FLOOD08610

• County: Statewide
• Basin: Statewide
• Title: Techniques of Water-Resources Investigations of the US Geological Survey Some Statistical Tools in Hydrology
• Date: 1/1/1969
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

30

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

s

y

x,

6

4

t: ~
'"
~
,...
-<
~ 0
<r
::>
()
~ -2
<r
~
~
~ -4
"
u;
'"
<r

-6

-s

-10

o ~

Figure ~1.-Multiple linear regression by the method of residuals.

4
X,

4. Lines of equal S are logarithmically spaced
at twice the logarithmic scale of the paper
and are parallel.
To solve, separate the regression into two
parts by introducing an intermediate variable
Q.., (to an arbitrary scale) so that

and

Q..,=j(A, P)

Q..,~j(S, QlO)'

Consider the first relation, For a fixed P, the
model would be

Q..,=KA",

in which K is the intercept on the Q.., scale
(at A= 1) and n is the slope of the line. In this
example, n= 1.28, which is the ratio of the
linear vertical to horizontal lengths. When
P= 1.20, the intercept K is 78. To obtain this
intercept graphically requires a long curve
extension. It is simpler to compute the intercept
from some other value of A than one. For

6

instance, for Q..,=I,OOO, A=7.3. Then

1,000=K(7.3)1.28, from which K=78.

Similarly, for Q..,= 10,000, A=44.5 and K =78.
Introducing P as a variable makes it neces-
sary to define the intercept K in terms of P
(because the intercept is different for each
value of P), The interval per tenth difference
in P when projected on the Q.., scale is 1.36,
that is, for each tenth increase in P, the inter-
cept increases 1.36 times (the intercept is on a
logarithmic scale). This increase can be meas-
ured for individual intervals or computed from
the total increase: For instance, for A= 10:

Q..,= 1,480 for P= 1.2,

and

Q..,=17,000 for P=2,O.

The increase for eight intervals is 17,000{
1,480=11.5, and (1.356)'=11.5.
Then

K=78(1.36)IO(P-l.",

.

s

.

.