Modeling of Extreme Precipitation Fields on the Territory of the European Part of Russia

Modeling of Extreme Precipitation Fields on the Territory of the European Part of Russia E. Yu. Shchetinin*, N. D. Rassakhan† * FGU “All-Russian research institute on problems of civil defence and emergencies of Emergency Control Ministry of Russia” 7 Davydkovskaya St., Moscow, 121352, Russian Federation † Department of Applied Mathematics Moscow State Technology University “STANKIN” 3a Vadkovsky Ln., Moscow, 127055, Russian Federation


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][2][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.

Extreme Value Theory
Extreme value theory is based on Fisher-Tippett-Gnedenko theorem [2] that states the existence of normalized maxima's 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   (   +   ) = () for  > 1 and   > 0 and   ∈ R,  ∈ R 1 .Such distributions can be written in alternative form where  + = max(, 0), −∞ <  < ∞ is location parameter,  > 0 is scale parameter and −∞ <  < ∞ is shape parameter.Last equation represents generalized extreme value (GEV) distribution [1] because it includes Weibull distribution ( < 0), Gumbel distribution ( = 0) and Frechet distribution ( > 0).Case  = 0 is interpreted as limiting  → 0. 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  > ,  +  > 0, −∞ <  < ∞ 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 .
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 doesn't 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.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

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.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].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 don't consider comparing Smith model [13,14] with presented ones because, despite being easy to understand and even easier to implement, it's 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.

Figure 1 .
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

Table 1
GEV parameters in observed locations (location , scale , shape )

Table 2
Comparison of 4 models of trend surfaces for GEV parameters.The best model is chosen by the least value of TIC . This is one of the directions for the further development of this work.