The long profile of rivers is shaped by the tectonic
history that acted on the landscape. Faster uplift produces steeper channel
segments, and knickpoints form in response to changes in the tectonic uplift
rates. However, when the fluvial incision depends non-linearly on the river
slope, as commonly expressed with a slope exponent of

Bedrock rivers that incise into tectonically active highlands are sensitive to changes in the tectonic conditions (Whipple and Tucker, 1999). Upon a change in the rock uplift rate with respect to a base level, the river steepness changes (Wobus et al., 2006; Whipple and Tucker, 2002), which in turn changes the local incision rate. In particular, an increase in uplift rate generates steeper slopes that facilitate faster incision, which can eventually lead to incision–uplift equilibrium. However, equilibrium is not achieved synchronously across the river long profile. Upon a change in the tectonic uplift rates, a knickpoint forms that divides the profile into reaches with different steepness and erosion rates (Rosenbloom and Anderson, 1994; Berlin and Anderson, 2007; Oskin and Burbank, 2007). Below the knickpoint, the steepness and erosion rate have already been shaped by the new tectonic conditions, and above the knickpoint, river steepness and erosion rate correspond to the previous conditions (Niemann et al., 2001; Kirby and Whipple, 2012). The erosion rate gradient across the knickpoint promotes knickpoint migration upstream, gradually changing the proportion of the channel that is equilibrated to the new tectonic conditions. For these dynamics, knickpoints are viewed as moving boundaries that separate channel reaches, recording different portions of the tectonic uplift history (e.g., Pritchard et al., 2009; Whittaker and Boulton, 2012).

Since the links between tectonic uplift history and river shape are mediated
by fluvial incision, resolving these links requires a fluvial incision
theory. The stream-power incision model (SPIM) is widely used to describe
detachment-limited vertical incision into channel bedrock, over
long timescales (commonly beyond millennial) and large length scales (Howard
and Kerby, 1983; Howard, 1994; Whipple and Tucker, 1999; Lague, 2014;
Venditti et al., 2019). The SPIM represents the rate of bedrock incision,

Equation (2) is a non-linear advection equation for the elevation, where

Previous general analytic exploration of Eq. (2) (e.g., Luke, 1972;
Weissel and Seidl, 1998; Prichard et al., 2009; Royden and Perron, 2013)
identified that upon a change in uplift rate that induces a long-profile
steepness change, portions of the solution, representing the river profile,
could form that are not strictly associated with the change in uplift rate,
and portions of the solution that hold tectonic information may be lost.
More specifically, when

While some field studies support the slope–incision linearity assumption
(e.g., Wobus et al., 2006; Ferrier et al., 2013; Schwanghart and Scherler,
2020), a growing body of work shows that

When

The current study addresses these questions by developing an analytic
description of the evolution of channel long profiles for the cases where
channel reaches may be consumed; namely

The SPIM model, Eq. (1), predicts that for channel segments that erode
at the uniform rate, the channel slope scales as a power-law function of
the drainage area:

Under steady-state conditions, when

A slope-break knickpoint occurs when there is an abrupt change in the slope and steepness index along a channel long profile (Wobus et al., 2006; Haviv et al., 2010). Within the scope of the SPIM, slope-break knickpoints are commonly associated with a step change in the rate of base level lowering. When the rate increases, the slope and steepness index downstream the knickpoint are greater, and the slope break is convex upward. When the rate decreases, the slope and steepness index below the knickpoint are smaller, and the slope break would appear as a concave kink along the overall concave channel profile. In this latter case, alluviation might occur below the knickpoint, and the assumption of detachment-limited conditions might be violated. This behavior is beyond the scope of the current analysis.

To predict the retreat rate of slope-break knickpoints, we develop a model based on long profile linearization in the proximity of the knickpoint as shown in Fig. 1.

Figure 1a shows the predicted channel profile evolution following a step
increase in the rock uplift rate from

Equations (8)–(14) are developed for the migration of a single knickpoint based on a Lagrangian perspective, i.e., in the reference frame of the migrating knickpoint. Accordingly, Eqs. (13)–(14) predict that knickpoint celerity and response time depend only on the steepness indices immediately above and below the knickpoint and are independent of the steepness indices at lower reaches below lower, newer knickpoints. This means that as long as knickpoints do not merge, as discussed in the following section, knickpoint celerity and response time are not affected by later changes in the tectonic uplift rate and channel steepness.

Equations (13)–(14) reveal that knickpoint dynamics depends on both the slope
exponent,

When more than a single knickpoint propagates upstream a channel profile
and

Figure 2 shows the results for convex-up consuming knickpoints (

The duration of convex knickpoint preservation as a function of slope exponent

The duration of concave knickpoint preservation as a
function of the slope exponent

For the case of concave-up consuming knickpoints (

We note that when

Upon knickpoint merging, only a single knickpoint propagates along the
channel, and the steepness indices above and below the merged knickpoint
correspond to

The elevation change of slope-break knickpoint,

Next, we combine Eq. (21), which is conditioned by knickpoint
preservation, with Eq. (16) that predicts the duration of preservation
to generate a piecewise solution for knickpoint elevation before and after
knickpoint merging. We consider the case of two knickpoints, kp

When deriving an analytic solution for the channel long profile as a
function of time, Eqs. (21)–(23) are used for knickpoint elevation,
Eqs. (24)–(25) are used for the knickpoint

Comparison between the analytic forward
model and a numerical model for the evolution of a river long profile, with
drainage area set by Hack's law,

.

Here, the analytic solution for knickpoint evolution is used to derive a
linear inverse model for retrieving the tectonic uplift history from the river
long profile. The inverse model relaxes the critical assumption of

Changes in

Difficulty may arise because

We propose the following three steps for the application of the inverse
model. First, the data of basins and tributaries are considered in the

Second, the

Segment division should ideally be based on division points that represent
true slope-break knickpoints. Several algorithms have been previously
proposed to identify slope-break knickpoints (e.g., Mudd et al., 2014).
Here, we suggest a different approach that relies on the simplicity and
efficiency of the inverse model. We propose running the inversion procedure
many times, while choosing the number and location of division points
randomly. The quality of the solution with a specific number and location of
division points could be evaluated based on an optimization criterion, such
as a misfit. Mudd et al. (2014) used the Akaike information criterion
(Akaike, 1974) to balance the goodness of fit against model complexity.
Here, we consider a simpler misfit function that penalizes models with more
knickpoints (more parameters) for their excess complexity:

The third step is introducing natural dimensions to the tectonic uplift
history by solving Eqs. (26)–(27) for

In the following, the inversion procedure is demonstrated for both numerical
data and natural data from the Dadu River basin. For the numerically based
demonstration, we use a low-resolution numerical model that solves Eq. (2). The model is used to generate 10 river profiles with variable channel
length and drainage area distribution with pre-chosen model parameters of

Inversion of numerical rivers with

As a second demonstration of the

Application of the inverse model to the Dadu River basin.

Two main tectono-geomorphic events were suggested to control the late Cenozoic erosional history of the Dadu River basin. First, a regional cooling event dated to the late Miocene was inferred based on synchronous rapid exhumation from Shimian and upstream as recorded by low-temperature thermochronology and was attributed to be a response to the regional tectonic uplift that initiated at about 9–12 Ma (e.g., Tian et al., 2015; Zhang et al., 2017; Yang et al., 2019). Second, a major capture event of the upper Dadu River that used to drain through the Anninghe and was redirected to the Yangtze River near Shimian (Clark et al., 2004) was dated to the early Pleistocene by using provenance analysis and thermal modeling (Yang et al., 2019) together with inversion of detrital apatite fission track (AFT) ages in the modern Anninghe River basin (Wang et al., 2021).

Ma et al. (2020) performed a linear inversion on all the streams of the Dadu
River basin while assuming

We inverted five long main trunks of the Dadu River basin, which all drain
to the same base level and generate a uniform trend in the

To introduce natural dimensions to the uplift rate history, the slope
exponent,

The analysis presented here explores river long profile evolution in
response to temporal step changes in the tectonic rock uplift rate

The analysis of merging knickpoints further emphasizes a critical property
of the links between tectonic and long profile evolution when

A basic assumption underlying the analytic derivation and particularly the
forward and inverse models is that the channel system experiences
space-invariant uplift (also consistent with a base-level fall). This
assumption, which is commonly referred to as block uplift conditions, is more
likely to hold over discrete, well-defined tectonic domains with relatively
little internal complexity rather than over large length scales (Goren et
al., 2022). However, larger domains could also experience spatial uniformity
in the uplift history. One way to test for this uniformity is to explore the

The analytic derivation lacks a process-based perspective of knickpoint
migration and instead relies on a simplified stream power parameterization
of knickpoint dynamics. Consequently, a second major assumption, with
specific impact on the inverse model, is that the natural knickpoints
analyzed for changes in the tectonic uplift history are indeed slope-break
knickpoints, which were formed following a change in the tectonic uplift
rate. Knickpoints may also form by autogenic processes (e.g., Scheingross and
Lamb, 2017) or due to spatial changes in the uplift rate (Wobus et al.,
2006), rock erodibility (Kirby and Whipple, 2012) or local hydrologic
conditions (Hamawi et al., 2022). However, when analyzing a branching
channel network, it is relatively easy to distinguish between migrating
slope-break knickpoints which were formed due to a regional uplift rate
change and locally controlled knickpoints. The migrating knickpoints share approximately similar

The current derivation focuses on particular combinations of tectonic uplift
histories and slope exponent with either increasing

When

With

To elucidate this idea, we revisit the simple case discussed in Sect. 4,
with

More generally, analytic solutions of river long profile evolution can
significantly expedite forward and inverse tectonic–fluvial landscape
evolution models. However, so far, analytic solutions were used in such
models only under the

We develop an analytic slope-break knickpoint retreat model under the
assumption of space-invariant uplift rate. The model is based on a
Lagrangian frame of reference and can deal with both convex- (

In this section, we show that two knickpoints formed with

Equations (26)–(27) express the scaled uplift rate history as a series of
values (

First, we prove the base case with a single knickpoint. For this case,
Eq. (21) predicts the knickpoint elevation

Then, assuming that Eq. (29) holds for knickpoint

This study has no complex codes or data sharing issues. All the figures can
be reproduced by solving the related equations. The DEM (digital elevation
model) data used for river profile inversion (Fig. 6) are from the 90 m Shuttle
Radar Topography Mission (SRTM) DEM (downloaded from

The supplement related to this article is available online at:

YW derived the theory and developed the forward and inverse analytic models with input from LG. YW designed the application of the inverse model to the Dadu River basin with input from LG. YW and LG wrote the manuscript. LG wrote the 1-D numerical code. DZ and HZ provided valuable suggestions and made some revisions.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank George E. Hilley for revising the earlier version of this paper many times with great patience and stimulating our inspiration in using the method of characteristics to solve the equation. We thank Fiona Clubb, Eitan Shelef and Sean F. Gallen for constructive instructions on an earlier version of this paper. Constructive suggestions from Philippe Steer and the anonymous reviewer and Simon Mudd (associate editor) helped to improve our paper.

This research has been supported by the National Science Foundation of China (grant no. 41802227) and by the Israel Science Foundation (grant no. 526/19).

This paper was edited by Simon Mudd and reviewed by Philippe Steer and one anonymous referee.