Corresponding author: Ildar Muratov (

Academic editor: Aleksandr I. Malov

Deciphering the satellite images of medium and high spatial resolution of the northern territories of Western Siberia has been carried out using geoinformation system ArcGIS 10.3. Images of medium resolution Landsat-8 and high resolution Kanopus-V were used. Kanopus-V images alluded to determine the number and areas of small lakes, which are considered as intensive sources of methane emission into the atmosphere from thermokarst lakes. Data on the spatial characteristics of thermokarst lakes were obtained. Based on the integration of images of medium and high spatial resolution, a synthesized histogram of the distribution of lakes in a wide range of sizes was constructed, taking into account small lakes. The obtained histogram was approximated by a lognormal distribution law by the Pearson criterion with a probability of 0.99. Based on the geo-simulation approach, a new model of the spatial structure of the fields of thermokarst lakes is presented, taking into account the lognormal law of the lake size-distribution. Algorithms for modeling the spatial structure of the fields of thermokarst lakes are described. An example of modeling the field of thermokarst lakes with a lognormal law of their size-distribution is given. The practical applicability of the previously developed model with an exponential distribution of lakes in size, based on data from Landsat images, has been experimentally confirmed. The results can be used to obtain predictions of the dynamics of methane emissions from the thermokarst lakes of the Arctic zone of Northern Eurasia for the coming decades in the context of climate changes.

The current global warming of the climate, most clearly manifested in the northern latitudes of the planet, accelerates the degradation of permafrost. Permafrost, being a storehouse of canned carbon in the vast frozen peat bogs of Northern Eurasia, can become a source of even more warming with the release of greenhouse gases, which will lead to the formation of new big challenges for the world community related to the violation of human-nature interaction. Indeed, carbon is currently in a bound state as an organic matter in a layer of permafrost in the northern territories of Eurasia and America. With the warming of the climate, a rise in temperature will lead to the melting of frozen rocks and the additional release of methane as a product of the vital activity of microorganisms recycling thawed organic matter, which can make an additional tangible contribution to climate warming.

The dominant role in the accumulation of methane of small thermokarst lakes (with areas less than 0.01-0.05 ha) was established (

According to

Different groups of authors have introduced the special terms for modeling spatial objects.

All the above types of modeling aim to study spatial objects using spatial data, analysis of which is based on spatial analysis methods implemented with the help of modern geo-information systems (GIS-analysis). In our opinion, the most suitable general term for the mentioned types of modeling (mathematico-cartographical, spatial, geo-information, geo-simulation et al.) is the term ‘geo-simulation modeling’ which is defined as a model creation and model application for objects with spatial structure. The spatial nature and high degree of complexity of the fields of thermokarst lakes as objects of modeling makes it necessary to use geo-simulation modeling.

The most important task is to develop a geo-simulation model of a field of thermokarst lakes, which is understood as a mathematical model that reproduces the spatial structure of fields of thermokarst lakes by simulating the shape, size and relative position of lakes in the study area taking into account the experimentally established statistical laws of their random location and size distribution. The development of such a model was considered in (

In connection with the above, the main goal of this work was to consider the issues of modeling the spatial structure of thermokarst lakes fields based on the integration of medium and high resolution images that take into account lakes of all sizes.

The informational basis for the experimental study of the properties of the fields of thermokarst lakes is the data of remote measurement of the areas of lakes from satellite images of the studied territory. The studies were carried out on the territory of all three permafrost zones in Western Siberia by remote method based on medium and high resolution satellite images taken in a relatively short period of time (2013–2015). All images were selected in a fairly short period of the summer season (end of June – August) to minimize the effect of seasonal fluctuations in the water level in the lakes. During this period, the ice cover on the lakes completely disappears, preventing them from being excreted when interpreting images.

Since the medium resolution images of Landsat (30 m) provided a complete coverage of the study area, a mosaic of these images was used for research, which allowed studying the properties of hundreds of thousands of lakes. A study on high resolution images of Kanopus-B (2.1 m) was carried out on a set of test sites, the map-layout of which in the different zones of permafrost in Western Siberia is shown in Fig.

Location of test sites

Test sites were chosen by us, as is customary in similar studies, in places where thermokarst lakes accumulate, i.e. in zones where conditions exist for the formation and development of foci of thermokarst processes (

The coastal boundaries of thermokarst lakes were determined from Landsat satellite images using the Fmask algorithm. This algorithm uses a number of parameters obtained from the spectral channels of Landsat satellite images to build assumptions about the nature of reflections of various geographical objects in different spectral ranges of Landsat-8. The algorithm is described in detail in (

The fragment of the deciphered space image, resulted on Fig.

Fragment of a decrypted Landsat-1 image (1973) with the image of field of thermokarst lakes

Creating a geo-simulation model of thermokarst lakes fields requires knowledge of the basic properties of these fields, which can be obtained experimentally from satellite images. A remote study of the shape of the boundaries of thermokarst lakes conducted in (

For the development of the model, the task of constructing histograms of the distribution of lakes by size (area), which would take into account all the lakes of the study area in a wide range of sizes – from tens of meters to tens of kilometers, becomes important. To construct such a histogram in (^{2}, 50-100 m^{2}, 100-200 m^{2}, etc. up to 200 km^{2}, which made it possible to present data on the distribution of lakes over intervals of their sizes rather compactly over a very wide range of changes in the areas of lakes.

Such histograms of the distribution of lake areas can be constructed only on the basis of the integration of data on the areas of water bodies obtained from satellite images of both medium and high resolution. The developed methodology for combining (synthesizing) data on the areas of lakes obtained from images of different spatial resolution in order to construct synthesized histograms of the distribution of areas of lakes in a very wide range of their sizes is described in (

In Fig.

The illustration of MR (Landsat-8, black columns) – HR (Kanopus-V, white columns) lake number (^{8} m² with and overlap in the range of 5×10^{3} to 10^{6} m². Vertical line marks the point of integration (stitching) of high and medium-resolution-based lake diagrams

The synthesized histograms of the distribution of the number of lakes and their total area in size (Fig.

The determination of the type of the law of the lake size-distribution was carried out on the basis of an approximation of the obtained sinthesized histogram of the distribution of lakes (Fig.

According to (

where

Based on the above, we can formulate the following fundamental principles that determine the essential properties of the model of the spatial structure of the thermokarst lake field:

1. The shape of the shoreline of the lake can be represented by the equation of a circle with the coordinates of the centers xi, yi and the area si (i is the serial number of the lake).

2. The coordinates of the centers of the circles are random variables, the distribution of which is determined by the law of uniform density.

3. The circles sizes are random variables whose distribution is determined by a lognormal law.

4. Spatial changes in the coordinates of the centers of the circles and their areas are statistically independent.

The geo-simulation model of a field of thermokarst lakes developed in accordance with these principles is a set of random circles (Fig.

Geometrical representation of fragment of the thermokarst lake field model as an aggregate of random circles

where _{k}_{k}

where

In a general case, mutual density of probabilities of random coordinates of centers and areas of circles imitating lakes in a mathematical model of random thermokarst lake fields can be presented in the form:

where

Consequently, the set of circles in the model of lake fields will be represented as a sequence of triples of random variables. In order to develop an algorithm for modeling thermokarst lake fields, it is necessary to take into account the type of x, y and s distribution laws and the statistical relationships between changes in the coordinates of lake centers and their areas, which, according to (

where f (x) and f (y) of the probability density of a uniform distribution.

Taking into account equation (5), the generation of a sequence of random numbers that determine the characteristics of the location of the centers of circles is carried out using a pseudo-random number generator distributed according to the law of uniform density. To simulate lakes with random sizes, the areas of which are distributed according to the lognormal law (1), sequences of pseudo-random numbers are generated that satisfy the lognormal distribution law, in accordance with the equation obtained in (

where r is a pseudo-random number distributed according to the normal law, calculated by the formula:

where q_{j} is a random variable uniformly distributed on the interval [0,1].

The implementation of algorithms for modeling the spatial structure of the fields of thermokarst lakes described above was carried out using the C # programming language (

General structure of the software package for geo-simulation modeling of thermokarst lakes fields

1) data entry tools required for modeling thermokarst lakes;

2) the module for generating pseudorandom number variables distributed according to the normal law in accordance with equation (7);

3) a module for generating pseudorandom number variables distributed according to a lognormal law in accordance with equation (1);

4) means of displaying simulation results in a spreadsheet format in the form of files with the extension .xlsx and in the graphical format like Fig.

Result of modeling the field of thermokarst lakes with the lognormal size-distribution law

The result of the work of the geo-simulation software system is a field of model lakes, the areas of which are distributed according to a lognormal law. For illustration inFig. 6 shows the fragment of a modeled image of a field of thermokarst lakes. In the simulation of this fragment, the number of model lakes 3000 and the parameters of the lognormal distribution M = 6.88 and D = 3.42, defined by experimental data based on satellite images for the permafrost zone of Western Siberia, were specified.

The data presented in the “Results” section allows us to experimentally substantiate the applicability of the previously developed (

The approximation of the truncated lakes distribution histogram, made using the Excel software package, showed that the empirical histogram of the distribution of lake areas in the size range 2-500 ha with a high coefficient of determination (r^{2} = 0.72) corresponds to an exponential law. This can be considered as an experimental confirmation of the practical applicability of a previously developed model with an exponential distribution of lakes in size, obtained from Landsat image data. Such a simplified model can be used, for example, to estimate water reserves accumulated in thermokarst lakes of the Arctic zone of Russia, or to study and predict the dynamics of areas of thermokarst lakes under conditions of climatic changes and other tasks.

However, the model with the exponential distribution of lakes does not take into account small lakes, which are considered intensive sources of methane emissions. These small lakes are not found on medium-resolution Landsat images and therefore are not involved in developing a model with an exponential distribution of lake sizes. Therefore, in modeling the fields of thermokarst lakes in the permafrost zone, which require consideration of small lakes, for example, in estimating methane emissions, it is necessary to use a model with a log-normal distribution of lake size based on the sharing of high-resolution satellite images.

A visual comparison of the graphs in Figs

The article presents an approach to modeling the spatial structure of the fields of thermokarst lakes based on a geo-simulation model representing a set of random circles with a uniform distribution of the coordinates of their centers and a log-normal distribution of lake areas. An experimental substantiation of the lognormal distribution of lake sizes is given on the basis of the results of research on the empirical distribution of areas of thermokarst lakes in a very wide range of their sizes in the permafrost zone based on the joint use of satellite images of different spatial resolution obtained for the northern territories of Western Siberia. The results of checking the compliance of this law with the empirical histogram showed that the log-normal law corresponds to the experimental data, according to the Pearson criterion, at a significance level of 0.99.

The procedure for modeling the field of thermokarst lakes is briefly described, where each model lake is characterized by a triple of numbers: the coordinates of the center and the area of the lake. A fragment of a simulated field of thermokarst lakes is presented. The statistical characteristics of the model field of the lakes were obtained using images of medium and high spatial resolution.

The results can be used to obtain predictions of the dynamics of methane emissions from the thermokarst lakes of the Arctic zone of Northern Eurasia for the coming decades in the context of climate summit decisions (Paris, 2015).

The reported research was funded by Russian Foundation for Basic Research and the government of the Tomsk region of the Russian Federation, grants № 18-47-700001, № 18-45-860002 and № 18-45-703001.