11-method of analysis

8/12/2019 11-Method of Analysis

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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


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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!.

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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''asAustin

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.

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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 (


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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 (


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(


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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 (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.*.

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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/tpand q/qphere, tpis 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

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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


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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

!(=!( = ** e*k+& .

(2.2!

The variables , 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 , 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.!( =


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

0

0


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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

pp

et

t

q

tq 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.

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;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

ttq

-5.?24.

-5.4-?.

00.50-.

!-5.?

(!433.(

33.2

!-5.4

(!4?.(

?3.0

!00.5

(!3.(

.00!(

+

=

. (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 tpparameter could be left variable

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

2?


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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!(

++ aba

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

unity. 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

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!(

!(

+=

*

eyt%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 !( +* 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

kt

n

e1

t

n1q =

!(

= , (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 &pand tpre&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?.f4J.2f2, (2.44!

20


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

time to peak tpin the folloing manner.

nnkt np =!(!=( = . (2.40!

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

np ktn

p

p

et

03

!=(!(

= , (2.43!

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

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

nttnnpp

n

pett00 =!!=(!((!=(= = . (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 pttt =R = , S&uation (2.4! may be ritten as

ntnn n

etq=!!((

RRR!( = . (2.2!

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

obtains

RRRln=!=ln!(lnln( tnqntnn += . (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

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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

51,t

q1,t

56,t

q6,t

57,t

q7,t

58,t

q8,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


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a%

At= . (2.2?!

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

!exp(!( tt5t5 = .

(2.2!

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

!exp(!exp(!(

t

t5t

A

t

ta%5tq == . (2.20!

;igure 2.5


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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%.

%5taa9q 4== . (2.23!

) mass balance for this model is

%5tadt

d5A 4= . (2.2!

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

reservoir model! is

!!(exp(!(4

t

t5t5 = . (2.5!

The discharge function is

!!(exp(4

!!(exp(4!( 4

4.

4

.

t

t

t

tA5

t

t%t5atq == . (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.

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;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


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

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

== pap%5ta9q . (2.54!

) mass balance for this model is

= pap%5tdt

d5A . (2.52!

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

reservoir model! is

!!(exp(!(

p

t

t5t5 = . (2.55!

The discharge function is

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

.. p

p

ppp

t

t

t

ptA5

t

t5ap%ttq ==

. (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?


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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.

titi a%5Aq ,, = . (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.

tititi a%5Aq5A ,,, =

.

(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 the8-th reservoir is expressed in S&uation 2.5.

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888

o

5t

5t

5

5t

5t

5

5t

5t

5

5t

55

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

t8

t

tA5q

8

8

t8

=

.

(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!

dt

t

t8

t

tAqdq

8

8

i !exp(!(

!(!(

.,

=

. (2.?!

The accumulated responses are given by

=

t

8

8

8 d

t

t

t8

t

tAqtq

.

. !exp(!(

!(!(!(

.

(2.?!

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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

dt:;t0

!(!(!( . (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(!(!(!(

!(!(

!(!(

.

tt

t8t

tAt:

q;

tqt0

8

8

8

=

==

(2.?2!

Fe call the kernel ( :'t


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

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

888

o

5t

t5

t

t5

5t

t5

t

t5

5t

t5

t

t5

5

t

t55

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

t8

t

t

tA5q

8

8

t8

=

, (2.?0!

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

=

t

8

8

i dt

t

t8

t

t

tAqtq

.

4

4

4

4

4.!

!(exp(

!!((

!!((!(4!(

, (2.?3!

This distribution is identical to


40/41

8

p

8

p

8

pp

pp

p

o

5t

tp5

t

tp5

5

t

tp5

t

tp5

5t

tp5

t

tp5

5t

tp55

2

4

2

4

4

=

=

=

=

(2.!

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

2.4,

!!(exp(!!((

!(

.,

p

8p

8p

p

p

t8 t

t

t8

t

t

tpA5q

=

. (2.4!

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

S&uation 2.2.

= t

p

p

8

8pp

i dt

t

t8

t

t

tpAqtq

.

. !!(

exp(!(

!!((!(!(!(

.

(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!. Ifpis greater than one, then decay is fast (steep falling limb!: if pis 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.

?


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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.

11-method of analysis

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