VLIDORT User's Guide

6.4. SLEAVE SupplementHere, the surface-leaving (SL) supplement is described. The SL supplement is a separate systemof VLIDORT-based software that provides an additional source of upwelling radiance at thelower boundary of the atmosphere. One familiar example is the "water-leaving" radiance over anocean boundary; another is the near infra-red (NIR) fluorescence signature from vegetation. ThisSL term is a diffuse upwelling radiance from the lower boundary, on top of the existing diffuseanddirect-beam reflected light. This is the first version of the SL supplement, and is currentlyconfigured for simple treatments of water-leaving radiance and NIR fluorescence. We note thatthe supplement also has the observational geometry facility like VLIDORT itself.In section 6.4.1, we give an overview of SL construction and discuss the available options. Thisis followed by a description of the supplement’s current development state in section 6.4.2. Next,a sample calling sequence for the supplement is given in section 6.4.3. The supplement inputsand outputs are then given in tables in section 6.4.4; in particular, we wish to fill up the SL inputsin Tables C4 and G4 in sections 6.1.1.3 and 6.1.1.7, respectively.6.4.1. SLEAVE formulationThe SL contribution output from the supplement consists of 3 terms which will in general dependon the solar illumination angle θ oand the relative azimuth angle ϕ − ϕo(think of the waterleavingradiance). These terms are (we have omitted the Stokes-component index for clarity):S ( μ , θ ); S ( γ, θ ); S ( γ, θ , ϕ− ϕ ) . (6.4.1)m i o m o exact o oHere, the Fourier component index m= 0,1,...,2N− 1, where N is the number of discreteordinates in the half-space. The first term is the (Fourier component of) upwelling radiance intothe polar discrete ordinate directions μ i, and this is required for the diffuse-scattering boundarycondition at the lower surface. The second term is the (Fourier component of) upwelling radianceinto user-defined stream directions γ, and this is required for post-processing of the discreteordinate solution (source function integration).The first two terms arise from the Fourier cosine-azimuth expansions of the full functionsS ( μ , θ , ϕ− ϕ ) and S ( γ , θ , ϕ− ϕ ) ; thus for example:exact i o oexact o oS ( μ , θ , ϕ− ϕ ) = S ( μ , θ ) + 2 ∑ S ( μ , θ )cos( ϕ−ϕ)(6.4.2)exact i o o o i o m i o om=1∞In the discrete-ordinate approximation we can only use 2N-1 components in the sum in Eq.(6.4.2). In the post-processing, it is more accurate to use the complete term Sexact ( γ , θo, ϕ− ϕo)inplace of the Fourier-series truncation, and this "correction" is implemented. This is akin to makea precise calculation of the direct-beam surface reflection, and indeed the "direct" surface termwill now include Sexact ( γ , θo, ϕ− ϕo) if desired. In this case, the Fourier terms Sm( γ , θo)are notneeded.116