Automated numerical estimation of meander length and amplitude

This paper details a novel numerical method for automatic estimation of stream meander length. It is based upon Langbein et al. (1966) and produces a single meander length and meander amplitude over a given reach that minimises the least-squares error. This algorithm has been applied to 1,456 reach segments, each approximately of 1 km flow path, throughout the Murray-Darling Basin. These estimates were used in the Physical Form theme of the Murray-Darling Basin Sustainable Rivers Audit (SRA). This paper details the method and presents results from one reach and then a summary from all 1,546 study reaches. The algorithm for estimation of meander length is based upon the theory of Langbein et al. (1966) and further applied and assessed by Williams (1986). The theory proposes that the angle between the flow direction at a given point and the regional stream flow path changes, with reach distance, to produce a sinusoidal relationship. By numerically fitting a sine function to this relationship, a wave length and thus the meander length can be estimated. However, a challenge in applying this method is in defining the regional stream flow path to which the observed angle is calculated. If it was to be applied to only a single reach, the regional flow path could simply be qualitatively estimated by the practitioner. However, implementation to the 1,456 reaches investigated herein required an automatic method for estimation of this regional flow path and estimation of the meander length. To achieve this, and after trialling a considerable number of alternatives, an algorithm was developed to rotate the reach, fit a k order polynomial to estimate the regional flow path, calculate the residuals to this path and then numerically fit the Langbein et al. (1966) sinusoidal model to the residuals. The fitting and rotation was undertaken by multi-start Trust-Region non-linear leastsquares regression. Overall, the algorithm developed provided a robust, objective and reproducible means for estimating meander length. Furthermore, those reaches for which the algorithm does not perform satisfactorily can easily be identified by the low coefficient of efficiency. The algorithm does, however, have some weakness. Most notably, the modelled meander length is a single best estimate, in a least-squares sense, to the entire one kilometre reach. If smaller or longer reach chainage were investigated it is likely that the estimate would differ. Therefore, the estimated meander length herein is very likely to be a function of the scale of the application.