that can be placed in the model output; that is, confidence is generally 

 inversely related to variance. 



Analysis of this variability is important in a management context to 

 establish error bounds on predictions. Output from these deterministic models 

 often influences decisions affecting many thousands of people socially and 

 economically (e.g., Vallentyne and Thomas 1978); yet quantitative confidence 

 limits are lacking for these models (Thomann and Barnwell 1980). In particular, 

 only qualitative evaluations of calibration and verification results have been 

 carried out to date, and experience with even these tests is limited. Because 

 eutrophication models are crude representations of highly variable, stochastic 

 systems, ignoring such important attributes often results in naive confidence 

 or unwarranted disbelief in the models' solutions. For these models to become 

 more generally accepted and effectively used, they must be placed in their 

 proper perspective. Evaluating the effects of input (forcing function and 

 parameter) variance on model output provides some of the needed perspective. 



Analysis of model variability is also important in research contexts where 

 a model's ability to simulate must be evaluated prior to investigation of specific 

 system properties and recognition of actual system variability is important. 

 Output from these models is often used to assess the relative importance of 

 various system compartments or processes and thus to focus additional effort on 

 key problems. Prior to using a model in this context, it is important to evaluate 

 its ability to function as a synthesizer or interpolator. Traditionally, this 

 evaluation is done by comparing model and measurement trajectories, with no 

 quantitative assessment of model or measurement variability. As discussed above 

 and in Scavia (1980b) comparison of modeled and measured state variables alone 

 is not sufficient for this purpose. Calculation of variance associated with 

 state variables and of correlations among state variables and parameters will 

 assist in evaluation of these models for use in research contexts. 



We (Scavia et al. 1981a, b) have used Monte Carlo and first-order variance 

 propagation analyses to explore impacts of uncertain parameters, loads, and 

 initial conditions on a relatively simple model of plankton seasonal dynamics 

 in Saginaw Bay, Lake Huron. In these analyses, we use estimates of natural 

 variability of the input properties as sources of uncertainty. For Saginaw Bay, 

 natural variability far outweighs uncertainty introduced by inaccurate measure- 

 ments . 



Treating input errors in that way does not strictly estimate error associ- 

 ated with the ability of the model to predict. To do this, one certainly must 

 examine errors introduced by the equations themselves and perform the analysis 

 over the time frame of the prediction, as has been done for some empirical and 

 simpler lake models (e.g., Reckhow 1979). However, because variance due to 

 measurement errors is small compared to natural variability in this system, 

 these variance estimates measure at least their contribution to prediction 

 variances . 



Using the input statistics and first-order variance propagation, state- 

 variable variance estimates were made for an annual simulation based on variances 

 of the initial conditions and given parameters. Resulting variances are 

 represented as model output plus or minus its standard deviation in figure 15. 



79 



