# Springer 2006
Environmental Modeling and Assessment (2006) 11: 297Y313 DOI: 10.1007/s10666-005-9029-z
Application of the Morris algorithm for sensitivity analysis of the REALM model for the Goulburn irrigation system R. D. Braddocka,* and S. Yu. Schreiderb a
Centre for Environmental Systems Research, Faculty of Environmental Sciences, Griffith University, Nathan Qld 4111, Australia E-mail:
[email protected] b School of Mathematical and Geospatial Sciences, RMIT, Melbourne Vic 3001, Australia
The REALM modelling shell is widely used in Australia as a water allocation modelling tool. It has been used to develop the Goulburn System Model (GSM) of the Goulburn, Broken, Loddon and Campaspe Rivers in northeastern Victoria. REALM represents the river and irrigation system as a network of storages and carriers. The model has been optimised to best represent the water harvesting and allocation for use by water management authorities. The model is analysed to assess the sensitivity of a subset of the model outputs, to a subset of the system parameters. The New Morris algorithm uses sampling paths generated in the space of the parameters, to generate points at which the model is run (to generate the model outputs). These model runs are then used to estimate the first and second-order effects of the parameters on the outputs. The results illustrate the mild linkage of the Goulburn and Broken systems, and the Broken system also shows differences between minimum and average outflows. The Goulburn is more sensitive to some of the numerical convergence parameters used in the allocation software, while the Broken is less sensitive to these factors. The numerical convergence factors also lead to important second-order effects. Keywords: sensitivity analysis, water allocation model, REALM
1. Introduction Water is an important commodity, especially in Australia which is generally recognised as the driest continent. The proper management of this resource is important to farming, community centres, and also the environment. Models have been developed for many of the Australian river catchments, and are an important tool in the management and allocation of this resource. Analysis of model sensitivities is an important component of the model-based assessment and management of catchment water resources. REALM (REsource ALlocation Model) is widely used to model water allocation systems throughout Australia [1]. Perera et al. [2] describe the general framework of this model and give case studies of its application. One of the most important applications of REALM is water allocation modelling in the Goulburn irrigation area which includes basins of the Goulburn, Broken, Campaspe and Loddon Rivers. These are four large tributaries to the Murray River, and their waters are heavily used for irrigation. The REALM model calibrated in this region for the 1993/94 level of land use development and infrastructure, is referred to below as the Goulburn Simulation Model (GSM). This period of calibration was selected because it corresponds to the period when the cap diversion constraints were established. The cap is an upper limit on the allowable diversion in the MurrayYDarling Basin (MDB) and was introduced in 1997 by the MDB Ministerial Council. The cap is defined as the amount of water which would have been diverted under * Corresponding author.
current climatic conditions but with the infrastructure that existed in 1993/1994 irrigation season [3]. The detailed description of the GSM calibration and implementation is presented in Perera and James [4]. The REALM framework has been applied to the modelling of this system, which has a total catchment area of more than 4.3 million hectares. This region is an important contributor to Australia’s rural industry with the total value of the agricultural production estimated from the GoulburnYBroken area, as AU$1.35 billion per annum. The water resources and use in the Goulburn System are summarised in table 1 (after [5]). The use of groundwater resources in the Goulburn System is negligible compared to that of surface water resources and is not considered in the GSM. For example, the Campaspe catchment has the largest portion of ground water resources used among all the catchments in the system but it constitutes less than 15,600 ML of water divertable per year, which is less than 6% of the mean annual discharge of this catchment. Sensitivity analysis is a fundamental tool in the construction, use and understanding of models. In many areas of research, parameter values are not well known, or are hard to measure [6]. The complexities of the sensitivity analysis needs for the models used in integrated catchment management are demanding and relate to both data inputs and accuracy, as well as to parameter values [7]. Sensitivity analysis can identify the most important parameters, or factors, within a model and can indicate the Frobustness_ of the model [8]. Sensitivity analysis is rapidly becoming a key tool in the development, verification and validation of a model. Sensitivity analysis can also be used to identify
298
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model Table 1 Water resources and use in the Goulburn system [5].
River Basin
Broken Goulburn Campaspe Loddon
Area (in thousand ha) and percent of total area of Victoria 772 1,619 418 1,532
Total mean annual streamflow (ML and percentage of total state mean in Victoria)
(3.4%) (7.1%) (1.8%) (6.8%)
325,000 3,040,000 280,000 250,000
(1.5%) (13.7%) (1.3%) (1.1%)
second-order interactions between parameters or factors, and hence identify sensitive non-linearities in the model. The information obtained can be used to identify weaknesses in the model, where effort needs to be spent in measuring parameters more precisely, or where care is needed in the use of the model. This paper continues the GSM sensitivity analysis work started by Schreider et al. [9], where the sensitivity analysis of this model in relation to the variation of single input factors was implemented. The aim of this paper is to analyse the full first- and second-order sensitivity properties of the REALM model of the Goulburn system.
2. Methodology 2.1. GSM input factors and output functions Details of the REALM shell and of its application to the GSM are given in Perera and James [4] and Perera et al.
Water use in Basin, including surface water imports (ML) 801,524 780,700 258,230 1,336,770
Irrigation (ML and % of total water use in the Basin) 741,804 739,020 238,730 1,275,200
(93%) (95%) (92%) (95%)
[2], and these details will not be repeated here. REALM falls into a class of network models which represent the study area as a set of supply (reservoirs, dams, river inflows) and diversion (irrigation and urban) nodes connected by the delivery carriers or arcs of the network. REALM uses a fast network linear programming algorithm for optimisation of water delivery (allocation) to diversion nodes in different regions, representing a network structure of the catchment system under study. A simplified version of the GSM nodal structure is presented in figure 1. The GSM incorporates a large number of parameters to describe the physical processes in managing and allocating water. It also uses numerical or computational parameters to control the numerical processes employed in the model. The computational load or number of model runs needed in a sensitivity analysis, depends on the number of parameters which are investigated. The analysis had to be restricted to only the few parameters that the modellers considered, on the basis of experience, to be the most influential. The list of output functions to be considered, was also reduced to
Figure 1. Simplified structure of the GSM REALM [9].
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
299
Table 2 List of key GSM input factors that were used in the sensitivity analysis of the GoulburnYBroken catchment [9].
1 2 3 4 5 6 7 8 9
Parameter (calibrated value)
Meaning of the parameter
Node or carrier type
Goulburn River transmission loss (0.06) Goulburn Weir forced spill Storage convergence criterion (0.1%) Carrier convergence criterion (5%) Absolute convergence criterion (100 ML/month) Boort entitlement Broken River transmission loss Broken River operational loss Waranga Basin evaporation loss (0.85)
Transmission loss as proportion of flow Forces spills to simulate recorded diversion efficiency Defines tolerance for acceptable LP solution Defines tolerance for acceptable LP solution Defines tolerance for acceptable LP solution Entitlement limit curve Transmission loss as proportion of flow Forces spills to simulate recorded operational efficiency Converts pan evaporation to lake evaporation
Acceptable range Basic
the essential ones. The identification of the most important input factors and output functions of the GSM was implemented previously by Schreider et al. [9]. The list of the nine input parameters used in this analysis and their ranges of variation are presented in table 2. The ninedimensional space bounded by the acceptable ranges of values of each parameter, as given in table 2, is referred to as the parameter space. The sensitivity analysis of the GSM was implemented for the five output functions listed in table 3. Three of these five functions (Goulburn River outflow, Broken River outflow and Goulburn seasonal allocation level) were computed on a monthly basis. The Goulburn seasonal allocation level function is defined only over the irrigation season from August to May: in June and July its value is always zero. The two other output functions (Goulburn and Broken cap diversions) were computed on an annual basis. The cap values to the end of June of each year were analysed. Therefore, thirty-six 110-year output time series were computed for each replicate run of GSM. These output vector functions were then analysed using the
Extended for the third experiment
Variable capacity carrier
0.05 to 0.07
0.04 to 0.08
Variable capacity carrier
j15 to +15%
j50 to +50%
Linear programming parameter
0.1 to 5.1%
0.1 to 5.1%
Linear programming parameter
1 to 51%
1 to 51%
Linear programming parameter
10 to 1,010 ML
10 to 1,010 ML
Demand node Variable capacity carrier
j15 to +15% j15 to +15%
j50 to +50% j50 to +50%
Variable capacity carrier
j15 to +15%
j50 to +50%
Reservoir node
0.7 to 0.85
0.5 to 1.0
average and maximum values over the 110 years. The extremes of these functions, minima especially, are more important characteristics because they indicate the security of the system. As the Goulburn system is relatively reliable (compared, say, with the WimmeraYMallee area located in northwest Victoria) the average allocations calculated for 110 years are always above 100% of entitlement. The most interesting information for the water authorities in the region is to understand how the minimum allocation levels occurring in very dry years, would change under different input factor fluctuations. Summarising the aforementioned, 36 2 = 72 output values were computed for each GSM run. The shortlist of the 16 most important output functions was formed for further sensitivity analysis using the Morris and the New Morris algorithms. Averages and minima of the Goulburn Rive outflow, the Broken river outflow and Goulburn allocation level were analysed for August (beginning of the irrigation season) and February (month when farmers make key decisions on irrigation strategy) which constitute 3 2 2 = 12 output functions. Averages and minima of the
Table 3 List of key GSM outputs that were used to assess sensitivity in the GoulburnYBroken catchment. Parameter The Goulburn River outflow The Broken River outflow Goulburn seasonal allocation The Goulburn River cap diversion The Broken River cap diversion
Meaning of the parameter
Node or carrier type
Outflow from Goulburn River to Murray River Outflow from Broken River Goulburn River to Murray River Percentage of basic water entitlement (water right) allocated for the irrigation season Total diversion from the Goulburn River over the financial year Total diversion from the Broken River over the financial year
Fixed capacity carrier Fixed capacity carrier Variable capacity carrier Variable capacity carrier Variable capacity carrier
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R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model Table 4 Reference number of 16 output functions used in the analysis. Reference number
Output function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Minimum Goulburn outflow for February Average Goulburn outflow for February Minimum Goulburn outflow for August Average Goulburn outflow for August Minimum Broken outflow for February Average Broken outflow for February Minimum Broken outflow for August Average Broken outflow for August Minimum Goulburn allocation for February Average Goulburn allocation for February Minimum Goulburn allocation for August Average Goulburn allocation for August Minimum Goulburn cap diversion Average Goulburn cap diversion Minimum Broken cap diversion Average Broken cap diversion
Goulburn and Broken rivers cap diversions give another four functions selected for the analysis. The reference numbers of these 16 outputs are presented in table 4. 2.2. The Morris and the New Morris algorithms The sensitivity analysis was conducted using the firstorder screening method of Morris [10] and the secondorder screening method of Campolongo and Braddock [11], as corrected by Cropp and Braddock [12]. These methods construct sets of paths through parameter space, to provide points in parameter space to sample the model output. The pathways are designed so as to provide estimates of the first-order effects, or first derivatives, of the output with respect to the input parameters (the Morris method) throughout the parameter space. The sensitivity analysis provides a mean value i for each first-order effect i, where the samples are taken across the parameter space. The corresponding standard deviation i, for each firstorder effect i, is also estimated throughout the parameter space. The new Morris method also designs a second set of
pathways so as to estimate the second-order effects, or second derivatives, of the output with respect to the input parameters [12]. This second-order sensitivity analysis provides a mean value i and a corresponding standard deviation i, for each pairwise interaction between factors for Output i, where the samples are taken over the parameter space. Both methods are based in graph theory, and use the optimal number of model evaluations to reduce computation time, for a given accuracy. For this study, the parameter space is nine-dimensional, using the nine factors listed in table 2, and some 16 output variables were calculated for use in the sensitivity analysis. The first- and second-order Morris methods were originally designed to operate with just one output variable. The extension of the screening methods from one output function to multiple output variables is obvious [12]. These changes were made to the code of Cropp and Braddock [12] so that all 16 output functions or variables were analysed using the same paths in parameter space. Three experiments were conducted on the nine selected factors of the REALM model for the 16 output variables
Table 5 Mean 4, standard deviation 4, and Euclidean distance "4 values for the first-order effects on Output 4. Factor 1 2 3 4 5 6 7 8 9
Mean first-order sensitivity, 4
Rank on mean
Standard deviation, 4
Rank on standard deviation
Euclidean distance, "4
Rank on Euclidean distance
2,261 3,072 3,783 269 526 345 486 260 943
3 2 1 8 5 7 6 9 4
1,289 1,437 2,350 956 1,768 276 86 66 598
4 3 1 5 2 7 8 9 6
2,602 3,392 4,453 993 1,845 442 494 268 1,117
3 2 1 6 4 8 7 9 5
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
301
Figure 2. Scatter plot of 4 and 4 for the first-order effects on Output 4. The numbers indicate the Factor.
using the smaller or basic range of parameter values given in table 2. The first experiment used two runs with a resolution of six in the sensitivity software. This was in the nature of a pilot experiment, used the smaller range of the parameter values, and the number of runs was less than that recommended by Cropp and Braddock [12]. This limited experiment used only 180 points in parameter space, and hence only 180 full runs of GSM. A second experiment was conducted on the basic range of parameter values, and used six runs and a resolution of eight. This falls within the range recommended by Cropp and Braddock [12], although their recommendation related to a relatively small theoretical model which was Bcheap to run^. Repeating their experiments with GSM would take an unacceptably large number of model runs. This experiment used 540 sample points in parameter space, as selected by the sensitivity software, and took some 18 hours on a moderate sized PC. A third experiment was conducted using the extended range of parameter values, (see table 2) using six for the number of runs and a resolution of eight, and hence a sampling at 540 points in parameter space. Again, this experiment took some 18 hours of computer time.
3. Results The output from the sensitivity analysis is copious, given that 16 function outputs are being analysed at once.
The results are mean i, and standard deviation i, values of the first-order effects (first derivatives) and mean i, and standard deviation i, values for the second-order effects (second derivatives) sampled across the parameter space. The results of the first experiment with 180 sampling points showed an unacceptably high error, and these results will not be given here. The results for the second experiment were far more reliable, and are presented in detail.
3.1. Nature of output for Output Function 4: second experiment Output Function 4 (average Goulburn outflow for August) for the second experiment was selected so as to illustrate the results from the sensitivity analysis. The results for the first-order effects are given in table 5 and figure 2, and the second-order interaction data is given in table 6 and figure 3. In table 5, the Euclidean distance "i ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2i þ i2 ;
for Output i = 4, where i and i (for i = 4) are the mean and standard deviation of Output Function 4 (the average Goulburn outflow for August) for the samples in parameter space, and are given rounded to integer values. The mean, standard deviation and Euclidean distance have also been assigned a rank from one (highest) to nine (lowest), in
302
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model Table 6 Mean 4, standard deviation 4 and Euclidean distance "4, values for the second-order effects on Output 4.
Factor interactions 1Y2 1Y3 1Y4 1Y5 1Y6 1Y7 1Y8 1Y9 2Y3 2Y4 2Y5 2Y6 2Y7 2Y8 2Y9 3Y4 3Y5 3Y6 3Y7 3Y8 3Y9 4Y5 4Y6 4Y7 4Y8 4Y9 5Y6 5Y7 5Y8 5Y9 6Y7 6Y8 6Y9 7Y8 7Y9 8Y9
Mean 4
Rank on 4
Standard deviation, 4
Rank on 4
Euclidean distance, "4
Rank on "4
7,928 8,131 443 20 5,023 2 16 5,037 18,431 202 26 3,334 3 2 2,220 18,323 36 6,387 1 2 8,350 41,990 21 1 31 1 10 2 2 9 199 2,691 2,658 579 1,306 703
6 5 17 24 9 31 25 8 2 18 22 10 28 33 13 3 20 7 35 30 4 1 23 34 21 36 26 32 29 27 19 11 12 16 14 15
5,776 7,524 695 29 5,059 2 32 4,550 13,285 371 24 1,018 3 2 1,640 16,254 50 6,031 1 1 2,633 21,399 0 0 27 0 15 3 3 13 468 2,422 2,304 329 3,196 1,682
6 4 16 22 7 31 21 8 3 18 24 15 29 30 14 2 20 5 32 33 10 1 36 35 23 34 25 28 27 26 17 11 12 19 9 13
9,809 11,078 824 35 7,129 3 36 6,788 22,720 423 36 3,486 4 3 2,760 24,494 61 8,785 2 2 8,756 47,128 21 1 41 1 18 3 4 16 508 3,621 3,517 666 3,453 1,823
5 4 16 24 8 31 23 9 3 19 22 12 28 32 14 2 20 6 34 33 7 1 25 35 21 36 26 30 29 27 18 10 11 17 13 15
Ranks based on 4, 4 and "4 are also given, with Rank 1 being the highest.
terms of the values of 4, 4 and "4. Note that the mean i is normally the accepted measure of sensitivity of the output to an input factor, and that ranking is more important than the values of i. Note that the storage convergence criterion parameter (Factor 3) has the highest rank with respect to 4, and that the average Goulburn outflow for August (Output 4) is most sensitive to this factor at first order. The Goulburn River transmission loss and the storage convergence criterion parameters (Factors 1, 2 and 3) have high values of 4 and represent the three more sensitive factors. On the other hand, there is a considerable gap down to the carrier convergence criterion, the absolute convergence criterion, the Boort entitlement, the Broken River transmission loss and the Broken River operational loss parameters (Factors 4, 5, 6, 7 and 8) which have low mean values, and the average Goulburn outflow for August (Output 4) is least sensitive to these factors.
The ranking for the standard deviation is an indication that the factor may be involved in second-order or twofactor interactions. The storage convergence criterion parameter (Factor 3) has the largest value of 4, and this indicates that this factor is likely to be involved in higherorder interactions. The absolute convergence criterion parameter (Factor 5) also has a large value of 4, and is the second ranked factor in terms of possible higher order interactions. However, it provides only a moderate firstorder mean value. The Euclidean distance is sometimes used in assessing sensitivity effects, and is given here for completeness. The data is also shown graphically in figure 2, which is a scatter plot of 4 against 4. The patterns referred to above are readily discernable. Note that the results for Boort entitlement, the Broken River transmission and operational loss parameters (Factors 6, 7 and 8) are affected mainly by errors in estimating the derivative from the sampling path. These mean and
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
303
Figure 3. Scatter plot of 4 and 4 for the second-order effects on Output 4. The numbers indicate the more important Factor interactions.
standard deviation values are Bdown in the error noise^ of the sensitivity analysis, and appropriate interpretations are needed. The second-order effects or interactions are given in table 6 for the average Goulburn output for August (Output 4), where the Euclidean distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ’i ¼ 2i þ 2i ; for i = 4, where i and i (for i = 4) are the mean and standard deviation of the samples of the second-order effects or derivatives. The values have been rounded to the nearest integer. The mean, standard deviation and Euclidean distance have also been assigned a rank, with Rank 1 being the highest. This ranking is assigned on the raw values, and before rounding, to avoid ties. Thus the interaction values 4 for the interacting pairs 3Y7, 4Y7 and 4Y9, are rounded to one, in table 6, but the rankings are different, being assigned on the raw values. The values of 4 show that the interaction between the carrier convergence criterion and the absolute convergence criterion parameters (Factors 4 and 5) is the strongest, and is more than twice as strong as the next highest interaction between the Goulburn Weir forced spill and the storage convergence criterion parameters (Factors 2 and 3). The higher standard deviations for the first-order effects, have successfully indicated which are the greater second-order interaction factors. Many of the interactions are small, with
the 5Y6 interaction having a mean of 10, and ranked at 26. Interactions at this level are insignificant compared with the highest ranked interactions, and are also Bdown in the noise^ associated with the estimation of the derivatives. The higher ranked interactions based on 4 indicate the potential for higher third-order interactions. This data is shown graphically in figure 3. Note that Factors 3, 4 and 5 are convergence factors relating to the numerical processes in the GSM model (table 2). The GSM is showing the highest first and secondorder sensitivities to numerical parameters in the model [for the average Goulburn outflow for August (Output 4)]. Of the physical parameters, the Goulburn Weir forced spill parameter (Factor 2) gives rise to the higher sensitivities for the first and second-order effects.
3.2. First-order effects: second experiment The analysis in section 3.1 can be repeated for the other 15 outputs from GSM. Such an approach would illustrate the detail, and would involve a long list of tables and figures. The separate outputs also are different physical variables, and there is no direct comparison between the means, standard deviations or Euclidean distances between the different outputs. For the GSM, the output functions relate to outflows, diversions and allocations, and each has its own natural scale. These scales may not be commensu-
304
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model Table 7 Factor ranks based on i for the outflows from the Broken River. Ranks by mean i (Broken River)
Factor
1 2 3 4 5 6 7 8 9
Output 5
Output 6
Output 7
Output 8
9 6 8 7 5 3 2 1 4
1 6 5 7 8 9 4 3 2
9 6 8 7 5 2 1 3 4
3 5 6 8 9 7 1 2 4
Sum of ranks 5Y8
Rank by i (Output 16)
22 23 25 29 27 21 8 9 14
j1 j1 j1 j1 j1 j1 2 1 j1
The ranks of Outputs 5Y8 are also summed. The ranking of j1 for Output 16 indicates zero sensitivity.
rate. For a more complex modelling system, the outputs may relate to totally different physical outputs and the direct comparison of values becomes impossible. The ranks are ordinal measures, can be compared, and are used in the presentation of the data. In tables 5 and 6, the use of the rank does hide the difference in values of or (say), but does preserve order and permits some compaction and comparison of the data. The ranks will be heavily used in the analysis of the data from these sensitivity runs. The minimum Goulburn outflow for February, and the minimum Broken cap diversion (Outputs 1 and 15) for all model runs were effectively constant. The minimum Goulburn outflow for February (Output 1) was zero, while the minimum Broken cap diversion (Output 15) remained constant at 9765 Ml. Thus the means and standard deviations of the first and second-order effects for these outputs were zero. These two output variables will be ignored in subsequent analyses. The remaining output functions fall into two groups: Outputs 5, 6, 7, 8 and 16 relating to the outflows from the Broken River; and Outputs 2, 3, 4, 9, 10Y14 relating to the Goulburn River.
Tables 7 and 8 show the rankings for the first-order i and i for the Broken River. In each case, the ranks for Broken River (Outputs 5Y8) are summed and given for each factor. The average Broken cap diversion (Output 16) only showed sensitivity for the Broken River transmission and operational loss parameters (Factors 7 and 8), with the other factors having zero sensitivity; this zero sensitivity is shown by a j1 in the ranking for this output. These results show that the outputs relating to the Broken River are more sensitive to the Boort entitlement and Broken River transmission and operational parameters (Factors 7, 8 and 9), and are less sensitive to Factors 3, 4 and 5 which are the numerical convergence factors in the water allocation model. Note the large differences in rank based on i for the Goulburn River transmission loss parameter (Factor 1) between Outputs 5 and 7 (minimum Broken River outflows for February and August) and Outputs 6 and 8 (average Broken River outflows for February and August), and also for the Boort entitlement (Factor 6), although the effect is reversed. This effect arises from the nature of the output functions with Outputs 5 and 7 being minimum outflows, and Outputs 6 and 8 being average outflows. There may
Table 8 Factor ranks based on i for the outflows for the Broken River. Ranks by i (Broken River)
Factor
1 2 3 4 5 6 7 8 9
Output 5
Output 6
Output 7
Output 8
9 6 8 7 5 2 3 4 1
2 3 4 5 6 9 8 7 1
9 7 8 6 5 1 4 3 2
4 5 6 7 8 9 1 3 2
Sum of ranks 5Y8
Rank by i (Output 16)
24 21 26 25 24 21 16 17 6
j1 j1 j1 j1 j1 j1 2 1 j1
The ranks of Outputs 5Y8 are also summed. The ranking of j1 for Output 16 indicates zero higher order interactions.
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
also be a seasonal effect for the Boort entitlement and Broken River transmission loss parameters (Factors 6 and 7), where Outputs 7 and 8 (relating to flows in August) are generally more highly ranked than Outputs 5 and 6 (relating to Broken River outflows in February). The columns giving the sum of the rankings do not show this level of detail, but clearly show the parameters to which the Broken River is sensitive. The above patterns are repeated for the rank by i, shown in table 8, where the Waranga Basin evaporation loss parameter (Factor 9) provides the highest rank, followed by the Broken River transmission and operational loss parameters (Factors 7 and 8). The storage convergence and carrier convergence criterion parameters (Factors 3 and 4) have the lowest ranked effects on the outflows for the Broken River. The minimum/average dichotomy is repeated for the Goulburn River transmission loss and the Boort entitlement parameters (Factors 1 and 6). Thus the Broken River outflows show a higher sensitivity for the physical factors relating to this system, with the indication of possible second-order interactions. The numerical convergence factors for the water allocation model provide the least sensitivity in the outflows of the Broken River system. There are nine outputs to be considered for the Goulburn River system, and tables 9 and 10 show the rankings by i and i for the Goulburn River. The output functions for the Goulburn River can also be further grouped, and the rankings for the Goulburn River outflows (Outputs 2Y4), have been summed and listed in these tables. Outputs 9Y12, the Goulburn irrigation allocations, and 13Y14, the Goulburn cap diversions, have been treated in the same way. These are Bnatural^ groupings of the output functions. Note that the Goulburn River transmission loss (Factor 1) arises as the most important factor in the sensitivity of first-order effects, of these output functions, particularly with respect to the mean i. The Goulburn River forced spill and the storage convergence criterion parameters (Factors 2 and 3) are the next most sensitive parameters on the outputs generally, while the Broken River transmission and operational loss parameters (Factors 7 and 8), have the least effect on the sensitivity. Further, any minimum/ average pattern is not very pronounced. For the Goulburn cap diversions (Outputs 13 and 14), the physical factors 1, 2 and 9 (the Goulburn River transmission loss, the Goulburn Weir forced spill and the Waranga Basin evaporation parameters) are highly ranked, as is the allocation convergence factor 3. The pattern based on ranking by is less definite (see table 10). The Broken River transmission and operational loss parameters (Factors 7 and 8) as well as the Waranga Basin evaporation loss parameter (Factor 9) are the lowest ranked using , indicating a low probability of higher order interactions. For the Goulburn River outflows (Outputs 2Y4), the Goulburn River transmission loss, the Goulburn Weir forced spill, and the storage convergence criterion
305
parameters (Factors 1, 2 and 3) indicate the highest probability of higher order interactions, with the carrier and absolute convergence criterion parameters (Factors 4 and 5) in the middle. For the Goulburn allocations (Outputs 9Y14), the carrier and absolute convergence criterion parameters (Factors 4 and 5) show out with the highest ranking, and indicate potential for higher order interactions. Once again, the numerical convergence factors are all showing potential for higher order interaction. There are two special cases which need to be noted. The first is the Broken River operational loss parameter (Factor 8), which is ranked one on the minimum Goulburn outflow for August (Output 3) for both 3 and 3, but which is ranked much lower, i.e. eighth or ninth for the other outputs (see tables 9 and 10). This high ranking on Output 3 arises from mean and standard deviation values which are more than double the values of the next ranked factors; it is not an artefact of the ranking system or of the sampling of parameter space. The second special case is the first ranking of the Waranga Basin evaporation loss (Factor 9) for the mean for the minimum Goulburn allocation for August (Output 11); generally this factor is ranked three to six. Here, all the values are small and tightly grouped, and the high ranking may be due to the ranking system or the sampling of parameter space.
3.3. Second-order interactions There are 36 possible non-directional interactions between the nine factors, and these are listed in tables 11 and 12 for the Broken River, and tables 13 and 14 for the Goulburn River. The minimum Goulburn outflow for February and the minimum Broken cap diversion (Outputs 1 and 15) showed that the first-order effects are zero, and hence give no second-order effects. The average Broken cap diversion (Output 16) showed that only two factors, 7 and 8 (the Broken River transmission and operational loss parameters), gave rise to a first-order effect, and hence the Interaction 7Y8 is the only possible second-order effect. These outputs have been omitted in the tables 11Y14. Further, there were several factor pairs which provided a zero value for the mean i and standard deviation i for various outputs i. Where this occurred, these were all assigned a rank of 36, while all the non-zero interactions were ranked from one (highest) to the lowest. Also given in tables 11 and 12 are the sums of the ranks of each factor for the Broken outflows (Outputs 5Y8). In tables 13 and 14, the sums of ranks for each factor are given for the blocks of output for the Goulburn outflows (Outputs 2Y4), the Goulburn allocations (Outputs 9Y12), and for the Goulburn cap diversions (Outputs 13Y14). Table 11 shows the ranks of the sensitivities of the pairwise factor interactions based on i for i = 5, 6, 7 and 8 for the Broken River; table 12 shows similar data based on the ranking by i. The interactions 7Y8 (Broken River transmission and operational loss parameters) and 3Y9 (the
1 2 4 6 7 5 8 9 3
Ave. (February)
2 6 3 7 8 4 9 1 5
Min. (August)
Outputs 2Y4
3 2 1 8 5 7 6 9 4
Ave. (August) 6 10 8 21 20 16 23 19 12
Sum of ranks 2Y4
The ranks for Outputs 2Y4, 9Y12 and 13Y14 are also summed and listed.
1 2 3 4 5 6 7 8 9
Factor
1 4 2 5 7 6 9 8 3
Min. (February) 1 3 2 4 7 6 9 8 5
Ave. (February) 2 3 5 6 7 4 9 8 1
Min. (August)
Outputs 9Y12
1 3 2 4 7 5 8 9 6
Ave. (August)
Table 9 Factor ranks based on i for the Goulburn River.
5 13 11 19 28 21 35 33 15
Sum of ranks 9Y12
1 4 2 5 7 6 9 8 3
Min.
1 3 2 5 6 7 9 8 4
Ave.
Outputs 13Y14
2 7 4 10 13 13 18 16 7
Sum of ranks 13Y14
306 R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
2 1 5 6 7 4 8 9 3
Ave. (February)
3 8 5 2 4 6 9 1 7
Min. (August)
Outputs 2Y4
4 3 1 5 2 7 8 9 6
Ave. (August) 9 12 11 13 13 17 25 19 16
Sum of ranks 2Y4
The ranks for Outputs 2Y4, 9Y12 and 13Y14 are also summed and listed.
1 2 3 4 5 6 7 8 9
Factor
6 4 3 1 2 5 9 8 7
Min. (February) 5 3 4 1 2 6 9 8 7
Ave. (February) 1 2 4 6 7 3 9 8 5
Min. (August)
Outputs 9Y12
5 3 4 1 2 6 8 9 7
Ave. (August)
Table 10 Factor ranks based on i for the Goulburn River.
17 12 15 9 13 20 35 33 26
Sum of ranks 9Y12
6 4 3 1 2 5 9 8 7
Min.
4 5 3 2 1 6 9 8 7
Ave.
Outputs 13Y14
10 9 6 3 3 11 18 16 14
Sum of ranks 13Y14
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model 307
308
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model Table 11 Pairwise factor interactions based on i for the Broken River.
Factor interaction 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8
2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 4 5 6 7 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9
Output 5 min. (February)
Output 6 ave. (February)
Output 7 min. (August)
Output 8 ave. (August)
Sum of ranks
36 13 23 3 5 22 12 26 34 2 4 10 11 16 28 33 9 14 15 31 7 35 19 30 24 20 17 25 32 18 29 6 21 1 8 27
6 13 36 36 18 11 20 1 2 36 36 12 10 20 5 4 36 14 21 16 7 3 36 36 36 36 36 36 36 36 17 36 22 8 15 9
35 25 23 6 7 15 18 22 34 2 1 17 12 11 30 36 10 9 13 29 5 32 14 20 26 16 24 21 33 19 31 3 27 8 4 28
1 13 36 36 18 14 15 7 5 36 36 11 10 19 6 9 36 12 17 16 3 8 36 36 36 36 36 36 36 36 22 23 21 4 20 2
78 64 118 81 48 62 65 56 75 76 77 50 43 66 69 82 91 49 66 92 22 78 105 122 122 108 113 118 137 109 99 68 91 21 47 66
The ranks for i = 5, 6, 7 and 8 are also summed.
storage convergence criterion and the Waranga Basin evaporation loss parameter) are the ones giving the greatest sensitivity for this group of outputs, with the interactions 2Y8, 7Y9, 1Y6, 3Y6 and 2Y6 also showing up as producing sensitive interactions for this group of outputs. The rankings in table 8, suggested that the Broken River transmission and operational loss and Waranga evaporation loss parameters (Factors 7, 8 and 9) may be involved in the second-order interactions. Note that Factors 7, 8 and 9 are physical parameters for the Broken Irrigation system, which Factor 3 is a convergence factor for the water allocation. All the interactions involving the carrier storage and allocation convergence criterion parameters (Factor 4 or Factor 5) show low levels of interaction. The table also shows marked differences in ranking between Outputs 5 and 7 (minimum Broken River outflows) and Outputs 6 and 8 (average Broken River outflows). For example, the interaction 2Y4 shows relatively high rankings for the minimum flow outputs (Outputs 5 and 7) and low rankings
for the average flow regimes. This effect is most pronounced with respect to the carrier and absolute convergence criterion parameters (Factors 4 and 5) interacting with other factors. Except for the interaction 4Y5, the interaction of Factor 5 with the other seven factors produces almost no sensitivity for the average Broken River outflows (Outputs 6 and 8), but the minimum Broken outflows (Outputs 5 and 7) are more highly ranked. The carrier convergence criterion parameter (Factor 4) generally shows the same pattern. With respect to the rankings based on i for the Broken River, note that the interaction 3Y9 (carrier storage convergence criterion and Waranga Basin evaporation parameters) is prominent and indicates possible third-order interactions. There is then a gap in terms of the sums of the rankings, to the interactions 1Y7, 7Y9 and 8Y9, with the interactions of the storage and absolute convergence criterion parameters (Factor 4, or 5), with other factors being the least sensitive. The pattern in the minimum/
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
309
Table 12 Pairwise factor interactions based on i for the Broken River. Factor interaction 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8
2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 4 5 6 7 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9
Output 5 min. (February)
Output 6, ave. (February)
Output 7 min. (August)
Output 8 min. (August)
Sum of ranks
36 22 28 7 9 3 11 24 34 5 10 20 19 15 27 33 14 26 12 30 6 35 17 4 25 18 13 23 32 16 2 8 21 31 1 29
4 15 36 36 22 13 16 3 6 36 36 9 10 17 1 7 36 12 20 14 2 5 36 36 36 36 36 36 36 36 19 36 21 11 18 8
35 24 21 3 6 15 22 27 34 2 1 16 14 17 31 36 12 11 18 5 9 33 19 23 28 20 29 26 13 25 32 4 30 10 8 7
3 15 36 36 18 12 13 5 6 36 36 10 9 16 4 8 36 14 21 20 2 7 36 36 36 36 36 36 36 36 22 23 19 11 17 1
78 76 121 82 55 43 62 59 80 79 83 55 52 65 63 84 98 63 71 69 19 80 108 99 125 110 114 121 117 113 75 71 91 63 44 45
The ranks for i = 5, 6, 7 and 8 are also summed.
average model outputs is repeated on approximately the same set of interacting pairs for i, as shown in table 8. Table 13 shows the interaction rankings based on i for the Goulburn River. The interactions 4Y5, 2Y3 and 3Y4, all of which involve some of the convergence criterion parameter, show the strongest second-order sensitivities, followed by 1Y2, 1Y3, 2Y6 and 3Y9, for the Goulburn outflows (Outputs 2, 3 and 4). The other interactions of the carrier and absolute convergence criterion parameters (Factor 4, or 5), with other factors, are generally lowly ranked. The minimum/average pattern is not apparent, but note that the minimum Goulburn outflow for February (Output 1) has zero sensitivity at first and second order. The results are also consistent across the groups of outputs. The group of outputs i = 9, 10, 11 and 12 (for the Goulburn allocations) show high second-order sensitivity (based on i) to the interactions 3Y4, 4Y5, 2Y3 and 3Y9, all of which include convergence parameters. Surprisingly,
nearly all the other interactions involving Factor 4, or 5, or 7 or 8, lead to very low second-order sensitivity rankings. The same pattern is repeated for the Goulburn cap diversions (Outputs 13 and 14), where the interactions 3Y4, 4Y5, 2Y3 and 3Y9 correspond to relatively high second-order sensitivities. All of these sensitive interactions include at least one of the convergence factors for the water allocation. Table 14 shows the ranking of the interactions based on i, and a high ranking is indicative of possible higher order interactions. For the Goulburn outflows (Outputs 2, 3 and 4), the interactions 4Y5, 3Y4, 2Y6, 1Y2 and 1Y6 are highly ranked and indicate possible third-order sensitivity effects. The other interactions of Factor 4, or 5 or 7, or 8, with the other factors are low in rank and indicate that third-order interactions involving these groupings are not likely. The pattern for the Goulburn allocations (Outputs 9, 10, 11 and 12) are similar, with the interactions 2Y3, 3Y4 and
2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 4 5 6 7 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9
2 5 36 36 11 13 23 6 1 36 36 8 15 17 9 4 36 10 12 22 7 3 36 36 36 36 36 36 36 36 14 16 19 18 21 20
Output 2 ave. (February)
14 12 36 36 15 36 1 13 7 36 36 6 36 36 11 5 36 10 16 36 9 4 36 36 36 36 36 36 36 36 36 3 8 17 36 2
Output 3 min. (August)
Group 1
6 5 17 24 9 31 25 8 2 18 22 10 28 33 13 3 20 7 35 30 4 1 23 34 21 36 26 32 29 27 19 11 12 16 14 15
Output 4 ave. (August) 22 22 89 96 35 80 49 27 10 90 94 24 79 86 33 12 92 27 63 88 20 8 95 106 93 108 98 104 101 99 69 30 39 51 71 37
Sum of ranks 12 5 36 36 11 36 36 10 3 36 36 9 36 36 8 1 36 7 36 36 4 2 36 36 36 36 36 36 36 36 36 36 6 36 36 36
Output 9 min. (February) 10 3 14 36 7 36 36 12 4 36 36 8 36 36 9 1 36 6 36 36 5 2 36 36 36 36 36 36 36 36 36 36 11 36 36 13
Output 10 ave. (February) 13 12 36 36 11 36 36 10 3 9 36 8 36 36 7 4 36 2 36 36 1 6 36 36 36 36 36 36 36 36 36 36 5 36 36 36
Output 11 min. (August)
Group 2
The ranks are summed by groups for i = 2, 3 and 4 (Group 1), i = 9, 10, 11 and 12 (Group 2) and i = 13 and 14 (Group 3).
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8
Factor interaction
Table 13 Pairwise factor interactions based on i for the Goulburn River.
7 3 19 36 10 36 36 8 4 14 18 11 13 36 12 1 17 5 36 36 6 2 16 36 36 36 22 36 36 21 15 36 9 36 36 20
Output 12 ave. (August) 42 23 105 144 39 144 144 40 14 95 126 36 121 144 36 7 125 20 144 144 16 12 124 144 144 144 130 144 144 129 123 144 31 144 144 105
Sum of ranks 10 4 24 36 7 16 26 12 3 36 36 8 30 23 11 1 36 6 27 18 5 2 13 29 15 28 14 36 36 22 25 17 9 21 19 20
Output 13 min. 7 6 16 21 9 19 33 4 2 15 23 10 13 17 8 3 24 11 29 22 5 1 20 28 30 25 27 36 32 26 18 31 12 34 35 14
Output 14 ave.
Group 3
17 10 40 57 16 35 59 16 5 51 59 18 43 40 19 4 60 17 56 40 10 3 33 57 45 53 41 72 68 48 43 48 21 55 54 34
Sum of ranks
310 R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 4 5 6 7 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9
6 2 36 36 10 12 17 4 11 36 36 1 21 19 7 8 36 9 14 20 3 5 36 36 36 36 36 36 36 36 13 16 23 22 18 15
Output 2 ave. (February)
Group 1
7 17 36 36 3 36 1 8 4 36 36 2 36 36 16 6 36 15 14 36 13 5 36 36 36 36 36 36 36 36 36 12 11 10 36 9
Output 3 min. (August) 6 4 16 22 7 31 21 8 3 18 24 15 29 30 14 2 20 5 32 33 10 1 36 35 23 34 25 28 27 26 17 11 12 19 9 13
Output 4 ave. (August) 19 23 88 94 20 79 39 20 18 90 96 18 86 85 37 16 92 29 60 89 26 11 108 107 95 106 97 100 99 98 66 39 46 51 63 37
Sum of ranks 12 1 36 36 11 36 36 10 3 36 36 9 36 36 8 2 36 7 36 36 6 4 36 36 36 36 36 36 36 36 36 36 5 36 36 36
Output 9 min. (February) 9 4 14 36 6 36 36 12 1 36 36 7 36 36 8 2 36 10 36 36 5 3 36 36 36 36 36 36 36 36 36 36 11 36 36 13
Output 10 ave. (February)
The ranks are summed by groups for i = 2, 3 and 4 (Group 1), i = 9, 10, 11 and 12 (Group 2) and i = 13 and 14 (Group 3).
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8
Factor interaction
13 11 36 36 11 36 36 10 3 9 36 8 36 36 7 4 36 2 36 36 1 6 36 36 36 36 36 36 36 36 36 36 5 36 36 36
Output 11 min. (August)
Group 2
Table 14 Pairwise factor interactions based on i for the Goulburn River.
6 4 22 36 10 36 36 5 1 21 22 7 19 36 11 2 18 9 36 36 8 3 17 36 36 36 16 36 36 15 14 36 12 36 36 13
Output 12 ave. (August) 40 20 108 144 38 144 144 37 8 102 130 31 127 144 34 10 126 28 144 144 20 16 125 144 144 144 124 144 144 123 122 144 33 144 144 98
Sum of ranks 9 1 30 36 7 29 18 12 3 36 36 8 28 14 10 2 36 5 20 16 6 4 27 26 25 24 23 36 36 22 15 13 11 17 19 21
Output 13 min. 4 5 16 22 10 19 31 9 3 15 23 11 13 17 6 1 25 8 32 20 7 2 21 33 30 24 26 36 29 27 18 28 12 34 35 14
Output 14 ave.
Group 3
13 6 46 58 17 48 49 21 6 51 59 19 41 31 16 3 61 13 52 36 13 6 48 59 55 48 49 72 65 49 33 41 23 51 54 35
Sum of ranks
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model 311
312
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
4Y5 indicating possible third-order sensitivities. For the Goulburn cap diversions (Outputs 13 and 14), the interactions 3Y4, 2Y3, 1Y3 and 4Y5 indicate the possibility of third-order interactions with other factors.
3.4. Effects of expanded parameter range Similar sensitivity analyses were conducted with the expanded parameter range as given in table 2. The full details of this output and analysis will not be given here, but the results are readily compared with those from the original parameter range. 3.4.1. First-order effects, expanded parameter range Output 1 (minimum Goulburn outflow) still remained constant, at zero, while Output 15, (maximum Broken River cap) did show some variation to Factor 8, but not for the other factors. The variation of the minimum Broken cap diversion (Output 15) with respect to the Broken River operational loss (Factor 8), occurred near the extremities of the extended range, i.e. where the operational loss parameter was T40% of the normal value. For the Broken River, Outputs 5 and 7 (minimum outflow for February and August) showed little change, of the order of 1%, in the values of i and i, compared to the original parameter range. The patterns in the rankings in the earlier results were retained. Outputs 6 and 8 (average Broken outflows for February and August) showed greater changes, of the order of 20%, in values of i and i. In terms of the rankings, the Goulburn River transmission loss parameter (Factor 1) decreased in importance to around a ranking of five, while the Boort entitlement parameter (Factor 6) moved up to the rankings of 1 and 2. The convergence criterion parameters (Factors 3, 4 and 5) were still registering relatively low in order of sensitivity. For the Goulburn outflows (Outputs 2, 3 and 4), the values of i and i changed by varying amounts. Some very low values increased several fold, while the larger ones changed by T20% in general. The rankings changed only a little, with the numerical convergence parameters moving up by one or two positions generally, and the physical transmission and operational loss parameters (Factors 1, 7 and 8) slipping down to compensate. For the Goulburn allocations and cap diversions (Outputs 9Y14), the changes in value of and were generally of the order of T20%, with some large relative changes among those with small values. The rankings changed only marginally. 3.4.2. Second-order effects, expanded parameter range The values of i and i for the expanded parameter range were more variable when compared to the values from the original parameter range, with changes of the order of T60%. There were some larger changes for the interactions which had low values for the original param-
eter range. The major changes in the ranking of the interactions relate to the convergence criterion parameters (Factors 3, 4 and 5), with the interactions between any two of these being the highest ranked for all outputs. These interactions were dominant, both by ranking and by value. They also had dominance in the standard deviations, suggesting the possibility of higher order interactions between them.
4. Discussion The REALM model is a major tool for assessing allocation level in the Goulburn system. Water authorities also actively use the GSM for formulating environmental flow rules, water quality (salinity) modelling and modelling other hydrological processes in the region [13]. Hence the sensitivity analysis for the GSM is a very topical task for hydrologists. This analysis was started by the work of Schreider et al. [9] where sensitivity to the variation of single input factors was analysed using the Monte Carlo method. However, this Monte Carlo simulation did not answer the question of how simultaneous variations of a group of factors affect the GSM outputs and how these input factors are related to each other. Answering these questions is the major challenge of the present study which makes sensitivity analysis of the GSM more comprehensive. The output functions have been grouped on a geographical basis of the Goulburn outflows, Broken outflows, Goulburn diversion and also the Goulburn cap. The GSM input factors also relate implicitly or explicitly to these outputs. Some of the parameters relate more directly to the Goulburn system, and play only an indirect role with respect to the Broken system; and vice versa. This relationship is apparent in the results. Output 1, the minimum Goulburn outflow in February, is always zero, a value which is expected for summer. The minimum Broken cap diversion, was also constant, at 9765, for both of the first and second experiments. Some variation did occur with respect to Factor 8, the Broken River operational loss parameter. This occurred at the ends of the extended range of values for this parameter, and agrees with the results obtained by Schreider et al. [9]. Other factors, such as the convergence parameters, play a direct (or indirect) role with respect to the full system. This general dependence is reflected in the results of the sensitivity analysis. In figures 2 and 3, relating to the average Goulburn outflow for August, the more sensitive first and second-order effects relate to Factor 2 (the Goulburn Weir forced spill), which is a physical factor relating to the Goulburn system, and to Factors 3, 4 and 5, which are numerical convergence criteria affecting the whole system. Those factors more explicitly relating to the Broken system, display much less sensitivity with respect to the average Goulburn outflow in August. Figure 1 does show that these systems are linked, and that
R.D. Braddock and S.Y. Schreider / Sensitivity of REALM Goulburn model
the Goulburn and Broken outflows do merge. Thus they do have some interaction potential. However, the sensitivities of these interactions are low, due to either a low level of interaction, or indicating the level of error in sampling of the parameter space. This feature is repeated across the results generally, with the Goulburn parameters providing low sensitivities for the Broken outputs, and vice versa. The numerical convergence factors (Factors 3, 4 and 5) provide very low first-order sensitivities for the Broken system (tables 7 and 8), while Factor 3 (and to a lesser extent 4 and 5) yield highly ranked sensitivity for the Goulburn system (tables 9 and 10). This indicates that the storage convergence criterion (Factor 3) for the Goulburn system, may need to be constrained more tightly. There is also a marked difference in the minimum and average outflows of the Broken River, and their sensitivity to the Goulburn transmission loss and the Boort entitlement parameters (Factors 1 and 6). The average outflows (Outputs 6 and 8) of the Broken system, are sensitive to Factor 1, the Goulburn River transmission loss. The minimum outflows of the Broken system are insensitive to this factor. The situation is reversed for the Boort entitlement (Factor 6) where the minimum outflows of the Broken River are sensitive to this factor. Figure 1 shows that outflows to the Murray River, can be provided from either the Goulburn or the Broken River. Thus there can be trade offs between the river systems in the provision of this outflow. Further, the Boort entitlement (Factor 6) is linked directly to the Goulburn system, but can also be the beneficiary of this trade off, particularly for the minimum flows (Outputs 5 and 7).
313
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