D. Koutsoyiannis, Encolpion of stochastics: Fundamentals of stochastic processes, doi:10.13140/RG.2.2.10956.82564, Department of Water Resources and Environmental Engineering – National Technical University of Athens, Athens, 2013.
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.
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Our works that reference this work:
|1.||D. Koutsoyiannis, Hydrology and Change, Hydrological Sciences Journal, 58 (6), 1177–1197, doi:10.1080/02626667.2013.804626, 2013.|
|2.||F. Lombardo, E. Volpi, D. Koutsoyiannis, and S.M. Papalexiou, Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology, Hydrology and Earth System Sciences, 18, 243–255, doi:10.5194/hess-18-243-2014, 2014.|
|3.||P. Dimitriadis, and D. Koutsoyiannis, Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes, Stochastic Environmental Research & Risk Assessment, 29 (6), 1649–1669, doi:10.1007/s00477-015-1023-7, 2015.|
|4.||D. Koutsoyiannis, Generic and parsimonious stochastic modelling for hydrology and beyond, Hydrological Sciences Journal, 61 (2), 225–244, doi:10.1080/02626667.2015.1016950, 2016.|
|5.||P. Dimitriadis, D. Koutsoyiannis, and K. Tzouka, Predictability in dice motion: how does it differ from hydrometeorological processes?, Hydrological Sciences Journal, 61 (9), 1611–1622, doi:10.1080/02626667.2015.1034128, 2016.|
|6.||P. Dimitriadis, D. Koutsoyiannis, and P. Papanicolaou, Stochastic similarities between the microscale of turbulence and hydrometeorological processes, Hydrological Sciences Journal, 61 (9), 1623–1640, doi:10.1080/02626667.2015.1085988, 2016.|
|7.||D. Koutsoyiannis, P. Dimitriadis, F. Lombardo, and S. Stevens, From fractals to stochastics: Seeking theoretical consistency in analysis of geophysical data, Advances in Nonlinear Geosciences, edited by A.A. Tsonis, 237–278, doi:10.1007/978-3-319-58895-7_14, Springer, 2018.|
Other works that reference this work (this list might be obsolete):
|1.||Vertommen, I., R. Magini and M. da Conceição Cunha, Scaling Water Consumption Statistics, J. Water Resour. Plann. Manage., 10.1061/(ASCE)WR.1943-5452.0000467, 04014072, 2014.|