VLIDORT User's Guide

In section 2.4.1, scattering was assumed to take place along the nadir, so that the scatteringgeometry Ω ≡ { μ0 , μ,φ − φ0}is unchanged along the vertical. For a slant line-of-sight path(Figure 2, lower panel), the scattering geometry varies along the path. For layer n traversed bythis path, the upwelling Stokes vector at the layer-top is (to a high degree of accuracy) given by:↑↑↑↑I ( Ωn−1)≅ I ( Ωn) T(Ωn) + Λn( Ωn) + Mn( Ωn) . (2.103)↑Here, I ( Ωn) is the upwelling Stokes vector at the layer bottom, T ( Ωn) the layer transmittance↑along the line of sight, and ( Ω↑Λn n) and Mn( Ω n) are the single- and multiple-scatter layersource terms respectively. The transmittances and layer source terms are evaluated withscattering geometries Ωn at positions V n . Equation (2.103) is applied recursively, starting with↑the upwelling Stokes vector IBOA( ΩNTOTAL)evaluated at the surface for geometry ΩNTOTAL , andfinishing with the field at top of atmosphere (n = 0). The single-scatter layer source terms↑Λn( Ωn) may be determined through an accurate single scatter calculation (cf. Eq. (2.102))allowing for changing geometrical angles along the line of sight. To evaluate the multiple scattersources, we run VLIDORT in “multiple-scatter mode” successively for each of the geometriesfrom ΩNTOTAL to Ω1, retaining only the appropriate multiple scatter layer source terms, and, forthe first VLIDORT calculation with the lowest-layer geometry ΩNTOTAL , the surface upwelling↑Stokes vector IBOA( Ω NTOTAL).For N TOTAL layers in the atmosphere, we require N TOTAL separate calls to VLIDORT, and this ismuch more time consuming that a single call with geometry ΩNTOTAL (this would be the default inthe absence of a line-of-sight correction). However, since scattering is strongest near the surface,the first VLIDORT call (with geometryΩNTOTAL ) is the most important as it provides the largest↑scattering source term MNTOTAL( Ω NTOTAL).An even simpler line-of-sight correction is to assume that all multiple scatter source terms aretaken from this first VLIDORT call; in this case, we require only the accurate single scattercalculation to complete I↑ TOA. This approximation is known as the “outgoing” sphericitycorrection; it requires very little extra computational effort compared to a single VLIDORT call.The sphericity correction can also be set up with just two calls to VLIDORT made with the startand finish geometries ΩNTOTAL and Ω1; in this case, multiple scatter source terms at othergeometries are interpolated at all levels between results obtained for the two limiting geometries.In the scalar case, accuracies for all these corrections were investigated in [Spurr, 2003].In VLIDORT 2.0, the facility for generating multiple layer source terms has been dropped, asthere has been little usage. However, the outgoing sphericity correction is important, and a newformulation has been developed for this release. This has been validated against the TOMRADcode and is applicable also to the vector VLIDORT model. We now describe this.2.4.4. A more accurate outgoing sphericity correctionIn this section, the exposition applies to the scalar intensity, but the treatment is the same for theVLIDORT implementation.33