## Uncertainty Propagation

Once the input/parameter uncertainties have been quantitatively characterized, various methods can be used to propagate the impact ofthese uncertainties through to the model output predictions ofinterest. The most appropriate propagation method depends on how the modeler wishes to describe the model prediction uncertainty. If a complete approximation of the model output PDF is desired, then some type of Monte Carlo simulation or sampling approach is needed. The propagation method also depends on whether the input/parameter uncertainties were conditioned to measured calibration data, and the dashed text boxes in Figure 3 highlight that a specific

Monte Carlo sampling experiment is required for propagation with the MCMC or GLUE uncertainty approaches. If no calibration data were used, then more traditional methods of uncertainty propagation can be utilized.

The most commonly used method for uncertainty propagation is Monte Carlo simulation. The purpose of Monte Carlo simulation is to obtain a distribution of the model output(s), given distributions of the inputs (e.g., forcing functions, model parameters, boundary conditions). This is achieved by sampling repeatedly from the input distributions and calculating the corresponding model output(s). As part of the sampling process, different realizations (e.g., combinations) of inputs are obtained, resulting in different model outputs. If this process is repeated for a sufficient number of iterations, very accurate distributions of model outputs are obtained. The number of iterations required varies, but is generally in the order of 5000-10 000. As part of the sampling process, the correlation structure between the inputs can be taken into account. The main drawback of Monte Carlo simulation is its computational inefficiency. The computational burden of Monte Carlo simulation can be reduced by using specialized, rather than random, sampling techniques, such as Latin hyper-cube and importance sampling.

When the distribution of outputs need not be fully approximated, there are alternatives to Monte Carlo sampling that can be more efficient for uncertainty propagation. For example, if approximating the moments of the model outputs is deemed sufficient, then first- or second-order approximations of the moments of the output distribution based on a Taylor series expansion can be used (Figure 3). Such approximations are typically much more computationally efficient than Monte Carlo simulation. First-order approximations of the model output mean and variance are often called firstorder error analysis or Gaussian approximation. Firstorder approximations are not usually appropriate for highly nonlinear ecological models. Approximate reliability analysis methods are another typically more efficient alternative to Monte Carlo simulation when modelers are only concerned with estimating whether model predictions are above or below some threshold value (this is equivalent to defining one point on the cumulative distribution function of the model output). Example methods include the mean and advanced first-and second-order second moment reliability methods. The accuracy of these approximate reliability methods is case-study dependent and often poor for highly nonlinear ecological models.

In the case where calibration data are used to describe input/parameter uncertainties, uncertainty propagation is usually achieved by sampling from the input/parameter vectors identified as part of the

GLUE or MCMC analysis. These vectors are generally a direct characterization of the joint posterior distribution of uncertain inputs/parameters, which cannot be described analytically for the vast majority of ecological models.

After uncertainty propagation, and depending on the methodology used, the sampled outputs can be described with empirical or fitted probability distributions and/or sample moments. The output distributions can also be used to estimate various risk-based system performance indicators, as discussed previously. 