N. Gournari, Probability distribution of the climacogram using Monte Carlo techniques, Diploma thesis, 108 pages, Department of Water Resources and Environmental Engineering – National Technical University of Athens, Athens, July 2017.
Many hydrological and geophysical phenomena cannot be adequately simulated using deterministic processes. The debate of long term dependence structure in geophysical processes rather than short term has raised the scientific interest in the last decades. In the current thesis, we analyze normally distributed processes that are of high significance in hydrology since the annual scale is often used (which is very close to normality basd on the central limit theorem) in water resources management. A simple scaling model known as the fractional Gaussian noise is chosen for the best description of the behavior of the geophysical processes. This stochastic model was devised to represent the Hurst phenomenon. In order to estimate the long term percistence a stochastic tool is used known as the climacogram, i.e., variance of the time-averaged process over averaging time scale. In this way, we can quantify the Hurst parameter since this tool compared to others (like the autocovariance or the power spectrum) appears to have the lowest statistical uncertainty, an important advantage in stochastic model building. The scope of this thesis is the identification of the statistical distribution of the climacogram in every scale of a Gaussian process. Moreover, the effect of the long term percistence in the statistical properties of the processes is examined, due to the bias and the dependence structure between the processes. The analysis is carried out using the Monte Carlo method for the generation of the climacograms and their distribution. It can be proved that in the case of 𝐻=0.5 (white noise), the distribution of the climacogram is the chi-square distribution whereas when the Hurst parameter increases, the skewness of the distribution is increasing approaching the gamma distribution. Furthermore, the results of various statistical properties are presented analytically, leading to the conclusion that the most probable value of the climacogram is converging to the Q-25 quantile, a useful conclusion in case of limited data. Finally, some applications are presented where this particular analysis could contribute for a more accurate simulation and prediction of a stochastic process.