Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)1789510.22363/2312-9735-2018-26-1-74-83Research ArticleModeling of Extreme Precipitation Fields on the Territoryof the European Part of RussiaShchetininE Yu<p>Shchetinin E. Yu. - professor, Doctor of Physical and Mathematical Sciences, Leading Researcher of FGU “All-Russian research institute on problems of civil defence and emergencies of Emergency Control Ministry of Russia”</p>riviera-molto@mail.ruRassakhanN D<p>Rassakhan N. D. - Master of Science of the Applied Mathematics Department, MSTU “Stankin”</p>rassahan@gmail.comFGU “All-Russian research institute on problems of civil defence and emergencies of Emergency Control Ministry of RussiaMoscow State Technology University “STANKIN”15122018261748328022018Copyright © 2018, Shchetinin E.Y., Rassakhan N.D.2018<p>Present work is devoted to the study and development of space-time statistical structures ofextreme type modeling with the use of the max-stable processes. The theory of one-dimensionalextremal values and its extension to the two-dimensional case are considered and for that max-stable processes are introduced and then the main parametric families of max-stable processes(Schlather, Smith, Brown-Resnick, and Extremal-t) are presented. By modifying the maximumlikelihood method, namely using the paired likelihood function, parameter estimates wereobtained for each of the models whose eﬃciency was compared using the Takeuchi informationcriterion (TIC).Resulting models are coherent with classical extreme value theory and allow consistenttreatment of spatial dependence of rainfall. We illustrate the ideas through data, based ondaily cumulative rainfall totals recorded at 14 stations in central European part of Russia forperiod 1966-2016 years. We compare ﬁts of diﬀerent statistical models appropriate for spatialextremes and select the model that is the best for ﬁtting our data. The method can be used inother situations to produce simulations needed for hydrological models, and in particular forthe generation of spatially heterogeneous extreme rainfall ﬁelds over catchments. It is shownthat the most successful model for the data we studied is the model from the extremal-t familywith the Whittle-Matern correlation function.</p>spatial modelingextreme rainfallmax-stable processesex-treme value theoryspatial structures of statistical dependencepairwise likelihood functionпространственное моделированиеэкстремальные осадкипроцессыустойчивых максимумовтеория экстремальных величинпространственные структурыстатистической зависимостипарная функция правдоподобия<p>1. Introduction The rapidly growing number of various natural and man-made disasters that previously were considered extremely rare indicates that the global climate change of the Earth is becoming obvious. Observable in various regions of the world and in particular in Russia, hurricanes, rainfalls and other natural disasters bring human casualties and substantial material damage to states and their economies. Therefore it is necessary to develop new methods of resisting the impacts of diﬀerent environmental disasters, including comprehensive measures for forecasting, preventing and adapting the population to extreme situations. The study of regional climate change peculiarities that take place in connection with global warming is a priority area of modern international research projects. Important place in this area is given to the study of changes in the frequency and intensity of extreme weather events, including extreme precipitation, as it often leads to serious economic, environmental and human losses. According to recent studies signiﬁcant increase in the frequency of extreme events including rainfall is expected as a result of global and regional climate change. The archives of long-term accumulated observations and numerical model calculations of hydrometeorological parameters make it possible to study general patterns of spatiotemporal variability of extreme precipitation in Russia, caused by both environmental and anthropogenic factors over the historical observation period and to calculate the projections of their possible future changes. Spatial modeling methods is a popular approach for studying extreme events in environmental applications. Numerous scientiﬁc publications (like [1 - 3]) on this subject are engaging extreme value theory (EVT) and extreme processes to the analysis of environmental problems. Present work is devoted to the study and development of precipitation models in European Russia for the period 1966-2016 with the aim of constructing a short-term precipitation forecast in a given region exceeding the normative indices. The study of regularities of long-term variability of extreme precipitation on the territory of Russia is aimed at the development of long-term forecasts. At the same time, such studies are important for the subsequent solution of many applied problems, including long-term planning of regional economic development. 2. Extreme Value Theory Extreme value theory is based on Fisher-Tippett-Gnedenko theorem [2] that states the existence of normalized maximas marginal distribution for sequence of i.i.d. random variables. If such distribution exists and is non-degenerate then it satisﬁes requirements of max-stable distributions Such distributions can be written in alternative form where is location parameter, is scale parameter and is shape parameter. Last equation represents generalized extreme value (GEV) distribution [1] because it includes Weibull distribution Gumbel distribution and Frechet distribution is interpreted as limiting Another approach to order statistic modeling known as the threshold approach is bound to previous one. Following Pickands theory [1] under suitable conditions and for a suﬃciently high threshold , the upper tail distribution of a wide class of random variables can be well approximated by where and Here is the probability that the threshold is exceeded, and and are respectively scale and shape parameters determining the distribution of exceedances corresponding to those of the limiting distribution of maxima. The parametrization of the generalized Pareto distribution (GPD), whose survivor function appears in the braces on the right part of equation is diﬀerent from the usual one and has the advantage that the parameters and do not depend on the choice of threshold . 3. Max-Stable Processes Using of max-stable processes [3] is an extension of extreme value theory applied to spatio-temporal precipitation ﬁelds. Let be a sequence of non-negative independent copies of stochastic process with continuous sample paths. If there are such continuous functions 0 and R that marginal process is non-degenerate, then is max-stable process. For the consistency of this theorem with one-dimensional case it is considered that marginal distribution in its condition must be distributed according to GEV. Their spectral representation [4] has the following form: where are points of the Poisson process at is a sequence of non-negative independent copies of stochastic process such that Points in the spectral characterization are radii while the stochastic processes are angles. Further, four main families of max-stable processes will be considered: 1) Smith process [5]: Here, the Brown-Resnick process is characterized by Gaussian stationary process with variogram It is possible to derive a formula for ﬁnite-dimensional distribution from spectral characterization. For each It fully describes the joint distribution and is called exponential function. It is obvious from the formula above that Function is called -dimensional extremal coeﬃcient [8] and represents total dependence measure between elements of random vector Due to independence of radial and angular components of the multidimensional extreme value the extremal coeﬃcient doesnt depend on the radius, that is, from in and shows relation we are interested in. We focus on the two-dimensional case and deﬁne the function of the extremal coeﬃcient: Extremal coeﬃcient function takes values in the interval [1, 2], where the smallest value corresponds to complete dependence, and the largest corresponds to complete independence. For these two cases we obtain It is important to note that the calculation of the exponential function for 2 can be diﬃcult [9], therefore the consideration of ﬁnite-dimensional distributions of max-stable processes is mostly reduced to the two-dimensional case. Example of spatial dependence measurement is shown in Fig. 1; here we plot pairwise f-madogram and extremal coeﬃcient to show how dependence changes with distance. Figure 1. Pairwise F-madogram (left panel) and extremal coeﬃcient (right panel) for the best ﬁtting max-stable process for our data (that will be shown below). Distance between stations can be calculated as Distance ℎ 111 km 4. Modeling of Extreme Precipitation Spatial Fields In this study precipitation data of the All-Russian Research Institute of Hydrometeorological Information - the World Data Center of the Russian Federation is used, which show monthly precipitation in 14 cities of the European part of Russia. The data is freely available (on the website http://aisori.meteo.ru/ClimateR ) and is represented by a set of tables (a separate table for each city); each table contains daily rainfall value for the period 1966-2016 years. Thus, we face not only the problem of analyzing the statistical properties of one-dimensional time series for each station, but also the problem of model development that contains spatial structure of the statistical relationships in various locations [10]. Preliminary analysis of empirical data distribution properties in observed locations showed signiﬁcant deviations of their statistical properties from the Gaussian distribution. It is for this reason that the use of GEV is justiﬁed, yet we need to evaluate the quality of ﬁtting our data with GEV models. Diagnostic plots that are shown in Fig. 2 help us with that. Figure 2. Diagnostic plots for GEV distribution in Ryazan (density plot and quantile plot) Then GEV parameters were found for each city, they are shown in Table 1. Developing trend surfaces for GEV parameters is important next step in our research because it might help us to estimate GEV parameters at any point of the ﬁeld under study. It is important to note that the form parameter should be constant since it is the one that determines the model behaviour; position and scale parameters depend on the spatial coordinates, therefore they include latitude, longitude and their joint contribution. Thus, selection is made among models described as Table 2 shows the results of calculations, the choice of the best model is made using the Takeuchi information criterion (TIC) [11]. Table 1 GEV parameters in observed locations (location , scale , shape ) Station (City) St. Petersburg 39.764 22.911 0.059 Pskov 40.072 22.882 0.077 Zheleznodorozhny 40.891 24.541 0.037 Smolensk 42.843 25.261 0.09 Bryansk 40.268 24.644 0.028 Kostroma 36.57 22.254 0.073 Pereslavl-Zalessky 36.252 21.984 0.092 Nizhny Novgorod 39.389 23.786 0.034 Mozhaysk 38.547 23.755 0.097 Moscow VDNH 43.327 25.309 0.017 Kolomna 33.606 20.851 0.098 Ryazan 33.546 20.801 0.107 Tambov 30.413 20.217 0.08 Penza 32.414 21.085 0.048 Table 2 Comparison of 4 models of trend surfaces for GEV parameters. The best model is chosen by the least value of TIC GEV Trend Surface TIC Finally, we compare the various models of max-stable processes [12]. The Table 3 below shows various families of processes and correlation functions are given in parentheses. The best model corresponds to the smallest value of TIC. We dont consider comparing Smith model [13, 14] with presented ones because, despite being easy to understand and even easier to implement, its quite ineﬀective in terms of modeling and ﬁtting real environmental problems. 5 out of 7 models belong to Schlather family that can be explained by its popularity in comparison with more complex Brown-Resnick and Extremal-t processes yet last ones show better results [15]. Their modeling and ﬁtting are still very consuming, both in terms of time and in terms of computing resources. The best model is an extremal-t process with the Whittle-Matern correlation function. Table 3 Results of max-stable processes parameters estimating Model Parameters TIC Brown-Resnick nugget = 0.4543 454911.7 range = 6.4889 smooth = 0.7099 Schlather nugget = 3.97 10 -5 455315.8 (Whittle-Matern) range = 7.569 10 smooth = 8.006 10 -2 Schlather nugget = 0.4543 455151.4 (Cauchy) range = 6.4889 smooth = 0.7099 Schlather nugget = 0.4679 455170.3 (Power Exponential) range = 9.6612 10 smooth = 2.0 Schlather nugget = 0.4655 455170.9 (Bessel) range = 0.4309 smooth = 120.2719 Schlather nugget = 0.4513 455155.1 (Generalized Cauchy) range = 6.9826 smooth = 1.5525 smooth2 = 2.0 Extremal-t nugget = 0.2552 454108.9 (Whittle-Matern) range = 2.7715 smooth = 96.3061 df = 3.2056 5. Discussion In this paper we propose using the extreme value theory methods for modeling daily maximum precipitation ﬁelds in the European part of Russia. Our approach consists in estimating parameters of one-dimensional extreme distributions (1) for each metering station and developing models of statistical dependence spatial structures with the use of max-stable processes for the entire measurement domain. Using the data of the All-Russian Scientiﬁc Research Institute of Hydrometeorological Information - World Data Center, the ﬁelds of precipitation of daily measurements converted into monthly maximum precipitation were studied in 14 cities of the European part of Russia for the period 1966-2016 years. Parameters of precipitation ﬁelds models were estimated using the censored method of pairwise maximum likelihood [16] which further allows us to simulate daily precipitation amount throughout the region. Various parametric families of max-stable processes are developed and their estimates are obtained. The best model is the t-extremal process with the parameters shown above (in Table 3). Interpolation of precipitation values in unobservable regions adjacent to the observed ones is usually solved by kriging [17] but despite the fact that this yields the optimal result for Gaussian processes, it can give erroneous forecasts for extreme values due to the unsuitability of the Gaussian model for the data. Approach that uses conditional max-stable simulation [18] is more suitable for these purposes. Proposed approach can be used in other areas where spatial modeling of extreme values and processes is required. The models of the max-stable processes used by us are also suitable for time scales in which precipitation measurements are stationary series. However the inﬂuence of estimation errors autocorrelation increases in case of more frequent measurements and then it is necessary to develop models of space-time dependence structures [19, 20]. This is one of the directions for the further development of this work.</p>[V. A. Akimov, A. A. Bykov, E. Y. 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