# On the Very High-Resolution Radar Image Statistics of the Exponentially Correlated Rough Surface: Experimental and Numerical Studies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental and Numerical Methods

#### 2.1. The Rough Surface Samples

**λ**/20 in the Ka band and sufficient for the imaging study. The main advantage of this technique is that it permits the usage of lossy material, so as to generate samples simulating the half space rough surface scattering scenario, such as the moist soil ground. In this work, the measured permittivity is 6.22-j2.86 at 32 GHz and that value was used in the FDTD computations.

#### 2.2. Experimental Configuration and Procedure

_{i}of 30°, 40°, 50° and even 60°. The target is placed on a low-scattering supporting cylinder made of the foam material, which is set on a motor driven rotation platform (for φ rotation as shown in Figure 3). Multiple image acquisitions should be achieved for the speckle statistics. More specifically, in each case of θ

_{i}, the imaging measurements for the rough surface sample are performed 20 times and after each time φ is changed with an equal angular interval, as shown in Figure 3. Therefore, 20 images are obtained for a specific θ

_{i}, then speckle statistical analysis can be conducted based on those results.

#### 2.3. Numerical Simulation Configuration and Procedure

_{i}are considered, namely, 30°, 40° and 50°. Furthermore, cases of different surface RMS heights h are also considered to support the extended discussions. The discretization cell size in the FDTD simulation is set to 1/30

**λ**at 32 GHz, which is sufficient in modeling the high frequency roughness of the exponentially correlated rough surface.

#### 2.4. Verification Results with AIEM

_{i}, clearly, good agreement can be observed in the Figure 6.

**λ**× 40

**λ**. It should be noted that, in common scattering simulations for rough surfaces [4], the aperture size of 32

**λ**× 32

**λ**is electrically large enough. Because of the huge computational burden, with one realization requiring approximately 1 h, only 50 realizations are performed to obtain the averaged scattering coefficients distributions over the upper space. Again, the AIEM results are utilized as reference and two sets of results are compared in Figure 7. Although the FDTD results failed in presenting a smooth contour due to the insufficiently large realization number, the results by numerical and analytical methods clearly agree well with each other in the distribution patterns.

#### 2.5. The Differences in Experimental and Numerical Studies

^{2}; while in the numerical study, the scattering coefficients without unit of m

^{2}, are computed without imaging process. To make a direct correspondence for the speckle results, the PDF of the measured image amplitudes and the simulated scattering amplitudes are normalized by the averaged intensity (RCS σ

_{RCS}in experimental images and scattering coefficient σ

_{0}in numerical simulations), respectively. The averaged σ

_{RCS}from the experimental images can also be normalized into σ

_{0}by using:

_{gr}and ρ

_{a}are the ground range resolution and azimuth resolution, respectively.

#### 2.6. Speckle Description

^{2}). The K-distribution, is given by

## 3. Analysis on the Experimental Imaging and Numerical Results

#### 3.1. Imaging at Different Resolutions and Incident Angles

_{i}. The scattering hot-spots at a VHR scale are observed to be fused into those at the lower resolution scales. In addition, clearly, as θ

_{i}increases, the RCS of those spots decrease. The presented images are obtained at one of the 20 φ positions and along with the results at other φ positions, they are used to produce the speckle results.

_{i}of 30°, the fitted α is the smallest and the PDF curve is with the tail in semi-log plot most away from the Rayleigh curve. This fact is also observed in Figure 10, the results in which are with a resolution cell size of 60 mm, or in correlation length, 1.25l. Meanwhile, as the resolution cell size enlarges from 30 mm to 60 mm, the fitted α of K-distribution at each θ

_{i}gets larger and the corresponding PDF is closer to the Rayleigh reference. That agrees with the common sense. On the other hand, it should be noted that the PDFs in case of resolution cell size 60 mm (Figure 10) is not fitting the K-dis as well as those in the case of resolution cell size 30 mm (Figure 9). That is due to the limited size of the sample (250 mm in diameter), as it becomes insufficient in providing with rich enough scatter returns with increasing resolution cell size.

_{RCS}, the averaged scattering coefficients σ

_{0}, the fitted α of the K-distribution and the computed γ are listed in the Table 4 for referencing purpose. Further, the computed γ in case of different θ

_{i}are plotted via resolution cell sizes in Figure 11. The coarse but not perfect trends can be well observed, that the γ curves are approaching to the γ

_{R}= 0.5227 with the rising of the resolution cell size. The imperfection in those trends is also due to the limited sample size. Also in Figure 11, the results of computed γ results without antenna pattern compensation conducted in the imaging process are also presented, showing the necessity of such a treatment in the chamber imaging experiments for speckle properties.

#### 3.2. Results of Numerical Simulations

_{i}= 30°, 40° and 50°. It can be clearly observed that, after averaged over 1600 realizations, the computed bi-static scattering coefficients converges to a smooth distribution.

_{i}= 30°, 40° and 50°, are compared in Figure 13, with a 3 dB footprint size of 30 mm in diameter. It is very clear that, from θ

_{i}= 30° to θ

_{i}= 50°, the larger θ

_{i}leads to larger fitted а and the amplitude PDF is closer to the Rayleigh distribution. That trend has also been observed in PDF results from measured images (Figure 9 and Figure 10). For reference, the averaged backscattering σ

_{0}, fitted α for K-distribution and computed γ are concluded in Table 5. As can be observed from the results in Table 5 and Table 6, both the experimental imaging and numerical simulation statistics show the same trend, that a larger θ

_{i}(from 30° to 50°) leads to the γ closer to the fully developed γ

_{R}. That is, as the θ

_{i}get larger in the moderate region, the observed VHR speckle of the exponentially correlated sample approaches toward the fully developed speckle.

## 4. Discussion

#### 4.1. Equivalent Scatterer Numbers Prediction

_{z}= k*cos(θ

_{i}), A

_{e}is the area of a resolution cell (not image pixel and n is 1 for the exponential correlation, 2 for the Gaussian correlation function, with t = 1 [11].

_{s}) and a relationship can be concluded as α = N*(α

_{s}). It is also important that in the rough surface scattering, the α

_{s}namely the K-distribution parameters for the “independent scatters” may vary according to surface roughness parameters and incident angle θ

_{i}. It is possible that in the radar image speckle from rough surfaces, if there is only one equivalent scatter in a resolution cell but the scatter is with Gaussian statistics (α

_{s}is very large), then the overall image speckle properties acts as fully developed. On the other hand, if there is a sufficiently large N so that N*(α

_{s}) is sufficiently large, then the overall image speckle properties also act nearly as fully developed.

_{s}keep unchanged, therefore the speckle properties approaches to the Rayleigh model.

#### 4.2. Similarity to Sea Surface Scattering-Scattering Scaling Effects

_{i}gets larger from 30° to 50° (moderate region), the speckle are approaching towards the fully developed Rayleigh description. The answer to this phenomenon, however, can also be found from the knowledge in scattering mechanisms from sea surface [22,23,24]. In Reference [24], the scattering mechanisms were discussed in the aspects of roughness scales for the two-scale or more precisely multi-scale sea rough surface. Specifically, the wavelength filtering effect is states that when the incident angle becomes larger, the dominating factor shifts from long-scale roughness towards short-scale roughness [22,23,24]. And for the vertical polarization, this effect is more notable than that of horizontal polarization [24]. If the effect of long-scale roughness is weakened enough due to the enlarging of θ

_{i}in the moderate region, then the multiplicative process is weakened because that the scattering from short-scale is taking the dominance of scattering mechanism, as well as the speckle properties. And the scattering from short scale roughness most possibly follows a Rayleigh PDF. It is interesting that, from the results of simulated sea backscattering in Reference [21], one can find that the speckle PDF results is also approaching toward Rayleigh when θ

_{i}get larger in the moderate region.

_{i}region, can be clearly explained: the multiplicative effect of scattering process is weaker when the θ

_{i}get larger from small to moderate, as the effects of carrier long-scale roughness gets weaker. In this case, because the scattering from short scale roughness gets more dominating as the θ

_{i}get larger and itself acts as Gaussian, the overall speckle distribution approaches towards the Rayleigh model.

#### 4.3. Further Discussions on the Two Factors

**λ**), 4 mm (0.43

**λ**), 8 mm (0.85

**λ**), 12 mm (1.28

**λ**@32 GHz) at θ

_{i}= 30°. For each cases of h, still 1600 realizations are computed for the speckle analysis. In Figure 14, the recorded Electric fields at 32 GHz in the incident plane cut of the computation domain in one specific realization were presented, as an intuitive exhibition for the difference of scattering process in case of different h. It seems that, when h = 4 mm, occasional specular reflection may contribute to the backscattering of θ

_{i}= 30°. Actually as shown in the scattering coefficient results of Figure 15, the backscattering at h = 4 mm is larger than those at h = 2 mm, h = 8 mm and h = 12 mm. In Figure 15, it is also interesting to observe that, as the h changes from 2 mm to 12 mm, the dominate bi-static scattering region gradually moves from the forward region (h = 2 mm) to the backward region (h = 12 mm). Also, the averaged computed backscattering coefficients are listed in Table 4 and a good agreement with the AIEM results can be observed.

_{i}= 30°. Keep in mind that the overall α is proportional to both the N and the statistical property of each independent scatters (α

_{s}), in the theoretic framework of equivalent number of scatterers. The rapid change of the estimated α

_{s}shown in the last line of Table 7 as the RMS h varies from 2 mm to 4 mm, is hard to be anticipated when using the equivalent number of scatters theory, if one does not consider other understandings on the scattering mechanisms. On the other hand, the long scale undulate gets more severe as the RMS height gets larger, the two scale multiplicative scattering process should be strengthened. It is also hard to understand that the counted α keeps getting larger as the RMS h varies from 4 mm to 12 mm (Table 7), if one only considers the explanation of the two scale multiplicative scattering process. Clearly, none of the two theories that produces monotonous trend prediction, can be independently used to explain the data results with such a non-monotonous trend.

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Surface roughness spectrum of the considered rough surface samples at 32 GHz (k

_{0}= 2π/

**λ**). (Exponentially correlated rough surface, correlation length l of 48 mm).

**Figure 2.**Geometry Configuration of the 3D model and the fabricated sample of the exponentially correlated rough surface. (

**a**): CAD model; (

**b**): Fabricated sample.

**Figure 3.**Configuration of the indoor imaging measurement (sketch and photo). The 3 small spheres were utilized to compensate the antenna pattern effect.

**Figure 4.**Image and range profiles of 3 small metal balls, showing the non-uniform illumination due to the antenna pattern effect. (

**a**): Image profile; (

**b**): Range profile (Two cut lines were obtained by performing 180° rotation of φ in the measurements).

**Figure 5.**Configuration of FDTD simulations for rough surface scattering (The presented profile is one of the realization conducted in the numerical simulations).

**Figure 6.**Comparison between measured scattering coefficients (σ

_{0}) of the exponentially correlated rough surface sample and the results by AIEM prediction, at 32 GHz. (The image extracted σ

_{0}are from the image results of 60 mm resolution cell size and are also listed in Table 4).

**Figure 7.**Comparisons between FDTD simulated upper-space bi-static scattering coefficients (σ

_{0}) and AIEM results, VV, 32 GHz, exponentially correlated rough surface, relative permittivity: 6.22-2.86j; (

**a**) and (

**b**): correlation length l = 48 mm, RMS height h = 4 mm; (

**c**) and (

**d**): l = 24 mm, h = 2 mm.

**Figure 8.**Image example of the exponentially correlated rough surface sample at different resolutions (30 mm, 60 mm and 90 mm) and different θ

_{i}(30°, 40°, 50° and 60°), VV.

**Figure 9.**The normalized amplitude PDF curves of measured images at different θ

_{i}, at the resolution cell size of 30 mm, VV, compared with Rayleigh distribution and K-distribution. (Sample parameters are shown in Table 1).

**Figure 10.**The normalized amplitude PDF curves of measured images at different θ

_{i}, at the resolution cell size of 60 mm, VV, compared with Rayleigh distribution and K-distribution. (Sample parameters are shown in Table 1).

**Figure 11.**The computed γ value versus resolution cell size in case of different θ

_{i}from the measured images, VV. “No Ant Correction” denotes for the γ values from images without the antenna pattern correction procedure. (Sample parameters are shown in Table 1).

**Figure 14.**Recorded total field (E-Field, maximum normalized, real part) in the incident plane cut from the exponentially correlated rough surface simulation, θ

_{i}= 30°. (

**a**) h = 2 mm, (

**b**) h = 4 mm, (

**c**) h = 8 mm.

**Figure 16.**PDF curves of FDTD-simulated backscattering amplitude at θ

_{i}= 30°, considering exponentially correlated surface with different h, (

**a**): h = 2 mm; (

**b**): h = 4 mm; (

**c**): h = 8 mm; (

**d**): h = 12 mm. VV, compared with referencing Rayleigh distribution and K-distribution, 1600 realizations. Sample and computation parameters are in Table 1 and Table 3(set 2).

Size | h | l | |
---|---|---|---|

Measured Sample | 250 mm in diameter | 4 mm | 48 mm |

(in λ@32 GHz) | 26.7 in diameter | 0.427 | 5.12 |

Numerical Samples | 125 mm × 125 mm | 2 mm, 4 mm, 8 mm, 12 mm | 48 mm |

(in λ@32 GHz) | 13.3 × 13.3 | 0.214, 0.427, 0.854, 1.281 | 5.12 |

**Table 2.**Imaging parameters for different resolution cell size (R = 1.4 m, ground range resolution, Kaiser windows (β = 2.5) applied in both domains).

Ground Resolution | Center Frequency | Aperture Length | Bandwidth (GHz) | |||
---|---|---|---|---|---|---|

θ_{i}: 30° | θ_{i}: 40° | θ_{i}: 50° | θ_{i}: 60° | |||

30 × 30 mm^{2} | 32.0 GHz | 240 mm | 11.00 | 8.25 | 6.60 | 6.15 |

45 × 45 mm^{2} | 160 mm | 7.35 | 5.50 | 4.40 | 4.10 | |

60 × 60 mm^{2} | 120 mm | 5.50 | 4.13 | 3.30 | 3.07 | |

75 × 75 mm^{2} | 96 mm | 4.40 | 3.30 | 2.65 | 2.45 | |

90 × 90 mm^{2} | 80 mm | 3.70 | 2.75 | 2.20 | 2.05 |

Set 1 | 3 dB Footprint Size | ds * (mm) | Domain Size (Yee Cells ^{+}) | Realization Number | θ_{i} (h = 4 mm) |

30 mm in diameter | 0.3123 | 400 × 400 × 180 | 1600 | 30°, 40°, 50° | |

Set 2 | 3 dB Footprint Size | ds * (mm) | Z Direction Size (Yee Cells) | Realization Number | h (mm) θ_{i} = 30° |

30 mm in diameter | 0.3123 | 160, 180, 220, 280 | 1600 | 2, 4, 8, 12 |

^{+}Yee cell is the mesh cell in the FDTD.

**Table 4.**Comparisons of the average intensity values and amplitude speckle values from the measured images at the resolution cell size of 30 mm and 60 mm. (Sample parameters shown in Table 1).

Resolution | θ_{i} (deg) | 30 | 40 | 50 | 60 |
---|---|---|---|---|---|

AIEM σ_{0} (dB) | −6.8 | −8.6 | −10.4 | −11.3 | |

30 mm × 30 mm | Average σ_{RCS} (dBsm) | −37.5 | −39.3 | −41.4 | −43.9 |

Average σ_{0} (dB) | −6.4 | −7.7 | −9.0 | −10.5 | |

α for K-dis | 4.5 | 5.2 | 6.3 | 6.0 | |

Computed γ | 0.598 | 0.576 | 0.571 | 0.564 | |

60 mm × 60 mm | Average σ_{RCS} (dBsm) | −32.0 | −33.5 | −35.1 | −38.8 |

Average σ_{0} (dB) | −6.9 | −7.9 | −8.7 | −11.4 | |

α for K-dis | 8.2 | 17.4 | 31.2 | 13.0 | |

Computed γ | 0.562 | 0.532 | 0.534 | 0.555 |

θ_{i} (deg) | 30 | 40 | 50 |
---|---|---|---|

AIEM σ_{0} (dB) | −6.8 | −8.6 | −10.4 |

Average σ_{0} (dB) | −7.3 | −9.0 | −10.2 |

α for K-dis | 4.6 | 7.4 | 33.4 |

Computed γ | 0.560 | 0.547 | 0.531 |

Method/θ_{i} (deg) | 30 | 40 | 50 |
---|---|---|---|

Imaging Measurement | 0.598 | 0.576 | 0.571 |

Numerical Simulation | 0.560 | 0.547 | 0.531 |

Fully Developed Ref | 0.5227 | 0.5227 | 0.5227 |

RMS Height h | 2 mm | 4 mm | 8 mm | 12 mm |
---|---|---|---|---|

AIEM σ_{0} (dB) | −10.4 | −6.8 | −11.8 | −18.2 |

Numerical σ_{0} (dB) | −10.0 | −7.3 | −12.0 | −17.7 |

Amplitude γ | 0.525 | 0.56 | 0.535 | 0.528 |

α for K-dis | 246.1 | 4.6 | 31.5 | 70.7 |

ENS Prediction N *^{1} | 2.4 | 43.4 | 719.9 | 3668 |

ENS Predicted α_{e} *^{2} | 0.24 | 4.34 | 72.0 | 366.8 |

α_{s} = α/N (sub-scatters) | 102.5 | 0.106 | 0.04 | 0.019 |

^{1}Equivalent number of scatterers is predicted using Equation (8). *

^{2}α

_{e}= N*α

_{s}and assuming that α

_{s}= 0.1.

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**MDPI and ACS Style**

Jin, M.; Chen, K.-S.; Xie, D. On the Very High-Resolution Radar Image Statistics of the Exponentially Correlated Rough Surface: Experimental and Numerical Studies. *Remote Sens.* **2018**, *10*, 1369.
https://doi.org/10.3390/rs10091369

**AMA Style**

Jin M, Chen K-S, Xie D. On the Very High-Resolution Radar Image Statistics of the Exponentially Correlated Rough Surface: Experimental and Numerical Studies. *Remote Sensing*. 2018; 10(9):1369.
https://doi.org/10.3390/rs10091369

**Chicago/Turabian Style**

Jin, Ming, Kun-Shan Chen, and Dengfeng Xie. 2018. "On the Very High-Resolution Radar Image Statistics of the Exponentially Correlated Rough Surface: Experimental and Numerical Studies" *Remote Sensing* 10, no. 9: 1369.
https://doi.org/10.3390/rs10091369