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**Project ID 262 - Mathematical Modeling of Rainfalls in Dobroudja**

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The general objective of this project is to increase the knowlege area in the field of the modeling the hydrological time series and the application of this knowledge to solve some concrete problems (the rainfalls in Dobroudja).

The most important problems that will be solved by this project concern the multiscale modeling, the long range dependence and the noise presence in the data series.

- Although well known, the time series decomposition in trend, seasonal component and random component is a method that was especially used in the economic sciences. We propose its application on additive and multiplicative models and the residuals study for the reasons that the rainfall data have a seasonal behaviour and the residual analysis can give an indication on the steps to be followed in the series modeling.

- The problem of the detection of the data dependence in time appears because the data are generally dependent ones to others and they have not the same expectation and variance. Box-Jenkins methods will be used to model the stationary rainfall series. It was used till now to model the river flows and offers the opportunity to give short time predictions. The long range dependence (LRD) is related to the presence of the statistical auto similarity that can be characterized by the auto similarity parameter, H. LRD is a statistical property difficult to deal with. It can be thought in the time domain and in the frequency domain. The series with LRD property are slowly convergent to the expectation. Hurst parameter is perfectly defined point of view of mathematics, but it is very difficult to be measured because the form of the spectral density must be a priori known. The best estimation methods are parametric (Whittle estimator, local Whittle). H estimated can be biased if the function of spectral density is not correct. As a consequence, we will try to improve the methods based on periodogramme, to estimate and to reduce the bias of H. We shall try to apply a method that represents the estimators of H as functions of a parameter that balances the displacement from the variance. The method will be tested on simulated signals and will be used in correlation with other methods of H determination, on the real data. A program that calculates H will be elaborated.

- In physics, geophysics, hydrology, meteorology etc. the research, the studies, the technological and operational developments blunder into a fundamental difficulty: the extreme variability on a large range of scales in time and space, derived by the nonlinear interactions between the scales and also between different phenomena. To surpass this difficulty, without artificial parametrization, we shall try to develop some specific instruments (in order to obtain simple theoretical representations) and some methods of quantitative evaluation of this variability, permitting to be repeated in a simple manner at different scales. It implies the use of the scale or fractal invariance, evolving a unifying approach in the mentioned domains.

- The noise presence is one of the impediments that affects the data accuracy and has an impact on the correctness of a mathematical model. Interesting methods for the noise detection and removal from stationary series are known. In the case of nonstationary series there are some recent approaches (based on the recurrence points, the analysis of cross-correlation sum) for the dynamical systems, in which the evolution lows are known. In our study we shall try to apply a new methods based on the determination of the curves of logarithmic displacements, applied on quasi - stationary parts of the series, taking into account that the evolution low is unknown. The hidden frequencies (relieved by Fourier analysis) will be determined, adapting an algorithm that use the recurrence points and that function on simulated cases.

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The project is divided into the following stages:

2007Objectives:

2008Objectives:

2009Objectives:

- analysis of short/long dependence of precipitations series,
- determination of ARIMA type models for the series that are stationary or can be stationarized,
- prediction of precipitation evolution,
- determination of FARIMA type models for the series with the long range dependence property,
- dissemination (Synthesis, Results).

2010Objectives:

The activities associated to the objectives of ach year and their realisation stages are presented in Execution Plan.

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** PROJECT DIRECTOR**

Assoc. Prof. Dr

.AlinaBĂRBULESCU- CV and Publications

** MEMBERS**

Prof. Dr. Eng. Carmen

MAFTEI- CV and PublicationsAssist. Prof. Dr. Elena

PELICAN- Web pageT.A.

PhD. Student CristinaGHERGHINA (marriedSERBAN)- CV and Publications

Dr. Eng.Dacian ConstantinTEODORESCU- CV and Publications

** THE ROLE OF
THE YOUNG RESEARCHERS **

** Pelican Elena: **

2007: The calculus of the statistical indices of data series (expectance, variance, variation coefficients, asymmetry coefficients etc.)

2008: The study of the errors in the prediction models and the parameters optimization in order to obtain the minimum of the errors.

2009: The realization of some programs that detect the long range dependence and the stationarity of the rainfall series.

2010: The realization of some programs that calculate the Hurst coefficient.

The title of the PhD. Thesis (2008) is: Contributions of inverse problems in mathematical physics.

** Gherghina (married Șerban) Cristina: **

2007: Building the data series.

2008: Choosing and fitting the freqvential models for the maximum rainfall in 24 hours.

2009: Prediction of the monthly rainfall using Holt and Winters methods.

2010: The interpretation of the results of the fractal model and the comparison with the results from the scientific literature.The title of the PhD. Thesis is: Web and Grid techniques for processing digital images

Teodorescu Dacian Constantin:

2007: Initial analysis of data series.

2008: Choosing and adjusting the freqvential models for the annual precipitations.

2009: Prediction of annual rainfall using Holt and Winter methods.

2010: Extreme events prediction using the multifractal model and comparison with the results from the scientific literature.The title of the PhD. Thesis (2009) is: The water resources of the continental Dobrogea space. Genesis, hydrological regime and exploitation degree.

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The researchproved that most of the hydrological series has common characteristics - the non-stationarity and the presence of the data perturbations, the absence of normality, of independence or of homoscedasticity of the residuals, that make difficult the elaboration of mathematical models. The aim of this project is to obtain models of the rainfall series in Dobroudja for the last 50 years, surmounting the cited difficulties. The rainfall analysis will be made in the frequency domain ( IDF curves etc.), as well as in the time domain (AR, MA, ARMA models etc) .

The proposed modelswill have at least a deterministic and a stochastic component, because:

- the pure deterministic approach is based on the scale homogeneity hypotheses, implies the use of partial differential equations, blunted and numerical integrated, being difficult to deal with;

- the pure stochastic approach suppose an unknown causality of phenomena.

If for stationary series, Box - Jenkins methods will be applied, for the non-stationary ones we propose FARIMA and SARIMA methods, the characterization using the fractal (Box-counting method) and multifractal methods, based on a new statistical approach of the time series that contains perturbations. The models of multifractal analysis (R/S, Hurst modified rescaled analysis, Roseta) will blot out the problems implied by the use of multi-scale spatial and spatial – temporal models that indefensible favors one of the scales.

In order to improve the calculs efficiency, some programs for statistical analysis, Box dimension and Hurst coefficient calculation will be elaborated. The obtained models can be used to predict the rainfall evolution, particularly the catastrophic ones, in Dobroudja.

The proposed subjectis interesting because such a study wasn't yet performed for Dobroudja. The prediction models can be used to plan the irrigation regime. The developed methods can also be applied to study the river flow and of the hydrographical basins.