<p>The historical adoption of Bayesian approaches was limited by two main impediments: 1) the requirement for subjective prior information, and 2) the unavailability of analytical solutions for all but a few simple model forms. However, water quality modeling has always been subjective; selecting point values for model parameters, undertaking some &#8220;judicious diddling&#8221; to adjust them so that model output more closely matches observed data, and declaring the model to be &#8220;reasonable&#8221; is a long-standing practice. Water quality modeling in a Bayesian framework can actually reduce this subjectivity as it provides a rigorous and transparent approach for model parameter estimation. The second impediment, lack of analytical solutions, has for many applications, been largely reduced by the increasing availability of fast, cheap computing and concurrent evolution of efficient algorithms to sample the posterior distribution. In water quality modeling, however, the increasing computational availability may be reinforcing the dichotomy between probabilistic and &#8220;process-based&#8221; models. When I was a graduate student we couldn&#8217;t do both process and probability because computers weren&#8217;t fast enough. However, current computers unimaginably faster and we still rarely do both. It seems that our increasing computational capacity has been absorbed either in more complex and highly resolved, but still deterministic, process models, or more structurally complex probabilistic models (like hierarchical models) that are still light process. In principal, Bayes Theorem is quite general; any model could constitute the likelihood function, but practically, running Monte Carlo-based methods on simulation models that require hours, days, or even longer to run is not feasible. Developing models that capture the essential (and best understood processes) and that still allow a meaningful uncertainty analysis is an area that invites renewed attention.</p>
Analytical Solutions to Fractional Fluid Flow and Oscillatory Process Models