Dielectric Properties of Lunar Materials at the Chang’e-4 Landing Site

: On January 3rd 2019, the Chang’e-4 mission successfully landed in the Von K á rm á n Crater inside the South Pole-Aitken (SPA) basin and achieved the ﬁrst soft landing on the farside of the Moon. Lunar penetrating radar (LPR) equipped on the rover measured the shallow subsurface structure along the motion path for more than 700 m. LPR data could be used to obtain the dielectric properties of the materials beneath the exploration area, providing important clues as to the composition and source of the materials. Although the properties of the upper ﬁne-grained regolith have been studied using various methods, the underlying coarse-grained materials still lack investigation. Therefore, this paper intends to estimate the loss tangent of the coarse-grained materials at depth ranges of ~12 and ~28 m. Stochastic media models with different rock distributions for the LPR ﬁnite-difference time-domain (FDTD) simulation are built to evaluate the feasibility of the estimation method. Our results show that the average loss tangent value of coarse-grained materials is 0.0104 ± 0.0027, and the abundance of FeO T + TiO 2 is 20.08 wt.%, which is much higher than the overlying ﬁne-grained regolith, indicating different sources.


Introduction
In January 2019, the Chang'e-4 lander successfully landed in the Von Kármán Crater on the lunar farside, becoming the first probe to land on the farside of the moon. The Yutu-2 rover is equipped with instruments including a multi-antenna, two channels (channel one and two) lunar penetrating radar (LPR) system. The central frequency of channel one is 60 MHz, which is utilized to detect the relatively deep part of the subsurface structure, specifically the basalt layered structure, basalt thickness, etc. The central frequency of channel two is 500 MHz, which is used to detect the shallow subsurface of the lunar regolith, including the thickness of the lunar regolith and the depth of the basalt layer and the physical parameters of the lunar regolith, such as the electrical constant, loss tangent and density [1,2].
Up to now, most lunar missions have been flyby or orbital explorations, and the more than 20 landing sites are all on the nearside of the Moon. Therefore, the successful implementation of the Chang'e-4 exploration mission provides an unprecedented opportunity to observe the subsurface structure of the lunar farside [3].
The landing site of the Chang'e-4 is located at 177.588 • E, 45.457 • S in the Von Kármán Crater, marked by a red triangle in Figure 1 [4,5]. The Von Kármán Crater is inside the South Pole-Aitken (SPA) basin, the largest and oldest impact crater on the Moon. According to the spectrum results and impact numerical simulation, it is generally believed that the Aitken Basin at the Lunar South Pole sputters the lower lunar crust and even the upper lunar mantle material on the lunar surface, so SPA Basin has important research value to understand the lunar interior and history [6]. The Von Kármán Crater has experienced marked by a red triangle; the yellow dashed lines show the directions of Finsen ejecta. (b) Enlarged view of the red rectangle in (a). The landing site of Chang'e-4 is marked by a red triangle. Red arrows mark the linear features in the northeast-southwest direction created by Finsen ejecta with elevated topography. The base map is of the SLDEM slope, obtained from Barker et al. (2016) [5]. The color bar is the degree of the SLDEM slope.

LPR Data
Channel two of LPR has one transmitting and two receiving antennas, which are deployed on the bottom of the Yutu-2 rover, named as 2A and 2B, respectively. In this study, we only analyzed LPR 2B data. The bandwidth of LPR channel 2B is 450 MHz, which indicates that the resolution is 0.3 m in a vacuum [22]. The Yutu-2 rover has been working on the Moon for more than two years, and the LPR has detected a more than 700 m long subsurface structures along its motion path. The first 15 days of LPR data were utilized in this work ( Figure 2). The Data File IDs were listed in Table S2.  [4]. The landing site of Chang'e-4 is marked by a red triangle; the yellow dashed lines show the directions of Finsen ejecta. (b) Enlarged view of the red rectangle in (a). The landing site of Chang'e-4 is marked by a red triangle. Red arrows mark the linear features in the northeast-southwest direction created by Finsen ejecta with elevated topography. The base map is of the SLDEM slope, obtained from Barker et al. (2016) [5]. The color bar is the degree of the SLDEM slope.

LPR Data
Channel two of LPR has one transmitting and two receiving antennas, which are deployed on the bottom of the Yutu-2 rover, named as 2A and 2B, respectively. In this study, we only analyzed LPR 2B data. The bandwidth of LPR channel 2B is 450 MHz, which indicates that the resolution is 0.3 m in a vacuum [22]. The Yutu-2 rover has been working on the Moon for more than two years, and the LPR has detected a more than 700 m long subsurface structures along its motion path. The first 15 days of LPR data were utilized in this work ( Figure 2). The Data File IDs were listed in Table S2.  [4]. The white arrow indicates north. The landing site of the Chang'e-4 mission is marked by a red triangle and the waypoints of the rover's path are marked as green dots. The number near the waypoint is the starting position of the corresponding lunar day.
Data preprocessing is necessary before interpreting radar data. Based on the characteristics of the Chang'e-4 LPR data, a set of preprocessing methods is proposed [13]. In this paper, the data processing scheme is modified to retain the original amplitude information. 1. Data reading: 2B processing-level data are utilized in the paper, which are generated with raw data after integration, conversion from unsigned integers to signed integers, normalization, removing direct current offset, and adding Data preprocessing is necessary before interpreting radar data. Based on the characteristics of the Chang'e-4 LPR data, a set of preprocessing methods is proposed [13]. In this paper, the data processing scheme is modified to retain the original amplitude information.

1.
Data reading: 2B processing-level data are utilized in the paper, which are generated with raw data after integration, conversion from unsigned integers to signed integers, normalization, removing direct current offset, and adding geometric coordinates.

2.
Redundant data removal: when LPR is operating, the Yutu-2 rover may stop to perform the operations of other instruments, which generated duplicate data. So, data collected during static sampling should be removed.

3.
Data file stitching: the files of different lunar days are stored separately. In this step, these data need to be stitched together.

4.
Adjust the time delay: the receiving antenna is turned on 28.203 ns earlier than the transmitting antenna, so the initial 28.203 ns of all the data traces were removed.

5.
Background removal: subtracting the average value from the LPR data to reduce the 'ringing' effects and periodic noises caused by the antenna-ground coupling. 6.
Band-pass filter: the finite impulse response (FIR) filter is used to reduce noise. Its cut-off frequencies are set as 150, 250, 750, and 850 MHz.

Loss Tangent Estimation Method
Grimm et al. proposed a method of measuring attenuation using ground penetrating radar [23]. The radar range equation, given in Skolnik [24], is: where P r is the received power, P t is the transmitted power, G is the system gain, λ is the wavelength in the medium, ξ is the backscatter cross section, e −4αR is the attenuation in the medium, and α is the spatial attenuation coefficient. Later, α can also be expressed as η (α in dB) in unit dB/m, η = 20 log 10 (e α ) = 8.686α. In general, α can be expressed as the following formula [25]: where ω is the angular frequency, µ is the magnetic permeability, ε is the dielectric permittivity, and tan δ is the loss tangent. For a lunar sample, tan δ 1, and µ is assumed to be the free space magnetic permeability. Then, where ε is the real part of the relative permittivity. Formula (1) indicates that the attenuation loss can be divided into two parts-the scattering loss and the intrinsic (or absorption) loss. If the scattering loss can be effectively compensated for, then the absorption loss can be estimated.
For different reflected targets, the ratio between the received power and the transmitted power is proportional to the reciprocal power of the propagation distance: In this case, three different models of reflected targets are considered: n = 2 for a planar and smooth reflected target; n = 3 for the Fresnel zone case; and n = 4 for discrete scatters. After compensating for the scatter signals, the attenuation η can be estimated by least squares fitting of the two-way distance versus the ratio of the transmitted power to received power in unit dB/m. After obtaining the attenuation η, tan δ can be estimated by Formula (3).

Stochastic Media Model
In order to verify the feasibility of the loss tangent estimation method, a simulation model was utilized. Previous studies have shown that the internal structure of the lunar regolith is close to the stochastic media model [26]. Hence, the stochastic media model with an ellipsoidal autocorrelation function was used in this study to simulate the lunar regolith structure [27]. The method is described as follows: 1. A two-dimensional stochastic equivalent medium model is established by using ellipsoidal autocorrelation function: where r is the fuzzy factor of the stochastic media model, and a and b represent the autocorrelation length in the X and Y directions, respectively.
2. The f (x, y) is transformed into the wavenumber domain (F k x , k y ) by the twodimensional Fourier transform equation, and then the power spectrum function R k x , k y is calculated.
3. Generate the independent and uniform random numbers φ k x , k y in the range of [0, 2π).
4. By adding the random number φ k x , k y into Formula (7), a new energy spectral density function is obtained: 5. The inversed two-dimensional Fourier transform is performed to transform F k x , k y from the wavenumber domain to the spatial domain to obtain a new f (x, y).

Simulation Model
In order to model different scenarios of lunar regolith structure, three stochastic media models were used. Model One is shown in Figure 3a. In the models (e.g., Figure 3a,b), the material of the first layer is free space. Model Two and Model Three have the same layered structure, but with different abundances and distribution of rocks. Five rocks are placed in the upper part of Model Two, while in Model Three, they are in the lower part. The autocorrelation lengths in the X and Y directions are both 2. The fuzzy factor r is 0 in this model. The model is 10 m in length, and 11 m in depth, and the spatial steps in the X (length) and Y (depth) direction are both 0.005 m. The time window is 150 ns, the time step is 0.0118 ns. The moving steps of both transmitter and receiver antenna are 0.05 m. The average permittivity of background material is 3.5, which is consistent with the result of Lai et al. [13] and Wang et al. [15]. The relative loss tangent is set as 0.02, which indicates the average conductivity of the model is about 2 × 10 −3 (e.g., Figure 3b). The transmitter antenna and receiver antenna are 0.3 m higher than the zero depth. The range between the transmitter antenna and the receiver antenna is 0.32 m, the same as the settings of the Chang'e 4 LPR. Remote Sens. 2021, 13, x FOR PEER REVIEW 7 of 22

Synthetic Data Results
The finite-difference time-domain (FDTD) simulation method is utilized to verify the correctness of the loss tangent estimation method, which is mentioned in Section 2.2. gprMax software, an open source software package for numerical modeling of ground penetrating radar (GPR), is used for simulation [28]. It is foreseeable that the reflection signal of the buried rock and the multiple scattering signals among the rocks will affect the results of the loss tangent estimation, so three models are used in this experiment to evaluate the influence of rocks. The base stochastic media model is the same for these

Synthetic Data Results
The finite-difference time-domain (FDTD) simulation method is utilized to verify the correctness of the loss tangent estimation method, which is mentioned in Section 2.2. gprMax software, an open source software package for numerical modeling of ground penetrating radar (GPR), is used for simulation [28]. It is foreseeable that the reflection signal of the buried rock and the multiple scattering signals among the rocks will affect the results of the loss tangent estimation, so three models are used in this experiment to evaluate the influence of rocks. The base stochastic media model is the same for these three models, except for locations of the rocks. Figure 4 shows the comparisons between the simulation results of these three models.  The method mentioned in Section 2.2 is utilized to estimate the loss tangent of the simulation model. In order to reduce the influence of noises, the mean square of each radar trace is calculated before estimating the loss tangent. Figure 5 shows the estimated attenuation result. In Figure 5, the red, blue and green lines represent the synthetic data after R 2 , R 3 , and R 4 backscatter/spreading correction, respectively. The dashed purple line indicates the least linear fitting results. The slope of the dashed purple line is equal to the attenuation η. Then, Formula (3) is used to calculate loss tangent parameters. Table 1 shows the attenuation's estimated result. The average loss tangent of Model One is 0.0229 ± 0.0084, which is consistent with the simulation model (0.02). The average loss tangents of Model Two and Model Three are 0.0273 ± 0.0061 and 0.0156 ± 0.0134, respectively. Thus, the presence of rocks introduces some errors to the results, but within a tolerable range. The error of results of Model Two and Model Three is 36.5% and 22%, respectively. The distribution of rocks also affects the inferred loss tangent. In this example, the rocks at the bottom result in fewer errors of the estimated result than that of rocks in the upper part. Next, the algorithm will be used in the Chang'e-4 LPR data.

LPR Results
The loss tangent estimation method introduced in Section 2.2 is used to calculate the loss tangent of the coarse material. In order to mitigate the influence of the rocks, we only select the depth range of relatively uniform materials (the sampling time exceeds 150ns for the 6th to the 12th lunar days in Figure 6 have been selected. After averaging the energy of adjacent 437 tracks of LPR data, which is obtained after data preprocessing described in Section 2.1, the original root mean square data are obtained (black line in Figure 7). Then, the spreading and backscatter correction factor (R 2 , R 3 , R 4 ) are utilized to compensate for the spreading and backscatter loss. Then, the least squares fitting method is used to fit the data after exponential function compensation (dashed purple lines in Figure 7). The estimated results of signal attenuation of LPR data are shown in Figure 7. The corresponding loss tangent results are given in Table 2. Remote Sens. 2021, 13, x FOR PEER REVIEW 13 of 22 Figure 6. The first fifteen lunar days' LPR data image after processing. The color bar is the amplitude of the LPR signal after processing. Groups a-g indicate the areas used to calculate the loss tangent.     Table 1 show estimated results of the loss tangent of the simulation models. For the simulation model, the R 3 correction result, which indicates that reflections come from a Fresnel-zone sized area, is plausible. The loss tangent of Model One after R 3 correction is 0.0229, compared to the parameter~0.02 used in the stochastic media model, and the error is less than 15%. Taking into account the result after over-compensation (R 4 correction) and under-compensation (R 2 correction), the real value is within the range of estimation values (0.0229 ± 0.0084). Therefore, the simulation results show that the estimation method can effectively obtain the loss tangent when little rock is present.
Model Two includes five stones in the upper part of layer two. It is expected that the reflection signal from the rocks and the scattering signal among the rocks will affect the radar signals. The locations of rocks are known in the simulation model. The deepest rock is placed at the propagation distance of about 10 m, so we selected the echo signal with a propagation distance between 14 and 19 meters to calculate the energy attenuation curve. The average loss tangent of Model Two is 0.0273 ± 0.0061, which is higher than the actual setting. Judging from the results, rocks cause scattering of the EM wave and result in a slightly larger loss tangent than the simulation parameter, as the interference of rocks increases the signal attentions. Therefore, avoiding rocks in the calculation area could improve the accuracy of the loss tangent. However, if rocks are ubiquitous, the result can also be considered as the upper limit of the loss tangent.
Model Three has five stones distributed on the bottom. The shallowest rock is about 10 m in propagation distance, so we selected the echo signal with a propagation distance between 6 and 8.5 m to calculate the energy attenuation curve. The average loss tangent of Model Three is 0.0231 ± 0.0144, which is consistent with the real setting and better than the case in which rocks are above the study region (Model Two). Hence, the result of the loss tangent can be improved by selecting the area of calculation appropriately. It is worth noting that although the results of Model One and Model Two are similar, the confidences of these two models (0.0084 vs. 0.0144) are different due to the different horizontal length of the calculation area. Increasing the calculation area can reduce the confidence interval. In addition, by comparing Model One and Model Three, it can be concluded that a longer calculation interval in depth could reduce the error bar.
In actual situations, if the selected area contains rocks in the lower part, the tail of the energy attenuation curve may go up, resulting in the estimated loss tangent being smaller than the actual situation. For example, in Model Three, the calculation area is expanded to 9.5 m, so the rocks are included, then the average loss tangent is changed to 0.0156 ± 0.0133. In this case, it can be considered that the calculated loss tangent is the lower limit of the actual situation. Figure 6 shows the LPR data after processing with the steps proposed by Lai et al. [13]. The significant feature of the LPR image is that there is a continuous horizontal reflection around the depth of~12 m. In order to calculate the loss tangent of coarse materials, a study area in the depth range of~146 to~273 m on the Yutu-2 rover's motion path was selected in this study. LPR data were divided into a set of groups along the horizontal direction, with each group being 15 m long. The total number of groups is eight, which are named as group a, b, . . . , h, respectively ( Figure 6). The attenuation of these eight groups was calculated separately, and the corresponding results are shown in Figure 7 and Table 2. Then, we utilized Formula (3) to estimate the loss tangent. From Figure 7 and Table 2, it can be seen that each data group used different depth ranges to calculate the loss tangent to reduce the noises caused by rocks. Table 2 shows that the estimated loss tangent of groups a, b, c, d, g and h are similar, the average value of which is 0.0105. The estimated loss tangent of group e is larger than the average result, while group f's value is smaller.

Loss Tangent of the Coarse Material
The main reason for deviations in the estimation results is that there are several energy bumps (Figure 7e) at bottom (40-43m) of the group e, probably due to rock scattering, which causes the result of group e to be relatively small. The group e can be regarded as the situation that the tail of the energy attention curve is affected by the rock reflection signal, which is similar to the situation of Model three, and it can be considered as the lower limit of the loss tangent.
From Figure 7, the energy attenuation curve of group f changes relatively smoothly compared to other groups, and the error bar is smaller than those of other groups. However, the loss tangent of group f is larger than the average value, which may indicate that some rocks might be distributed above the selected area. Similar to the results of Model Two, scattering caused by rocks affects the estimation of loss tangent. In this situation, the upper limit of the loss of coarse materials can be obtained.
In groups a, b, c and d, the maximum value of the estimated loss tangent after R 3 correction is 0.0112 and the minimum value is 0.0101. The difference between these two values is about 10%. To improve the confidence of the results, the sampled data on a longer motion path are used to calculate the energy attenuation curve and then estimate the loss tangent. In this study, we use the energy attenuation curve of adjacent 1700 tracks of LPR data to re-estimate the loss tangent, and obtain Figure 8. The corresponding result of Figure 7 is calculated in Table 3. Table 3. The attenuation estimated result of adjacent 1700 tracks of LPR data. The corresponding result of Figure 7 is calculated in Table 3. From Figure 8, we can conclude that increasing the size of the calculation area can effectively improve the R-square value, and can fit the curve better to reduce the error bars. The average value of the loss tangent is 0.0104, which is consistent with the average value of the loss tangent of group a, b, c, d, g and h (0.0105).
Based on the previous analysis, it is concluded that the estimated average loss tangent of the coarse materials in the calculated area is 0.0104 ± 0.0027. If empirical results of the simulation model are used, the upper limit of the loss tangent is 0.0120, and the lower limit is 0.087.

The Geological History of the Chang'e-4 Landing Site
In radargrams, there are continuous horizontal reflectors inside the coarse-grained materials [13][14][15][16][17]. Zhang et al. (2020) argued that there are 4-5 layers of impact ejecta blanket coverage between 12 and 40 meters under the motion path of the Yutu-2 rover [16]. Xu et al. (2021) found four impact craters (the thickness of the local materials and ejecta deposits are more than 1 meter) that can have a greater impact on the landing site through the model [29]. In order to better analyze the subsurface structure of the landing site, the migration method and topographic correction are applied to the radargram in Figure 2. Figure 9a shows the radar image after processing. There are some horizontal reflectors around~210,~270 and~360 ns, which are marked by red lines in Figure 9b. These continuous interface reflections appear on both sides of the loss tangent estimated area (Figures 6 and 9). The interruption of the continuity is because of a bowl-shaped area marked by the black dotted line in Figure 9b.  and Zhou et al. (2021) believed that this area is a buried impact crater, and the continuous reflective interfaces were destroyed by the subsequent impact event [30,31]. The materials under the impact crater were mixed and are relatively uniform, so the loss tangent estimation model can be applied. The estimated loss tangent is the loss tangent of the mixed multi-layered coarse materials.
Considering that the central frequency of the Yutu-2 high-frequency radar is 500 MHz, the empirical formula based on the data points only representing measurements at 450 MHz is utilized [32]. tan δ = 10 (0.038×(%TiO 2 +%FeO)−2.746) Substitute the average loss obtained by the previous calculation (0.0104) into Formula (8), and then we can obtain the total FeO T and TiO 2 concentrations of the coarse materials at the CE-4 landing site as 20.08 wt.%. Additionally, with the upper and lower loss tangent bounds (0.0120 and 0.0087), the total FeO T + TiO 2 concentrations will be 21.7 and 18.0 wt.%, respectively. The FeO T + TiO 2 concentration of the surface fine materials around the Chang'e-4 landing site is lower than 15 wt.%, and the TiO 2 content of which is between 1 and 2 wt.% [9,[33][34][35][36]. This value is below even the lowest boundary estimates based on Equation (8) of the coarse materials. This may indicate that the surface fine material has much lower FeO T content than that of the underlying coarse substance, and may share a different source. Qiao et al. (2019) proposed that dozens of craters (about 330 m-1.1 km in diameter) have relatively high FeO T in the crater interiors and ejecta blankets [36]. The high FeO T + TiO 2 concentrations of the coarse materials may be caused by the iron-rich nature of the ejecta blankets.

Conclusions
In this study, we use signal attenuation to estimate the loss tangent value of the coarse material in lunar regolith beneath the motion path of the Chang'e-4 rover. Three stochastic

Conclusions
In this study, we use signal attenuation to estimate the loss tangent value of the coarse material in lunar regolith beneath the motion path of the Chang'e-4 rover. Three stochastic media models are utilized to verify the estimation method. The simulation results demonstrate that the estimation is close to the real setting when the propagation medium is uniform with little scattering, and the error is about 15%. If, however, scattering targets are present in the study area, the results will be affected. For example, the result becomes larger when rocks appear above the calculation area. In this case, the estimated result can be considered as the upper limit of the loss tangent. When the rocks appear below the calculation area, the estimated result is smaller than the actual value. In addition, increasing the calculation area could improve the confidence of the estimation results.
The average loss tangent value of the coarse materials in the Chang'e-4 landing site is about 0.0104; considering the influence of the rocks, the upper limit of the loss tangent is 0.0120, and the lower limit is 0.087. Based on the empirical model of Apollo samples, the total FeO T and TiO 2 content of coarse-grained materials was estimated to be 20.08 wt.%, higher than that of the upper fine-grained materials, which indicates different origins of these two parts.