Application of hybrid multi-site stochastic model in South
Africa for water resource optimisation
ALBERT JELENI
Report submitted in partial fulfilment of the requirements for the degree
MASTER OF SCIENCE (INDUSTRIAL SYSTEMS)
In the
FACULTY OF ENGINEERING, BUILT ENVIRONMENT, AND INFORMATION
TECHNOLOGY
UNIVERSITY OF PRETORIA
October 2007
i
ACKNOWLEDGEMENTS
To my supervisor Professor Yadavalli, thank you for your guidance and patience to
making this project a success, and much appreciation to Mr Pieter Van Rooyen for
his inputs.
I would also like to express my gratitude to my wife, Faith and my daughter, Tinyiko
and the rest of my family and friends for their unconditional support and
understanding.
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Application of hybrid multi-site stochastic model in South Africa for water resource optimisation
Albert Jeleni Supervisor: Prof. V.S.S. Yadavalli
Department of Industrial and Systems Engineering
Master of Science (Industrial Systems)
EXECUTIVE SUMMARY
The long-term water resource management in South Africa is established on the
basis of the so-called Probabilistic Management by accounting for the hydrologic
uncertainty using stochastic simulation. The model currently in use is a Monthly
Multi-Site Stochastic Streamflow referred to as STOMSA (Stochastic Model of South
Africa), and is effectively based on widely used Periodic Parametric Models. In the
context of stochastic modelling of streamflows, a major limitation of the periodic
parametric models is their inability to simultaneously reproduce summary statistics
and dependence structure at different temporal levels. To circumvent this, linear
disaggregation models were developed. However, these models are not
parsimonious, and in addition they require empirical adjustments in order to restore
summability of the disaggregated flows to the aggregate flows, in the event of
normalizing transformations being applied. For this purpose, a multivariate
streamflow generation model called the multivariate contemporaneous PAR(1)NT-
hybrid model was proposed and applied to a multisite monthly streamflow
generation problem for the Vaal, Bloemhof, Delangesdrift, Welbedacht, and Katse
catchments. The proposed model was then compared with a multivariate STOMSA
model. This study showed that the proposed model reproduces the mean, variance,
and standard deviation comparative with the STOMSA and the historical data.
Further, the proposed model reproduces cross-correlations between the last month
of the previous year and the first month of the current year well. The study also
developed a conceptual model for the inclusion of this proposed model into the
South African water industry. The rational for the conceptual model is to ensure that
if a new model is to be introduced or the current models are to be improved on,
current knowledge should not be lost. Keywords: stochastic hydrology, nonparametric models, periodic parametric models, water resource
management, STOMSA.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS............................................................................................. I
EXECUTIVE SUMMARY ............................................................................................. II
TABLE OF CONTENTS.............................................................................................. III
LIST OF ABBREVIATIONS.........................................................................................V
LIST OF TABLES .......................................................................................................VI
LIST OF FIGURES.....................................................................................................VII
CHAPTER 1 .....................................................................................................1
INTRODUCTION..............................................................................................1
1.1 Background...................................................................................1 1.2 Research Statement .....................................................................3 1.3 Literature Review..........................................................................4
1.3.1 Theoretical background .................................................4 1.3.3.1 Stochastic processes and time series ..................4 1.3.3.2 Synthetic streamflow generation...........................6 1.3.3.3 Stochastic simulation ...........................................14
1.3.2 History of stochastic simulation of streamflow.........16 1.3.3 South African situation ................................................21
CHAPTER 2 ...................................................................................................30
METHODOLOGY...........................................................................................30
2.1 Application of Hybrid model.......................................................30 2.2 Conceptual model for implementation......................................32
CHAPTER 3 ...................................................................................................33
APPLICATION OF THE HYBRID MODEL....................................................33
3.1 Analysis.......................................................................................33 3.2 Conceptual model for implementation .....................................36
3.2.1 Development guidelines and framework....................36 3.2.1.1 Pointers from the current modelling environment.
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36 3.2.1.2 Conceptual Model .................................................38 3.2.1.3 Stochastic streamflow generation process in
STOMSA.................................................................41 3.2.1.4 Stochastic streamflow generation process in the
Hybrid Model .........................................................42 3.2.1.5 Proposed generation process incorporating both
models....................................................................43
CHAPTER 4 ...................................................................................................44
CONCLUSION ...............................................................................................44
REFERENCES........................................................................................................... 46
APPENDIX A: MODEL INPUT DATA ....................................................................... 49
APPENDIX B: MODEL’S RESULTS COMPARISON............................................... 54
APPENDIX C: MATLAB CODE................................................................................. 76
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LIST OF ABBREVIATIONS
STOMSA Stochastic model of South Africa
LP Linear parametric
NPD Nonparametric disaggregation
DDM Dynamic disaggregation model
k-NN k-nearest neighbour
NP Nonparametric
ISM Index sequential method
MBB Moving block bootstrap
MABB Matched block bootstrap
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LIST OF TABLES
Table 3.1 : Runoff characteristics for the selected sub-catchments.........................33 Table 3.2: Bloemhof Catchment Results comparison ..............................................35
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LIST OF FIGURES
Figure 1: Diagrammatic representation of the conceptual model.............................40 Figure 2: Illustration of influence of coefficient of variation on firm yield (Basson et al)
.................................................................................................................................40 Figure B-3: Comparison of the mean for the Bloemhof Catchment .........................54 Figure B-4: Comparison of Coefficient of Variance for the Bloemhof Catchment ....54 Figure B-5: Comparison of the Standard Deviations of the Bloemhof Catchment ...55 Figure B-6: Serial correlation of month 1 and 12 from the Hybrid model for the
Bloemhof Catchment................................................................................................55 Figure B-7: Comparison of the Variances for the Bloemhof Catchment ..................56 Figure B-8: Comparison of the mean flows for the Delangesdrift Catchment ..........57 Figure B-9: Comparison of the standard deviations for the Delangesdrift Catchment
.................................................................................................................................57 Figure B-10: Comparison of the Variances for the Delangesdrift Catchment ..........58 Figure B-11: Comparison of the coefficients of Variations for the Delangesdrift
Catchment ................................................................................................................58 Figure B-12: Box plot of the serial correlation between month 1 current year and 12
previous year for the Delangesdrift Catchment ........................................................59 Figure B-13: Comparison of the Mean flows for the Katse Catchment ....................59 Figure B-14: Comparison of the Standard Deviations for the Katse Catchment ......60 Figure B-15: Comparison of the Coefficients of Variations for the Katse Catchment
.................................................................................................................................60 Figure B-16: Boxplot of the Serial correlation between moth 1 of the current year
and month 12 of the previous year for the Katse Catchment ...................................61 Figure B-17: Comparison of the Mean flows for the Vaal Catchment ......................62 Figure B-18: Comparison of the Standard Deviations for the Vaal Catchment ........62 Figure B-19: Comparison of the Coefficients of Variations for the Vaal Catchment.63 Figure B-20: Boxplot of the Serial Correlation between month 1 of current year and
month 12 of the previous year for the Vaal Catchment ............................................63 Figure B-21: Comparison of the Mean flows for the Welbedacht Catchment ..........64 Figure B-22: Comparison of the Standard Deviations of the Welbedacht Catchment
.................................................................................................................................64 Figure B-23: Comparison of the Coefficients of Variations for the Welbedacht
Catchment ................................................................................................................65
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Figure B-24: Boxplot of Serial correlations between month 1 of current year and
month 12 of previous year for the Welbedacht catchment .......................................65 Figure B-25: Bloemhof Catchment's Boxplots of Mean Streamflows.......................66 Figure B-26: Bloemhof Boxplot of Mean streamflows from STOMSA......................66 Figure B-27: Bloemhof Boxplot of Standard Deviations from the Hybrid Model.......67 Figure B-28: Bloemhof Boxplot of Standard Deviations from STOMSA...................67 Figure B-29 : Delangesdrift Catchment's Boxplots of Mean Streamflows................68 Figure B-30: Delangesdrift Catchment Boxplot of Mean streamflows from STOMSA
.................................................................................................................................68 Figure B-31 : Delangesdrift Catchment’s Boxplot of Standard Deviations from the
Hybrid Model ............................................................................................................69 Figure B-32: Delangesdrift Catchment Boxplot of Standard Deviations from
STOMSA ..................................................................................................................69 Figure B-33: Katse Catchment's Boxplots of Mean Streamflows.............................70 Figure B-34: Katse Catchment Boxplot of Mean streamflows from STOMSA .........70 Figure B-35: Katse Catchment’s Boxplot of Standard Deviations from the Hybrid
Model........................................................................................................................71 Figure B-36: Katse Catchment Boxplot of Standard Deviations from STOMSA ......71 Figure B-37: Vaal Catchment's Boxplots of Mean Streamflows...............................72 Figure B-38: Vaal Catchment Boxplot of Mean streamflows from STOMSA ...........72 Figure B-39: Vaal Catchment’s Boxplot of Standard Deviations from the Hybrid
Model........................................................................................................................73 Figure B-40: Vaal Catchment Boxplot of Standard Deviations from STOMSA ........73 Figure B-41: Welbe Catchment's Boxplots of Mean Streamflows............................74 Figure B-42: Welbe Catchment Boxplot of Mean streamflows from STOMSA ........74 Figure B-43: Welbe Catchment’s Boxplot of Standard Deviations from the Hybrid
Model........................................................................................................................75 Figure B-44: Welbe Catchment Boxplot of Standard Deviations from STOMSA .....75
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CHAPTER 1
INTRODUCTION
1.1 Background
In hydrology, stochastic models are widely used for simulation of streamflows and
other hydro-climatic variables. They have been proven useful for various problems
related to planning and management of water resources systems. Typical examples
include determining the storage capacity of reservoirs, assessment of risk and
reliability of water resources system operation under various potential hydrologic
scenarios and the analysis of critical droughts. In general, the need for stochastic
hydrology originated from the requirement to estimate the assurance of supply, at
say a recurrence interval of a failure of 1:200 years, when the available recorded
streamflow data rarely exceeds 40 years and that, through rainfall-runoff simulation,
a limited period length, in some cases a maximum of only 80 years, can be derived.
Stochastic hydrology provides the capability to synthetically increase the available
data in order to evaluate the behaviour of water resource systems using alternative,
but statistically plausible, streamflow conditions. This gives the opportunity to
assess the probability of occurrence of critical periods that can be as long as nine
years, which is difficult given the relative short historical time series.
In South Africa, Stochastic hydrology is a standard technique that has been applied
to determine the reliability of supply of water resource systems by the Department of
Water Affairs and Forestry since the early nineteen eighties. The model currently in
use is a Monthly Multi-Site Stochastic Streamflow referred to as STOMSA
(Stochastic Model of South Africa), and is effectively based on widely used Periodic
Parametric Models. Since the first version of the stochastic streamflow model was
developed, a number of refinements were introduced to incorporate, amongst
others, the following:
• “warm up” procedures to ensure that generated flows are independent
of the seed values obtained from random number generators;
• Extension of the serial correlation modelling feature to allow for larger
dimensions of the Auto Regressive Moving Average Model. Currently
up to nine possible time-series algorithms are available;
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• Improved modelling of streamflow sequences that incorporate zero
annual flows;
• Incorporation of basic yield-capacity relationship characteristics to
improve criteria for the selection of appropriate time-series algorithms.
In the context of stochastic modelling of streamflows, a major limitation of the
periodic parametric models is their inability to simultaneously reproduce summary
statistics and dependence structure at different temporal levels. To circumvent this,
linear disaggregation models were developed. However, these models are not
parsimonious, and in addition they require empirical adjustments in order to restore
summability of the disaggregated flows to the aggregate flows, in the event of
normalizing transformations being applied.
The increasing awareness of the need to model nonlinearity and non-stationarity
has spurred the growth of nonparametric methods in several areas of hydrology.
This has gained from the development and use of nonparametric methods in more
general time series analysis and has resulted in a development of a nonparametric
disaggregation (NPD) model. This model is data-driven and relatively automatic,
consequently is able to model the nonlinearity inherent in the dependence structure
of observed flows reasonably well as well as to provide a good amount of smoothing
in synthetic simulations.
While the conventional parametric models require assumptions regarding the
marginal distribution of flows and the order of dependence, the nonparametric
methods are, in general, data-driven and can capture the linear and nonlinear
dependence of observed flows without any prior assumptions. While parametric
models provide considerable smoothing and extrapolation in the simulations,
nonparametric bootstrap methods such as the moving block bootstrap and k-
nearest-neighbor bootstrap cannot. They simply mimic the marginal distribution of
observed flows, because flow values are resampled from the historic data. Such
parsing of the data defeats the purpose of synthetic streamflow simulation.
Considering the relative merits and demerits of both simple low-order linear periodic
parametric models and the nonparametric bootstrap methods, Srinivas and
Srinivasan (2001) introduced simulations from a novel method that blends the merits
of both parametric and nonparametric methods. In Srinivas and Srinivasan (2005),
this technique is further improved by using the matched-block bootstrap in lieu of
moving block bootstrap. In both these hybrid methods, periodic streamflows are
3
partially pre-whitened using a parsimonious linear periodic
autoregressive/autoregressive moving average model and residuals are extracted.
The non-overlapping within-year blocks formed from the residuals are conditionally
resampled using the block/matched-block bootstrap to obtain innovations, which are
then post-blackened to generate synthetic replicates.
They reported that their hybrid model can preserve basic statistics and the
correlation structure of the historical data. The major advantage of the
nonparametric approach in hydrologic time series modeling is that the historical data
do not need to be transformed to satisfy the assumption of normality. Furthermore,
the hybrid time series model can preserve various correlation structures of the
original data by the proper selection of the length of the moving blocks even though
the data have a complex dependence structure.
1.2 Research Statement
The stochastic stream-flow generation techniques contained in STOMSA is
effectively based on parametric models with an annual time step and monthly
disaggregation features, consequently its inability to model the flows in the first
month of a year so as to follow from the flows in the last month of the previous year.
Van Rooyen & Mckenzie (2004), recommended that although the model is
appropriate for a wide variety of hydrological conditions experienced in South Africa,
careful consideration should be given in cases where the critical period of the water
resource system is less than a year, and in such cases it may be found that a
stochastic model based on monthly flows rather than annual flows is required.
Therefore, the objective of this project is to develop and test a Monthly Multi-Site
Stochastic Streamflow model based on the hybrid model of Srinivas & Srinivasan
(2001, 2005) by:
• extending the hybrid model to a multi-site streamflow generation model
• testing the model on one of the South African rivers/ water system,
comparing the results to those of STOMSA where appropriate,
and to develop a conceptual model for the implementation of the methodology in
South Africa, taking cognisance of the current models and the abundant knowledge
available based on the current models.
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1.3 Literature Review
1.3.1 Theoretical background
1.3.3.1 Stochastic processes and time series
Historical records of rainfall or streamflow at a particular site are a sequence of
observations called a time series. In a time series, the observations are ordered by
time, and it is generally the case that the observed value of the random variable at
one time influences one’s assessment of the distribution of the random variable at
later times. This means that the observations are not independent. Time series are
conceptualised as being a single observation of a stochastic process, which is a
generalisation of the concept of a random variable.
Describing stochastic processes
A random variable whose value changes through time according to probabilistic
laws is called a stochastic process. An observed time series is considered to be one
realisation of a stochastic process, just as a single observation of a random variable
is one possible value the random variable may assume. In the development here, a
stochastic process is a sequence of random variables {X(t)} ordered by a discrete
time variable t = 1, 2, 3, . . . , n. The properties of a stochastic process must
generally be determined from a single time series or realisation. To do this, several
assumptions are usually made. First, one generally assumes that the process is
stationary. This means that the probability distribution of the process is not changing
over time. In addition, if a process is strictly stationary, the joint distribution of the
random variables X(t1), . . . , X(tn) is identical to the joint distribution of X(t1 + t), . . . ,
X(tn + t) for any t; the joint distribution depends only on the differences ti − tj between
the times of occurrence of the events at time ti and tj.
For a stationary stochastic process, one can write the mean and variance as
(1.1) )]([ tXEX =µ
and
(1.2) )]([2 tXVarX =σ
respectively.
Both are independent of time t. The autocorrelations, the correlation of X with itself,
5
are given by
(1.3) )](),([)( 2X
XktXtXCovk
σρ +
=
for any positive integer time lag k. These are the statistics most often used to
describe stationary stochastic processes. When one has available only a single time
series, it is necessary to estimate the values of Xµ , 2Xσ and )(kXρ from values of
the random variable that one has observed. The mean and variance are generally
estimated essentially as follows:
(1.4) )(1ˆ1∑=
==T
tX tX
TXµ
and
(1.5) ])([1ˆ1
22X ∑
=
−=T
tXtX
Tσ
respectively, while the autocorrelations )(kXρ for any time lag k can be estimated
as:
(1.6) . ))((
))()()(()(ˆ
1
2
1
∑
∑
=
−
=
−
−−+== T
t
kT
tkX
XtX
XtXXktXrkρ
The sampling distribution of these estimators depends on the correlation structure of
the stochastic process giving rise to the time series. In particular, when the
observations are positively correlated as is usually the case in natural streamflows
or annual benefits in a river basin simulation, the variances of the estimated x and 2ˆ Xσ are larger than would be the case if the observations were independent. It is
sometimes wise to take this inflation into account. All of this analysis depends on the
assumption of stationarity; only then the quantities defined in Equations (1.1) to (1.3)
have the intended meaning. Stochastic processes are not always stationary. Urban
development, deforestation, agricultural development, climatic variability, and
changes in regional resource management can alter the distribution of rainfall,
streamflows, pollutant concentrations, sediment loads and groundwater levels over
time. If a stochastic process is not essentially stationary over the time span in
question, then statistical techniques that rely on the stationary assumption do not
apply and the problem generally becomes much more difficult.
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1.3.3.2 Synthetic streamflow generation
This section is concerned primarily with ways of generating sample streamflows
data in water resource systems simulation studies. Generated streamflows have
been called synthetic to distinguish them from historical observations. The activity
has been called stochastic hydrologic modelling. More detailed presentations can be
found in Marco et al. (1989) and Salas (1993).
River basin simulation studies can use many sets of streamflow, rainfall,
evaporation, and/or temperature sequences to evaluate the statistical properties of
the performance of alternative water resources systems. For this purpose, synthetic
flows and other generated quantities should resemble, statistically, those sequences
that are likely to be experienced during the planning period.
Use of only the historical flow or rainfall record in water resource studies does not
allow for the testing of alternative designs and policies against the range of
sequences that are likely to occur in the future. We can be very confident that the
future historical sequence of flows will not be the historical one, yet there is
important information in that historical record. That information is not fully used if
only the historical sequence is simulated. By fitting continuous distributions to the
set of historical flows and then by using those distributions to generate other
sequences of flows, all of which are statistically similar and equally likely, gives one
a broader range of inputs to simulation models. Testing designs and policies against
that broader range of flow sequences that could occur more, clearly identifies the
variability and range of possible future performance indicator values. This in turn
should lead to the selection of more robust system designs and policies.
The use of synthetic streamflows is particularly useful for water resource systems
having large amounts of over-year storage. Use of only the historical hydrologic
record in system simulation yields only one time history of how the system would
operate from year to year. In water resource systems having relatively little storage
so that reservoirs and/or groundwater aquifers refill almost every year, synthetic
hydrologic sequences may not be needed if historical sequences of a reasonable
length are available. In this second case, a 25-year historic record provides 25
descriptions of the possible within-year operation of the system. This may be
sufficient for many studies.
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Generally, use of stochastic sequences is thought to improve the precision with
which water resource system performance indices can be estimated, and some
studies have shown the evidence this in (Vogel and Shallcross, 1996; Vogel and
Stedinger, 1988). In particular, if system operation performance indices have
thresholds and shape breaks, then the coarse descriptions provided by historical
series are likely to provide relative inaccurate estimates of the expected values of
such statistics.
On the other hand, if one is only interested in the mean flow, or average benefits
that are mostly a linear function of flows, then use of stochastic sequences will
probably add little information to what is obtained simply by simulating the historical
record. After all, the fitted models are ultimately based on the information provided in
the historical record, and their use does not produce new information about the
hydrology of the basin. If in a general sense one has available n years of record, the
statistics of that record can be used to build a stochastic model for generating
thousands of years of flow. These synthetic data can then be used to estimate more
accurately the performance of the system, assuming, of course, that the flow-
generating model accurately represents nature. But the initial uncertainty in the
model parameters resulting from having only n years of record would still remain
(Schaake and Vicens, 1980).
An alternative is to run the historical record (if it is sufficient complete at every site
and contains no gaps of missing data) through the simulation model to generate n
years of output. That output series can be processed to produce estimates of
system performance. So the question is: is it better to generate multiple input series
based on uncertain parameter values and use those to determine average system
performance with great precision, or is it sufficient to just model the n-year output
series that results from simulation of the historical series? The answer seems to
depend upon how well behaved the input and output series are. If the simulation
model is linear, it does not make much difference. If the simulation model were
highly nonlinear, then modelling the input series would appear to be advisable. Or if
one is developing reservoir operating policies, there is a tendency to make a policy
sufficiently complex that it deals very well with the few droughts in the historical
record giving a false sense of security and likely misrepresenting the probability of
system performance failures.
8
Another situation where stochastic data generating models are useful is when one
wants to understand the impact, on system performance estimates, of the parameter
uncertainty stemming from short historical records. In that case, parameter
uncertainty can be incorporated into streamflow generating models, so that the
generated sequences reflect both the variability that one would expect in flows over
time as well as the uncertainty of the parameter values of the models that describe
that variability (Valdes et al., 1977; Stedinger and Taylor, 1982a,b; Stedinger, Pei
and Cohn, 1985; Vogel and Stedinger, 1988).
If one decides to use a stochastic data generator, the challenge is to use a model
that appropriately describes the important relationships, but does not attempt to
reproduce more relationships than are justified or that can be estimated with
available data sets.
Two basic techniques are used for streamflow generation. If the streamflow
population can be described by a stationary stochastic process, a process whose
parameters do not change over time, and if a long historical streamflow record
exists, then a stationary stochastic streamflow model may be fitted to the historical
flows. This statistical model can then generate synthetic sequences that describe
selected characteristics of the historical flows. However, the assumption of
stationarity is not always plausible, particularly in river basins that have experienced
marked changes in runoff characteristics due to changes in land cover, land use,
climate, or the utilization of groundwater during the period of flow record.
Similarly, if the physical characteristics of a basin will change substantially in the
future, the historical streamflow record may not provide reliable estimates of the
distribution of future unregulated flows. In the absence of the stationarity of
streamflows or a representative historical record, an alternative scheme is to
assume that precipitation is a stationary stochastic process and to route either
historical or synthetic precipitation sequences through an appropriate rainfall-runoff
model of the river basin.
Streamflow generation models
The first step in the construction of a statistical streamflow generating model is to
extract from the historical streamflow record the fundamental information about the
joint distribution of flows at different sites and at different times. A streamflow model
should ideally capture what is judged to be the fundamental characteristics of the
9
joint distribution of the flows. The specification of what characteristics are
fundamental is of primary importance. One may want to model as closely as
possible the true marginal distribution of seasonal flows and/or the marginal
distribution of annual flows. These describe both how much water may be available
at different times and also how variable is that water supply. Also, modelling the joint
distribution of flows at a single site in different months, seasons, and years may be
appropriate. The persistence of high flows and of low flows, often described by their
correlation, affects the reliability with which a reservoir of a given size can provide a
given yield (Fiering, 1967; Lettenmaier and Burges, 1977a,b; Thyer and Kuczera,
2000). For multi-component reservoir systems, reproduction of the joint distribution
of flows at different sites and at different times will also be important.
Sometimes, a streamflow model is said to resemble statistically the historical flows if
the streamflow model produces flows with the same mean, variance, skew
coefficient, autocorrelations, and/or cross correlations as were observed in the
historic series. This definition of statistical resemblance is attractive because it is
operational and requires that an analyst need only find a model that can reproduce
the observed statistics. The drawback of this approach is that it shifts the modelling
emphasis away from trying to find a good model of marginal distributions of the
observed flows and their joint distribution over time and over space, given the
available data, to just reproducing arbitrarily selected statistics. Defining statistical
resemblance in terms of moments may also be faulted for specifying that the
parameters of the fitted model should be determined using the observed sample
moments, or their unbiased counterparts.
Other parameter estimation techniques, such as maximum likelihood estimators, are
often more efficient. Definition of resemblance in terms of moments can also lead to
confusion over whether the population parameters should equal the sample
moments, or whether the fitted model should generate flow sequences whose
sample moments equal the historical values. The two concepts are different
because of the biases in many of the estimators of variances and correlations
(Matalas and Wallis, 1976; Stedinger, 1980, 1981; Stedinger and Taylor, 1982a).
For any particular river basin study, one must determine what streamflow
characteristics need to be modelled. The decision should depend on what
characteristics are important to the operation of the system being studied, the
available data, and how much time can be spared to build and test a stochastic
10
model. If time permits, it is good practice to see if the simulation results are in fact
sensitive to the generation model and its parameter values by using an alternative
model and set of parameter values. If the model’s results are sensitive to changes,
then, as always, one must exercise judgment in selecting the appropriate model and
parameter values to use.
Reproducing the marginal distribution
Most models for generating stochastic processes deal directly with normally
distributed random variables. Unfortunately, flows are not always adequately
described by the normal distribution. In fact, streamflows and many other hydrologic
data cannot really be normally distributed because of the impossibility of negative
values. In general, distributions of hydrologic data are positively skewed having a
lower bound near zero and, for practical purposes, an unbounded right-hand tail.
Thus they look like the gamma or lognormal distribution.
The asymmetry of a distribution is often measured by its coefficient of skewness. In
some streamflow models, the skew of the random elements yV is adjusted so that
the models generate flows with the desired mean, variance, and skew coefficient.
Multivariate models
If long concurrent streamflow records can be constructed at the several sites at
which synthetic streamflows are desired, then ideally a general multi-site streamflow
model could be employed. O.Connell (1977), Ledolter (1978), Salas et al. (1980)
and Salas (1993) discuss multivariate models and parameter estimation.
Unfortunately, identification of most appropriate model structure is very difficult for
general multivariate models.
For example, the multi-site generalisation of the annual AR(1) or autoregressive
Markov model following the approach taken by Matalas and Wallis (1976), can be
further extended to generate multi-site/multi-season modelling procedures, by, for
example, employing what have been called disaggregation models or using the
hybrid method.
Multi-season, multi-site models
In most studies of surface water systems it is necessary to consider the variations of
flows within each year. Streamflows in most areas have within-year variations,
11
exhibiting wet and dry periods. Similarly, water demands for irrigation, municipal,
and industrial uses also vary, and the variations in demand are generally out of
phase with the variation in within-year flows; more water is usually desired when
streamflows are low and less is desired when flows are high. This increases the
stress on water delivery systems and makes it all the more important that time
series models of streamflows, precipitation and other hydrological variables correctly
reproduce the seasonality of hydrological processes. This section discusses two
approaches to generating within-year flows. The first approach is based on the
disaggregation of annual flows produced by an annual flow generator to seasonal
flows. Thus the method allows for reproduction of both the annual and seasonal
characteristics of streamflow series. The second approach generates seasonal flows
in a sequential manner using the combination of Parametric and NP method
(Hybrid).
Disaggregation Model
The disaggregation model proposed by Valencia and Schaake (1973) and extended
by Mejia and Rousselle (1976) and Tao and Delleur (1976) allows for the generation
of synthetic flows that reproduce statistics both at the annual level and at the
seasonal level. Subsequent improvements and variations are described by
Stedinger and Vogel (1984), Maheepala and Perera (1996), Koutsoyiannis and
Manetas (1996) and Tarboton et al. (1998).
Disaggregation models can be used for either multi-season single-site or multisite
streamflow generation. They represent a very flexible modelling framework for
dealing with different time or spatial scales. Annual flows for the several sites in
question or the aggregate total annual flow at several sites can be the input to the
model (Grygier and Stedinger, 1988). These must be generated by another model,
such as those discussed in the previous sections. These annual flows or aggregated
annual flows are then disaggregated to seasonal values.
Let TN
yyy ZZ ),...,( 1=Z
be the column vector of N transformed normally distributed annual or aggregate
annual flows for N separate sites or basins. Next, let
TnTy
nyTyyTyyy XXXXXX ),...,,...,,...,,,...,( 1
221
111=X
12
be the column vector of nT transformed normally distributed seasonal flows styX for
season t, year y, and site s = 1, ..., n. Assuming that the annual and seasonal
series, sty
sy XZ and , have zero mean (after the appropriate transformation), the basic
disaggregation model is
(1.7) , yyy BVAZX +=
where Vy is a vector of nT independent standard normal random variables, and A and B are, respectively, nT x N and nT x nT matrices. One selects values of the
elements of A and B to reproduce the observed correlations among the elements of
Xy and between the elements of Xy and Zy. Alternatively, one could attempt to
reproduce the observed correlations of the untransformed flows as opposed to the
transformed flows, although this is not always possible (Hoshi et al., 1978) and often
produces poorer estimates of the actual correlations of the flows (Stedinger, 1981).
When flows at many sites or in many seasons are required, the size of the
disaggregation model can be reduced by disaggregation of the flows in stages. Such
condensed models do not explicitly reproduce every season-to-season correlation
(Lane, 1979; Stedinger and Vogel, 1984; Gryier and Stedinger, 1988; Koutsoyiannis
and Manetas, 1996). Nor do they attempt to reproduce the cross correlations among
all the flow variates at the same site within a year (Lane, 1979; Stedinger et
al.,1985). Contemporaneous models, like the Markov model, are models developed
for individual sites whose innovation vectors Vy have the needed cross-correlations
to reproduce the cross-correlations of the concurrent flows (Camacho et al., 1985).
Grygier and Stedinger (1991) describe how this can be done for a condensed
disaggregation model without generating inconsistencies.
Hybrid Model (HM)
This section presents the algorithm for generating synthetic seasonal streamflows
by the hybrid model proposed by Srinivas and Srinivasan (2001), which uses the
postblackening approach suggested by Davison and Hinkley (1987).
Let the observed (historical) streamflows be represented by the vector τν ,Q , where
ν is the index for year (ν =1,…, N) and τ denotes the index for season (period)
within the year (τ = 1,…, ω ); N refers to the number of years of historical record,
and ω represents the number of periods within the year. The modelling steps
13
involved are as follows:
1. Standardize the elements of the vector τν ,Q as
τ
ττντν s
qqy
−= ,
, , (1.8)
where τq and τs are the mean and standard deviation, respectively, of the
observed streamflows in period τ . Note that the historical streamflows are
not transformed to remove skewness.
2. Pre-whiten the standardized historical streamflows, τν ,Y using a simple
periodic autoregressive model of order one (PAR(1)) and extract the
residuals τνε , . Take 01,0 =y :
(1.9) , 1,,1,, −−= τνττντν φε yy
where τφφ ,11,1 ,..., , are the periodic autoregressive parameters of order one. It
is to be noted that the residuals τνε , may possess some weak dependence
(since the parameters are estimated from a simple PAR(1) model). Srinivas
and Srinivasan (2001) mentioned that bootstrap schemes like the moving
block bootstrap (MBB) (Künsch, 1989) can serve as reliable tools for
modelling the weak linear dependence, if any, in the residuals.
3. Obtain the simulated innovations *,τνε by bootstrapping τνε , using the moving
block bootstrap (MBB) method. The monthly residuals resulting from the
PAR(1) model are divided into (possibly) overlapping blocks Bi with block
size L taken as an integral multiple of the number of periods (ω ) within the
year. It is to be noted that each of the overlapping blocks starts with the first
period in a hydrological water year. This is done with a view to capturing the
within-year correlations for a significant number of lags. For example, the
block sizes of residuals in monthly streamflow modelling context would be
12, 24, 36, and so on (abbreviated as ωωω 3 ,2 , === LLL , and so on).
Note that when the block length L is n years long, the overlap is (n - 1) years,
so that when it is 1 year long there is no overlap. In general, the ith block
with size ωmL = , may be written as
(1.10) , ),...,( ,11, ωεε −+= miiiB
where i = 1,…,q and q = N – m + 1. For example, if 12 and 3 == ωωL , the
fourth block is written as ),...,( 12,61,44 εε=B . The block size L, to be selected
14
for resampling the residuals, would primarily depend on the amount of
unextracted weak dependence present in the residuals. Bootstrapped
innovations *,τνε are generated by resampling the overlapping blocks Bi at
random, with replacement from the set ),...,( 1 qBB and pasting them end-to-
end. It is to be noted that each of the (possibly) overlapping blocks has equal
probability (1/q) of being resampled.
4. The bootstrapped innovation series *,τνε is then postblackened by reversing
Equation (1.9) to obtain the sequence τν ,Z
*,1,,1, τντνττν εφ += −zz . (1.11)
The synthetic generation process is started with 00,1 =z . The “burn-in” or
“warm-up” period is chosen to be large enough to remove any initial bias.
The values of τν ,Z are then inverse standardized (using Equation (1.12)) to
obtain the synthetic streamflow replicate τν ,X :
τττντν qszx +×= )( ,, . (1.12)
It is to be noted that no normalizing transformation is applied in the case of
the hybrid model. In this context it should be noted that when the number of
data points in the historical record is limited (as in case of annual streamflow
modelling), the mean of residuals recovered from the partial pre-whitening
stage need not be necessarily equal to zero. In such a case, the residuals
are to be re-centred to zero before proceeding with resampling them for
generating the innovation series, see Davison and Hinkley (1997). However,
when the data points are relatively plentiful (as in case of periodic streamflow
modelling), it is found that the sum of residuals recovered from the partial
prewhitening stage tends to zero, and hence one need not re-centre the
residuals.
1.3.3.3 Stochastic simulation
This section introduces stochastic simulation. Simulation is the most flexible and
widely used tool for the analysis of complex water resources systems. Simulation is
trial and error. One must define the system being simulated, both its design and
operating policy, and then simulate it to see how it works. If the purpose is to find the
15
best design and policy, many such alternatives must be simulated and their results
must be compared. When the number of alternatives to simulate becomes too large
for the time and money available for such analyses, some kind of preliminary
screening, perhaps using optimization models, may be justified.
As with optimisation models, simulation models may be deterministic or stochastic.
One of the most useful tools in water resource systems planning is stochastic
simulation. While optimisation can be used to help define reasonable design and
operating policy alternatives to be simulated, simulations can better reveal how each
such alternative will perform. Stochastic simulation of complex water resources
systems on digital computers provides planners with a way to define the probability
distributions of multiple performance indices of those systems.
When simulating any system, the modeller designs an experiment. Initial flow,
storage, and water quality conditions must be specified if these are being simulated.
For example, reservoirs can start full, empty, or at random representative conditions.
The modeller also determines what data are to be collected on system performance
and operation and how they are to be summarized. The length of time the simulation
is to be run must be specified and, in the case of stochastic simulations, the number
of runs to be made must also be determined. These considerations are discussed in
more detail by Fishman (2001) and in other books on simulation.
The simulation model
The simulation model is composed primarily of continuity constraints and the
proposed operating policy. The volume of water stored in the reservoir at the
beginning of seasons 1 (winter) and 2 (summer) in year y are denoted by S1y and
S2y. The reservoir’s winter operating policy is to store as much of the winter’s inflow
Q1y as possible. The winter release R1y is determined by the rule
⎪⎩
⎪⎨
⎧
+
≥−+≥
>−+−+
=
, otherwise S(1.13) 0 if
if
11y
min11min
min1111
1
y
yy
yyyy
y
QRQSKR
KRQSKQS
R
where K is the reservoir capacity of 4 x 107 m3 and Rmin is 0.50 x 107 m3, the
minimum release to be made if possible. The volume of water in storage at the
beginning of the year’s summer season is
16
(1.14) 1112 yyyy RQSS −+=
The summer release policy is to meet each year’s projected demand or target
release Dy, if possible, so that
⎪⎩
⎪⎨
⎧
+≤+≤
>−+−+=
otherwise 1.15) (K D-Q0 if
if
22
y2y2
22112
2
yy
yy
yyyyyy
y
QSSD
KDQSRQSR
Therefore, the volume of water in storage at the beginning of the next winter season
is
(1.16) . 22211 yyyy RQSS −+=+
1.3.2 History of stochastic simulation of streamflow
In the past four decades, since the pioneering work of Fiering (1964), a number of
studies have addressed the application of parametric models to stochastic
simulation of multi-season streamflows. Considerable effort has gone into analysis
and development of methods ranging from linear parametric models (Box and
Jenkins, 1976; Salas et al., 1980; Bras and Rodrı ´guez-Iturbe, 1985; Salas, 1993)
to nonlinear parametric models (e.g., Bendat and Piersol, 1986; Tong, 1990), and
from linear disaggregation models (e.g., Valencia and Schaake, 1973; Grygier and
Stedinger, 1988) to nonlinear disaggregation models (e.g., Koutsoyiannis, 1992;
Koutsoyinannis and Manetas, 1996). And in the beginning of the 21st century,
parametric methods that couple stochastic models of different time scales
(Koutsoyiannis, 2001) have also been proposed.
In the linear parametric (LP) modelling framework, it is necessary to identify an
appropriate normalizing transformation to transform the time series to Gaussian (or
near-Gaussian). These normalising transformations may have some ill effects as
identified by Srinivas and Srinivasan (2000). Further, in case of short hydrologic
records often encountered, the errors arising from parameter estimation can easily
overwhelm issues of model choice (Stedinger and Taylor, 1982). Moreover, the
linear form of LP methods restricts their ability to reproduce nonlinearities inherent in
the observed hydrologic sample. Consequently, these methods fail to simulate
historical trend of critical and mean run characteristics effectively (Srinivas &
Srinivasan, 2000). Lall (1995), Tarboton et al. (1998) and Srinivas & Srinivasan
(2000) amongst others have addressed the drawbacks of parametric models. Even
17
though nonlinear parametric models (Bendat and Piersol, 1986; Tong, 1990) can be
used instead of LP models to model time series that exhibit nonlinearity, it is
essential to specify the form of nonlinear dependence, which may not be easy for
the practitioner.
Further, the need to preserve statistical properties at different time and space scales
directed the development of disaggregation models (Valencia and Schaake, 1973;
Mejia and Rousselle, 1976; Grygier and Stedinger, 1988). These models simulate
flow values at higher-level (e.g., annual) by typical LP models such as
autoregressive (AR) or autoregressive moving average (ARMA), which are
subsequently divided into flow values at lower time scale (e.g., monthly, weekly
etc.). The conventional disaggregation models consider the issue of parsimony by
explicitly modelling only a selected set of relationships among the seasonal flows.
In the 1990s, Koutsoyiannis (1992) developed a parsimonious nonlinear multi-
variate dynamic disaggregation model (DDM) that followed a stepwise approach for
simulation of hydrologic time series. This consisted of two parts: (i) a linear step-by-
step moments determination and (ii) an independent nonlinear partitioning. This
model was shown to treat the skewness of the lower level variables explicitly,
without loss of additive property. Koutsoyiannis and Manetas (1996) proposed
another simpler multivariate disaggregation method that retained the parsimony in
model parameters for lower level variables as in DDM and implemented accurate
adjusting procedures to allocate the error in the additive property, followed by
repetitive sampling to improve the approximations of the statistics that are not
explicitly preserved by the adjustment procedures.
In 2000, Koutsoyiannis (2000) proposed a generalised mathematical framework for
stochastic hydrological simulation and forecasting problems, where, a generalised
autocovariance function is introduced and is implemented within a generalised
moving average generating scheme that yields a new time-symmetric (backward–
forward) representation. A notable highlight of this model framework is that unlike in
the traditional stochastic models, the number of model parameters, the type of
generation scheme and the type of autocovariance function can be decided
separately by the modeller. This framework is shown to be appropriate for stochastic
processes with either short-term or long-term memory. Koutsoyiannis (2001) also
proposed a methodology for coupling stochastic models of hydrologic processes
that apply to different time scales. It is noted that DDM and the further developments
(Koutsoyiannis and Manetas, 1996; Koutsoyiannis, 2000, 2001) perform reasonably
18
well at the verification stage. These models were developed to reproduce long-term
dependence and have been validated for practical water resources use through
application to the management of two major multireservoir hydrosystems of Greece
(Koutsoyiannis et al., 2002).
Despite a plethora of studies in the area to date, there is dearth of attempts that
quantify the effect of bias in preservation of the various statistical attributes on the
prediction of the more important validation statistics such as reservoir storage
capacity, critical and mean run characteristics of streamflows. Hence one cannot
justify explicitly selecting a set of statistics and relationships to be modelled by the
disaggregation models.
On this premise, the need to develop data-driven parsimonious models that mimic
various features of the underlying distribution of historical time series, have gained
prominence. In the 1990s, and generally in parallel with developments in
nonparametric (NP) time series analysis in statistics, data-driven NP methods have
gained recognition in hydrology (Lall, 1995; Lall et al., 1996; Lall and Sharma, 1996;
Sharma et al., 1997; Tasker and Dunne, 1997; Tarboton et al., 1998; Rajagopalan
and Lall, 1999; Kumar et al., 2000; Sharma and O’Neill, 2002). Unlike traditional
parametric models, the NP models do not make assumptions regarding the form of
the probability density function of hydrologic data. The NP methods are increasingly
recognised for their ability to model nonlinearity inherent in the underlying dynamics
of the geophysical processes (Helsel and Hirsch, 1992; Lall, 1995). Since these
models are data-driven in nature, they simulate the skewness and other
distributional features (including multi-modality) of the historical flows efficiently.
Bootstrap is a simple NP technique for simulating the distribution of a statistic or a
specific feature of the distribution by resampling data. The use of bootstrap methods
in time series analysis has been receiving considerable attention in recent times
(e.g., Künsch, 1989; Efron and Tibshirani, 1993; Davison and Hinkley, 1997;
Carlstein et al., 1998; Politis et al., 1999; Politis, 2003). Moving block bootstrap
(MBB, Ku¨nsch, 1989) consists of dividing the data into blocks of observations and
resampling the blocks randomly with replacement. The blocks may be non-
overlapping or overlapping. In MBB, although the original dependence structure is
maintained within the blocks, it gets lost at boundaries between the blocks. As a
result, the adjoining blocks appear independent in the synthetic replicates. The
number of blocks available for resampling should be large enough to ensure a good
19
estimate of the distribution of the statistic (Davison and Hinkley, 1997). For a time
series with strong dependence, resampling small size moving blocks leads to poor
preservation of the same in simulations. If the block size is increased in an effort to
capture the dependence structure, the number of blocks that could be formed from a
given time series drop, thus affecting the variety in simulations from MBB. Lahiri
(1993) brought out the drawback of MBB in capturing long-range dependence.
Srinivas and Srinivasan (2000, 2001) addressed the inefficiency of MBB in
simulating streamflows at annual and periodic time scales. Lall and Sharma (1996)
introduced k-nearest neighbour (k-NN) bootstrap in hydrology for resampling
dependent hydrologic data (Sharma et al., 1997). Multivariate nearest neighbour
probability density estimation provides the basis for the resampling scheme. It uses
a discrete kernel to resample from the successors of k-nearest neighbours of the
conditioning vector (Rajagopalan and Lall, 1999; Sharma and Lall, 1999; Kumar et
al., 2000). The nearest neighbour bootstrap and its variations may be preferable if
the data are plentiful, as in case of daily streamflow modeling (Lall and Sharma,
1996). Srinivas and Srinivasan, (2001a) showed that for historical time series with
strong dependence, the k-NN model is ineffective in simulating higher lag serial
correlations, cross-year serial-correlations and autocorrelation at aggregated annual
level. Consequently, the performance of the model in simulating run characteristics
(validation statistics according to Stedinger and Taylor, 1982) at periodic time scale
is not satisfactory.
A limitation of the aforementioned NP methods is that simulations from these
resampling methods can neither fill in the gaps between the data points in the
observed record nor extrapolate beyond the observed extrema.
In the mid 1990s, kernel-based nonparametric methods have been developed for
streamflow simulation (Sharma et al., 1997), streamflow disaggregation (Tarboton et
al., 1998) and for generation of multivariate weather variables (Rajagopalan et al.,
1997) to alleviate the limitation of the bootstrap methods. However, these methods
demand considerable computational effort for the estimation of bandwidth in higher
dimensions. Moreover, the kernel methods suffer from severe boundary problems,
especially in higher dimensions, that can bias the simulations (Prairie, 2002).
Despite the many studies undertaken, none of the methods seem to have gained
universal acceptability among practicing engineers for various water resources
20
applications. This may either be due to lack of confidence in the existing models, or
the inability to adopt models proposed in the literature because of their complexity or
both. Consequently, the practising hydrologists have resorted to simple techniques
that may not model the data adequately. Thus, there is a pressing need for
identification of simulation models that are efficient and at the same time
computationally simple to be readily adopted by practising hydrologists in river basin
simulation and reservoir operation studies.
Davison and Hinkley (1997), introduced post-blackening approach for stochastic
modelling of streamflows that exhibit complex dependence, and further explored by
Srinivas and Srinivasan (2000, 2001a,b). This approach suggests using a
parsimonious linear parametric model for partial pre-whitening of the observed
streamflows. The structure in the residuals extracted from the partial prewhitening
stage is simulated using MBB to generate innovations that are then post-blackened
to synthesize the replicates of the observed flows. This model is referred to as
Hybrid MBB (HMBB) by Srinivas and Srinivasan (2001). They mentioned that HMBB
(like NP models), does not make assumptions regarding the form of the probability
density function of hydrologic data. Izzeldin and Murphy (2000) have suggested the
use of this model for obtaining finite sample critical values of modified rescaled
range, which is used to detect long memory in financial, economic and hydrologic
time series.
Preservation of the complete dependence structure (both linear and nonlinear) of
streamflows is essential for the efficient prediction of reservoir storage capacity and
modelling critical run characteristics. For the effective preservation of these
statistics, the HMBB model needs resampling of long blocks of residuals (Block size
L = 36, 48 months etc.), particularly when the cross-year dependence is strong. This
is owing to the aforementioned limitations of MBB, which is used by HMBB for
synthesizing innovations through resampling of blocks formed from the residuals
extracted at the pre-whitening stage. The variety and the smoothing in the
simulations diminish with increase in the size of blocks being resampled, which
affects the validation performance of the model in the form of poor variability in
simulated critical run characteristics and reservoir storage capacity. The variability in
preservation of a statistic is measured in terms of the interquartile range of the box-
plots depicting the statistic. Adopting a stochastic model with poor validation
performance affects the design decisions.
21
Srinivas and Srinivasan’s (2001) motivation for their work came from a desire to
identify a potential bootstrapping strategy for synthesizing innovation series in a
post-blackening approach. In other words, they believed a viable alternative to MBB
for resampling residuals extracted from the partial pre-whitening stage of a post-
blackening model, would enhance the validation performance of the model.
Therefore, they found the matched block bootstrap (MABB) method presented by
(Hesterberg, 1997; Carlstein et al., 1998) to be useful. The MABB was proposed
with a view to improve the performance of MBB in modeling dependence structure
through matching rules for resampling moving blocks. Out of a few matching rules
recommended by Carlstein et al. (1998), the rank matching rule was found to be the
most accurate and generally satisfactory (Hesterberg, 1997). In a rank matching
procedure, the blocks are matched using a single value at the beginning or the end
of a block. In the proposed model, periodic streamflows are partially pre-whitened
using a parsimonious linear Periodic AR/ARMA model and residuals are extracted.
Non-overlapping within-year blocks formed from the residuals are conditionally
resampled using the rank matching procedure to obtain innovations. The
innovations are then post-blackened to synthesize replicates of the observed flows.
The proposed model was shown to provide efficient simulation of multi-season
streamflows that display strong dependence structure, and as a result, is able to
reproduce the critical drought statistics and predict the storage–performance– yield
relationships effectively.
1.3.3 South African situation
In South Africa, stochastic hydrology is a standard technique that has been applied
to determine the reliability of supply of water resource systems by the Department of
Water Affairs and Forestry since the early nineteen eighties. This section provides a
description of the basic procedures for the generation of stochastic streamflows in
South Africa. Note however that a detailed account of the underlying mathematical
and statistical principles and approaches is not included since extensive information
in this regard can be obtained from existing study reports and papers that have been
published and presented around the world (DWAF, 1986).
Stochastic streamflow generation process
According to Van Rooyen & Mckenzie (2004), the foundation for the generation of
acceptable stochastic streamflow is sound historical naturalised streamflow data that
22
is derived through rigorous hydrological assessments. The first step in the process
of stochastic streamflow generation is to capture the various statistical properties
inherent to the natural historical streamflow sequence of each incremental sub-
catchment under investigation. This, in the case of STOMSA, is achieved by
selecting the appropriate statistical distribution models and parameter sets that best
describe:
• The characteristics of the marginal distribution of the annual flows.
The aim is to find a distribution that can be used most successfully to
transform the annual flows to a normal distribution;
• The time-series distribution that best represents the serial correlation
exhibited by the normalised annual flows. The result is used to
determine the normalised residual annual flows ;
• The cross-correlation between the normalised residual annual flows
from multiple catchments.
Based on the selected statistical distribution models and parameter sets, annual
stochastic flow values are generated for a particular sub-catchment by following
basically the same steps as outlined for parameter estimation above, but undertaken
in reverse order. It starts with random number generation, followed by the
introduction of cross-correlation and then serial correlation characteristics, after
which the marginal distribution model is applied. Monthly stochastic flows, in turn,
are generated based on the annual stochastic flows, disaggregating into 12
corresponding monthly values.
Marginal distribution
The marginal distribution for a historical streamflow sequence refers to the
relationship between the total annual flows when ranked according to magnitude. it
depicts annual flows (in units of volume) plotted against probability of exceedance
(as a percentage). The marginal distribution can also be presented on a transformed
graph, with the probability of exceedance plotted in terms of standard deviations
from the mean.
There are three alternative marginal distribution models used in South Africa. These
are the 3-parameter Log-normal (LN3), 2-parameter Log-normal (LN2), 4-parameter
Bounded (SB4) and 3-parameter Bounded (SB3) distributions respectively.
23
The Log-normal distribution is defined as follows:
Y = γ + δ ln (X - ξ) (1.17)
and the Bounded distribution is defined as follows:
Y = γ + δ ln (X - ξ) / (λ + X - ξ) , (1.18)
where:
• X is an annual streamflow variate;
• Y is the transformed variate;
• ξ < X < λ; and
• γ (Gamma), δ (Delta), ξ (Xi) and λ (Lambda) are parameters.
The aim is to find a marginal distribution that can be used most successfully to
transform the annual historical streamflows to a normal distribution. The selection is
made based on various statistical criteria as described by the so-called Hill
Algorithm which is based on the Johnson Transform Suite (Hill et. al., 1976). More
information in this regard can be found in the publication Stochastic Modelling of
Streamflow (DWAF, 1986).
Serial correlation
Using the normalised annual historical streamflows for the sequence under
consideration, a determination needs to be made of the time-series model and
associated parameter set that best represent the serial correlation exhibited by the
data. The serial correlation characteristics of a particular sequence are illustrated by
means of a graphical representation called a correlogram.
The sequence of normalised annual historical streamflows is analysed by means of
the Auto Regressive Moving Average Model, based on nine possible ARMA(φ,θ)
time-series model types. The most appropriate model type is selected based on a
selection criteria and can be ARMA (0,0), ARMA (0,1), ARMA (0,2), ARMA (1,0),
ARMA (1,1), ARMA (1,2), ARMA (2,0), ARMA (2,1) or ARMA (2,2).
The ARMA(φ,θ) time series model is defined as follows:
24
X(t) - φ1 X(t – 1) - φ2 X(t – 2) = a(t) - θ1 a(t – 1) - θ2 a(t – 2), (1.19)
where:
• X(1), X(2), … X(n) is a stationary sequence of centred (zero mean)
normal variates;
• a(t) is a sequence of independent random variables with a normal
distribution having zero mean and constant variance (white noise);
• φ1 and φ2 (Phi 1 and 2) are auto-regressive model parameters; and
• θ1 and θ2 (Theta 1 and 2) are moving average model parameters.
Once an appropriate time-series model has been selected, the model is applied to
the normalised annual historical streamflow data for the purpose of “removing” its
serial correlation characteristics. This results in a corresponding set of normalised
residual annual historical streamflows.
Cross-correlation
When generating stochastic streamflow data for more than one sub-catchment
simultaneously, the inherent inter-dependence between flows that occur in the
catchments must be preserved. This is required to generate sequences that exhibit
the same correlating properties between adjacent catchment, which is particularly
important for yield analysis of water resource systems with inter-basin transfers.
The cross-correlation that occurs between flows from multiple catchments is
determined based on the normalised residual annual historical streamflows, using a
technique called Singular Value Decomposition. The result of the process is a set of
matrices that are used to re-generate the cross-correlation dependencies among all
the runoff sequences considered for a water resource system. These matrix
parameters together with the results of the marginal distribution and serial
correlation analyses are written to a stochastic parameter file generally referred to
as the PARAM.DAT file. The parameter file is used together with sophisticated
computational routines in the process of generating stochastic streamflows. More
information in this regard can be found in the publication Stochastic Modelling of
Streamflow (DWAF, 1986).
Monthly disaggregation
Over the course of the development of the stochastic model (during the early
25
nineteen eighties), various approaches were considered for the generation of
monthly flow values. Finally, the approach that was adopted is based on a technique
by which each annual stochastic flow is disaggregated into 12 corresponding
monthly values. This method was found to result in realistic monthly flow values
without the necessity of developing a complex monthly stochastic flow generator. A
description of the process of disaggregating annual flows into monthly flows is
provided below.
The disaggregation of the generated annual flow totals to monthly flow values are
undertaken based on a user defined set of so-called key gauges. If a total of say 40
sub-catchments are to be included in the streamflow generation process, 10 of
these might be considered the most important and will therefore be selected as the
key gauges. Using the generated annual flows for each key gauge, the historical
streamflow time series is analysed to identify the year for which the total flow is
closest to the generated annual flow value. If there are 10 key gauges, then 10 such
years will be identified. Some of the years may be the same, for example the year
1956 may be selected for four of the 10 gauges, although it is not unusual for all of
the 10 years to differ. After having identified the 10 key years, a simple
least squares fit-analysis is undertaken to select the single year for which the
difference between the historical and the generated annual flow values is the
smallest for the group of 10 key gauges.
Using the single key historical year identified in this manner, the monthly distribution
for that year is used to distribute the generated annual flows of all catchments. In
other words, if 1956 is selected, the distribution for 1956 in catchment A is used to
disaggregate the annual flows in catchment A, while the distribution for 1956 in
catchment B is used to disaggregate the annual flows in catchment B and so on.
Verification and validation
The primary objective when undertaking stochastic streamflow generation is to
provide realistic alternative sequences of flow data that can be used to determine
the assurance of supply from a water resource system. What is important to note is
that rigorous assessments of the validity of the stochastic streamflow sequences
have to be undertaken to ensure the yield results are reliable realistic and plausible.
Two different classes of tests are used when checking stochastically generated
26
streamflow data:
• Verification tests involve the re-sampling of various statistics from the
generated sequences to ensure that the model can reproduce the
statistics from the historical sequence within reasonable boundaries.
Comparison of the mean and standard deviation are examples of
verification tests;
• Validation tests involve testing certain features of the generated
sequences that were not directly employed as part of the generation
process. All tests in this category relate to the role of reservoir storage
and include the maximum deficit, duration of maximum deficit, duration
of longest depletion and yield-capacity relationship tests. Note that such
tests are always undertaken assuming zero evaporation losses from the
reservoir water surface.
Any one of the above tests is undertaken by generating a number of stochastic
streamflow sequences and calculating, for each sequence, the value of the
characteristic under consideration (e.g. mean, maximum deficit, etc.). The result is a
range of values that are represented as a distribution by means of a so-called
box-and-whisker plot. The box-and-whisker plot is evaluated by comparison with the
corresponding value from the historical data and generally the results are deemed
acceptable if the historical value lies between the 25 and 75 percentiles.
In cases where the historical value lies outside the normally accepted limits, it is the
responsibility of the analyst to decide whether or not there is a problem with either
the historical naturalised data or a shortcoming in the stochastic model. It should be
remembered that no stochastic model is perfect, particularly one in which stochastic
sequences are generated simultaneously for multiple catchments. Errors or
anomalies should be evaluated individually to ensure that they are not large enough
to have a significant influence on the overall results of an analysis. The time and
effort required to address a possible problem should also be compared to the
expected benefit. This model is considered to be one of the most robust available
and has been thoroughly tested over a number of years. It is, however, not
necessarily applicable to every water resources system and modifications may be
required in certain cases.
27
Distribution of normalised annual flows
The first step in generating stochastic flow sequences for a particular catchment is
to select a marginal distribution for the purpose of normalising the annual historical
streamflows. Each distribution has its strengths and weaknesses with the result that,
careful checking needs to be undertaken to ensure that realistic and meaningful
results are produced. For this purpose the annual streamflows are normalised using
the marginal distribution that has been selected and the results plotted on a graph.
Note that in this case a standardised graph is used, which means that both the
normalised annual streamflows and the probability of exceedance are plotted in
terms of standard deviations from the mean. In general the result is considered
acceptable if the trend of the plotted values approximates a straight line.
Correlation of normalised residual annual flows
As discussed in, a time-series model is selected for the purpose of removing the
serial correlation characteristics of the normalised annual streamflows, resulting in a
corresponding set of residual annual flows. In order to evaluate the selected
time-series model, the serial correlation of the normalised residual annual
streamflows is illustrated using a correlogram. This correlogram can then be
compared with that of the normalised annual streamflows before application of the
time-series model.
Monthly and annual means
The first and most basic verification test carried out on stochastically generated
streamflow sequences involves comparing the monthly and annual means of each
generated sequence with that of the historical sequence. The distribution of monthly
and annual means for stochastic sequences is depicted in the form of
box-and-whisker plots.
Monthly and annual standard deviations
The second verification test involves the assessment of the monthly and annual
standard deviations (SDs) of the stochastic and historical streamflow sequences.
Annual SDs are of particular importance in water resource analyses where yield
calculations are involved. The yield from a reservoir will generally be significantly
greater for a low annual SD compared to that obtained when the SD is high.
28
Minimum run-sums
The minimum run-sum is defined for a given streamflow sequence as the lowest
flow to occur during the complete sequence for a specified number of consecutive
months. Minimum run-sums are usually plotted for a variety of time period, such as
12 months, 24 months, 36 months and so on. This is a validation test since the run-
sum characteristics of the historical streamflow sequence are not used in any way to
generate the stochastic flows.
Maximum deficits and deficit durations
The maximum deficit and deficit duration are validation tests undertaken for a
particular generated streamflow sequence by imposing various target water
requirements on an imaginary reservoir, starting full. The maximum deficit is
calculated as the minimum reservoir storage (in units of volume) required to provide
an uninterrupted supply of requirements of 40%, 50%, 60%, 70% and 80% of the
mean annual runoff (MAR) for the sequence in question.
The maximum deficit duration represent the drought event causing the maximum
deficit and is calculated as the period (in months) over which the reservoir level
drops from full supply to the maximum deficit and then recovers again. Note that the
deficit duration can never exceed the total length of the sequence analysed.
Longest depletion durations
This validation test is undertaken in the same way as that for the maximum deficit
duration. In this case the duration of the longest depletion is determined for a
particular streamflow sequence (in months) caused by the supply of requirements of
40%, 50%, 60%, 70% and 80% of the MAR for the sequence in question. Note that
generally the same drought event causes both the maximum deficit and longest
depletion, but that this is not true in all cases.
Yield-capacity relationship
The yield-capacity relationship validation test is undertaken by estimating, for a
particular streamflow sequence, the minimum reservoir storage (in units of volume)
required to provide an uninterrupted supply for a range of target water requirements.
Requirements of 20%, 40%, 60%, 80% and 100% of the MAR of the historical
streamflow sequence are analysed.
29
In this regard it is important to note that the analysis undertaken for the yield-
capacity relationship test is similar to that for the maximum deficit, but differs in that
water requirements are expressed in terms of the MAR of the historical streamflow
sequence, whereas in the case of the maximum deficit, the MAR of the stochastic
sequence being analysed is used.
Cross-correlation test
Finally, an additional test can be undertaken for the purpose of evaluating the
cross-correlation that occurs between monthly and annual flows from selected pairs
of catchment flow time series. Similar to the other tests, the cross correlation is
calculated for each of the generated sequences and the distribution of these values
is compared with the cross-correlations calculated for the pair of historical
sequences.
Correlation tests can be undertaken as described above for various pairs of
sub-catchments, depending on the physical relational characteristics of the
catchments under consideration and the particular requirements of the analyst.
30
CHAPTER 2
METHODOLOGY
The project has two objectives, i.e., to apply the hybrid model to the South African
river basins and to develop a conceptual model for the implementation of the model
in South Africa, of which the approaches to be followed are discussed in the
following sub-sections.
2.1 Application of Hybrid model
The hybrid model proposed in this study uses a simple multivariate
contemporaneous PAR(1) (Salas et al., 1980) as a parametric constituent of the
model and the residual resampling scheme based on the moving block bootstrap as
a nonparametric constituent. The modelling steps are as follows:
(a) Standardize the series τν ,Q to remove the periodicity,
τντττν ,, YsqQ += , (2.1)
where τq and τs are the (n×1) vectors representing periodic mean and
standard deviation of season τ , respectively, τν ,Q is a (n×1) vector of the
original seasonal data and τν ,Y is the standardized data.
(b) Pre-whiten the series at each site with an univariate PAR(1)NT-hybrid
model,
ν,τν,τ,τν,τ εYΑY += −11 , (2.2)
where τ,1A is an (n×n) matrix of lag-1 autoregressive coefficient and τν ,ε is
an (n×1) vector of residuals. Equation (2.2) can be re-written as follows:
31
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
)(,
)2(,
)1(,
)(1,
)2(1,
)1(1,
)(,1
)2(,1
)1(,1
)(,
)2(,
)1(,
00
0000
nnnn y
yy
a
aa
y
yy
τν
τν
τν
τν
τν
τν
τ
τ
τ
τν
τν
τν
ε
εε
, (2.3)
where n is the site number and )(,1na τ is a lag-1 autoregressive coefficient for
site n and season τ .
The autoregressive coefficient can be estimated by
( )( )
( )∑
∑−
=
−
=−
−
−−== 1
1
2,
1
11,,
)(,1
)(,1 N
tt
N
ttt
nn
yy
yyyya
ττ
ττττ
ττ γ , (2.4)
where )(,1nτγ is a lag-1 autocorrelation coefficient for site n and season τ , and
N is the length of the time series.
(c) Generate the innovations (or residuals) by moving block bootstrap. To
maintain the cross dependence between sites, the length of moving blocks
should be same at each site.
1,,1,, −−= τνττντν YAYε (2.5)
(d) Generate the series τν ,Y from equation (2.2) using the simulated innovation
τν ,ε and then generate the series τν ,Z by the inverse standardization of
equation (2.1).
The proposed model will be applied to monthly streamflow series of 5 sub-
catchments of the Vaal River basin, using the incremental runoffs files which were
used in the testing of the STOMSA model, i.e the Bloemhof Dam (BLOEM9.INC),
Delangesdrift (DELA9.INC), Katse Dam (KAT9.INC), Vaal Dam (VAAL9.INC) and
Welbedacht Dam (WELB9.INC) sub-catchments. Coding will be done in matlab.
To test performance of the proposed model will be compared as far as possible to
the results of the STOMSA, taking into consideration that STOMSA is based on
Fortran and the application in this study is based on Matlab, as follows:
32
• Verification
Comparison of the mean, standard deviation and skewness will be
performed in the verification tests.
• Validation
Validation tests will involve testing certain features of the generated
sequences that were not directly employed as part of the generation
process. All tests in this category relate to the role of reservoir storage and
include:
o maximum deficit,
o duration of maximum deficit,
o duration of longest depletion,
o cross-correlation test, and
o yield-capacity relationship.
These tests will be undertaken assuming zero evaporation losses from the reservoir
water surface.
2.2 Conceptual model for implementation
The approach will be to develop guidelines to be used in the implementation of the
method, taking cognisance of:
• the current modelling environment,
• the required environment for the operation of the hybrid method,
• envisaged analysis procedure, and
• system layout.
33
CHAPTER 3
APPLICATION OF THE HYBRID MODEL
3.1 Analysis
The proposed model was applied to monthly streamflow series of five selected
incremental sub-catchments, i.e Bloemhof Dam, Delangesdrift, Katse Dam, Vaal
Dam, and Welbedacht Dam sub-catchments. Time-series data files of monthly
natural historical streamflows for each of the sub-catchments are provided in
Appendix A and a summary of the associated characteristics in Table 3.1. The data
exists for at least 68-year period from 1920 to 1994. The data was taken as was
used in the testing of the STOMSA during its development, thus no preliminary data
analysis was undertaken.
Table 3.1 : Runoff characteristics for the selected sub-catchments
Description Data file name Start year (hydrological)
End year (hydrological)
Record period length (years)
Mean annual runoff
(106 m3)
Bloemhof Dam BLOEM9.INC 1920 1994 75 154
Delangesdrift DELA9.INC 1920 1994 75 249
Katse Dam KAT9.INC 1920 1995 76 546
Vaal Dam VAAL9.INC 1920 1994 75 519
Welbedacht Dam WELB9.INC 1920 1987 68 630
To test performance of the proposed model, this study compared two generation
models: STOMSA and the MCPAR(1)-hybrid models. Each model generated 100
sets of 76-year monthly series for each of the selected sub-catchments and the
results were used for undertaking verification tests by comparing the mean,
standard deviation, variance and CV, and further showing the performance of the
MCPAR(1) in producing the correlation between the last month of the previous year
with the first month of the current year, and the very same sequences can be used
for validation, although that was not undertaken in this project. MATLAB was used
for calibrating the MPAR(1)-hybrid while the results from the development tests on
the STOMSA were used for comparison purposes.
34
Table 3.2 shows comparison of the Bloemhof results between STOMSA,
MCPAR(1)NT-hybrid and historical data, and it can be seen that the results are
comparable to the historical data for both models. See appendix B for the rest of
the results.
35
Table 3.2: Bloemhof Catchment Results comparison
mean Variance Coeficient fo Variance Standard deviation month STOMSA Historical MPAR(1) STOMSA Historical MPAR(1) STOMSA Historical MPAR(1) STOMSA Historical MPAR(1)
1.00 6.50 6.32 6.28 156.25 151.56 134.17 1.92 1.95 1.84 12.50 12.31 11.582.00 14.50 14.59 14.77 841.00 861.32 802.93 2.00 2.01 1.92 29.00 29.35 28.343.00 17.00 16.93 16.53 462.25 468.07 424.98 1.26 1.28 1.25 21.50 21.63 20.614.00 26.80 26.84 26.62 2401.00 2449.85 2100.73 1.83 1.84 1.72 49.00 49.50 45.835.00 28.00 28.05 27.71 1936.00 1955.00 1772.80 1.57 1.58 1.52 44.00 44.22 42.106.00 31.80 31.75 29.74 3422.25 3494.57 2951.27 1.84 1.86 1.83 58.50 59.11 54.337.00 13.00 13.24 12.62 400.00 398.55 338.37 1.54 1.51 1.46 20.00 19.96 18.398.00 5.50 5.47 5.35 132.25 115.60 98.93 2.09 1.96 1.86 11.50 10.75 9.959.00 2.50 2.46 2.42 6.25 8.44 7.64 1.00 1.18 1.14 2.50 2.90 2.76
10.00 2.00 1.91 1.93 2.25 3.70 3.24 0.75 1.00 0.93 1.50 1.92 1.8011.00 2.20 2.07 2.21 12.25 14.18 11.33 1.59 1.82 1.52 3.50 3.77 3.3712.00 4.00 4.05 4.03 182.25 190.93 152.46 3.38 3.42 3.06 13.50 13.82 12.35
36
3.2 Conceptual model for implementation
From the literature review chapter, it is clear that there is a need for a parsimonious
model that will be able to reproduce the statistics on temporal level, and it has been
shown that the hybrid model is capable of doing so. However it is also clear that
although there is numeral literature on alternative models, there has not been
implementation or adoption for inclusion to the current modelling environment of
these new modelling methodologies. Therefore this section explores ways in which
this hybrid model can be introduced to the South African water industry and present
a conceptual model that will ensure that current knowledge and experience is
transferred and maintained in the new model.
Therefore, the purpose of this section is to describe conceptually, a Hybrid model for
water resource analysis using the existing network simulation models and analysis
techniques that are applied in South Africa. The section begins by providing
guidelines used to direct the design of the conceptual model and give an
explanation of the framework in which the conceptual model was developed. This is
followed by the description of how the streamflows are stochastically modelled and
validated in the STOMSA and then presents the proposed structure of the Hybrid
model and the connections with STOMSA and finally the proposed analysis
procedure is presented showing how the hybrid model will interact with the
STOMSA.
3.2.1 Development guidelines and framework
This section describes the guidelines that were applied in the design of the
conceptual hybrid model and serve as a means of presenting the thought process
and background that led to the particular model description.
3.2.1.1 Pointers from the current modelling environment.
Introducing a new procedure into the water resource analysis environment, where a
large pool of knowledge is founded on present technologies, requires careful
planning. The aim is to implement the Hybrid model so that high degree of the
current knowledge is maintained and future analysis requirements are recognised.
37
To this end, relevant guidelines are listed below.
• The hybrid model should be developed for the existing water resource
models that are used in practice for the management of most of the countries
water resources. These models are the STOMSA, and the accompanying
Water Resources Yield Model and the Water Resources Planning Model.
The advantage of using the existing models is that the existing pool of
knowledge that reside in water resource analysts, who are currently applying
these models, can be extended further by adding a monthly model.
• Cognisance should be taken of the analysis techniques that are used in
practice in South Africa. Currently the streamflows are modelled to asses the
reliability of the water systems, both regulated and unregulated, therefore,
the hybrid model should be designed for such environment, such that its
performance can be checked against that of the currently used model and/or
the historical record.
• Due to the interconnectedness of the systems, there is a need for the model
to extend to joint modelling of several streamflow sequences simultaneously.
• While there are other benefits to the introduction of the hybrid model in South
Africa, the primary objective should be to overcome the difficulties
encountered in the current models, and in particular, the inability to model
the flows in the first month of a year so as to follow from the flows in the last
month of the previous year and that in the generation. Consequently, the
hybrid model should not be designed to be the replacement of the current
models, but to aid and improve where current models cannot.
• In order to build confidence in the application of the hybrid model it is
deemed essential to introduce (develop and verify) the hybrid model in
phases. The intention is to improve the value of the hybrid model by using
basic stochastic analysis and relatively simple systems as a first step.
• In testing the model, the current model should be used as a benchmark, as
the model have been tested over 20+ years and has been refined where
necessary. Figure 2 shows and example where coefficient of variation
influences the slope of the probabilistic firm yield line. Such things if not
controlled and benchmarked against the existing model, new decision could
have an impact on the water resource system.
• The basic design of the hybrid model should be to maintain the flexibility and
generic structure that is provided by the current models.
38
3.2.1.2 Conceptual Model
It is imperative to have a high-level perspective when designing a conceptual model
for the inclusion of the Hybrid model into the STOMSA for water resource systems
optimisation. It is understood that the Hybrid model is, but one of the many
alternative models that can be incorporated into STOMSA to deal with some of the
short comings, and it is also understood that advances in technology and/or
research will, in the future, necessitate improvement to the whole set of models
packaged in STOMSA/or whatever package it will be called after improvements.
Thus, it is important to have a holistic view of the problem and solution to ensure
that the individual models can merge to provide the maximum benefit.
The conceptual model consists of five main modules, namely: a use interface,
control module, synthetic data generating module, statistical data analysis module
and stochastic model fitting module. The proposed model is shown diagrammatically
in Figure 1.
The user controls the modelling process via the “user interface module”. Allowance
should be made for selecting the type of analysis (i.e. monthly or annual) to be
performed and the type of stochastic model to be used (i.e. disaggregation or
hybrid).
The “control module” handles the analysis process, from statistical data analysis
to generation of synthetic data: it takes input from the user via the user interface,
links with the required analysis and stochastic models to generate synthetic data
and then returns output via the user interface. This control module allows for other
types of the stochastic models and/or other changes to the models to be added as
they are developed, and are linked to allow more than one type of analysis to be
performed simultaneously, which is ideal for comparison purposes.
The “Statistical data analysis module” consists of data plotting, checking the
normality of the data, data transformation, and data statistical characteristics.
Plotting the data helps in detecting trends, shifts, outliers, or errors in the data.
Probability plots are used for verifying the normality of the data. The data can be
transformed to normal by using different transformation techniques. A number of
statistical characteristics of the data can be determined in this module including the
39
basic statistics such as mean, standard deviation, skewness, serial correlations (for
annual data), season-to-season correlations (for seasonal data), annual and
seasonal cross-correlations for multisite data, and drought, surplus, and storage
related statistics. These statistics are important in investigating the stochastic
characteristics of the data.
“Stochastic model fitting module” or parameter estimation allows the user to
perform parameter estimation using different models and choose the best fit, this will
include marginal distribution estimation, selection of a time series distribution from
combinations of ARMA(2,2) and determination of cross correlation. There are
however other processes to be undertaken in parameter estimation depending on
the type of modelling to be performed.
“Synthetic data generation module”, this allows the user to generate stochastic
streamflow sequences based on the results of the parameter estimation. The main
philosophy behind synthetic data generation is that synthetic samples are generated
which preserve certain statistical properties that exist in the natural hydrologic
process. As a result, each generated sample and the historic sample are equally
likely to occur in the future. The historic sample is not more likely to occur than any
of the generated samples. The model should allow the user to generate synthetic
data and eventually compare important statistical characteristics of the historical and
the generated data. Such comparison is important for checking whether the model
used in generation is adequate or not. If important historical and generated statistics
are comparable, then one can argue that the model is adequate.
40
User Interface
Statistical Data
Analysis Module
Data generation module
Control Module
Stochastic
Model fitting Module
Figure 1: Diagrammatic representation of the conceptual model
Exceedance probability of base yield as a percentage of sequences analysed
0 20 40 60 80 100
Dra
ft (%
of m
axim
um d
raft)
0
20
40
60
80
100
0
20
40
60
80
100
Bas
e yi
eld
(% o
f max
imum
yie
ld)
CV = 1.8
CV = 0.43
CV = 0.4
1/20
yea
r
1/50
yea
r
1/10
0 ye
ar
1/20
0 ye
ar
Figure 2: Illustration of influence of coefficient of variation on firm yield (Basson et al)
41
3.2.1.3 Stochastic streamflow generation process in STOMSA
Streamflow
RANK
File name .INC File name| RNK
ANNUAL
File name .COR
File name . ANS
File name .YER
CROSSYR
CROSSYR.ANS Param.DAT
GENTST
GENTST.PIN GENTST.PIN2 GENTST.ANS
STPLOT
42
3.2.1.4 Stochastic streamflow generation process in the Hybrid Model
START
Standardise data, choose the model for pre-whitening
Better performance model obtained
Extract residuals and obtain innovations by bootstrapping
Post-blacken the innovations and inverse standardise to obtain synthetic streamflows
STOP
STOMSA’s Crossys extracts croscorrelation information
43
3.2.1.5 Proposed generation process incorporating both models
STOMSA
ANNUAL
CROSSYR
GENTST
Hybrid model
44
CHAPTER 4
CONCLUSION
A new hybrid stochastic model that effectively blends the merits of the parsimonious
parametric model (PAR(1)NT) and simple moving block bootstrap (nonparametric)
model has been adopted, and applied in simulating multi-season multi-site
streamflows. The models ability was demonstrated through stochastic simulations
performed using monthly streamflows of the Bloemhof Dam, Vaal Dam,
Delangesdrift Dam, Welbedacht Dam and Katse Dam Catchments. The application
focused only on the verification of the model and developing a conceptual model for
validation and incorporation to the South African water industry. Comparison was
drawn between the results from the Hybrid Model, the STOMSA and the historical
data in simulating historical monthly streamflows of the afore mentioned catchments.
This hybrid model is shown to offer alternative and/or even better simulations than
its own constituents, by acquiring certain properties that are characteristic of either
of these models. The preservation of cross-year serial correlations is due to the
hybrid effect. The hybrid model ensures annual-to-monthly consistency, thus
averting the adjustments to monthly or annual flows and the associated problems
that surface in the case of linear parametric disaggregation models.
The mean and the standard deviation of observed stream- flows are well reproduced
by the hybrid model at monthly levels, as presented on figures (B-1) to (B-22) in
Appendix B. Being a data-driven model, the hybrid model reproduces skewness of
flows at monthly levels (see figures B-1 to B-22). It should be noted that in the case
of Hybrid Model, no normalizing transformation is applied to the historical data, and
hence skewness of historical streamflows is apparently retained in the residuals that
are extracted from the partial prewhitening stage. The skewness contained in these
residuals is well reproduced in the bootstrapped innovations. The hybrid model is
found to inherit the characteristic of capturing the salient features of the marginal
distribution (asymmetry, peakedness, and multimodality) of observed flows from its
nonparametric constituent (Bootstrap) and is able to provide some smoothing and
limited extrapolation, owing to its parametric constituent. Modeling monthly serial
correlations across water years is important for the efficient simulation of the critical
water use (validation) statistics (especially when such correlations are significant). It
45
is observed from figures (B-1) to (B-22) that the hybrid model is able to preserve the
serial correlations between adjoining water years, owing to the hybrid effect.
On average, the hybrid model performs better (with reference to the mean, standard
deviations, and coefficient of variations) during the low flow months (i.e May, June,
July and August) as compared to the STOMSA model. Figures (B-23) to (B-42) draw
comparisons of the mean and the standard deviations of the catchments between
the STOMSA and the Hybrid model using box plots. It is observed that the Hybrid
model performs better, as in most cases the mean is on the 50 percentile level.
The conceptual model developed for the incorporation of the Hybrid Model into the
South African Water Industry ensures that current knowledge and experience
acquired through the use of the parametric models is not lost. The Conceptual
model also indicates the importance of using the two models conjunctively, as they
both have areas that they are more suitable for. Further, the conceptual model
indicates that while stochastic simulation is intended to give a wide variety of
possibilities of what might occur, it is imperative that a chosen model provides
realistic alternative sequences of flow data that can be used to determine the
assurance of supply from a water resource system, thus rigorous assessments of
the validity of the stochastic streamflow sequences have to be undertaken to ensure
the yield results are reliable, realistic and plausible, and thus the development of the
conceptual model to be used as a guide.
46
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49
APPENDIX A: MODEL INPUT DATA
Bloemhof Dam sub-catchment (BLOEM9.INC)
Monthly natural historical streamflows (million m3) 1920 6.29 2.64 1.33 3.46 9.91 234.34 80.28 1.87 1.46 1.24 1.07 0.95 344.84 1921 0.87 24.68 102.70 41.73 4.05 1.20 0.96 0.79 0.82 0.84 0.99 0.89 180.52 1922 1.11 23.26 11.66 21.72 55.11 17.86 1.35 1.20 1.15 1.15 1.06 0.87 137.50 1923 0.73 1.02 0.69 5.96 3.17 34.24 12.44 1.07 0.93 0.88 0.84 1.20 63.17 1924 4.28 25.97 19.67 15.93 9.97 269.12 97.29 5.36 2.62 1.78 1.33 1.12 454.44 1925 0.98 1.53 1.00 3.43 10.66 7.95 2.30 0.87 0.86 0.87 0.82 0.75 32.02 1926 1.33 1.14 4.46 5.78 4.16 23.80 8.66 1.00 0.88 1.11 1.11 0.87 54.30 1927 1.95 1.09 1.38 43.09 20.10 8.56 3.28 1.20 1.05 0.96 0.91 0.98 84.55 1928 1.46 7.10 3.26 24.40 9.58 9.55 4.16 1.33 1.57 1.56 1.26 11.69 76.92 1929 5.03 19.80 26.66 34.08 12.65 6.45 3.23 1.47 1.17 1.06 0.97 0.84 113.41 1930 0.69 0.67 3.68 32.84 13.92 5.27 41.46 14.40 1.11 1.00 0.91 0.75 116.70 1931 1.46 20.73 7.48 0.69 17.03 8.84 1.87 0.97 0.87 0.82 0.78 0.70 62.24 1932 0.62 0.93 12.36 4.55 0.76 1.18 1.00 0.82 0.74 0.71 0.69 0.62 24.98 1933 0.55 75.73 51.98 367.63 123.81 5.63 2.74 12.90 5.46 1.53 1.38 1.06 650.40 1934 4.25 106.46 44.87 3.88 1.51 28.39 10.73 1.32 1.03 0.93 0.84 0.74 204.95 1935 0.66 2.77 6.47 6.72 11.91 73.03 25.13 3.61 2.59 1.59 1.21 0.99 136.68 1936 0.97 199.21 68.31 39.20 21.16 5.40 1.77 1.05 0.91 0.84 0.78 0.69 340.29 1937 0.62 0.55 15.60 23.93 17.62 4.47 4.94 2.52 1.15 1.09 1.11 0.93 74.53 1938 10.70 4.07 15.71 32.36 50.86 26.54 5.61 1.47 1.29 1.79 1.74 1.30 153.44 1939 1.74 11.63 5.30 3.26 2.08 5.58 3.20 1.44 1.24 1.14 1.00 1.15 38.76 1940 0.95 4.66 15.63 28.95 22.36 5.69 2.26 1.52 0.98 0.87 0.79 0.73 85.39 1941 1.66 0.93 1.33 38.69 17.77 11.61 4.65 1.31 1.06 0.96 0.97 0.88 81.82 1942 5.95 5.30 77.66 32.02 3.35 5.34 48.25 57.70 15.45 2.58 2.23 1.54 257.37 1943 21.03 68.89 72.23 24.13 230.19 82.24 2.26 1.33 4.72 3.21 1.63 2.26 514.12 1944 6.54 11.83 4.13 2.29 2.62 75.78 27.11 1.44 1.17 1.06 0.93 0.79 135.69 1945 0.66 0.57 0.69 22.51 14.75 46.75 16.42 1.64 1.33 1.15 1.00 0.84 108.31 1946 4.32 2.49 1.60 8.26 8.33 11.68 8.16 2.77 1.24 1.05 0.91 0.80 51.61 1947 0.73 1.03 21.49 16.99 4.06 190.77 67.00 2.44 1.42 1.15 1.00 0.84 308.92 1948 0.97 4.95 2.06 12.03 4.63 1.71 1.15 0.79 0.80 0.84 0.80 0.73 31.46 1949 1.57 9.10 40.79 17.85 7.18 36.16 34.41 46.86 15.09 2.07 1.56 1.18 213.82 1950 0.96 0.79 43.03 21.81 4.09 3.73 4.12 2.46 1.65 1.44 1.28 1.02 86.38 1951 7.95 3.26 1.26 2.64 8.57 3.41 0.96 0.88 0.82 1.11 1.09 0.83 32.78 1952 0.87 15.42 29.61 9.00 51.14 20.85 3.06 1.66 1.23 1.07 0.96 0.82 135.69 1953 3.12 5.09 3.88 6.30 15.17 17.50 5.36 1.20 1.11 1.05 0.93 0.82 61.53 1954 0.73 3.08 7.02 70.75 92.03 25.49 6.36 3.28 1.62 1.33 1.11 0.91 213.71 1955 1.80 2.75 8.52 3.17 45.98 38.08 8.46 1.69 1.44 1.12 0.95 0.84 114.80 1956 6.93 3.52 61.71 57.52 15.40 5.39 2.54 1.18 1.66 2.22 2.08 103.24 263.39 1957 54.13 9.09 15.03 94.58 31.22 1.68 2.90 2.68 2.02 1.55 1.29 1.28 217.45 1958 1.12 2.93 11.67 7.65 2.59 1.20 4.83 3.35 2.04 1.51 1.06 0.73 40.68 1959 3.15 1.74 20.65 5.01 7.86 11.94 5.36 3.50 1.90 1.51 1.37 1.23 65.22 1960 1.23 1.79 72.86 29.04 2.50 2.21 30.80 14.65 0.70 3.98 1.97 0.80 162.53 1961 0.32 37.55 12.93 0.00 17.76 14.73 12.12 5.09 1.71 0.91 0.37 0.00 103.49 1962 0.00 12.60 7.42 37.91 14.13 0.18 0.00 1.48 2.78 1.98 1.06 0.00 79.54 1963 0.23 19.70 10.74 2.24 1.06 17.70 3.91 1.33 1.56 0.31 0.00 0.00 58.78 1964 52.09 9.77 6.58 14.56 4.61 0.00 2.46 1.79 1.17 1.14 0.42 0.00 94.59 1965 0.00 0.00 0.00 7.20 58.98 25.61 1.15 0.73 0.98 0.58 0.00 0.00 95.23 1966 1.02 1.52 13.59 158.05 120.07 42.58 56.79 20.62 2.59 1.71 1.32 1.09 420.95 1967 0.93 1.54 1.50 0.78 0.56 14.25 7.02 1.93 1.21 0.96 0.84 0.73 32.25 1968 0.40 1.23 4.63 1.72 0.57 2.66 1.54 18.90 6.98 0.78 0.60 0.48 40.49 1969 7.87 3.97 4.23 7.21 2.77 0.84 0.79 0.82 0.84 0.97 0.94 0.84 32.09 1970 0.61 2.35 15.07 16.51 5.12 1.42 5.34 2.72 1.02 0.78 0.66 0.57 52.17 1971 0.00 0.00 11.58 47.00 13.14 19.92 5.16 0.00 0.00 0.67 1.46 0.84 99.77 1972 0.38 0.78 0.60 1.35 14.61 5.29 2.00 1.20 0.63 0.54 0.55 0.99 28.92 1973 2.44 7.01 21.73 51.34 81.03 11.12 38.63 4.10 2.80 2.23 1.17 4.34 227.94 1974 0.45 31.07 19.34 2.37 160.13 80.16 29.56 8.86 5.59 6.58 5.94 5.57 355.62 1975 0.00 0.00 50.41 150.07 165.82 178.83 32.64 53.93 7.98 7.83 7.65 3.28 658.44 1976 60.02 29.45 0.78 0.00 101.98 22.73 6.58 2.24 6.92 6.17 4.16 7.33 248.36 1977 12.63 7.09 10.39 2.02 78.41 60.14 65.66 14.19 10.50 11.24 8.68 8.24 289.19 1978 18.11 5.15 6.69 8.10 10.19 2.50 2.83 4.21 5.17 6.07 30.65 13.89 113.56 1979 0.00 0.09 4.79 0.00 3.54 19.81 4.36 3.13 3.31 3.61 3.37 6.32 52.33 1980 3.97 11.61 41.77 33.61 24.26 71.27 7.71 3.12 4.31 4.91 3.76 4.89 215.19 1981 3.65 6.47 26.93 19.08 4.10 3.47 31.93 11.26 3.65 3.34 3.80 2.78 120.46 1982 16.92 9.28 1.99 2.44 1.84 0.00 1.33 1.02 4.41 2.77 1.31 0.00 43.31 1983 9.42 22.28 9.05 2.06 0.78 3.83 1.75 0.61 0.61 0.65 0.84 0.71 52.59 1984 5.89 5.68 2.13 14.71 12.05 6.21 2.11 0.97 0.91 0.87 0.82 0.73 53.08 1985 5.67 2.35 9.61 6.05 1.57 2.44 3.22 1.59 0.95 0.91 1.00 0.96 36.32 1986 3.78 22.18 9.17 13.05 0.36 0.00 0.00 0.00 0.00 0.00 2.08 62.88 113.50 1987 32.93 10.01 11.70 8.31 0.00 295.93 17.69 7.20 1.88 3.73 3.70 3.89 396.97 1988 42.06 39.82 9.31 77.80 110.32 35.43 10.88 13.07 5.57 5.12 7.35 2.02 358.75 1989 3.25 11.66 13.17 9.13 9.86 18.22 26.79 9.84 4.15 2.21 5.06 8.02 121.36 1990 0.76 0.00 3.53 42.63 37.72 17.17 3.10 0.00 1.65 3.74 3.77 2.81 116.88 1991 10.94 4.58 2.66 2.62 0.00 0.00 0.15 0.00 0.00 0.00 0.00 4.50 25.45 1992 0.00 86.33 7.47 5.21 12.41 2.05 0.60 0.00 0.48 0.00 0.00 0.00 114.55 1993 3.98 0.00 0.00 32.81 38.73 13.25 2.50 2.47 2.54 3.70 4.57 1.84 106.39 1994 0.60 0.91 0.99 9.55 3.73 15.25 5.87 1.02 0.93 0.84 0.78 0.71 41.18
50
Delangesdrift sub-catchment (DELA9.INC)
Monthly natural historical streamflows (million m3) 1920 45.49 35.20 11.23 13.69 11.96 45.01 16.78 4.55 1.93 1.38 0.88 4.22 192.32 1921 2.08 123.68 140.52 40.80 26.85 10.19 3.89 1.99 2.61 1.49 2.73 1.54 358.37 1922 64.87 184.39 102.50 213.19 102.83 22.84 7.24 4.47 3.35 3.54 1.85 1.09 712.16 1923 0.61 5.24 5.43 60.88 38.56 30.93 9.40 4.12 2.13 1.65 1.26 1.81 162.02 1924 6.01 42.29 33.30 22.37 15.63 80.34 73.24 112.44 15.42 8.99 5.52 5.31 420.86 1925 18.94 45.07 6.56 8.88 18.07 12.07 4.97 2.21 3.50 1.73 1.21 13.78 136.99 1926 3.01 6.12 17.48 11.12 47.08 35.79 18.68 3.48 1.96 3.07 4.49 1.31 153.59 1927 15.46 10.03 21.29 35.27 15.04 34.60 9.43 3.88 1.66 1.27 1.40 5.13 154.46 1928 4.13 2.89 7.75 20.53 12.29 51.01 9.26 2.34 3.47 2.38 1.56 22.36 139.97 1929 80.21 111.92 155.74 101.80 29.99 9.13 5.12 2.66 2.04 1.58 1.52 1.05 502.76 1930 1.70 1.90 7.64 33.06 46.99 4.79 6.53 1.27 0.87 1.44 0.63 0.27 107.09 1931 0.33 4.95 2.13 5.27 28.01 29.50 8.68 8.22 3.49 1.58 1.05 0.58 93.79 1932 2.24 2.15 7.91 5.71 13.61 10.36 2.90 0.66 0.58 0.59 0.34 0.26 47.31 1933 1.28 45.69 100.46 130.39 61.44 17.00 7.63 4.22 3.05 4.28 5.95 1.60 382.99 1934 2.57 20.01 66.66 15.70 14.84 19.24 7.76 4.16 2.09 1.55 1.02 0.92 156.52 1935 0.54 1.25 3.69 109.95 23.25 7.66 19.47 29.36 22.51 6.64 2.83 1.67 228.82 1936 1.66 121.84 29.73 132.94 211.01 24.85 7.07 3.34 2.45 2.02 1.51 1.58 540.00 1937 10.97 3.85 120.27 16.55 9.00 7.99 4.16 1.50 1.63 1.56 1.69 0.69 179.86 1938 41.20 7.08 35.98 40.54 308.86 64.99 15.44 8.26 4.85 26.18 10.86 6.37 570.61 1939 7.46 93.32 64.38 31.40 23.66 15.68 8.56 42.33 20.16 10.06 4.74 3.65 325.40 1940 6.51 52.08 66.91 54.39 50.41 22.56 57.17 8.79 4.23 3.73 1.96 1.00 329.74 1941 3.17 0.80 39.33 25.50 20.17 22.89 10.43 3.99 8.86 2.54 2.05 3.30 143.03 1942 23.45 24.35 129.90 53.84 12.07 34.36 42.06 95.38 15.21 105.15 45.65 48.34 629.76 1943 34.53 105.08 78.84 90.72 329.47 60.86 16.75 6.15 15.05 6.01 3.21 21.96 768.63 1944 61.37 25.67 7.22 3.14 9.78 26.42 5.05 2.57 1.84 1.47 1.15 0.71 146.39 1945 0.36 0.57 0.54 14.58 39.90 59.19 13.10 3.51 1.93 1.51 1.07 0.49 136.75 1946 8.73 16.42 12.68 18.16 30.13 10.04 16.71 5.59 5.51 3.31 2.08 0.81 130.17 1947 0.82 7.09 14.15 33.34 10.18 5.75 3.79 1.35 0.85 0.74 0.61 1.16 79.83 1948 2.02 11.69 3.29 10.75 30.43 9.65 2.95 7.74 2.04 1.26 0.82 0.74 83.38 1949 2.35 20.56 58.31 63.99 8.77 28.57 14.05 7.59 2.80 1.67 2.14 1.23 212.03 1950 3.08 1.83 11.07 16.23 24.50 9.07 3.09 2.23 1.07 0.98 0.62 0.97 74.74 1951 16.62 3.78 14.05 23.60 24.97 3.39 9.12 1.37 1.34 5.08 1.73 0.75 105.80 1952 0.42 45.87 15.95 11.89 177.84 58.20 32.95 8.31 3.79 2.57 2.03 1.77 361.59 1953 2.13 5.19 25.12 18.35 23.00 22.01 7.27 0.14 2.62 2.59 1.42 0.92 110.76 1954 4.32 21.81 13.59 12.05 119.67 62.00 10.76 5.52 3.70 2.23 1.69 0.91 258.25 1955 4.29 4.68 19.07 61.27 12.69 21.01 23.42 16.27 5.29 2.97 1.60 2.57 175.13 1956 4.08 49.52 138.54 137.84 21.64 16.77 5.98 2.97 1.87 37.24 12.66 223.36 652.47 1957 297.68 88.54 61.29 183.54 39.33 14.75 26.24 7.26 5.06 3.57 2.29 6.40 735.95 1958 7.55 39.45 51.44 13.68 8.16 4.83 0.12 8.88 5.05 3.17 1.46 1.49 145.28 1959 9.37 131.20 43.24 13.50 11.80 9.32 7.84 8.91 3.12 2.22 2.24 1.26 244.02 1960 8.21 24.91 128.05 46.02 12.32 15.53 45.04 9.91 9.11 4.58 2.40 3.34 309.42 1961 10.42 17.91 41.39 23.94 11.96 6.62 1.41 1.29 0.88 0.92 0.67 0.69 118.10 1962 1.36 13.38 27.68 59.20 16.25 3.87 2.66 1.02 1.32 57.64 7.19 2.36 193.93 1963 3.21 39.66 16.56 54.23 19.86 12.56 5.32 1.93 1.59 1.38 1.58 1.45 159.33 1964 97.30 276.61 50.82 39.48 27.23 4.04 3.66 1.66 1.55 1.60 1.00 1.26 506.21 1965 1.03 10.24 12.38 1.76 12.29 3.27 0.69 0.36 0.35 0.37 0.37 0.55 43.66 1966 1.79 1.79 23.24 39.34 140.38 35.18 27.89 8.49 6.14 6.02 4.56 2.51 297.33 1967 3.95 17.36 26.44 9.25 3.42 5.08 2.64 1.40 1.04 0.94 0.82 1.05 73.39 1968 0.63 4.26 11.39 22.95 6.82 20.83 21.02 11.05 4.70 2.26 1.58 0.77 108.26 1969 23.63 11.75 31.72 13.50 26.28 2.73 0.40 0.75 0.60 0.43 0.86 1.09 113.74 1970 6.05 9.73 0.26 6.89 9.67 3.36 46.72 2.67 0.33 0.35 1.24 2.54 89.81 1971 1.40 22.84 88.58 26.55 10.47 56.25 4.69 0.89 0.76 1.09 1.36 0.77 215.65 1972 0.67 2.11 6.39 2.48 26.71 4.84 21.59 0.42 1.65 0.37 13.27 1.42 81.92 1973 8.53 19.97 21.87 65.07 79.76 20.74 14.48 14.52 9.92 7.09 2.87 3.60 268.42 1974 2.50 41.56 78.79 94.51 409.70 76.13 27.22 11.70 6.51 3.93 3.10 14.32 769.97 1975 22.45 26.82 190.40 117.55 165.54 53.61 33.68 34.76 8.50 5.67 3.31 1.85 664.14 1976 27.31 23.64 25.39 31.72 196.52 25.32 16.75 4.15 2.79 2.25 1.56 2.17 359.57 1977 2.67 8.63 21.89 217.32 61.24 45.36 8.63 4.62 2.83 2.65 1.94 4.13 381.91 1978 13.52 5.33 4.22 3.51 4.33 4.88 1.19 0.48 0.33 0.96 2.28 6.55 47.58 1979 5.34 4.77 14.20 18.88 77.03 34.93 1.10 0.92 1.12 1.02 0.75 0.70 160.76 1980 1.27 2.82 12.54 9.63 31.89 48.91 2.72 2.01 1.48 1.17 1.18 3.66 119.28 1981 1.70 0.75 3.41 9.15 0.76 0.66 0.49 0.36 0.32 0.41 0.40 0.38 18.79 1982 1.82 9.71 1.04 4.07 1.04 0.80 0.97 0.41 0.43 0.50 0.54 1.47 22.80 1983 14.44 56.47 69.24 34.68 13.95 16.84 26.44 2.95 1.84 2.02 1.52 11.58 251.97 1984 12.81 4.70 4.31 2.15 24.30 14.56 1.87 1.07 0.72 0.47 0.35 0.37 67.68 1985 1.30 13.73 16.47 3.91 11.80 3.61 1.71 1.09 0.67 0.79 0.53 0.41 56.02 1986 0.99 4.14 16.94 45.76 6.16 23.77 3.29 1.30 0.84 0.66 1.01 12.30 117.16 1987 106.54 123.92 176.16 111.50 41.05 81.79 9.11 4.04 2.49 5.45 2.37 2.81 667.23 1988 76.98 36.51 58.01 95.04 159.80 85.84 10.89 9.13 11.70 5.08 3.14 1.62 553.74 1989 1.81 53.00 45.78 23.21 17.44 55.27 19.68 18.96 4.01 2.78 1.96 1.14 245.04 1990 0.68 1.42 1.64 37.86 69.08 24.48 6.33 2.67 1.91 1.67 1.06 0.79 149.59 1991 6.06 2.70 7.02 2.90 7.81 3.41 0.89 0.44 0.32 0.28 0.30 0.31 32.44 1992 0.43 1.02 3.46 4.80 10.30 7.76 1.46 1.07 0.66 0.49 0.29 0.36 32.10 1993 59.87 43.98 17.40 60.12 74.45 16.48 10.16 2.63 1.55 1.34 0.90 0.62 289.50 1994 1.41 2.58 0.93 9.21 9.52 3.58 2.93 2.79 1.30 0.65 0.48 0.44 35.82
51
Katse Dam sub-catchment (KAT9.INC)
Monthly natural historical streamflows (million m3) 1920 65.58 27.56 29.18 39.14 81.93 106.63 36.59 5.40 1.95 1.05 0.70 18.29 414.00 1921 18.48 70.18 100.66 108.03 59.14 39.77 10.34 4.51 8.74 4.31 9.34 5.54 439.04 1922 48.43 112.97 71.55 72.88 95.64 45.23 26.32 13.41 34.21 19.61 6.25 4.51 551.01 1923 16.32 35.96 35.43 59.30 50.68 129.15 40.87 1.93 1.55 1.61 1.63 51.58 426.01 1924 70.24 130.18 131.46 63.77 67.06 200.72 106.81 25.21 5.75 2.37 1.46 25.23 830.26 1925 40.88 72.99 34.62 56.00 59.37 83.99 24.30 2.04 2.90 2.00 0.90 41.48 421.47 1926 35.80 39.60 54.80 52.90 71.80 32.40 6.80 2.50 1.10 11.20 17.10 5.30 331.30 1927 45.50 24.60 77.70 98.50 65.50 48.30 14.10 4.00 2.90 1.80 4.20 5.60 392.70 1928 18.50 50.60 47.40 60.50 36.40 94.20 15.60 21.20 59.40 30.30 10.00 79.30 523.40 1929 51.90 49.00 64.70 57.20 55.60 131.20 75.10 13.40 4.10 3.60 9.70 37.70 553.20 1930 28.40 12.10 32.60 112.80 67.90 69.40 163.00 23.00 1.50 20.90 5.30 1.10 538.00 1931 14.30 47.80 21.60 22.70 127.60 96.70 14.50 4.40 2.60 2.00 1.20 3.50 358.90 1932 4.40 60.00 40.80 10.80 35.70 41.80 10.40 3.70 2.70 3.60 2.40 1.70 218.00 1933 2.90 150.90 162.80 244.10 123.30 100.20 79.00 74.70 31.60 54.70 42.20 7.10 1073.50 1934 46.20 205.00 112.70 33.90 35.70 67.30 28.20 12.70 5.00 2.40 5.30 3.70 558.10 1935 10.70 14.90 33.30 30.90 59.80 75.50 12.70 48.40 8.50 2.30 1.40 1.40 299.80 1936 54.00 244.80 69.60 176.20 183.70 83.80 12.00 2.00 1.00 1.80 1.20 0.90 831.00 1937 9.50 10.90 22.50 149.80 147.70 21.90 61.20 12.70 40.00 14.30 51.10 14.10 555.70 1938 54.30 17.00 74.80 86.60 213.60 46.20 5.90 24.20 6.40 17.90 17.40 8.80 573.10 1939 52.70 130.40 51.90 54.50 57.50 54.30 88.00 75.40 13.00 3.20 2.20 45.40 628.50 1940 10.50 75.70 96.00 108.90 97.80 29.60 52.70 9.10 1.20 4.10 2.80 8.30 496.70 1941 83.80 15.30 8.90 77.60 99.70 111.10 60.30 11.70 2.70 6.10 27.40 11.30 515.90 1942 65.10 89.30 111.50 109.70 29.30 74.50 78.40 82.30 15.80 150.00 68.00 10.60 884.50 1943 206.50 157.40 145.90 67.50 123.90 48.00 6.10 5.40 69.20 11.90 1.90 59.50 903.20 1944 65.00 44.70 10.40 11.60 68.40 186.20 44.80 12.20 4.60 1.90 1.30 0.90 452.00 1945 4.80 9.50 10.20 81.00 53.70 95.70 19.60 32.70 6.90 2.70 1.60 1.50 319.90 1946 140.70 95.00 24.50 18.90 68.10 62.10 61.10 10.20 4.90 5.20 2.70 26.20 519.60 1947 57.20 62.80 90.50 73.50 52.40 244.20 78.80 9.80 1.60 1.30 1.50 2.00 675.60 1948 30.20 9.60 6.60 64.30 35.80 97.30 70.60 14.60 3.00 2.40 2.50 4.80 341.70 1949 21.50 40.30 53.30 46.70 93.10 165.80 105.20 55.90 8.60 17.00 90.60 15.70 713.70 1950 11.30 11.70 74.20 64.00 74.50 49.10 43.30 12.40 5.60 2.90 7.80 6.30 363.10 1951 158.60 27.40 14.40 63.90 154.30 68.00 23.70 4.80 2.10 23.50 22.30 7.50 570.50 1952 7.60 40.70 39.20 29.90 158.40 30.80 19.00 6.50 2.60 1.50 5.20 3.50 344.90 1953 63.40 27.50 54.00 56.40 60.70 66.70 14.50 10.50 5.50 2.60 1.50 3.60 366.90 1954 4.40 30.20 26.30 165.60 204.70 37.30 33.00 24.30 5.70 3.30 2.20 1.40 538.40 1955 11.10 65.60 81.40 34.70 144.50 74.80 30.70 26.80 5.50 2.30 1.90 4.30 483.60 1956 74.80 120.00 222.70 118.20 48.30 92.20 26.10 4.40 9.20 21.10 29.70 240.60 1007.30 1957 269.60 72.90 45.50 175.50 37.30 24.50 66.60 32.20 6.60 1.90 1.20 15.40 749.20 1958 17.70 71.00 41.90 18.40 44.70 18.50 96.50 191.50 25.70 44.20 9.20 1.90 581.20 1959 103.70 70.90 96.90 49.50 123.40 101.20 46.50 9.80 3.20 4.00 13.30 8.90 631.30 1960 61.60 78.50 90.40 85.70 24.90 78.20 111.30 53.60 31.90 8.80 4.50 7.20 636.60 1961 2.80 105.60 59.60 61.80 177.40 65.40 44.20 8.20 1.50 1.00 2.30 3.50 533.30 1962 6.20 65.10 30.00 176.90 85.50 92.00 99.40 18.80 18.20 11.50 4.50 2.00 610.10 1963 33.30 65.90 42.50 108.80 38.70 142.70 36.70 4.90 26.40 5.80 4.80 6.80 517.30 1964 176.90 37.60 60.20 72.70 14.40 21.40 60.20 10.60 16.50 8.60 16.40 5.60 501.10 1965 10.10 36.10 12.50 166.30 85.60 21.70 7.40 4.20 3.10 2.00 2.80 2.70 354.50 1966 9.50 32.60 44.10 247.30 150.20 97.40 76.70 29.00 6.30 3.20 4.80 2.60 703.70 1967 1.80 91.60 68.70 16.30 5.90 25.50 25.00 83.00 8.80 6.00 4.00 6.00 342.60 1968 12.20 11.30 58.90 6.10 5.20 27.20 55.00 29.20 24.50 5.00 4.00 3.00 241.60 1969 75.60 29.70 39.60 40.00 46.20 3.60 2.10 1.70 1.00 1.00 1.00 17.80 259.30 1970 89.40 41.30 76.60 103.90 54.60 42.90 69.10 20.20 3.70 3.00 1.50 0.70 506.90 1971 1.60 10.70 25.50 106.10 111.80 152.10 20.80 27.90 7.00 2.80 1.60 1.50 469.40 1972 15.00 50.10 3.80 0.70 103.00 31.80 34.90 5.10 2.20 1.00 51.00 28.60 327.20 1973 30.80 31.00 71.50 107.20 115.20 24.00 41.10 15.20 16.90 6.30 11.00 11.50 481.70 1974 3.60 228.30 46.90 117.50 142.40 148.80 19.10 10.30 3.30 5.10 2.00 50.80 778.10 1975 77.40 252.20 113.30 204.20 190.60 113.50 56.90 32.80 48.20 9.20 4.30 18.20 1120.80 1976 225.20 211.40 13.30 63.30 120.70 190.50 20.90 10.00 3.50 2.00 1.40 6.40 868.60 1977 120.10 39.30 14.70 232.20 51.80 46.30 223.90 14.80 4.20 2.70 2.80 64.10 816.90 1978 60.40 26.30 164.30 33.40 26.90 27.60 3.10 8.70 10.30 31.10 157.00 92.30 641.40 1979 128.70 59.00 121.40 30.80 76.50 18.60 4.70 2.40 1.60 0.80 0.70 18.10 463.30 1980 24.00 73.00 52.10 247.90 64.50 96.00 63.80 24.00 35.70 4.40 33.80 33.90 753.10 1981 7.90 65.20 116.40 11.90 10.40 13.60 104.30 18.80 3.70 4.10 2.20 4.00 362.50 1982 53.60 158.70 14.50 5.10 10.40 18.00 5.80 17.10 8.60 2.60 4.60 3.10 302.10 1983 20.20 30.60 46.20 58.10 5.90 12.60 13.70 20.70 2.10 1.70 1.10 20.60 233.50 1984 14.80 44.30 22.20 11.30 102.90 55.00 11.90 1.90 0.80 0.70 0.20 0.10 266.10 1985 21.00 91.00 198.40 20.10 66.40 13.20 37.60 7.20 21.00 2.60 10.00 32.40 520.90 1986 133.20 240.40 22.60 19.70 20.10 23.90 61.70 4.30 1.30 1.00 44.70 271.70 844.60 1987 178.20 124.00 108.00 72.00 71.20 355.60 59.70 22.30 22.70 22.10 8.90 89.60 1134.30 1988 84.70 104.90 153.30 76.50 236.10 54.00 34.40 46.20 99.40 27.60 8.50 3.80 929.40 1989 15.30 139.10 52.30 16.60 54.90 47.90 124.10 36.80 7.90 14.30 26.90 8.30 544.40 1990 7.00 5.40 14.40 104.30 141.10 78.10 7.20 2.30 1.70 1.30 0.30 2.90 366.00 1991 140.20 50.30 24.60 8.70 2.40 5.20 3.10 0.80 0.10 0.10 0.60 7.60 243.70 1992 16.40 129.10 16.20 11.80 47.70 37.00 82.10 9.10 3.20 2.00 3.70 1.90 360.20 1993 142.50 58.00 93.70 117.50 167.40 45.70 81.70 30.20 4.50 2.80 2.20 1.50 747.70 1994 1.50 9.70 3.80 19.40 43.80 20.40 21.30 25.60 4.40 2.60 2.20 1.90 156.60 1995 11.40 27.90 106.50 108.10 115.90 132.10 10.60 10.80 10.90 20.00 17.40 8.70 580.30
52
Vaal Dam sub-catchment (VAAL9.INC)
Monthly natural historical streamflows (million m3) 1920 680.82 145.28 3.78 2.62 5.72 549.69 147.51 2.26 1.34 0.96 0.71 0.91 1541.60 1921 4.98 422.04 115.31 8.25 5.98 7.00 2.63 1.32 1.39 1.21 3.61 2.54 576.26 1922 217.37 354.78 83.01 476.46 127.60 0.91 1.39 1.22 1.17 1.95 1.50 0.62 1267.98 1923 1.46 5.88 0.35 46.76 76.35 15.44 29.47 10.42 7.00 5.84 4.72 0.44 204.13 1924 12.77 90.11 21.71 84.08 38.28 598.40 118.76 147.94 26.84 24.84 12.70 10.26 1186.69 1925 19.79 26.87 21.78 0.46 32.79 27.56 18.74 7.97 0.16 7.43 6.00 2.51 172.06 1926 12.87 8.23 35.78 42.01 125.24 193.98 64.40 4.92 5.97 3.29 12.07 3.72 512.48 1927 1.71 48.43 29.86 74.25 29.69 55.11 16.90 8.93 5.48 2.94 2.12 0.45 275.87 1928 28.48 20.29 17.65 47.76 72.38 228.68 24.04 13.98 22.45 17.62 7.06 69.31 569.70 1929 67.91 96.40 331.25 58.05 0.37 0.44 0.38 0.35 0.34 0.30 7.50 5.34 568.63 1930 0.81 16.11 2.10 105.20 93.01 20.54 29.97 7.88 1.37 3.70 6.10 1.77 288.56 1931 0.50 22.88 11.34 4.44 45.78 31.96 11.42 8.31 3.93 5.83 3.67 2.26 152.32 1932 7.88 17.38 15.09 21.74 10.65 57.78 6.43 4.44 1.24 1.23 1.49 1.18 146.53 1933 0.06 190.75 130.99 212.96 318.66 5.90 30.82 3.50 21.09 9.57 50.25 16.41 990.96 1934 23.42 202.02 245.70 50.01 19.73 62.34 39.71 23.06 10.57 9.19 7.53 5.82 699.10 1935 5.27 4.79 0.45 135.22 54.40 3.72 88.49 137.67 46.89 15.99 10.87 0.93 504.69 1936 7.94 492.75 72.18 145.56 330.36 32.88 25.24 7.61 7.88 5.85 4.04 1.07 1133.36 1937 20.40 8.71 223.57 80.15 45.15 17.17 7.62 0.58 9.99 8.93 0.91 15.05 438.23 1938 85.28 22.81 66.63 163.16 335.98 67.44 3.61 7.58 10.57 34.32 21.72 7.90 827.00 1939 19.19 184.32 146.20 46.54 0.56 9.56 7.43 6.82 7.81 9.32 9.47 18.69 465.91 1940 2.88 105.32 211.04 129.65 103.27 25.17 58.52 4.62 0.54 14.22 6.02 6.66 667.91 1941 9.75 12.81 11.82 45.94 29.57 20.93 23.48 6.33 10.59 11.54 13.97 2.94 199.67 1942 57.42 21.28 109.53 0.88 0.67 52.52 35.51 135.24 0.50 304.90 66.22 106.60 891.27 1943 89.00 108.86 219.94 150.25 828.86 59.14 0.43 0.22 4.66 19.87 18.06 81.79 1581.08 1944 153.44 46.04 0.24 4.91 83.34 220.89 14.50 1.70 14.21 15.28 6.07 6.40 567.02 1945 0.00 0.00 4.75 78.79 159.64 138.36 9.79 10.18 4.23 10.70 0.99 0.00 417.43 1946 10.16 50.80 53.32 78.01 37.41 1.45 40.64 9.65 7.12 8.34 5.12 2.62 304.64 1947 13.45 30.20 11.32 73.52 9.51 32.33 55.83 8.18 1.84 4.83 4.63 5.98 251.62 1948 1.28 17.16 22.97 28.56 66.75 27.77 1.73 4.80 0.48 0.49 2.51 0.16 174.66 1949 36.45 142.18 213.30 32.79 15.14 95.20 15.55 16.39 0.59 4.75 16.05 14.75 603.14 1950 16.02 2.90 4.56 83.53 57.02 12.71 11.94 4.95 2.82 3.17 6.49 7.02 213.13 1951 150.06 18.40 41.39 47.08 18.52 16.28 12.01 0.83 6.32 0.57 6.68 0.16 318.30 1952 3.85 143.16 25.37 13.33 172.87 120.94 46.55 7.37 0.63 3.39 2.99 0.00 540.45 1953 3.39 14.70 68.14 39.63 39.18 8.30 6.20 0.65 1.08 2.15 2.62 0.38 186.42 1954 0.65 26.91 63.89 1.40 569.63 215.63 9.92 0.60 0.64 10.47 5.43 8.47 913.64 1955 23.03 27.75 119.95 89.33 2.83 75.73 40.27 4.82 9.73 11.53 7.69 1.27 413.93 1956 11.23 175.21 391.30 156.47 29.22 69.77 17.60 0.84 0.53 82.81 17.42 272.71 1225.11 1957 434.71 95.05 21.08 46.83 20.76 23.89 29.38 3.58 3.81 13.33 17.88 33.54 743.84 1958 13.28 45.05 47.88 124.91 66.13 12.11 0.82 24.09 11.95 17.15 12.76 17.78 393.91 1959 18.97 63.57 66.47 10.54 93.45 61.80 2.00 32.13 5.80 9.97 17.89 7.37 389.96 1960 31.10 24.84 169.80 83.32 34.82 17.55 62.14 8.28 23.07 0.69 20.63 5.85 482.09 1961 18.06 8.63 48.08 11.93 39.59 31.91 1.05 1.95 1.98 4.32 5.23 29.62 202.35 1962 17.89 80.94 78.24 32.21 36.21 0.98 11.12 3.98 1.13 50.35 9.06 15.34 337.45 1963 6.21 28.81 71.14 154.07 42.69 36.16 21.22 8.67 0.64 11.65 9.85 9.68 400.79 1964 95.38 231.42 230.36 113.21 67.57 7.65 1.37 3.85 0.61 5.02 0.40 22.52 779.36 1965 0.18 7.74 7.61 30.88 68.12 1.30 0.18 1.13 2.22 0.51 5.28 2.92 128.07 1966 4.67 12.47 114.51 180.94 596.26 49.43 101.66 8.41 4.42 0.57 3.90 4.99 1082.23 1967 3.00 32.91 119.98 17.68 1.16 3.49 1.58 5.56 0.98 0.48 0.68 1.28 188.78 1968 2.26 4.19 3.96 21.62 6.26 33.91 18.70 14.29 6.40 0.81 0.65 2.03 115.08 1969 77.20 48.24 81.17 37.77 23.64 16.27 0.89 0.74 0.85 3.07 1.66 9.90 301.40 1970 15.01 98.92 1.63 46.20 41.99 3.74 140.90 13.68 1.06 0.96 3.93 4.09 372.11 1971 1.31 124.33 118.73 164.12 10.31 21.87 16.15 8.82 1.80 4.56 10.04 8.80 490.84 1972 13.60 22.51 8.33 0.69 61.27 7.09 1.84 1.24 0.80 0.44 14.80 6.29 138.90 1973 3.73 55.24 205.46 346.90 141.94 1.58 11.84 1.62 0.80 0.70 1.96 7.35 779.12 1974 0.69 123.31 211.08 200.87 711.90 123.59 48.66 12.28 8.55 7.13 9.93 10.32 1468.31 1975 28.92 132.75 182.31 168.90 510.02 3.56 11.39 113.82 10.87 12.65 16.75 15.73 1207.67 1976 130.87 51.37 69.16 68.85 400.42 15.63 14.45 0.71 4.83 6.93 9.46 13.75 786.43 1977 15.51 17.76 91.77 468.76 61.59 77.03 25.28 3.97 6.43 6.85 3.20 4.95 783.10 1978 98.89 18.47 43.73 20.11 2.59 12.83 1.40 0.79 0.53 4.61 28.70 21.95 254.60 1979 23.09 66.70 62.16 69.75 178.80 18.78 0.85 0.62 0.58 7.13 7.43 5.62 441.51 1980 10.94 23.37 80.94 83.06 53.32 163.27 1.78 0.82 0.74 2.42 1.11 15.96 437.73 1981 5.67 27.46 22.55 26.43 0.00 3.08 1.28 7.63 0.53 6.68 13.67 21.45 136.43 1982 38.83 25.96 9.74 17.06 6.82 10.16 0.64 0.52 0.70 4.74 16.55 1.84 133.56 1983 10.20 5.58 139.99 63.09 0.77 15.39 8.13 4.27 2.10 5.07 11.69 7.26 273.54 1984 10.60 16.97 11.93 21.86 3.24 17.26 2.58 2.69 4.21 11.51 18.66 13.62 135.13 1985 25.16 57.07 31.29 60.10 25.18 12.98 8.91 2.80 1.00 6.13 10.33 8.92 249.87 1986 13.26 35.78 31.34 76.76 37.92 9.62 7.30 0.87 3.55 3.41 6.33 98.59 324.73 1987 243.00 144.56 235.26 2.82 5.68 49.16 1.34 2.71 0.96 1.01 16.31 15.74 718.55 1988 271.46 80.97 2.19 64.89 551.39 80.98 1.75 12.42 14.77 4.97 18.94 9.38 1114.11 1989 11.02 16.94 16.33 5.75 17.08 16.00 18.85 34.00 3.66 7.49 9.24 12.69 169.05 1990 12.13 5.62 31.16 100.42 93.25 6.65 2.87 5.41 1.72 2.41 15.16 7.13 283.93 1991 5.78 7.47 9.43 16.41 12.89 9.90 5.79 2.12 1.95 4.86 8.90 6.43 91.93 1992 7.88 63.28 16.44 40.80 11.61 20.53 2.86 1.22 3.15 5.69 11.71 11.66 196.83 1993 92.78 15.78 66.98 42.35 23.19 14.71 2.62 1.72 1.03 1.29 5.78 4.76 272.99 1994 5.13 6.58 6.56 4.55 18.57 24.26 23.67 5.86 1.41 1.41 1.75 1.32 101.07
53
Welbedacht Dam sub-catchment (WELB9.INC)
Monthly natural historical streamflows (million m3) 1920 29.90 9.60 1.60 55.00 104.40 88.70 79.50 10.20 2.90 0.60 0.40 1.20 384.00 1921 1.10 56.70 145.10 122.40 14.00 4.30 2.70 0.80 4.60 1.20 0.60 0.40 353.90 1922 10.90 54.60 19.70 64.50 162.30 26.50 19.80 14.40 15.20 11.80 4.60 4.10 408.40 1923 1.60 33.60 14.10 36.00 75.70 214.70 27.10 3.20 1.90 1.10 0.60 16.70 426.30 1924 41.60 279.50 140.90 72.80 96.50 495.90 191.40 91.90 22.40 6.40 3.90 4.40 1447.60 1925 17.80 17.00 10.30 8.90 29.20 82.30 23.10 2.10 1.80 1.30 0.80 3.10 197.70 1926 8.10 52.20 37.60 34.90 35.80 139.00 36.60 2.00 1.10 1.20 2.60 0.80 351.90 1927 29.10 10.20 61.20 171.60 80.90 34.40 17.60 1.90 1.20 0.90 0.60 0.30 409.90 1928 27.80 37.70 58.40 61.20 20.80 88.10 25.10 4.00 11.50 12.60 2.60 64.90 414.70 1929 42.70 32.40 139.10 72.80 35.80 66.50 106.40 14.10 3.00 2.00 1.40 1.30 517.50 1930 10.30 1.90 10.30 75.80 75.60 50.40 199.30 18.90 3.10 5.00 2.20 0.50 453.30 1931 11.40 165.00 17.40 47.10 53.20 53.20 15.40 2.00 1.20 0.80 0.60 1.80 369.10 1932 1.20 14.30 21.30 10.70 15.20 20.70 7.80 0.10 1.20 0.40 0.10 0.00 93.00 1933 0.00 254.10 242.50 854.90 184.40 108.60 83.60 48.20 41.60 14.70 24.30 5.30 1862.20 1934 26.50 187.40 131.50 26.20 20.00 93.10 43.60 31.90 7.40 4.30 2.20 2.50 576.60 1935 5.00 27.00 11.70 66.00 33.70 19.10 40.70 9.80 6.20 0.80 0.40 0.20 220.60 1936 26.40 185.10 81.50 243.00 255.70 76.60 31.00 6.30 2.70 1.40 0.80 0.80 911.30 1937 1.70 8.20 20.50 27.30 143.60 39.20 24.80 15.40 4.00 3.70 4.10 7.20 299.70 1938 23.60 27.50 50.90 115.30 272.30 65.10 5.90 7.10 2.30 2.90 5.90 4.50 583.30 1939 108.80 136.30 22.50 8.30 26.40 41.40 31.40 38.90 3.40 1.70 1.00 37.80 457.90 1940 16.10 93.00 66.70 73.30 233.80 56.50 35.70 11.90 1.30 1.30 0.80 2.20 592.60 1941 49.60 12.70 1.00 78.20 60.70 161.60 33.90 10.80 1.90 1.00 5.20 5.40 422.00 1942 47.40 98.80 216.10 108.30 41.80 32.20 42.00 167.70 29.90 35.00 66.10 113.50 998.80 1943 155.00 503.20 445.00 160.80 137.60 69.70 11.20 4.60 24.60 7.80 2.80 8.60 1530.90 1944 42.90 36.60 14.00 6.90 25.00 110.50 19.90 6.50 3.20 1.40 0.80 0.70 268.40 1945 2.50 4.90 15.10 140.40 83.00 51.30 44.10 14.10 6.00 0.90 0.80 0.70 363.80 1946 64.20 40.30 22.80 36.90 88.90 26.40 35.50 40.20 1.50 0.90 0.80 5.10 363.50 1947 56.60 35.80 126.80 70.50 27.20 316.70 62.20 23.40 5.30 2.60 1.50 0.70 729.30 1948 6.60 6.10 0.60 41.30 49.60 29.00 2.90 2.20 0.70 0.40 0.20 0.40 140.00 1949 2.60 101.50 111.00 73.20 70.60 243.30 364.50 241.20 38.80 15.40 78.40 43.90 1384.40 1950 9.20 5.70 108.00 234.30 75.10 46.70 39.30 13.90 5.20 3.00 1.50 1.70 543.60 1951 209.80 79.80 11.10 32.30 90.50 76.20 11.10 1.70 1.50 6.00 2.60 2.70 525.30 1952 2.10 79.40 67.20 31.40 108.40 33.60 76.00 19.10 2.80 0.60 0.50 0.40 421.50 1953 57.70 57.50 144.50 73.40 68.00 238.90 72.90 16.60 11.90 7.00 0.40 0.40 749.20 1954 0.40 11.80 29.20 259.20 506.50 61.80 23.10 32.90 8.60 6.80 4.50 0.50 945.30 1955 9.90 35.20 189.40 59.70 191.20 169.90 144.80 18.50 7.90 3.20 2.50 1.80 834.00 1956 13.50 97.60 451.90 159.40 168.30 87.30 39.60 5.40 3.50 7.10 9.30 239.60 1282.50 1957 500.50 189.30 147.50 275.20 92.80 32.40 51.40 44.80 11.00 2.70 1.70 3.80 1353.10 1958 2.10 175.10 47.20 42.30 55.40 30.60 18.70 121.30 13.90 12.10 4.50 1.90 525.10 1959 18.90 65.40 133.80 50.20 58.80 62.20 65.50 12.00 3.20 3.00 4.80 3.10 480.90 1960 23.40 34.50 38.80 70.30 29.00 57.80 99.50 31.90 42.30 8.90 5.20 2.40 444.00 1961 1.00 80.50 141.50 25.40 360.10 105.70 17.80 5.90 2.30 1.80 1.30 1.10 744.40 1962 2.10 74.40 17.50 119.00 76.40 94.20 241.00 39.90 5.20 6.90 3.60 1.50 681.70 1963 2.10 89.20 81.60 22.00 16.50 55.80 120.50 4.40 4.30 1.60 1.90 1.00 400.90 1964 105.30 53.50 37.10 40.30 17.10 0.20 17.30 4.60 1.60 0.90 1.00 1.00 279.90 1965 1.00 8.60 5.70 186.00 178.70 5.00 1.10 1.10 1.00 0.60 0.40 0.30 389.50 1966 0.70 18.90 47.20 81.90 338.40 43.10 152.80 44.50 25.50 4.80 2.60 1.60 762.00 1967 6.10 42.70 17.50 7.10 2.20 23.70 31.70 39.10 4.30 2.30 1.30 1.20 179.20 1968 3.10 1.60 39.70 4.30 35.50 74.90 79.70 15.10 6.50 1.50 1.70 0.60 264.20 1969 44.70 17.50 15.70 22.00 28.50 1.10 0.10 0.10 0.20 0.60 0.40 4.90 135.80 1970 33.90 22.90 91.80 91.70 53.40 16.60 46.80 10.70 2.30 1.00 0.80 1.50 373.40 1971 0.90 1.60 14.50 267.30 295.90 240.20 46.50 23.10 3.20 1.30 0.50 2.10 897.10 1972 8.70 20.50 6.60 0.30 97.40 23.80 9.20 0.40 0.30 0.20 13.80 7.80 189.00 1973 4.00 7.50 57.70 325.60 376.20 90.20 31.20 4.70 2.30 1.50 2.10 1.20 904.20 1974 0.60 110.90 64.30 99.70 251.30 283.10 22.10 6.00 2.80 3.40 2.60 3.40 850.20 1975 11.60 107.10 144.10 671.70 780.60 397.10 236.90 43.80 14.10 7.30 4.30 20.20 2438.80 1976 321.50 128.60 23.50 69.40 159.40 326.70 41.80 5.60 3.60 2.30 2.00 33.70 1118.10 1977 83.60 36.40 30.80 310.80 127.00 93.60 473.30 54.10 7.80 5.20 4.00 6.30 1232.90 1978 11.10 4.60 294.70 33.20 43.70 25.00 2.60 2.60 1.40 6.40 57.80 18.50 501.60 1979 58.00 52.50 45.00 19.80 27.00 18.10 3.60 0.40 0.90 1.00 0.90 12.00 239.20 1980 8.60 29.50 53.90 266.60 195.50 140.00 25.80 8.70 8.70 2.60 42.30 16.40 798.60 1981 2.60 50.30 67.20 26.60 27.30 7.90 166.40 26.50 3.60 2.90 1.20 2.00 384.50 1982 40.30 162.90 25.10 3.00 6.10 3.60 3.50 1.90 1.50 5.30 1.70 0.70 255.60 1983 17.30 40.50 61.10 113.00 8.20 17.10 8.50 19.70 1.30 0.70 2.20 4.40 294.00 1984 8.90 45.80 34.10 27.40 58.30 56.20 4.50 0.20 2.60 0.40 0.30 0.50 239.20 1985 9.10 124.70 158.50 33.20 47.20 27.10 9.10 1.80 4.60 0.60 1.90 7.80 425.60 1986 55.20 283.10 25.60 3.90 14.30 16.20 39.60 2.40 0.30 0.40 11.40 161.60 614.00 1987 80.60 81.90 113.40 27.60 392.90 729.10 96.90 28.40 9.70 8.20 4.10 22.90 1595.70
54
APPENDIX B: MODEL’S RESULTS COMPARISON
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
1 2 3 4 5 6 7 8 9 10 11 12
month
Stre
amflo
w (m
illio
n cu
bicm
eter
s/a)
STOMSA meanHybrid MeanHistorical Mean
Figure B-3: Comparison of the mean for the Bloemhof Catchment
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
1 2 3 4 5 6 7 8 9 10 11 12
Month
Coe
ficie
nt o
f Var
ianc
e
Historical Hybrid ModelSTOMSA
Figure B-4: Comparison of Coefficient of Variance for the Bloemhof Catchment
55
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
1 2 3 4 5 6 7 8 9 10 11 12
Month
Stan
dard
dev
iatio
n
STOMSA StdevHybrid StdevHistorical Stdev
Figure B-5: Comparison of the Standard Deviations of the Bloemhof Catchment
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
lag -1 lag 0 lag 1
lag
Seria
l cor
rela
tion
Figure B-6: Serial correlation of month 1 and 12 from the Hybrid model for the Bloemhof Catchment
56
0.00
400.00
800.00
1200.00
1600.00
2000.00
2400.00
2800.00
3200.00
3600.00
1 2 3 4 5 6 7 8 9 10 11 12
Month
Varia
nce STOMSA Variance
Hybrid VarianceHistorical Variance
Figure B-7: Comparison of the Variances for the Bloemhof Catchment
57
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12
Month
Stre
amflo
w (m
illio
n cu
bicm
eter
s/a)
STOMSA meanHybrid MeanHistorical Mean
Figure B-8: Comparison of the mean flows for the Delangesdrift Catchment
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7 8 9 10 11 12
Month
Stan
dard
dev
iatio
n
STOMSA StdevHybrid StdevHistorical Stdev
Figure B-9: Comparison of the standard deviations for the Delangesdrift Catchment
58
0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
7000.00
1 2 3 4 5 6 7 8 9 10 11 12
Month
Varia
nce
of th
e st
ream
flow
dat
a
STOMSA VarianceHistorical VarianceHybrid Variance
Figure B-10: Comparison of the Variances for the Delangesdrift Catchment
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
1 2 3 4 5 6 7 8 9 10 11 12
Months
Coe
ffien
ts o
f Var
iatio
n
STOMSA CVHistorical CVHybrid CV
Figure B-11: Comparison of the coefficients of Variations for the Delangesdrift Catchment
59
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Lag -1 Lag 0 Lag 1
Lag
Seria
l Cor
rela
tion
Figure B-12: Box plot of the serial correlation between month 1 current year and 12 previous year for the Delangesdrift Catchment
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7 8 9 10 11 12
Months
Mea
n of
str
eam
flow
s (M
illio
n cu
bicm
eter
s/a)
STOMSA MeanHybrid MeanHistorical Mean
Figure B-13: Comparison of the Mean flows for the Katse Catchment
60
0
10
20
30
40
50
60
70
1 2 3 4 5 6 7 8 9 10 11 12
Months
Stan
dard
Dev
iatio
ns
STOMSA StdevHybrid StdevHistorical Stdev
Figure B-14: Comparison of the Standard Deviations for the Katse Catchment
0.00
0.50
1.00
1.50
2.00
2.50
1 2 3 4 5 6 7 8 9 10 11 12
Months
Coe
ffici
ents
of V
aria
nce
STOMSA CVHybrid CVHistorical CV
Figure B-15: Comparison of the Coefficients of Variations for the Katse Catchment
61
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Lag -1 Lag 0 Lag 1
Lag
Seria
l Cor
rela
tion
Figure B-16: Boxplot of the Serial correlation between moth 1 of the current year and month 12 of the previous year for the Katse Catchment
62
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10 11 12
Months
Mea
n of
str
eam
flow
s (M
illio
n cu
bicm
eter
s/a)
STOMSA MeanHistorical MeanHybrid Mean
Figure B-17: Comparison of the Mean flows for the Vaal Catchment
0
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9 10 11 12
Months
Stan
dard
Dev
iatio
ns o
f str
eam
flow
s
STOMSA StdevHybrid StdevHistorical Stdev
Figure B-18: Comparison of the Standard Deviations for the Vaal Catchment
63
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
1 2 3 4 5 6 7 8 9 10 11 12
Months
Coe
ffici
ents
of V
aria
tions
STOMSA CVHybrid CVHistorical CV
Figure B-19: Comparison of the Coefficients of Variations for the Vaal Catchment
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Lag -1 Lag 0 Lag 1
Lag
Seria
l Cor
rela
tion
Figure B-20: Boxplot of the Serial Correlation between month 1 of current year and month 12 of the previous year for the Vaal Catchment
64
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10 11 12
Months
Mea
n flo
ws
(Mill
ion
cubi
cmet
ers/
a)
STOMSA MeanHybrid MeanHistorical Mean
Figure B-21: Comparison of the Mean flows for the Welbedacht Catchment
0
20
40
60
80
100
120
140
160
1 2 3 4 5 6 7 8 9 10 11 12
Months
Stan
dard
Dev
iatio
ns
STOMSA StdevHybrid StdevHistorical Stdev
Figure B-22: Comparison of the Standard Deviations of the Welbedacht Catchment
65
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 11 12
Months
Coe
ffici
ents
of V
aria
tions
STOMSA CVHistorical CVHybrid CV
Figure B-23: Comparison of the Coefficients of Variations for the Welbedacht Catchment
-0.2
0
0.2
0.4
0.6
0.8
1
Lag -1 Lag 0 Lag 1
Lag
Seria
l cor
rela
tions
Figure B-24: Boxplot of Serial correlations between month 1 of current year and month 12 of previous year for the Welbedacht catchment
66
0
5
10
15
20
25
30
35
40
45
50
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Months
Mea
n St
ream
flow
(Mill
ion
cubi
cmet
ers)
Figure B-25: Bloemhof Catchment's Boxplots of Mean Streamflows
Figure B-26: Bloemhof Boxplot of Mean streamflows from STOMSA
67
0
10
20
30
40
50
60
70
80
90
100
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Months
Stan
dard
Dev
iatio
ns o
f str
eam
flow
s (M
illio
n cu
bicm
eter
s)
Figure B-27: Bloemhof Boxplot of Standard Deviations from the Hybrid Model
Figure B-28: Bloemhof Boxplot of Standard Deviations from STOMSA
68
0
10
20
30
40
50
60
70
80
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Mea
n st
ream
flow
(Mill
ion
cubi
cmet
ers)
Figure B-29 : Delangesdrift Catchment's Boxplots of Mean Streamflows
Figure B-30: Delangesdrift Catchment Boxplot of Mean streamflows from STOMSA
69
0
20
40
60
80
100
120
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Months
Stan
dard
Dev
iatio
ns o
f Str
eam
flow
s (M
illio
n cu
bicm
eter
s)
Figure B-31 : Delangesdrift Catchment’s Boxplot of Standard Deviations from the Hybrid Model
Figure B-32: Delangesdrift Catchment Boxplot of Standard Deviations from STOMSA
70
0
20
40
60
80
100
120
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Mea
n St
ream
flow
s (M
illio
n cu
bicm
eter
s)
Figure B-33: Katse Catchment's Boxplots of Mean Streamflows
Figure B-34: Katse Catchment Boxplot of Mean streamflows from STOMSA
71
0
10
20
30
40
50
60
70
80
90
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Stan
dard
Dev
iatio
ns o
f str
eam
flow
s (M
illio
n cu
bicm
eter
s)
Figure B-35: Katse Catchment’s Boxplot of Standard Deviations from the Hybrid Model
Figure B-36: Katse Catchment Boxplot of Standard Deviations from STOMSA
72
0
20
40
60
80
100
120
140
160
180
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Mea
n St
ream
flow
(Mill
ion
Cub
icm
eter
s)
Figure B-37: Vaal Catchment's Boxplots of Mean Streamflows
Figure B-38: Vaal Catchment Boxplot of Mean streamflows from STOMSA
73
0
50
100
150
200
250
300
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Stan
dard
Dev
iatio
ns o
f str
eam
flow
s (M
illio
n cu
bicm
eter
s)
Figure B-39: Vaal Catchment’s Boxplot of Standard Deviations from the Hybrid Model
Figure B-40: Vaal Catchment Boxplot of Standard Deviations from STOMSA
74
0
20
40
60
80
100
120
140
160
180
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Mea
n St
ream
flow
(Mill
ion
cubi
cmet
ers)
Figure B-41: Welbe Catchment's Boxplots of Mean Streamflows
Figure B-42: Welbe Catchment Boxplot of Mean streamflows from STOMSA
75
0
50
100
150
200
250
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Month
Stan
dard
Dev
iatio
ns o
f Str
eam
flow
(Mill
ion
cubi
cmet
ers)
Figure B-43: Welbe Catchment’s Boxplot of Standard Deviations from the Hybrid Model
Figure B-44: Welbe Catchment Boxplot of Standard Deviations from STOMSA
76
APPENDIX C: MATLAB CODE
c=0; for i=1:5 c=c+1; if (c>=1 & c<2) load Vaal g=Vaal; elseif (c>=2 & c<3) load Welbe g=Welbe; elseif (c>=3 & c<4) load Dela g=Dela; elseif (c>=4 & c<5) load bloem g=bloem; else load Katse g=Katse; end %load bloem %g=bloem; %obtain the dimention of data and store in vs vs=size(g,1); Y=zeros(vs,12); %standardise original data g, into Y for i=2:13 for j=1:vs Y(j,i-1)=(g(j,i)-mean(g(1:vs,i)))/std(g(1:vs,i)); end end %calculate the coefiecints using autocorrelation coef=zeros(5,12); for i=2:13 coef(1:5,i-1)=autocorr(Y(1:vs,i-1),4); end %transform Y to a vector of (1 X 961), with the first value being zero Yp=zeros(1,vs*12+1); l=0; for j=1:vs for i=1:12 l=l+1; Yp(1,l+1)=Y(j,i); end end %calculate the residual e=zeros(vs*12,1); l=0; for j=1:vs for i=1:12 l=l+1; e(l,1)=Yp(l+1)-coef(2,i)*Yp(l); end end %apply bootstrap vl=76; K=zeros((12*vl*99),1); E=[e;K];
77
a = 1; b = (vl-1); B=zeros((12*vl*100),1); r=(12*vl*100)/24; for i=1:r x = round(a + ((b-a) * rand(1)));%generate random numbers between 1 and 79, uniformly distributed B((i*24-23):(i*24),1)=E((x*12-11):(x*12+12),1); end %postblanketing Z=zeros((12*vl*100),1); Zp=[0;B]; l=0; r=(vl*100); for j=1:r for i=1:12 l=l+1; Z(l)=coef(2,i)*Zp(l)+ Zp(l+1); end end %inverse standardise X=zeros((12*vl*100),1); l=0; r=vl*100; for j=1:r for i=2:13 l=l+1; X(l)=Z(l)*std(g(1:vs,i))+mean(g(1:vs,i)); if (X(l)<0) X(l)=0; end end end %convert the synthetic data to a matrix of a number of years 76 X100 those of the original data M=zeros((vl*100),12); l=1; for i=1:(vl*100) for j=1:12 M(i,j)=X(l,1); l=l+1; end end %Calculate autocorrelation/correlation coeficients to compare with data of origina data coefsyth=zeros(5,12); for i=1:12 coefsyth(1:5,i)=autocorr(M(((vl*100)-vl):(vl*100),i),4); end %calculate the matrices of stsistics (Mean, skew, Stdev, Autocorr) KK=[M,transpose(sum(transpose(M)))];%make totals %MM(1:7600,c)=transpose(sum(transpose(M)); %Mean MeanMatrix=zeros(100,13); l=0; lm=0; for i=1:100 l=l+vl; MeanMatrix(i,1:13)=mean(KK((1+lm):l,1:13)); lm=lm+vl; end %StDev StDevMatrix=zeros(100,13); l=0; lm=0; for i=1:100
78
l=l+vl; StDevMatrix(i,1:13)=std(KK((1+lm):l,1:13)); lm=lm+vl; end %Skewness SkewMatrix=zeros(100,13); l=0; lm=0; for i=1:100 l=l+vl; SkewMatrix(i,1:13)=skewness(KK((1+lm):l,1:13)); lm=lm+vl; end % Maximum deficit which is the lowest storage level for each % sequence for r=1:5 l=0; lm=0; for i =1:100%number of sequences M_Stor =0; l=l+vl; for j=1:vl%number of years in a sequences for rj=1:12 B_Storage(1,rj)=M_Stor-KK(lm+j,rj)+(((9-r)/10)*(mean(g(1:vs,14))))/12; Monthly_Storage(1,rj)=max(0,B_Storage(1,rj)); M_Stor=Monthly_Storage(1,rj); end Storage(j,i)=sum(transpose(Monthly_Storage)); end lm=lm+vl; end Maxd=max(Storage); Maxdef(r,1:100)=Maxd; end %{ % Run Sums 6,12,24,36,48,60,72,84,96,108/ R=[6,12,24,36,48,60,72,84,96,108]; l=0; lm=0; for i=1:100 l=l+912; RunsMatrix=skewness(X((1+lm):l,1)); for j=1:6 r=R(j); t=912-r+1; for k=1:t sumall=sum(RunsMatrix(k:r+k-1)); sums(k)=sumall(k); end minruns=min(sums); Minsums(i,j)=minruns end lm=lm+912; end %} %Serial correlation between month 1 and moth 12 l=0; lm=0;
79
for j=1:100 l=l+vl; AutocorSyth1=KK((1+lm):l,1); AutocorSyth12=KK((1+lm):l,12); SerialS=crosscorr(AutocorSyth1(1:76,1),AutocorSyth12(1:76,1),1); SerialcorrSyth(j,1:3)=transpose(SerialS(1:3,1)); lm=lm+vl; end AutoData1=(g(1:vs,2)); AutoData2=(g(1:vs,13)); SerialData=transpose(crosscorr(AutoData1,AutoData2,1)); SerialSyth=mean(SerialcorrSyth); %storing the results into different matrices if (c>=1 & c<2)% store the results for the Vaal %VautocorrMatrix=AutocorrMatrix; VAutocorr=coefsyth; VMmeanmatrix=MeanMatrix; VStDevMatrix=StDevMatrix; VSkewMatrix=SkewMatrix; VMmean=mean(MeanMatrix); VMeandata=mean(g(1:vs,2:13)); VMeansyth1=mean([M,transpose(sum(transpose(M)))]); VMeansyth=VMeansyth1(1:12); %remove the total from meansyth1 VStddata=std(g(1:vs,2:13)); VStdsyth1=std([M,transpose(sum(transpose(M)))]); VStdsyth=VStdsyth1(1:12); %remove the total from Stdsyth1 VSkdata=skewness(g(1:vs,2:13)); VSksyth1=skewness([M,transpose(sum(transpose(M)))]); VSksyth=VSksyth1(1:12); %remove the total from Sksyth1 VMn=transpose([VMeandata;VMeansyth]); VSt=transpose([VStddata;VStdsyth]); VSk=transpose([VSkdata;VSksyth]); VSythData=KK; VSerialcorr1_12Syth=SerialcorrSyth; VSerialcorr1_12Data=SerialData; VSerialcorSythmean=SerialSyth; %Write data to files csvwrite('Vaalmean.csv',VMn); csvwrite('VaalSTdev.csv',VSt); csvwrite('VaalSkew.csv',VSk); csvwrite('VaalMeanMatrix.csv',VMmeanmatrix); csvwrite('VaalStdevMatrix.csv',VStDevMatrix); csvwrite('VaalSkewMatrix.csv',VSkewMatrix); csvwrite('VaalAutocorr.csv',VAutocorr); csvwrite('Vaal_Sytheticdata.csv',KK); csvwrite('Vaal_Maxdef.csv',transpose(Maxdef)); csvwrite('VSerialcorrSyth.csv',VSerialcorr1_12Syth); csvwrite('VSerialcorrData.csv',VSerialcorr1_12Data); csvwrite('VSerialcorSythmean.csv',VSerialcorSythmean); elseif (c>=2 & c<3)% store the results for the Welbe WAutocorr=coefsyth; WMmeanmatrix=MeanMatrix; WStDevMatrix=StDevMatrix; WSkewMatrix=SkewMatrix; WMmean=mean(MeanMatrix); WMeandata=mean(g(1:vs,2:13)); WMeansyth1=mean([M,transpose(sum(transpose(M)))]); WMeansyth=WMeansyth1(1:12); %remove the total from meansyth1 WStddata=std(g(1:vs,2:13)); WStdsyth1=std([M,transpose(sum(transpose(M)))]); WStdsyth=WStdsyth1(1:12); %remove the total from Stdsyth1
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WSkdata=skewness(g(1:vs,2:13)); WSksyth1=skewness([M,transpose(sum(transpose(M)))]); WSksyth=WSksyth1(1:12); %remove the total from Sksyth1 WMn=transpose([WMeandata;WMeansyth]); WSt=transpose([WStddata;WStdsyth]); WSk=transpose([WSkdata;WSksyth]); WSythData=KK; WSerialcorr1_12Syth=SerialcorrSyth; WSerialcorr1_12Data=SerialData; WSerialcorSythmean=SerialSyth; %Write data to files csvwrite('Welbemean.csv',WMn); csvwrite('WelbeSTdev.csv',WSt); csvwrite('WelbeSkew.csv',WSk); csvwrite('WelbeMeanMatrix.csv',WMmeanmatrix); csvwrite('WelbeStdevMatrix.csv',WStDevMatrix); csvwrite('WelbeSkewMatrix.csv',WSkewMatrix); csvwrite('WelbeAutocorr.csv',WAutocorr); csvwrite('Welbe_Sytheticdata',KK); csvwrite('Welbe_Maxdef.csv',transpose(Maxdef)); csvwrite('WSerialcorrSyth.csv',WSerialcorr1_12Syth); csvwrite('WSerialcorrData.csv',WSerialcorr1_12Data); csvwrite('WSerialcorSythmean.csv',WSerialcorSythmean); elseif (c>=3 & c<4)% store the results for the Dela DAutocorr=coefsyth; DMmeanmatrix=MeanMatrix; DStDevMatrix=StDevMatrix; DSkewMatrix=SkewMatrix; DMmean=mean(MeanMatrix); DMeandata=mean(g(1:vs,2:13)); DMeansyth1=mean([M,transpose(sum(transpose(M)))]); DMeansyth=DMeansyth1(1:12); %remove the total from meansyth1 DStddata=std(g(1:vs,2:13)); DStdsyth1=std([M,transpose(sum(transpose(M)))]); DStdsyth=DStdsyth1(1:12); %remove the total from Stdsyth1 DSkdata=skewness(g(1:vs,2:13)); DSksyth1=skewness([M,transpose(sum(transpose(M)))]); DSksyth=DSksyth1(1:12); %remove the total from Sksyth1 DMn=transpose([DMeandata;DMeansyth]); DSt=transpose([DStddata;DStdsyth]); DSk=transpose([DSkdata;DSksyth]); DSythData=KK; DSerialcorr1_12Syth=SerialcorrSyth; DSerialcorr1_12Data=SerialData; DSerialcorSythmean=SerialSyth; %Write data to files csvwrite('Delamean.csv',DMn); csvwrite('DelaSTdev.csv',DSt); csvwrite('DelaSkew.csv',DSk); csvwrite('DelaMeanMatrix.csv',DMmeanmatrix); csvwrite('DelaStdevMatrix.csv',DStDevMatrix); csvwrite('DelaSkewMatrix.csv',DSkewMatrix); csvwrite('DelaAutocorr.csv',DAutocorr); csvwrite('Dela_Sytheticdata',KK); csvwrite('Dela_Maxdef.csv',transpose(Maxdef)); csvwrite('DSerialcorrSyth.csv',DSerialcorr1_12Syth); csvwrite('DSerialcorrData.csv',DSerialcorr1_12Data); csvwrite('DSerialcorSythmean.csv',DSerialcorSythmean); elseif (c>=4 & c<5)% store the results for the Bloem BAutocorr=coefsyth; BMmeanmatrix=MeanMatrix; BStDevMatrix=StDevMatrix; BSkewMatrix=SkewMatrix; BMmean=mean(MeanMatrix); BMeandata=mean(g(1:vs,2:13)); BMeansyth1=mean([M,transpose(sum(transpose(M)))]); BMeansyth=BMeansyth1(1:12); %remove the total from meansyth1
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BStddata=std(g(1:vs,2:13)); BStdsyth1=std([M,transpose(sum(transpose(M)))]); BStdsyth=BStdsyth1(1:12); %remove the total from Stdsyth1 BSkdata=skewness(g(1:vs,2:13)); BSksyth1=skewness([M,transpose(sum(transpose(M)))]); BSksyth=BSksyth1(1:12); %remove the total from Sksyth1 BMn=transpose([BMeandata;BMeansyth]); BSt=transpose([BStddata;BStdsyth]); BSk=transpose([BSkdata;BSksyth]); BSythData=KK; BSerialcorr1_12Syth=SerialcorrSyth; BSerialcorr1_12Data=SerialData; BSerialcorSythmean=SerialSyth; %Write data to files csvwrite('Bloemmean.csv',BMn); csvwrite('BloemSTdev.csv',BSt); csvwrite('BloemSkew.csv',BSk); csvwrite('BloemMeanMatrix.csv',BMmeanmatrix); csvwrite('BloemStdevMatrix.csv',BStDevMatrix); csvwrite('BloemSkewMatrix.csv',BSkewMatrix); csvwrite('BloemAutocorr.csv',BAutocorr); csvwrite('Bloem_Sytheticdata',KK); csvwrite('Bloem_Maxdef.csv',transpose(Maxdef)); csvwrite('BSerialcorrSyth.csv',BSerialcorr1_12Syth); csvwrite('BSerialcorrData.csv',BSerialcorr1_12Data); csvwrite('BSerialcorSythmean.csv',BSerialcorSythmean); else %store the results for Katse KAutocorr=coefsyth; KMmeanmatrix=MeanMatrix; KStDevMatrix=StDevMatrix; KSkewMatrix=SkewMatrix; KMmean=mean(MeanMatrix); KMeandata=mean(g(1:vs,2:13)); KMeansyth1=mean([M,transpose(sum(transpose(M)))]); KMeansyth=KMeansyth1(1:12); %remove the total from meansyth1 KStddata=std(g(1:vs,2:13)); KStdsyth1=std([M,transpose(sum(transpose(M)))]); KStdsyth=KStdsyth1(1:12); %remove the total from Stdsyth1 KSkdata=skewness(g(1:vs,2:13)); KSksyth1=skewness([M,transpose(sum(transpose(M)))]); KSksyth=KSksyth1(1:12); %remove the total from Sksyth1 KMn=transpose([KMeandata;KMeansyth]); KSt=transpose([KStddata;KStdsyth]); KSk=transpose([KSkdata;KSksyth]); KAnnual=transpose(sum(transpose(M))); KSythData=KK; KSerialcorr1_12Syth=SerialcorrSyth; KSerialcorr1_12Data=SerialData; KSerialcorSythmean=SerialSyth; %Write data to files csvwrite('Katsemean.csv',KMn); csvwrite('KatseSTdev.csv',KSt); csvwrite('KatseSkew.csv',KSk); csvwrite('KatseMeanMatrix.csv',KMmeanmatrix); csvwrite('KatseStdevMatrix.csv',KStDevMatrix); csvwrite('KatseSkewMatrix.csv',KSkewMatrix); csvwrite('KatseAutocorr.csv',KAutocorr); csvwrite('Katse_Sytheticdata.csv',KK); csvwrite('Katse_Maxdef.csv',transpose(Maxdef)); csvwrite('KSerialcorrSyth.csv',KSerialcorr1_12Syth); csvwrite('KSerialcorrData.csv',KSerialcorr1_12Data); csvwrite('KSerialcorSythmean.csv',KSerialcorSythmean); end % end storring the results %MM(1:7600,c)=transpose(sum(transpose(M)); %csvwrite('Annual_for_all.csv',MM); end
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% CROSS CORRELATIONS % 1: Vaal and Welbe l=0; lm=0; for j=1:100 l=l+vl; VSyth=VSythData((1+lm):l,1:13); WSyth=WSythData((1+lm):l,1:13); for i=1:13 XCFS = crosscorr(VSyth(1:76,i),WSyth(1:76,i),4); VWcrosscorSyth(1:9,i)=XCFS(1:9,1); end %sort VWlag14(j,1:13)=VWcrosscorSyth(1,1:13); VWlag13(j,1:13)=VWcrosscorSyth(2,1:13); VWlag12(j,1:13)=VWcrosscorSyth(3,1:13); VWlag11(j,1:13)=VWcrosscorSyth(4,1:13); VWlag0(j,1:13)=VWcrosscorSyth(5,1:13); VWlag1(j,1:13)=VWcrosscorSyth(6,1:13); VWlag2(j,1:13)=VWcrosscorSyth(7,1:13); VWlag3(j,1:13)=VWcrosscorSyth(8,1:13); VWlag4(j,1:13)=VWcrosscorSyth(9,1:13); lm=lm+vl; end for i=1:13 XCFD = crosscorr(Vaal(1:68,i+1),Welbe(1:68,i+1),4); VWcrosscorData(1:9,i)=XCFD(1:9,1); end VWlag14Mean=mean(VWlag14); VWlag13Mean=mean(VWlag13); VWlag12Mean=mean(VWlag12); VWlag11Mean=mean(VWlag11); VWlag0Mean=mean(VWlag0); VWlag1Mean=mean(VWlag1); VWlag2Mean=mean(VWlag2); VWlag3Mean=mean(VWlag3); VWlag4Mean=mean(VWlag4); VWcrosscorSythMean=[VWlag14Mean;VWlag13Mean;VWlag12Mean;VWlag11Mean;VWlag0Mean;VWlag1Mean;VWlag2Mean;VWlag3Mean;VWlag4Mean]; csvwrite('VWcrosscorData.csv',VWcrosscorData); csvwrite('VWcrosscorSyth.csv',VWcrosscorSythMean); Mmeanmatrix=mean(MeanMatrix(1:100,1:12)); %make summary statistics Meandata=mean(g(1:vs,2:13)); Meansyth1=mean([M,transpose(sum(transpose(M)))]); Meansyth=Meansyth1(1:12); %remove the total from meansyth1 Stddata=std(g(1:vs,2:13)); Stdsyth1=std([M,transpose(sum(transpose(M)))]); Stdsyth=Stdsyth1(1:12); %remove the total from Stdsyth1 Skdata=skewness(g(1:vs,2:13)); Sksyth1=skewness([M,transpose(sum(transpose(M)))]); Sksyth=Sksyth1(1:12); %remove the total from Sksyth1 Mn=transpose([Meandata;Meansyth]); St=transpose([Stddata;Stdsyth]); Sk=transpose([Skdata;Sksyth]); Maencomp=[Mmeanmatrix;Meansyth]; figure
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boxplot(KMmeanmatrix) title('Katse MeanMatrix') figure boxplot (KStDevMatrix) title('Katse Standard deviation matrix') figure boxplot (KSkewMatrix) title('Katse Skewness Matrix') figure plot(St) title('Standard deviation') xlabel({'Raw';'Syth'}) figure plot(Mn) title('Mean') xlabel({'Raw';'Syth'}) figure plot(Sk) title('Skewness') xlabel({'Raw';'Syth'}) gd=g(1:vs,2:13); figure boxplot(gd) title('gd') hold on %boxplot(M) plot(Mn) title('Syth') hold off figure boxplot(transpose(Maxdef)) title('Maxdef') figure plot(transpose(Storage(1:vl,1))) title('Storage trajectory')
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