# A Compressive Sensing-Based Approach to Reconstructing Regolith Structure from Lunar Penetrating Radar Data at the Chang’E-3 Landing Site

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## Abstract

**:**

## 1. Introduction

_{2}content based on the 500-MHz LPR data. Feng et al. [15] derived the regolith’s permittivity distribution laterally and vertically by processing the 500-MHz LPR data. In the analysis and evaluation of LPR data, the response signal caused by discrete reflectors beneath the lunar surface provides very useful information [4,6,7,14,15,16]. For example, the hyperbolic signatures produced by these targets are small with respect to radar wavelength, whose axes and vertices are functions of their position and relative dielectric characteristics [16,17]. In the lunar regolith, the most common subsurface materials are fine-grained regolith and basalt debris [1,4], and the layered reflection is not obvious [7]. Moreover, there is extensive clutter and noise in LPR data images [15], such as the coupling between antennas and the lunar surface, electromagnetic interference, etc. [4,14], and these can partially or totally hide or distort the response signal of discrete reflectors in the regolith [18]. Although many corrections have been applied to LPR data, such as background removal [14], amplitude compensation [14,15], band-pass filtering [13], and bi-dimensional empirical mode decomposition filtering [7], only a few reflections can be clearly identified [7,14,15]. Therefore, improving the capability to identify response signals of the discrete reflectors from LPR data is necessary.

## 2. Methodology and Preliminary Numerical Tests

#### 2.1. Signal model

#### 2.2. CS Algorithm Using Random Fourier Series

#### 2.3. Numerical Analysis

## 3. Algorithm Verification Using 2D the Random Regolith Model

- (1)
- (2)
- A Gaussian random field was used to model the regolith. A great deal of natural science data display marked Gaussian characteristics [33,34,35]. We built the lunar regolith relative permittivity model with clipped Gaussian random field theory [36]. A relative permittivity that is set from 2.5–3.5 satisfies the Gaussian random distribution.
- (3)
- The formation of lunar regolith indicates the existence of detritus [6]. In the LPR data, most reflections are formed by basalt debris [7]. Three strongly-reflecting debris materials (A: a square with a side length of 0.1 m; B: a circle with a radius of 0.1 m; C: a square with a side length of 0.2 m; Figure 6a) were incorporated into the 2D regolith model with the same relative permittivity as basalt (${\epsilon}_{b}=6$).

## 4. Processing and Analyzing LPR Data

#### 4.1. Preprocessing

#### 4.2. Parameter Estimation Using the CS Algorithm

#### 4.3. Results

- The top layer (depth < 1 m) cannot extract reflection parameters at all. This is the fine-grained regolith part of the lunar regolith. In [6], this part was interpreted as a reworked zone. Even though the fine-grained regolith was composed of numerous layers, the layer thickness was typically on the order of several centimeters [37], which is much smaller than the LPR range resolution [4]. Therefore, it is difficult to extract the reflected signal from this layer.
- The middle layer (buried from 1–7 m) had the most signal reflectors beneath the lunar surface. It is the paleoregolith of mare basalt with much debris. After the processing step in the CS-based approach, the reflective response signal, which is difficult to extract from the original LPR data, became clear (Figure 13). The size of the reflection curve is often proportional to the volume of the debris [7]. On the basis of the continuity and the absolute value of the estimated amplitude, some reflection response signals are marked by a red curve in Figure 14. Obviously, this is a good improvement for evaluating the LPR data.
- The last part is the basalt base. Since there is no obvious reflective interface between the regolith and mare basalt, one can distinguish this region by the distribution of a strong reflection response signal. Obviously, from Figure 13, there were no reflection signals ($\mid {\left\{{a}_{j}\right\}}_{j=1}^{L}\mid >1.5$). The estimated reflection response at [I] (Figure 13) is a false reflection, considering the unusually persistent periodicity of this part of the LPR data.

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LPR | lunar penetrating radar |

CS | compressive sensing |

GPR | ground-penetrating radar |

LRS | Lunar Radar Sounder |

ALSE | Apollo Lunar Sounder Experiment |

CE-3 | Chang’E-3 |

UWB | ultra-wideband |

NAOC | National Astronomical Observatories, Chinese Academy of Sciences |

AGC | automatic gain control |

TV | total-variation |

SDF | semidefinite program |

FDTD | finite-difference time-domain |

AWGN | additive white Gaussian noise |

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**Figure 1.**(

**a**) The 1D lunar regolith model (Rx: receive antenna; Tx: transmitting antenna) and (

**b**) the LPR response signal of the 1D lunar regolith model with the direct waves removed ($x\left(t\right)$).

**Figure 2.**The results of the first numerical experiment. (

**a**) The continuous time Fourier transformation of $g\left(t\right)$. (

**b**) Estimated parameters by the CS algorithm in Bandwidth 1 (B1). (

**c**) Estimated parameters by the CS algorithm in B2. (

**d**) Estimated parameters by the CS algorithm in B3.

**Figure 3.**Estimated value of a2 from B1 obtained by executing the CS algorithm 60 times for numerical experiments. The average value (AVG) is 0.2553, and the standard deviation ($\sigma $) is 0.0025.

**Figure 4.**LPR data with periodic noise and parameters estimated by the CS algorithm. (

**a**) The frequencies of the sine interference are 200 and 800 MHz; (

**b**) the frequencies of the sine interference are 450 and 550 MHz).

**Figure 5.**LPR data with Gaussian noise and the parameters estimated by the CS algorithm. (

**a**) Noise level: −30 dB; (

**b**) noise level: −20 dB; (

**c**) noise level: −10 dB.

**Figure 6.**(

**a**) The 2D random regolith model. (

**b**) Permittivity trend at 5 m (black dotted line in (

**a**)).

**Figure 7.**(

**a**) A snapshot of a radar wave in 28.8 ns at 5 m. (

**b**) Simulation results of the 2D regolith model.

**Figure 8.**(

**a**) A single trace in 5 m (red line in Figure 7b) and the parameters estimated by the CS algorithm. (

**b**) Image of the estimated absolute amplitudes of the simulated LPR profile.

**Figure 10.**(

**a**) The original LPR data extracted from the raw data, (

**b**) the preprocessed LPR data, (

**c**) a signal trace of the LPR data at 7.5 m (black line), and (

**d**) a signal trace of the preprocessed LPR data at 7.5 m.

**Figure 14.**Interpretation of the LPR data from C to D (Figure 9).

Parameter | Value | Direction |
---|---|---|

Height of antenna | 0.3 m | d2 in Figure 1a |

Offset of Tx and Rx | 0.32 m | d1 in Figure 1a |

Transmitted Waveform | Ricker | $g\left(t\right)$ |

Center frequency | 500 MHz | |

Absorbing boundary | C-PML | |

Thickness of absorbing boundary | 0.1 m | 10 PML layers |

Discrete grid | 0.01 × 0.01 m | the size of the grid cells |

Time step | 0.03125 ns | |

Time window | 70 ns |

Amplitude (a) | Value | Time Delay ($\mathit{\tau}$) | Value (ns) |
---|---|---|---|

a1 | 0.9421 | $\tau $1 | 3.7500 |

a2 | 0.2546 | $\tau $2 | 26.5625 |

a3 | −0.0092 | $\tau $3 | 49.6875 |

Type | Laptop |

Operating system | Microsoft Windows 10 |

CPU | Intel(R) Core(TM) i7-6820HQ |

RAM | 16 GB |

Frequency Band (MHz) | Amplitudes (a), Time Delays ($\mathit{\tau}$) | Value | err_rel (%) | Standard Deviation ($\mathit{\sigma}$) | Time Cost (s) |
---|---|---|---|---|---|

B1[400–600] | a1 | 0.9421 | 0 | 0.0006 | 160 |

a2 | 0.2553 | 0.3 | 0.0025 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

B2[800–1000] | a1 | 0.9420 | 0.1 | 0.004 | 159 |

a2 | 0.2570 | 1.0 | 0.0045 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

B3[1200–1400] | a1 | 0.8325 | 11.63 | 0.2 | 160 |

a2 | 0 | ∼ | ∼ | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 6.0 ns | 60 | 3 | ||

$\tau $2 | ∼ | ∼ | ∼ | ||

$\tau $3 | ∼ | ∼ | ∼ |

Frequency Band (MHz) | Amplitudes (a), Time Delays ($\mathit{\tau}$) | Value | err_rel (%) | Standard Deviation ($\mathit{\sigma}$) | Time Cost (s) |
---|---|---|---|---|---|

B4[425–575] | a1 | 0.9421 | 0 | 0.0002 | 113 |

a2 | 0.2550 | 0.3 | 0.0015 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

B5[375–625] | a1 | 0.9420 | 0.1 | 0.004 | 265 |

a2 | 0.2565 | 0.9 | 0.0036 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

B6[350–650] | a1 | 0.9420 | 0.1 | 0.005 | 402 |

a2 | 0.2568 | 0.9 | 0.0062 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ |

Number of Fourier Series | Amplitudes (a), Time Delays ($\mathit{\tau}$) | Value | err_rel (%) | Standard Deviation ($\mathit{\sigma}$) | Time Cost (s) |
---|---|---|---|---|---|

10 | a1 | 0.9323 | 1 | 0.1049 | 152 |

a2 | 0.2050 | 20.1 | 0.0473 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0.01 | ||

$\tau $2 | 26.5625 ns | 0 | 0.01 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

60 | a1 | 0.9421 | 0 | 0.0001 | 192 |

a2 | 0.2550 | 0.1 | 0.0002 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ |

Frequency of Sine Wave (MHz) | Amplitudes (a), Time Delays ($\mathit{\tau}$) | Value | err_rel (%) | Standard Deviation ($\mathit{\sigma}$) | Time Cost (s) |
---|---|---|---|---|---|

200,800 | a1 | 0.9749 | 3 | 0.001 | 162 |

a2 | 0.2414 | 5 | 0.001 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

450,550 | a1 | 0.98531 | 5 | 0.05 | 169 |

a2 | 0.2358 | 7 | 0.03 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ |

AWGN Noise Level (dB) | Amplitudes (a), Time Delays ($\mathit{\tau}$) | Value | err_rel (%) | Standard Deviation ($\mathit{\sigma}$) | Time Cost (s) |
---|---|---|---|---|---|

30 | a1 | 0.8802 | 7 | 0.002 | 169 |

a2 | 0.2450 | 0.3 | 0.003 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

20 | a1 | 1.125 | 20 | 0.05 | 170 |

a2 | 0.2315 | 9 | 0.06 | ||

a3 | 0 | ∼ | ∼ | ||

$\tau $1 | 3.2500 ns | 12 | 0 | ||

$\tau $2 | 26.5625 ns | 0 | 0 | ||

$\tau $3 | ∼ | ∼ | ∼ | ||

10 | a1 | 1.022 | 0.8 | 0.2 | 165 |

a2 | ∼ | ∼ | ∼ | ||

a3 | ∼ | ∼ | ∼ | ||

$\tau $1 | 3.7500 ns | 0 | ∼ | ||

$\tau $2 | ∼ | ∼ | ∼ | ||

$\tau $3 | ∼ | ∼ | ∼ |

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## Share and Cite

**MDPI and ACS Style**

Wang, K.; Zeng, Z.; Zhang, L.; Xia, S.; Li, J.
A Compressive Sensing-Based Approach to Reconstructing Regolith Structure from Lunar Penetrating Radar Data at the Chang’E-3 Landing Site. *Remote Sens.* **2018**, *10*, 1925.
https://doi.org/10.3390/rs10121925

**AMA Style**

Wang K, Zeng Z, Zhang L, Xia S, Li J.
A Compressive Sensing-Based Approach to Reconstructing Regolith Structure from Lunar Penetrating Radar Data at the Chang’E-3 Landing Site. *Remote Sensing*. 2018; 10(12):1925.
https://doi.org/10.3390/rs10121925

**Chicago/Turabian Style**

Wang, Kun, Zhaofa Zeng, Ling Zhang, Shugao Xia, and Jing Li.
2018. "A Compressive Sensing-Based Approach to Reconstructing Regolith Structure from Lunar Penetrating Radar Data at the Chang’E-3 Landing Site" *Remote Sensing* 10, no. 12: 1925.
https://doi.org/10.3390/rs10121925