# Effects of Plant Crown Shape on Microwave Backscattering Coefficients of Vegetation Canopy

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

#### 2.1. The Modeling of Crown Shape

- The vegetation canopy comprises a mixture of leaves and branches (or stalks), and there is no obvious stratification inside the vegetation canopy.
- Without loss of generality, we assumed that the volume density of all vegetation components in the canopy is the same. The volume density is denoted as ${\rho}_{v}$, and its unit is cm
^{−3}. - For a given vegetation canopy, all vegetation components are uniformly distributed.
- Among the same kind of vegetation component in the canopy, the geometrical features and the physicochemical properties, such as the geometrical shape, the size and the moisture of the component, remain the same.
- In order to simplify the representation of the vegetation canopy, all leaves are horizontally arranged, and all branches or stalks are vertically aligned inside the crown envelope surface.

#### 2.2. The Modified Tor Vergata Model

^{−3}) is the volume density of the j-th kind of scatterer.

^{−5}cm

^{−3}in the simulation. The vegetation canopy scene was assumed to be irradiated by a homogeneous plane wave. The scattering phase matrix of disc leaves, needle leaves and stalks (or branches) were calculated using physical optics approximation [32], Rayleigh–Gans approximation [33] and infinite length approximation [34], respectively. We simulated the variation in the canopy backscattering coefficients at different canopy heights for VV, HH, VH and HV polarizations of L (1.2 GHz), C (5.3 GHz) and X (9.6 GHz) bands. The simulation results are shown in Section 3.1.

#### 2.3. Vegetation Canopy Modeling and Simulation with FEKO

- Modeling 3D geometrical vegetation canopies, as shown in Figure 6.
- Setting parameters, which include the media properties (such as structure and permittivity), frequencies, wave sources, polarizations and solution requests. The computational complex dielectric constants of canopies A and B are shown in Equations (16) and (17). The frequency was set at 5.3 GHz, the incident angle was 43°, the azimuthal angles ranged from 0° to 360° with an interval of 45°, and the wave source was set as a plane wave at VV and VH polarizations.
- Creating mesh, which is related to the solution methods, dielectric properties, frequencies, geometry curvature, etc. FEKO provides three mesh options to automatically determine the appropriate mesh size for the model: coarse, standard or fine mesh can be selected based on the required accuracy and the available computational resources. In addition, a custom and local mesh size can also be set without applying the automatic meshing algorithm. In this study, the coarse mesh was applied to the vegetation canopies for the sake of computation efficiency.
- Choosing the solution method, which depends mainly on the model mesh size and computational efficiency. Because the maximum mesh of the 3D vegetation canopy models is more than two million, the MLFMM was chosen in this study, and the box size in wavelength was set at 0.21 to improve the convergence.
- Running the solver and analyzing results. The computation was conducted on a server with four Intel(R) Xeon(R) Gold 6252 CPUs. Every CPU has 24 cores in the server; the CPU clock speed is 2.10 GHz, and the total memory is 2 TB. For the maximum mesh size, it takes about six hours to compute a single result (for a single frequency, single incident angle and single azimuthal angle).

## 3. Results and Discussion

#### 3.1. MTVM Simulation Results

^{−5}cm

^{−3}), and the crown heights of the four canopies remain the same (H = 80–300 cm).

- Cylinder: a = 0, b = 0, c = 1/3;
- Cone: a = 1, b = −2, c = 1;
- Inverted Cone: a = 1, b = 0, c = 0;
- Ellipsoid: a = −2, b = 2, c = 0.

#### 3.2. FEKO Simulation Results

#### 3.3. Comparative Analysis

## 4. Conclusions

- Using the Modified Tor Vergata Model, the backscattering coefficients of the cylinder, cone, inverted cone and ellipsoid canopies with different crown heights (H = 80–300 cm) were simulated for canopies A and B at VV, HH, VH and HV polarizations in L (1.2 GHz), C (5.3 GHz) and X (9.6 GHz) bands. However, the backscattering coefficients of the four canopies with different crown shapes and heights were simulated for canopies A and B at only VV and VH polarizations in the C (5.3 GHz) band with FEKO because of the long computational time and huge memory cost. The FEKO simulation establishes a good foundation to explore and develop applications of computational electromagnetic methods in microwave scattering domain of vegetation.
- The simulation results show that, for canopy A or B, different crown shapes possess significant differences in backscattering coefficients, of which the mean relative differences due to variations in crown shape are as high as 127%. Therefore, it can be demonstrated that the crown shape has a non-negligible influence on microwave backscattering coefficients of vegetation canopies. In turn, this also suggests that investigating the crown shape may have the potential to improve the simulation accuracy of microwave scattering models of vegetation, especially in canopies where volume scattering is the predominant mechanism.
- Regardless of whether canopy A’s or B’s parameters are set as model inputs, the backscattering coefficients of vegetation canopies with different crown shapes almost all gradually increase as the canopy height increases from 80 cm to 300 cm when simulated by either MTVM or FEKO. Taking MTVM for example, the exception is the cone canopy at VV, HV and VH polarizations in the L band (1.2 GHz) and at VV polarization in the X band (9.6 GHz), for which the MTVM simulation results when using canopy A’s parameters as inputs slightly decrease with the increase in crown height.
- For each specified model or method, the backscattering coefficients and their relative differences for canopy A are larger than those for canopy B for a given crown shape in the same band and polarization, which indicates that vegetation canopies with different components possess different backscattering characteristics. It also suggests that the crown shape effect acts synergistically with the effects of the vegetation component parameters, which mainly include the geometrical and physical parameters of the vegetation components.
- In preliminary experiments with MTVM and FEKO, a large discrepancy can be observed between the results of the three crown shapes studied and the reference cylinder. Overall, the ranking of the backscattering coefficients of the four vegetation canopies is ${\sigma}_{\mathit{cone}}^{0}>{\sigma}_{\mathit{cylinder}}^{0}>{\sigma}_{\mathit{ellipsoid}}^{0}>{\sigma}_{\mathit{inv}-\mathit{cone}}^{0}$, which is approximately the same order of ranking as that of the volume fraction of the lower half of the vegetation canopy, for which the fractions are cone (75%) > cylinder (50%) = ellipsoid (50%) > inverted cone (25%). This correspondence can possibly be attributed to the attenuation effects of the upper canopy components; in other words, a lower volume fraction produces less attenuation, increasing the backscattering coefficients of the lower part of the canopy. However, a quantitative explanation of this result requires further analysis.
- Specifically, at VV and VH polarizations of the C band, the simulation results of FEKO are higher than those of MTVM. The reasons for the large difference may lie in different unreasonable assumptions and simplifications or the parameter settings of the two simulations. The actual reasons need to be further explored by analytical simulation and experimental measurement, which will be the focus of our future work.

## 5. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Crown shapes in nature [22]. Vegetation crown shapes are written in bold; the corresponding typical vegetation types are written in normal font and are: (

**a**–

**e**).

**Figure 2.**Tridimensional geometry representation of the ideal crown shape in the cylindrical coordinate system $\left(\rho ,\varphi ,h\right)$, where an ellipse represents a leaf, and a cylinder represents a branch; $H$ is crown height, and $\rho (\varphi ,h)=F(\varphi ,h)$ is the crown envelope equation.

**Figure 3.**Geometric representations of the three specific crown shapes. The crown shapes, from left to right, are (

**a**) cone, (

**b**) inverted cone and (

**c**) ellipsoid.

**Figure 4.**The process of using the Matrix Doubling Algorithm to calculate the multiple scattering of unit incident power between two close thin layers.

**Figure 5.**Two different structures of vegetation canopy components: (

**a**) the component structure of canopy A; (

**b**) the component structure of canopy B.

**Figure 7.**The MTVM simulation results of the backscattering coefficients for different crown shapes in L band (1.2 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 1 (canopy A).

**Figure 8.**The MTVM simulation results of the backscattering coefficients for different crown shapes in C band (5.3 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 1 (canopy A).

**Figure 9.**The MTVM simulation results of the backscattering coefficients for different crown shapes in X band (9.6 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 1 (canopy A).

**Figure 10.**The MTVM simulation results of the backscattering coefficients for different crown shapes in L band (1.2 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 2 (canopy B).

**Figure 11.**The MTVM simulation results of the backscattering coefficients for different crown shapes in C band (5.3 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 2 (canopy B).

**Figure 12.**The MTVM simulation results of the backscattering coefficients for different crown shapes in X band (9.6 GHz) at (

**a**) VV, (

**b**) HH, (

**c**) VH and (

**d**) HV polarizations. The model input parameters are from Table 2 (canopy B).

**Figure 13.**The feasible region of parabola factors a and b. The blue line represents the feasible region, the cross symbols represent the integer feasible solutions, and the bold dots with coordinates denote the solutions of a and b, which correspond to the four crown shapes.

**Figure 14.**The squares of the parabola equations with different integer feasible solutions, which correspond to the cross symbols in Figure 13. The yellow, green, red and blue represent the square of the cylinder, cone, inverted cone and ellipsoid equations, respectively, and correspond to the bold dots in Figure 13.

**Figure 15.**Backscattering coefficient simulation results of different crown shapes with discrete a and b in the feasible region; the bold dots correspond to the four crown shapes.

**Figure 16.**Rendering graphs of five transition crown shapes from the inverted cone to the cone. The parabola factors are (

**a**) a = 1, b = 0; (

**b**) a = 1, b = −0.5; (

**c**) a = 1, b = −1; (

**d**) a = 1, b = −1.5; and (

**e**) a = 1, b = −2.

**Figure 17.**The simulated and fitted backscattering coefficients of vegetation canopies when using canopy A’s parameters as inputs at different canopy heights.

**Figure 18.**The averaged FEKO simulation results of the backscattering coefficients for different crown shapes at VV and VH polarizations of C band (5.3 GHz) when using canopy A’s parameters as inputs. (

**a**) VV polarization; (

**b**) VH polarization.

**Figure 19.**The averaged FEKO simulation results of the backscattering coefficients for different crown shapes at VV and VH polarizations of C band (5.3 GHz) when using canopy B’s parameters as inputs. (

**a**) VV polarization; (

**b**) VH polarization.

Symbol | Value | Unit | Description |
---|---|---|---|

$f$ | 5.3 | GHz | Radar Frequency |

$\theta $ | 43 | Deg | Incidence Angle |

$H$ | 80–300 | cm | Canopy Height |

${r}_{\mathit{leaf}}$ | 3 | cm | Leaf Disc Radius |

${d}_{\mathit{leaf}}$ | 0.02 | cm | Leaf Disc Thickness |

${w}_{\mathit{leaf}}$ | 0.85 | 100% | Leaf Volumetric Moisture |

${r}_{\mathit{stalk}}$ | 1.25 | cm | Stalk Cylinder Radius |

${l}_{\mathit{stalk}}$ | 10 | cm | Stalk Cylinder Length |

${w}_{\mathit{stalk}}$ | 0.85 | 100% | Stalk Volumetric Moisture |

Symbol | Value | Unit | Description |
---|---|---|---|

$f$ | 5.3 | GHz | Radar Frequency |

$\theta $ | 43 | Deg | Incidence Angle |

$H$ | 80–300 | cm | Canopy Height |

${r}_{\mathit{leaf}}$ | 0.3 | cm | Needle Leaf Radius |

${l}_{\mathit{leaf}}$ | 4 | cm | Needle Leaf Thickness |

${w}_{\mathit{leaf}}$ | 0.6 | 100% | Leaf Volumetric Moisture |

${r}_{\mathit{branch}}$ ^{1} | 0.2 | cm | Branch Cylinder Radius |

${l}_{\mathit{branch}}$ | 30 | cm | Branch Cylinder Length |

${w}_{\mathit{branch}}$ | 0.6 | 100% | Branch Volumetric Moisture |

^{1}For information about the grading of branches, refer to the Michigan Microwave Canopy Scattering Model (MIMICS) [3].

**Table 3.**The number of layers in the four canopies with different crown shapes and the number of vegetation components in each layer.

Item | Symbol | Canopy A | Canopy B | |
---|---|---|---|---|

Layer Number | ${N}_{l}$ | $\lfloor \frac{H}{{l}_{\mathit{Stalk}}}+0.5\rfloor $ | $\lfloor \frac{H}{{l}_{\mathit{Branch}}}\rfloor $ | |

Vegetation Component Number in i-th Layer | Cylinder | ${N}_{i}$ | $\lceil \frac{{\rho}_{v}{V}_{h}}{{N}_{l}}\rceil $ | $\lceil \frac{{\rho}_{v}{V}_{h}}{{N}_{l}}\rceil $ |

Cone | $\lceil {\rho}_{v}{V}_{i}\rceil $ | $\lceil {\rho}_{v}{V}_{i}\rceil $ | ||

Inverted Cone | ||||

Ellipsoid |

**Table 4.**$\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$ of the simulation results between the three crown shapes studied and the reference cylinder crown shape using canopy A’s parameters as inputs.

Band | Polarization | Cone | Inverted Cone | Ellipsoid |
---|---|---|---|---|

L (1.2 GHz) | VV | 1.32 $\downarrow $ ^{1} | 13.64 | 5.70 |

12.13 | 126.68 $\uparrow $ | 53.05 | ||

HH | 3.16 | 13.76 | 4.89 | |

16.62 | 72.95 | 25.97 | ||

VH | 3.43 | 14.55 $\uparrow $ | 5.27 | |

11.68 | 50.09 | 18.20 | ||

HV | 3.36 | 14.45 | 5.23 | |

11.17 $\downarrow $ | 48.45 | 17.56 | ||

C (5.3 GHz) | VV | 3.32 | 11.80 | 3.83 |

17.14 | 61.25 | 19.93 | ||

HH | 3.59 | 11.78 | 3.60 | |

19.60 | 64.87 $\uparrow $ | 19.89 | ||

VH | 4.53 | 11.79 | 3.18 | |

16.01 | 42.02 | 11.39 $\downarrow $ | ||

HV | 4.80 | 11.82 $\uparrow $ | 3.17 $\downarrow $ | |

19.27 | 47.88 | 12.92 | ||

X (9.6 GHz) | VV | 4.40 | 11.76 $\uparrow $ | 3.19 |

15.65 | 42.75 | 11.61 | ||

HH | 4.52 | 11.70 | 3.15 | |

19.28 | 50.26 $\uparrow $ | 13.61 | ||

VH | 4.82 | 11.61 | 2.97 $\downarrow $ | |

16.02 | 38.93 | 9.99 $\downarrow $ | ||

HV | 4.81 | 11.66 | 2.97 $\downarrow $ | |

16.05 | 39.61 | 10.11 |

^{1}The notation $\uparrow $ denotes the maximum values of $\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$, and $\downarrow $ denotes the minimum values.

**Table 5.**$\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$ of the simulation results between the three crown shapes studied and the reference cylinder crown shape when using canopy B’s parameters as inputs.

Band | Polarization | Cone | Inverted Cone | Ellipsoid |
---|---|---|---|---|

L (1.2 GHz) | VV | 3.76 | 11.60 $\uparrow $ | 3.45 |

9.54 | 29.41 $\uparrow $ | 8.74 | ||

HH | 3.90 | 11.56 | 3.38 | |

7.85 | 23.29 | 6.80 | ||

VH | 5.65 | 11.10 | 2.38 | |

7.16 | 14.07 | 3.02 | ||

HV | 5.67 | 11.09 | 2.37 $\downarrow $ | |

6.90 | 13.50 | 2.88 $\downarrow $ | ||

C (5.3 GHz) | VV | 2.63 $\downarrow $ | 13.74 $\uparrow $ | 5.01 |

6.31 $\downarrow $ | 33.09 | 12.08 | ||

HH | 4.07 | 13.62 | 4.43 | |

10.35 | 34.61 $\uparrow $ | 11.27 | ||

VH | 5.28 | 13.48 | 3.76 | |

10.70 | 27.39 | 7.65 | ||

HV | 5.30 | 13.47 | 3.75 | |

9.83 | 25.02 | 6.97 | ||

X (9.6 GHz) | VV | 2.59 $\downarrow $ | 11.70 $\uparrow $ | 3.98 |

6.94 | 31.40 | 10.69 | ||

HH | 3.51 | 11.65 | 3.58 | |

9.89 | 32.90 $\uparrow $ | 10.11 | ||

VH | 4.89 | 11.51 | 2.86 | |

11.10 | 26.22 | 6.53 | ||

HV | 4.95 | 11.48 | 2.82 | |

10.36 | 24.11 | 5.94 $\downarrow $ |

**Table 6.**$\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$ of FEKO simulation results between the three crown shapes studied and the reference cylinder crown shape at VV and VH polarizations of C band (5.3 GHz).

Canopy | Polarization | Cone | Inverted Cone | Ellipsoid |
---|---|---|---|---|

A | VV | 2.29 | 4.31 | 1.84 $\downarrow $ |

16.17 | 31.65 $\uparrow $ | 13.50 | ||

VH | 2.34 | 5.76 $\uparrow $ | 2.52 | |

10.48 $\downarrow $ | 27.22 | 11.97 | ||

B | VV | 2.70 | 4.17 $\uparrow $ | 2.37 |

16.10 | 26.56 $\uparrow $ | 14.93 | ||

VH | 2.54 | 3.79 | 2.08 $\downarrow $ | |

12.18 | 19.71 | 10.77 $\downarrow $ |

**Table 7.**$\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$ of the MTVM simulation results for the same crown shapes between canopies A and B.

Polarization | Item | Cylinder | Cone | Inverted Cone | Ellipsoid |
---|---|---|---|---|---|

VV | $\overline{{\mu}_{a}}$ | 22.18 $\downarrow $ | 22.88 | 24.12 $\uparrow $ | 23.36 |

$\overline{{\mu}_{rA}}$ | 115.14 | 143.28 $\uparrow $ | 77.53 $\downarrow $ | 101.01 | |

$\overline{{\mu}_{rB}}$ | 53.45 | 58.83 $\uparrow $ | 43.65 $\downarrow $ | 50.22 | |

VH | $\overline{{\mu}_{a}}$ | 21.13 | 20.38 $\downarrow $ | 22.82 $\uparrow $ | 21.71 |

$\overline{{\mu}_{rA}}$ | 75.06 | 86.12 $\uparrow $ | 57.03 $\downarrow $ | 69.18 | |

$\overline{{\mu}_{rB}}$ | 42.85 | 46.25 $\uparrow $ | 36.31 $\downarrow $ | 40.88 |

**Table 8.**$\overline{{\mu}_{a}}$ and $\overline{{\mu}_{r}}$ of the FEKO simulation results for the same crown shapes between canopies A and B.

Polarization | Item | Cylinder | Cone | Inverted Cone | Ellipsoid |
---|---|---|---|---|---|

VV | $\overline{{\mu}_{a}}$ | 2.71 | 2.55 | 2.47 $\downarrow $ | 3.09 $\uparrow $ |

$\overline{{\mu}_{rA}}$ | 20.22 | 22.83 $\uparrow $ | 13.71 $\downarrow $ | 21.50 | |

$\overline{{\mu}_{rB}}$ | 15.90 | 17.48 $\uparrow $ | 11.72 $\downarrow $ | 16.55 | |

VH | $\overline{{\mu}_{a}}$ | 2.38 $\downarrow $ | 3.54 | 4.59 $\uparrow $ | 3.25 |

$\overline{{\mu}_{rA}}$ | 10.21 $\downarrow $ | 15.91 | 15.98 $\uparrow $ | 13.39 | |

$\overline{{\mu}_{rB}}$ | 12.59 $\downarrow $ | 20.15 | 20.66 $\uparrow $ | 16.21 |

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**MDPI and ACS Style**

Liu, X.; Shao, Y.; Liu, L.; Li, K.; Wang, J.; Li, S.; Wang, J.; Wu, X.
Effects of Plant Crown Shape on Microwave Backscattering Coefficients of Vegetation Canopy. *Sensors* **2021**, *21*, 7748.
https://doi.org/10.3390/s21227748

**AMA Style**

Liu X, Shao Y, Liu L, Li K, Wang J, Li S, Wang J, Wu X.
Effects of Plant Crown Shape on Microwave Backscattering Coefficients of Vegetation Canopy. *Sensors*. 2021; 21(22):7748.
https://doi.org/10.3390/s21227748

**Chicago/Turabian Style**

Liu, Xiangchen, Yun Shao, Long Liu, Kun Li, Jingyuan Wang, Shuo Li, Jinning Wang, and Xuexiao Wu.
2021. "Effects of Plant Crown Shape on Microwave Backscattering Coefficients of Vegetation Canopy" *Sensors* 21, no. 22: 7748.
https://doi.org/10.3390/s21227748