A Back‐Trace Numerical Method for Calculating the Numerical Solution of the True Total Contributing Area for Real‐World Terrains

Total Contributing Area (TCA) has been extensively accepted as a crucial terrain attribute in digital terrain analysis and geological simulations. However, existing flow direction algorithms work poorly in the accuracy of TCA estimation due to the irrationality of their empirically designed strategies. To solve this problem, our work proposes a novel method for TCA calculation in a fundamentally different way from existing algorithms. Firstly, general equation of Specific Contributing Area (SCA) along one‐dimensional slope line is deduced based on the constitutive relations between SCA and plane curvature. Its finite difference is revised to enable the calculation of the SCA by just an upslope trace of slope line with only two temporary variables. Secondly, Bicubic B‐spline (BBS) surface is constructed to approximate the terrain surface represented by a digital elevation model (DEM). Finally, the Back‐Trace Numerical (BTN) method is proposed for calculating the TCA of a DEM pixel based on the mathematical relations between pixel‐scale TCA and point‐scale SCA. Particularly note that the outcome of BTN method is the numerical solution of true TCA, which is fundamentally different from the TCAs estimated by existing algorithms based on empirical strategies. Three cases are designed to assess the performances of the BTN method on various terrain surfaces. If fine BTN parameters are adopted, BTN TCAs show extremely high accuracy on all synthetic surfaces with the mean absolute relative errors of <0.20%. Meanwhile, various basin characteristics (e.g., river, valley and ridge lines) could be accurately recognized based on the BTN TCAs for the DEMs of real‐world terrains.