Random differential equations in river water quality modeling

A one‐dimensional steady state probabilistic river water quality model is developed to compute the joint and marginal probability density functions of BOD and DO at any point in a river. The model can simultaneously consider randomness in the initial conditions, inputs, and coefficients of the water quality equations. Any empirical or known distribution can be used for the initial conditions. Randomness in each of the water quality equation inputs and coefficients is modeled as a Gaussian white noise process. The joint probability density function (pdf) of BOD and DO is determined by numerically solving the Fokker‐Plank random differential equation. In addition, moment equations are developed which allow the mean and variance of BOD and DO to be calculated independently of their joint pdf. The probabilistic river water quality model is examined through sensitivity study and an application to a hypothetical river system.