An efficient method for applying a differential equation to deriving the spatial distribution of specific catchment area from gridded digital elevation models

Deriving the spatial distribution of specific catchment area (SCA) from a gridded digital elevation model (DEM) is one of the most important issues in digital terrain analysis. Conventional methods usually estimate SCA for each cell using a flow direction algorithm, but the results obtained are often unsatisfactory. Recently, Gallant and Hutchinson (2011, Water Resources Research, 47(5), W05535) proposed a differential equation which quantifies the change of SCA along a slope line, and thus the numerical solution of SCA at any point on a surface can be calculated accurately by integrating the differential equation. However, obtaining the numerical SCA solution based on this differential equation is so computationally intensive that it is too time-consuming to use it to derive the overall SCA spatial distribution from a gridded DEM. In this study, we developed a parallel algorithm based on OpenMP to make the numerical SCA solution based on Gallant and Hutchinson (2011)'s differential equation practical to derive the spatial distribution of SCA from a gridded DEM. Experiments based on two artificial surfaces with theoretical SCA and a more complex real terrain surface demonstrated that the proposed parallel algorithm obtained satisfactory acceleration performance and a much lower error than the MFD-md algorithm, which is a representative of conventional grid-based flow direction algorithms. Due to the speedup effects of the proposed parallel algorithm, we analyzed the effects of the DEM grid size and integration step length on the numerical SCA solution in detailed experiments. The experimental results suggested that the proposed algorithm performed best normally at the resolution of 5 m. A step ratio of 0.5 is suitable in applications of the proposed parallel algorithm.