# A New Methodology for Estimating Surface Albedo in Heterogeneous Areas from Satellite Imagery
## 4 Methods
### Atmospheric Correction in Planetary Albedo  
The surface albedo is calculated according to Equation (1) [37,51,52]:  
\[\alpha=\frac{\alpha_{TOA}-\alpha_{path\_radiance}}{\tau_{SW}{}^{2}}\, \tag{1}\]  
being  
\(\alpha\): surface albedo (BOA, Bottom of Atmosphere) or albedo with atmospheric correction.  
\(\alpha_{path\_radiance}\): atmospheric albedo.  
\(\tau_{SW}\): atmospheric transmittance.  
\(\alpha_{TOA}\): Top Of Atmosphere (TOA) albedo or planetary albedo, defined as (Equation (2)):  
\[\alpha_{TOA}=\sum(\omega_{\lambda}\times\rho_{\lambda})\, \tag{2}\]  
where:  
\(\omega_{\lambda}\): weight of each spectral band, computed as in Equation (3):  
\[\omega_{\lambda}=\frac{ESIN_{\lambda}}{\sum ESIN_{\lambda}}\, \tag{3}\]  
Figure 2: Methodologies used in this article for computation of surface albedo from satellite imagery. These methodologies are classified into original (existing) and combined (new methodologies). The original ones are distinguished into indirect methods from [PERSON] et al. 2011 [36] (V01 with atmospheric transmittance from [PERSON] et al. 1998 [51] and V02 with atmospheric transmittance from [PERSON] 1984 [52]), which perform a first surface albedo calculation and then correct it atmospherically, and direct methods from [PERSON] et al. 2014 [31] (V03 and V04), which perform the atmospheric correction of the bands previously to the surface albedo calculation. Within these subcategories, the methodologies differ in some equations and/or parameters: V01 and V02 use different atmospheric transmittance equations and V03 and V04 use different regression coefficients. The combined methodologies use several of the above-mentioned methodologies depending on the NDVI of the ground.  
\(\rho_{\lambda}\): monochromatic reflectance of each spectral band (Equation (4))  
\[\rho_{\lambda}=\frac{\pi\cdot L_{\lambda}\cdot d^{2}}{ESUN_{\lambda}\cdot cos\phi}\, \tag{4}\]  
\(L_{\lambda}\): TOA spectral radiance (W\(\cdot\)m\({}^{-2}\).sr\({}^{-1}\cdot\)um\({}^{-1}\)).  
\(ESUN_{\lambda}\): mean exo-atmospheric solar spectral irradiance (W\(\cdot\)m\({}^{-2}\).sr\({}^{-1}\cdot\)um\({}^{-1}\)).  
\(d\): Earth-Sun distance measured in astronomical units.  
\(\theta\): solar zenith angle in sexagesimal degrees, which corresponds to \(\theta\) = 90\({}^{\circ}\) - \(\theta e\), where \(\theta e\) is the solar elevation.  
For the case of Landsat 8 (satellite for which the equation was designed), TOA spectral radiance and mean exo-atmospheric solar spectral irradiance are defined as stated by Equations (5) and (6):  
\[L_{\lambda}=M_{L}\cdot Qcal_{\lambda}+A_{L}\, \tag{5}\]  
\[ESUN_{\lambda}=\pi\cdot d^{2}\cdot\frac{RADIANCE_{MAXIMUM_{\lambda}}}{REFLECTANCE_{ MAXIMUM_{\lambda}}}\, \tag{6}\]  
Specifically:  
\(\rho_{\lambda}\): bands 2-7 from Landsat 8 are used.  
\(M_{L}\): Band-specific multiplicative rescaling factor, determined from the metadata.  
\(Q_{cal\lambda}\): quantized and calibrated standard product pixel values (DN). This value refers to each pixel in one of the bands.  
\(A_{L}\): Band-specific additive rescaling factor from the metadata.  
\(RADIANCE_{MAXIMIMIM}\): maximum radiance value in each band. Values are provided in the image metadata file.  
\(REFLECTANCE_{MAXIMUM_{\lambda}}\): maximum reflectance value in each band. Values are provided in the image metadata file.  
The value of atmospheric transmittance has yet to be defined. This value distinguishes methods denoted as V01 and V02 from one another.  
It should be stated that in the case of working with Landsat 9 imagery, the TOA spectral radiance and mean exo-atmospheric solar spectral irradiance are defined by the same equations. The calibration coefficients for Equation (6) are also available in the image metadata file.