Optimization Approach for 4-D Natural Landscape Evolution

This paper presents a unique optimization method developed for landscape evolution problems. An existing hypothesis of the optimal channel network states that fluvial landscape evolution can be characterized as the procedure that follows minimum total energy expenditure. Previous studies have tested this hypothesis by solving an optimization problem, i.e., finding landscapes that satisfy the minimum total energy expenditure criterion, and showed that such optimized landscapes are similar to natural landscapes in many respects. These studies have approximated a 3-D landscape as a 2-D river network. While this network-based approach has greatly simplified the formulation of the optimization problem, this approximation limits the investigation of features such as longitudinal profiles, since their representation requires the gravitational direction-wise dimension. Here, an alternative technique is devised to fully handle the optimization of 3-D landforms over time. The proposed idea is to break down the time domain and to apply an optimization algorithm sequentially for discrete time steps. For the optimization part, a heuristic algorithm motivated from adaptation strategies of natural systems (here landscape formation) is used. This method is applied to a theoretical landscape with the condition that the balance between tectonic uplift and sediment lost is satisfied. It is found that landscapes of minimum total energy expenditure exhibit the Hack's law and the power-law in the exceedance probability distribution of drainage area, which are the characteristics found in natural river networks. However, they demonstrate no systematic pattern in longitudinal profiles.