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At present, an investigation of the lunar ground at great depths is of paramount importance. This investigation can be carried out using decameter and meter waves. This article aims to analyze the variations of the reflectioncoefficient at decametric, meter and decimeteric bands. A possibility of determination of lunar ground characteristics by bistatic radar using powerful ground-based transmitters at VHF and UHF bands and a receiver aboard a Moon’s satellite is analysed. Appropriate algorithms are considered for determinationof the regolith layer thickness, dielectric permittivity, loss tangent, and density of the regolith and bedrocks. Expected results of measurements have been presented for a two-layer model of lunar ground, consisting of an upper layer with the loose porous rocks (regolith), and the rocks situated more deeply. Revealed regularities are a basis for determining the distribution of the permittivity in subsurface layer.

In 2007, the Russian program of the Moon and near-lunar space exploration started. The space research of the composition and physicochemical properties of the lunar regolith is a special branch of this program. The determination of the regolith density and thickness is of interest both of a practical and of a geological point of view. To understand the origin and evolution of the Moon, it is important to study the depth structure and composition of the soil. This information can be obtained by sounding the lunar ground using radar located on a spacecraft. Polar-orbiting satellite for an appropriate time provides the study of the Moon entire surface as both seen from the Earth, and the reverse side. Similar experiments are planned to carry out in the framework of the mission Luna-Glob using modulated radio signals in the frequency bands of 20 MHz and 200 MHz [

The investigation of ground on the visible side of the Moon up to great depth is possible by method of bistatic sounding using powerful ground-based transmitters operating in the HF, VHF and UHF frequency bands, and a multichannel receiver aboard a lunar satellite.

The soil explorations are fulfilled now by NASA’s Mini-RF monostatic radar installed on the Lunar Reconnaissance Orbiter (LRO), which used also the Arecibo transmitter (emitting 200 kW at wavelength 13 cm) for bistatic studying of the lunar regolith [

The idea of bistatic location of the Moon and planets using of high-power ground-based transmitters decameter and natural sources of radio emission has been proposed in [

The study of the Moon ground at the great depth, which can be carried out at present by radar-tracking methods using decameter and meter waves, became relevant. The purpose of this article is the analysis of possibilities of sounding ground on the big depth by method of bistatic radar using powerful ground-based transmitters of meter and decameter radio waves and a receiver located on the lunar satellite. It is necessary to define the reflection coefficient for several wavelengths for the layered model of the ground depending on the position of the satellite and to solve a direct problem of bistatic sounding the lunar ground. For definition of ground characteristics, it is necessary to develop a method of solving an inverse problem, i.e. to find the horizontal and vertical distribution of the dielectric permittivity and density on a basis of the experimental values of the signal characteristics.

as

It is suggested that the primary model of the lunar ground is the regolith layer lying on the bedrocks. The regolith density and the thickness of its layer depend on geologically activities of the certain region.

We will present the medium model as layered structure with complex dielectric permittivity_{a}, angles of incidence

The real part of dielectric ground permittivity

The data about absorption coefficient of radio-waves by regolith are the most ambiguous, this characteristic is expressed by a known relation through a loss tangent

has values from 10^{−3} up

where ^{−3}.

The thickness of the regolith layer l has been estimated according to seismic data from space vehicle impact against the lunar surface and registration of ground oscillations by registration of seismic sensors after falling spacecraft on the Moon surface. For dark sites of the surface―“lunar mare” it was found out l » 2 − 5 m, and for light areas―“continents” l » 8 − 15 m. It is supposed, that flat areas of the big craters can have a little thickness of regolith, and in some areas, the thickness of the regolith layer changes smoothly along the surface. The specified data allow to accept following possible limits of variable parameters of the ground model: for regolith

We introduce the reflection coefficient F. It equals to the ratio of the radio wave field strength between tracks “Earth - reflected site - satellite A” and “Earth - satellite”. F is represented by two factors:

where T is the reflection coefficient of the field strength from a smooth sphere with a high conductivity, M is a module of the complex reflection coefficient of a radio wave from the plane boundary. This boundary is the tangent to the sphere in reflecting point. Here we discuss horizontally polarized wave. The first factor in the expression (3) is determined with increasing the cross sectional area when beam is reflected from the sphere. Тhe coefficient T depends only on the incidence angle

Derivation of this expression can be found in some books, e.g. in [

Here

In relationships (6) and (7) we introduce the complex refractive indices

We are interested in the module of the reflection coefficient of a radio wave field strength in the site of the lunar surface, which is determined by the following expression with taking relationships (5) and (8) into account:

Components of the formula (9) are given by relations (6)-(8) and the following expressions:

The Fresnel coefficients arguments in (6) and (7) are denoted as

Equation (9) has two asymptotic solutions. At _{1}(η), i.e. radio wave is reflected from the regolith surface, and the presence of bedrock is not shown. Putting l_{max} = 15 m and tan d_{1} = 4 ´ 10^{−2}, we find that this is possibly only at case of centimeter radio waves. Other asymptotic solution is fairly obviously: at _{2}(η), i.e. wave reflected from the rock, not feeling the regolith. In all other cases, the dependence M(η) is an oscillating function. The position of the local extremes of the function M(η) and their number when changing η in the range from 0˚ to 90˚ is given by formula

where m―the integer 1, 2, …It follows from (12) that the reflection coefficient is a multi-valued function, even for fixed values of λ,

Using the model of the lunar ground, and the expressions (6) - (11) we analyze dependence of factor F on the layer thickness regolith l, the wavelength

According to (9), the upper

with the formulas:

Values

radio wave incidence angle oscillations dependence

The regolith layer thickness, the density and dielectric parameters of a rocks must be are reconstructed using the experimental values of the reflection coefficient F. Formulae (3) and (4) show that the measured values of the reflection coefficient allow to calculate M = F/T, which is the reflection coefficient from plane-layered ground. The reflection coefficient M is connected with regolith and bedrock parameters by the complex relationship (9). Using a digital map of the Moon we choose a relatively flat region, for which the Rayleigh law is correct:

where

We assume that long-term operation of the lunar satellite receiver creates a large array of reflection coefficient measurements at different incidence angles. Such array can be obtained for three radio bands: in decameter, meter and decimeter bands.

Let us now consider the method for determination of the dielectric constant

Using this value

We present a procedure for determining the thickness of the regolith l . For this aim, we use the results of measurements of the reflection coefficient of radio waves in meter band and value

connected with the functions

Fresnel reflection coefficients from the boundary between vacuum and regolith

which depends on two unknown X and l. l is a multiplier in

Here L_{1},

From the equations system (18) we obtain:

Substituting (19) into the first equation of system (18) gives the equation relating

This equation depends on one unknown l , it is solved numerically. The solution is the minimal value, when (20) is correct.

In order to determine the imaginary part of the regolith refractive index

where the unknown

The reflection coefficient from the boundary “regolith - bedrock”

The density of the bedrock

For decameter waves, the interference changes of the reflection coefficient can be observed in some regions with a horizontal gradient of the regolith thickness

Here

Registration of oscillations of reflection coefficient for meter or decimeter radio waves in the region with the horizontal gradient

Discussed principles of the inverse problem solving of sensing subsurface layer are the basis for the creating a multi-stage algorithm for determining

The method of bistatic subsurface sounding allows measuring the reflection coefficient of the radio waves with high accuracy. It is possible because the reflection coefficient is a ratio of the field strengths of the direct and reflected from the Moon radio waves. The relative measurement of the amplitudes of these signals may be due to their frequency separation. In this case, the possibly variations of the transmitter power, receiver gain and the effects of the Earth’s ionosphere do not affect the accuracy of the measurements. The transmitter location on the Earth provides much greater signal power compared with transmitter located at the satellite. It provides a deeper penetration of radio waves in the subsurface. The sizes of reflecting spots on the lunar surface depend on satellite height, wavelength and an incidence angle of radio waves. For height

Using both bistatic and monostatic modes of radio location improves the accuracy and reliability of the research results. The possibilities of a monostatic location of the subsurface with receiver and transmitter located on the lunar satellite are given in papers [

We thank A.I. Efimov, whose comments helped improve this manuscript. The work has been supported by the Program № 9 “Experimental and theoretical studies of objects in the solar system and planetary systems of stars” of the Presidium RAS and partially by Grant of Russian Fund for Basic Research RAS № 13-02-00526.

Oleg I.Yakovlev,Olga V.Yushkova,Stanislav S.Matyugov,Alexander G.Pavelyev,Vladimir M.Smirnov, (2015) Determination of the Lunar Ground Characteristics Using Bistatic Radar. International Journal of Geosciences,06,1267-1276. doi: 10.4236/ijg.2015.612101