On different types of adjustment usable to calculate the parameters of the stream power law

Model parameterization through adjustment to field data is a crucial step in the modeling and the understanding of the drainage network response to tectonic or climatic perturbations. Using as a test case a data
set of 18 knickpoints that materialize the migration of a 0.7-Ma-old erosion wave in the Ourthe catchment
of northern Ardennes (western Europe), we explore the impact of various data fitting on the calibration of
the stream power model of river incision, from which a simple knickpoint celerity equation is derived. Our
results show that statistical least squares adjustments (or misfit functions) based either on the streamwise distances between observed and modeled knickpoint positions at time t or on differences between observed and modeled time at the actual knickpoint locations yield significantly different values for the m and K
parameters of the model. As there is no physical reason to prefer one of these approaches, an intermediate
least-rectangles adjustment might at first glance appear as the best compromise. However, the statistics of
the analysis of 200 sets of synthetic knickpoints generated in the Ourthe catchment indicate that the timebased adjustment is the most capable of getting close to the true parameter values. Moreover, this fitting
method leads in all cases to an m value lower than that obtained from the classical distance adjustment
(for example, 0.75 against 0.86 for the real case of the Ourthe catchment), corresponding to an increase in
the non-linear character of the dependence of knickpoint celerity on discharge