530 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 3, JULY 2010
The Modeling Analysis of Microwave EmissionFrom Stratified Media of Nonuniform Lunar CrateredTerrain Surface for Chinese Chang-E 1 Observation
Ya-Qiu Jin, Fellow, IEEE, and Wenzhe Fa
Abstract—In China’s first lunar exploration project, Chang-E 1,the multichannel (3.0, 7.8, 19.35, 37 GHz) microwave radiome-ters were aboard the satellite, with the purpose of measuringmicrowave brightness temperature from the lunar surface andsurveying the global distribution of lunar regolith layer thickness,and global evaluation of 3He content. To analyze the modelingof microwave radiative transfer from three-layered media of thelunar surface, some factors, such as the cratered lunar surfaceroughness and scattering of regolith particulate medium withtemperature profile, are discussed. The three-layer model makesthe predominance of the parameters, such as the regolith layerthickness and stratified structures, to be studied.
Index Terms—Lunar cratered terrain, nonuniform tempera-ture profile, radiative transfer, randomly rough surface, stratifiedmedia.
I. INTRODUCTION
THE PHYSICAL structure and composition of stratifiedlunar media are the main tasks of lunar exploration. In
China’s first lunar exploration project, namely, Chang-E 1(CE-1), multichannel (3.0-, 7.8-, 19.35-, and 37-GHz) mi-crowave radiometers were aboard the satellite, with the pur-poses of measuring microwave brightness temperature fromthe lunar surface, surveying the global distribution of lunarregolith layer thickness, and globally evaluating 3He content.The spatial resolution of radiometer observation is 30–50 km,with an accuracy of 0.5 K [1]. To meet the scientific missionbased on CE-1 technology, a theoretical three-layer model, i.e.,composed of lunar soil, regolith layer, and underlying rockmedia, has been proposed [2]–[4].
To analyze the modeling of microwave radiative transferfrom three-layer media of the lunar surface, based on the sta-tistics of the lunar cratered terrain, e.g., population, dimension,and shape of craters [5]–[7], and using the Monte Carlo (MC)method, the cratered lunar surfaces are numerically generated.The triangulated network [8], [9] is utilized to divide the undu-lated lunar surface into discrete triangle meshes with 10-m sizefor the spatial resolution (30 km × 30 km) of CE-1 radiometerobservation. The reflectivities of each plane mesh are calcu-
Manuscript received October 27, 2009; revised December 6, 2009. Date ofpublication March 15, 2010; date of current version April 29, 2010. This workwas supported in part by National Science Foundation of China under Projects60971091 and 40637033 and in part by the State Key Laboratory of RemoteSensing Science under Grant 2009KFJJ011.
The authors are with the Key Laboratory of Wave Scattering and RemoteSensing Information (Ministry of Education), Fudan University, Shanghai200433, China (e-mail: [emailprotected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LGRS.2010.2040802
lated, and then, the average reflectivity for all MC-realized lunarsurface and the emissivity are obtained. The model of parallel-stratified lunar media makes the predominance of parameters,such as regolith layer thickness and stratified structures, to bestudied. Using the radiative-transfer equation of stratified mediawith dense particulate scatterers, the scattering coefficient ofthe regolith layer is found negligible, and the emission ismainly governed by the absorptive property of the medium.The brightness temperatures of multilayered media, i.e., lunarsoil, regolith layer with temperature profile, and underlyingrock media, are derived and calculated from the radiative-transfer equation, and all relevant factors affecting modelingand emission simulation are analyzed.
II. NUMERICAL GENERATION OF LUNAR CRATERED
SURFACE AND CALCULATION OF MICROWAVE EMISSIVITY
During the long geological history of the Moon, the lunar sur-face has been affected by volcanism and high-velocity impactsof large and small meteoroids, and there are many linear andcrater structures on the lunar surface, such as maria, mountains,cliffs, rilles, and craters [5]. The uppermost layer of the Moonsurface is known as lunar regolith, which consists of fragmentedmaterials such as surface-layer dust, unconsolidated rock mate-rial, breccia, glassy fragments, and so on. The average thicknessof the regolith layer was estimated as about 4–5 m for mariaand about 10–15 m for highlands. Knowledge of the structure,composition, and distribution of the lunar regolith providesimportant information concerning lunar geology and resourcesfor future lunar exploration.
It has been known that the Moon surface is classified intotwo types: maria (singular: mare) and highlands (or terrae).Maria, as broad smooth plains, take about 17% of the totallunar surface, while rugged highlands, with an altitude of about2–3 km higher than that of maria, take the rest of 83% in total[5]. Generally, surface undulation makes a small slope gradientof about 3◦ at the meter scale and low concentration of meter-scale rock inclusions for most of the lunar surface.
A. Generation of Lunar Cratered Surface
The crater diameter varies from tens of centimeters to hun-dreds of kilometers, and statistics shows that there are totallyabout 33 000 craters with a diameter larger than 1 km [5]. Togenerate the cratered lunar-surface terrain, the statistical resultsof the cumulative crater population and crater-depth–diameterrelation are employed [6], [7]. Because the impact probabil-ity of small craters is larger than that of larger craters, a
1545-598X/$26.00 © 2010 IEEE
JIN AND FA: MODELING ANALYSIS OF MICROWAVE EMISSION FROM STRATIFIED MEDIA 531
Fig. 1. Cratered lunar surfaces randomly generated by the MC method.
Fig. 2. Triangulated network to divide a crater with 0.5-km radius.
power-law relation between the cumulative number Ncum ofcaters and crater diameter D was given as follows [5]–[7]:
log Ncum ∝ −2 · log D. (1)
The generation process of craters is assumed to be impactcratering on the simulation plane due to meteorite bombard-ments. The position of an impact point is randomly chosenusing the Monte Carle method [10], and the parameterizedcrater dimension and shape are chosen randomly under thestatistics of cumulative number of craters [5]–[7].
Using the MC method to randomly generate 4000 craterson the area of 30 km × 30 km, which is equivalent to thespatial resolution of CE-1 radiometers. Thus, the crater densityis 4.45/km2. The minimum radius of the crater is Rmin =50 m. Fig. 1 shows two lunar surfaces generated by two MCrealizations. Totally, there are ten MC realizations in this letter.The lunar topography is mainly described by the crater shape,dimension, and number.
B. Triangulated Network for Emissivity Calculation
To numerically calculate the microwave emissivity of theundulated lunar surface, a triangulated network (Fig. 2) is em-ployed to divide the lunar surface into multitriangular meshes[8], [9]. As good computation is required from a computer,all meshes with 10-m size each are composed to generate thedigital surface topography. Thus, there would be totally 1.8 ×107 meshes (= 2 × (30 km/10 m) × (30 km/10 m) meshes) ina lunar surface of 30 km × 30 km, which is equal to the spatial
Fig. 3. Calculation of the reflectivities of triangular meshes.
Fig. 4. Scattering coefficient of dense particular medium (εs = (3. +i0.03)). (a) 3 GHz. (b) 37 GHz. (Solid line) a = 0.15 cm. (Dashed line)a = 0.10 cm. (Dash-dotted line) a = 0.5 cm.
resolution of CE-1 radiometers. Fig. 3 shows an example of acrater (the size is 0.5 km) using a triangulated network.
As shown in Fig. 4, using the coordinates of the end pointsof each mesh, the local normal vector of each mesh nl iscalculated, and its angle with the z-direction is θl = cos−1(z ·nl), l = 1, . . . , L = 1.8 × 107.
Under a spatial resolution of 30 km2, to see how muchthe roughness of the cratered lunar surface can affect the mi-crowave emission, the emissivity can be calculated as [10]–[12]
ep(θ) = 1 − rp(θ)
= 1 − 14π
2π∫0
dφ′π∫
dθ′ sin θ′ [γpp(θ, θ′, φ′) + γpq(θ, θ′, φ′)]
(2)
where ep(θ) and rp(θ) are the p (= v, h)-polarized emissivityand reflectivity, respectively, and γpp(θ, θ′, φ′) and γpq(θ, θ′, φ′)are the copolarized and cross-polarized bistatic scatteringcoefficients, respectively. Because the observation direction ofthe CE-1 radiometer is θ = 0◦, no polarization is specified inthe following discussion.
For a lunar surface generated by one MC realization, thereflectivity of each plane mesh is easily written as
rl(m) =14π
∫∫Sl(m)
dθ′ sin θ′dφ′(γvv + γvh) (3)
which is the reflectivity of the lth mesh plane in the mth MCrealization. Such reflectivity of a plane mesh can simply becalculated as
rl =
∣∣∣∣∣cos θl −√
ε1 − sin2 θl
cos θl +√
ε1 − sin2 θl
∣∣∣∣∣2
(4)
where ε1 is the dielectric constant of the lunar surface. Thus, itgives the average reflectivity of the lunar surface with the size
532 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 3, JULY 2010
30 km × 30 km in the mth MC realization as
〈rm〉 =1L
L∑l=1
rl(m). (5)
Letting ε1 = 2.7 + i0.01, the average reflectivities for eachMC realized lunar surface, i.e., 〈rm〉 (m = 1, . . . , 10), are,respectively, summarized as follows:
0.0620339 0.0620372 0.0617953 0.0618913 0.06204610.0622154 0.0616440 0.0614814 0.0615608 0.0624106.
(6)
Thus, the average reflectivity of ten MC realizations in total is
〈r〉 =1M
M∑m=1
〈rm〉 = 0.0619116. (7)
Correspondingly, the reflectivity of a flat surface is easilyobtained as
r0 =∣∣∣∣1 −√
ε1
1 +√
ε1
∣∣∣∣2
= 0.0592122 (8)
The difference between results (7) and (8) is only 0.0026994.Supposing that the physical temperature of the lunar surface is350 K, the reflectivity difference might cause 0.9 K of bright-ness temperature. Therefore, the roughness of the undulatedlunar surface is not as significant as one might expect undera spatial resolution of 30 km2. It is good enough to modelthe cratered lunar surface as a flat surface, and it reduces theunknown parameters to be retrieved.
III. BRIGHTNESS TEMPERATURE OF STRATIFIED LUNAR
MEDIA WITH TEMPERATURE PROFILE
The lunar regolith consists of fragmented materials suchas surface-layer dust, unconsolidated rock material, breccia,glassy fragments, and so on, and its thickness varies fromseveral meters to tens of meters. As the lunar surface becomesmore mature due to exposure to solar wind and galactic cosmicrays, the regolith particulates become smaller and more dense.It makes the regolith as a uniformly dense clustered particulatemedium. There is a soil dust layer overlapped on the regolithwith a thickness of several centimeters, which has great changeof physical temperature in lunar day and night, and heat conduc-tion makes the regolith layer with a temperature profile, whichwas described as an exponential distribution [13].
A. Scattering Coefficient of Dense Medium and EffectiveDielectric Constant
Scattering and absorption of the regolith medium are relatedwith the scatterer fraction, size and shape, dielectric properties,etc. The real part of the dielectric constant of the regolith ismainly dependent on its volumetric density ρ, and the imag-inary part (loss tangent tan δ ≡ ε′′2/ε′2) is particularly relatedto the content S of FeO + TiO2. Suppose that the regolith is alayer of random spherical particles with equal radius a, whosefractional volume is fv and dielectric constant is εs. The scatter-ing and extinction coefficients, i.e., κs and κe, respectively, canbe derived from effective wavenumber K using dense scatter
Fig. 5. Emission from a layer of the uniform medium with temperature profileT (z) and thickness d.
media theory with the Percus–Yevick pair distribution function[10]–[12]. It has been obtained that
K2 = k20 + fv
k2s − k2
1 + (k2s − k2
0) (1 − fv)/(3K2)
×[1+i
29Ka3 k2
s − k20
1 + (k2s − k2
0) (1 − fv)/(3K2)
(1 − fv)4
(1 + 2fv)2
](9)
where k0 is the wavenumber of free space and ks =k0√
εs. From (9), it yields the extinction coefficient κe =2K ′′ = κs + κa.
Fig. 5 shows the scattering coefficient κs of dense scatterermedium for 3 and 37 GHz. As the scatterers are densely distrib-uted (fv > 0.1), scattering from random scatterers becomes co-herent. The thin lines in the figure denote independent Rayleighscattering, which are much deviated from the correspondingcurves as fv keeps increasing. It can be seen that as fv >0.5 (the regolith porosity becomes very small), the particulatemedium tends a uniform body, and scattering actually becomesnegligible. Thus, the regolith medium can be seen as a uniformabsorptive medium without scattering (in the figure, the particleradius is chosen in the millimeter order due to clustering).
Of course, if there are some significant interruptions instratification or, for example, distribution of large rock stones,scattering might become significant. This scattering effect inthe penetration depth should be taken into account, particularlyfor active radar remote sensing with much high spatial re-solution [13].
To take account of possible nonuniform particulates, as afirst-order approximation in the strong fluctuation approach[10]–[12], the permittivity εg of the regolith medium can beseen as an effective permittivity and is calculated by the ensem-ble average of dielectric fluctuation [ε(r) ∈ (εs, εs0)] as⟨
ε(r) − εg
ε(r) + 2εg
⟩= 0
i.e., fv
(εs − εg
εs + 2εg
)+ (1 − fv)
(ε0 − εg
ε0 + 2εg
)= 0 (10a)
or, explicitly
εg =14
{√[εs − 2ε0 − 3fv(εs − ε0)]
2 + 8εsε0
− [εs − 2ε0 − 3fv(εs − ε0)]}
. (10b)
As fv becomes very large or very small, εg tends to thealgebraic average.
JIN AND FA: MODELING ANALYSIS OF MICROWAVE EMISSION FROM STRATIFIED MEDIA 533
Fig. 6. Model of three-layer lunar surface media.
B. Radiative Transfer of Stratified Lunar Media
Consider a uniform medium with temperature profile T (z)and thickness d, as shown in Fig. 6. The upgoing emission indirection θ from a thin layer dz at z is written as
dTB = ka(z)T (z) sec θe−
z∫0
ka(z′) sec θ dz′
dz. (11)
Note that the variable z in integral is taken as positive forsimplicity. Moreover, the downgoing emission in π − θ iswritten as
dTB = ka(z)T (z) sec θe−
d∫z
ka(z′) sec θ dz′
dz. (12)
It is reflected by the underlying surface at z = −d (the reflec-tivity is r23(θ)) and returned upgoing through layer attenuationand finally comes out as
dTB = r23(θ)e−
d∫0
κa(z) sec θ dz
ka(z)T (z)
× sec θe−
d∫z
ka(z′) sec θ dz′
dz. (13)
Thus, from (12) and (13), the upgoing emission from the mediaof Fig. 5 is written as [10]–[12]
TBp(θ, z = 0)
= e23p(θ)T3e−
d∫0
κa(z) sec θ dz
+ r23p(θ)e−
d∫0
κa(z) sec θ dz
×d∫
T (z)κa(z) sec θe−
d∫z
κa(z′) sec θ dz′
dz
+
d∫0
T (z)κa(z) sec θe−
z∫0
κa(z′) sec θ dz′
dz. (14)
Finally, the p(= v, h)-polarized brightness temperature ob-served in region 1 is obtained as
TBp(θ1) = t12p(θ)TBp(θ, z = 0)= [1 − r12p(θ)] TBp(θ, z = 0) (15)
where θ and θ1 satisfy the Fresnel law, t12p(= 1 − r12p) isthe p-polarized transmittivity (i.e., emissivity), κa(z) is theabsorption coefficient of the medium, and T (z) is the temper-ature profile. Given ε(z), it can be obtained that κa = 2K ′′ =
Fig. 7. Brightness temperature of stratified media with temperature profileT1 = 390 K, T3 = 250 K, εg1 = 2. + i0.02, εg2 = 3. + i0.03, ε3 = 8. +i0.08, θ = 0, and d2 = 500 cm. (a) T1 = 390 K. (b) T1 = 150 K. (Linewith triangle) d1 = 0.01 m, β = 0.1. (Line with inverted triangle) d1 =0.01 m, β = 5.0. (Line with square) d1 = 0.05 m, β = 0.1. (Line with circle)d1 = 0.05 m, β = 5.0.
(2πν/c)(ε′′g/√
ε′g) (where ν and c are the frequency and lightspeed in free space, respectively; εg = ε′g + iε′′g is seen as aneffective permittivity of the layer). As T (z) is given, (14) canbe numerically calculated for the brightness temperature ofmedia with both nonuniform temperature profile and volumetricdensity.
Now, consider a three-layer model, as shown in Fig. 7. Theinterface between the dust and regolith layers is located at z = 0in these coordinates. Supposing that the physical temperaturesof layered media are T1, T2(z) = Ae−βz + B, and T3 andsatisfy
T2(z = 0) = A + B ≡ T1
T2(z = −d2) = Ae−βd2 + B ≡ T3 (16)
it yields
A =T1 − T3
1 − e−βd2B = T1 − T1 − T3
1 − e−βd2. (17)
Following the radiative transfer described in (12) and (13), thebrightness temperature of three-layer hom*ogeneous media iswritten as
TB = TB1 + TB2 + TB3 (18)
where the contribution from the first layer is
TB1 = (1 − r01)(1 − e−κa1d1 sec θ)(1 + r12e−κa1d1 sec θ)T1.
(19a)
The contribution coming from the second layer, i.e., the regolithlayer, is written as (19b), shown at the top of the next page.The contribution from the underlying media, i.e., the thirdlayer, is
TB3 = (1 − r01)(1 − r12)(1 − r23)e−κa1d1 sec θe−κa2d2 sec θT3.(19c)
Note that each θ in (19a)–(19c) is not the same (satisfying theFresnel law). However, for simplicity and θ = 0◦, as chosen inCE-1, the difference is not specified.
If β → 0, it gives T2(z) = A + B = T1, which means thatthe temperatures of layers 1 and 2 are the same.
If β � 1, and writing T2(z) = B = T3 ≡ T2, (18) and(19a) return to the model of three-layer media with uniform
534 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 3, JULY 2010
TB2 = (1 − r01)e−κa1d1 sec θ(1 − r12)
⎡⎣κa2 sec θ
d2∫0
T2(z)e−κa2z sec θdz + r23e−κa2d2 sec θκa2 sec θ
d2∫0
T2(z)e−κa2(d2−z) sec θdz
⎤⎦
= (1 − r01)e−κa1d1 sec θ(1 − r12)
×⎡⎣κa2 sec θ
d2∫0
(Ae−βz + B)e−κa2z sec θdz + r23e−2κa2d2 sec θκa2 sec θ
d2∫0
(Ae−βz + B)eκa2z sec θdz
⎤⎦
= (1 − r01)e−κa1d1 sec θ(1 − r12)
{Aκa2 sec θ
[1 − e−(κa2 sec θ+β)d2
κa2 sec θ + β+ r23e
−κa2 sec θd2e−βd2 − e−κa2 sec θd2
κa2 sec θ − β
]
+ B(1 + r23e−κa2d2 sec θ)(1 − e−κa2d2 sec θ)
}(19b)
temperatures (T1, T2, and T3) at θ = 0◦ as
TB = (1 − r01)(1 − e−κa1d1)(1 + r12e−κa1d1)T1+(1 − r01)
× (1 − r12)(1 − e−κa2d2)(1 + r23e−κa2d2)e−κa1d1T2
+ (1 − r01)(1 − r12)(1 − r23)e−κa1d1e−κa2d2T3. (20)
Fig. 7 shows the brightness temperature of three-layer lunarmedia with temperature profile T2(z), where β = 0.1, 5. Theline --� -- is for the case of half space with T1, and the line--•-- is for the case of half space with T3. It can be seen that thebrightness temperatures of different frequency channels duringdaytime and nighttime are quite different. Furthermore, TB19
and TB37 of high-frequency channels are very sensitive to thetop layer. Thus, d1 and β are the main factors affecting TB19
and TB37, and TB7 and, in particular, TB3 of low-frequencychannels are mainly contributed by TB2 and TB3 of (19b) and(19c) due to the large penetration depth.
With the chosen d1, (20) can be applied to the calculation ofthe brightness temperature of three-layer lunar media.
IV. CONCLUSION
To simulate and retrieve the information of Chinese CE-1radiometer observation, a three-layer model has been devel-oped. To analyze the modeling of microwave radiative transferfrom the three-layer model, based on the statistics of the lunarcratered terrain and using the MC method, the cratered lunarsurfaces have been numerically generated. The triangulatednetwork has been utilized to divide the undulated lunar surfaceinto discrete triangle meshes for calculations of reflectivity andemissivity. It is found that under a spatial resolution of 30 km ×30 km of CE-1 radiometer observation, the lunar surface canbe well modeled as a flat surface. It makes the predominanceof parameters for information retrieval, such as regolith layerthickness and stratified structures.
Using the radiative-transfer equation of stratified media withdense scatterers, the scattering coefficient of the regolith layerhas been found negligible, and the emission is mainly governedby the absorptive property, which is related to the local contentof FeO + TiO2 in the lunar surface.
The brightness temperatures of multilayered media, i.e., lu-nar soil, regolith layer with temperature profile, and underlying
rock media, have been derived and calculated. It can be seenthat high-frequency channels, i.e., 19 and 37 GHz, are sensitiveto the temperature profile of the top media and that low-frequency channels, i.e., 7 GHz and, in particular, 3 GHz, aresensitive to the whole regolith layer. The emissions are mainlycontributed by the regolith and underlying rock media. Themodeling of three-layer media is feasible to applications ofsimulation of lunar-surface emission, inversion of regolith layerthickness, evaluation of 3He content, and data validation ofCE-1 microwave emission observation [4].
REFERENCES
[1] J. S. Jiang, “Microwave Moon,” Sci. Chin. (D), vol. 39, no. 8, p. 1028,2009.
[2] W. Fa and Y. Q. Jin, “Simulation of brightness temperature of lunarsurface and inversion of the regolith layer thickness,” J. Geophys. Res.,Planet, vol. 112, p. E05 003, 2007, DOI: 10.1029/2006JE002751.
[3] W. Fa and Y. Q. Jin, “Quantitative estimation of helium-3 spatial dis-tribution in the lunar regolith layer,” Icarus, vol. 190, no. 1, pp. 15–23,Sep. 2007.
[4] W. Fa and Y. Q. Jin, “Analysis of microwave brightness temperature oflunar surface and inversion of regolith layer thickness: Primary resultsfrom Chang-E 1 multi-channel radiometer observation,” Icarus, 2010,doi:10.1016/j.icarus.2009.11.034.
[5] G. H. Heiken, D. T. Vaniman, and B. M. French, Lunar Source-Book: AUser’s Guide to the Moon. London, U.K.: Cambridge Univ. Press, 1991.
[6] R. B. Baldwin, “Lunar crater count,” Astrophys. J., vol. 69, pp. 377–391,1964.
[7] R. J. Pike, “Depth/diameter relations of fresh lunar craters: Revisionfrom spacecraft data,” Geophys. Res. Lett., vol. 1, no. 7, pp. 291–294,Nov. 1974.
[8] R. A. Dwyer, “A faster divide and conquer algorithm for constructingDelaunay triangulations,” Algorithmica, vol. 2, no. 1–4, pp. 137–151,Nov. 1987.
[9] W. Fa, F. Xu, and Y. Q. Jin, “Image simulation of SAR remote sensingover inhom*ogeneously undulated lunar surface,” Sci. Chin. (F), vol. 52,no. 4, pp. 559–574, Apr. 2009.
[10] Y. Q. Jin, Theory and Approach of Information Retrievals From Electro-magnetic Scattering and Remote Sensing. Berlin, Germany: Springer-Verlag, 2005.
[11] L. Tsang, J. A. Kong, and B. Shin, Microwave Remote Sensing Theory.New York: Wiley, 1985.
[12] Y. Q. Jin, Electromagnetic Scattering Modelling for Quantitative RemoteSensing. Singapore: World Scientific, 1994.
[13] A. R. Vasavada, D. A. Paige, and S. E. Wood, “Near-surface temperatureon Mercury and the Moon and the stability of polar ice deposits,” Icarus,vol. 141, no. 2, pp. 179–193, Oct. 1999.
[14] Y. Q. Jin, F. Xu, and W. Z. Fa, “Numerical simulation of polarimetricradar pulse echoes from Lunar regolith layer with scatter inhom*ogeneityand rough interfaces,” Radio Sci., vol. 42, no. 3, pp. RS3 007-1–RS3 007-10, May 2007, DOI: 10.1029/RS2006003523.
