ORIGINAL PAPER A stochastic differential equation model for assessing drought and flood risks Koichi Unami • Felix Kofi Abagale • Macarius Yangyuoru • Abul Hasan M. Badiul Alam • Gordana Kranjac-Berisavljevic Published online: 26 November 2009 Ó Springer-Verlag 2009 Abstract 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 sto- chastic 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 sto- chastic differential equation. A numerical model is devel- oped to compute that probability, using the finite element method with an effective upwind discretization scheme. Applicability of the model is demonstrated at three ra- ingauge sites located in Ghana, where rainfed subsistence farming is the dominant practice in a variety of tropical climates. Keywords Point rainfall  Dry spell  Mean-reverting Ornstein–Uhlenbeck process  Backward Kolmogorov equation  Ghana 1 Introduction Droughts and floods are two opposite natural hazards, but both fundamentally stem from precipitation irregularity. In contrast to physical hydrology, stochastic hydrology applies probability theory to represent the variability of precipitation for engineering purposes. Rainfall at a par- ticular site, i.e., point rainfall, is the most basic stochastic hydrological quantity used to characterize floods and droughts. Standard methodologies are well established to deal with point rainfall data in terms of the relationship between duration and intensity in rainfall events, the return period of high intensity storms or dry spells, and time series patterns of storms (Elliot 1995). However, these conventional approaches do not consider point rainfall as a continuous stochastic process. Since the 1990s, the Bartlett–Lewis rectangular pulse model has become the prevalent method to describe the statistical structure of continuous point rainfall over the entire time domain on a wide range of scales (Onof et al. 2000; Koutsoyiannis and Mamassis 2001), but it still considers the arrival of a storm and K. Unami (&)  A. H. M. Badiul Alam Graduate School of Agriculture, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan e-mail: unami@adm.kais.kyoto-u.ac.jp A. H. M. Badiul Alam e-mail: alam@kais.kyoto-u.ac.jp F. K. Abagale  G. Kranjac-Berisavljevic Faculty of Agriculture, University for Development Studies, P. O. Box TL 1882, Tamale, Ghana e-mail: fabagale@yahoo.com G. Kranjac-Berisavljevic e-mail: gordanak@gmail.com M. Yangyuoru Institute of Agricultural Research, University of Ghana, P. O. Box 68, Accra, Ghana e-mail: macarius_y@yahoo.com 123 Stoch Environ Res Risk Assess (2010) 24:725–733 DOI 10.1007/s00477-009-0359-2