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I I I I I I I I, I I I I I I I I I I I discussed in Section 3.3.2, which has provided empirical evidence concerning the shape of the intermediate part of the rainfall frequency curve, a region of particular relevance to the assessment of adequacy of many existing spillways. The reduclion in relative uncertainty, however, is likely to have its greatest impact on the estimation of Rare to Extreme floods. This is a region where "the computation of hydrologic probabilities is based on arbitrary assumptions about the probabilistic behaviour of hydrologic processes rather than on empirical evidence or theoretical knowledge and understanding of these processes" (Klemes, 1993). Improving the consistency of the manner in which such assumptions are applied in practice will thus minimise the potential for differences in the results obtained by different hydrologists. The main strategy available for reducing the impact of relative uncertainty is to ensure that the practitioners undertaking the work are appropriately qualified and supervised. In addition, prescriptive procedures relating to the estimation of floods beyond the credible limit of extrapolation are justifiable as without empirical evidence or scientific justification there can be little rational basis for departing from a consensus approach. 5,6.2 Evaluation of Uncertainty There are several techniques to assess model uncertainty. The techniques most commonly used to assess the uncertainty of extreme flood estimates are sensitivity analysis and Monte Carta simulation, though other techniques may also be suitable. The available literature on uncertainty analysis is extensive and highly mathematical. While there are many papers dealing with the theoretical aspects of different techniques and comparative analyses, fewer papers have been written on techniques that are applicable to practical design problems. Thus, while the importance of uncertainty analysis is widely recognised, there are few tools available to the practitioner that can be used routinely for design. The foliowing sections outline the characteristics of a few methods which have the potential to be readily applied to practical problems. (a) Sensitivity analysis Sensitivity analysis is a simple technique used to identify the parameters that have the greatest impact on the outcome of interest. It does not quantify uncertainty in an absolute sense, but rather is used to indicate the degree to which uncertainty in certain parameters impacts upon the results. The analysis involves the perturbation of inputs or model parameters from a base value, and then using some criterion (such as flood peak and/or volume) to measure the change in model response. The sensitivity of the resuUs to the most influential parameters should be investigated and quantified in all design situations. If there are significant differences in outcome within the range of parameter uncertainly then the likely range of consequences should be explicitiy considered when developing mitigation strategies. Uncertainties can be simplistically quantified by undertaking a separate assessment for the most sensitive input(s), and weighting \he resuUs based on the asses~ probabilities of their occurrence. For example, Laurenson and Kuczera (1998) recommend that the uncertainty associated with the AEP of the PMP be directly incorporated into a risk analysis by performing an assessment for each of the AEPs in Table 5 and weighting tf\e results using the associated $lJbjeellve probabilities. While accurate estimates of probabilities for different class intervals are usually not available, the adoption of ............" VI - ........".,"'"....,...... .......,l:f......... L..^.........,... I ,........... subjective probabilities in this manner does provide an objective basis for incorporating judgement into the outcome. (b) Monte Carlo Methods Monte Carlo analysis is perhaps the most commonly used method in practice. The most important input parameters are treated as variables having a specified distribution rather than being fixed values. Numerous computer simulations are conducted where each simulation uses a set of input parameters that are selected at random from continuous (or discrete) probability distributions. Different probability distributions may be used. for the various parameters, and the characteristics of the distributions are selected to be representative of the uncertainty of each input (as well as preserving the dependencies between the parameters). In each simulation run of the Monte Carlo analysis the results are stored, and by generating many replicates one obtains samples that describe the probability distribution of the outputs. The characteristics of the resulting sample (e.g. the mean, standard deviation, or percentile statistics) can be used to describe and evaluate the decision i{l1plications of uncertainty in the model inputs. Examples of application of this approach to rainfall-runoff models can be found in Krajeweski et al. (1991), Kuczera and Williams (1992), and Haan et al. (1995). The uncertainty regarding the selection of the distributions used to characterise the inputs is one of the major difficulties with the method. An assessment of the distributions may be based on judgement and experience, or from the literature. Available evidence (e.g. Haan et aI., 1988) suggests that good estimates of the means and variances of the input parameters are of greater importance than the actual form of the distribution adopted. Thus while there may be uncertainty regarding the distributional characteristics of the input variables, it is considered that the method does provide an objective basis for assessing uncertainly. Latin Hypercube sampling (McKay, 1988) is a stratified sampling approach that is more efficient than traditional Monte Carlo simulation. An example of its application to the HEC-1 flood event model is provided in Melching (1995). Thompson et al. (1997) discuss other sampling strategies that increase the efficiency of the simple Monte Carta method. (c) Other Techniques While Monte-Carta techniques are robust, they can be computationally intensive, especially for low probability outcomes. As a result, there has been considerable interest in alternative procedures which could offer computational savings. Alternative techniques that have been used with flood event models include point estimate methods (Rogers et aI., 1985; Binley et aI., 1991), and first and second order analysis techniques (Haan et aI., 1995; Melching et aI., 1990; Melching, 1992). A useful review of the different methods is provided by Melching (1995). While these other methods have the potential to be more efficient, they are generally applicable to a more restricted set of conditions. Several studies have been conducted on comparing the efficacy of different techniques (e.g. see Melching, 1995), and it may generally be concluded that at this stage Monte Carlo techniques are perhaps the only approach that can be relied upon to provide estimates of uncertainty in systems that are highly non-liJ\ear or Iligllly variable.