Table 1 Formulas used for atmospheric corrections and remote sensing indices.

Top-of-atmosphere (TOA) radiance

\(L_{\lambda } = M_{L} \times Q_{{{\text{cal}}}} + A_{L}\)

(1)

Converts digital number (DN) values to spectral radiance using sensor-specific rescaling factors \(M_{L}\) and A_L

⁴³

TOA reflectance

\(\rho_{\lambda }{\prime} = M_{\rho } \times Q_{{{\text{cal}}}} + A_{\rho }\)

(2)

Converts DN to reflectance using reflectance rescaling factors Mρ and Aρ

⁴⁴

TOA reflectance (sun elevation correction)

\(\rho_{\lambda } = \frac{{\rho_{\lambda }{\prime} }}{{\sin \left( {\theta_{{{\text{SE}}}} } \right)}}\)

(3)

Corrects TOA reflectance for solar elevation angle θ_SE

⁴⁵

Brightness temperature (BT)

\(BT = \frac{{K_{2} }}{{\ln \left( {\frac{{K_{1} }}{{L_{\lambda } }} + 1} \right)}}\)

(4)

Converts radiance to brightness temperature using thermal calibration constants K₁ and K₂

⁴⁶

Land surface emissivity (ε)

\(\varepsilon = 0.004\,P_{v} + 0.986\)

(5)

Estimates emissivity based on fractional vegetation proportion P_v

⁴⁷

Land surface temperature (LST)

\(LST = \frac{BT}{{1 + \left( {\frac{\lambda \,BT}{\rho }} \right)\ln \left( \varepsilon \right)}}\)

(6)

Computes surface temperature incorporating emissivity and effective wavelength λ = 10.895 μm

⁴⁸

Resultant LST

\(LST_{{{\text{res}}}} = \frac{{\left( {LST_{1990} - \sigma_{1990} } \right) + \left( {LST_{2000} - \sigma_{2000} } \right) + \left( {LST_{2010} - \sigma_{2010} } \right) + \left( {LST_{2020} - \sigma_{2020} } \right)}}{4}\)

(7)

Mean standardized LST across four periods (1990–2020)

⁴⁹

Normalized difference vegetation index (NDVI)

\(NDVI = \frac{NIR - R}{{NIR + R}}\)

(8)

Measures vegetation vigor from red (R) and near-infrared (NIR) bands

⁵⁰

Vegetation proportion (Pᵥ)

\(P_{v} = \left( {\frac{{NDVI - NDVI_{min} }}{{NDVI_{max} - NDVI_{min} }}} \right)^{2 }\)

(9)

Fractional vegetation cover derived from NDVI range

⁵¹

Topsoil grain size index (TGSI)

\(TGSI = \frac{R - B}{{R + B + G}}\)

(10)

Assesses topsoil coarseness using red (R), blue (B), and green (G) reflectance

¹⁸

Normalized difference salinity Index (NDSI)

\(NDSI = \frac{R - NIR}{{R + NIR}}\)

(11)

Detects soil salinity using red (R) and near-infrared (NIR) reflectance

⁵²

Resultant NDVI

\(NDVI_{{{\text{res}}}} = \frac{{\left( {NDVI_{1990} - \sigma_{1990} } \right) + \left( {NDVI_{2000} - \sigma_{2000} } \right) + \left( {NDVI_{2010} - \sigma_{2010} } \right) + \left( {NDVI_{2020} - \sigma_{2020} } \right)}}{4}\)

(12)

Composite standardized NDVI for multi-temporal vegetation assessment

⁵³

Resultant TGSI

\(TGSI_{{{\text{res}}}} = \frac{{\left( {TGSI_{1990} - \sigma_{1990} } \right) + \left( {TGSI_{2000} - \sigma_{2000} } \right) + \left( {TGSI_{2010} - \sigma_{2010} } \right) + \left( {TGSI_{2020} - \sigma_{2020} } \right)}}{4}\)

(13)

Multi-temporal standardized TGSI representing cumulative soil coarseness

–

Resultant NDSI

\(NDSI_{{{\text{res}}}} = \frac{{\left( {NDSI_{1990} - \sigma_{1990} } \right) + \left( {NDSI_{2000} - \sigma_{2000} } \right) + \left( {NDSI_{2010} - \sigma_{2010} } \right) + \left( {NDSI_{2020} - \sigma_{2020} } \right)}}{4}\)

(14)

Mean standardized NDSI indicating long-term salinity variation

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