# Evaluation of Linear Kernel-Driven BRDF Models over Snow-Free Rugged Terrain

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. RTLSR Kernel-Driven Model

#### 2.2. TCKD Model

#### 2.3. KDST-TCKD Model

## 3. Materials and Methods

#### 3.1. Simulated Multi-Angle Reflectance of Rough Terrain with 3-D LESS

#### 3.2. Multi-Angle Reflectance Data from the Terrain Sandbox

#### 3.3. MODIS Satellite Observations

#### 3.4. Evaluation Methods

## 4. Result

#### 4.1. Evaluation of Kernel Shapes

_{iso}, K

_{geo}, and K

_{vol}for the three models under the three topographies of Figure 1. First, Figure 6a–c shows that the isotropic kernel value of the RTLSR model is set to 1, because the RTLSR model assumes a flat and homogeneous surface. However, even if the interior of the rugged terrain is Lambertian, the entire scene is anisotropic due to shadowing effects and uneven distribution of solar radiation [23]. Therefore, the K

_{iso}of the TCKD and KDST-TCKD models varies with the view geometry and changes more significantly with the increase of the roughness of the terrain. Secondly, the K

_{vol}in the RTLSR model cannot describe the hotspot effect in the PP because the correlation between the solar and sensor angles is not considered [36]. However, the volumetric-scattering kernels in the TCKD and KDST-TCKD models are larger in the near-hotspot region (fewer terrain shadows) due to the addition of topographic factors, as is shown in Figure 6e,f. In addition, the volumetric-scattering kernels of the TCKD and KDST-TCKD models show similar kernel shapes. This is because the RossT radiative-transfer model [37] is equivalent to the geometric rotation of the reflectance of the slope with the reflectance of the horizontal plane. Third, Figure 6g–i shows that, when compared to the TCKD model and RTLSR model, the KDST-TCKD model presents a great difference in the kernel shape due to a great improvement in the geometric optical kernel. Compared with the RTLSR model in the PP, the K

_{geo}values of the TCKD and the KDST-TCKD models are generally larger in the forward direction and smaller in the backward direction.

_{iso}is correlated with VZA due to shadowing effects and the uneven distribution of incident energy. Furthermore, as is shown in Figure 7a,b, the difference of K

_{iso}between the PP and CPP indicates that K

_{iso}is also associated with relative azimuth angle (RAA). Figure 7e,f indicates that the geometric–optical kernel has a dome shape and exhibits a distinct hot-spot effect. With the increase of the average slope, the K

_{geo}increases as a whole in the PP. While the hot-spot value is basically unchanged, the width of the hot spot increases. Wu et al. [29] explained that this was due to the increase in slope, which resulted in a decrease in the relative height of the canopy center from the slope [36]. Figure 7c,d shows that, similar to the K

_{geo}, the value of the K

_{vol}increases with the average slope in the PP and the K

_{vol}kernel shape becomes asymmetric in the CPP.

#### 4.2. Model Comparisons with 3-D LESS Simulations

#### 4.3. Model Comparisons with Sandbox Measurements

#### 4.4. Model Comparisons with MODIS Observations

## 5. Discussion and Conclusions

_{vol}characterizes a bowl-shaped curve in the PP, and it cannot depict the hot-spot effects due to the neglect of the correlation between the solar illumination and sensor observation. However, the volumetric-scattering kernels of TCKD and KDST-TCKD models are large in the near-hot-spot region because fewer shadows can be observed. Compared with the RTLSR model, the K

_{geo}values of the TCKD and the KDST-TCKD models in the PP are generally larger in the forward direction and smaller in the backward direction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three 1.2 km simulated rugged terrain scenes with different mean slopes and normal distribution of elevations ((

**a**) $\mathsf{\alpha}\text{}=9.13$°; (

**b**) $\mathsf{\alpha}\text{}=22.83$°; and (

**c**) $\mathsf{\alpha}\text{}=33.40$°).

**Figure 2.**BRFs of three rugged terrains in LESS simulations along the principal plane, (SZA = 45°, SAA = 0°; (

**a**) NIR Band and (

**b**) Red Band) The negative and positive values on the abscissa represent the back- and forward-scattering directions in the PP, respectively. The SZA, SAA, VZA, and VAA of each rugged terrain scene are the same; only the average slope is different.

**Figure 3.**Four typical terrain sandboxes ((

**a**,

**b**) are both normally distributed terrains. Their mean slopes are 8.4° and 24.7°, respectively; (

**c**) is a concave valley with an average slope of 25.7°; and (

**d**) is a convex ridge with an average slope of 30.36°).

**Figure 4.**Hemispheric distributions of the NIR reflectance observed for the four sandboxes. In polar plots, the radii orient with the relative azimuth angles and the concentric circles corresponding to the view zenith angles.

**Figure 5.**The blue box is the geographic location of the study area H25V05, which is located in the hinterland of the Qinghai−Tibet Plateau and contains a large amount of rugged terrain.

**Figure 6.**The kernel shape of the RTLSR, TCKD and KDST−TCKD (abbreviated as ‘Model’) models in the PP under different terrains (the three terrains in Section 3.1, (

**a**–

**c**) are the kernel shapes of the isotropic kernel, (

**d**–

**f**) are the volumetric-scattering kernel shapes, and (

**g**–

**i**) are the geometric–optical kernel). Columns from left to right are the three rugged terrains with mean slopes of 9.13°, 22.83°, and 33.40°, respectively.

**Figure 7.**The kernel shape of the KDST-TCKD model in the PP and CPP under different terrain ($\alpha $= 0°, $\alpha $ = 9.13°, $\alpha $ = 22.83°,$\mathrm{and}\text{}\alpha $ = 33.40°). (

**a**,

**b**) are the kernel shapes of the isotropic kernel, (

**c**,

**d**) are the volumetric−scattering kernel shapes, and (

**e**,

**f**) are the geometric–optical kernel in the PP and CPP, respectively.

**Figure 8.**Comparison of performance of the RTLSR model, TCKD model and KDST-TCKD model in red and NIR bands. The abscissa was divided into three groups, representing the three terrains of Figure 1 with mean slopes of 9.13°, 22.83°, and 33.40°, respectively. The width of the violin plot represents the frequency of the RMSE distribution.

**Figure 9.**Comparison of performance of RTLSR model, TCKD model and KDST-TCKD model in red and NIR band over four different sandboxes. The width of the violin plot represents the frequency of the RMSE distribution. Columns from left to right are the four rugged terrains (Sandbox1: α = 8.4°, normal distribution; Sandbox2: α = 24.7°, normal distribution; Sandbox3: α = 25.7°, concave valley; Sandbox4: α = 30.36°, convex ridge). The first row are results for NIR band, and the second for red band.

**Figure 10.**Comparison of the NIR BRF retrieved by the RTLSR, TCKD, and KDST−TCKD models with MODIS BRFs under different terrains (all land types are included). The orange lines are the lines of best fit. The colors correspond to the point density from the lowest (blue) to highest (yellow). From the first row to the fourth row are different mean slopes, and columns from left to right are the RTLSR, TCKD, and KDST−TCKD models.

**Table 1.**Accuracy statistics of RTLSR model, TCKD model and KDST model in the red and NIR bands over three different terrains.

Models | DEM1 | DEM2 | DEM3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

NIR | Red | NIR | Red | NIR | Red | |||||||

RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | |

RTLSR | 0.0358 | 7.859% | 0.0342 | 23.791% | 0.0392 | 9.665% | 0.0446 | 39.195% | 0.0471 | 15.7869% | 0.0516 | 61.869% |

TCKD | 0.0366 | 7.992% | 0.0337 | 23.337% | 0.0257 | 6.245% | 0.0324 | 28.049% | 0.0252 | 8.8421% | 0.0292 | 30.511% |

KDST-TCKD | 0.0192 | 4.409% | 0.0269 | 17.711% | 0.0167 | 4.522% | 0.0224 | 18.407% | 0.0169 | 5.1521% | 0.0180 | 18.930% |

**Table 2.**Accuracy statistics of RTLSR model, TCKD model and KDST model in the red and NIR bands over four different sandboxes.

Band | Models | Sandbox1 | Sandbox2 | Sandbox3 | Sandbox4 | ||||
---|---|---|---|---|---|---|---|---|---|

RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | ||

NIR | RTLSR | 0.0147 | 4.198% | 0.0346 | 14.797% | 0.0174 | 5.247% | 0.0265 | 12.343% |

TCKD | 0.0144 | 4.129% | 0.0298 | 14.002% | 0.0171 | 5.194% | 0.0251 | 11.699% | |

KDST-TCKD | 0.0137 | 3.878% | 0.0175 | 8.402% | 0.0133 | 4.441% | 0.0234 | 11.111% | |

Red | RTLSR | 0.0085 | 3.542% | 0.0156 | 18.666% | 0.0126 | 5.495% | 0.0165 | 12.023% |

TCKD | 0.0086 | 3.582% | 0.0127 | 17.866% | 0.0125 | 5.443% | 0.0154 | 11.297% | |

KDST-TCKD | 0.0082 | 3.368% | 0.0079 | 11.124% | 0.0116 | 5.167% | 0.0149 | 11.075% |

**Table 3.**Comparisons of the retrieval accuracy of RTLSR model, TCKD model, and KDST-TCKD model in the NIR band under different land-cover types and different $\alpha $.

Model | Broad | Needleleaf | Savannas | Shrub | Glasslands | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | ||

a < 10° | RTLSR | 0.0309 | 6.161% | 0.0601 | 15.405% | 0.0603 | 16.128% | 0.0250 | 5.875% | 0.0300 | 6.261% |

TCKD | 0.0307 | 6.011% | 0.0600 | 15.285% | 0.0599 | 15.974% | 0.0231 | 5.268% | 0.0293 | 6.057% | |

KDST-TCKD | 0.0302 | 5.905% | 0.0580 | 14.721% | 0.0589 | 15.679% | 0.0223 | 5.058% | 0.0270 | 5.693% | |

10°–20° | RTLSR | 0.0360 | 7.443% | 0.0703 | 18.080% | 0.0682 | 16.792% | 0.0296 | 6.575% | 0.0340 | 7.413% |

TCKD | 0.0309 | 6.154% | 0.0649 | 16.446% | 0.0692 | 16.296% | 0.0248 | 5.124% | 0.0297 | 6.069% | |

KDST-TCKD | 0.0286 | 5.667% | 0.0602 | 14.964% | 0.0590 | 14.274% | 0.0202 | 4.641% | 0.0249 | 5.430% | |

20°–30° | RTLSR | 0.0421 | 10.684% | 0.0801 | 22.129% | 0.0677 | 17.138% | 0.0569 | 15.719% | 0.0396 | 9.480% |

TCKD | 0.0287 | 6.485% | 0.0691 | 18.817% | 0.0613 | 15.924% | 0.0465 | 12.289% | 0.0332 | 7.581% | |

KDST-TCKD | 0.0218 | 5.112% | 0.0633 | 17.360% | 0.0575 | 14.952% | 0.0377 | 10.518% | 0.0300 | 6.825% | |

a > 30° | RTLSR | -- | -- | 0.0842 | 27.130% | 0.0718 | 19.038% | 0.0602 | 19.127% | 0.0514 | 14.240% |

TCKD | -- | -- | 0.0843 | 26.971% | 0.0450 | 14.929% | 0.0498 | 14.558% | 0.0409 | 10.833% | |

KDST-TCKD | -- | -- | 0.0765 | 24.059% | 0.0384 | 11.752% | 0.0420 | 12.544% | 0.0340 | 8.625% |

**Table 4.**Comparisons of the retrieval accuracy of RTLSR model, TCKD model and KDST-TCKD model in the red band under different land-cover types and different $\alpha $.

Model | Broad | Needleleaf | Savannas | Shrub | Glasslands | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | ||

a < 10° | RTLSR | 0.0250 | 6.642% | 0.0544 | 19.478% | 0.0549 | 19.452% | 0.0209 | 6.587% | 0.0233 | 7.422% |

TCKD | 0.0243 | 6.343% | 0.0543 | 19.288% | 0.0542 | 19.260% | 0.0202 | 6.170% | 0.0224 | 6.909% | |

KDST-TCKD | 0.0238 | 6.147% | 0.0530 | 18.742% | 0.0531 | 18.721% | 0.0196 | 5.986% | 0.0216 | 6.684% | |

10°–20° | RTLSR | 0.0282 | 9.161% | 0.0557 | 21.029% | 0.0690 | 21.201% | 0.0213 | 7.632% | 0.0333 | 10.116% |

TCKD | 0.0238 | 7.221% | 0.0515 | 18.514% | 0.0634 | 18.849% | 0.0164 | 5.420% | 0.0274 | 7.849% | |

KDST-TCKD | 0.0221 | 6.621% | 0.0484 | 17.358% | 0.0595 | 17.188% | 0.0157 | 5.136% | 0.0236 | 7.008% | |

20°–30° | RTLSR | 0.0643 | 14.828% | 0.0664 | 24.813% | 0.0695 | 21.954% | 0.0416 | 17.608% | 0.0322 | 11.468% |

TCKD | 0.0579 | 11.395% | 0.0609 | 22.068% | 0.0566 | 17.112% | 0.0336 | 14.103% | 0.0272 | 9.016% | |

KDST-TCKD | 0.0470 | 9.869% | 0.0547 | 19.154% | 0.0536 | 16.064% | 0.0297 | 12.137% | 0.0247 | 8.014% | |

a > 30° | RTLSR | -- | -- | 0.0692 | 27.268% | 0.0774 | 23.279% | 0.0609 | 35.147% | 0.0429 | 19.369% |

TCKD | -- | -- | 0.0601 | 25.493% | 0.0533 | 19.120% | 0.0453 | 24.785% | 0.0334 | 14.436% | |

KDST-TCKD | -- | -- | 0.0584 | 25.733% | 0.0523 | 18.318% | 0.0358 | 19.428% | 0.0283 | 11.548% |

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**MDPI and ACS Style**

Zhu, W.; You, D.; Wen, J.; Tang, Y.; Gong, B.; Han, Y.
Evaluation of Linear Kernel-Driven BRDF Models over Snow-Free Rugged Terrain. *Remote Sens.* **2023**, *15*, 786.
https://doi.org/10.3390/rs15030786

**AMA Style**

Zhu W, You D, Wen J, Tang Y, Gong B, Han Y.
Evaluation of Linear Kernel-Driven BRDF Models over Snow-Free Rugged Terrain. *Remote Sensing*. 2023; 15(3):786.
https://doi.org/10.3390/rs15030786

**Chicago/Turabian Style**

Zhu, Wenzhe, Dongqin You, Jianguang Wen, Yong Tang, Baochang Gong, and Yuan Han.
2023. "Evaluation of Linear Kernel-Driven BRDF Models over Snow-Free Rugged Terrain" *Remote Sensing* 15, no. 3: 786.
https://doi.org/10.3390/rs15030786