ABSTRACT
MODEL EQUATIONS To illustrate the application of the decomposition teclulique, consider a steady-state model designed to simulate the concentration of any single substance in river waters. Such a model can be suitable for evaluating conservative substances, such as major ions, metals or organic compounds. It can also be used for nonconservative substances driven by volatilization, biodegradation and sedimentation. In the latter case the basic equation has the form
~~t) = -aX(t)- bX(t)- cX(t) + d (I)
where X(t) is substance concentration, in mg/L; a is biodegradation or time decay, per day; b is sedimentation, per day; c is volatilization, per day; d is diffuse (nonpoint) source input, in mg/L/day; and t is travel time, in days. Assunte that the uncertainties in the predictions obtained by Equation (1) are caused by uncertainties of the model parameters and the input only. If, for the sake of clarity of this demonstration, all three parameters can be lumped together, and both the new parameter and the input are random variables with known distributions, then the appropriate random differential equation equivalent to Equation (1) can be written as
where /c is a random variable (RV) representing the new, combined reaction-rate coefficient, in mg/L; dis a random diffuse source input, in mg/L/day. Equation (2) is
Computer Techniques and Applications 41
a stochastic differential equation with constant coefficient. Random quantities k and d can be expressed as
(3)
(4)
where m• and mJ are the mean values of RV's k and d, respectively, and lC and 6 are zero-mean RV's representing fluctuating parts, or in other words, variations of k and d from the means.
