(1) S(τ) = S0(τ) + 0∫τv dτ' S(τ') . K1(|τ'-τ|, |t'(τ')-t(τ)|)where τ is the vertical optical depth variable for the scatterers (τv being the total optical depth), t the corresponding optical depth for the absorbers, S0(τ) the initial source function, and the kernel function K1 gives the probability for a photon to traverse unscattered and unabsorbed between τ' and τ and be (isotropically) scattered at the latter level. In case of a continuous (i.e. frequency-independent) scattering/absorption cross section, K1 is simply determined by the usual exponential absorption law (i.e. for a plane-parallel atmosphere by the first exponential integral E1), but for the case of a spectral line it is given by
(2) K1(τ,t)= 1/2/√π.-∞∫∞dx Φ2(x).E1(τ.Φ(x)+t) ,where Φ(x) is the line profile function (which is normalized such that the integral over it (from -∞ to +∞) is equal to √π (e.g. in case of a pure Gaussian (Doppler broadened) profile Φ(x)=e-x2, or in case of combined Doppler and natural broadening Φ(x)=H(a,x) (the Voigt function)).
(3) S(τj) = S0(τj) + Σ S(τk) .[ | K2(|τk+1-τj| , |tk+1-tj| , αk) - K2(|τk-τj|, |tk-tj| , αk) | ] ,
(4) K2(τ,t,α) = 0∫τ dτ' K1(τ',t'(τ')) = 1/2/√π.-∞∫∞dx Φ2(x)/(Φ(x)+α) .E2(τ.Φ(x)+t) .with E2 the second exponential integral, and the optical depth of the absorbers assumed to change proportionally to the optical depth of the scatterers in each interval, i.e.
(5) t'(τ') = tk + αk.(τ'- τk) ; τk≤τ'≤τk+1Although Eq.(3) could be iterated as such, it is better to strictly solve it for S(τj) (which also appears on the right hand side of the equation (for k=j)). This yields
(6) S(τj) = [ S0(τj) + Σ S(τk) .dK2(j,k) ] / [1-dK2(j,j) ] ,
(7) dK2(j,k) = | K2(|τk+1-τj| , |tk+1-tj| , αk) - K2(|τk-τj|, |tk-tj| , αk) | .Eq.(6) represents now a system of N equations for the N unknowns S(τj), which can be solved by iteration, using the given initial source function S0(τj) as starting values.
(8) Iup(τj,μ) = 1/μ. 0∫τj dτ' S(τ') . T( (τj-τ')/μ , (tj-t')/μ ) )where
(9) T(τ,t) = 1/√π.-∞∫∞dx Φ(x).e-(τ.Φ(x)+t) :Discretization and integration of Eq.(8) yields then
(10) Iup(τj,μ) = Σ S(τk) .[ Z(τj-τk)/μ , (tj-tk)/μ , αk) - Z(τj-τk+1)/μ , (tj-tk+1)/μ , αk) ] ,
(11) Z(τ,t,α) = 0∫τ dτ' T(τ',t'(τ')) = 1/√π.-∞∫∞dx Φ(x)/(Φ(x)+α).[e-(t-ατ) -e-(τ.Φ(x)+t)] .
(12) Idown(τj,μ) = Σ S(τk) .[ Z(τk+1-τj)/μ , (tk+1-tj)/μ , αk) - Z(τk-τj)/μ , (tk-tj)/μ , αk) ] .