Quantitative tectonic geomorphology hinges on the analysis of longitudinal
river profiles. The model behind almost all approaches in this field originates
from an empirical relationship between channel slope and catchment size, often
substantiated in the form of the stream-power model for fluvial incision.
Significant methodological progress was recently achieved
by introducing the

The vast majority of the approaches used to derive information on
tectonic processes from topography are based on the
analysis of longitudinal river profiles. The fundamental relationship between
channel slope

Understanding and quantitative interpretation of Eq. (

In the simplest version of the stream-power
approach it is assumed that the erosion rate

Physically based models of bedrock incision suggest that the concavity index

A range of

Compared to the concavity index

Using Eq. (

The simplest interpretation of Eq. (

The most interesting applications of the stream-power erosion equation
(Eq.

Recently, the so-called

The most striking property of the

Since a clear distinction requires the consideration of a large
number of tributaries, the inherent limitation of the stream-power
approach to the fluvial regime also limits the

The transition from a fluvial regime at large catchment sizes to a
regime dominated by hillslope processes is explored by an example from Taiwan
in Fig.

Relationship between mean channel slope and catchment size for the
topography of Taiwan. Channel slopes and catchment sizes were derived from
the SRTM1 and SRTM3 DEMs, and mean slopes were obtained from logarithmic bins with a
factor of

On the other hand, the number of nodes with a catchment size of

In the following we present two extensions of the basic relationship
between channel slope and catchment size (Eq.

The approaches presented in the following are intended to be as simple as
possible. First, we aim at a representation by a uniform equation without
distinguishing different regimes, although the domain below (concerning
catchment size) the region
where Flint's law holds is sometimes described as the debris flow regime

The data shown in Fig.

The respective modification of the

As an alternative approach, a constant value can be added to the term

Equation (

The respective modified

As shown by the red and green lines in Fig.

Each of the approaches contains one or two
adjustable parameters (

In the simplest situation, a steady-state topography under homogeneous uplift and
erodibility, the

Including small catchment sizes in the analysis even facilitates the determination
of the adjustable parameters since the collinearity of a large bunch of
lines in the

Due to the problems with the rank correlation coefficients discussed above,
we suggest an alternative criterion for assessing the collinearity of all
rivers in the

Some attention should be paid to pairs of identical elevation values occurring
frequently in integer-valued DEMs. Here we suggest to assume that
all

In the following we compare the different approaches

In order to get a sufficient number of catchments of similar sizes where each
catchment contains a significant portion in the fluvial regime, a procedure to automatically delineate
catchments with a size

Map of the 89 considered catchments in Taiwan with catchment sizes

Cumulative distribution of the

The mountainous catchment in Taiwan with the lowest

A catchment in Taiwan with a rather high

The topography of Taiwan yields 89 catchments meeting
these criteria, with each
of them containing between 6464 and 27 732 valid SRTM1 DEM nodes.
These catchments are shown in Fig.

Figures

Taking into account the width of the

The relevance of spatial heterogeneity to the

If only a smaller range of catchment sizes is considered, the differences between
the methods partly disappear. Figure

Map of elevation (coded by color) and

Cumulative distribution of the

Cumulative distribution of the

It is also immediately recognized that
restricting the lower limit of catchment size reduces the absolute
values of the

As a second example we consider the European Alps as an orogen that was heavily affected
by glacial erosion in the past. For simplicity, we define
the region as the domain inside the 600 m elevation contour line as previously
done by

Beyond the goodness of the fit expressed by the

Cumulative distribution of the concavity index

We have presented and investigated several concepts of extending Flint's law and the

Among the approaches considered in this study, an extension of Flints's law
similar to an equation originally suggested for debris flow channels

Minimizing the

The authors would like to thank W. Schwanghart and J. Jasiewicz for their constructive reviews and G. Sofia for the editorial handling. Edited by: G. Sofia