Determination of the Lunar Ground Characteristics Using Bistatic Radar

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 reflection coefficient 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 determination of 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.


Introduction
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 [1].
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 [2].Experiences of bistatic sounding the Moon realised by means of the first lunar missions, have shown, that frequency selection of signals accepted on the Earth allows us to separate the reference and reflected signals and to define the reflection coefficient and dielectric properties of surface rocks [3] [4].The main regularities of the radio waves scattering by a rough lunar surface have been studied in papers [5]- [15], where it is shown, that this method allows estimating density of the surface rocks and statistical characteristics of the relief irregularities.The layered structure of the ground was not considered for the interpretation of experiment results.In reviews [16] [17] there are presented the results of the first stage of developing the method for bistatic sounding of the heavenly bodies.
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 [18].
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.

The Scheme of Measurements and Ground Model
Figure 1 shows the scheme of the bistatic location of the Moon in the observation plane.The Moon center is denoted as point B and the satellite is denoted as point A. The lunar satellite moves in a circular orbit shown by a dotted line.The satellite height is H, projection of the velocity vector on the observation plane is V.The unit vectors K and a K indicate the direction of propagation of the incident wave on the surface and reflected one from the surface respectively.The satellite position is set by the angle θ and its distance from the Moon center as BA a H R = + = , where a is the Moon radius.The position of point D on the surface is the center of reflected spot, where the incidence angle η is equal to the angle of reflectiion.The distance from the satellite to this point D is equal AD r = .From the experiment geometry and a condition of equality of incident angle and re- flected one the formula for the angles θ , η and the distance r was obtained: ( ) ( ) 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 ( ) , which depends on depth z.The lunar ground consists of the top layer of loose rocks of small density-regolith and underlying on denser bedrock.Figure 2 shows a small area of ground around the point D. On the top, the layer is situated regolith 1, and deeper-basic rock 2. The unit vectors of beam lines of incidence K and reflected wave K a , angles of incidence η on the regolith, on the bedrock γ and a projection of the velocity vector of satellite V are specified.We will result rough parameters of ground model.Possible values of , ε ε ′ ′′ , density of ground ρ and thickness l of a regolith layer are presented in [17]- [23].
The real part of dielectric ground permittivity ε ′ , averaged for the central part of the visible side of the ε ′ = ± .This behavior of ε ′ can be explained by increase of penetration depth of radio-waves into the subsurface; values of ε ′ measured in centimeter and decimeter range corresponds to regolith, and joint influence of the regolith and bedrock, respectively.In the meter range this characteristic corresponds to influence of bedrock.Dielectric permittivity and density of rocks have been defined also by the laboratory measurements of the lunar ground samples delivered by missions "Apollo" [19]- [22].Measurements of several samples have shown, that for loose samples 2 3 ε ′ ≈ − , and for breccia where ρ has dimension g•сm −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

The Reflection Coefficient
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 η and is described by the relationship: Derivation of this expression can be found in some books, e.g. in [17].The second factor depends on the parameters of a stratified ground model and wavelength λ .For the calculation of the radio wave complex reflec- tion coefficient from a stratified medium shown to Figure 2 the well-known expression is used: Here 1 M is the Fresnel complex reflection coefficient from boundary between the vacuum and the regolith: ) M is the Fresnel complex reflection coefficient from boundary between the regolith and the bedrock: ) In relationships ( 6) and (7) we introduce the complex refractive indices n n in ′ ′′ = + .The angle of radio wave incidence on the bedrock γ is related with the inci- dence angle on the regolith η by the Snell's law as shown in Figure 2: 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: , 2 cos , 2 cos , q q i q q l n q l n γ λ γ λ The Fresnel coefficients arguments in ( 6) and ( 7) are denoted as ( ) ( ) , where M ′′ are the imaginary ones.Numerical analysis shows that in (11) the expressions in the brackets are so small, that 1 2 2 2πq ϕ ϕ = = .The factor 2 q indicates the phase relations of radio waves reflected from the boundaries between the media 0 and 1, 1 and 2, and the value of 1 q is connected with the absorption of waves in a rego- lith.
Equation ( 9) has two asymptotic solutions.At  factor M(η) is approximated by M 1 (η), i.e. ra- dio wave is reflected from the regolith surface, and the presence of bedrock is not shown.Putting l max = 15 m and tan δ 1 = 4 × 10 −2 , we find that this is possibly only at case of centimeter radio waves.Other asymptotic solution is fairly obviously: at  M(η) converges to M 2 (η), i.e. wave reflected from the rock, not feel- ing 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, 1 ε and η.The value M in this case depends on the regolith thickness and losses therein.
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 λ and the angle η for 100 km satellite altitude.In our case, when reflection of radio waves from the two boundaries of subsurface is significant, the dependence F on the ratio of regolith layer thickness and the wavelength gets the oscillatory character.Figure 3 shows the dependence on the reflection coefficient F as function of the regolith layer thickness for 1 2.7 ε ′ = with the formulas: Values max F and min F depend on the loss tangent 1 tan δ and wavelength λ .In the decimeter range, with increasing layer thickness l, the difference between max F and min F decreases (Figure 4(c)) and when l = 10 − 15 m oscillations disappear.If the thickness of the regolith layer in the direction of motion of the reflecting point is changed, a periodic fluctuation of the reflected signal level should appear.The period of such variations ( ) , where d V is velocity of moving point D on the surface.At variations of a radio wave incidence angle oscillations dependence ( ) F η should also be observed.Figure 4 shows the de- pendences of ( ) and the same ground parameters tan δ , as in Figure 3.The reflection coefficient F decreases rapidly when 80 η >  due to the influ- ence of the factor T.

Determination of Ground Parameters
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 z ∆ is the standard deviation of the heights of surface roughness from the plane.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 1 ε of the regolith.For the deci- meter waves 3 λ the oscillations of M will be weakly or absents.It indicates that there is the strong radio wave absorption.In this case, the influence of the medium can be neglected 3(c)) and the average experimental value mid M corresponds to the expression (6) for the radio waves reflection from the boundary between the medium 0 and 1.If the regolith dielectric permittivity 1 Using this value 1 ε ′ and the formula (2) we may determine the regolith density 1 ρ .
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 1 ε ′ defined as described above.We choose a pair of experimental values are calculated with the formula (6) using the value 1 ε ′ for angles 1 η and 2 η .For the accepted model of ground the numerical analysis showed that, the function ( ) exp 2π M q changed little when the incidence angles changed in the range of 10˚.For the selected angles 1 η and 2 η this function can be considered as constant.
We denote the function η η as X.Taking into account the above notation, the formulae ( 9)-( 11) and the condition 1 which depends on two unknown X and l. l is a multiplier in 2 q .The equations system ( 17) is reduced to the form: Here L 1 , 2 L , 3 L , 1 N , 2 N and 3 N are the values depending on ( ) M η as follows: 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 1 n′′ , it is necessary to take two meas- ured values in the decameter band additional for the pair of radio wave reflection coefficients of meter range.The two sets must satisfy the conditions of the described above procedure.For each pair the equations ( 18) must be solved.For reconstruction 1 n′′ the solutions of the system (18)  1 M и 2 M , and after them from Fresnel formulas to restore 1 ε ′ and 2 ε ′ .
Registration of oscillations of reflection coefficient for meter or decimeter radio waves in the region with the horizontal gradient d d l x permits to determine the absorption of radio waves in the regolith.This method of the absorption detection is based on the analysis of decreasing function max

M
with increasing l, as shown at Figure 3 by curves A and B. If we define that the number of registered maximums of the reflection coefficient is equal m and take into account that, according to (12), the total change of the thickness of the regolith in probing site is m l ∆ , then the absorption coefficient in the regolith may be determined.Discussed principles of the inverse problem solving of sensing subsurface layer are the basis for the creating a multi-stage algorithm for determining 1 ε ′ , 2 ε ′ , 1 ε ′′ , 1 ρ , 2 ρ and l.Testing real algorithm for determining ground parameters requires the use of experimental data of the reflection coefficient of radio waves for different angles of incident η .

Conclusion
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 100 km H = , 2 2 m λ = and 30 η ≈  in a plane of incidence radio wave and in a perpendicular direction they are approximately 2 km x y ∆ ≈ ∆ ≈ , and when η ≈ 50˚ − 70˚ they increase to 5 km.In this work, the features of the reflection of radio waves from the lunar ground have been discussed.The method of ground parameters reconstruction has been described in detail.
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 [1], [24].The monostatic mode allows exploring the ground parameters both on visible and internal side of the Moon and-what is important-in polar areas.Radar, combined in one hardware complex and using the two methods, would provide detailed high-precision measurements for determination of the distribution of ground parameters in various regions of the Moon.

Figure 1 .
Figure 1.Geometry of the experiment.

Figure 2 ..
Figure 2. Medium model and beam lines of radio waves.

Figure 3 .
Figure 3. Dependences of reflection coefficient of radio waves F on a thickness l of the regolith layer.
η for three wavelengths: curves B and C correspond to 2 the curve A is given for 3 37.5 m λ =.It is seen that in the meter band the oscillations ( ) F η are great, and decameter oscillations absent.These curves are drawn at 7 m l =

Figure 4 .
Figure 4. Reflection coefficient F in dependence on an incidence angle η for the regolith layer thickness l = 7 m.
the system of two equations: