VLIDORT User's Guide

m⎛ P⎞l( μ)0 0 0⎜⎟mmm ⎜ 0 Rl( μ)− Tl( μ)0 ⎟Pl( μ)= ⎜mm⎟ . (2.25)⎜0 − Tl( μ)Rl( μ)0⎟m⎝ 0 0 0 Pl( μ)⎠The “Greek matrices” Β l for 0 ≤ l ≤ LM contain the sets of expansion coefficients that define themmscattering law. The Pl(μ)matrices contain entries of normalized Legendre functions Pl(μ)andfunctions R ml(μ)and mlTl(μ)which are related to Pmn(μ)(for details, see for example [Siewert,2000b]).2.1.3. Boundary conditionsDiscrete ordinate RT is pure scattering theory: in a multilayer medium, it is only necessary tospecify the layer total optical thickness values Δ n , the layer total single scatter albedo ω n , and thelayer 4 x 4 matrices Β nl of expansion coefficients (l being the moment number) for the totalscattering. To complete the calculation of the radiation field in a stratified multilayer medium,we have the following boundary conditions:(Ι) No diffuse downwelling radiation at TOA. Thus for the first layer we have:+In( 0, μ,φ)= 0 . (n = 1) (2.26)(ΙΙ) Continuity of the upwelling and downwelling radiation fields at intermediateboundaries. If N TOTAL is the number of layers in the medium, then:±( ) ±I n − 1Δ n −1 = I n(0) . (n = 2,…N TOTAL ) (2.27)(ΙΙΙ) A surface reflection condition relating the upwelling and downwelling radiation fieldsat the bottom of the atmosphere:−+I ( , μ , φ)= ( μ,φ;μ′, φ′) ( , μ′, φ′nΔnR InΔn) . (n = N TOTAL ) (2.28)Here, reflection matrix R relates incident and reflected directions.The convention adopted here is to use a “+” suffix for downwelling solutions, and a “−” suffixfor upwelling radiation. Conditions (I) and (II) are obeyed by all Fourier components in theazimuthal series. For condition (III), it is necessary to construct a Fourier decomposition of theBRDF operator R to separate the azimuth dependence; we return to this issue in section 2.5.4.The Lambertian case (isotropic reflectance) only applies for Fourier component m = 0 and Eq.(2.28) then becomes:1⎡⎤−I = ⎢ ( − ) + ∫ + ′ ′ ′n( Δn,μ)2δm, 0R0E1μ0I0Tn−1exp λnΔnIn( Δn,μ ) μ dμ⎥ . (2.29)⎣0⎦Here, R 0 is the Lambertian albedo, E 1 = diag{1,0,0,0}, and T n− exp( − λnΔn)1 is the wholeatmosphereslant path optical depth for the solar beam.18