Slope–area analysis and the integral approach have both been widely used in stream profile analysis. The former is better at identifying changes in concavity indices but produces stream power parameters with high uncertainties relative to the integral approach. The latter is much better for calculating channel steepness. Limited work has been done to couple the advantages of the two methods and to remedy such drawbacks. Here we show the merit of the log-transformed slope–area plot to determine changes in concavities and then to identify colluvial, bedrock and alluvial channels along river profiles. Via the integral approach, we obtain bedrock channel concavity and steepness with high precision. In addition, we run bivariant linear regression statistic tests for the two methods to examine and eliminate serially correlated residuals because they may bias both the estimated value and the precision of stream power parameters. We finally suggest that the coupled process, integrating the advantages of both slope–area analysis and the integral approach, can be a more robust and capable method for bedrock river profile analysis.

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):

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).

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 (

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

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 natural river usually consists of different channel substrates, for
example colluvial, bedrock and alluvial channels. In spite of their complex
formation processes, we can identify them from a log-transformed slope–area
plot (Fig. 1; Snyder et al., 2000). The colluvial channel, characterized by
steep channel slope (> 20

Via the integral approach (Perron and Royden, 2013), we derive concavities
of bedrock channels. Based on a reference concavity index (Hu et al., 2010;
Kirby et al., 2003, 2007; Perron and Royden et al., 2013; Snyder et al.,
2000; Wobus et al., 2006), the

The coupled process does provide a better way to perform stream profile
analysis. Indeed, both the slope–area analysis and integral approach are
bivariant linear regression methods. Statistically, some tests must be done
to meet two critical conditions – i.e., the residuals are independent and
homoscedastic (Cantrell, 2008; Kirchner, 2001). Perron and Royden (2013)
noticed that the precision in steepness derived from the integral approach
would be overestimated due to auto-correlated residuals. Mudd et al. (2014)
proposed a statistical framework to quantify spatial variation in channel
gradients and calculated Durbin–Watson statistics in their code
(

Stream profile analysis of Juan stream.

Range of Durbin–Watson statistic and the related meaning.

We took the integral approach for an example and rewrote Eq. (5) into
another form:

We first calculated the self-correlation coefficient of residuals via Eq. (8):

Then, the DW statistic was derived as DW

We then examined whether the residuals were auto-correlated according to Table 1.

To eliminate the self-correlation, new variables were generated as Eq. (9):

where the slope of a linear fit to revised relative elevation,

Correlation coefficients derived from slope–area analysis and the
integral approach. The slope was derived from the smoothed (horizontal
distance of 300 m) and re-sampled (elevation interval of 20 m) elevation data.
The correlation coefficients of

To evaluate whether the variance of residuals is a constant, we utilized the Spearman
rank correlation coefficient test (Choi, 1977; York, 1968):

Via a linear regression of

We sorted the

The Spearman rank correlation coefficient, rs, and the

When the

Statistic tests for the integral approach

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)

Revised relative elevation and

Correlation coefficients of

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

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)

Nevertheless, for the integral approach, it is difficult to recognize
bedrock and alluvial channels along a river profile. When computing

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

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 D

We also calculated

Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the

Correlation coefficients of

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

Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the

Even though it gives highly uncertain channel steepness values, slope–area
plots make no assumptions about

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

Correlation coefficients of

In most cases, the

We extracted the stems and tributaries of streams Singley, Davis, Fourmile
and Cooskie (Fig. 9a), based on

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

Nevertheless,

In most cases, a somewhat higher constant critical area (e.g., 1 or 5 km

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

Concavity values that maximize the co-linearity of the main stem
with its tributaries. Black thick lines in the

Erosional parameters in the stream power model (e.g.,

Uplift histories inferred from the stream profiles.

Log-transformed slope–area plots of streams in the high-uplift

Since the

The slope–area data and

We utilized variable erodibility (

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

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

We chose 0.1–3 km

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;

Concavity values that maximize the co-linearity of the main stem
with its tributaries.

The map of

The case study has disadvantages of including only short (< 10 km
long; < 20 km

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,

Usually, the method of best linearizing a

We extracted all the tributaries of the Mattole River and calculated their

Based on

In this contribution, we coupled the advantages of slope–area analysis and the integral approach to steady-state bedrock river profile analysis. First, we identified colluvial, bedrock and alluvial channels from a log-transformed slope–area plot. Utilizing the integral approach, we then derived concavity and steepness indices of a bedrock channel. Finally, via the Durbin–Watson statistic test, we examined and eliminated serial correlation of linear regression residuals, which produced more reliable and robust estimates of uncertainties in stream power parameters.

The DEM (digital elevation model) data we used in the manuscript are free. The
1 arcsec SRTM DEM can be downloaded at

Yizhou Wang did river profile analysis and the statistical tests and wrote the paper. Huiping Zhang and Dewen Zheng helped analyze some slope–area data and revise the manuscript. Jingxing Yu, Yan Ma and Jianzhang Pang analyzed the tectonic and climatic settings of the study area.

The authors declare that they have no conflict of interest.

We thank Eric Kirby (Oregon State University) and Liran Goren (Ben-Gurion University of the Negev) for significant suggestions on slope–area analysis and the integral approach, respectively. We thank Amanda McDowell (Oregon State University) and Qi Ou (University of Cambridge) for revision in English writing. We are grateful for the grants from the National Science Foundation of China (41622204, 41272215, 41272196, 41590861, 41661134011), State Key Laboratory of Earthquake Dynamics (LED2014A03) and the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB03020200). Yizhou Wang is also funded by state special supporting plan. Edited by: S. Mudd Reviewed by: two anonymous referees