The dishonest method in stream temperature modeling

The unidimensional form of the thermal energy conservation principle as a quasi‐linear partial differential equation (PDE) has been shown accurate for point temperature forecasts on completely mixed streams and river‐run reservoir systems. Two methods of solution are presented which lead to a unique solution. Yet this nonrandom solution is contrary to the behavior of nature, and so a philosophic change is introduced. From the PDE subsidiary differential system a stochastic differential equation (SDE) is obtained with random forcing function from the meteorologic and forebay‐depth analyses. With random initial conditions (time‐averaged components of the in situ water temperature vector) the SDE is then recast as a random nonlinear Volterra integral equation (RIE). Then a solution to the random‐initial‐conditions RIE (fixed points of the expectations), by the so‐called dishonest method (Keller, 1962), is compared with a solution obtained by repetitively solving the PDE as a random equation (expectations of the fixed points). Of course, these solutions, obtained by ‘dishonest and honest’ methods, are not necessarily unique, but they may be sufficiently close in some sense. Hence proper interpretation and use of this RIE model would enable water resource planners economically to determine stream temperatures in probability during critical climatic or river flow conditions. And then intelligent planning and scheduling could avoid some catastrophic aquatic event which might occur as a result of extreme water temperatures.