# A Refined Four-Stream Radiative Transfer Model for Row-Planted Crops

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}, cloud physics and forest research works on horizontal radiation calculated the reflectance from the perspective of the unitless viewing probability of the photons in the radiative transfer. The differences in the physical mechanism in the modeling process imply that some modeling techniques for horizontal radiative transfer in cloud physics and forests cannot be used directly in the four-stream radiative transfer equations.

## 2. Methodology

#### 2.1. Four-Stream Radiative Transfer Equations for Continuous Crops

#### 2.2. Refined Four-Stream Radiative Transfer Equations for Row-Planted Crops Considering Horizontal Radiative Transfer Equations

#### 2.3. Sum of the Directional Reflectance Factor of Row-Planted Crops

#### 2.3.1. Directional Reflectance Factor of the Canopy Closure

#### Single Scattering of the Canopy Closure

#### Multiple Scattering of the Canopy Closure

#### 2.3.2. Directional Reflectance Factor of the Between Row Area

#### Single Scattering of the Between Row Area

#### Multiple Scattering of the Between Row Area

#### 2.4. Input Parameters of the MFS Model

- (1)
- Geometrical parameters: solar zenith angle (${\theta}_{s}$), solar azimuth angle (${\phi}_{s}$), viewing zenith angle (${\theta}_{o}$), viewing azimuth angle (${\phi}_{o}$), and row azimuth angle (${\phi}_{r}$);
- (2)
- Canopy parameters: height of the canopy ($h$), row width (${A}_{1}$), distance between rows (${A}_{2}$), leaf area index ($L$) and effective leaf area index (${L}_{E}$), average leaf inclined angle (${\theta}_{l}$), and canopy dimension parameter (${l}_{L}^{*}$);
- (3)
- Biochemical leaf parameters: chlorophyll content (${C}_{ab}$), carotenoid content (${C}_{ar}$), brown pigment content (${C}_{brown}$), equivalent water thickness (${C}_{w}$), leaf mass per unit leaf area (${C}_{m}$), and structure coefficient ($N$);
- (4)
- Canopy radiative parameters: the fraction of incoming diffuse radiation ($skyl$).

## 3. Data

#### 3.1. Computer-Simulated Data

#### 3.1.1. Generation of Computer Abstract Scenes

#### 3.1.2. Input and Output Settings for Computer-Simulated Validation

#### Input Setting

_{ab}= 40.29 ug·cm

^{−2}, C

_{ar}= 8.54 ug·cm

^{−2}, C

_{brown}= 0, N = 1.52, Cw = 0.014 cm

^{−2}, C

_{m}= 0.004 g·cm

^{−2}) were also obtained (see Section 3.2.1 for details). The soil reflectance was obtained by measurement, in which the red band with 670 nm was 0.242 and the near-infrared band with 850 nm was 0.314. According to data from the Yingke Oasis, the sun zenith angle was 25° and the sun azimuth angle was 130°. To keep the flux density in both the MFS model and RGM model consistent, we changed the measured flux density to the normalized flux density in the validation. In the MFS model, this took into account the diffuse incidence (Equation (33)) at the top of the canopy that was ignored by the previous row model [40], for which the fraction of incoming diffuse radiation ($skyl$) for both models was set to 0.1. Combined with Equation (C-22) in Supplementary Material C, ${n}_{\Delta}$, ${l}_{*\u25b3}$, and ${w}_{*\u25b3}$ in Table 1 were used to calculate the canopy dimension parameter (${l}_{L}^{*}$) in the MFS model. Finally, for the row structure and in the MFS model, we used the measured values of the Yingke Oasis (i.e., the values of the abstract scene in Table 1).

#### Output Setting

#### 3.2. In Situ Data

#### 3.2.1. Measurement Experiment

#### Directional Reflectance Factor (DRF)

#### Canopy Structural and Leaf Biochemical Parameters

^{−2}[69]. The carotenoid content (${C}_{ar}$), brown pigment content (${C}_{brown}$), and structure coefficient ($N$) were determined using the Lopex1993 database [70].

#### 3.2.2. Input and Output Settings for In Situ Validation

#### Input Setting

^{−2}, Car = 11.41 ug·cm

^{−2}, Cbrown = 0, N = 1.52, Cw = 0.014 cm

^{−2}, and Cm = 0.004 g·cm

^{−2}. To keep the leaf inclination distribution function (LADF) in the comparison consistent, we replaced the LADF of the ellipse distribution in the SAIL model and the MFS model with the LADF based on the graphic method [52], as in the DRM model. Considering that the DRM model does not consider the clumping index issue, we divided the MFS model into two types: one version where the clumping index was considered (MFS

_{1}), and one version where the clumping index was not considered (MFS

_{2}). In MFS

_{1}, the canopy dimension parameter (${l}_{L}^{*}$) used the equation proposed in the DRM model, (i.e., Equation (C-9) in Supplementary Material C). In MFS

_{2}, the canopy dimension parameter (${l}_{L}^{*}$) used Equation (C-18) in Supplementary Material C. Finally, Table 2 showed other input parameters in the three models.

#### Output Setting

## 4. Results

#### 4.1. Validation of MFS Model Using Computer-Simulated Data

#### 4.1.1. Results of the Qualitative Analysis of DRFs in the Red Band

^{−3}and 10

^{−2}, and the difference between the two model simulations is not more than 16.69% (Table 3). Hence, the difference is very small. From Figure 5a,b to Figure 5g,h, the high-value area generated by the soil background shrinks into hotspots as the crop grows. In the cross-sectional view of the DRF fields, i.e., Figure 4, the difference in the contours of the fields in Figure 3 is very small. For the four viewing modes in the red band, DRFs simulated by the MFS model and DRFs simulated by the RGM model are more consistent, except for the simulation at a viewing zenith angle greater than 80° along row plane (AR) mode (Figure 4c).

#### 4.1.2. Results of the Qualitative Analysis of DRFs in the NIR Band

#### 4.1.3. Quantitative Analysis Results

#### 4.2. Validation of the MFS Model Using In Situ Data

_{1}and MFS

_{2}are highly consistent with the measured data. The DRFs simulated by MFS

_{2}are better than the DRFs simulated by MFS

_{1}, and their correlation coefficients (R) are higher than 0.985 with RMSEs of less than 0.023 (Table 4). The DRF simulated by the DRM model is seriously overestimated when the zenith angle is greater than 40°, especially in the NIR band (Figure 11a–f). Comparatively, the DRF simulated by MFS

_{1}and MFS

_{2}at the zenith angle of 60° is slightly better than that at the zenith angle of 50°. For the continuous crops, the performance of MFS and SAIL models is comparable, and DRFs simulated by the two models have the same R and RMSE values (Figure 11g).

## 5. Discussion

#### 5.1. Systematic Deviation of the DRFs in the NIR Band at Zenith Angles Larger than 40°

_{1}and MFS

_{2}) and the in situ measurement was maintained to the greatest extent and solved the systematic deviation of the DRM model in the NIR band when the zenith angle was greater than 40° (Figure 11a–f). Compared with MFS

_{1}and MFS

_{2}, the clumping index and the canopy dimension parameter (${l}_{L}^{*}$) did not obviously improve the overall accuracy (Figure 11a–f and Figure 12a–f). Therefore, when the zenith is larger than 40°, the large systematic deviation of the DRF in the NIR band simulated by the DRM model is not caused by these two factors. A previous study on the DRM model pointed out that the systematic deviation of the DRFs in the NIR band at zenith angles larger than 40° may be caused by the coefficient calculation method SAIL model used by the four-stream radiative transfer equations [40]. In this study, the calculation method for the coefficients ($k$, $K$, ${s}^{\prime}$, $s$, $\sigma $, $w$, $v$, ${v}^{\prime}$, and $a$ in Equations (1)–(4) and Equations (6)–(9), i.e., SAIL model [19]) was used to solve the modified four-stream radiative transfer equations, but the systematic deviation of the DRFs in the NIR band did not occur when the zenith angle was larger than 40° (Figure 11a–f and Figure 12a–f). These results also showed that when the zenith angle is greater than 40°, the large systematic deviation of the DRFs in the NIR simulated by the DRM model comes from other aspects. We further analyzed the core equations in the DRM model, but the horizontal radiative transfer was not considered. For the equation of the canopy closure for multiple scattering in the DRM model (i.e., Equation (13) in [40]), this was derived from an approximate radiative transfer equation [52] (i.e., Equation (4) in this study; the derivation process is detailed in Appendix A in [40]) instead of all of the four-stream radiative transfer equations (i.e., Equations (1)–(4)). These mathematical derivations imply that the original four-stream radiative transfer equations are subjected to a second mathematical approximation. Therefore, the calculation accuracy of the DRFs is reduced in the multiple scattering of the canopy closure. Based on the above analysis, the systematic deviation of the DRFs in the DRM model (Figure 11a–f) may come from the multiple scattering equation for the canopy closure and the lack of horizontal radiative transfer. The MFS model did not make any assumptions or simplifications in the original four-stream radiative transfer equations, directly calculated the exact solutions for the differential equations, solved the MFS radiative transfer equations (Section 2.2 and Supplementary material D), and then considered the DRF for between rows (Supplementary materials E) to build the MFS model, in which it was applied in the multi-angle simulation of DRF fields in row-planted crops.

#### 5.2. Differences between MFS Model and Computer Simulation

#### 5.3. The Relationship between the MFS Model and SAIL Model

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of radiative transfer in continuous crops and row-planted crops: (

**a**) cross-section of row-planted crops; (

**b**) scene of row-planted crops with nadir view; (

**c**) cross-section of the continuous vegetation; (

**d**) the three-dimensional map of row-planted crops. Here, A

_{1}is the row width, A

_{2}is the distance between rows, and h is the height of the canopy.

**Figure 2.**The abstract scenes of row-planted crops and canopy closures: (

**a**) leaf area index (LAI) = 0.58 for row-planted crops; (

**b**) LAI = 1.75 for row-planted crops; (

**c**) LAI = 3.51 for row-planted crops; and (

**d**) LAI = 5.11 for continuous crops.

**Figure 3.**Polar plots of the DRF field in the red band simulated by the 3D radiosity graphics model (RGM) model (left panels) and MFS model (right panels): (

**a**,

**b**) stage_rv1, (

**c**,

**d**) stage_rv2, (

**e**,

**f**) stage_rv3, (

**g**,

**h**) stage_cv. Here, points in the polar plots refer to the interpolation points, N represents the north direction, the blue line represents the principal plane (PP) mode, and the shaded annulus lies at a zenith angle of 40°.

**Figure 4.**The distribution of directional reflectance factors (DRFs) in the red band simulated by the RGM model and the MFS model in four viewing modes: (

**a**) principal plane (PP) mode, (

**b**) orthogonal plane (OP) mode, (

**c**) along row plane (AR) mode, and (

**d**) orthogonal (OR) mode. Here VZA = viewing zenith angle.

**Figure 5.**Polar plots of the DRF field in the NIR band simulated by the RGM model (left panels) and MFS model (right panels): (

**a**,

**b**) stage_rv1, (

**c**,

**d**) stage_rv2, (

**e**,

**f**) stage_rv3, (

**g**,

**h**) and stage_cv. Here, points in the polar plots of DRFs refer to the interpolation points, N represents the north direction, the blue line represents the PP mode, and the shaded annulus lies at a zenith angle of 40°.

**Figure 6.**The distribution of directional reflectance factors (DRFs) in the NIR band simulated by the RGM model and MFS model in four viewing modes: (

**a**) principal plane (PP) mode, (

**b**) orthogonal plane (OP) mode, (

**c**) along row plane (AR) mode, and (

**d**) orthogonal (OR) mode. Here VZA = viewing zenith angle.

**Figure 7.**Polar plots of the DRF field in the single scattering of the NIR band simulated by the RGM model (left panels) and MFS model (right panels): stage_rv1 (

**a**,

**b**), stage_rv2 (

**c**,

**d**), stage_rv3 (

**e**,

**f**), and stage_cv (

**g**,

**h**). Here, points in the polar plots of DRFs refers to the interpolation points, N represents the north direction, the blue line represents the PP mode, and the shaded annulus lies at a zenith angle of 40°.

**Figure 8.**The distribution of directional reflectance factors (DRFs) in the single scattering of the NIR band simulated by the RGM model and MFS model in four viewing modes: (

**a**) principal plane (PP) mode, (

**b**) orthogonal plane (OP) mode, (

**c**) along row plane (AR) mode, and (

**d**) orthogonal (OR) mode. Here VZA = viewing zenith angle.

**Figure 9.**Polar plots of the DRF field in the multiple scattering of the NIR band by the RGM model (left panels) and MFS model (right panels): stage_rv1 (

**a**,

**b**), stage_rv2 (

**c**,

**d**), stage_rv3 (

**e**,

**f**), and stage_cv (

**g**,

**h**). Here, points in the polar plots of DRFs refers to the interpolation points, N represents the north direction, the blue line represents the PP mode, and the shaded annulus lies at a zenith angle of 40°. The white shaded annulus in Figure 9. (

**g**,

**h**) is used to distinguish it from the DRF field.

**Figure 10.**The distribution of directional reflectance factors (DRFs) in the multiple scattering of the NIR band simulated by the RGM model and MFS model in four viewing modes: (

**a**) principal plane (PP) mode, (

**b**) orthogonal plane (OP) mode, (

**c**) along row plane (AR) mode, and (

**d**) orthogonal (OR) mode. Here VZA = viewing zenith angle.

**Figure 11.**Comparison of the simulated and measured DRFs at wavelengths ranging from 400 to 2500 nm with a viewing zenith angle of −60° in the (

**a**) principal plane (PP), (

**c**) orthogonal plane (OP) and (

**e**) orthogonal row (OR) on lateral “wall” A (left panels); and with a viewing zenith angle of 60° in the (

**b**) PP, (

**d**) OP, and (

**f**) OR on lateral “wall” B (right panels). (

**g**) Vertical observation of continuous crops. The in situ measurements in the spectrum impacted by water vapor are removed. Here LW_A = lateral wall A, LW_B = lateral wall B, DRM = a spectral directional reflectance model, SAIL = light scattering by leaf layers with application to canopy reflectance modeling, MFS

_{1}= modified four-stream radiative transfer model with considering clumping index, MFS

_{2}= modified four-stream radiative transfer model without considering clumping index.

**Figure 12.**Comparison of the simulated and measured DRFs at wavelengths ranging from 400 to 2500 nm, with a viewing zenith angle of −50° in the (

**a**) principal plane (PP), (

**c**) orthogonal plane (OP), and (

**e**) orthogonal row (OR) on lateral “wall” A (left panels); with a viewing zenith angle of 50° in the (

**b**) PP, (

**d**) OP,and (

**f**) OR on lateral “wall” B (right panels). The in situ measurements in the spectrum impacted by water vapor are removed. Here LW_A = lateral wall A, LW_B = lateral wall B. MFS

_{1}= modified four-stream radiative transfer model with considering clumping index, MFS

_{2}= modified four-stream radiative transfer model without considering clumping index.

Scene1 | Figure 2 | A_{1}(m) | A_{2}(m) | h (m) | L (m•m ^{−1}) | n_{△}(-) | w*_{△}(m) | l*_{△}(m) | θ_{l}(°) | φ_{r}(°) |
---|---|---|---|---|---|---|---|---|---|---|

Stage_rv1 | (a) | 0.4 | 0.6 | 0.48 | 0.58 | 10,419 | 0.02 | 0.02 | 44 | 0 |

Stage_rv2 | (b) | 0.5 | 0.5 | 0.88 | 1.75 | 25,840 | 0.02 | 0.02 | 44 | 0 |

Stage_rv3 | (c) | 0.75 | 0.25 | 1.28 | 3.51 | 48,325 | 0.02 | 0.02 | 44 | 0 |

Stage_cv | (d) | 1 | 0 | 1.58 | 5.11 | 40,262 | 0.02 | 0.02 | 44 | 0 |

^{2}to 00003m

^{2}in the scenes; the shape of the soil area is square, with an area of 0.0025 m

^{2}; and the leaf inclined angle is randomly distributed in the zenith and azimuth directions. The symbols in Table 1 are: A

_{1}= row width, A

_{2}= distance of between-rows, h = the height of the canopy, L = leaf area index, n

_{△}= the number of triangular leaves, w*

_{△}= one of the short sides of the horizontal triangle leaf, l*

_{△}= the other short sides of the horizontal triangle leaf, θ

_{l}is average leaf inclined angle. φ

_{r}is row azimuth angle.

**Table 2.**Values of input parameters in the MFS, DRM, and SAIL models for simulation of the DRFs of corn.

Vegetation Type | Model | L (m•m ^{−1}) | L_{E}(m•m ^{−1}) | LIDFa | LIDFb | A_{1}(m) | A_{2}(m) | h (m) | skyl (-) | θ_{o}(°) |
---|---|---|---|---|---|---|---|---|---|---|

RC | MFS_{1} | 2.51 | 2.51 | 0.38 | −0.02 | 0.46 | 0.46 | 0.98 | 0.1 | 60 |

MFS_{2} | 2.51 | 1.73 | 0.38 | −0.02 | 0.46 | 0.46 | 0.98 | 0.1 | 60 | |

DRM | 2.51 | - | 0.38 | −0.02 | 0.46 | 0.46 | 0.98 | 0.1 | 60 | |

CC | MFS | 3.99 | 3.99 | 0.38 | −0.02 | 1 | 0 | 1.54 | 0.1 | 0 |

SAIL | 3.99 | - | 0.38 | −0.02 | - | - | - | 0.1 | 0 |

_{E}= effect leaf area index, A

_{1}= row width, A

_{2}= distance of between-rows, h = the height of the canopy, skyl = fraction of incoming diffuse radiation, and θ

_{o}= viewing zenith angle, RC = row-planted crops and CC = continuous crops. LIDFa and LIDFb are the controlling parameters from the graphical method, which controls leaf inclination angle distribution function. LIDFa and LIDFb are obtained from the average leaf inclinated angle and the variance of the leaf inclinated angle in the measurement, and their equation are found in [71].

**Table 3.**Statistics for comparison of the DRFs simulated by the MFS model versus DRFs derived from the RGM model for four scenes.

Scenes. | Statistics | DRF_red | DRF_NIR | DRF_NIR_1 | DRF_NIR_m |
---|---|---|---|---|---|

Stage_rv1 | R | 0.9842 | 0.9900 | 0.9881 | 0.9846 |

RMSE | 0.0018 | 0.0082 | 0.0038 | 0.0063 | |

Average difference | −16.69% | −7.49% | 9.41% | −22.64% | |

Stage_rv2 | R | 0.9788 | 0.9807 | 0.9667 | 0.9864 |

RMSE | 0.0017 | 0.0141 | 0.0087 | 0.0079 | |

Average difference | −13.76% | −11.69% | −9.99% | 3.35% | |

Stage_rv3 | R | 0.9341 | 0.9352 | 0.9714 | 0.9610 |

RMSE | 0.0008 | 0.0186 | 0.0065 | 0.0112 | |

Average difference | 4.80% | −0.48% | −1.02% | −0.10% | |

Stage_cv | R | 0.9662 | 0.8207 | 0.9895 | 0.9084 |

RMSE | 0.0005 | 0.0083 | 0.0035 | 0.008 | |

Average difference | −5.25% | 1.53% | 1.16% | −3.02% |

**Table 4.**Summary of statistics for the comparison of the simulated DRFs versus in situ measurements.

RC(PP) | RC(OP) | RC(OR) | CV | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

lateral wall A | lateral wall B | lateral wall A | lateral wall B | lateral wall A | lateral wall B | ||||||||||

R | RMSE | R | RMSE | R | RMSE | R | RMSE | R | RMSE | R | RMSE | R | RMSE | ||

DRM | 0.997 | 0.012 | 0.997 | 0.011 | 0.990 | 0.026 | 0.998 | 0.010 | 0.996 | 0.017 | 0.999 | 0.008 | SAIL | 0.998 | 0.012 |

MFS_{1}(60) | 0.994 | 0.013 | 0.997 | 0.008 | 0.989 | 0.020 | 0.998 | 0.009 | 0.996 | 0.017 | 0.998 | 0.006 | MFS | 0.998 | 0.012 |

MFS_{2}(60) | 0.996 | 0.010 | 0.998 | 0.007 | 0.990 | 0.018 | 0.999 | 0.006 | 0.996 | 0.011 | 0.998 | 0.006 | |||

MFS_{1}(50) | 0.991 | 0.016 | 0.994 | 0.012 | 0.996 | 0.012 | 0.995 | 0.013 | 0.996 | 0.011 | 0.996 | 0.011 | |||

MFS_{2}(50) | 0.991 | 0.018 | 0.990 | 0.016 | 0.990 | 0.019 | 0.986 | 0.022 | 0.991 | 0.018 | 0.987 | 0.021 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ma, X.; Wang, T.; Lu, L.
A Refined Four-Stream Radiative Transfer Model for Row-Planted Crops. *Remote Sens.* **2020**, *12*, 1290.
https://doi.org/10.3390/rs12081290

**AMA Style**

Ma X, Wang T, Lu L.
A Refined Four-Stream Radiative Transfer Model for Row-Planted Crops. *Remote Sensing*. 2020; 12(8):1290.
https://doi.org/10.3390/rs12081290

**Chicago/Turabian Style**

Ma, Xu, Tiejun Wang, and Lei Lu.
2020. "A Refined Four-Stream Radiative Transfer Model for Row-Planted Crops" *Remote Sensing* 12, no. 8: 1290.
https://doi.org/10.3390/rs12081290