For over a century, geomorphologists have attempted to unravel information
about landscape evolution, and processes that drive it, using river profiles.
Many studies have combined new topographic datasets with theoretical models
of channel incision to infer erosion rates, identify rock types with
different resistance to erosion, and detect potential regions of tectonic
activity. The most common metric used to analyse river profile geometry is
channel steepness, or

Geomorphologists have been interested in understanding controls on the
steepness of river channels for centuries. In his seminal “Report on
the Henry Mountains”,

Channels do not yield such information easily, however. Any observer of
rivers or mountains will note that headwater channels tend to be steeper than
channels downstream. Declining gradients along the length of the channel
lead to river longitudinal profiles that tend to be concave up. Therefore,
the gradient of a channel cannot be related to erosion rates in isolation;
some normalising procedure must be performed. Over a century ago,

A number of studies

The noise inherent in

Equation (

The choice of the reference concavity index is important in determining the
relative

Sketch illustrating the effect of choosing different reference
concavities. The data can be well fitted with a single regression, suggesting
that all parts of the channel network have similar values of

Extracting a reliable reference concavity indices from slope–area data on
real landscapes is challenging: topographic data can be noisy, leading to a
wide range of channel gradients for small changes in drainage area. The
branching nature of river networks also results in large discontinuities in
drainage areas where tributaries meet, resulting in significant data gaps in

A typical slope–area plot. This example is from a basin near Xi'an,
China, with an outlet at approximately 34

In order to circumvent these problems with

Although the collinearity test does not assume any linearity of profiles in

Noting that vertical celerity is simply the horizontal celerity multiplied by
the local slope after disturbance

Thus, if we assume spatially homogeneous uplift and constant erodibility
(i.e. channels with the same slope and drainage area erode at the same rate),
then the vertical celerity propagating up the principal stream and all
tributaries will be a constant. Equation (

In this study, we revisit commonly used methods for estimating the concavity index using both slope–area analysis and collinearity methods based on integral analysis. Our objective is to determine the strengths and weaknesses of established methods alongside several new methods developed for this study, as well as to quantify the uncertainties in the concavity index. We present these methods in an open-source software package that can be used to constrain concavity indices across multiple landscapes. This information may give insight into the physical processes responsible for channel incision into bedrock, which are as yet poorly understood.

For slope–area analysis, in this paper, we forgo initial smoothing of the DEM
and use a fixed elevation drop along a D8 drainage pathway implemented using
the network extraction algorithm of

Sketch illustrating the methodology of the

Here, we present two new methods of identifying collinear tributaries in

For a given drainage basin, we can multiply the MLE for each tributary to get
the total MLE for the basin, and we can do this for a range of concavity
indices to calculate the most likely value of

There are two disadvantages to using Eq. (

For each iteration of the bootstrap method, we create a template of points in

Sketch showing how we compute residuals for our

We repeat these calculations over many iterations, and for each concavity
index we compute the median MLE, the minimum and maximum MLEs, and the first
and third quartile MLEs. We approximate the uncertainty range by first taking
the most likely concavity index (having highest median MLE value amongst all
concavity indices tested). We then find the span of concavity indices whose
third quartile MLE values exceed the first quartile MLE value of the most
likely concavity indices (Fig.

One complication of using collinearity to calculate the most likely concavity
index is that occasionally one may find a hanging tributary

We also implement a disorder statistic

The disorder metric relies on the use of all the data in a tributary network,
meaning that only one value of

In real landscapes, we can only approximate the concavity index based on
topography. However, we can create simulations where we fix a known concavity
index and see if our methods reproduce this value. To do this, we rely on
simple simulations driven by the general form of the stream power incision
model, first proposed by

We have chosen this model because it can be related to the concavity index
and therefore can be used to test the different methods under idealised
conditions. We can relate the stream power incision model to
Eq. (

Comparing Eqs. (

The stream power incision model also makes predictions about how tectonic
uplift can be translated into local erosion rates

To test the relative efficacy of our methods for calculating concavity
indices, we first run each method on a series of numerically simulated
landscapes in which the

In order to test the methods' ability to identify the correct

Shaded relief
plots of the model runs with temporally varying uplift, with drainage basins
plotted by the best-fit

We initialised the model runs using a low relief surface that is created
using the diamond–square algorithm

Each model was run with a baseline uplift rate of 0.5 mm yr

We analysed these model runs using each of the methods of estimating the
best-fit

Drainage basins were selected by setting a minimum and maximum basin area,

Plots showing the predicted best-fit

Figure

Alongside these temporally transient scenarios, we also wished to test the
ability of each method to identify the correct

Results of the model runs with spatially varying erodibility (

For the spatially varying uplift run, we varied uplift in the N–S direction
by modelling it as a half sine wave:

Inherent in each collinearity-based method of quantifying the most likely

Figure

Example

Our numerical modelling results suggest that the integral profile analysis is
most successful in identifying the correct concavity index out of the entire
range of

Exploration of the most likely concavity indices in the Loess
Plateau, China, Universal Transverse Mercator (UTM) zone 49 N. Basins with
the most likely concavity index determined by the disorder method are
displayed in panel

In order to demonstrate the ability of the methods to constrain concavity
indices in a relatively homogeneous landscape, we first analyse the Loess
Plateau in northern China. The channels of the Loess Plateau are incising
into wind-blown sediments that drape an extensive area of over
400 000 km

We ran each of the methods on an area of the Loess Plateau approximately
11 000 km

As well as determining the best-fit concavity index for the landscape as a
whole, we can also examine the channel networks in individual basins:
Fig.

Many studies analysing the steepness of channel profiles are focused in areas
where external factors, such as lithology or tectonics, are not uniform.
Here, we select an example of a landscape with two dominant lithologic types in a
location along the Oregon coast near the town of Waldport, Oregon
(Fig.

Exploration of the most likely concavity index near Waldport,
Oregon, UTM zone 10 N. Basin numbers and the underlying lithology is
displayed in panel

We find that whereas basins developed on basalt have a relatively uniform
concavity index of approximately 0.7, the most likely concavity indices in
the sandstone show considerably more scatter
(Fig.

Basins analysed near the Gulf of Evia, Greece, UTM zone 34 N, that
interact with active normal faults previously studied by

The steepness of channel profiles and presence of steepened reaches
(knickpoints) in tectonically active areas can reveal spatial patterns in the
distribution of erosion and/or uplift

Steep, smaller catchments tend to drain across the footwalls of these faults,
whilst larger catchments drain the landscape behind the faults, through the
relay zones between fault segments. We derived the best-fit concavity index
for each catchment following each of the five methods
(Fig.

The predicted best-fit

Profile

A number of authors have suggested that in both highly transient and rapidly
eroding landscapes processes other than fluvial plucking or abrasion become
important in shaping the channel profile, such as debris flows and plunge
pool erosion

Our aim here is not to provide a comprehensive examination of the topography
and tectonic evolution of the Sperchios Basin

Finally, it is recognised that transient landscapes are likely settings for
drainage network reorganisation

Spatial distribution of the

Our analysis of the topography in the Sperchios Basin, whilst not exhaustive,
highlights that river profiles and the resulting concavities (and/or

For over a century, geomorphologists have sought to link the steepness of
bedrock channels to erosion rates, but any attempt to do so requires some
form of normalisation. This normalisation is required because in addition to
topographic gradient, the relative efficacy of incision processes is thought
to correlate with other landscape properties that are a function of drainage
area, such as discharge or sediment flux. Theory developed over the last four
decades suggest that the concavity index may be used to normalise channel
gradient, and over the last two decades many authors have compared the
steepness of channels normalised to a reference concavity index derived from
slope–area data

In this contribution, we have developed a suite of methods to quantify the
most likely concavity index using both slope–area analysis and the integral
method. In addition to traditional slope–area methods, we also present
methods of analysing

We find that

Code used for analysis is located at

The supplement related to this article is available online at:

SMM and FJC wrote the code for the analysis, performed the numerical modelling and wrote the visualisation software. SMM, FJC, BG, and MDH performed the analysis on the real landscapes. SMM wrote the paper with contributions from other authors.

The authors declare that they have no conflict of interest.

We thank Rahul Devrani, Jiun Yee Yen, Ben Melosh, Julien Babault,
Calum Bradbury, and Daniel Peifer for beta testing the software. Reviews by
Liran Goren and Roman DiBiase substantially improved the
paper. This work was supported by Natural Environment Research Council grants
NE/J009970/1 to Simon M. Mudd and NE/P012922/1 to Fiona J. Clubb.
Boris Gailleton was funded by European Union Initial Training grant 674899 –
SUBITOP. All topographic data used for this study were SRTM 30 m data
obtained from