CONTRIBUTION C. L. Winter Stochastic hydrology: practical alternatives exist Question 1: Why have there not been many real-world applications of stochastic theories and approaches, despite the significant progress in developing such rigorous theo- ries and approaches for studying fluid flow and solute transport in heterogeneous media? First, let me define some terms and admit a bias. A stochastic model is a random version of the governing equations of flow or transport in porous media. The randomness may arise from uncertainty about bound- ary conditions, initial conditions, forcing functions, or system variables like hydraulic conductivity. Uncer- tainty may be quantified by geostatistical sampling or by obtaining opinions from expert geologists and engineers. A solution to a stochastic model is a proba- bilistic estimate of the system state. In practice, solutions often consist of the statistics of the system state – often the mean and covariance of pressure or of concentration – but they may also be extreme value statistics or intervals. Sometimes a solution is a worst- case analysis. Solutions are usually obtained via Monte Carlo simulation or through moment equations. Broader definitions can be proposed, but these will suffice for my discussion. As to my bias, it is that all significant applications of hydrogeology are instrinsically uncertain, whether we like to admit it or not. The parameters of hydrogeologic systems can vary greatly in space and time, but they are usually sparsely sampled. Our knowledge of system parameters is therefore partial at best, and the most we can usually do is to quantify our uncertainty through stochastic, or related, models. Indeed, the only settings in which deterministic models are actually appropriate are primarily academic: in the classroom where the additional complications of uncertainty modeling can distract students from learning the basic science of groundwater and in carefully controlled laboratory experiments. Yet, most hydrogeologists continue to avoid stochastic models. Why? Out of the many reasons one might advance, I want to single out two: (1) ad hoc alternatives exist for dealing with uncertainty and (2) until recently, most stochastic models have been too simple to be relevant to real problems. Engineering and the legal system have each developed alternative methods for dealing with uncertainty that short-circuit the need for stochastic models in some settings. Uncertainty has been a fact of engineering life since the time of the Pharoahs. Engineers evolved a pragmatic method for dealing with uncertainty long before the theory of probability had been defined. The method consists, essentially, of over-designing engi- neering solutions. This works well when the object of an analysis is a classical engineering goal like providing drinking water supplies. It is often much cheaper to over-design a well field than it is to gather the data needed for a stochastic model, and in many cases, the data gathering effort would not result in much tighter uncertainty bounds than expert opinion. The degree of over-design is based on experience, the level of uncer- tainty, and the costs of failure. As a sort of Poor Man’s worst-case analysis, over-design is fundamentally prob- abilistic, although this is not always recognized. It is based on subjective assessment of probability, rather than on probabilities and statistics obtained from extensive observation. Matters are more challenging when the problem is to localize risk or to assess responsibility for contamination. Then seat-of-the-pants estimates are usually too subjective to base decisions on and stochastic models are in order. The legal system has evolved a method for deter- mining responsibility in the face of uncertainty that uses science but is not, itself, scientific. Since many hydrog- eologists eventually find their way into court as expert witnesses, the legal system’s standards have affected their approach to uncertainty. Hydrogeologists quickly learn in court that the law does not encourage experts to have doubt. It is the job of a judge or a jury to resolve uncertainty by weighing the counter-claims of Stoch Envir Res and Risk Ass (2004) 18: 271 – 273 Ó Springer-Verlag 2004 DOI 10.1007/s00477-004-0198-0 C. L. Winter National Center for Atmospheric Research, P.O.Box 3000 Boulder, CO 80307-3000, USA