# Modern research

The mathematical formalism of all permafrost models is based on a solution of the Stefan problem. The peculiarity of the problem is the existence of the moving boundaries between the freezing and thawing ground which are the moisture phase change boundaries. The most perfect dynamic (process-based?динамические) models make it possible to calculate the consecutive temporal evolution of many permafrost parameters considering the thermal inertia of permafrost. However, usually it is impossible to realize the advantages of this kind of models for the large territories because of the limited data.

Statistical models are still the most useful and optimal models concerning the quality and spatial accuracy of the available data on climate, snow cover, vegetation and soil. In a canonical model version the soil temperature is calculated by several stages with a successive taking into account the influences of snow cover, vegetation and the thermal shift due to different thermal conductivity of freezing and thawing soils.

**Probabilistic-statistical models**

In the probabilistic-statistical models temporal and spatial large-scale changes of parameters influencing permafrost are explicitly specified, and their stochastic fluctuations, that lead to the ensemble of different permafrost states, are given statistically. This method allows considering the variability of soil properties, vegetation, snow cover and topography which cause stochastic variations of temperature and active-layer depth at local scales. Calculated ensemble characteristics make it possible to evaluate average permafrost parameters as far as their variability at local territories considered to be homogeneous by determinative approach. The significance of such variability is demonstrated by grid measuring at the CALM sites (1 x 1 km) where active layer thicknesses are measured on every 100 m (121 values).

The algorithm of the method of ensemble averaging is that in every cell of spatial grid several calculations with different combinations of parameters describing snow cover, vegetation and soil, varying round their average values, are set. The result is a sample of values of the analyzed parameters (e.g. soil temperature or maximal active-layer depth) that allows calculating average value, dispersion as far as their distribution function.

**Figure 1** shows the example of calculation based on the modern climatic data for the Russian cryolithozone with use of ensemble model. Every of the four maps of the modern climate represent the probability that seasonal thaw in that very point is in the range of values indicated (stated) by the map. Boundaries and the number of such ranges can be specified manually before making the probabilistic maps. The same way forecast maps based on different climatic scenarios can be made.