h e o b j e c t i v e f u n c t i o n , each r e q u i r i n g s i m u l a t i o n o f watershed response. The u n i - v a r i a t e technique, w i t h c e r t a i n h e u r i s t i c r u l e s f o r d e a l i n g w i t h v i o l a t e d c o n s t r a i n t s and w i t h good i n i t i a l estimates o f t e n s e l e c t s near-optimal, acceptable parameter estimates w i t h approximately the same number o f func- t i o n evaluations. Furthermore, the random search1 techniques does n o t e x p l o i t t h e knowledge o f good i n i t i a l estimates o f t h e parameters. Experience a t HEC i n d i c a t e s t h a t t h i s i s c r i t i c a l because these estimates o f t e n reduce s u b s t a n t i a l l y t h e e f f o r t t o c a l i b r a t e and because automatic c a l i b r a t i o n schemes may chose an unreasonable, f a l s e optimum o t h e r - wise.

The least-squares o b j e c t i v e function, although w i d e l y accepted f o r a p p l i c a t i o n i n model c a l i b r a t i o n , i n many cases i s i n s e n s i t i v e t o v a r i a t i o n s i n the parameters o f HEC-1 and causes premature t e r m i n a t i o n o f t h e search.

This d i f f i c u l t y i s r e l a t e d a l s o t o i n t e r a c t i o n among t h e v a r i a b l e s o f t h e watershed model.

However, acceptance o f any m o d i f i c a t i o n s t o the model i s n o t l i k e l y , so a l t e r n a t i v e objec- t i v e f u n c t i o n s are being evaluated. These f u n c t i o n s i n c l u d e those suggested by Manley (1978) and o t h e r f u n c t i o n s suggested through research a t HEC.

A d d i t i o n a l tasks t o be completed i n t h e area o f parameter e s t i m a t i o n f o r t h e HEC-1 water- shed model i n c l u d e the f o l l o w i n g : ( 1 ) pro- gramming and t e s t i n g o t h e r n o n l i n e a r program- ming algorithms f o r parameter estimation;

( 2 ) programming and t e s t i n g a1 t e r n a t i v e o b j e c t i v e f u n c t i o n s f o r c a l i b r a t i o n ; ( 3 ) f u r t h e r comparing t h e e x i s t i n g parameter e s t i m a t i o n a l g o r i t h m w i t h t h e a l t e r n a t i v e a1 g o r i thms; and ( 4 ) a p p l y i n g t h e know1 edge gained from parameter e s t i m a t i o n research w i t h program HEC-1 t o o t h e r programs devel- oped and supported by the HEC.

SUMMARY

To s a t i s f y the need f o r a p r e c i p i t a t i o n - r u n o f f model f o r a p p l i c a t i o n i n water resour- ces planning and management by t h e Corps o f

Engineers, computer program HEC-1 was devel- oped. This model includes algorithms to accomplish the following tasks necessary to simulate watershed response:

1 . Determine effective precipitation.

2 .

Compute the subarea runoff due to the

effective precipitation.

3. Route and combine the subarea runoff hydrographs.

In addition, HEC-1 includes the capability to determine automatically the parameters of the functions employed in the simulation. This is accomplished using Newton's technique to minimize a weiqhted least-squares objective function. Currently, a1 ternative optimi- zation techniques and alternative objective functions are being evaluated.

The parameter estimation capability of pro- gram HEC-1 has been employed in a variety of studies at the Hydrologic Engineering Center. These applications have focused on modelling the impact of basin modifications, of channel improvements, of various flood- control measures, and on developing fre- quency curves for ungaged watersheds.

The parameter estimation technique of HEC-1 has been extended recently to update sequen- tially parameter estimates for flood fore- casting. In these applications, computed reservoir inflows and observed mean areal precipitation available at the time of fore- cast are used to estimate model parameters.

These parameters are used then to estimate future reservoir inflows, using the simula- tion capability of the program. The results of the applications are satisfactory for appl

i

cation to fl ood-control reservoir oper- ation.

REFERENCES

Box, M. J. (1965). A new method of con- strained optimization and a comparison with other methods. Tth~Computer Journal, 8, 42-52.

Chu, C. S., and C. E. Bowers (1978). An optimization technique for a mathematical urban runoff model. In Proceedings of International Symposium on Urban Storm

- Water Management. University of Kentucky, Lexington, KY, pp. 95-102.

Clark, C. 0. (1945). Storage and the unit hydrograph. Transactions, American Society of Civil Engineers,---

1419-1488.

Hydrologic Engineering Center (1 973). HEC-1 Flood Hydrograph Package Users M a n u a l . -

U. S. Army Corps of Engineers, Davis, CA.

Hydrologic Engineering Center (1976). Upper Oconee River Basjn Hydrologic Study, Special Projects Memo No. 456, U. S.

Army Corps of Engineers, Davis, CA.

Hydrologic Engineering Center (1 976). Hydro- logic Study Associated with Maurice River - Salem and Cumberland Counties, New Jersey, S ecial Pro'ects Memo No. 459.

U.

S. Army CErps of E n i i ~ r s , Davis,m.

Hvdroloaic Enaineerina Center (1976).

_'

~ c h ; kill+~iver asi in ~ o d e i ~ e v ; ~

~ d r a t i o n Studies, Special Projects Memo No. 465.

U.

S. Army Corps of Engineers, Davis, CA

Hydrologic Engineering Center (1978). Hydro- logic Study of Lehigh River Basin, Pennsylvania, Special Projects Memo No.

- 78-1. U. S. Army Corps of Engineers,

- Davis, CA.

Johnston, P. R., and 0. H. Pilgrim (1976).

Parameter ootimization for watershed models. water ResourcesResearch, 3,

477-486.

Manley, R. E. (1978). Calibration of hydro- logical model using optimization tech- nique. Journal of the Hydraulics, Division, American Society of Civil - Engineers, 104, 189-202.

-

Nat

Nelder, J. A., and R. Mead (1965). A Simplex method for function minimization. The

Computer Journal , L, 308-31 3.

Sherman. L. K. (1932). Streamflow from rain- fali by the 'unitigraph method. Engineer- ing News Record, 108, 501-505.

Slocum, A. H., and R. A. Dandekar (1975).

Evaluation of Streamflow Routinq Tech-

niques with Special Emphasis on Deter-

mining lionlinear Routinq Criteria by-

Optimization. Anderson-Nichols and Co.,

Inc., Boston, MA.

S o i l Conservation Service ( 1 972). S o i l R a i n f a l l - R u n o f f Models. Engineering Conservation Service National E-eeri nq Systems Dept., School o f Engineering and Handbook, Section 4: Hydrology.

U.

S. Applied Science, U n i v e r s i t y o f C a l i f o r n i a , Department o f A g r i c u l t u r e , Washington, Los Angeles, CA.

D. C.

Snyder, F. F. (1938). Synthetic unitgraphs.

Sorooshian, S., and J. A. Dracup (1978). Transactions, American Geophysical Union, Considerations o f Stochastic P r o p e r t i e s

-

1

,

447-454.

i n Parameter Estimation o f Hydrologic

STR KR ---

+

DLTKR

-4

RTIOL = A/B 1.4 rnmJ Loss Rate

Coefficient (AK)

(Arith. scale)

Q

-

Accumulated Lass (CUML)

Fig. 1. HEC-1 exponential l o s s r a t e f u n c t i o n

Discharge

STRTQ

Time

I I

k 1 6 ~ t i

Fig. 2. Base f l o w and recession parameters

Technical Paper Series

TP-1 Use of Interrelated Records to Simulate Streamflow TP-2 Optimization Techniques for Hydrologic

Engineering

TP-3 Methods of Determination of Safe Yield and Compensation Water from Storage Reservoirs TP-4 Functional Evaluation of a Water Resources System TP-5 Streamflow Synthesis for Ungaged Rivers

TP-6 Simulation of Daily Streamflow

TP-7 Pilot Study for Storage Requirements for Low Flow Augmentation

TP-8 Worth of Streamflow Data for Project Design - A Pilot Study

TP-9 Economic Evaluation of Reservoir System Accomplishments

TP-10 Hydrologic Simulation in Water-Yield Analysis TP-11 Survey of Programs for Water Surface Profiles TP-12 Hypothetical Flood Computation for a Stream

System

TP-13 Maximum Utilization of Scarce Data in Hydrologic Design

TP-14 Techniques for Evaluating Long-Tem Reservoir Yields

TP-15 Hydrostatistics - Principles of Application TP-16 A Hydrologic Water Resource System Modeling

Techniques

TP-17 Hydrologic Engineering Techniques for Regional Water Resources Planning

TP-18 Estimating Monthly Streamflows Within a Region TP-19 Suspended Sediment Discharge in Streams TP-20 Computer Determination of Flow Through Bridges TP-21 An Approach to Reservoir Temperature Analysis TP-22 A Finite Difference Methods of Analyzing Liquid

Flow in Variably Saturated Porous Media TP-23 Uses of Simulation in River Basin Planning TP-24 Hydroelectric Power Analysis in Reservoir Systems TP-25 Status of Water Resource System Analysis TP-26 System Relationships for Panama Canal Water

Supply

TP-27 System Analysis of the Panama Canal Water Supply

TP-28 Digital Simulation of an Existing Water Resources System

TP-29 Computer Application in Continuing Education TP-30 Drought Severity and Water Supply Dependability TP-31 Development of System Operation Rules for an

Existing System by Simulation

TP-32 Alternative Approaches to Water Resources System Simulation

TP-33 System Simulation of Integrated Use of Hydroelectric and Thermal Power Generation TP-34 Optimizing flood Control Allocation for a

Multipurpose Reservoir

TP-35 Computer Models for Rainfall-Runoff and River Hydraulic Analysis

TP-36 Evaluation of Drought Effects at Lake Atitlan TP-37 Downstream Effects of the Levee Overtopping at

Wilkes-Barre, PA, During Tropical Storm Agnes TP-38 Water Quality Evaluation of Aquatic Systems

TP-39 A Method for Analyzing Effects of Dam Failures in Design Studies

TP-40 Storm Drainage and Urban Region Flood Control Planning

TP-41 HEC-5C, A Simulation Model for System Formulation and Evaluation

TP-42 Optimal Sizing of Urban Flood Control Systems TP-43 Hydrologic and Economic Simulation of Flood

Control Aspects of Water Resources Systems TP-44 Sizing Flood Control Reservoir Systems by System

Analysis

TP-45 Techniques for Real-Time Operation of Flood Control Reservoirs in the Merrimack River Basin TP-46 Spatial Data Analysis of Nonstructural Measures TP-47 Comprehensive Flood Plain Studies Using Spatial

Data Management Techniques

TP-48 Direct Runoff Hydrograph Parameters Versus Urbanization

TP-49 Experience of HEC in Disseminating Information on Hydrological Models

TP-50 Effects of Dam Removal: An Approach to Sedimentation

TP-51 Design of Flood Control Improvements by Systems Analysis: A Case Study

TP-52 Potential Use of Digital Computer Ground Water Models

TP-53 Development of Generalized Free Surface Flow Models Using Finite Element Techniques TP-54 Adjustment of Peak Discharge Rates for

Urbanization

TP-55 The Development and Servicing of Spatial Data Management Techniques in the Corps of Engineers TP-56 Experiences of the Hydrologic Engineering Center

in Maintaining Widely Used Hydrologic and Water Resource Computer Models

TP-57 Flood Damage Assessments Using Spatial Data Management Techniques

TP-58 A Model for Evaluating Runoff-Quality in Metropolitan Master Planning

TP-59 Testing of Several Runoff Models on an Urban Watershed

TP-60 Operational Simulation of a Reservoir System with Pumped Storage

TP-61 Technical Factors in Small Hydropower Planning TP-62 Flood Hydrograph and Peak Flow Frequency

Analysis

TP-63 HEC Contribution to Reservoir System Operation TP-64 Determining Peak-Discharge Frequencies in an

Urbanizing Watershed: A Case Study

TP-65 Feasibility Analysis in Small Hydropower Planning TP-66 Reservoir Storage Determination by Computer

Simulation of Flood Control and Conservation Systems

TP-67 Hydrologic Land Use Classification Using LANDSAT

TP-68 Interactive Nonstructural Flood-Control Planning TP-69 Critical Water Surface by Minimum Specific

Energy Using the Parabolic Method