\
JERRY E. MUELLER DepaT/menJ oj Gt'ogrophy. Univusily of Wyoming, Laramit>, Wyoming 82070
Re-evaluation of the Relationship of Master
Streams and Drainage Basins: Reply
Mosley and Parker (1973) have raised a
number of points which have varying degrees
of validity and certainly require further
daboration. My basic contention was that the
ponent in my original equation
L=4.499A"·'"
L ~ 1.7
(1)
ggcsts that moderate- and large-size basins
'den faster than they lengthen, as long as
e is willing to substitute space for lime and
believe that large basins grow from smaller
·os. Obviously, the lower the value of the
ponent, the stronger the suggestion that
rge basins will have greater width-length
ratios than do smaller basins.
Smart and Surkan (1967), as reported by
Mosley and Parker (1973), found thar in small
basins in the eastern United States, sinuosity
of the mainstream could inflate somewhat the
value of 'he exponent in Hack's (1957) equa-
tion of
L=IAAO.,.
decreases, whereas floodplain sinuosity (meandering) lends 10 increase. The absolute
sinuosily may change very little. Preliminary
work on third- and fourth-order arroyo complexes in the EI Paso area indicates that
(2)
Inasmuch as my theoretical arguments arc
based on an exponent that is already less than
5, it would appear that subtracting the influence of sinuosity from my exponent would
y enhance, rather chan detract from, my
riginal conclusion. Also, Smart and Surlwn
1967, p. 966) stated rhat " ... we can only
y a priori that it appears more likely that the
·ostream sinuosity increases with increasing
rder and consequently with increasing length
area." My work, both published (1968)
unpublished, has indicated that the range
sinuosity, if defined in terms of total deviao from a straight-line course, may not be as
t as that envisioned by Mosley and Parker
Smart and Surkan. As one proceeds downstream through a drainage net from lowcst to
highest order, lOpographic sinuosity usually
AI·' ,
(3)
and that eliminating the influence of sinuosity
transforms the equation to
L
=
1.2.11°.&9&.
(4)
In other words, in the case of the arroyos. the
influence of sinuosity is almost totally absorbed
by the cocOicielH and nol by the cXIXJncnL
As a point of clarification, ~ losley and
Parker (19i3) stated that the exponent in
Equation 1 " . . . might in fact be considerably less than 0.466," owing to the influence of sinuosity. Sinuosity is part of length
and, therefore, must be included in any equation that predicts stream length from basin
area. Their statement, when viewed in light
of their subsequent comments, must be based
on the erroneous assumption that L in Equation 1 is b."1sin length, whcn, in fact, that equa,ion was described (~lucllcr, 1972, p. 3472) as
"A General Equation for predicting the Icngth
of a master stream in modcrate- and large-size
drainage basins. . . ." i\losley and Parkcr's
figures and equations are developed from basin
lengths; minc arc based on strCam Icngths.
Another point raised by ~Iosley and Parker
is whether my exponent of 0.466 is significantly
diffcrent [rom rhe square root. Il should bc
evidcnt that the significance of this diO'crcnce
incrcases as basin area increases, if in fact we
want to predict the length of thc master
stream. The Yenisie, one o[ the world's largest
drainage systems, has a basin area of approximately 1,000,000 sq mi and a mainstream
length of2,566 mi. ~Iy equation for moderate-
Geological Society of America Bulletin, v. 84, p. 3127-3130, I fig., September 1973
.3127
3128
J.
E. /I.·fUELLER
100,000 sq mi are best-fitted by the equation
and large-size basins, using the exponent of
0.466, yields a length of 2.814 mi, or an error
of 9.7 percellt. If. instead, we lise 0.5 as the
exponent, the predicted Icngrh of the Ycnisic
is approximately 4,500 mi (several hundred
miles longer than the Nile or .\m:lZon), yield~
ing an error of 75.3 percent.
:-.. rosley and Parker also suggested that,
owing to rhe change in the c:xponcnr [rom 0.6
[or slllall :Hcas to 0.466 or less lor large areas,
it is likely that the exponent is continuously
changing through a full array of clata [rom vcry
small to vcry large basins, and the length-area
relation may best be described as curvilinear.
The same son of reasoning led me and an
assistant to compile several thousand sets of
length-area clata during 1972 in an attempt to
plot the full range of values :lnd to derive a
parabolic-type, best-fit equation. Figure 1,
based on a str:ltified, random s.'lInple of 250
basins between 0.0 I and 3.000,000 sq mi, docs
not suggest a continuously changing exponent.
Inste:ld. the distribution is best described as
having three straight-line segments along each
of which rhe exponent remains fairly rigid.
Hack's Equation 2 defines exceedingly well the
length-area rclation in basins with less than
8.000 sq mi. whereas basins between 8,000 and
L
~
3.0 AO.' ,
(5
and basins larger than 100,000 sq mi are st
best-fitted by Equation 1. Equation 5 sugges
that. during a portion of basin deveiopmen
thc rates of widening and lengthening arc esse
tially rhe same; therefore, we cannot as y
discount my tentativc condusion 4 (1972.
3473) as Mosley and Parker would have us d
Figurc I does not indicate what appear'
be fairly rapid changes in the exponent ?-t t
8,000- and 100,000-sq-mi limits, owing to t
small scale at which the graph was rcproduc
The deflections to the right wcre, howey
readily apparent on the oversize worksh
from which Figurc I was compiled. Lastly,
onc is willing to sacrifice a bit of accuracy fi
simplicity, rhe full array of data in Figure
can be fitted by a simple, straight· line equati
of
L ~ 1.6386
A"."" .
Obviously, the basic arguments on rh
significance of equations that describe length
arca relations assume that the initial data wer
reliable. In this respect, the validity of th
data in Mosley and Parker's Figure I can be
10' ~---+---+---+_---I----+---+-~~",,:::c.:"::"'---.j!-----d
...,'
~ 10 1 1,----t----t-----t----+_--..;--t;.;>."'."---+---+---+--___d
.. ~\
z
-
..
'
10 1,----t----t----+r7""-~+_---+---+---+---+--___d
1
"
"
~
l
100
= 1.6386 AO,SS36
~---I----,,.·,,·:;;·~::+f.-..- '---+----+----+------<----I------"'----d
10' r--+-----j---+---f----+---l---+---l---d
10'
AREA
IN
Figure 1. Relation between master-Stream length
and basin arca for 250 sets of modornly select/:d data
SQUARE
MILES
from various parts of the world.
(D~;:~~
MASTER STREAMS AND DRAINAGE BASINS: REPLY
uestionoo. Their basin areas were extracted
rom various encyclopedias which, during my
ta colleerion in 1972, proved fO be the least
liable sources of data. The most aberrant
lot in all my analyses to date has come [com
basin with a drainage area of 70,115 sq mi,
though an encyclopedia claims 167,500.sq mi
r the same basin! Certain federal and state
gcncies, regional planning commissions, largc~
Ie maps, and field measurements provjdc
occ reliable data. Even U.S Gealnulcal
urvey ~ra ~inal utility, for river
ngth aiiOl5iSln area arc usually given only
r the area upstream [rom gaging nations.
The validity of the basin-length data that
osley and Parker extracted from straight
nes drawn on atlas maps can also be ques·oned.lfbasin data are taken from maps, their
curacy will depend on the map's inherent
eralization and deformation of shape or
. The measurement of length across a given
on a map demands that shapes and areas
represented in projX)rtion to their characterics on the globe where scale is the same at
JXlinrs and in all directions around a point.
or a given map projection, it is impossible
retain both area and shape; that is, these
o properties are mutually exclusive of one
other, never appearing together on the same
p (Robinson and Sale, 1969). On large·seale
ps the problem is negligible, but on smallIe maps, such as those in atlases, areas may
exaggerated or condensed several hundred
rcent. Also, the only tcue distance between
river's mouth and source is along a great·
de route, and usually this is not a straight
e on most maps. The only map projection
t shows all great circles as straight lines is
gnomonic, but this projection is never
in modern reference atlases 3S a base for
tinental maps, except for an occasional map
:Antarctica. These criticisms of technique
not directed at Mosley and Parker, but,
tead, should serve as general words of
ution for those who attempt to extract the
metric properties of drainage basins from
.metric maps.
There a"re other problems with defining basin
gth in terms of a straight line drawn be·
n a river's mouth and source. In cases
here basins have irregular shapes, such a line
uld depart enough from the main axis o[ the
·n so that it might actually cross part of an
jacent basin. I tested this, using the Prentice·
all World Atlas (1963), and found
the
angtze cuts across the Hwang-Ho, the Sal-
,ha'
3129
wccn bisects both lhe Brahmaputra and Irra·
waddy Basins, the trace of lhe Euphrates would
be drawn along the axis of the Tigris Valley,
the Tennessee cuts across the Cumberland, and
so forth. It would seem that a true measure of
basin length should be somc natural parametcr
whose length parallels the main axis of the
basin. Lengths derivcd by using thc course of
the mastcr stream, thc axis of the main vallcy,
or the trace of the thalwcg appear more mean·
ingful, although, it is conccded, these may be
marc diflicult to measure than a straight linc.
Thc distinction madc by ~Iosley and Parker
bctween size and growth relations is important.
It is based on the fact that wc use static data
to try to interprct long-term dynamic changes
in basin geometry. If one does not accept that
larger basins have grown [rom smaller basins
not too unlike those in the present landscape,
then aU that can be stated with certainty is
that larger basins havc greater width-length
ratios than smaller basins. On the other hand,
if onc believes sizc·related differences arc also
growth-related differences, then, based on
Equations I and 2, small basins lengthen faster
than they widen, whereas large·size basins
widen faster than they lengthen.
There is some evidence in the litcrature that
clearly indicates a predominance of basin
widening ovcr basin lengthening with the
passage of timc. Ruhc's work (1952) on dated
tills in Iowa showcd that basins on older tills
have, owing to tributary development, denser
drainage networks and greater width-length
ratios than basins on younger tills. Leopold
and others (1964), in reviewing Ruhe's work,
noted that the consistem relation over time
between channel length and channel number
for the Iowa basins was similar to that for
b..,sins in widely separated areas, and concluded
that "This supports the hypothesis that the
spatial distribution is in fact a 'growth'
model." A cursory glance at the several thousand sets of length-area data that I have compiled [rom the glaciated midwestern states has
temativcly led me to the same conclusion.
\Vhat we do not know, and may never know,
is whether these growth· related changes in
basin geometry have been prevalent in the
larger range of basin sizes.
ACKNOWLEDGMENTS
i\ University Research Grant [rom the
University of Texas at £1 Paso provided support for this work. Ted £. :\pley and Earl
~f. P. Lovejoy reviewed the manuscript, and
.'
.
3130
J. E. MUELLER
Vernon Kennedy, Eugene F. Schuster, and
rhe University Computation Center olTered
technical assist:lnce.
REFERENCES CITED
Hack, ]. T., 1957, Studies of longitudinal stream
profiles in Virginia and Maryland: U.S. Cool.
Survey Prof. Paper 294- B, p. B45-897.
Leopold, L. B., Wolman, M. G., and Miller, ].
P., 1964. Fluvial processes in geomorphology:
San Francisco, \Y. H. Freeman Co., 522 p.
Mosley, M. P., and Parker, R. S., 1973, Re-evalua~
tion of the relationship of master streams and
dl"Jinage basins: Discussion: Geol. Soc. Amer·
iC3 Bull., v. 84, no. 9, p. 3123-3126.
Mueller, J. E., 196B, An introduction to the hy.
dcaulic and topographic sinuosity incle.xes:
Assoc. Am. Geographers Annals, v. 58. p.
371-385.
- - 1972. Rc·cvaluation of the relationship of
master streams and drainage basins: Cool.
Soc. America Bull., v. 83, p. 3471-3473.
Prentice-Hall World Atlas, 1963: EnglewoOO Cliffs,
N. J., Prentice-HaJJ, Inc.
Robinson, A. H., and Sale, R. D., 1969, E1emen
of cartography: New York, John Wiley
Sons, Inc., 415 p.
Ruhe, R. V., 1952, Topographic discontinuities
the Des Moines lobe: Am. Jour. Sci., v. 25
p_ 46--56.
Smart,]. S., and Surkan, A. J., 1967, The relati
between mainstream length and area in drain
age basins: Water Resources Research, v.
p.963-973.
MANUSCRIPT RECEIVED BY TIlE SOCIETY FEBRUAR
23, 1973