8 Receptor impact model prediction (stage 7)

This stage applies the receptor impact model methodology to predict the response of the receptor impact variables to simultaneous changes in one or more of the hydrological response variables. The general framework allows for the receptor impact model to be applied either at a single or at multiple receptor locations, for example multiple locations that are considered to be representative of a landscape class within a bioregion or subregion. The receptor impact modelling elicitations, however, are designed for all locations within a landscape class given the same hydrology. The uncertainty in the elicited responses therefore represents the natural variability that one would expect in receptor impact variables that experience the same hydrological conditions at different locations in the landscape class, together with the experts’ uncertainty about this response.

The hydrology modelling produces simulated values from stochastic groundwater and surface water models that describe the uncertain impact of coal resource development pathways (CRDPs) on the hydrological response variables at a particular location (see companion product M09 (as listed in Table 1) for propagating uncertainty through models (Peeters et al., 2016)). The uncertainty from the hydrology modelling is propagated through the receptor impact model at each receptor location to deliver the predicted distribution of receptor impact variables at different time points for the two futures considered by BA (baseline and CRDP). Integrating over all of these receptors produces the overall predicted response of the receptor impact variable for the landscape class given the choice of the BA future. These landscape class results are summarised in product 3-4 (impact and risk analysis) for each bioregion or subregion (Figure 3). The method used for receptor impact model predictions is detailed below.

BAs assume that the only difference between the CRDP and baseline development pathway are those differences captured by the joint distribution of hydrological response variables provided by the stochastic hydrology modelling. All other variables that might also influence the receptor impact variables are assumed to behave in an identical fashion in a future world under baseline conditions, and a future world under the CRDP. To predict how receptor impact variables will respond to future changes in the hydrological response variables, the receptor impact modelling draws surface water and groundwater simulations from the joint distribution function, , where represents the hydrological predictions under the baseline pathway () and represents hydrological conditions under the CRDP (). For some hydrological response variables, such as surface water hydrological response variables that are aggregated to 30-year periods (Figure 11), the stochastic hydrological model output varies depending on the period of interest, , and . Other hydrological response variables, however, are defined over the entire future period, such as the maximum depth of groundwater drawdown in the future period, in which case the values are the same for the short and long period, .

Figure 14 shows the temporal dependence in the hydrological response variables across years given a choice of development pathway. This is depicted, for example, by the connecting arrows in the first row of Figure 14, which shows that the future hydrological response variables for the baseline development pathway depend on what has occurred in the past. The second row shows that same temporal dependence for hydrological response variables in the coal resource development pathway. Within a particular period, the hydrological response variables may also depend on common factors that are shared across the two development pathways. Thus, Figure 14 shows between-year dependence and within-year dependence, for example, between hydrological response variables in the reference period for the two development pathways.

Figure 14 A directed acyclic graph that shows dependencies among hydrological response variables (h) and receptor impact variables (y)

The choices of development pathways are and for assessment years 2012 (ref), 2042 (short) and 2102 (long). The parameter describes how the receptor impact variable in the current assessment year relates to hydrological response variables in the current assessment year and, for the future assessment years, the receptor impact variable in the 2012 assessment year.

Define the design point , which depends on the model structure, the known hydrological response variable values , the known value for the receptor impact variable in the reference year , and the assessment year . In the reference year, the value of is fixed at zero and absorbed into the intercept (Section 5.1.2). Conditional on a set of known hydrological response variable values, and , the joint distribution of the receptor impact variables in the reference assessment year for the two development pathways (Figure 14) is given by:

where is the inverse link function, is the elicited prior and the normalising constant is .

The joint distribution of the receptor impact variables for both development pathways in the short-term assessment year is conditioned on the hydrological response variable values in the short-term assessment period, and also the receptor impact variables in the reference assessment year (Figure 14). This joint distribution conditional on the hydrological response variables is given by:

Similarly, the distribution of the receptor impact variable in the long-term assessment year conditional on hydrological response variables is given by:

Realisations from the joint distributions in Equations 38, 39 and 40 are obtained using Monte Carlo simulation (Section 8.2). During these simulations, the BAs impose perfect positive dependence in the samples drawn from between development pathways within an assessment year in accordance with the assumption of BAs that, after accounting for the effect of hydrological response variables, receptor impact variables behave in an identical fashion under the baseline and CRDP.

Consider predictions for a particular location, or ‘assessment unit’ (defined as a geographic area that is used to partition the entire assessment extent into square polygons that do not overlap), denoted . The predicted distributions for the assessment and future years are given by:

The distributions of the hydrological response variables in Equation 41 depend on both the choice of development pathway and the assessment unit. Simulated values from the joint distribution of the hydrological response variables for each assessment unit, are provided by the surface water and groundwater models.

In addition to the above predictions, it is also of interest to consider functions of these unknowns. Two possibilities are considered for the future period: the actual change, , and the relative change, for . These predictions depend on the hydrological response variables and are thus also spatially explicit, that is, dependent on the choice of the assessment unit:

for the future assessment with .

8.1 Prediction to landscape class

The companion submethodology M10 (as listed in Table 1 and Figure 3) for analysing impacts and risks (Henderson et al., 2018) lays out the definition of a receptor impact model in terms of what it needs to predict and the appropriate conditioning and uncertainty that it should incorporate. As discussed in companion submethodology M10 (Henderson et al., 2018), a receptor impact model is defined with respect to a particular landscape classification. It is associated with a particular receptor impact variable and hydrological response variables chosen from the conceptual modelling output. It is explicit about the temporal definition of the receptor impact variable. It describes how the receptor impact variable changes as the hydrological response variables change across the entire landscape class.

The receptor impact model predicts the value of the receptor impact variable at all locations across the landscape class that experience specific hydrological conditions. The uncertainty in these predictions represents the natural variability in the value of the receptor impact variable that would occur at different locations within the landscape class that experience the same hydrological conditions, and the experts’ uncertainty about how receptor impact variables respond to different hydrological conditions. The predicted value of a receptor impact variable is thus not the predicted response at a particular location within a landscape, which would depend on many additional localised factors such as land use.

The receptor impact models are designed to facilitate prediction to the landscape class level while integrating across the spatially explicit hydrological response variables. This integration requires a quantifiable estimate of the amount of each landscape class within each assessment unit. Thus, a non-negative weight is associated with each of the assessment units, , The weight is defined by the amount of the landscape class within each assessment unit. For linear landscape classes defined along stream reaches, it is the length of reach assigned to the landscape class within each assessment unit. For areal landscape classes, it is the area assigned to the landscape class within each assessment unit.

Since it is assumed that the receptor impact variable predictions for each assessment unit are conditionally independent given the hydrological response variables (Chapter 8), the landscape class averaged predictions are given by the weighted averages:

where , and denotes dependence for the landscape class average on all assessment units, , , that form the landscape class. Aggregating to the landscape class requires conditioning on all assessment units in the landscape class.

The landscape class averaged actual change and relative change are obtained by:

which is a weighted average of the individual assessment units. Relative change can be expressed as percentage change by multiplying the relative change by 100.

8.2 Monte Carlo approximation

The realisations of the unknown coefficients are independent of the hydrological response variables (Figure 14). In particular, the realisations of are independent across the assessment units. Spatial dependence may nevertheless be introduced into the receptor impact variable predictions across assessment units to the extent that spatial dependence is captured by the hydrological response variables.

Monte Carlo approximations to the above joint distributions are available through the method of composition (Tanner, 1996). To sample from (Equation 41):

Repeat the above steps times to obtain simulations and store all values. Samples from the marginal are obtained by considering only the simulations .

Next, to sample from :

Repeat times.

Similarly, to sample from :

Repeat times for each assessment unit .

Given the simulated values of the jointly dependent receptor impact variable , Monte Carlo approximations of any function of , are also available. For example, simulated values of the actual change and relative change for the short-term assessment year are given by:

and likewise for the long-term assessment year, Predictions are thus available for all assessment units.

Aggregated predictions to the landscape class are also available with simulated values of the landscape class level weighted averages. Given simulated values of a quantity for each assessment unit, such as a prediction of the receptor impact variable or some other function such as the actual or relative change, the landscape class prediction is given by:

For each landscape class and for each assessment unit, a set of quantiles, which includes the 5th, 10th, 15th, 20th, 25th, 30th, 35th, 40th, 45th, 50th, 55th, 60th, 65th, 70th, 75th, 80th, 85th, 90th, and 95th quantiles, was estimated through the above Monte Carlo approach for each unknown quantity of interest that depends on the receptor impact variable. Let be the inverse cumulative distribution function for . The qth quantile is given by, for . Let the empirical cumulative distribution function of the samples of be given by . A Monte Carlo estimate of the qth quantile is then given by:

Extreme quantiles are more sensitive to approximation error by the Monte Carlo method, and larger sample sizes reduce Monte Carlo error. The number of receptor impact variable simulations can be made arbitrarily large. However, the composition sampling that preserves the joint dependence between the receptor impact variables and the hydrological response variables is limited by the number of realisations available from the hydrological modelling simulations.

Rather than quantiles, it may also be of interest to report the average of the above unknown quantities either at the level of assessment units or landscape class or both. A mean estimate for an unknown quantity is trivially obtained by taking the sample mean of the above Monte Carlo simulations.

8.3 Conclusion

The Monte Carlo simulations permit the probabilistic prediction of receptor impact variables. The receptor impact models include the ability to capture direct, indirect and cumulative impacts through the hydrological response variables. In addition, ecological lags imposed through the receptor impact variable are also captured by the above method. For example, the long-term response for a long-lived terrestrial species such as large woody riparian vegetation may be more sensitive to the previous history of the receptor within an assessment unit compared to a short-lived aquatic species. The approach can flexibly estimate a function of the receptor impact variable, such as actual or relative change, quantiles or averages. The range or distribution of receptor impact variable outcomes will be summarised for baseline (all years) and for the change due to additional coal resource development for future assessment years at landscape class level. The change due to additional coal resource development may be expressed in relative terms or actual terms and may depend on the specific receptor impact variable. These responses can be aggregated from the assessment units to the landscape class level to assess overall impacts of development within the bioregion or subregion.