Proposal of a facet-to-facet multiple flow direction algorithm based on geometrical and mathematical analysis of physical dispersion over triangle facet

Flow direction algorithms have been widely used to extract crucial terrain attributes of great hydrological and geomorphological significance. However, essential distinctions between the empirically-designed strategies of typical algorithms and the natural rules of physical dispersions bring various problems (e.g. parallel channel, artificial dispersion), leading to the low size and extent precisions of estimated results. In this work, geometrical and mathematical analysis is conducted on the inherent characteristics of physical dispersions along slope lines on local terrains. On each 3 × 3 window of digital elevation model (DEM), center pixel is divided into eight non-overlapping sub-facets. Necessary and sufficient (NS) conditions of size relationships between the elevations of adjacent pixels are summarized to directly identify the receiving facets of a sub-facet. Then, strict mathematical relations are derived between slope direction of a sub-facet and flow proportions allocated to receiving facets. A strategy is designed to re-adjust receiving facets and flow proportions for the boundary flow of adjacent facets. Lastly, a multiple-flow-direction algorithm called TFGA is proposed with the NS condition of size relationships, mathematical relation of slope direction with flow proportion, and adjustment strategy of boundary flow. Case studies are conducted for investigating the total contributing areas (TCA) and specific contributing areas (SCA) estimated by TFGA. Results reveal all-side superiorities of TFGA to typical algorithms in spatial patterns, error indicators and statistic characteristics of estimated TCAs and SCAs. Particularly, TFGA improves size precision of estimated results by approximately one order of magnitude. In a conclusion, we highly recommend TFGA for digital elevation analysis.