A non-Markovian model of rill erosion

Stochastic processes with reinforcement are inherently non-Markovian and therefore may model geophysical processes with memory, for instance patterns of rill erosion, more realistically than Markovian models. Reinforcement provides a bias to a system that is equivalent to infinite memory, making a system more likely to occupy a given state the more often the state is visited. Some well-studied examples in applied mathematics include variations on the urn of P'olya and reinforced random walks. Many natural phenomena exhibit similar behavior: for instance, an overall pattern of rills is relatively stable once it is established, although small details of the pattern may change frequently and catastrophes that permanently alter it may occasionally occur. To model the phenomenology of rill erosion, we propose a simple discrete time, infinite-memory random process defined on the nodes and edges of an oriented diagonal lattice. Lattice models have often been used to investigate the morphology of natural drainage networks, but our focus is as much on the dynamics of network formation as it is on morphology. The lattice in our model starts out smooth in the sense that it has no edges initially, but it sprouts edges everywhere the instant the process starts, much as rain can start soil erosion everywhere on a hillslope at once. Exactly one edge (rill segment) descends from each node, and it points either left or right. Sediment loads travel along networks of edges and are accumulated at nodes. At every node and at every time step, a simple two parameter reinforcing law randomly determines the direction of the node’s output and then is updated. The degree of reinforcement is set by comparing the node's current sediment load to the load history of the entire network above it and is governed by two system parameters representing respectively rainfall intensity and the soil’s resistance to change. The current pattern of connections among nodes represents the present state of the process, and the pattern’s stability - measured by the tendency of the same state (or one similar to it) to occur on subsequent iterations of the process - represents the pattern’s strength as a memory. At any given moment the current pattern is a collection of dendritic networks that appears similar to drainage networks found in nature.