A stochastic differential equation model for assessing drought and flood risks

Droughts and floods are two opposite but related hydrological events. They both lie at the extremes of rainfall intensity when the period of that intensity is measured over long intervals. This paper presents a new concept based on stochastic calculus to assess the risk of both droughts and floods. An extended definition of rainfall intensity is applied to point rainfall to simultaneously deal with high intensity storms and dry spells. The mean-reverting Ornstein-Uhlenbeck process, which is a stochastic differential equation model, simulates the behavior of point rainfall evolving not over time, but instead with cumulative rainfall depth. Coefficients of the polynomial functions that approximate the model parameters are identified from observed raingauge data using the least squares method. The probability that neither drought nor flood occurs until the cumulative rainfall depth reaches a given value requires solving a Dirichlet problem for the backward Kolmogorov equation associated with the stochastic differential equation. A numerical model is developed to compute that probability, using the finite element method with an effective upwind discretization scheme. Applicability of the model is demonstrated at three raingauge sites located in Ghana, where rainfed subsistence farming is the dominant practice in a variety of tropical climates.