On the mathematical description of glaciated valley cross sections

Source abstract

Abstract Morphometric analyses of glaciated valleys typically involve attempts to fit empirical data from valley cross‐profiles to quadratic (parabolic) or power‐law equations, with the choice of equation depending on the goal of the analysis. In assessing the relative merits of these two types of equations, some confusion has arisen because of apparent variations in coordinate system datums between studies which have made use of the power‐law equation. However, we show that this confusion relates to a simple mathematical error, rather than any true difference in methodology. Of more significance, however, is the fact that in fitting a power‐law equation to observed profiles, significant bias is introduced by the use of a logarithmic data transformation. Because of this implicit bias, power‐law equation parameters are influenced most strongly by data points close in elevation to the centre of the valley, and so failure to account for the depositional elements which frequently occur in these areas can lead to significant errors if the aim of the analysis is to characterize the form of the erosional profile.