The non-linear reliance of channel steepness on erosion rates can be reconciled by the stochastic-threshold incision model that incorporates river incision threshold and discharge probability distribution into erosion efficiency. Here, we explored the usage of the model in river longitudinal profile inversion, by assuming time-dependent tectonic forcing and a linear exponent that relates channel incision to slope. We developed an analytical solution to the model equation and an inverse scheme to retrieve relative uplift rate history, whose validity was based on the theoretical demonstration on knickpoint preservation. Application of the inverse scheme to the main trunks of the Dadu River basin in the eastern Tibetan Plateau produced a history with two-phase increases in the uplift/incision rates, which is similar to the results from low-temperature thermochronology. Thus, our analytical procedures provide new insights into the link of tectonic uplift and river profile evolution, when channel steepness depends on erosion rates non-linearly. 

The long profile of rivers is widely considered as a recorded of tectonic uplift rate. Knickpoints form in response to rate changes and faster rates produce steeper channel segments. However, when the exponent relating fluvial incision to river slope, n, is not unity, the links between tectonic rates and channel profile are complicated by channel dynamics that consume and form river segments. Here, we explore non-linear cases leading to channel segment consumption and develop a Lagrangian analytic model for knickpoint migration. We derive a criterion for knickpoint preservation and merging, and develop a forward analytic model that resolves knickpoint and long profile evolution before and after knickpoint merging. We further propose a linear inverse scheme to infer tectonic history from river profiles when all knickpoints are preserved. Our description provides a new framework to explore the links between tectonic uplift rates and river profile evolution when n!=1.