A stochastic coverage model for erosion events caused by the intersection of burnt forest and convective thunderstorms

Abstract

Soil erosion events following fire can wash sediment and ash into streams and reservoirs, contaminating water supplies for cities and towns. These risks are real, yet difficult to quantify, constraining the optimal selection of preventative and remedial options such as prescribed burning and investment in water treatment infrastructure. Non-stationary climate and fire regimes resulting from climate change add to this difficulty. What is the chance of a water supply becoming unusable due to fire? Will this increase with climate change? Will prescribed burning increase or decrease this risk? Answering these questions is challenging because both fire and rainfall regimes are already complex processes to model individually. Considering the interaction between these two processes substantially increases the complexity of the modeling problem. The model outlined in this paper is based on the premise that highmagnitude erosion events following fire result from the spatial and temporal intersection of burnt areas and high-intensity rainfall events. In this new model we consider fires and storms as independent stochastic processes with properties of spatial extent, temporal duration, and frequency of occurrence. This is illustrated in Figure 1, where we have superimposed realizations of fire and storm processes. Here the (x,y)-axes give the spatial extent and the vertical axis gives duration. The volume of intersection of the two processes (shown by the overlap of the large and small discs) gives a measure of hazard of high-magnitude erosion events, and we can quantify how it changes in response to changing fire and climate regimes. Let the set Ωrepresent the catchment for a single year, then we are interested in the "risk set" R=Ω ∩ burnt area × duration ∩ stormy area × durationHere R has dimensions Km 2×.yeras The duration of a fire is the time it takes for the vegetation to recover (a couple of years), rather than the time the fire is active (a couple of days). The volume of R, that is ∥R∥, represents the annual area where burn and rainfall satisfy the conditions required for high-magnitude erosion events to occur in a catchment where post-fire response thresholds are known. Let λ = fire event rate (per unit area and unit time); μ = storm event rate (per unit area and unit time); α = E∥fire event∥ (in Km 2× years); and β = E∥rainfall event∥ (in Km 2× years). Given these definitions, we use the mathematics of coverage processes to show that E∥R∥= ∥Ω ∥(1-e -λα) (1-e -μβ) In addition to deriving this result we obtain estimates for, and, and consider the effect on of changing these parameters.