## 2 Methodology  
The theoretical foundation of an IR remote sensing forward model is [PERSON]'s equation of radiative transfer (RT). In a non-scattering atmosphere under local thermodynamic equilibrium, the following equation governs the transfer of emitted TIR radiance (I\({}_{\lambda}\)(t)) at nadir that reach the top of the atmosphere (TOA) cf. [31] at a given wavenumber, \(\lambda\):  
\[\mathrm{I}_{\lambda}(t)=\mathrm{I}_{\lambda}(z_{0})\tau(z_{0},z)+\int_{z_{0}}^ {z}\mathrm{J}_{\lambda}\left(z^{\prime}\right)\left(\frac{\mathrm{d}\tau(z^{ \prime},z)}{\mathrm{d}z^{\prime}}\right)\mathrm{d}z^{\prime} \tag{1}\]  
where, z\({}_{0}\) and z are atmospheric height at the surface and TOA respectively; and z\({}^{\prime}\) is the in between two, J\({}_{\lambda}\) is the emission from a particular layer and \(\tau\) is transmissivity, I\({}_{\lambda}\)(z\({}_{0}\)) represents the radiation at the surface and \(\tau\)(z\({}_{0}\), z) is total transmissivities between the surface and TOA. A Fourier transform spectroscopic instrument typically works in the IR region and measures the radiance at a finite number of spectral points with equidistant wavenumbers. Therefore, a suitable discretization process is used over the integrals in Equation (1). There are many different discretization possibilities: simple classic quadrature method, collocation points and nodes, degenerate kernel approximations (by Eigen functions or by orthonormal systems or approximation by Taylor series or interpolation) and projection methods (Galerkin moment or least squares). We have employed GENSPECT [32] for our forward modeling, which is a line-by-line (LBL) RT model (RTM) that uses a degenerate kernel function. The discretization process of RT equations leads to a set of nonlinear system of equations, which are in a Hilbert space. To solve the above-mentioned Hilbert function, the quadratic constrained RTLS problem stated in [13; 33; 34; 35; 36; 37] is used as follows:  
\[\frac{min}{\Delta\mathbf{x}\in\mathbf{X}}\phi(\mathbf{x},\mathbf{y})\ \ \frac{\|\mathbf{K}\Delta\mathbf{x}-\Delta\mathbf{y}\|^{2}}{1+\|\Delta \mathbf{x}\|^{2}}\ \ \ \ \ \text{subject to}\ \ \|\mathbf{L}\Delta\mathbf{x}\|^{2}\leq \delta^{2} \tag{2}\]  
the term \(\mathbf{L}\) denotes the regularization operator, \(\phi(\mathbf{x},\mathbf{y})\) is the cost function and \(\delta\) is infinitesimal. \(\mathbf{K}\) is the Jacobian, \(\Delta\mathbf{x}\) is the update of the state space at the \(i\)th iteration and \(\Delta\mathbf{y}\) is the residual. The detailed derivation of applied RTLS for current problem is available in [13; 16] and the final form is  
\[\Delta\mathbf{x}=\Big{(}\mathbf{K}^{T}\,\mathbf{K}-\mathbf{g}(\mathbf{x}) \mathbf{I}+\boldsymbol{\alpha}^{2}\mathbf{L}^{T}\mathbf{L}\Big{)}^{-1} \mathbf{K}^{T}\Delta\mathbf{y}_{\delta} \tag{3}\]  
where, \(\mathbf{g}(\mathbf{x})=\frac{\|\mathbf{y}_{\delta}-\mathbf{K}\Delta\mathbf{x} \|^{2}}{1+\|\Delta\mathbf{x}\|^{2}}\), \(\boldsymbol{\alpha}\) is regularization strength, \(\Delta\mathbf{y}_{\delta}\) is the residual including measurement noise and \(\mathbf{I}\) is the identity matrix. Although fully explained in [13], the working mechanism of RTLS method is reiterated in this paragraph for the sake of completeness and for the convenience of the readers. For the calculation of \(\mathbf{g}(\mathbf{x})\), an update of \(\mathbf{x}\) is required, which is obtained as \(\Delta\mathbf{x}=0\) for the first iteration and the retrieved \(\Delta\mathbf{x}\) for successive iterations. Even though considering \(\Delta\mathbf{x}=0\) for the first iteration, the second regularization term of RTLS stabilizes the solution by the value of \(\boldsymbol{\alpha}\), which is calculated using the same value of \(\Delta\mathbf{x}\). Thus, any under- or over-estimate of \(\Delta\mathbf{x}\) is compensated by the value of \(\boldsymbol{\alpha}\) and it is stated as a dual-regularized method.