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Impact of hydroclimate parameter uncertainty on system yield.

1. Introduction

System yield is commonly determined by Monte Carlo simulation using synthetic climate data. This involves simulating the behaviour of the water resource system over many years of synthetic climate and determining how often specified events occur. In these calculations, yield is defined as the amount of water that can be supplied each year for one or more given risk thresholds. A possible definition of yield could, for example, be the quantity of water that can be supplied per year without restrictions needing to be imposed in more than 1 out of every 10 years (Berghout 2009; Erlanger et al. 2005).

Synthetic data, generated by a stochastic procedure, require calibration of parameters such as the mean, standard deviation and serial autocorrelation of streamflow and rainfall, to observed records. These parameters are hereafter referred to as hydroclimate parameters. The brevity of observed records is a source of uncertainty in the estimated hydroclimate parameters (Stedinger and Taylor 1982; Thyer et al. 2006a). This study investigates the impact of the uncertainty in model parameters on the resulting system yield.

Stedinger and Taylor (1982) investigated the effect of hydroclimate parameter uncertainty on the sizing of a reservoir at the outlet of the upper Delaware River basin at a site where 50 years of historic flow information was available. They showed that uncertainty in the parameters describing the distribution of annual flows (i.e. the mean and standard deviation) could have a significant impact on the required reservoir capacity.

In the approach of Stedinger and Taylor (1982), the required reservoir capacity was calculated for each of 500 synthetic replicates, thereby producing 500 estimates of the required reservoir volume. In the base case (the case without hydro climate-parameter uncertainty), all 500 climate replicates were produced using the sample mean and standard deviation of the historic data. In the case that incorporated parameter uncertainty, new values of mean and standard deviation were sampled for each climate replicate. In both cases, therefore, 500 estimates of required reservoir capacity were derived.

In the analysis by Stedinger and Taylor (1982), the incorporation of parameter uncertainty into the calculation effectively altered the range in the 500 estimates of required reservoir capacity compared with the situation where all 500 replicates were based on a single estimate of mean and standard deviation. The results of the Stedinger and Taylor (1982) experiment have been reproduced qualitatively in Figure 1. It is clear from Figure 1 that inclusion of sampling variability in the selection of mean and standard deviation for each streamflow replicate has an appreciable impact on the variability between synthetic climate sequences, and therefore on the uncertainty in the required reservoir capacity. This is especially noticeable in the increased spread of the middle 50% of the results range.

In contrast to Stedinger and Taylor (1982), this paper explores the impact of hydroclimate parameter uncertainty on a contemporary calculation of system yield, which involves the calculation of the average annual quantity of water that can be supplied (i.e. the yield) from a single or multiple source headworks system for given risk thresholds. This investigation reflects the approach that is generally used for urban water supply planning in Australia to investigate how much water can be supplied from an existing system or to test the impacts on yield from possible changes to the system.

There is a limited literature on how uncertainty in hydroclimate parameters influences yield. The exception is Thyer et al. (2006b) who demonstrate that allowing for hydroclimate parameter uncertainty leads to a more conservative (or lower) estimate of the expected yield. Their approach, however, does not fully communicate the implications of parameter uncertainty.

The main contribution of the present study is to present and illustrate a methodology that determines the probability distribution of yield arising from uncertainty in key hydroclimate parameters. This approach has not previously been presented in the literature.

The paper is organised as follows: Section 2 presents a description of the case study site, the synthetic climate generation method, the definition of yield, the method used to determine yield, and a description of the methods used to sample the hydroclimate parameters and then determine the impact of hydroclimate parameter uncertainty on yield. The results are then presented in Section 3, first reporting on hydroclimate parameter uncertainty and then on the uncertainty in the yield using various methods to include hydroclimate parameter uncertainty. Sections 4 and 5 conclude the paper with a discussion and conclusions.

2. Methodology

2.1. Case study description

The impact of hydroclimate parameter uncertainty on uncertainty in the yield is illustrated for an urban water supply case study from the east coast of Australia. The case study system is the water supply system for the urban areas of the lower Hunter Valley, including the cities of Newcastle, Lake Macquarie and Maitland. This system harvests water from two surface water reservoirs, Chichester Dam and Grahamstown Dam, and a groundwater system, the Tomago Sandbeds. The key hydroclimate input data for the simulation model for this system are streamflow time series data for eight surface water sub-catchments and rainfall data for three locations:

(1) Chichester Dam catchment streamflow

(2) Williams River at Tillegra catchment streamflow

(3) Lostock Dam catchment streamflow

(4) Allyn River at Halton catchment streamflow

(5) Paterson River at Gostwyck residual catchment streamflow

(6) Williams River at Glen Martin residual catchment streamflow

(7) Grahamstown Dam catchment streamflow

(8) Williams River at Seaham Weir residual catchment streamflow

(9) Chichester Dam rainfall

(10) Grahamstown/Tomago rainfall

(11) Lostock Dam rainfall

One hundred and thirteen years of historic streamflow and rainfall data are available for the 11 sites using observed data with gaps in-filled by correlation with neighbouring sites and/or rainfall to run-off simulation.

The simulation model is a custom-built FORTRAN mass balance model that runs at a daily time step. The model simulates all key operating rules within the system, the key inputs and outputs of each reservoir (for example, environmental releases, pipe supply, spills, and evaporation), a climate- and time-of-year dependent demand model, an aquifer model, and a water use restriction strategy that reduces the demand according to set rules when storage levels are low.

2.2. Synthetic climate generation method

The probability model used to generate synthetic climate data is the well-established annual multi-site Markov model (Matalas 1967). The Matalas model is

implemented as follows. Let [q.sub.t] be the vector of annual hydroclimate values at m sites for year t; that is, [q.sub.t] = {[q.sub.ti],i = 1,..,m}. For each site i, the annual value [q.sub.ti] is transformed using the Box-Cox transformation to give the transformed value [z.sub.ti].

[mathematical expression not reproducible] (i)

The objective of the transformation is to select a value for the parameter [lambda] which makes the marginal distribution of z closely approximate a Gaussian or normal distribution. The vector of transformed values is described by the Markov model.

[Z.sub.t] = [mu] + A([z.sub.t-1] - [mu]) + [[epsilon].sub.t] (2)

where [mu] is the mean of [z.sub.t], A is an m x m lag-one matrix and [[epsilon].sub.t] is an m-vector of normally distributed disturbances with mean 0 and covariance matrix [SIGMA].

The annual synthetic hydroclimate data are disaggregated into daily data using the method of fragments (Svanidze 1960). In this particular application of the method of fragments, a key site nearest neighbour approach is used in the selection of the year of historic data used to supply the fragments for any given synthetic year. The synthetic climate generation code is based on that developed for WATHNET (Kuczera 1997).

2.3. Definition of yield

System yield is assessed by determining how much water can be supplied per year (i.e. the yield) with respect to specified system performance constraints. For the purpose of this investigation, two system performance constraints are used to define yield:

(1) No more than 1 in 10 years of the analysis shall contain restrictions. In this example, the restriction trigger is assumed to be 60% storage, and the frequency of restrictions being imposed is assessed by counting the number of years during which the lowest storage reached is below the restriction trigger of 60%.

(2) No more than 1 in 1000 years of the analysis shall have a storage that falls below 20%.

The first constraint is commonly used in Australia when setting acceptability limits on the frequency that water use restrictions can be imposed for urban water supply schemes. It reflects the expectation that unrestricted water supply will be reasonably reliable.

The second constraint is typical of the style used when setting acceptability limits on reaching a low-storage situation, and reflects the expectation that supply must be reliable during severe droughts. Together, the two constraints establish storage performance standards for the spectrum of high- and low-frequency events.

2.4. Determination of yield

System performance is assessed against the yield criteria by simulating storage behaviour for a large number of synthetic climate replicates and for a range of demand scenarios. In this experiment, each of the 5000 synthetic climate replicates is the same length as the historic climate record, which is 113 years. Statistics relating to storage behaviour are not calculated during a warm-up period of five years at the start of each replicate. The purpose of the warm-up period, the length of which is selected to suit the system that is being analysed, is to allow storage performance characteristics to be determined without being influenced by the initial storage situation. Storage behaviour statistics are thereby gathered from 108 years of storage simulation for each of 5000 replicates, thus representing 540,000 years of simulated storage behaviour for a given demand scenario.

The key statistic that is collected in each year of the analysis (after the warm-up period) is the lowest storage level that is reached during that year. This approach thereby produces 540,000 samples of the lowest annual storage for a given demand scenario. These samples can be used to determine the 1 in n year storage situation for that demand.

In order to estimate the system yield, it is necessary to repeat this process for a sufficiently wide range of demand scenarios to detect event frequencies above and below the yield thresholds. Once the storage risk statistics have been collected for a range of demand scenarios it becomes possible to assemble storage risk curves, which show the relationship between the 1 in n year storage level and demand. These curves can then be used to interpolate between demand scenarios to estimate the maximum annual demand that can be met with respect to each yield criterion.

In this experiment, storage risks are determined for demand scenarios ranging from 50GL/year up to 105 GL/year, in increments of 5GL/year, in the expectation that the yield will lie somewhere in that range.

2.5. Estimating the hydroclimate parameters

The calibration of a hydroclimate stochastic model involves estimating its parameters [theta] using a sample of historic observations denoted by the vector q. The parameters are estimated using an estimator which is some function of the observations [??] (q). If the observations are resampled K times, there would be K different sets of data {[q.sub.1],...,[q.sub.K]} and hence K different parameter estimates {[mathematical expression not reproducible]}. This is called sampling variability. As K becomes large, the K estimates define the sampling distribution of [theta] denoted as p([theta]|[??]). This describes the uncertainty about the parameters. For example, suppose k samples {[x.sub.1]...,[x.sub.k]} are drawn from a normal distribution with mean [mu] and standard deviation a. A well-known estimator of the mean y is the arithmetic mean [bar.x] = 1/k [k.summation over (i=1)] [x.sub.i], Its sampling distribution can be shown to be p([mu]|[bar.x]) = N ([bar.x], [[sigma].sup.2]/k) which is a normal distribution with mean x and standard deviation [sigma]/[square root of (k)] . For example, if y = 1 and [sigma] = 1, the sampling distribution of [bar.x], based on 100 observations, has a standard deviation equal to 0.1. The 95% confidence limits on the mean would therefore be {[bar.x] - 0.196, [bar.x] + 0.196}. This means the true mean is known within [+ or -]20%, a rather substantial uncertainty. While this is a hypothetical example, it does convey the important notion that even with 100 years of observed climate there can remain considerable uncertainty about the parameters of the hydroclimate probability model. This is particularly true if there is considerable natural variability and non-stationary climate influences.

For the probability model used in the case study, the calibration was performed using a two-step procedure:

(1) For each site the transformation parameter [lambda] is selected so that the L skew of the transformed values z is zero;

(2) The parameters y, E and A are estimated using a maximum likelihood approach.

The uncertainty in the parameters is approximated by considering the uncertainty in the parameters [theta] = {[mu], [SIGMA]} assuming A = 0. When A = 0, the probability model in Equation (2) reduces to an independent multi-normal distribution. The sampling or posterior distribution for this model is well known (DeGroot 1970). The disturbance covariance [SIGMA] is distributed as a Wishart distribution and the mean y is conditionally distributed as a multi-normal with covariance equal to [SIGMA]/[n.sub.h] where [n.sub.h] is the number of observed years of data. It is noted this approach underestimates the true uncertainty but is deemed reasonable as A is typically close to zero.

2.6. Assessing the impact of hydroclimate parameter uncertainty

The impact of hydroclimate parameter uncertainty is assessed in terms of its impact on the yield and on the risk of reaching a specified low-storage level for a given level of demand. The fundamental steps in evaluating the yield are summarised by Equations (3) and (4):

[Q.sup.NxM.sub.[theta]] [left arrow] p(q|[theta]) (3)

where [Q.sup.NxM.sub.[theta]] xMis the set of synthetic flows for N years and M replicates sampled from the streamflow probability model p(q|[theta]) conditioned on the hydroclimate parameters 9

[mathematical expression not reproducible] (4)

where Y(x, [theta]) is the yield for the given hydroclimate parameters [theta] and g(x, [Q.sup.NxM.sub.[theta]], D) is the vector of system performance metrics (such as frequency of restrictions and unacceptable storage) conditioned on the synthetic flows [Q.sup.NxM.sub.[theta]], decisions x (such as the drought contingency plan and operating rules) and annual demand D, and [g.sub.min] are the minimum acceptable performance metrics (e.g. restrictions no more than 1 in every 10 years).

The impact of hydroclimate parameter uncertainty is assessed against a base case in which yield and storage performance are determined using synthetic climate replicates that do not incorporate hydroclimate parameter uncertainty. In the base case, the hydroclimate parameters are taken to be the maximum likelihood (or most likely) values, denoted by [??]. Formally the maximum likelihood values are

[mathematical expression not reproducible] (5)

where [q.sup.his] is the historical data. Then 5000 113-year synthetic climate replicates are generated using the same climate parameters [theta] from which the traditional (or base case) yield Y (x, [theta]) is obtained.

Two different methods are used to assess the impact of hydroclimate parameter uncertainty on yield. Both of these methods require sampling hydroclimate parameters from the posterior or sampling distribution p([theta]|[q.sub.his]).

The first approach is similar to the approach described in Thyer et al. (2006b), whereby the only difference from the base case approach is that instead of using the most likely hydroclimate parameters for all replicates, a new set of hydroclimate parameters is sampled and used to generate each replicate. The yield is determined by

[mathematical expression not reproducible] (6)

where E(Y (x)) can be thought of as the expected yield in the sense that it is averaged over 5000 samples from the posterior distribution p([theta]|[q.sup.his]).

The second approach involves substantially more computational effort than the first approach. Under the second approach, a full yield assessment is undertaken for each of 500 sets of hydroclimate parameters. This approach effectively involves the calculation of 500 different storage risk curves, one for each set of hydroclimate parameters, and then the determination of 500 different possible realisations of yield.

[mathematical expression not reproducible] (7)

This approach has not been described previously in the literature.

The impact of hydroclimate parameter uncertainty is also investigated in terms of its impact on the assessed risk of reaching a specified low-storage level for a given demand scenario. This analysis involves the collection of storage risk data from 5000 replicates for a single demand scenario for each of 500 sets of climate parameters. The demand scenario used for this analysis is the system yield from the base case, which is around 81GL/ year.

Table 1 presents a summary of the approaches used by Stedinger and Taylor (1982) as well as the approaches used in this investigation to assess the impact of hydroclimate parameter uncertainty on the estimation of yield.

3. Results

3.1. Hydroclimate parameter distributions

The distributions of the hydroclimate parameters and derived statistics are presented in Figure 2 for the 11 data sites used in the case study. These plots are based on 500 samples.

Figure 2(a) presents scatter plots of the transformed mean [[mu].sub.i] and disturbance standard deviation [square root of ([[SIGMA].sub.ii])] for the 11 sites. A notable feature is the absence of correlation between the parameters.

A more interpretable result can be obtained by deriving the mean and standard deviation of annual values associated with each of the 500 parameters - this is accomplished by generating for each parameter a very long record and estimating the mean and standard deviation of the site annual values. Figure 2(b) presents scatter plots of the derived annual mean and standard deviation for each site. The correlation between the mean and standard deviation is strong for the streamflow sites and less so for the rainfall sites. This behaviour is due to the varying strength of the Box-Cox transformation to make the transformed values normally distributed.

Taking the average result from the eight streamflow sites, the 10th and 90th percentile samples of the mean are 2.2% below and 2.7% above the 50th percentile, respectively, and the 10th and 90th percentile samples of standard deviation are 9% below and 11% above the 50th percentile, respectively. In contrast, taking the average result from the three rainfall sites, the 10th and 90th percentile samples of mean are 0.4% below and 0.4% above the 50th percentile, respectively, and the 10th and 90th percentile samples of standard deviation are 8% below and 9% above the 50th percentile standard deviation, respectively.

3.2. Yield without hydroclimate parameter uncertainty (base case)

In the base case, the estimated system yield is 83GL/year for the first criterion (1 in 10-year risk of the storage level falling below the restriction trigger of 60%) and 81GL/ year for the second yield criterion (1 in 1000-year risk of the storage level falling below 20%). The storage risk curves and estimates of yield for the base case are shown in Figure 3.

3.3. Yield with hydroclimate parameter uncertainty

The results from the first approach to determining the impact of hydroclimate parameter uncertainty are presented in Figure 4. It is noted that the change in yield associated with sampling a different set of hydroclimate parameters for each replicate is barely perceptible for the 1- in 10-year risk threshold. The change in yield associated with the 1 in 1000-year risk threshold, however, is visible, with this yield dropping from approximately 81GL/year to 79GL/year.

For the second approach, a different estimate of yield is derived for each of the 500 sets of hydroclimate parameters. Summary results using this approach are presented in Figure 5. The dark lines in Figure 5 are the 1 in 10-year and 1 in 1000-year water storage risk curves calculated without considering parameter uncertainty (as in Figure 3), and the pale lines above and below the dark lines are the corresponding 10th percentile and 90th percentile water storage risk curves when parameter uncertainty is taken into account.

The derived yields are also shown as box and whisker plots, illustrating the spread in the estimated yield of the system at the nominated storage risk thresholds. The yield values shown in the box and whisker plots are the smallest estimate (of 500 samples), 10th, 25th, 50th (median), 75th and 90th percentile and largest estimate.

The distribution of yield is presented as a smoothed density plot in Figure 6. Both densities are negatively skewed with a longer left tail.

3.4. Impact of parameter uncertainty on storage risk

The impact of parameter uncertainty on the frequency that storage falls below 20% is assessed for a demand of 81GL/year, which is the yield for the base case against the second yield criterion. The risk of reaching 20% storage is assessed by taking the average result from 5000 replicates. By definition, therefore, the base case risk of reaching 20% storage is the same as the yield risk threshold, which is 1 in 1000.

The impact of hydroclimate parameter uncertainty is introduced by repeating the storage risk calculation using storage risk results from the 500 sets of climate parameters, which leads to 500 estimates of the storage risk being calculated. The distribution of the estimated risk of storage falling below 20% is presented in Figure 7 where it can be seen that the risk is positively skewed towards higher risks. The smallest and largest risks are calculated to be around 1 in 15,000 and 1 in 160, respectively. The 10th percentile and 90th percentile risks are 1 in 2400 and 1 in 430, respectively, and the 25th percentile and 75th percentile risks are 1 in 1600 and 1 in 700, respectively.

4. Discussion

Results from two methods for exposing the impact of hydroclimate parameter uncertainty on the determination of yield have been presented. The first is similar to the approach used in previous studies (Thyer et al. 2006a, 2006b), in which a new set of hydroclimate parameters is sampled before each replicate is generated within a single Monte Carlo assessment of storage behaviour.

The problem with this approach is that, like the base case situation which uses a single set of hydroclimate parameters, the results from all replicates are pooled before extracting the required system performance data. While some sets of hydroclimate parameters will tend to produce drier or wetter replicates than the base case, all results are mingled, and the net effect is that, while a slightly wider variability in climate is sampled, the true impact of particular sets of hydroclimate parameters remains hidden. It is shown for the case study that this approach has negligible impact on the estimated yield in terms of the first criterion, and a 2% reduction in yield in terms of the second criterion. While this approach correctly integrates out the parameter uncertainty to produce expected risks, it does not communicate the full impact of parameter uncertainty on system performance.

The alternative approach presented in this study allows the full impact of uncertainty in the hydroclimate parameters to be expressed in the results of the analysis. In the case study, a separate yield is determined for each sampled set of hydroclimate parameters, and a similar approach is used to explore the estimated risk of reaching a particular storage level.

It can be seen from the results in Figures 5 and 6 that the impact of parameter uncertainty has an appreciable impact on the yield, with this impact being somewhat greater for the lower frequency yield criterion than for the higher frequency yield criterion. For the 1 in 1000year storage risk threshold, the 10th and 90th percentile values of yield are 10% below and 7% above the most likely (or base case) yield, respectively. This means that there is a 10% chance that the true yield is less than 90% of the most likely yield and a 10% chance it is more than 7% greater than the most likely yield.

In terms of its impact on the risk of reaching a particular storage, it is found for the case study that there is a 10% chance that the true risk of reaching 20% storage is more than 2.4 times the most likely (or base case) risk, and a 10% chance that the true risk is less than 1/2.4 of the most likely risk.

The computational burden to determine the yield distributions is considerable. This study did not consider methods to reduce this burden. However, noting the near-symmetry of the yield distributions, the use of first-order methods to estimate the standard deviation of the yield is expected to produce results that are practically useful at a fraction of the computational cost.

5. Conclusion

It is natural for water resource planners to get caught up finessing their simulation models, their demand forecasts, and their analysis of climate change projections. Indeed we can become quite enthused about enhancements, such as adding more replicates, to make our storage risk projections look smoother and the yield estimates more accurate.

The sobering finding of this study, however, is that there are fundamental uncertainties that stem from the available streamflow and rainfall data that we simply cannot get around. In the case study, this uncertainty leads to error bands on the yield that are in the order of plus or minus 10% of the most likely yield, not even considering other model deficiencies. This uncertainty is likely to surpass, potentially by an order of magnitude or more, the influence of many choices made in both the modelling and operation of water resource systems. And when it comes to working out the risk of reaching a low-storage level, the risk could easily be more than double what we think.

So, what should we do? Ignoring this issue is not an option, particularly if decision-makers are risk averse and/or the consequences of under design are catastrophic. The challenge is to ensure that the uncertainty in our calculations is quantified and that it is communicated effectively so that sound decisions can be made, mindful of the uncertainty. This problem is simply another element of the technical communication challenge that surrounds water resource planning processes.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Brendan Berghout is a senior water resources engineer at Hunter Water Corporation. He holds a BE (Civil) and PhD from the University of Newcastle. Berghout started working with Hunter Water Corporation in 1987 and has extensive experience in various forms of headworks system modelling, drought planning and headworks system operation.

Benjamin J. Henley is a research fellow in hydrology and climate science at the School of Earth Sciences at the University of Melbourne. He is currently working on an ARC Linkage project investigating decadal climate variability and the likelihood and impacts of severe drought in Australia. At Hunter Water Corporation, he performed hydrological modelling and water resource system assessments for regional water planning.

George Kuczera is recognised as a world authority on the theory and application of Bayesian statistical methods in hydrology and water resources. His research addresses the fundamental problem in application of hydrology to water engineering, namely limited predictive ability arising from large errors in data and model errors arising from limited understanding of dynamics and complexity. His work has focused on developing methods that make the best use of limited information and quantifying uncertainty to inform the decision-making process.

https://doi.org/10.1080/13241583.2017.1404550

ARTICLE HISTORY

Received 9 August 2017

Accepted 2 November 2017

References

Berghout, B. 2009. "Incorporating Drought Management Planning into the Determination of Yield." Australian Journal of Water Resources, Australia 13: 103-112.

DeGroot, M. H. 1970. Optimal Statistical Decisions, 489. New York: Wiley.

Erlanger, P. D., B. Neal, and S. K. Merz. 2005. Framework for Urban Water Resource Planning. Melbourne: Water Services Association of Australia.

Kuczera, G. 1997. Wathnet: Generalised Water Supply Headworks Simulation using Network Linear Programming. Australia: Department of Civil, Surveying and

Environmental Engineering, University of Newcastle. Matalas, N. C. 1967. "Mathematical Assessment of Synthetic Hydrology." Water Resources Research 3: 937-945.

Stedinger, J. R., and M. R. Taylor. 1982. "Synthetic Streamflow Generation: 2. Effect of Parameter Uncertainty." Water Resources Research 18: 919-924.

Svanidze, G. G. 1960. Mathematical Modelling of Hydrologic Series (translated from Russian). Fort Collins: Water Resources Publications.

Thyer, M., A. Frost, and G. Kuczera. 2006a. "Parameter Estimation and Model Identification for Stochastic Models of Annual Hydrological Data: Is the Observed Record Long Enough?" Journal of Hydrology 330: 313-328.

Thyer, M., A. Frost, G. Kuczera, and R. Srikanthan. 2006b. "Stochastic Modelling of (Not-so) Long-term Hydrological Data: Current Status and Future Research." 30th Hydrology & Water Resources Symposium: Past, Present & Future, Launceston, 321-326.

Brendan Berghout (a), Benjamin J. Henley (a,b) and George Kuczer (a,c)

(a) Water Planning, Hunter Water Corporation, Newcastle, Australia; (b) School of Earth Sciences, University of Melbourne, Parkville, Australia; (c) School of Engineering (Environmental Engineering), University of Newcastle, Callaghan, Australia

CONTACT Brendan Berghout [mail] brendan.berghout@hunterwater.com.au

Caption: Figure 1. Required reservoir storage capacity from Stedinger and Taylor (1982) with and without parameter uncertainty. Adapted from Figure 1 in Stedinger and Taylor (1982).

Caption: Figure 2. Distributions of the hydroclimate parameters an d derived statistics for streamflow and rainfall at case study sites (a) transformed mean [[mu].sub.i]. and disturbance standard deviation [square root of ([[SIGMA].sub.ii])], and (b) generated stochastic data.

Caption: Figure 3. Diagram showing the relationship between storage risks and demand for the base case.

Caption: Figure 4. Impact of sampling a different hydroclimate parameter set for each replicate on the estimation of yield.

Caption: Figure 5. Impact of hydroclimate parameter uncertainty on the estimation of yield. The pale lines show the 10th and 90th percentile estimates of storage risks. The box and whisker plots show the impact of parameter uncertainty on estimated yield for the two example yield thresholds.

Caption: Figure 6. Distribution of estimated yields resulting from uncertainty in the hydroclimate parameters.

Caption: Figure 7. Distribution of the calculated risk of storage falling below 20% when demand is 81GL/year.
Table 1. Summary of past and present hydroclimate
parameter uncertainty experiments.

                   Stedinger and Taylor (1982)

                     Without          With          Without
                    parameter      parameter       parameter
                   uncertainty    uncertainty     uncertainty

Number of sites    1             1                11
Length of          50            50               113
  simulation
  (years)
Number of          1             1                5000
  replicates
  per yield
  calculation
Number of yield/   500           500              1
  reservoir
  capacity
  calculations
Number of sets     1             500 (1 per       1
  of climate                       capacity
  parameters                       calculation)
Number of          1             1                12
  demand
  scenarios

                   This investigation

                    Parameter       Parameter
                   uncertainty:    uncertainty:
                   1st approach    2nd approach

Number of sites    11             11
Length of          113            113
  simulation
  (years)
Number of          5000           5000
  replicates
  per yield
  calculation
Number of yield/   1              500
  reservoir
  capacity
  calculations
Number of sets     5000 (1 per    500 (1 per
  of climate         replicate)     yield
  parameters                        calculation)
Number of          12             12
  demand
  scenarios
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Author:Berghout, Brendan; Henley, Benjamin J.; Kuczer, George
Publication:Australian Journal of Water Resources
Article Type:Report
Date:Oct 1, 2017
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