Water Resources in Bulgaria during the drought period - quantitative investigations

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Method of composition applied for selected
karstic springs in Bulgaria
Elena Kirilova Bojilova
National Institute of Meteorology and Hydrology -
Bulgarian Academy of Sciences
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Structure of the study

• Introduction of method of composition
• Optimal conditions for application of the method
• Method of composition
• function of two arguments
• correlation between two variables
• Conclusions
• Acknowledgements

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Introduction

• The method of composition for specifying
  frequency curves is possible to use in three
  cases
• Composition in case of function between two
  arguments and
• Correlation between two and
• Correlation between three variables.
• The object of the study is an application of
  method of composition in case of function between
  two arguments and correlation between two
  variables for selected karstic springs.

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Method of composition

• The method of composition is executed and is
  possible to apply in existence of the following
  conditions functional relationship between study
  random variable and its arguments and
  independence between the arguments.
• The method gives the possibility in case of
  functional relationship between the study set and
  l-independent components (sets) to obtain
  effective estimation of probability curve. The
  number of numerical realizations form n real
  numbers rises to Nnl.

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Method of composition can follow the next steps

• Increasing (decreasing) order of the set of
  functional time series X0 and its components X1,
  X2, X3,.., Xl
• Calculation of Nnl values of the function
  X0f(X1, X2, X3,..,Xl) applying to step by step
  arrangement of every argument Xi with all others
  X1,X2,Xi-1, Xi1,..,Xl
• Ordering of all N calculated values of X0 in
  increasing (decreasing) order
• Building of the empirical composed probability
  curve Px0PX0(M)ltx0 using one of existing
  methods (graphical, analytical or
  graphanalytical).

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Optimal conditions

• From independence of the arguments for their
  probability of exceedance (or non-exceedance) for
  X0 P(X0)P(X1). P(X2).
• Due to this we can obtain n2 possible
  realizations between the two independent
  variables in case of existing period of
  observation for both sets in the some time with
  the single length n.

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• Gerassimov (1978 and 1988) introduced for the
  first time the so-called volume of independent
  outgoing information (NH - reduced volume of
  outgoing information for given correlative
  function f(X1,X2)).
• The maximum effectiveness of the composition is
  obtained for r02r010.707. According to
  Gerassimov maximum volume of independent
  information can has its highest value NH1.293n
  in the case of equal correlation coefficients
  (r02r010.707) of the orthogonalysed relation.

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Optimal conditions
Table Effect of composition using Ek
(effectiveness)
Source Prof. Gerassimov
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Fig. Composed probability curve of non-exceedance
for annual discharge

• Table Composition for function of two
  arguments, Spring Kotel, n43 years.
• M is a number of possible compositions P,
  is an empirical probability.

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Fig. Composed probability curve of non-exceedance
for annual discharge for Spring Beden, n38 years
Composition for function of two arguments
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Method of composition Correlation between two
variables
Table Composition for Spring Jazo (N59), n41
years M is a number of possible
compositions P, is an empirical probability.
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Fig. Composed probability curve of exceedance for
annual flow volumes spring Jazo (Q, x106m3), n41
years
Method of composition Correlation between two
variables
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Method of composition Correlation between two
variables

• Table Composition for spring Malko Tarnovo
  (N63), n41 years

Fig. Composed probability curve of non-exceedance
of annual flow volumes (Q, 106m3)
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Results of method of composition using two
different cases
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Conclusions

• The method of composition is proved itself to be
  very powerful for extending probability curve of
  the empirical distribution.
• The empirical points are smoothed very well with
  one composed curve.

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Conclusions

• With method of composition the statistical
  parameters of distribution curves can be
  estimated without bias.
• With method of composition is possible to
  estimate the quantiles in the range of low
  probability of occurrence up to 1/n2 (were n is
  the length of the recorded stations).

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Acknowledgement

• The paper is based on the PhD (2003) thesis of
  the author. I want to express my deep sense of
  gratitude and appreciations to Prof. M.J. Hall
  (IHE-Delft) and Prof. Str. Gerasimov
  (NIMH-Sofia).
• I want to acknowledge the financial support from
  project 875.797.3 UVO-ROSTE Venice Office.
• I want to acknowledge the financial support from
  the BALWOIS Secretariat.

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Dear colleagues ,thank you very much for your
kind attention!! Elena