# Algorithm for the Reconstruction of the Ground Surface Reflectance in the Visible and Near IR Ranges from MODIS Satellite Data with Allowance for the Influence of Ground Surface Inhomogeneity on the Adjacency Effect and of Multiple Radiation Reflection

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## Abstract

**:**

## 1. Introduction

## 2. Algorithm for Retrieval of the Surface Reflectance

#### 2.1. Assumptions and Problem Formulation

- The atmosphere is a scattering and absorbing aerosol-gas medium.
- The atmosphere is cloudless and vertically stratified into 32 uniform layers.
- The “atmosphere-ground surface” system is spherical, and refraction is ignored. The boundaries of the atmospheric layers are spheres.
- The source of radiation is the sun. There are no other sources.
- The ground surface is non-uniform and reflects radiation according to the Lambert law.
- The ground surface is uniform within a pixel.
- Local topography is ignored.
- The change in the illumination of the ground surface due to a change in the solar zenith angle is negligibly small.
- The radiative transfer is considered in the monochromatic approximation.

#### 2.2. System of Equations to Be Solved

#### 2.3. Additional Simplifications to Reduce the Computation Time

#### 2.3.1. Use of Isoplanar Zones

#### 2.3.2. Use of the Adjacency Effect Radius

#### 2.3.3. Use of the Radius of Effect of Single Reflection on Ground Surface Illumination

#### 2.3.4. Use of Approximations for ${I}_{atm}$

#### 2.4. Block Diagram of the Algorithm

- Formation of the block of input data. Input data are the following: radiance received in the MODIS band ${I}_{r,ij}$ (i is the pixel line number, j is the pixel column number); aerosol optical depth (AOD) of the atmosphere; vertical profiles of temperature $T\left(z\right)$ and pressure $P\left(z\right)$; cloud mask ${n}_{ij}$; information about the mutual positions of observed pixels, the sun, and the satellite (pixel coordinates (${\phi}_{N,ij},{\lambda}_{N,ij}$), direction to the sun (${\theta}_{sun,ij},{A}_{sun,ij}$), direction to the satellite (${\theta}_{d,ij},{A}_{d,ij}$)). These data can be borrowed from MODIS thematic products MOD021_L2, MOD03_L2, MOD07_L2, MOD35_L2, and MOD08_D3.
- Construction of the atmospheric model. Satellite measurements of AOD, $T\left(z\right)$, and $P\left(z\right)$ formed the basis for constructing the atmospheric model. Profiles of the aerosol extinction and scattering coefficients are set based on MODTRAN models [39] closest in the aerosol optical depth to MODIS data. Profiles of the molecular scattering coefficients are set based on the temperature and pressure profiles and the values of the molecular scattering coefficients from [40]. Profiles of the molecular absorption coefficients are constructed based on the vertical temperature and pressure profiles, the MODTRAN model of the gas composition of the atmosphere for mid-latitude summer, and absorption cross-sections of atmospheric gases from the HITRAN database [41]. The atmospheric models can be found in the Supplementary Materials [37]. The algorithm for construction of these models is described in Appendix A.
- Calculation of the areas ${S}_{ij}$. The image under consideration was divided into sections with respect to the closeness to the pixel centers. The algorithm for calculating the areas is described in the Supplementary Materials [37].
- Calculation of the radiance ${I}_{atm,ij}$ for the radiation non-interacting with the ground surface.
- Determination of the number l and angles of the boundaries ${\theta}_{l}$ of isoplanar zones (zones in which one PSF of the AE h can be used).
- Calculation of direct transmittance at the “observed pixel–receiver” path ${\tau}_{ij}$.
- Calculation of the AE radii for isoplanar zones ${R}_{l}$.
- Calculation of $h({\rho}_{w},{\alpha}_{w},{\mu}_{l})$ for each isoplanar zone and its integral over the entire ground surface $H\left({\mu}_{l}\right)$.
- Estimation of the number of pixels in an image (in image lines $N{x}_{ij}$ and columns $N{y}_{ij}$) within the AE radius ${R}_{l}$ for each pixel.
- Solution of system of linear algebraic Equation (8) for luminosity of observed pixels ${Q}_{ij}$.
- Calculation of ground surface irradiance neglecting the reflected radiation ${E}_{0}$.
- Calculation of the radii ${R}_{S}$ of additional irradiance of the ground surface by surface pixels.
- Calculation of ${h}_{1}\left({\rho}_{w}\right)$ and its integral ${\gamma}_{1}$.
- Estimation of the number of pixels in an image (in image lines $M{x}_{ij}$ and columns $M{y}_{ij}$) within the radius of additional irradiance formation ${R}_{S}$ for each pixel.
- Solution of system of nonlinear Equation (9) for surface reflectance ${r}_{surf,ij}$.

#### 2.5. Algorithm Reliability

## 3. Algorithm Validation against Ground-Based Measurements

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Description of Atmospheric Models

## Appendix B. Monte Carlo Algorithms

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**Figure 1.**Optical fluxes forming the received radiation (according to [6]): (

**a**) radiation that does not interact with the ground surface; (

**b**) direct solar radiation that interacts with the surface fragment under study; (

**c**) diffuse solar radiation that interacts with the surface fragment under study; (

**d**) adjacency effect (AE) due to single reflected radiation; (

**e**) adjacency effect due to multiple reflected radiation.

**Figure 2.**Block diagram of the algorithm for retrieval of surface reflectance. Units employing the Monte Carlo method are shown in black.

**Figure 3.**Test surface area with the coordinates 38.4–39.4${}^{\xb0}$N, 8.25–9.25${}^{\xb0}$W. The red dot is for the test site with the coordinates 38.829${}^{\xb0}$N, 8.791${}^{\xb0}$W, for which the results of ground-based measurements are reported in [28]. A red line is drawn from the pixel with coordinates 38.351${}^{\xb0}$N, 8.660${}^{\xb0}$W to the pixel with coordinates 38.438${}^{\xb0}$N, 9.247${}^{\xb0}$W.

**Figure 4.**Ground-based measurements of reflectances of the surface area with coordinates 38.829${}^{\xb0}$N, 8.791${}^{\xb0}$W [28]: (

**a**) 1 April 2016; (

**b**) 25 April 2016; (

**c**) 20 May 2016; (

**d**) 3 June 2016. Numbers 1–4 are the numbers of the MODIS bands; thick lines are the average reflectances; thin lines are their standard deviations.

**Figure 5.**Comparison of averaged ground-based measurements [28] and the results provided by the three algorithms for the TERRA data: (1) averaged ground-based measurements and their standard deviations (gray), (2) proposed algorithm, (3) MOD09, (4) without atmospheric correction; MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 6.**Comparison of averaged ground-based measurements [28] and the results provided by the three algorithms for the AQUA data: (1) averaged ground-based measurements and their standard deviations (gray), (2) proposed algorithm, (3) MOD09, (4) without atmospheric correction; MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 7.**Comparison of reflectances provided by MOD09 (plotted along the Ox axis), the proposed algorithm (symbols 1), and the algorithm without atmospheric correction (symbols 2) (plotted along the Oy axis) for TERRA; MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 8.**Comparison of reflectances provided by MOD09 (plotted along the Ox axis), the proposed algorithm (symbols 1), and the algorithm without atmospheric correction (symbols 2) (plotted along the Oy axis) for AQUA; MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 9.**Example of reflectances calculated by the proposed algorithm for four MODIS bands (TERRA, 1 April 2016). The values below 0 correspond to negative reflectance. MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 10.**Example of reflectances calculated by the MOD09 algorithm for four MODIS bands (TERRA, 1 April 2016). The values below 0 correspond to negative reflectance. MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 11.**Example of reflectances calculated by the algorithm without atmospheric correction for four MODIS bands (TERRA, 1 April 2016). MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

**Figure 12.**Values of reflectance obtained by the three algorithms for line 879 of the TERRA image of the test area for 1 April 2016 from the pixel with coordinates 38.351${}^{\xb0}$N, 8.660${}^{\xb0}$W to the pixel with coordinates 38.438${}^{\xb0}$N, 9.247${}^{\xb0}$W. Negative values of ${r}_{surf}$ are omitted. The distance to the pixel with coordinates 38.351${}^{\xb0}$N, 8.660${}^{\xb0}$W is plotted along the Ox axis: (1) proposed algorithm; (2) MOD09 algorithm; (3) without atmospheric correction; MODIS band 1 ($\lambda $ = 0.62–0.67 $\mathsf{\mu}$m) (

**a**), 2 ($\lambda $ = 0.841–0.876 $\mathsf{\mu}$m) (

**b**), 3 ($\lambda $ = 0.459–0.479 $\mathsf{\mu}$m) (

**c**), and 4 ($\lambda $ = 0.545–0.565 $\mathsf{\mu}$m) (

**d**).

Factor | Authors/Reference | |||||||
---|---|---|---|---|---|---|---|---|

Putsay | Tanre | Berk | Vermote | Lyapustin | Reinersman | Katkovskiy | Shi | |

[7] | [6] | [8] | [9] | [10] | [11] | [12] | [13] | |

Surface | Non- | Non- | Non- | Non- | ||||

model | Lamb | Lamb | Lamb | Lamb | Lamb | Lamb | Lamb | Lamb |

Adjacency | ||||||||

effect | accurate | accurate | approx. | approx. | accurate | accurate | approx. | approx. |

Multiple | ||||||||

reflection | No | approx. | approx. | approx. | approx. | approx. | approx. | approx. |

Molecular | ||||||||

absorption | accurate | accurate | accurate | accurate | accurate | accurate | approx. | accurate |

Polarization | No | No | No | Yes | Yes | No | No | No |

Topography | No | No | No | No | No | No | No | Yes |

**Table 2.**Ground-based measurements for average wavelengths of MODIS bands and their standard deviations (SD).

MODIS Band | ||||||||
---|---|---|---|---|---|---|---|---|

Date | 1 | 2 | 3 | 4 | ||||

${\mathit{r}}_{\mathit{surf}}$ | SD | ${\mathit{r}}_{\mathit{surf}}$ | SD | ${\mathit{r}}_{\mathit{surf}}$ | SD | ${\mathit{r}}_{\mathit{surf}}$ | SD | |

1 April 2016 | 0.0425 | 0.0060 | 0.4295 | 0.0616 | 0.0228 | 0.0030 | 0.0686 | 0.0059 |

25 April 2016 | 0.0414 | 0.0092 | 0.3818 | 0.0533 | 0.0192 | 0.0051 | 0.0573 | 0.0077 |

20 May 2016 | 0.0659 | 0.0128 | 0.3175 | 0.0521 | 0.0275 | 0.0057 | 0.0661 | 0.0092 |

3 June 2016 | 0.0823 | 0.0176 | 0.2857 | 0.0512 | 0.0336 | 0.0098 | 0.0737 | 0.0127 |

**Table 3.**Discrepancy of the averaged ground-based measurements [28] and the results provided by the three algorithms for the TERRA data.

MODIS Band | Date | Proposed Algorithm | MOD09 Algorithm | Algorithm without Atmospheric Correction |
---|---|---|---|---|

1 | 1 April 2016 | 0.015 | 0.015 | 0.027 |

1 | 25 April 2016 | 0.014 | 0.016 | 0.044 |

1 | 20 May 2016 | −0.020 | 0.007 | 0.032 |

1 | 3 June 2016 | 0.008 | 0.007 | 0.031 |

2 | 1 April 2016 | −0.195 | −0.197 | −0.196 |

2 | 25 April 2016 | 0.003 | −0.027 | −0.032 |

2 | 20 May 2016 | 0.067 | 0.024 | 0.014 |

2 | 3 June 2016 | 0.052 | 0.044 | 0.033 |

3 | 1 April 2016 | 0.008 | 0.006 | 0.080 |

3 | 25 April 2016 | 0.013 | 0.011 | 0.152 |

3 | 20 May 2016 | −0.014 | 0.012 | 0.154 |

3 | 3 June 2016 | 0.015 | 0.016 | 0.144 |

4 | 1 April 2016 | −0.015 | −0.013 | 0.012 |

4 | 25 April 2016 | 0.014 | 0.016 | 0.065 |

4 | 20 May 2016 | −0.023 | 0.014 | 0.061 |

4 | 3 June 2016 | 0.005 | 0.017 | 0.060 |

**Table 4.**Discrepancy of the averaged ground-based measurements [28] and the results provided by the three algorithms for the AQUA data.

MODIS Band | Date | Proposed Algorithm | MOD09 Algorithm | Algorithm without Atmospheric Correction |
---|---|---|---|---|

1 | 1 April 2016 | 9.00 $\times {10}^{-5}$ | −4.80 $\times {10}^{-4}$ | 0.020 |

1 | 25 April 2016 | 0.023 | 0.024 | 0.040 |

1 | 20 May 2016 | 0.013 | 0.014 | 0.027 |

1 | 3 June 2016 | 0.009 | 0.003 | 0.023 |

2 | 1 April 2016 | −0.114 | −0.140 | −0.142 |

2 | 25 April 2016 | −0.064 | −0.059 | −0.064 |

2 | 20 May 2016 | −0.006 | −0.028 | −0.035 |

2 | 3 June 2016 | 0.055 | 0.034 | 0.026 |

3 | 1 April 2016 | −0.001 | 2.20 $\times {10}^{-4}$ | 0.086 |

3 | 25 April 2016 | 0.011 | 0.017 | 0.103 |

3 | 20 May 2016 | 0.008 | 0.015 | 0.092 |

3 | 3 June 2016 | 0.016 | 0.013 | 0.112 |

4 | 1 April 2016 | −0.013 | −0.014 | 0.018 |

4 | 25 April 2016 | 0.010 | 0.016 | 0.045 |

4 | 20 May 2016 | 0.005 | 0.012 | 0.037 |

4 | 3 June 2016 | 0.011 | 0.013 | 0.047 |

**Table 5.**Pearson correlation r, average discrepancies $|\overline{\Delta {r}_{surf}}|$, and standard deviations (SD) of MOD09 results (1) from the results of the proposed algorithm (2) and algorithm without atmospheric correction (3) for the considered AQUA and TERRA data.

MODIS | No Correction | Proposed Algorithm | ||||
---|---|---|---|---|---|---|

Band | r | SD | $\overline{\Delta {\mathit{r}}_{\mathit{surf}}}$ | r | SD | $\overline{\Delta {\mathit{r}}_{\mathit{surf}}}$ |

AQUA | ||||||

1 | 0.997 | 0.008 | 0.014 | 0.997 | 0.007 | 0.006 |

2 | 0.999 | 0.006 | 0.009 | 0.994 | 0.009 | 0.017 |

3 | 0.975 | 0.014 | 0.081 | 0.987 | 0.012 | 0.009 |

4 | 0.993 | 0.009 | 0.025 | 0.994 | 0.007 | 0.007 |

TERRA | ||||||

1 | 0.986 | 0.010 | 0.010 | 0.984 | 0.006 | 0.004 |

2 | 0.999 | 0.005 | 0.003 | 0.985 | 0.015 | 0.007 |

3 | 0.734 | 0.052 | 0.060 | 0.948 | 0.009 | 0.005 |

4 | 0.948 | 0.018 | 0.020 | 0.963 | 0.008 | 0.006 |

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## Share and Cite

**MDPI and ACS Style**

Tarasenkov, M.V.; Belov, V.V.; Engel, M.V.; Zimovaya, A.V.; Zonov, M.N.; Bogdanova, A.S.
Algorithm for the Reconstruction of the Ground Surface Reflectance in the Visible and Near IR Ranges from MODIS Satellite Data with Allowance for the Influence of Ground Surface Inhomogeneity on the Adjacency Effect and of Multiple Radiation Reflection. *Remote Sens.* **2023**, *15*, 2655.
https://doi.org/10.3390/rs15102655

**AMA Style**

Tarasenkov MV, Belov VV, Engel MV, Zimovaya AV, Zonov MN, Bogdanova AS.
Algorithm for the Reconstruction of the Ground Surface Reflectance in the Visible and Near IR Ranges from MODIS Satellite Data with Allowance for the Influence of Ground Surface Inhomogeneity on the Adjacency Effect and of Multiple Radiation Reflection. *Remote Sensing*. 2023; 15(10):2655.
https://doi.org/10.3390/rs15102655

**Chicago/Turabian Style**

Tarasenkov, Mikhail V., Vladimir V. Belov, Marina V. Engel, Anna V. Zimovaya, Matvei N. Zonov, and Alexandra S. Bogdanova.
2023. "Algorithm for the Reconstruction of the Ground Surface Reflectance in the Visible and Near IR Ranges from MODIS Satellite Data with Allowance for the Influence of Ground Surface Inhomogeneity on the Adjacency Effect and of Multiple Radiation Reflection" *Remote Sensing* 15, no. 10: 2655.
https://doi.org/10.3390/rs15102655