7.4 Directions for Future Work

7.4.1 Improvements to the Stochastic Response Surface Method

In the present form, the SRSM includes modules for the transformation of probability distributions commonly used to represent uncertainties in the inputs of environmental and biological models. The transformations are listed in Table 3.1. However, this list is not extensive, since model inputs can follow one of several other distributions. In order to extend this list, the methods outlined for transforming random variables that follow empirical distributions (Section 3.2.3) and for transforming random variables that are correlated (Section 3.2.4) can be incorporated into the SRSM tool.

The SRSM calculates probability density functions (pdfs) to quantify the uncertainties in model outputs. Additionally, several other statistical metrics, such as the cumulative density function, moments of the output metrics, and percentile estimates, can be calculated. This can be accomplished by following the techniques outlined in Section 3.5.

Finally, the methods for calculating correlations between outputs and between an input and an output, presented in Section 3.5, can also be incorporated into the SRSM tool. This would facilitate the identification of individual contributions of model inputs to the uncertainties in model outputs, in a more rigorous manner than that presented in relation to the case study in Chapter 5 (Table 5.2).

These improvements could further enhance the already wide range of applicability of the SRSM. Furthermore, the same improvements translate into the improvements in the SRSM-ADIFOR method, and even in the wed-based SRSM tool.

7.4.2 Further evaluation of the SRSM and the SRSM-ADIFOR

Further rigorous, statistically based evaluation of the SRSM and the SRSM-ADIFOR, that would be based not only on the pdfs of model outputs, but also on the moments, cumulative densities, percentiles and correlation coefficients, would aid in the wider acceptance of this method. Additionally, application of the SRSM and the SRSM-ADIFOR to more complex models could also aid in the identification of areas where these methods could be further improved.

7.4.3 Addressing uncertainty propagation under constraints

The case study for the evaluation of the SRSM with a groundwater model showed certain limitations in the application of the SRSM to models with discontinuous probability distributions. Transport-transformation models sometimes have constraints on the values the model inputs can assume; often, the constraints are based on the values of other model inputs. For instance, the sample value of random input may affect the range from which another random input is sampled. While many constraints are defined in terms of joint pdfs, in some cases, the constraints could follow a discontinuous pattern. For example, the diameter and porosity of a particle may not assume certain combinations in the EPACMTP model (Chapter 5).

The Monte Carlo method can address such constraints by following the rules listed below:

• generate sample point by randomly sampling all inputs from their respective distributions ignoring constraints,
• ignore the sample point if any constraints are not satisfied, and repeat the above step till an acceptable sample point is obtained,
• run the model at the accepted sample point,
• repeat the above procedure till required number of sample points are obtained, and
• statistically analyze the model outputs corresponding to all the sample points.

It must be noted that samples are drawn from the given pdfs, and they also obey the constraints. That is an advantage of using Monte Carlo methods in a brute-force manner.

In the SRSM, such an approach does not appear to be readily applicable, because the inputs are represented as algebraic functions of the srvs. This means that the inputs have pdfs that are continuous and well behaved (as opposed to the actual pdfs which are discontinuous). One possible approach that could be followed is suggested here:^7.1

• express inputs in terms of srvs, ignoring constraints,
• approximate the outputs using the polynomial chaos expansions, and
• generate the sample points for the srvs without considering the constraints,
• estimate the unknown coefficients in the series expansion through regression on the model outputs at these sample points, and
• calculate the statistical properties of the model outputs from the srvs using the methods listed in Chapter 3.5 and in the process rejecting the combinations of srvs that do not obey the constraints.

One potential focus of the future efforts could be on the evaluation of this approach, in addition to the identification of other techniques to address uncertainty propagation under constraints.

7.4.4 Random processes and random fields

The SRSM, in its present form addresses only random variables, i.e., random quantities that do not vary with time or space. From the perspective of the uncertainties that occur in the nature, the random variables are analogous to points in a multi-dimensional space. Random processes and random fields are generalizations of random variables to multi-dimensional spaces.

A random process can be considered as a function of time in the random space, as

$${\ensuremath{\textstyle\mbox{\boldmath ${y}$}} }(t,\zeta)$$

where $t$ denotes the time dimension, and $\zeta$ is used to denote the randomness. Here, a particular realization of can be considered as a deterministic function ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }(t,\zeta_i)$, where $\zeta_i$ denotes one realization of the various possible functions that the random process can assume. Further, at a given time, $t_i$, reduces to a random variable. The relationship between random processes, random variables, deterministic functions, and deterministic variables can be summarized as follows [160]:

• $(t,\zeta$) is a stochastic process, or a family of time functions,
• $(t_{i}$,$\zeta$) is a random variable,
• $(t,\zeta_i$) is a single time function, and
• $(t_{i}$,$\zeta_i$) is a single number.

Similarly, random fields are random functions of spatial coordinates and time. Random processes and random fields are common in environmental and biological systems: for example, the flow rates or emissions from a point source are random processes, and emissions from area sources, wind patterns over a domain are examples of random fields. Further research could focus on extending the SRSM so that it can address uncertainty propagation involving model inputs defined by random processes and random fields.

7.4.5 Uncertainties Associated with Evaluation Data

Characterization of uncertainty associated with the evaluation data is an important component of a comprehensive uncertainty analysis. The uncertainties in the evaluation data may sometimes in fact impact the selection of an appropriate model from a set of alternative models. Comparisons of parametric and evaluation data uncertainties can provide insight into where available resources must be focused (input data versus evaluation data). Various techniques (such as those used in the spatio-temporal analysis of environmental data, e.g., Vyas and Christakos [36,210]) could be explored for their potential to improve the characterization of uncertainties in environmental databases that are used to evaluate transport-transformation models.

Footnotes

... here:^7.1

This approach has not been tested, and its applicability is still unknown.