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Steepness Indices and Concavity
steeper
gentler
k sn normalized steepness
observed channel concavity
(θ)
reference channel concavity
Fig. 9.10 River channel steepness indices and concavity.
Forced regressions through slope-area data using a reference concavity (0.45 in this example) define differences
in normalized channel steepness ( k sn ). For these channels, the observed concavity ( q ) is similar to the reference
concavity, but normalized steepness varies five-fold. Note the difference in channel slope between the channels
for any given drainage area. Modified after Kirby et al. (2007).
of the Main Central Thrust (MCT; Fig. 9.9). This
study helped promote a new view of this
collisional orogen in which rates of rock uplift,
large-scale crustal structure, and regional
topography became inextricably linked.
With the widespread availability of digital
topography and of programs based on geo-
graphic information systems (GIS) that are
designed to analyze such topography, the use
of stream-gradient indices has been largely
replaced by use of a steepness index, k s (Wobus
et al. , 2006a). Recall from the previous chapter
that S = k s A q (Eqn 8.7), where S is slope, k s is the
steepness index, A is upstream catchment area,
and q is concavity (see Fig. 8.4). Although
concavity varies among rivers, it typically ranges
between 0.4 and 0.7 (Whipple, 2004). For the
sake of comparison among different rivers, the
steepness index can be normalized, k sn , by
using  the same reference concavity, q ref , for all
channels being analyzed, such that
the normalized steepness, k sn , when a regression
with a fixed reference concavity (commonly 0.45)
is forced through channel slope versus area data
to yield the best fit (Fig. 9.10). Combined Matlab
and GIS programs that analyze a DEM and make
a spatial map of variations in normalized steep-
ness can currently be downloaded from http://
www.geomorphtools.org. Such maps can permit
ready identification of river reaches or whole
regions characterized by anomalous steepness.
Analyses of river profiles on a rapidly deforming
fold (Kirby and Whipple, 2001) have shown that
high concavities characterize rivers whose head-
waters are uplifting more rapidly than more
downstream reaches (Fig. 8.24), but that concavi-
ties are commonly normal for a channel exposed
to a uniform uplift rate, irrespective of whether it
is rapid or slow. We might then wonder what
happens to concavity and steepness when the
rate of uplift changes at a given site. From a theo-
retical perspective, we can predict that, if the
relative uplift rate doubles (for example, by
accelerated fault slip at the channel's outlet), a
knickpoint will develop that propagates upstream
and that, downstream of the knickpoint, the
channel will become steeper in order to erode at
a rate that balances the new uplift rate (Fig. 9.11).
(9.2)
kkA qq
−−
(
)
=
ref
sn
s
cent
where A cent is the area upstream of the mid-point
of the reach being analyzed in the DEM (Wobus
et  al. , 2006a). In practice, this approach finds
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