Introduction
In an evolving landscape, information about tectonics, climatic change, and
lithology can be recorded by the bedrock river profiles (Fox et al., 2014,
2015; Goren et al., 2014; Harkins et al., 2007; Royden and Perron, 2013;
Snyder et al., 2000). How to retrieve such details has long been a focus in
both geologic and geomorphologic studies (Flint, 1974; Wobus et al.,
2006; Rudge et al., 2015). Most of these studies are based on a well-known
power-law relationship between local channel gradient and drainage area
(Flint, 1974; Hack, 1973; Howard and Kerby, 1983):
dzdx=ksA-θ,ks=(U/K)1/n,θ=m/n,
where z is elevation, x is horizontal upstream distance, U is bedrock uplift
rate, K is an erodibility coefficient, A is drainage area, and m and n are
constants. Parameters θ and ks are the concavity and
steepness indices, respectively. The power-law scaling holds only for
drainage areas above a critical threshold, Acr, which is the transition
from divergent to convergent topography or from debris-flow to fluvial
processes (Montgomery and Foufoula-Georgiou, 1993; Tarboton et al., 1989;
Wobus et al., 2006). A growing number of studies have quantitatively related
steepness to rock uplift (Hu et al., 2010; Kirby and Whipple, 2012; Kirby et
al., 2003, 2007; Tarboton et al., 1989). Assuming a steady-state river
profile under constant rock uplift rates and erodibility in time and space,
two forms of solutions to Eq. (1) are derived:
logUK1/n+-mnlogA=logdzdx,z=zb+(UKA0m)1/nχ,χ=∫0xA0A(x′)m/ndx′,
where zb is the channel elevation at x= 0 (river outlet). This is a
boundary condition to Eq. (1). A0 is an area-scale factor.
Schematic of a steady-state river profile consisting of colluvial,
bedrock and alluvial channels, revised from Fig. 7a and b in Snyder et al. (2000). (a) Stream profile. (b) Log-transformed slope–area plot.
The slope–area analysis, as shown in Eq. (4), yields concavity and steepness
indices by a linear fit to the log-transformed slope–area plot. Concavity
changes with different channel substrate properties, which can be reflected
and extracted from the slope–area data directly. Then, one can discriminate
channel properties according to variable concavity indices. For example,
available studies indicate that the colluvial, bedrock and alluvial channels
can be directly identified from the log-transformed slope–area plot (Kirby
et al., 2007; Snyder et al., 2000, 2003; Wobus et al., 2006). However,
estimates of slope obtained by differentiating and resampling noisy
elevation data are even noisier (Perron and Royden, 2013). Differentiation
leads to considerable scatter in slope–area plots, making it challenging to
identify a power-law trend with adequate certainty (Perron and Royden,
2013). In addition, the derived channel steepness suffers from high
uncertainty due to error propagation (Perron and Royden, 2013; see Sect. 3
for details).
Streams in the Mendocino Triple Junction (MTJ) region of northern
California, USA. Streams are from Snyder et al. (2000). The elevation data
are from a 1 arcsec SRTM DEM (http://earthexplorer.usgs.gov/).
The integral approach, based on an integration of Eq. (1), was proposed by
Royden et al. (2000) to alleviate such problems by avoiding calculating
channel slope. As shown in Eqs. (5) and (6), the transformed variable χ can be determined directly from drainage area data by simple numerical
integration. Based on a proper concavity, the steady-state river profile can
be converted into a straight line. The slope of the line is steepness (we assume
A0=1 m2 throughout the paper). As the best-fit value of θ
is not known a priori, we can compute χ-z plots for a range of θ
values and test for linearity (Perron and Royden, 2013). Thus, the integral
method provides an independent constraint on both θ and ks
(Perron and Royden, 2013). Nevertheless, the χ transformation contains
an assumption of a single concavity, which is distinctly different from
slope–area analysis. In fact, concavity can change. In places where there is
spatially varying concavities (because channels may go from bedrock to
alluvial), the integral approach may show a break in the χ-z plot.
Methods of separating areas of different concavities from a χ-z plot
have not been suggested. Despite a very noisy method compared to the
integral approach, slope–area analysis is a more direct measure of
concavity, because unlike the integral method one does not have to set an
m/n ratio but rather measures this ratio directly from topographic data. In
addition, the uncertainty in ks will be underestimated using the
integral method, because the transformed profile (χ-z plot) is a
continuous curve, and therefore the residuals of the linear fit are serially
correlated (Perron and Royden, 2013).
Based on the analysis above, coupling the advantages of the two methods can
make up for their individual drawbacks and provide a better way to constrain
stream power parameters. We also run bivariate linear regression statistic
tests for the two methods to evaluate whether the residuals of linear fit are
homoscedastic and serially correlated. In this paper, we take streams,
located in the Mendocino Triple Junction (MTJ) region of northern California
(Fig. 1), for example, to illustrate the process.
Stream profile analysis of Cooskie stream. (a) Log-transformed slope–area
plot. The slope was derived from the smoothed (horizontal distance of 300 m)
and re-sampled (elevation interval of 20 m) elevation data. (b) The full
river profile (without any smoothing or re-sampling) of Cooskie stream. (c) The
correlation coefficients, R, as a function of θ for least-squares
regression based on Eq. (5). The maximum value of R, which corresponds to
the best linear fit, occurs at θ= 0.45 (dotted line and black
arrow). (d) χ-z plot of the bedrock channel profile, transformed
according to Eq. (3) with θ= 0.45, Acr= 0.1 km2, and
A0= 1 m2.
Case study: Mendocino Triple Junction (MTJ) region
Based on a 1 arcsec SRTM DEM (digital elevation model), we extracted 15
streams in Mendocino Triple Junction (MTJ) region (Fig. 2). Here we first
took streams Cooskie and Juan, for example, to illustrate the advantages and
disadvantages of slope–area analysis and the integral approach, as well as
to explain the reason of coupling the two methods.
Channel concavity and steepness indices can be derived from either
slope–area analysis or the integral approach. For the same river profile,
both methods should yield identical results (Scherler et al., 2014). We
divided the profile of Cooskie stream into colluvial and bedrock channels
from the log-transformed slope–area plot by eye (Fig. 3a). The area of
process transition along a river profile can be determined using a number of rigorous
methods. For example, Mudd et al. (2014) used a segmentation algorithm and
Clubb et al. (2014) used a two-segment method for first-order channels to find
the area of process transition. Nevertheless, Fig. 3a shows a very simple
log-transformed slope–area plot, from which the colluvial (nearly constant
log(slope) ∼ -1) and fluvial (decreasing channel gradient)
sections can be discriminated just by eye. The elevation and area of the
dividing point are ∼ 500 m (Fig. 3b) and 0.1 km2
(critical area, Acr). The concavity of bedrock channel is
∼ 0.47 ± 0.05. We also computed the correlation
coefficients between bedrock channel elevation and χ values based on a
range of θ values. The best linear fit corresponds to θ= 0.45. Both of them are similar to the result (0.43 ± 0.12) of
Snyder et al. (2000), but they are slightly higher than the result (0.36) of Perron
and Royden (2013), which may be attributed to the difference in DEM
resolution or choosing different critical areas.
Revised relative elevation and χ values. Gray lines and
numbers are revised data and steepness index with uncertainty estimates.
Correlation coefficients of χ-z plots as a function of
θ. (a) Location of the streams. (b–e) Correlation coefficients of
χ-z plots based on a range of θ values. Black thick lines
indicate stems.
Although the concavities derived from the two methods are in agreement,
uncertainties (dividing the estimated value by error) in channel steepness
differ a lot. The uncertainty from slope–area analysis is ∼ 40 % (ks= 79.16 ± 29.35; Fig. 3a), but the integral approach
gives only ∼ 0.5 % (ks= 62.81 ± 0.39; Fig. 3d).
In addition to smoothing and re-sampling of elevation data, we attribute
such large uncertainty to error propagation. The natural logarithmic value
of steepness from slope–area analysis is 4.37 ± 0.37, which results in
a ks value of 79.16 ± 29.35. This indicates that the steepness
indices will have large uncertainties even for high linear correlation of
the log-transformed slope–area plot. Hence, the integral approach is much
better for calculating channel steepness.
Concavity indices usually vary along river channels where different
substrates outcrop (e.g., alluvium, and bedrock). For example, along the Juan stream, we identified colluvial (log(slope) ∼ -1, drainage
area < 0.16 km2, elevation > 700 m), bedrock
(decreasing channel gradient) and alluvial (channel slope decreases in a
much higher concavity, drainage area > 8.89 km2,
elevation < 150 m) channels from the log-transformed slope–area plot
for their variable concavities (Fig. 4a and b). As shown in Fig. 4a, these
channels are characterized by different concavities, consistent with
estimates from Snyder et al. (2000). According to the concavity of the
bedrock portion of the river (θ= 0.52, derived from the integral
approach, Fig. 4c), the bedrock channel profile is converted into a straight
line (Fig. 4d).
Nevertheless, for the integral approach, it is difficult to recognize
bedrock and alluvial channels along a river profile. When computing χ-z plots (Acr= 0.16 km2, for the whole fluvial channel including
both bedrock and alluvial portions) based on a series of concavity values,
the best-fit θ is 0.72 (Fig. 4e). As shown by the transformed
profile (Fig. 4f), a knickpoint (at elevation of ∼ 400 m)
occurs on the channel. Below the knickpoint, the alluvial and bedrock
portions share the same slope (ks= 3354 ± 20 m1.44) despite
the different channel substrates. Above the knickpoint, the ks value is
1667 ± 15 m1.44. Variations in the slope of χ-z plot may be
treated as spatially or temporarily variant rock uplift rates (Goren et al.,
2014; Perron and Royden, 2013; Royden and Perron, 2013). However, no
knickpoint occurs on stream Juan because the river has been controlled by
uniform rock uplift and under steady state (Snyder et al., 2000; see Sect. 4.2 for discussion). Thus, a χ-z plot generated by a single concavity
may lead to misestimates in stream power parameters. We should recognize
changes in concavities from slope–area space.
According to the log-transformed slope–area plots, we identified bedrock
channels of the 15 streams. Concavity indices were then calculated via both
slope–area analysis and the integral approach. As shown in Fig. 5, both
methods yielded similar concavities. Based on a mean θ value of
0.45 ± 0.10 (1σ), we computed χ-z plots and normalized
steepness indices (ksn) with uncertainty estimates (Fig. 6). The
uncertainties in steepness indices (no statistical test) are nearly lower
than 1.0 % (Fig. 6).
We run statistic tests (Durbin–Watson test and Spearman rank correlation
coefficient test) for the integral approach and slope–area analysis. For the
integral approach, all the DW statistics are lower than DL (Fig. 7a),
indicating serially correlated residuals. Then, we revised the elevation and
χ data according to Eq. (9) (Fig. 8). The DW statistics of revised
χ-z plots are all between DU and 4-DU (Fig. 7a), indicating
independent residuals. The results of linear fit are shown in Fig. 8. The
uncertainties in steepness indices (revised by Durbin–Watson test) are about
2.4–9.9 % (Fig. 8), which are much higher than those without
statistical test (lower than 1 %, Fig. 6). In addition to uncertainty
estimate, auto-correlated residuals can also bias the regression
coefficient, steepness. The channel steepness values of streams Fourmile,
Kinsey and Hardy are 57.01, 103.90 and 58.78 m0.9 (Fig. 6). When
revised by the Durbin–Watson test, these values are 36.33, 82.16 and 76.65 m0.9, respectively (Fig. 8).
Thus, steepness varies
about 25.6–58.3 % (dividing the difference of the two kinds of steepness indices by
the values revised by Durbin–Watson test). Due to the influence of
auto-correlated residuals on both the estimated value and precision of
steepness, the Durbin–Watson test is necessary when applying the integral
approach. For slope–area analysis, the DW statistics are all between DU
and 4-DU (Fig. 7b), showing no auto-correlation.
We also calculated t statistics for both slope–area analysis and the
integral approach (Fig. 7a and b). All the results are less than 2,
indicating homoscedastic residuals. Despite no heteroscedasticity being found in
our study area, we suggest that Spearman rank correlation coefficient test
should also be done because the test is a part of linear regression
(statistically).
Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the χ-z plots are stems.
Correlation coefficients of χ-z plots as a function of
θ. (a) Location of the streams. (b–e) Correlation coefficients of
χ-z plots based on a range of θ values. Black thick lines
indicate stems.
In addition to statistic tests, another way proposed by Perron and Royden (2013) to estimate uncertainty in steepness is to make multiple independent
calculations of different river profiles. From Fig. 6, the mean ksn
of the high-uplift zone (U= 4 mm yr-1) is 104.40 ± 14.06, and that of the low-uplift
zone (U= 0.5 mm yr-1) is 71.25 ± 10.08. The standard errors of the mean
ksn among profiles are considerably larger than that for individual
streams. However, for multiple profiles under similar geological and/or
climatic settings, this approach should provide more meaningful estimates of
uncertainty.
Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the χ-z plots are stems.
Discussion
Even though it gives highly uncertain channel steepness values, slope–area
plots make no assumptions about θ and are therefore more sensitive
than χ analysis for detecting spatially varying concavities.
Slope–area analysis is thus useful to identify difference in substrates
along a river (e.g., bedrock, alluvium), which can be used as regression
limits to apply the integral method. The integral approach yields
better-constrained values of θ and ksn. Combining these methods
with statistical tests provides more reliable results when applied to
perform stream profile analysis. In the following sections, we will discuss
the parameter uncertainty and steady assumption to better illustrate this
method.
Uncertainty of channel concavity
Perron and Royden (2013) considered that the uncertainty in channel
concavity derived from a linear regression of the log-transformed
slope–area plot described how precisely one can measure the slope of the
plot, not how precisely the parameter is known for a given landscape. They
suggested that the difference between θ values that best linearize
the main stem profile and that maximize the co-linearity of the main stem
with its tributaries could be an estimate of uncertainty in θ for an
individual drainage basin.
Correlation coefficients of χ-z plots as a function of
θ. (a) Location of the streams. Streams are extracted with a
critical area of 0.5 km2. (b–e) Correlation coefficients of χ-z
plots based on a range of θ values. Black thick lines indicate stems.
In most cases, the θ value that collapses the main stem and its
tributaries is often used as a reference concavity (Mudd et al., 2014;
Perron and Royden, 2013; Willett et al., 2014; Yang et al., 2015). In fact,
supposing a drainage basin under uniform geologic and climatic settings,
this kind of θ value can be compared with the mean value of
concavities of the stem and its tributaries. We thereafter refer to these two
concavities as θCo (derived from the co-linearity test) and
θmR (from averaging the concavity values of all the streams
within a catchment), respectively.
We extracted the stems and tributaries of streams Singley, Davis, Fourmile
and Cooskie (Fig. 9a), based on Acr of 0.1–0.16 km2 (Fig. 5). We
calculated the correlation coefficients of χ-z plots based on a range
of θ (Fig. 9b–d). The θmR of catchments are 0.45,
0.48, 0.43, and 0.55. We also calculated θCo that collapses the
stem and tributaries. The θCo values are 0.45, 0.45, 0.45, and 0.55 (Fig. 10). Both θCo and θmR are similar to the stem concavities: 0.50,
0.42, 0.50, and 0.45 (Fig. 5). Hence, for steady-state bedrock channels
under uniform lithologic and climatic settings, all three kinds of
concavities should be similar. Thus, the difference between these θ
values could be an estimate of concavity uncertainty.
However, concavity varies in streams consisting of both bedrock and alluvial
channels. We extracted the stems and tributaries of streams Hardy, Juan,
Howard and Dehaven (Fig. 11a). The θmR values of them are 0.57,
0.68, 0.73, and 0.73 (Fig. 11b–e), similar to the stem concavities (0.63,
0.70, 0.72, and 0.75; Fig. 11b–e), but larger than the θCo
(0.45, 0.45, 0.45, and 0.55; Fig. 12). In such case, differences between
θCo and θmR are not random errors and cannot be
estimates of concavity uncertainty.
Nevertheless, θCo values (0.45, 0.45, 0.45, and 0.55) are
similar to the concavities of bedrock reaches of stems (0.55, 0.52, 0.55,
and 0.40; Fig. 5). Thus, the differences between θCo and
concavities of bedrock reaches may be estimates of uncertainties in
θ. Hence, the reference concavity collapsing the stem and its
tributaries works well even for all the profile data consisting of both
bedrock and alluvial channels.
In most cases, a somewhat higher constant critical area (e.g., 1 or 5 km2) is assumed to calculate χ values of fluvial channels (Goren
et al., 2014, 2015; Willett et al., 2014; Yang et al., 2015). Here we
extracted streams of four drainages (Fig. 13a) – Hardy, Juan, Howard, and
Dehaven – based on a critical area of 0.5 km2 (3 to 4 times the
actual values). We then derived the concavities that best linearize stems
(0.73, 0.78, 0.82, and 0.84; Fig. 13b–e), θmR (0.60, 0.80,
0.75, and 0.75; Fig. 13b–e), and θCo (0.40, 0.50, 0.45, and
0.55; Fig. 14). All the results are similar to those based on
actual critical areas (Figs. 11 and 12). Hence, choosing a uniform
Acr somewhat different to the actual values might be reasonable and
would not have a significant influence.
Steady-state assumption of streams in the MTJ region
River shape may not be diagnostic of equilibrium conditions. In
some places, recent work on inversion of drainage patterns for uplift rate
histories indicates that river profile shapes are controlled by
spatiotemporal variations in uplift rate moderated by erosional processes
(Pritchard et al., 2009; Roberts and White, 2010; Roberts et al., 2012).
In the MTJ region, the uplift rates determined by marine terraces are
variable in space and time (0–4 mm yr-1; Merritts and Bull, 1989). However, in
the low-uplift zone (streams Hardy to Dehaven), uplift rates have been
approximately constant for at least 0.33 Ma (Merritts and Bull, 1989). The
bedrock-channel reaches are probably not affected by sea-level fluctuations
(Snyder et al., 2000). These streams thus can be in or near equilibrium.
Nevertheless, disequilibrium conditions are likely in regions of high-uplift
rate (e.g., the rivers north of 40∘ N). To test the steady-state
assumption, we modeled the uplift rate histories.
Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the χ-z plots are stems.
Erosional parameters in the stream power model (e.g., m and n) and uplift
histories can be determined from joint inversion of drainage network
(Glotzbach, 2015; Goren et al., 2014; Pritchard et al., 2009; Rudge et al.,
2015). Here, we utilized the method of Goren et al. (2014). For spatially
variant rock uplift, the study area is divided into four distinct zones,
from north to south, the north transition zone (streams Singley to Cooskie),
the King Range high-uplift zone (streams Randall to Buck), the
intermediate-uplift zone (stream Horse Mtn), and the low-uplift zone
(streams Hardy to Dehaven; Fig. 15a–d; Snyder et al., 2000). Within each
zone, we assumed spatially invariant rock uplift for small drainage areas
and similar uplift rates determined from marine terraces (Merritts and Bull,
1989). Snyder et al. (2000) suggested n ∼ 1 and variable K between
the high- and low-uplift zones. According to the linear inversion model of
Goren et al. (2014), the present river channel elevation is determined by
both rock uplift rate and response time, τ(x) (time for perturbations
propagating from the river outlet, at x= 0, to a point x along the channel):
z(x)=∫-χ(x)0U∗(t∗)dt∗,U∗=U/(KA0m),t∗=KA0mt.
For the linear model (n= 1) and A0= 1 m2, response time τ(x)=χ(x)/K. The scaled time t* has the same unit of χ, and U* is
dimensionless rock uplift rate.
Uplift histories inferred from the stream profiles. (a–d) The
map of streams in the four zones: the north transition zone (a), King Range
high-uplift zone (b), intermediate-uplift zone (c), and low-uplift zone (d).
(e–h) The χ-z plots of the streams within each zone (A0= 1 m2). The black line indicates an average result. (i) Scaled U* as a
function of scaled time t*. (j) Inferred relative uplift rate as a function
of time before the present. The bottom-left black axes show the results of
the north transition, high-uplift and intermediate-uplift zones. The top-right
gray axes show the result of low-uplift zone.
Log-transformed slope–area plots of streams in the high-uplift
(a–c) and low-uplift (d–f) zones. Streams within the same zone are
composited. The slope data are calculated via different methods: a 300 m
smoothing window and 20 m contour sampling interval (a, d), a 300 m
smoothing window and 10 m contour sampling interval (b, e), and a 100 m
smoothing window and 10 m contour sampling interval (c, f). Elevation
data are from 1/3 arcsec USGS DEM (downloaded from
https://catalog.data.gov/dataset/national-elevation-dataset-ned-1-3).
Since the χ-z plot may be affected by other factors (e.g., climate and
lithology), we extracted all the fluvial channels and calculated a mean
χ-z plot for each zone (Fig. 15e–h). We defined z1, z2, …,
zN and χ1, χ2, …, χN to be the elevations and
χ
values of N data points along a fluvial channels network (N= 10 here). Thus, based on Eq. (12), the dimensionless rock uplift histories for the four
zones are shown in Fig. 15i. For the low-uplift zone, alluvial channels in
the lower reaches were excluded for them being affected by sea-level
fluctuations.
The slope–area data and χ-z plot of the stem of Mattole
river. (a) Map of the stem and its tributaries in the Mattole drainage
basin. Elevation data are from 1/3 arcsec USGS DEM (downloaded from
https://catalog.data.gov/dataset/national-elevation-dataset).
(b) The log-transformed slope–area plot of the stem (a 300 m smoothing window
and 20 m contour sampling interval). A knickpoint is detected from the plot
with variant ksn along the channel. (c) River profile of the stem. The
gray arrow indicates the dividing point (∼ 450 m) between the
colluvial and fluvial portions. The black arrow shows the knickpoint
(∼ 280 m) on the stem. (d) The correlation coefficients
between elevation and χ values as a function of θ. The maximum
value of R, which corresponds to the best linear fit, occurs at θ= 0.30 (gray dashed line). (e) The χ-z plot of the stem, transformed
according to Eq. (3) with θ= 0.30, Acr= 0.1 km2, and
A0= 1 m2.
We utilized variable erodibility (K=U/ksn) values to calculate rock
uplift rates. The K values for transition, high-uplift, intermediate and
low-uplift zones are 6.17, 3.82, 3.38, and
0.37 × 10-5 m0.1 a-1, respectively. According to the
inferred uplift histories (Fig. 15j), the maximum response time (the
perturbations migrating from the river outlet to water head) differs
significantly from low- (0.43 Myr) to high-uplift (0.16 Ma) zones. The rock
uplift rates in the low- and intermediate-uplift zones have been constant
(∼ 0.3–0.4 mm yr-1 since 0.4 Ma, and ∼ 2–2.5 mm yr-1
since 0.16 Ma, respectively). The north transition and high-uplift zones
both experienced increases in the uplift rates (from ∼ 2.5 to 3.3, and from ∼ 3.7 to 4.3 mm yr-1,
respectively) starting about 0.12 Ma. However, the increase ratios are
much lower. Considering the maximum response time (∼ 0.16 Myr),
the uplift rates have been constant for a relatively long period. In
addition, no large knickpoints are found along the rivers. All of these
indicate that the rivers have been reshaped by the recent tectonic
activities and have reached steady state.
In the recent 0.02 Myr, the rock uplift rates seem to be a bit lower (Fig. 15j). That may be due to variant channel concavities. The reaches downstream
are usually characterized by rapidly decreasing gradient (higher
concavities). Then, lower U* will be produced when using a reference concavity
(0.45). As a result, the modeled rock uplift rates will be low. The
variance in channel concavity may indicate difference in river substrate
(e.g., sedimentation affected by sea-level fluctuations) rather than
tectonics (Snyder et al., 2000).
Influence of elevation data uncertainty
Roberts et al. (2012) noticed that the slope–area methodology might produce
unstable results because small amounts of randomly distributed noise added
to river profile will cause significant change in channel gradient. In spite
of little knowledge about the elevation data uncertainty here, we utilized
different datasets and various data handling methods (data smoothing and
sampling) to calculate channel slope with different uncertainties. Then, to
some extent, the influence of data uncertainty can be tested.
In the analysis above, the channel slope is derived from 1 arcsec SRTM
DEM via a 300 m smoothing window and 20 m contour sampling interval. We
reanalyzed the streams in high- and low-uplift zones based on 1/3 arcsec
USGS DEM (downloaded from
https://catalog.data.gov/dataset/national-elevation-dataset-ned-1-3).
We calculated the channel slope via a 300 m smoothing window and 20 m contour
sampling interval (Fig. 16a and d), a 300 m smoothing window and 10 m contour
sampling interval (Fig. 16b and e), and a 100 m smoothing window and 10 m
contour sampling interval (Fig. 16c and f), respectively. To get average
values, slope–area data from all the streams within the same zone were
composited.
We chose 0.1–3 km2 as regression limits for the high-uplift zone and
0.2–8 km2 for the low-uplift area. The channel concavity and steepness
(ksn) were calculated by linear regression of the log-transformed
slope–area data and χ-z plots (θref= 0.45), respectively.
The stream concavity indices in the high-uplift zone (0.41 ± 0.05) and
low-uplift region (0.48 ± 0.03) are similar to or within error of the
estimates reported by this study (0.45 ± 0.10, 1 arcsec SRTM DEM),
Wobus et al. (2006; 0.57 ± 0.05, 10 m pixel USGS DEM), and Snyder et al. (2000; 0.43 ± 0.11, 30 m USGS DEM). All the error estimates are
characterized by 1σ. Mean ksn values of 109 and 60 m0.9 in the high- and low-uplift zones, respectively, yield a ratio of
ksn (high)/ksn (low) of ∼ 1.82, which mirrors the
findings of both Snyder et al. (2000) and Wobus et al. (2006). We find no
distinct difference in concavity and channel steepness indices when using
different datasets and data handling methods.
Utilizing different datasets may cause some differences in parameter
estimate for an individual catchment. For example, when using the integral
approach, the resulting channel concavity of stream Cooskie is 0.45 (in
Sect. 3; 1 arcsec SRTM DEM) but 0.36 in Perron and Royden (2013; 1/3 arcsec USGS DEM). However, for averaged results (as done in Sect. 4.3),
uncertainty in elevation data may not cause distinct differences in
parameter estimates in this study area (e.g., θand ksn).
Concavity values that maximize the co-linearity of the main stem
with its tributaries. (a–d) The χ-z plots of the stem (black line) and
its tributaries (gray lines) using different values of θ
(Acr= 0.1 km2, and A0= 1 m2). (e) The elevation scatter
of the χ-z plots showing that minimum scatter is achieved with
θ= 0.45.
The map of ksn (θ= 0.45, elevation interval of 100 m) of the Mattole drainage basin. The black circle indicates the knickpoint
on the stem. Low values are shown along the whole stem and its tributaries
above the knickpoint. High ksn values are distributed along the upstream
of the tributaries below the knickpoint.
Disequilibrium circumstances in large rivers
The case study has disadvantages of including only short (< 10 km
long; < 20 km2 area) and steady streams. In many landscapes,
especially large rivers, this steady assumption will not be met (Harkins et
al., 2007; Wobus et al., 2006; Yang et al., 2015). To explore the effect of
landscape transience, we analyzed Mattole River, a large river in the MTJ
region (Fig. 17a). Here, 1/3 arcsec USGS DEM was used.
Using a 300 m smoothing window and 20 m contour sampling interval, we
derived a log-transformed slope–area plot of the stem (Fig. 17b). We
recognized the critical threshold of drainage area, Acr, ∼ 0.1 km2 and at the elevation of ∼ 450 m (Fig. 17b and c)
from the slope–area plot by eye. A knickpoint was detected by the scaling
break in the slope–area data and then marked in the shaded-relief map (Fig. 17a) and the river profile (Fig. 17c). The knickpoint is located at an
elevation of ∼ 280 m. The concavity indices above (0.61 ± 0.01) and below (0.58 ± 0.07) the knickpoint are nearly the same. To
compare with the adjacent streams, a reference concavity θref= 0.45 was used to calculate the channel steepness. Using the
integral approach and two statistic tests, we derived the ksn above
(10.81 ± 0.86 m0.9) and below (17.44 ± 1.16 m0.9) the
knickpoint. However, in the adjacent streams (e.g., Davis, Fourmile), the
ksn values are much larger than 60 m0.9. In addition to spatial
variations in Holocene uplift rates of marine platforms (Merritts, 1996),
these steepness indices suggest that other variables (e.g., sediment flux and
lithology) may affect channel steepness. This might limit our ability to
quantitatively relate steepness indices to uplift rates in this field
setting, as noticed by Wobus et al. (2006).
Usually, the method of best linearizing a χ-z plot is used to compute
θ for a steady-state bedrock river profile (Perron and Royden,
2013). However, a channel may be transient, in which case previous authors have
suggested either segmentation of χ profiles (Mudd et al., 2014) or
interpretation through inversion methods (e.g., Goren et al., 2014). We
computed the correlation coefficients between the channel elevation and
χ values (Acr= 0.1 km2, A0= 1 m2) of the stem
based on a range of θ (Fig. 17d). The best linear fit corresponds to
θ= 0.30 (ks= 1.45 ± 0.06 m0.6, R= 0.985, Fig. 17e),
which is distinctly different from the result of slope–area analysis.
We extracted all the tributaries of the Mattole River and calculated their
χ-z plots based on a range of θ values (Fig. 18a–d). The
elevation scatters of the χ-z plots are plotted against θ
values (Fig. 18e). The θ value that collapses the main stem and its
tributaries is 0.45, showing the reasonability of using 0.45 as a reference
concavity to calculate the stem ksn. As shown by Fig. 18c, the
knickpoint (with an elevation of about 280 m) can also be detected from the
χ-z plot of the stem. Both the slope–area data (Fig. 17b) and the χ-z plot based on a θ value derived from co-linearity test (Fig. 18c)
detect the unsteady signal on the trunk stream of the Mattole River, despite
the best linearity for the integral approach (Fig. 17e). We can find
that a river may be in disequilibrium condition in spite of a linear
relationship in the χ-z plot. In some cases, uplift can be inserted
along rivers, which makes values of χ difficult to interpret (Czarnota
et al.,2014; Paul et al., 2014; Pritchard et al., 2009; Roberts and White,
2010; Roberts et al., 2012; Wilson et al., 2014).
Based on θref= 0.45, we calculated the map of channel
steepness with an elevation interval of 100 m. The channel steepness values
range from 1 to 273. As shown in Fig. 19, the lower ksn values are along
the whole stem and its tributaries (low elevation) above the knickpoint,
while higher values are along the upstream (high elevation) of tributaries
below the knickpoint. Among the tributaries in the west of the stem, channel
steepness decreases from the central part (near streams Big to Shipman,
high-uplift zone) towards both north (close to stream Fourmile, north
transition zone) and south (near streams Horse Mtn and Telegraph,
intermediate-uplift zone). Both the spatial pattern of ksn and the
positive relationship between ksn and elevation may indicate a tectonic
control on channel steepness despite other potential variables.