Systems analysis techniques are used to construct a mathematical model of the variation of dissolved oxygen (DO) in a body of water. The application of these techniques leads to a set of ordinary differential equations that can be solved for the DO response in one section due to an input in any other section. The function that expresses this input-output relationship is shown to be dependent on the hydrodynamic nature of the stream or estuary. Successive use of a linear systems analysis leads to an analytical relationship between the effluent BOD and the resulting DO in the stream.
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Robert V. Thomann (1963) studied this question.