Fractal-based techniques have opened new avenues in the analysis of geophysical data. On the other hand, there is often a lack of appreciation of both the statistical uncertainty in the results, and the theoretical properties of the stochastic concepts associated with these techniques. Several examples are presented which illustrate suspect results of fractal techniques. It is proposed that concepts used in fractal analyses are stochastic concepts and the fractal techniques can readily be incorporated into the theory of stochastic processes. This would be beneficial in studying biases and uncertainties of results in a theoretically consistent framework, and in avoiding unfounded conclusions. In this respect, a general methodology for theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, is proposed. Some important modelling issues are discussed with focus on model identification and fitting, often made using inappropriate methods. The theoretical framework is applied to several processes, including turbulent velocities measured every several microseconds, and wind and temperature measurements. The applications shows that several peculiar behaviours observed in these processes are easily explained and reproduced by stochastic techniques.
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... First, the climacogram stochastic tool [22] (variance of the average process vs. temporal scale) is employed to analyze longterm dependence, addressing variability across multiple timescales and handling discretization and bias effects in wind speed modeling. This is followed by applying the Symmetric-Moving-Average (SMA) stochastic simulation algorithm [80], which preserves all key statistical properties such as intermittency and long-term persistence through marginal moments [16,23]. The third step accounts for wind speed's double periodicity (diurnal and seasonal) using an implicit scheme. ...
... First, the climacogram stochastic tool [22] (variance of the average process vs. temporal scale) is employed to analyze long-term dependence, addressing variability across multiple timescales and handling discretization and bias effects in wind speed modeling. This is followed by applying the Symmetric-Moving-Average (SMA) stochastic simulation algorithm [80], which preserves all key statistical properties such as intermittency and long-term persistence through marginal moments [16,23]. The third step accounts for wind speed's double periodicity (diurnal and seasonal) using an implicit scheme. ...
... The GHK long-term dependence model is expressed via the climacogram, γ (m 2 /s 2 ) (in Equation (1)) [16,25,80]. As a result, the simulated process (SMA) is expressed as the sum of the products of coefficients, and the long-term persistent behavior of wind is expressed by the standardized GHK model as ...
Siting an offshore wind project is considered a complex planning problem with multiple interrelated objectives and constraints. Hence, compactness and contiguity are indispensable properties in spatial modeling for Renewable Energy Sources (RES) planning processes. The proposed methodology demonstrates the development of a raster-based spatial optimization model for future Offshore Wind Farm (OWF) multi-objective site-prospecting in terms of the simulated Annual Energy Production (AEP), Wind Power Variability (WPV) and the Depth Profile (DP) towards an integer mathematical programming approach. Geographic Information Systems (GIS), statistical modeling, and spatial optimization techniques are fused as a unified framework that allows exploring rigorously and systematically multiple alternatives for OWF planning. The stochastic generation scheme uses a Generalized Hurst-Kolmogorov (GHK) process embedded in a Symmetric-Moving-Average (SMA) model, which is used for the simulation of a wind process, as extracted from the UERRA (MESCAN-SURFEX) reanalysis data. The generated AEP and WPV, along with the bathymetry raster surfaces, are then transferred into the multi-objective spatial optimization algorithm via the Gurobi optimizer. Using a weighted spatial optimization approach, considering and guaranteeing compactness and continuity of the optimal solutions, the final optimal areas (clusters) are extracted for the North and Central Aegean Sea. The optimal OWF clusters, show increased AEP and minimum WPV, particularly across offshore areas from the North-East Aegean (around Lemnos Island) to the Central Aegean Sea (Cyclades Islands). All areas have a Hurst parameter in the range of 0.55–0.63, indicating greater long-term positive autocorrelation in specific areas of the North Aegean Sea.
... Stochastic modelling in hydrology has made significant advancements, including the utilization of the Bayesian Generalized Likelihood Uncertainty (Beven et al., 2007;Beven & Binley, 1992;Freer et al., 1996) and Markov Chain Monte Carlo techniques (Fu & Jaime G omez-Hernández, 2009;Kuczera & Parent, 1998) with advanced methods like Adaptive Metropolis (AM) algorithms (Braak, 2006;Haario et al., 2001Haario et al., , 2006Laloy & Vrugt, 2012;Smith & Marshall, 2008;Vrugt et al., 2008;Xu et al., 2018). Stochastic models hold great value for hydrological applications and decisionmaking (Benke et al., 2008;Hrachowitz et al., 2013;Koutsoyiannis, 2010Koutsoyiannis, , 2016Koutsoyiannis et al., 2009Koutsoyiannis et al., , 2018Montanari & Koutsoyiannis, 2012). However, the models require a high level of statistical sophistication and computational resources (Dotto et al., 2012;Marshall et al., 2004;Zhang et al., 2020). ...
Runoff prediction is crucial for effective water resource management and risk mitigation. However, predicting these catchment responses is challenging due to their unique characteristics and the randomness of hydrological processes. This manuscript explores two different types of modelling frameworks (deterministic and stochastic) and aims to answer questions regarding the reliance on stochastic simulation based on deterministic simulations, the suitability of simple deterministic models, and the influence of catchment characteristics on the results. A simple deterministic rainfall‐runoff model (with only one model parameter) was used to feed the Brisk Local Uncertainty Estimator for Hydrological Simulations and Predictions (Bluecat) framework, exploring the whole range of values of the model parameter. Our findings showed that Bluecat enhanced the Kling‐Gupta Efficiency (KGE) outcomes in arid and semi‐arid regions as well as high‐altitude catchments. Additionally, using the mean of the confidence band of the stochastic simulation as the simulated discharge, rather than the median, resulted in improved KGE values for all catchments. Hysteresis between S‐KGE (stochastic KGE) and D‐KGE (deterministic KGE) was observed, indicating a non‐monotonic relationship between the two variables, and therefore, S‐KGE optimisation can be achieved even when D‐KGE is not optimal. Bluecat showed exceptional performance extended to arid and semi‐arid regions, as well as high‐altitude areas, making it a promising alternative for rainfall‐runoff simulations in these challenging locations.
... We also highlight that the Weibull distribution has been adopted in this analysis as a typical and widely adopted model for the sole purpose of evaluating how the estimation process works. The same efficiency might be achieved when dealing with other wind speed distributions, such as those examined in Section 3 or others available in the relevant literature [35,[49][50][51]. The adoption of a double-period probability model is also worth investigating for future studies, to address the typical double periodicity of wind [52,53]. ...
In the present paper, the process of estimating the important statistical properties of extreme wind loads on structures is investigated by considering the effect of large variability. In fact, for the safety design and operating conditions of structures such as the ones characterizing tall buildings, wind towers, and offshore structures, it is of interest to obtain the best possible estimates of extreme wind loads on structures, the recurrence frequency, the return periods, and other stochastic properties, given the available statistical data. In this paper, a Bayes estimation of extreme load values is investigated in the framework of structural safety analysis. The evaluation of extreme values of the wind loads on the structures is performed via a combined employment of a Poisson process model for the peak-over-threshold characterization and an adequate characterization of the parent distribution which generates the base wind load values. In particular, the present investigation is based upon a key parameter for assessing the safety of structures, i.e., a proper safety index referred to a given extreme value of wind speed. The attention is focused upon the estimation process, for which the presented procedure proposes an adequate Bayesian approach based upon prior assumptions regarding (1) the Weibull probability that wind speed is higher than a prefixed threshold value, and (2) the frequency of the Poisson process of gusts. In the last part of the investigation, a large set of numerical simulations is analyzed to evaluate the feasibility and efficiency of the above estimation method and with the objective to analyze and compare the presented approach with the classical Maximum Likelihood method. Moreover, the robustness of the proposed Bayes estimation is also investigated with successful results, both with respect to the assumed parameter prior distributions and with respect to the Weibull distribution of the wind speed values.
... Περιπτώσεις ασάφειας είναι συχνές στην κλιματολογία (Koutsoyiannis, 2021(Koutsoyiannis, , 2023, αλλά φτάνουν ακόμη και στα μαθηματικά (Koutsoyiannis et al., 2018). ...
Μέρος Α: Νείλος και η γέννηση της επιστήμης.
Μέρος Β: Επίκαιρα μαθήματα του Νείλου απ’ τα βάθη των αιώνων.
Μέρος Γ: Μερικές σκέψεις με αφορμή τον Νείλο.
[Part A: The Nile and the birth of science.
Part B: Current lessons of the Nile from the depths of the centuries.
Part C: Some thoughts inspired by the Nile.]
... This section describes the construction of a confidence curve for the parameter of a one parameter copula. In this article random variables (RVs), random vectors, and random matrices will be marked by an underscore (see also Hemelrijk, 1966;Koutsoyiannis et al., 2017). The confidence curve will be constructed from a given observed time series of length n. ...
... The next step of the analysis consists in the investigation of common and other marginal distribution functions to examine the goodness of fit for each wind wave process. For this test, we select the lognormal, Weibull, gamma, generalized-gamma, generalized Pareto and Pareto-Burr-Feller (PBF, Eq. (1): a, b shape parameters, c scale parameter; see also Koutsoyiannis et al., 2018) distributions. ...
A combination of stochastic and deterministic models is applied for the study of ocean wind waves. Timeseries of significant wave height and mean zero up-crossing period, obtained from globally scattered floating buoys, are analyzed in order to construct a double periodic model, and select an optimal marginal distribution and dependence function for the description of the stochastic structure of wind waves. It is concluded that wind waves, in contrast to the atmospheric wind speed process, are mostly governed by the seasonal periodicity rather than the diurnal periodicity, which is often weak and can be neglected. Also, the Pareto-Burr-Feller distribution is found to be a fair selection among other common three-parameter marginal distributions. The dependence function is simulated through the Hurst-Kolmogorov (HK) dynamics using the climacogram (i.e., variance of the averaged process in the scale domain), a stochastic tool that can robustly estimate both the short-term fractal and long-range dependence behaviors both apparent at the wind wave process. To test the validity of the model, a stochastic synthesis of the wind wave process is performed through the Symmetric Moving Average scheme, focused on the explicit preservation of the probabilistic and the dependence structures. Finally, the stochastic model is applied for simulation to an offshore station southeast of Australia having one of the largest record lengths. The energy potential is also estimated through the significant wave height and the mean zero up-crossing period of both the synthetic and observed timeseries, and the effectiveness of the model is further discussed.
Hurst’s paper on the Nile’s flow variability marked a pivotal moment in hydrology and beyond by introducing
what was called the Hurst phenomenon. Independently, Kolmogorov developed a mathematical model
describing this behaviour a decade earlier. The Hurst-Kolmogorov dynamics (HKd) is used to express this phenomenon
physically and mathematically, which is characterised by high uncertainty and persistence (across
spatial and temporal scales) and challenges traditional analytical frameworks, particularly in water resources-
related topics and implications in engineering designs. Given the importance of HKd, a bibliometric analysis
of it in water resources is helpful to trace its historical development, current state, and (possible) future trajectories.
The latter intends to offer a comprehensive perspective on HKd, serving as a guide for new readers
seeking an entry point into this field. Using the Web of Science database, 617 publications from 1974 to 2023 are
analysed, revealing a consistent growth trend in research outputs up to 2018. Collaborative efforts among researchers
worldwide have been prominent, with the USA and China leading in international collaborations.
High-impact journals on topics related to water resources and geosciences are primary outlets for research
related to the HKd. Interestingly, only two journals published the 20 most cited papers on this topic. A clear
pattern from “groundwater” to “streamflow” to “soil moisture” to “precipitation” was observed from the past to
the present. Overall, this analysis provides a comprehensive overview of the past, present, and future trends in
HKd research, and highlights its contribution to the scientific literature of water resources.
In this study, catchments are considered as complex systems, and information-theoretic measures are used to capture temporal streamflow characteristics. Emergence and self-organization are used to quantify information production and order in streamflow time series, respectively. The measure complexity is used to quantify the balance between emergence and self-organization in streamflow variability. The complexity measure is found to be effective in distinguishing streamflow variability for high and low snow-dominated catchments. The state of persistence-reflecting the memory of streamflow time series, is shown to be related to the complexity of streamflow. Moreover, it is observed that conventional causal detection methods are constrained by the state of persistence, and more robust methods are needed in hydrological applications considering persistence.
In any statistical investigation, we deal with the applications of probability theory to real problems, and the conclusions are inferences based on observations. To obtain plausible inferences, statistical analysis requires careful understanding of the underlying probabilistic model, which constrains the extraction and interpretation of information from observational data, and must be preliminarily checked under controlled conditions. However, these very first principles of statistical analysis are often neglected in favor of superficial and automatic application of increasingly available ready-to-use software, which might result in misleading conclusions, confusing the effect of model constraints with meaningful properties of the process of interest. To illustrate the consequences of this approach, we consider the emerging research area of so-called ‘compound events’, defined as a combination of multiple drivers and/or hazards that contribute to hydro-climatological risk. In particular, we perform an independent validation analysis of a statistical testing procedure applied to binary series describing the joint occurrence of hydro-climatological events or extreme values, which is supposed to be superior to classical analysis based on Pearson correlation coefficient. To this aim, we suggest a theoretically grounded model relying on Pearson correlation coefficient and marginal rates of occurrence, which enables accurate reproduction of the observed joint behavior of binary series, and offers a sound simulation tool useful for informing risk assessment procedures. Our discussion on compound events highlights the dangers of renaming known topics, using imprecise definitions and overlooking or misusing existing statistical methods. On the other hand, our model-based approach reveals that consistent statistical analyses should rely on informed stochastic modeling in order to avoid the proposal of flawed methods, and the untimely dismissal of well-devised theories.
An extension of the symmetric-moving-average (SMA) scheme is presented for stochastic synthesis of a stationary process for approximating any dependence structure and marginal distribution. The extended SMA model can exactly preserve an arbitrary second-order structure as well as the high order moments of a process, thus enabling a better approximation of any type of dependence (through the second-order statistics) and marginal distribution function (through statistical moments), respectively. Interestingly, by explicitly preserving the coefficient of kurtosis, it can also simulate certain aspects of intermittency, often characterizing the geophysical processes. Several applications with alternative hypothetical marginal distributions, as well as with real world processes, such as precipitation, wind speed and grid-turbulence, highlight the scheme’s wide range of applicability in stochastic generation and Monte-Carlo analysis. Particular emphasis is given on turbulence, in an attempt to simulate in a simple way several of its characteristics regarded as puzzles.
Most things are uncertain. Stochastics is the language of uncertainty. I believe the gospel of stochastics is the book by Papoulis (1991). However, as Papoulis was an electrical engineer, his approach may need some additions or adaptations in order to be applied to geophysical processes. The peculiarities of the latter are that (a) their modelling relies more on observational data because geophysical systems are too complex to be studied by theoretical reasoning and deduction, and theories are often inadequate; (b) the distinction signal vs. noise is meaningless; (c) the samples are small; (d) they are often characterized by long term persistence, which makes classical statistics inappropriate. Having studied several hydroclimatic processes, I have derived in handwritten notes some equations useful for such processes. Having repeated such derivations several times, because I had forgotten that I had produced them before or lost the notes, I decided to produce this document. Some of the equations and remarks contained here can be found in other texts, particularly in Papoulis, but some other cannot. I believe they can be useful to other people, researchers and students.
Long-term persistence or Hurst-Kolmogorov behaviour has been identified in many hydroclimatic records. Such time series are intriguing because they are the hallmark of multi-scale dynamical processes that govern the system from which they arise. They are also highly relevant for water resource managers because these systems exhibit persistent, for example, multi-decadal, mean shifts or extremes clustering that must be included into any long-term drought management strategy. During recent years the growing number of palaeoclimatic reconstructions has allowed further investigation of the long-term statistical properties of climate and an understanding of their implications for the observed change. Recently, the consistency of the proxy data for precipitation was strongly doubted, when their persistence property was compared to the corresponding estimates of instrumental records and model results. The latter suggest that droughts or extremely wet periods occur less frequently than depicted in the palaeoclimatic reconstructions. Here, we show how this could be the outcome of a varying scaling law and present some evidence supporting that proxy records can be reliable descriptors of the long-term precipitation variability.
The book provides the applied scientist, engineer or statistician with an introduction to geostatistics stressing the multivariate aspects. Geostatistics offers a variety of models, methods and techniques for the analysis, estimation and display of multivariate data distributed in space or time. The book presents a brief review of statistical concepts, a detailed introduction to linear geostatistics, and an account of three basic methods of multivariate analysis. Moreover, it presents an advanced presentation of linear models for multivariate spatial or temporal data, including the recent bilinear model of coregionalization, and an introduction to non stationary geostatistics with a special focus on the external drift method.
A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.