3.2 THE PITMAN MODEL
3.2.5 Parameter Constraining Procedure
In the initial approach proposed by Kapangaziwiri and Hughes (2008) either a priori parameter estimation using physical basin properties or simple estimation of likely parameter ranges or distributions were used. During the model run each parameter value is sampled independently but groups of sub-basins with similar expected responses are sampled from the same parameter space, such that high (or low) values of a specific parameter will be similarly high (or low) in all the sub-basins in the same group. The reasoning behind this approach is that if all sub-basins are considered independent, then the degree of uncertainty substantially reduces as more and more sub-basins are combined at downstream locations. This reduction in uncertainty occurs as a result of different sub-basins generating different relative responses to the climate inputs such that the upstream uncertainties tend to be largely cancelled out at downstream sub-basins. Allowing groups of similar sub-basins to have parameters sampled from a similar space has been demonstrated to better preserve the upstream uncertainties in downstream sub-basins (Hughes, 2013).
During the initial approach, the ensemble outputs of the model runs were examined on the basis of expected constraints on runoff response (such as mean monthly runoff volume, recharge depth, slope of flow duration curves) and a decision is made about which ensembles
51 are behavioural and which are not. However, the fundamental problem with this approach is that the selected behavioural ensembles at downstream sub-basins can be made up of a mixture of behavioural and non-behavioural ensembles in all of the upstream sub-basins (Hughes, 2013; Tumbo and Hughes, 2015). This could mean that ensemble outputs that are selected for further use in water resources management could be spatially inconsistent in terms of their representativeness relative to the known (or expected) ranges of hydrological response.
In the revised approach (Tumbo and Hughes, 2015) that was finally used in this study, not only are the parameter ranges defined as model inputs, but also the likely output constraints. The regional constraint bounds are therefore an additional input into the model. The constraint bounds are mainly based on observed streamflow within the region and reflect uncertainty in the available knowledge about the hydrological response of the different parts of the total basin and can be narrow if there are sufficient observation data available. The revised approach adopts a two-step procedure; the first step is aimed at identifying behavioural parameter sets (only for the natural hydrological response) for each sub-basin, while the second step involves using these parameter sets to simulate the basin response as a whole and can include uncertainty sampling of the parameters associated with water use and other anthropogenic impacts.
Step 1 of the approach is aimed at constraining only the parameters related to natural runoff in each sub-basin. The constraints apply to the incremental runoff in each sub-basin and step 1 does not simulate the cumulative flows in downstream sub-basins. The sub-basin response characteristics used for constraining or for deciding behavioural ensembles are: mean monthly runoff (MMQ), high (Q10), medium (Q50) and low (Q90) flow volumes of the flow duration curve relative to mean monthly flow, mean monthly groundwater recharge and percentage of time with zero flow. The mean monthly runoff and the flow duration curve characteristics are typically obtained from stream flow gauging data from catchments with similar physiographic characteristics to the ones being studied. Groundwater recharge estimates are established in South Africa from the Groundwater Resources Assessment study (GRA II- DWAF, 2005).
For each sub-basin the model is run up to 100 000 times with an independent Monte Carlo sampling procedure from pre-determined parameter distributions. The outputs are assessed against the constraints during each run of the model. If all constraints are satisfied the full parameter set is saved for further analysis. The model continues until a pre-defined number of parameter sets (2 000 to 5 000) are saved or until the maximum number of model runs (50 000 to 100 000) is reached.
52 A facility is available within SPATSIM (Figure 3.3) to examine the distributions of parameter values and constraints in the saved results to guide any decisions about whether to change the input parameter distributions or re-evaluate the constraint ranges to achieve the required number of behavioural parameter sets.
Figure 3.3 Illustration of the tool designed to help with determining appropriate parameter bounds.
Figure 3.4 illustrates three possible outcomes of the first step. Figure 3.4A shows a situation where the parameter bounds and constraint bounds are compatible, and the constraint bounds are compatible with each other. The required number of parameter sets is found and less than 100 000 test runs of the model are required. The results evaluation method (Figure 3.3) can be used to determine if the results are reasonably well distributed within the constraint bounds or whether the initial parameter bounds could be changed to achieve the required number of behavioural parameter sets more efficiently. In Figure 3.4B the constraint bounds are not compatible with each other and no ensembles meet all of the behavioural requirements. In this situation the constraint bounds have to be adjusted to ensure compatibility. Figure 3.4C illustrates a situation where all of the model simulations are inconsistent with the constraints and therefore either the constraints need to be re-evaluated, or the ranges of some of the parameters have to be modified to match the expected response (defined by the constraints).
53 There can be a number of intermediate situations between Figures 3.4B and 3.4C where some behavioural parameter sets are found but not enough before the total number of model runs is reached. The facility illustrated in Figure 3.3 is then used to identify which parameters require their ranges to be adjusted. In the example it can be seen that the Q10 constraint is always at the lower end of the input range and that ZMIN and ZMAX (surface runoff parameters) are also at the low ends of their input ranges. Shifting the range of one or both of these parameters downwards will generate more surface runoff, solving the problems with the Q10 constraint and increasing MMQ which is also under-simulated in the example provided.
Figure 3.4 Step 1 in the revised approach to uncertainty estimation with the Pitman model (Source: Tumbo and Hughes, 2015).
Step 2 is only initiated after all of the sub-basins have the desired number (typically 2 000 or 5 000) of saved parameter sets. At the beginning of step 2 the saved behavioural parameter sets are sorted according to the 6 constraint values from wetter to drier conditions. The sub- basins are then grouped and random samples (typically 10 000) are drawn from the parameter sets using the same sub-basin group sampling dependency referred to in the description of the earlier uncertainty method. In this step all the model parameters are sampled and used with all sub-basins linked to generate cumulative streamflow simulations. While the parameters related to the natural runoff are randomly sampled from the saved parameter sets, the others (e.g.
54 those related to water use and land use changes) are sampled independently from their pre- defined input distributions. The main advantage with this approach is that, unlike the previous methods, it assures that all the incremental catchments have behavioural simulations relative to the constraints.
For sub-basins with stream flow records, a summary output file provides evaluations of all the ensembles based on four objective functions: Nash-Sutcliffe coefficient of efficiency and percentage bias in mean monthly flow for normal and natural log-transformed data.