Modeling Tools for Environmental Engineers and Scientists

y setting the reaction rate constant of a contaminant to zero, the model may beeasier to solve algebraically and the output may be more easily compared withthe case of a conservative substance, which may be readily obtained. Otherinformal validation tests can include running the model under a wide range ofparameters, input variables, boundary conditions, and initial values and thenplotting the model outputs as a function of space or time for visual interpretationand comparison with intuition, expectations, or similar case studies.For formal validation, a “testing” data set from the real system, either historicor generated expressly for validating the model, can be used as a benchmark.The calibrated model is run under conditions similar to those of thetesting set, and the results are compared against the testing set. A model canbe considered valid if the agreement between the two under various conditionsmeets the goal and performance criteria set forth in Section 2.2.1. Animportant point to note is that the testing set should be completely independentof, and different from, the training set.A common practice used to demonstrate validity is to generate a parity plotof predicted vs. observed data with associated statistics such as goodness offit. Another method is to compare the plots of predicted values and observeddata as a function of distance (in spatially varying systems) or of time (in temporallyvarying systems) and analyze the deviations. For example, the numberof turning points in the plots and maxima and/or minima of the plots andthe locations or times at which they occur in the two plots can be used as comparisoncriteria. Or, overall estimates of absolute error or relative error over arange of distance or time may be quantified and used as validation criterion.Murthy et al. (1990) have suggested an index J to quantify overall error indynamic, deterministic models relative to the real system under the sameinput u(t) over a period of time T. They suggest using the absolute error or therelative error to determine J, calculated as follows:J = T e(t) T e(t)dt or J = T ẽ(t) T ẽ(t)dtooe(t)where e(t) = y s (t) – y m (t) or ẽ(t) = y s(t)y s (t) = output observed from the real system as a function of time, ty m (t) = output predicted by the model as a function of time, t2.2.5 SUMMARY OF THE MATHEMATICAL MODELDEVELOPMENT PROCESSIn Chapter 1, physical modeling, empirical modeling, and mathematicalmodeling were alluded to as three approaches to modeling. However, as couldbe gathered from the above, they complement each other and are appliedtogether in practice to complete the modeling task. Empirical models are used© 2002 by CRC Press LLC