FLOOD02408

• County: Statewide
• Community: Nationwide
• Basin: Statewide
• Title: Design Hydrographs for Very Small Wathersheds from Rainfall
• Date: 7/1/1962
• Prepared By: USDA Hydrograph Laboratory, Civil Engineering Section Colorado State University
• Floodplain - Doc Type: Educational/Technical/Reference Information

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MATHEMATICAL TECHNIQUES

Chapter 2

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In this chapter some of the mathematical
equations basic to this analysis of the hyclrograph
will be developed. The application of certain common
statistics to the description of a rainstorm will be
discussed. Finally a brief description will be pre-
sented of the regression analysis technique, which
will be applied to the data in the next two chapters.

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Hydrograph Model

Equation and its integration - It was decided
to pursue the assumption of Yevdjevich (27) and
attempt to fit the three parameter Pearson type III
function to the discharge hydrographs. For the pur-
pose of the present study this is expressed by the
equation

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i I

m

t
-0

[1+ : t

. . . . . (1)

q. qo e

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where the symbols are explained in Fig. 1. * Inte-
grating gives the total volume of runoff,

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m

fOO qo e

t
-0

[1 +~]G

dt.

w.

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-m

Let t + m = u I then dt = du

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m

and w.qoe~ fOO e- ~ [;:,t duo

o

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u
Lety=O'

o Y :II U and G dy II: du I

and

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m

00 m
GI e -y y G dy.

o

m
W = qo :G[~]

G

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m

~[G]G
w=qoe in or

[1 + ~] . , . . . (2)

J 00 e -y yX dy
o

Whence

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because by definition r (1 + x)

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*Besides their definition in the text, frequently used
symbols are listed along with their units and dimen~
sions in Table 14 of the Appendix.

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Apart from the ordinate q and abscissa t
equation (1) contains parameters qo I m and G.

Three parameters must therefore be specified to
determine the particular shape of this model. It
therefore possesses more flexibility with which it
may be fitted to observed hydrographs than do the
two parameter models employed by earlier workers
(6, 9),

Interch eability of time parameters, G
and m - Equation 2 shows that if the volume of
runoff. W I is given as well as q I then G and
o
m are uniquely interrelated. Consequently m may
be omitted and G used as the final definitive para-
meter I along with Wand q . This change is de-
o
sirable in practice I since m is difficult to ascertain
and 0 has a physical significance.

Graphical determination of hydrograph para-
meters - Equations (1) and (2) uniquely determine q
as a function of t I for each set of values of q
o
G and W . Complexity of the mathematical rela-
tions precludes an explicit solution, but the following
graphical solution was found satisfactory:

Equation (2) may be rewritten as

" =

[{-]

G

(3)

where

m

"=e~[~]G r[1+~] ......

(4)

Waver qo is grouped within parentheses in equa-

tion (3) since it represents a fixed quotient for each
hydrograph of known volume and peak. Various
values of G were tried and each will specify a dif-

ferent hydrograph for the particular W ratio.
qo

The selection of each G fixes the value of
O! from equation (3). This. in turn. assigns a unique

value to ~ according to equation (4). To obviate

I the repeated trial and error solution for the latter
from the tables. the relationship was expressed
graphically as illustrated in Fig. 2. Thus for each

G selected only one ratio of ~ satisfied equations

(3) and (4) simultaneously. In summary t many paired
values of G and m are available which satisfied
equation (2) for each hydrograph of fixed W and q .
o

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