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CHAPTER 3

METHOD OF ANALYSIS

In this research the goal is to determine an IUH from observed rainfall-runoff

data. This research assumes that an IUH exists, and that it is the response function to a

linear system, and the research task is to find the parameters (unknon coefficients! of

the transfer function.

To accomplish this task a database must be assembled that contains appropriate

rainfall and runoff values for analysis. "nce the data are assembled, the runoff signal is

analy#ed for the presence of any base flo, and this component of the runoff signal is

removed. "nce the base flo is removed, the remaining hydrograph is called the direct

runoff hydrograph ($%H!. The total volume of discharge is determined and the rainfall

input signal is analy#ed for rainfall losses. The losses are removed so that the total

rainfall input volume is e&ual to the total discharge volume. The rainfall signal after this

process is called the effective precipitation. 'y definition, the cumulative effective

precipitation is e&ual to the cumulative direct runoff.

If the rainfall-runoff transfer function and its coefficients are knon a-priori, then

the $%H signal should be obtainable by convolution of the rainfall input signal ith the

IUH response function. The difference beteen the observed $%H and the model $%H

should be negligible if the data have no noise, the system is truly linear, and e have

selected both the correct function and the correct coefficients.

If the analyst postulates a functional form (the procedure of this thesis! then

searches for correct values of coefficients, the process is called de-convolution. In the

present ork by guessing at coefficient values, convolving the effective precipitation



signal, and comparing the model output ith the actual output, e accomplish de-

convolution. ) merit function is used to &uantify the error beteen the modeled and

observed output. ) simple searching scheme is used to record the estimates that reduce

the value of a merit function and hen this scheme is completed, the parameter set is

called a non-inferior (as opposed to optimal! set of coefficients of the transfer function.

3.1. Database Construction

U*+* small atershed studies ere conducted largely during the period spanning

the early /s to the middle 0/s. The storms documented in the U*+* studies can be

used to evaluate unit hydrographs and these data are critical for unit hydrograph

investigation in Texas. 1andidate stations for hydrograph analysis ere selected and a

substantial database as assembled.

Table 2. is a list of the 33 stations eventually keypunched and used in this

research. The first to columns in each section of the table is the atershed and sub

atershed name. The urban portion of the database does not use the sub atershed

naming convention, but the rural portion does. The third column is the U*+* station I$

number.

This number identifies the gauging station for the runoff data. The precipitation data

is recorded in the same reports as the runoff data so this I$ number also identifies the

precipitation data. The last numeric entry is the number of rainfall-runoff records

available for the unit hydrograph analysis. The details of the database construction are

reported in )s&uitn et. al (45!.

4



Table 2..*tations and 6umber of *torms used in *tudy

Watershed Sub-Shed Station ID #Events Watershed Sub-Shed Station ID #Events

BartonCreek 08155200 5 AshCreek 0805720 5

BartonCreek 0815500 8 Ba!h"anBran!h 08055700 1

BearCreek 08158810 8 CedarCreek 08057050

BearCreek 08158820 2 Coo"bsCreek 08057020 7

BearCreek 08158825 2 CottonWoodCreek 0805710 $

Bo%%&Creek 08158050 10 Du!kCreek 080$1$20 8

Bo%%&SouthCreek 08158880 1 E'a"Creek 0805715 8

Bu''Creek 0815700 1 (ive)i'eCreek 0805718 7

*itt'eWa'nutCreek 0815880 2 (ive)i'eCreek 0805720 10

+nionCreek 08158700 $ ('o&dBran!h 080571$0 8

+nionCreek 08158800 2 ,oesCreek 08055$00 1

Shoa'Creek 0815$$50 1 e.tonCreek 080575

Shoa'Creek 0815$700 1$ /rairieCreek 080575 8

Shoa'Creek 0815$750 1 ushBran!h 0805710 5

Shoa'Creek 0815$800 2 South)esuite 080$120

S'au%hterCreek 0815880 South)esuite 080$150 1

S'au%hterCreek 081588$0 2 S3ank&Creek 08057120

Wa''erCreek 08157000 0 4urt'eCreek 0805$500 2

Wa''erCreek 08157500 8 Wood&Bran!h 0805725 1

Wa'nutCreek 08158100 15Wa'nutCreek 08158200 17 Watershed Sub-Shed Station ID #Events

Wa'nutCreek 0815800 10 Dr&Bran!h 0808550 25

Wa'nutCreek 08158500 1 Dr&Bran!h 0808$00 27

Wa'nutCreek 08158$00 22 *itt'e(ossi' 0808820 20

WestBou'dinCreek 08155550 10 *itt'e(ossi' 0808850 2

Wi'bar%erCreek 0815150 2 S&!a"ore 0808520 2

Wi''ia"sonCreek 0815820 1 S&!a"ore 080850 28

Wi''ia"sonCreek 081580 18 S&!a"ore 080850 2

Wi''ia"sonCreek 0815870 1$ S&!a"ore SSSC 21

Watershed Sub-Shed Station ID #Events Watershed Sub-Shed Station ID #Events

A'aanCreek 0817800 0 BrasosBasin Co.Ba&ou 080$800 8

*eonCreek 08181000 10 BrasosBasin 6reen 080000 28

*eonCreek 0818100 15 BrasosBasin /ond-E'" 080800 1

*eonCreek 0818150 2 BrasosBasin /ond-E'" 08108200 21

+'"osCreek 08177$00 12 Co'oradoBasin Dee3 081000 27

+'"osCreek 08177700 2 Co'oradoBasin Dee3 0810000 28

+'"osCreek 08178555 10 Co'oradoBasin )uke.ater 081$00 22

Sa'adoCreek 08178$00 1 Co'oradoBasin )uke.ater 0817000 8

Sa'adoCreek 08178$0 10 Co'oradoBasin )uke.ater 0817500

Sa'adoCreek 08178$5 5 SanAntonioBasin Ca'averas 0818200 2

Sa'adoCreek 08178$0 SanAntonioBasin Es!ondido 08187000 1

Sa'adoCreek 081787$ 12 SanAntonioBasin Es!ondido 0818700 21

4rinit&Basin E'"(ork 08050200

4rinit&Basin one& 08057500 1

4rinit&Basin one& 08058000 2

4rinit&Basin *itt'eE'" 08052$0 2

4rinit&Basin *itt'eE'" 08052700 58

4rinit&Basin orth 0802$50 1

4rinit&Basin orth 0802700 5$

4rinit&Basin /in+ak 080$200

(ort Worth

Da''as Austin

S"a''ura'ShedsSan Antonio

3.. Data Pre!aration

)n additional processing step used in this thesis is the interpolation of the

observed data into uniformly spaced, one minute intervals.

2



3..1. "ase F#o$ Se!aration

Hydrograph separation is the process of separating the time distribution of base

flo from the total runoff hydrograph to produce the direct runoff hydrograph (7c1uen

3!. 'ase flo separation is a time-honored hydrologic exercise termed by

hydrologists as 8one of the most desperate analysis techni&ues in use in hydrology9

(Helett and Hibbert 0! and 8that fascinating arena of fancy and speculation9

()ppleby 0: 6athan and 7c7ahon !. Hydrograph separation is considered more

of an art than a science ('lack !. *everal hydrograph separation techni&ues such as

constant discharge, constant slope, concave method, and the master depletion curve

method have been developed and used. ;igure 2. is a sketch of a representative

hydrograph that ill be used in this section to explain the different base flo separation

methods.

$ischarge (<2=T!

Time (T!

$ischarge (<2=T!

Time (T!

;igure 2. %epresentative Hydrograph

5



1onstant-discharge method

The base flo is assumed to be constant regardless of stream height (discharge!.

Typically, the minimum value immediately prior to beginning of the storm is pro>ected

hori#ontally. )ll discharge prior to the identified minimum, as ell as all discharge

beneath this hori#ontal pro>ection is labeled as 8base flo9 and removed from further

analysis. ;igure 2.4 is a sketch of the constant discharge method applied to the

representative hydrograph. The shaded area in the sketch represents the discharge that

ould be removed (subtracted! from the observed runoff hydrograph to produce a direct-

runoff hydrograph.

$ischarge (<2=T!

Time (T!

$ischarge (<2=T!

Time (T!

;igure 2.4. 1onstant-discharge base flo separation.

The principal disadvantage is that the method is thought to yield an extremely

long time base for the direct runoff hydrograph, and this time base varies from storm to

storm, depending on the magnitude of the discharge at the beginning of the storm

?



(<insley et, al, 5!. The method is easy to automate, especially for multiple peak

hydrographs.

1onstant-slope method

) line is dran from the inflection point on the receding limb of the

storm hydrograph to the beginning of storm hydrograph, as depicted on ;igure 2.2. This

method assumes that the base flo began prior to the start of the current storm, and

arbitrarily sets to the inflection point.

$ischarge (<2=T!

Time (T!

inflection point identified as

location here second derivative

of the hydrograph passes through #ero

$ischarge (<2=T!

Time (T!




;igure 2.2. 1onstant-slope base flo separation.

The inflection point is located either as the location here the second derivative

passes through #ero (curvature changes! or is empirically related to atershed area. This

method is also relatively easy to automate, except multiple peaked storms ill have

multiple inflection points.



1oncave method

The concave method assumes that base flo continues to decrease hile stream

flo increases to the peak of the storm hydrograph. Then at the peak of the hydrograph,

the base flo is then assumed to increase linearly until it meets the inflection point on the

recession limb.

;igure 2.5 is a sketch illustrating the method applied to the representative

hydrograph. This method is also relatively easy to automate except for multiple peak

hydrographs hich, like the constant slope, method ill have multiple inflection points.

$ischarge (<2

=T!

Time (T!




$ischarge (<2

=T!

Time (T!




;igure 2.5 1oncave-method base flo separations

$epletion curve method

This method models base flo as discharge from accumulated groundater

storage. $ata from several recessions are analy#ed to determine the basin recession

constant. The base flo is modeled as an exponential decay term !exp(!( , kt qt q obb −= .

The time constant, k , is the basin recession coefficient that is inferred from the recession

portion of several storms.

0



Individual storms are plotted ith the logarithm of discharge versus time. The

storms are time shifted by trial-and-error until the recession portions all fall along a

straight line. The slope of this line is proportional to the basin recession coefficient and

the intercept ith the discharge axis at #ero time is the value for obq , . ;igure 2.?

illustrates five storms plotted along ith a test storm here the base flo separation is

being determined. The storm ith the largest flo at the end of the recession is plotted

ithout any time shifting. The recession is extrapolated from this storm as if there ere

no further input to the groundater store. The remaining storms are time shifted so that

the straight line portion of their recession limbs come tangent to this curve. 'y trial-and-

error the master depletion curve can be ad>usted and the storms time shifted until a

reasonable agreement of all storms recessions ith the master curve is achieved.

1

10

100

0 100 200 00 #00 500 $00

4i"e hours9

D i s ! h a r % e

8 ! : s 9

)aster;De3'etion;Curve 4est;Event #;11;#1

#;11;2$ #;1;#$ ;2;7

2;2;#0

;igure 2.? 7aster-$epletion 1urve 7ethod

($ata from 7c1uen, 3, Table -4, pp 53!

3



"nce the master curve is determined, then the test storm is plotted on the curve

and shifted until its straight-line portion come tangent to the master curve, and the point

of intersection is taken as the base flo value for that storm. In the example in ;igure

2.?, the base flo for the test event is approximately 9.1 cfs, the basin recession constant

is 0.0045/hr , and the base flo at the beginning of the recession is 17 cfs. "nce the base

flo value is determined for a particular test event, then base flo separation proceeds

use the constant discharge method.

The depletion curve method is attractive as it determines the basin recession

constant, but it is not at all easy to automate. ;urthermore, in basins here the stream

goes dry (such as much of Texas!, the recession method is difficult to apply as the first

storm after the dry period starts a ne master recession curve. "bserve in ;igure 2.? the

storms used for the recession analysis span a period of nearly 5 years, and implicit in the

analysis is that the basin recession constant is time invariant and the storms are

independent.

The folloing ;igure 2. is a multiple peak storm event from $allas )sh1reek

station3?024. To automate the rest of data set using this method ill be a challenge

because of the change of master recession curve for different peaks.



File : #IUH_1_sta08057320_1977_0327.dat

Dallas AshCreek

0<00E=00

5<00E-0

1<00E-02

1<50E-02

2<00E-02

2<50E-02

0 500 1000 1500 2000 2500

4i"e 8"inutes9

A ! !

u " < D e 3 t

#A4E;/ECI/ #A4E;>+((

;igure 2. 7ultiple peak storms from $allas module

Se#ection o% Met&o' to E(!#o)

The principal criterion for method selection as based on the need for a method

that as simple to automate because hundreds of events needed processing. )ppleby

(0! reports on a base flo separation techni&ue based on a %icatti-type e&uation for

base flo. The general solution of the base flo e&uation is a rational functional that is

remarkably similar in structure to either a <a@lace transform or ;ourier transform.

Unfortunately the paper omits the detail re&uired to actually infer an algorithm from the

solution, but it is useful in that principles of signal processing are clearly indicated in the

model.

6athan and 7c7ahon (! examined automated base flo separation

techni&ues. The ob>ective of their ork as to identify appropriate techni&ues for

determination of base flo and recession constants for use in regional prediction

e&uations. To techni&ues they studied in detail ere a smoothed minima techni&ue and

a recursive digital filter (a signal processing techni&ue similar to )pplebyAs ork!. 'oth

techni&ues ere compared to a graphical techni&ue that extends pre-event runoff (>ust

before the rising portion of the hydrograph! ith the point of greatest curvature on the

4



recession limb (a constant-slope method, but not aimed at the inflection point!. They

concluded that the digital filter as a fast ob>ective method of separation but their paper

suggests that the smoothed minima techni&ue is for all practical purposes

indistinguishable from either the digital filter or the graphical method. ;urthermore the

authors ere vague on the constraint techni&ues employed to make the recursive filter

produce non-negative flo values and to produce peak values that did not exceed the

original stream flo. @ress et.al. (3! provide convincing arguments against time-

domain signal filtering and especially recursive filters. 6evertheless the result for the

smoothed minima is still meaningful, and this techni&ue appears fairly straightforard to

automate, but it is intended for relatively continuous discharge time series and not the

episodic data in the present application.

The constant slope and concave methods are not used in this ork because the

observed runoff hydrographs have multiple peaks. It is impractical to locate the recession

limb inflection point ith any confidence. The master depletion curve method is not

used because even though there is a large amount of data, there is insufficient data at each

station to construct reliable depletion curves. %ecursive filtering and smoothed minima

ere dismissed because of the type of events in the present ork (episodic and not

continuous!. Therefore in the present ork the discharge data are treated by the constant

discharge method.

The constant discharge method as chosen because it is simple to automate and

apply to multiple peaked hydrographs. @rior researchers (e.g. <aurenson and "A$onell,

: 'ates and $avies, 33! have reported that unit hydrograph derivation is

insensitive to base flo separation method hen the base flo is not a large fraction of

4



the flood hydrograph B a situation satisfied in this ork. The particular implementation

in this research determined hen the rainfall event began on a particular day: all

discharge before that time as accumulated and converted into an average rate. This

average rate as then removed from the observed discharge data, and the result as

considered to be the direct runoff hydrograph.

The candidate models ill be run in to cases ith or ithout base flo

separation, so one can compare ho much the separation ould effect the runoff

prediction.

3... E%%ecti*e Preci!itation

The effective precipitation is the fraction of actual precipitation that appears as

direct runoff (after base flo separation!. Typically the precipitation signal (the

hyetograph! is separated into three parts, the initial abstraction, the losses, and the

effective precipitation.

Initial abstraction is the fraction of rainfall that occurs before direct runoff.

"perationally several methods are used to estimate the initial abstraction. "ne method is

to simply censor precipitation that occurs before direct runoff is observed. ) second

method is to assume that the initial abstraction is some constant volume (Ciessman,

3!. The 6%1* method assumes that the initial abstraction is some fraction of the

maximum retention that varies ith soil and land use (essentially a 16 based method!.

<osses after initial abstraction are the fraction of precipitation that is stored in the

atershed (depression, interception, soil storage! that does not appear in the direct runoff

hydrograph. Typically depression and interception storage are considered part of the

initial abstraction, so the loss term essentially represents infiltration into the soil in the

44



atershed. *everal methods to estimate the losses includeD @hi-index method, 1onstant

fraction method, and infiltration capacity approaches (HortonAs curve, +reen-)mpt

model!.

@hi-index model

The φ-index is a simple infiltration model used in hydrology. The method

assumes that the infiltration capacity is a constant φ E(in=hr!. Fith corresponding

observations of a rainfall hyetograph and a runoff hydrograph, the value of φ can in many

cases be easily guessed. ;ield studies have shon that the infiltration capacity is greatest

at the start of a storm and that it decreases rapidly to a relatively constant rate. The

recession time of the infiltration capacity may be as short as to ? minutes. Therefore,

it is not unreasonable to assume that the infiltration capacity is constant over the entire

storm duration. Fhen the rainfall rate exceeds the capacity, the loss rate is assumed to

e&ual the constant capacity, hich is called the phi (φ! index. Fhen the rainfall is less

than the value of φ, the infiltration rate is assumed to e&ual to the rainfall intensity.

7athematically, the phi-index method for modeling losses is described by

;(t!G I(t!, for I(t! φ (2.!

;(t!G φ ,for I(t! φ, (3.2)

here ;(t! is the loss rate, I(t! is storm rainfall intensity, t is time, and φ is a constant.

If measured rainfall-runoff data are available, the value of φ can be estimated by

separating base flo from the total runoff volume, computing the volume of direct

runoff, and then finding the value of φ that results in the volume of effective rainfall

being e&ual to the volume of direct runoff. ) statistical mean phi-index can then be

42



computed as the average of storm event phi values. Fhere measured rainfall-runoff data

are not available, the ultimate capacity of HortonAs e&uation, f c, might be considered.

HortonAs model

Infiltration capacity (f p! may be expressed as

f p G f c J (f o B f c! e-Kt, (2.2!

here f o G maximum infiltration rate at the beginning of a storm event and reduces to a

lo and approximately constant rate of f c as infiltration process continues and the soil is

saturated K G parameter describing rate of decrease in f p.

;actors assumed to be influencing infiltration capacity, soil moisture storage,

surface-connected porosity and effect of root #one paths follo the e&uation

f G a*a.5J f c, (2.5!

here f G infiltration capacity (in=hr!,

a G infiltration capacity of available storage ((in=hr!=(in!.5!

(Index of surface connected porosity!,

*a G available storage in the surface layer in inches of ater e&uivalent ()-hori#on in

agricultural soils - top six inches!.

;actor f c G constant after long etting (in=hr!.

The modified Holton e&uation used by U* )gricultural %esearch *ervice is

f G +Ia *a.5 Jf c, (2.?!

here +I G +roth index - takes into consideration density of plant roots hich assist

infiltration (. - .!.

+reen-)mpt 7odel

45



+reen L )mpt (! proposed the simplified picture of infiltration shon in

;igure 2.0.

;igure 2.0.Cariables in the +reen-)mpt infiltration model. The vertical axis is the

distance from the soil surface: the hori#ontal axis is the moisture content of the soil.

(*ourceD Applied Hydrology by 1ho=7aidement=7ays 33!

The wetting front is a sharp boundary dividing soil belo ith moisture content θ i

from saturated soil done ith moisture content θ i above. The etting front has penetrated

to a depth L in time t since infiltration began. Fater is ponded to a small depth h on the

soil surface. The method computes total infiltration rate at the end of time t , ith the

folloing e&uation,

;(t! G Mt J NO lnP J ;(t!=( NO!Q, (2.!

here

M G Hydraulic conductivity,

4?



t G time in hrs,

;(t! G Total infiltration at the end of time t,

! " Fetting front soil suction head, and

NO G increase in moisture content in time t.

Unlike the *1* curve method, this method gives the total amount of infiltration in

the soil at the end of a particular storm event. $epending on this value and the total

amount of precipitation, e can easily calculate the amount of runoff.

1onstant ;raction 7odel

The constant fraction model simply assumes that some constant ratio of

precipitation becomes runoff: the fraction is called a runoff coefficient. )t first glance it

appears that it is a rational method disguise, but the rational method does not consider

storage and travel times. Thus in the rational method, if one doubles the precipitation

intensity, and halved the duration, one ould expect the peak discharge to remain

unchanged, hile in a unit hydrograph such changes should have a profound effect on the

hydrograph. )s a model, the method is simple to apply, essentially

∫ ∫ =

=

dt t #$H dt t Ap

t p%rpt p

e

rawe

!(!(

!(R!(,

(2.0!

here %rpG the runoff coefficient,

e p G the effective precipitation,

raw p G the ra precipitation,

) G drainage area.

4



The first e&uation states that the effective precipitation is a fraction of the ra

precipitation, hile the second states that the total effective precipitation volume should

e&ual the total direct runoff volume.

3.3. Su((ar) o% Data Pre!aration

'ase flo separation as accomplished using the constant discharge method

because it is amenable to automation. Fe analy#ed the data ith and ithout a

separation to test hether separation as necessary in our data set. Sffective

precipitation as alays modeled using the constant fraction model, because of the need

to automate and also because of the sheer magnitude of the dataset, but the fraction as

left as a fitting constant. Ideally, the fitted result should preserve the re&uired mass

balance (precipitation volume G runoff volume!.

)n important detail in this research as the conversion of the original data into

8pseudo data9 for IUH analysis. The time-step length used in the research as one-

minute. This time length as chosen because it is the smallest increment that can be

represented in the current $)TSTI7S format in the database. It should be noted that

there are very fe actual one-minute intervals in the original data, so linear interpolation

as used to convert the cumulative precipitation into one-minute intervals, then

numerical differentiation is performed to obtain the rainfall rates. The resulting units are

inches per minute.

;igure 2.3 is a sketch shoing the incremental rate and the cumulative depth

relationship. The cumulative depth scale is the left vertical scale and the incremental rate

scale is the right vertical scale. 7athematically the cumulative rainfall distribution is the

integral of the incremental rainfall distribution (or rainfall density! over the entire rainfall

40



event. S&uation 2.3 expresses this relationship: integration over the entire number line is

intended to indicate the entire lifetime of the individual rainfall event.

∫ ∞

∞−

= dt t pt & !(!( . (2.3!

;igure 2.3. 1umulative @recipitation and Incremental @recipitation %elationship

In ;igure 2.3 the cumulative precipitation, &'t(, is indicated by the open circles,

hile the rate, p't !, is indicated by the open s&uares. In practice only the cumulative

depth is recorded as a function of time: so to determine the rate e simply differentiate

the cumulative precipitation.

Q!(P!(

!( ∫ ∞

∞−

== dt t p

dt

d

dt

t d& t p . (2.!

The present ork used a simple centered differencing scheme, except at the first

and last time interval, hen forard and backard differencing ere used, respectively.

t

t t & t t & t p

∆∆−−∆+

≈4

!(!(!( . (2.!

0

0<5

1

1<5

2

2<5

0 20 #0 $0 80 100

4i"e "in<9

C u " u ' a t i v e / r e ! i 3 < 8 i n < 9

0

0<01

0<02

0<0

0<0#

0<05

0<0$

0<07

0<08

/ r e ! i 3 <

0 a t e 8 i n < ? "

i n < 9

A!!/re!i3 Inst/re!i3

43

0

0<5

1

1<5

2

2<5

0 20 #0 $0 80 100

4i"e "in<9

0

0<01

0<02

0<0

0<0#

0<05

0<0$

0<07

0<08



$etails of the 8pseudo data9 conversion ere reported by 1leveland et. al, (42!.

The -minute data for roughly 54 storms are located on a University of Houston server

and can be publicly accessed at the U%< associated ith this citation.

3.+. NRCS ,nit H)'ro-ra!&

The 6atural %esources 1onservation *ervice (6%1*!, formerly the *oil

1onservation *ervice, developed a unit hydrograph (UH! in the ?s. This UH as used

to develop storm hydrographs and peak discharges for design of conservation measures

on small agricultural atersheds.

7ockus (?! discussed development of the standard 6%1* unit hydrograph

and the peak rate e&uation,

& pGM)=T p, (2.!

here the peak discharge rate & p is a function of drainage area ), direct runoff

volume , factor M, and time to peak of the unit hydrograph T p. He indicated that the

peak rate factor (@%;! of M is e&ual to

MG4.=(JH!, (2.4!

here H is the ratio of the time of recession to the time peak (Tr = T p!. He also indicated

that M as a function of the UH shape and that 2=3 of the storm runoff volume in the

rising limb and ?=3 in the recession limb ere typical of small agricultural atersheds. M

also includes a conversion factor to make the e&uation dimensionally correct.

7ockus used the triangular UH shape in development of above to e&uations. It

appears that 7ockus analy#ed many flood hydrographs to >ustify the selection of the peak

rate factor M of 535. ) UH ith @%; of M of 535 as felt to be representative of small

agricultural atersheds in the U.*.

4



NRCSD,H /0a((a a!!roi(ation2

The 6%1* $imensionless Unit Hydrograph (U*$), 3?! used by the 6%1*

(formerly the *1*! as developed by Cictor 7ockus in the late 5As. The *1*

analy#ed a large number of unit hydrographs for atersheds of different si#es and in

different locations and developed a generali#ed dimensionless unit hydrograph in terms

of t/t p and q/q p here, t p is the time to peak. The point of inflection on the unit graph is

approximately .0 the time to peak and the time to peak as observed to be .4 the base

time (hydrograph duration! () b!.

The functional representation is presented as tabulated time and discharge ratios,

and as a graphical representation. Table 2.4 is the tabulation of the 6%1* $UH from

the 6ational Sngineering Handbook.

Table2.4. %atios for dimensionless unit hydrograph and mass curve

Time ratios

(t=Tp!

$ischarge ratios

(&=&p!

7ass 1urve %atios

(=p!

. . .

. .2 ..4 . .

.2 . .4

.5 .2 .2?

.? .50 .?

. . .0

.0 .34 .2

.3 .2 .443

. . .2

. . .20?

. . .5?

.4 .2 .?44

.2 .3 .?3.5 .03 .?

.? .3 .0

. .? .0?

.0 .5 .0

.3 .2 .344

. .22 .35

4. .43 .30

4.4 .40 .3

2



4.5 .50 .25

4. .0 .?2

4.3 .00 .0

2. .?? .00

2.4 .5 .35

2.5 .4 .3

2. .4 .2

2.3 .? .?

5. . .0

5.? .? .

?. . .

CS Di"ension'ess >nit &dro%ra3h and )ass Curve

0

0<1

0<2

0<

0<#

0<5

0<$

0<7

0<8

0<

1

0 1 2 # 5 $

t?t3

1 ? 1 3

0

0<1

0<2

0<

0<#

0<5

0<$

0<7

0<8

0<

1

@ ? @ a

Dis!har%e atios ?39 )ass Curve atios @?@39

;igure2.. @lot of $UH and 7ass 1urve

;igure 2. is a plot of these ratios. This figure is identical to ;igure . in the

6ational Sngineering handbook (except this figure is computer generated!.

2



The IUH analysis assumed that the hydrograph functions are continuous and the

database as analy#ed using discrete values calculated from continuous functions.

%ather than use the 6%1* tabulation in this ork, the fit as tested of a function of the

same family (the gamma distribution! as the IUH function and this function as used in

place of the 6%1* tabulation. ) similar approach as used by *ingh (4! to express

common unit hydrographs (*nyderAs, *1*, and +rayAs! by a gamma distribution.

The gamma function used to fit the tabulation is

!(=!( η λ λ η η Γ = −− *

* e *k + & .

(2.2!

The variables λ , η and k are unknon, and ere determined by minimi#ation of the sum

of s&uared errors beteen the tabulation and the model (the function! by selection of

numerical values for the unknon parameters. 8Sxcel solver9 as used to perform the

minimi#ation. The values for parameters λ , η and k ere 2.33, 5.3 and .4

respectively. *o the 6%1* $UH approximation is

. (2.5!

;igure 2. is a plot of the model and tabulation, the variable * in the e&uations is

the dimensionless time. ualitatively the fit is good. The maximum residual(s! occur

early in dimensionless time and at V of the runoff duration, but the magnitudes are

&uite small, and thus this model of the 6%1* $UH is deemed acceptable for use.

24

*

* e * + & 33.23.23.5

33.2R4.!( −=



CS Curve-(ittin% >sin% 6a""a :un!tion

0

0<1

0<2

0<

0<#

0<5

0<$

0<7

0<8

0<

1

0 1 2 # 5 $

t?43

1 ? 1 3

+bserved )ode'ed

;igure2.. @lot of Tabulated and +amma-7odel $UH

) 1hi-s&uare fitness test as performed to further support the decision to use the

model in lieu of the tabulation. The test statistic for the chi-s&uare test as calculated as

i

k

i

ii% , , - =!(

44

∑=

−= χ . (2.?!

The test statistic is .?242. ;or to degrees of freedom and V confidence limits the

value as . hich is greater than the test statistic (.?242! therefore the hypothesis

(model! represent the observed values.

The 6%1* $UH as presented in the 6SH integrates to a little over .5 and thus it

is not a true unit hydrograph as presented. It is likely that it originally as a UH: then it

as ad>usted procedurally so that the peak value of the dimensionless distribution is .

(thus the factor that scales the integral correctly is imbedded in the & p value!. The

research assumes that all unit graphs and the accompanying functional representations of

22



IUHs integrate to one: so in this ork the 6%1* $UH approximation is ad>usted by

dividing by the integral of the original $UH, in this case the value is .42. Therefore

the final approximation to the 6%1* $UH as a functional representation useful in IUH

analysis is

*e * + &* 33.23.23.533.2!( −= .

(2.!

"r ith all the constants evaluated and simplified and expressed in the 6%1*

terminology the 6%1* $UH (as an IUH function! is

pt

t

p p

et

t

q

t q 33.23.2

!(?230.23!( −= . (2.0!

3.. Co((ons H)'ro-ra!&

1ommons (54! developed a dimensionless unit hydrograph for use in Texas,

but details of ho the hydrograph ere developed are not reported. The labeling of axes

in the original document suggests that the hydrograph is dimensionless. ;or the sake of

completeness in this ork, an approximation as produced for treatment as another

transfer function by fitting a three-gamma summation model. Sssentially there ere

three integrated gamma models ith different peaks and eights to reproduce the shape

of 1ommonsA hydrograph. The 1ommons hydrograph is &uite different in shape after the

peak than other dimensionless unit hydrographs in current use (i.e. 6%1* $imensionless

Unit Hydrograph! B it has a very long time base on the recession portion of the

hydrograph.

25



;igure 2.. Hydrograph developed by trial to cover a typical flood

1ircles are tabulation from digiti#ation of the original figure. 1urve is a *mooth

1urve )pproximation. ;igure 2. is 1ommonsA hydrograph reproduced from a manual

digiti#ation. The smooth curve is given by the folloing e&uation that as fit by trial and

error.

p

p

p

t

t

p

t

t

p

t

t

p

et

t

et

t

et

t t q

-5.?24.

-5.4-?.

00.50-.

!-5.?

(!433.(

33.2

!-5.4

(!4?.(

?3.0

!00.5

(!3.(

.00!(

−

−

−

Γ −

Γ +

Γ =

. (2.3!

The numerical values are simply the result of the fitting procedure. The time axis

as reconstructed (in the fitting algorithm! so that the t p parameter could be left variable

for consistency ith the other hydrograph functions. The tabulated function integrates to

2?



approximately : thus the function above is divided by this value to produce a unit

hydrograph distribution.

3.4. 0a((a S)nt&etic H)'ro-ra!&s

The gamma distribution is given in the e&uation

ab * *.e * f

=!(

−= . (2.!

In the e&uation 1 e&uals!(

+Γ + aba

to make the area enclosed by the curve e&ual to

unity. Γ is called the gamma function. Calues can be found tabulated in mathematical

handbooks. The gamma distribution is similar in shape to the @oisson distribution that is

given the form asW

!( *

e/ * f

/ * −

= .The curve starts at #ero hen the variable x is #ero,

rises to a maximum, and descends to a tail that extends indefinitely to the right. The

values that the variable x can take on are thus limited by on the left. Calues can extend

to infinity on the right.

The gamma distribution differs from the @oisson distribution is that it has to

parameters instead of the single parameter of the @oisson. This allos the curve to take

on a greater variety of shapes than the @oisson distribution. The parameter a is a shape

parameter hile b is a scale parameter. The shape of the +amma distribution is similar to

the shape of a unit hydrograph, so many researchers started looking for the application of

the +amma distribution into hydrograph prediction. This first started ith Sdson (?!,

ho presented a theoretical expression for the unit hydrograph assuming to be

proportional to yt *et −

2



!(

!(

+Γ =

−

*

e yt %Ay0

yt *

, (2.4!

here G discharge in cfs at time t: )G drainage area in s&uare miles: x and y G

parameters that can be represented in terms of peak discharge: and !( +Γ * is the

gamma function of (xJ!. 6ash (?! and $ooge (?!, based on the concept of n

linear reservoirs ith e&ual storage coefficient M, expressed the instantaneous UH (IUH!

in the form of a +amma distribution as

k t

n

e 1

t

n 1 q =

!(

−−

Γ = , (2.4!

in hich n and MG parameter defining the shape of the IUH: and &Gdepth of runoff per

unit time per unit effective rainfall. These parameters have been referred to as the 6ash

model parameters in the subse&uent literature.

@revious attempts to fit a +amma distribution to a hydrograph ere by 1roley(3!,

)ron and Fhite (34!, Hann et al. (5!, and *ingh (3!. The procedure given by

1roley (3! to calculate n for knon values of & p and t p re&uires programming to

iteratively solve for n. 1roley also proposed procedures to obtain a UH from other

observable characteristics. The method by )ron and Fhite (4! involves reading the

values from a graph, in hich errors are introduced. 'ased on their methods, 7c1uen

(3! listed a step-by-step procedure to obtain the UH, hich maybe briefly described

by the folloing e&uations,

nG.5?J.?fJ?.f 4J.2f 2, (2.44!

20



in hich A

t 0 f

p p= , here p is in cubic feet per second, t p is in hours, and ) is in

acres. These to e&uations re&uire careful attention for the units, and these cannot be

used as such hen pt p is re&uired to be computed for a value of n knon from other

sources. Hann et al. (5! gave the folloing expression to calculate n,

4.!(?.-

2

t 0n

p p+= , (2.42!

here CGtotal volume of effective rainfall. )n e&uation provided by *ingh (3! to

obtain the value of n may be ritten,

4.--5.. β β ++=n , (2.45!

here p pt q=β (dimensionless!, in hich pq is the peak runoff depth per unit time per

effective rainfall. *ingh observed that the error in n obtained from the e&uationD

4.--5.. β β ++=n (2.4?!

is .?2V hen 4?.=β and .?V hen .=β . The error in n calculated decreases

ith increasing values of β .

3.5. 6eibu## Distribution

Historically a to-parameter Feibull distribution is employed to define the

configuration of a natural hydrograph of direct runoff and is given in the folloing forms

as (1anavos, 35!

nk t n e 3t 0 !=( −−= , (2.4!

here is the discharge ordinate of the natural hydrograph corresponding to the time t

after the commencement of direct runoff, n is the dimensionless shape factor, and k is the

23



storage time constant. 'oth n and k reflect the basin characteristics and are related to the

time to peak t p in the folloing manner.

nnk t n

p =!(!=( −= . (2.40!

The constant of proportionality ' in S&uation (2.4! is evaluated as

n p k t n

p

p

et

0 3

!=(!( −−

= , (2.43!

here p is the peak discharge and e is the base of the natural logarithms. 1ombining

S&uation (2.4!, (2.40! and (2.43! yields

nt t nn p p

n

pet t 00 =!!=(!((!=(= −−−= . (2.4!

S&uation (2.4! is the desired form of the dimensionless Feibull distribution as used in

this study. )nalytical formulation of the parameter n can be developed as follos.

$esignating p00q =R = , and pt t t =R = , S&uation (2.4! may be ritten as

nt nn n

et q=!!((

RRR!( −−−= . (2.2!

Taking natural logarithms of both sides of the above e&uation and solving for n, one

obtains

RRRln=!=ln!(lnln( t nqnt nn −−+= . (2.2!

The value of n can be obtained from S&uation (2.2! through graphical means. "nce the

value of n has been ascertained properly, the value of k can then be determined from

S&uation (2.2! conveniently.

3.7. Reser*oir E#e(ents

)n alternative ay to construct the hydrograph functions is to model the

atershed response to precipitation as the response from a cascade of reservoirs. The

response function is developed as the response to an impulse of input, and the response to

2



a time series of inputs is obtained from the convolution integral. The end result is the

same, a function that is a distribution function, but the parameters have a physical

interpretation. The kernel (response function! to an impulse in this ork is an

instantaneous unit hydrograph (IUH!. The conceptual approach for a cascade of

reservoirs is ell studied and orks ell for many unit hydrograph analyses (e.g. 6ash,

?3: $ooge, ?: $ooge, 02: 1roley, 3!. In this ork the cases are examined

here a +amma, %ayleigh, and Feibull distribution govern the individual reservoir

element responses, respectively. In addition, e have also converted the 6%1*-$UH

into its on response model (a special case of gamma!.

water4hed 4y4te/

re4pon4e /odel

5 1,t

q1,t

5 6,t

q6,t

5 7,t

q7,t

5 8,t

q 8,t

e*%e44 pre%ipitation 'depth(

ob4er9ed dire%t r:noff 'depth(

A

;igure 2.4 1ascade of %eservoir Slements 1onceptuali#ation

;igure 2.4 is a schematic of a atershed response conceptuali#ed as a series of

identical reservoirs ithout feedback. The outflo of each reservoir is related to the

5



accumulated storage in the reservoir. The behavior of the individual reservoir elements

determines hether the model becomes a +amma, %ayleigh, or Feibull distribution.

;igure 2.2 is a schematic of a reservoir response element. In the sketch, the element

area is A, the accumulated excess storage is , and the outlet flo area is a.

5 a q"a9

A

pe

5 a q"a9

A

pe

;igure 2.2 %eservoir Slement 7odel

The outlet discharge is the product of the outlet area, a, and the flo velocity . The

input is pe.

3.7.1 Linear /0a((a2 Reser*oir E#e(ent

The first response model is a linear reservoir model, here the reservoir discharge

is proportional to the accumulated depth of input. The constant of proportionality is %.

The discharge e&uation is

a%5 a9q == . (2.24!

) mass balance of the reservoir is

a%5 dt

d5 A −= . (2.22!

The input pe is applied over a very short time interval: so the resulting depth, before

outflo begins is o. The solution to this "$S ("rdinary $ifferential S&uation! is

!exp(!( t

A

a% 5 t 5 −= . (2.25!

The ratio A/a% is called the residence time of the linear reservoir.

5



a%

At = . (2.2?!

Thus in terms of residence time the accumulated depth in a linear reservoir is

!exp(!( t t 5 t 5 −= .

(2.2!

The discharge rate is the product of this function and the constant of proportionality

!exp(!exp(!(

t

t 5 t

A

t

t a%5 t q −=−= . (2.20!

;igure 2.5 <inear %eservoir 7odel 4=t , AG, 0G1

This particular atershed model has the folloing propertiesD

• 1umulative discharge is related to accumulated time.

• Instantaneous discharge is inversely related to accumulated time.

• The peak discharge is proportional to the precipitation input depth, and occurs at

time #ero.

• The peak discharge is proportional to the atershed area.

54



3.7. Ra)#ei-& Reser*oir E#e(ent

The next response model assumes that the discharge is proportional to both the

accumulated excess precipitation (linear reservoir! and the elapsed time since the impulse

of precipitation as added to the atershed (translation reservoir!. The constant of

proportionality in this case is 6%.

%5t aa9q 4== . (2.23!

) mass balance for this model is

%5t adt

d5 A 4−= . (2.2!

The solution (using the same characteristic time re-parameteri#ation as in the linear

reservoir model! is

!!(exp(!(4

t

t 5 t 5 −= . (2.5!

The discharge function is

!!(exp(4

!!(exp(4!( 4

4.

4

.

t

t

t

t A5

t

t %t5 at q −=−= . (2.5!

This result is a %ayleigh distribution eighted by the product of atershed area

and the initial charge of precipitation (hence the name %ayleigh reservoir!. The discharge

function for unit area and depth integrates to one: thus it is a unit hydrograph, and it

satisfies the linearity re&uirement, thus it is a candidate IUH function.

52



;igure 2.?. %ayleigh %eservoir Fatershed 7odel. 4=t , AG, 0G1

"f particular interest, the %ayleigh model &ualitatively looks like a hydrograph

should, ith a peak occurring some time after the precipitation is applied (unlike the

linear reservoir! and a falling limb after the peak ith an inflection point. Sxamination of

the discharge function includes the folloing relationshipsD

• 1umulative discharge is proportional to accumulated time.

• Instantaneous discharge is proportional to accumulated time until the peak, then

inversely proportional afterards.

• The peak discharge is proportional to the precipitation input depth, and occurs at

some non-#ero characteristic time.

• The peak discharge is proportional to the atershed area.

3.7.3 6eibu## Reser*oir

The Feibull response model assumes that the discharge is proportional to both

the accumulated excess precipitation (linear reservoir! and the elapsed time raised to

55



some non-#ero poer since the impulse of precipitation as added to the atershed

(translation reservoir!. The constant of proportionality in this case is p%.

−== pap%5t a9q . (2.54!

) mass balance for this model is

−−= pap%5t

dt

d5 A . (2.52!

The solution (using the same characteristic time re-parameteri#ation as in the linear

reservoir model! is

!!(exp(!(

p

t

t 5 t 5 −= . (2.55!

The discharge function is

!!(exp(!!(exp(!(

.. p

p

p p p

t

t

t

pt A5

t

t 5 ap%t t q −=−=

−− . (2.5?!

This result is a Feibull distribution eighted by the product of atershed area

and the initial charge of precipitation (hence the name Feibull reservoir!. The discharge

function for unit area and depth integrates to one, thus it is a unit hydrograph, and it

satisfies the linearity re&uirement, thus it is a candidate IUH function.

These three models constitute the reservoir element models used in this research.

3.8. Casca'e Ana#)sis

;igure 2.4 is the schematic of a cascade model of atershed response. In our

research e assumed that the number of reservoirs 8internal9 to the atershed could

range from to J ∞ . "ur initial theoretical development assumed integral values, but

others have suggested fractional reservoirs can be incorporated into the theory. To

develop the cascade model(s!, start ith the mass balance for a single reservoir element,

and the discharge from this reservoir becomes the input for subse&uent reservoirs and e

5?



determine the discharge for the last reservoir as representative of the entire atershed

response.

3.8.1. 0a((a Reser*oir Casca'e

S&uation 2.5, here i represents the accumulated storage depth, a% is the

reservoir discharge coefficient, qi is the outflo for a particular reservoir, and A is the

atershed area, represents the discharge functions for a cascade of linear reservoirs that

comprise a response model. The subscript, i , is the identifier of a particular reservoir in

the cascade.

t it i a%5 Aq

,, = . (2.5!

S&uation 2.50 is the mass balance e&uation for a reservoir in the cascade. In

S&uation 2.5, the first reservoir receives the initial charge of ater, o over an

infinitesimally small time interval, essentially an impulse, and this impulse is propagated

through the system by the drainage functions.

t it it i a%5 Aq 5 A ,,, −= −

.

(2.50!

The entire atershed response is expressed as the system of linear ordinary

differential e&uations, S&uation 2.53, and the analytical solution for discharge for this

system for the 8 -th reservoir is expressed in S&uation 2.5.

5



8 8 8

o

5 t

5 t

5

5 t

5 t

5

5 t

5 t

5

5 t

5 5

242

44

−=

−=

−=

−=

−

. (2.53!

The result in e&uation 2.53 is identical to the 6ash model (6ash ?3! and is

incorporated into many standard hydrology programs such as the 1"**)%% model

(%ockood et. al. 04!. The factorial can be replaced by the +amma function (6auman

and 'uffham, 32! and the result can be extended to non-integer number of reservoirs.

!exp(!W(

.,t

t

t 8

t

t A5 q

8

8

t 8 −

−

⋅= −

−

.

(2.5!

To model the response to a time-series of precipitation inputs, the individual

responses (S&. 2.5! are convolved and the result of the convolution is the output from

the atershed. If each input is represented by the product of a rate and time interval

( o't( " qo't( dt ! then the individual response is (note the +amma function is substituted

for the factorial!

τ τ τ

τ τ

d t

t

t 8

t

t Aqdq

8

8

i !exp(!(

!(!(

.,

−−

Γ −

⋅=

−

−

. (2.?!

The accumulated responses are given by

∫ −

−

Γ −

⋅= −

−t

8

8

8 d t

t

t 8

t

t Aqt q

.

. !exp(!(

!(!(!( τ

τ τ τ .

(2.?!

50



S&uation 2.? represents the atershed response to an input time series. The

convolution integral in 1hapter 0 in 1ho, et al (33!, an overvie of that ork, is

repeated as S&uation 2.?4,

∫ −=t

d t : ; t 0

!(!(!( τ τ τ . (2.?4!

The analogs to our present ork are as follos (1hoAs variable list is shon on

the left of the e&ualities!D

!exp(!(!(!(

!(!(

!(!(

.

t t

t 8 t

t At :

q ;

t qt 0

8

8

8

τ τ τ

τ τ

−−

Γ −

⋅=−

==

−

−

(2.?2!

Fe call the kernel ( :'t < τ ( ! for the linear reservoir a gamma response because

the kernel is essentially a gamma probability distribution. The reason for representing the

function as being derived from a cascade is that this derivation provides a 8physical9

meaning to the distribution parameters.

The analysis is repeated for the %ayleigh and Feibull distributions.

3.8.. Ra)#ei-& Reser*oir Casca'e

) %ayleigh response is developed in the same fashion as the gamma, except the

%ayleigh reservoir element is used instead of the linear (gamma! response. The discharge

and mass balances for the %ayleigh case are given as S&uations 2.?5 and 2.??,

respectively,

t it i a%t5 Aq ,, 4= , (2.?5!

t it it i a%t5 Aq 5 A ,,, 4−= − . (2.??!

53



The entire atershed response is expressed as the system of linear ordinary

differential e&uations in S&uation 2.?.

8 8 8

o

5 t

t 5

t

t 5

5 t

t 5

t

t 5

5 t

t 5

t

t 5

5

t

t 5 5

44

44

44

4

242

44

−=

−=

−=

−=

−

(2.?!

The analytical solution for any reservoir is expressed in S&uation 2.?0.

!!(exp(!!((

!(4 4

4

4

4.,t

t

t 8

t

t

t A5 q

8

8

t 8 −

Γ

⋅=

−

−

, (2.?0!

S&uation 2.?3 gives the convolution integral using this kernel.

∫ −

−

Γ −

−⋅=

−

−t

8

8

i d t

t

t 8

t

t

t Aqt q

.

4

4

4

4

4. !!(

exp(!!((

!!((!(4!( τ

τ τ τ τ , (2.?3!

This distribution is identical to <einhardAs 8hydrograph distribution9 (<einhard,

04! that he developed from statistical-mechanical analysis.

3.8.3. 6eibu## Reser*oir Casca'e

) Feibull response is developed in the same fashion as the gamma by

substitution of the Feibull reservoir element in the analysis. The discharge and mass

balances are given as S&uations 2.? and 2., respectively,

t it i a%t5 Aq ,, 4= , (2.?!

t it it i a%t5 Aq 5 A ,,, 4−= − . (2.!

The entire atershed response is again expressed as a system of linear ordinary

differential e&uations: S&uation 2..

5



8

p

8

p

8

p p

p p

p

o

5 t

t p 5

t

t p 5

5

t

t p 5

t

t p 5

5 t

t p 5

t

t p 5

5 t

t p 5 5

2

4

2

4

4

−

−

−

−−

−−

−

−=

−=

−=

−=

(2.!

The analytical solution to this system for any reservoir is expressed in S&uation

2.4,

!!(exp(!!((

!(

.,

p

8 p

8 p

p

p

t 8 t

t

t 8

t

t

t pA5 q −

Γ

⋅=

−

−−

. (2.4!

The accumulated responses to a time series of precipitation input are given by

S&uation 2.2.

∫ −−

Γ −

−⋅= −

−−t

p

p

8

8 p p

i d t

t

t 8

t

t

t pAqt q

.

. !!(

exp(!(

!!((!(!(!( τ

τ τ τ τ .

(2.2!

The utility of the Feibull model is that both the linear cascade (exponential! and the

%ayleigh cascade are special cases of the generali#ed Feibull model, thus if e program

a Feibull-type model as the IUH, e can investigate other models by restricting

parameter values. The parameters have the folloing impacts on the discharge functionD

. The poer term controls the decay rate of the hydrograph (shape of the falling

limb!. If p is greater than one, then decay is fast (steep falling limb!: if p is less

than one then the decay is slo (long falling limb!.

4. The t term controls the scale of the hydrograph. It simultaneously establishes

the location of the peak and the magnitude of the peak.

?



2. The reservoir number, 8 , controls the lag beteen the input and the response, as

ell as the shape of the hydrograph.

The next chapter describes ho the distribution parameters are determined from

observations.

Documents

11-Method of Analysis.doc