As noted in [1], approximately 75% - 90% of all information contains coordinates or spatial locations of objects. Analysis of this type of information allows us to obtain qualitatively new geographic information and discover previously unknown patterns. Modern methods of collecting information, such as ground-based, cartographic and remote methods, are widely used in practice. Remote data collection methods can be implemented using spacecraft, manned and unmanned aerial vehicles. Effective planning of the operation of unmanned aerial vehicles (UAVs) allows us to obtain high-resolution remote information in a short period of time. The use of UAVs to obtain remote information helps to avoid some problems and dangers. previously encountered when obtaining information about certain objects or processes [2][3].

As noted in the work [4], at present, UAVs equipped with appropriate equipment are widely used to study the condition of forests and agricultural areas, to control various emergency situations, to monitor various pipelines, and for other purposes that require obtaining accurate and timely information for making operational decisions.

According to [5], UAVs can be successfully used to validate remote sensing data obtained from satellites, including data obtained from small satellites flying in formation. Formation flight technologies are also being applied to UAVs [6]. Formation flights of UAVs allow one to avoid limitations inherent to unmanned aerial vehicles, such as short flight times, small payloads, and small volumes of received data. It is obvious that the use of a group of UAVs to obtain reliable and accurate data on certain processes or objects requires their optimal placement at different flight altitudes and distances till data receive center in real atmospheric conditions, which are often characterized by high pollution, turbulence, and thermal heterogeneity. In general, unfavorable atmospheric conditions may create some problems for the successful implementation of a planned mission of a formation flight of several UAVs. As it was well-known [1]-[4] UAV formation flight can be classified into three catecories: position based: displacement placad and distance placed interaction topology. Major factor negatively affected on volume of information obtained by a group of UAVs, is the atmospheric attenuation of electromagnetic signals transmitted from the UAV to the information receiving unit from different altitudes. But the task on optimization of said tiopological factors concerning the interrelation of them linked with information losses was not researched.

Further in this article, we analyze the condition for avoiding extreme information losses that occur during group flights of UAVs at different altitudes transmitting information to a common mobile information receiving node.

A graphical illustration of the operation of several UAVs performing a group flight for remote sensing purposes is shown in Figure 1.

As shown in the work [7], the Beer-Lambert law can be taken as a basic model for the passage of electromagnetic radiation through the atmosphere, as applied to the case when the flight is carried out at a height h and the information is transmitted over a distance d till the data receiving center. According to this law

where η атм ( h ) is the atmospheric transmittance.

According to the work [8], the function γ ( h ) is defined as follows:

Figure 1. Graphical interpretation of group flights of UAVs in quantity n at different altitudes, performing remote sensing of the earth, transmitting data to the receiving center occupying positions at different altitudes. The numbers show: 1 i ( i = 1 , n ¯ ) -UAV at altitude h i , transmitting data to the center located at a distance d i 2 i -positions of the UAV data reception center.

Taking into account (1) and (2) we obtain

Taking into account the monotonic nature of the exponential function, we will investigate the expression found in the exponent of the first exponent, i.e. the function f ( α 0 , h , h 0 , d ) in the form

Obviously, to achieve maximum transmission, we must have the maximum value of f ( α 0 , h , h 0 , d ) , i.e., the maximum − d α 0 exp ( − h h 0 ) . To solve this problem, we will generalize it to the case of and the number of UAVs performing remote sensing at altitudes h i , i = ( 1 , n ¯ ) and transmitting information over a distance d j ; j = ( 1 , n ¯ ) .

Let us consider the total value β of functions f ( α 0 , h , h 0 , d ) in the form

To study β the group flight of UAVs, we introduce for consideration the optimized function

Further, in relation to group flights of UAVs, we can consider the problem of calculating the total value β in the form

Taking into account (6) and (7), moving to a continuous model of the process under consideration, we can formulate the following objective functional F 1 in the form

Continuous model (8) upon growth of n can be calculated using well-known Simpson formulae used for numerical calculation of integrals. Concerning the considered case this formulae comprise Formulae (7) multiplied to determined constant linked with increment of d. So that the inverse procedure i.e. transition from discrete model (7) to continuous model (8) is may be considered as correct. To determine the optimal form of the function, in which F 1 the extreme value would be reached, we impose the following restrictive condition on the function ψ ( d ) .

The condition (9) means that the sum of UAVs positions altitude should be limited. If not the energy consumption during formation flight may be extremely high. From another side the flight altitude is related with noisy images taken by them and big losses upon transfer of them to data receiving center. Taking into account expressions (8) and (9), we will compose the following target functional F 0 of variational optimization

where is λ the Lagrange multiplier; λ = c o n s t .

The solution of problem (10) according to Euler’s method satisfies the following condition:

From condition (11) we obtain

From expression (12) we find

Taking the logarithm of expression (15) we obtain

Thus, when solving (14), the objective functional (10) reaches an extreme value. Applying the Lagrange criterion, it is easy to determine that this extremum is a maxima, since the derivative of expression (12) with respect to the function ψ ( d ) turns out to be a negative value, which swings the factor λ, t₀, and the value can be determined using expressions (9) and (14). Taking into account expressions (8) and (14), we obtain

where

Thus, the average value of atmospheric transmission according to expressions (3) and (15) reaches its maximum value when solving (3)

To determine the reliability of the obtained results, we will conduct a model study.

In the obtained solution (14), to simplify the model study we will adopt the following relationships for simplicity and clarity.

In this case, expression (14) takes the following form

The functional (8) takes the following form

The graph of the function ψ ( d ) defined by expression (19) is shown in Figure 2.

In the interval d min = (5.7 - 15.4), curves 2.3 satisfying condition (11) were constructed graphically. The values were calculated F 1 using Formula (22), which are given in Table 1.

As can be seen from the data given in the table with solution (21), the greatest value is observed F 1 , i.e. the maximum total transmission of the atmosphere can be achieved at all heights occupied by free-riders [8].

Figure 2. Graphs of the function ψ ( d ) and alternative functions f ( d ) satisfying φ ( d ) condition (9). The numbers indicate: 1-graph of the function ψ ( d ) ; 2-graph of the function f ( d ) ; 3-graph of the function φ ( d ) .

Table 1. Magnitudes of the function F₁.

1) The problem of optimizing the remote sensing of the earth’s surface by a group of UAVs occupying positions at various altitudes and at various locations away from a mobile center for receiving information from them, which receives information from the UAV upon reaching the same altitude, has been formulated and developed.

2) The optimal type of function for the relationship between the position altitude and the distance between the UAV and the mobile information reception center has been determined, at which the total value of atmospheric transmission reaches a maximum.