A probabilistic river water quality model is developed with the capability of determinging the joint and marginal probability density function of biochemical oxygen demand (BOD) and dissolved oxygen (DO) at any point in a river. The one dimensional steady-state model can be applied to a river system with any reasonable number of point loads and diversions and lateral surface and subsurface inflow. The model can simultaneously consider randomness in the intital conditions, inputs, and coefficients of the water quality equations. Any empirical or known distribution can be used for the initial condition. The randomness in 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 equation. Moment equations are developed which allow the mean and variance of the marginal distrubution of BOD and DO to be calculated independently of the joint pdf. An upper limit on the coefficient noise variance parameter is presented for which the BOD-DO covariance matrix will be asymptotically stable. The probabilistic river water quality model is applied to two problems, a sensitivity problem and a hypothetical problem. The sensitivity problem and a hypothetical problem. The sensitivity problem is used to gain familiarity with the simulation model and determine the sensitivity of the model responses to changes in the standard deviation parameter of the input and coefficient noise. The standard deviation parameter of the input noise if varied between zero and 30 percent of the respective input, while the standard deviation parameter of the coefficient noise is varied between zero and 50 percent of the respective coefficeitn. The model responses are foind to be fairly sensitive to changes in the standard deviation parameter of the coefficient noise but relatively insensitive to changes in the standard deviation parameter of the input noise. The possibility of using the moment equations and a normal approximation in lieu of calculating the joint pdf of BOD and DO is discussed. The accuracy of the numerical solution technique for the Fokker-Plank equation is also discussed. The hypothetical problem is used to evaluate the performance of the model in simulating a more complex river system (which included two point loads) and to evaluate the numerical quandrature algorithm used to determine the joint pdf of BOD and DO immediately downstream of a point load. The numerical solution technique used to determine the joing pdf of BOD and DO remained stable throughout the simulation and the computational costs are judged to be reasonable for a problem of this complexity. The quadrature algorithm was judged to have performed adequately for both the pont loads.
Finney, Brad A.; Bowles, David S.; and Windham, Michael P., "Random Differential Equations in Water Quality Modeling" (1979). Reports. Paper 491.