JERRY E. MUELLER Department ofGeological Sciences, University ofTexas at El Paso, El Paso, Texas
79968
Re-evaluation of the Relationship of
Master Streams and Drainage Basins
ABSTRACT
A specific equation describing the relationship of stream length to drainage area was
originally formulated for very small rivers and
basins in localized areas of the eastern United
States. Use of the same equation for larger
rivers and basins on a world-wide scale produces
gross error of prediction. A general or best-fit
equation is offered which describes more accurately the length-area relationship for moderate
and large-size drainage systems. The theoretical
implications of the general equation are just
the inverse of those inherent in the original
equation.
INTRODUCTION
Hack (1957), in his paper on stream profiles
in Virginia and Maryland, formulated equations which accurately describe the relationship
of several sets of fluvial variables. In one case,
he found that drainage area and stream length
are interdependent, and can be described by
the equation
L = 1.4 AO.6.
Here L is stream length in miles and A is the
drainage area in square miles. Very importantly, Hack's measurement of L includes not
only the channel length, but also the horizontal
distance between the stream's source and the
drainage divide directly above.
Because he was working with data from just
two small drainage basins in the eastern United
States, Hack also tested his equation against
measurements taken in Arizona and South
Dakota. Significantly, he discovered that
stream lengths in the latter areas were proportional to the 0.7 power of the drainage
area. He cautioned that the relationship described by his equation " ... can be considered
valid only for the region under discussion in
this report" (Hack, 1957, p. 64). Also of
importance, Hack discovered that the coefficient, and not the exponent, was highly influenced by varying geologic conditions within
his study area.
PROBLEM
Hack's equation has become ingrained in the
geomorphic literature since its inception, but
unfortunately, most of the caution and restrictions its author emphasized have been ignored.
For instance, L has been interpreted as absolute
channel or stream length, rather than the very
special length Hack measured to the drainage
divide. Thus, Hack's channel length exceeds
actual length in all cases. Also, the equation
has been loosely interpreted as defining the
relationship between stream length and drainage area on a world-wide basis, even for drainage
systems immensely larger than those analyzed
by Hack. In other words, we have carelessly
allowed the specific to become the general
(Encyclopedia Americana, 1970, p. 552).
Keeping in mind that Hack's equation is
undeniably accurate in limited areas, it is
possible to test for predictive error by collecting data on a few of the world's longest streams
and largest drainage basins. The procedure used
in this study was to program Hack's equation
and selected drainage areas into a computer,
and have the machine solve for predicted
stream length. Areas chosen ranged from 5 to
5,000,000 sq mi, and were arranged in tabular
form on the printout. The Amazon River is
3,915 rni long, and has a drainage area of
2,722,000 sq mi. Its predicted length using
Hack's equation is 10,164 mi. The difference
between predicted and actual lengths is 6,249
mi, an error of 159.6 percent (predicted-actual/
actual). Data on eight rivers with drainage
areas of more than one million square miles
each was tested against the computer lengths.
The mean, unweighted error of prediction for
the eight rivers is 117.9 percent.
Geological Society of America Bulletin, v. 83, p. 3471-3474, 1 fig., November 1972
3471
3472
J. E. MUELLER
SOLUTION
In order to develop a general equation that
describes more precisely the relationship between actual stream length and drainage area,
data were collected on 65 river systems distributed over five continents and located in areas
of diverse climatic and geologic conditions. The
selected drainage basin areas ranged from 5,000
sq mi for the Thames to nearly 3,000,000 sq mi
for the Amazon. The distribution plotted on
log-log paper is represented by an equation of
the type
L(A) ~ CAB.
Obviously, the accuracy of such an equation
will be determined by the values calculated
for the constants of C and B. Because any
positive number can be written as an exponent
of e, the equation can be rewritten in the form
L(A) ~ eaA B .
Taking natural logarithms of both sides, we
have
1nL(A)
=
a
+ BlnA .
If we replace InL(A) by Yand InA by X, the
problem is reduced to an equation of a straight
line:
y~ a+BX.
The straight line or best-fit equation for an
array of data possessing a log-log relationship
can be solved using natural logarithms and the
well-known method of least squares. In this
study, the statistical procedures were programmed in Fortran IV, and the computer
calculated the best-fit equation (Fig. 1). The
computer also transformed the final equation
from natural logarithmic notation back to
standard notation (base 10). This transformation yields an equation of the type formulated
by Hack, that is, produces the coefficient and
exponent that define the length-area relationship. A General Equation for predicting the
length of a master stream in moderate- and
large-size drainage basins is
L
~
4.499 A°.466 •
The computer also calculated the mean error
of prediction on the 65 stream lengths using
Hack's equation and the General Equation.
Hack's equation yields a mean error of 70.12
percent; the General Equation yields 21.77
percent. If one were to weigh the mean errors,
the discrepency between the two figures would
be much greater. This is indicated by the fact
that Hack's equation was superior in only 13
of 65 sets of river data, and none of the 13
rivers had a drainage area in excess of 500,000
sq. mi. Nevertheless, Hack's equation has
proven to be more accurate than the General
Equation for the very small drainage basins
for which it was originally intended.
THEORETICAL IMPLICATIONS
It should be obvious that if a drainage basin
lengthened as fast as it widened, the length of
the master stream would be proportional to the
square root of the drainage area. In Hack's
study area, the exponent of his equation is
0.6, suggesting that, in fact, basins do lengthen
faster than they widen; hence the tear-shaped
or pear-shaped basins of mature drainage systems. Inasmuch as we cannot observe significant
growth in natural drainage basins, theoretical
implications are based not only on the exponent, but also on the fact that we are substituting space for time.
If the General Equation is as accurate as
first comparisons suggest, it carries with it profound theoretical implications because its exponent is only 0.466, and streams would therefore lengthen at a rate less than proportional
to the square root of the drainage area. This
indicates that as area increases, drainage basins
widen Jaster than they lengthen. In other words,
the General Equation has inherent theoretical
implications which are just the inverse of
those contained in Hack's equation.
SUMMARY AND CONCLUSION
Hack's equation cannot be applied on a
world-wide basis without inducing intolerable
error. Also, the predictive advantage of the
General Equation over Hack's equation is selfevident. Finally, the theoretical implications
of both equations can be summarized as follows:
1. On a world-wide basis, the General Equation suggests that moderate and large-size drainage basins widen faster than they lengthen.
2. On a local or regional scale, Hack's equation suggests that smaller drainage basins
lengthen faster than they widen.
3. Because the General Equation is best in
very large areas and Hack's equation is best in
very localized areas, perhaps drainage basins
lengthen faster than they widen when they
are small, and widen faster than they lengthen
when they grow larger. This might indicate
that basins predominantly lengthen initially
3473
RELATIONSHIP OF MASTER STREAMS AND DRAINAGE BASINS
10_..-----------r-----------r------------,
L=1.4Ao. 6
HACK
EQUATION
\.
~
0
""
",,""
0
0
.;'
o
g
~
0
/0"'"
l!: ,_'I-------------,j,,-".;''------,''''''''!i--._""'""''-=::-:-:=-:-:-+==:-~_,:_----0
GENERAL EQUATION
:t:
tilZ
__1
L=4.499A0.466
o
...w
_
o
o
'OC:'::_::----'--..J.-..L-..L-J.....J.~!-=J..:::",..--.....I--..J.-.l.-.l.-J.....J...J....._L,,_"=--...l--...I.-....l......l......l....J...J..u
2
DRAINAGE AREA IN MIlES
Figure 1. A comparison of Hack's equation and
the General Equation for 65 moderate and Iarge.size
rivers and drainage basins. Note the increasing error
from left to right when using Hack's equation.
owing to headward erosion and extension of
the master stream, then widen at an increasing
rate once the master stream encounters increased geologic constraints (J. T. Hack, 1972,
written commun.).
4. Because the General Equation's exponent
of 0.466 is quite close to the square root, perhaps the rates at which a basin widens and
lengthens are essentially the same throughout
much of the basin's development, and nence,
are derived from some equilibrium condition.
gratefully acknowledged. Vernon Kennedy collected data and wrote the FORTRAN IV program. Eugene F. Schuster assisted in the
statistical procedures, and Earl M. P. Lovely
and John I. Hack reviewed the manuscript.
ACKNOWLEDGMENTS
Financial support by a University Research
Grant of the University of Texas at El Paso is
REFERENCES CITED
Encyclopedia Americana, 1970, New York, Americana Corporation, v. 23.
Hack, J. T., 1957, Studies of longitudinal stream
profiles in Virginia and Maryland: U.S. Goo!.
Survey Prof. Paper 294-B.
MANUSCRIPT RECEIVED BY THE SOCIETY JANUARY
31, 1972
REVISED MANUSCRIPT RECEIVED APRIL
11, 1972