Forward and inverse transformations between Cartesian and channel-fitted coordinate systems for meandering rivers
Legleiter, C.J. and Kyriakidis, P.C.
Mathematical Geology, Vol. 38 Issue 8 pp. 927-958
The spatial referencing of river channels is complicated by their meandering planform, which dictates that Euclidean distance in a Cartesian reference frame is not an appropriate metric. Channel-fitted coordinate systems are thus widely used in application-oriented geostatistics as well as theoretical fluid mechanics, where flow patterns are described in terms of a streamwise axis s along the channel centerline and an axis n normal to that centerline. A means of transforming geographic (x, y) coordinates to their equivalents in the (s, n) space and vice versa is needed to relate the two frames of reference, and this paper describes a pair of transformation algorithms that are explicitly intended for reach-scale studies of modern rivers. The forward transformation from Cartesian to channel-fitted coordinates involves parametric description of the centerline using cubic splines, calculation of centerline normal vectors and curvature using results from differential geometry, and an efficient local search to find in-channel data points and compute their (s, n) coordinates. The inverse transformation finds the nearest vertices of a discretized centerline and uses a finite difference approximation to the streamwise rates of change of the centerline's Cartesian coordinates to obtain the geographic equivalent of a point in the (s, n) space. The performance of these algorithms is evaluated using: (i) field data from a gravel-bed river to examine the effects of initial centerline digitization and subsequent filtering; and (ii) analytically-defined centerlines and simulated coordinates to assess transformation accuracy and sensitivity to centerline curvature and discretization. Any discrepancy between a point's known coordinates in one frame of reference and the coordinates produced via transformation from the other coordinate system constitutes a transformation error, and our results indicate that these errors are 2–4% and 0.2–0.5% of the channel width for the field case and simulated centerlines, respectively. The primary sources of transformation error are the initial digitization of the centerline and the relationship between centerline curvature and discretization.