VLIDORT User's Guide

For the plane-parallel case, we have:VkTnLk[ λ n] = 0 ( ∀ k,∀n); Lk[ Tn] = − ( k < n); k[ ] = 0μL T n ( k ≥ n). (2.100)2.4.2. Exact single scatter solutions0In VLIDORT, we include an exact single-scatter computation based on the Nakajima-Tanaka(NT) procedure [Nakajima and Tanaka, 1988]. The internal single scatter computation inVLIDORT will use a truncated subset of the complete scatter-matrix information, the number ofusable Legendre coefficient matrices B l being limited to 2N −1 for N discrete ordinate streams.A more accurate computation results when the post-processing calculation of the truncated single− −scatter contribution (the term Qn( μ)En( x,μ)in Eq. (2.86) for example) is suppressed in favor ofan accurate single scatter computation, which uses the complete phase function. This is the socalled TMS procedure [Nakajima and Tanaka, 1988]. This N-T correction procedure appears inthe DISORT Version 2.0 [Stamnes et al., 2000] and LIDORT [Spurr, 2002] codes. A relatedcomputation has been implemented for the doubling-adding method [Stammes et al., 1989].The (upwelling) post-processed solution in stream direction μ is now written (c.f. Eq. (2.86)):I−nQ−−−( Z ( μ)+ Q ( μ)) ( x,μ)−−(Δ−x) / μ −( x,μ)= In( Δ , μ)e + Hn( x,μ)+nn,exactEn, (2.101)ω( I−nn, exactμ)=Πn( μ,μ0,φ − φ0)4π(1 − ωnfn)320. (2.102)Note the presence of in the denominator of the expression ( 1− ω nf )n which is required when thedelta-M approximation is in force; f n is the truncation factor (see section 3.4.1). From section2.1.1, Πn is obtained from the scattering matrix F n (Θ) through application of rotation matrices.There is no truncation: Πn can be constructed to any degree of accuracy using all availableunscaled Greek matrices B nl .Linearization. Chain-rule differentiation of Eq. (2.102) yields the linearization of the exact single−scatter correction term. Linearization of the multiplier En( x,μ)has already been established.Since the elements of Πn consist of linear combinations of Bnl , the linearization Ln( Πn) isstraightforward to write down in terms of the inputs Ln( Bnl) .2.4.3. Sphericity along the line-of-sightFor nadir-geometry satellite instruments with wide-angle off-nadir viewing, one must considerthe Earth’s curvature along the line of sight from the ground to the satellite. This applies toinstruments such as OMI on the Aura platform (swath 2600 km, scan angle 114° at the satellite)[Stammes et al., 1999] and GOME-2 (swath 1920 km) [EPS/METOP, 1999]. Failure to accountfor this effect can lead to errors of 5-10% in the satellite radiance for TOA viewing zenith anglesin the range 55-70° [Spurr, 2003; Rozanov et al., 2000; Caudill et al., 1997]. For LIDORT, asimple correction for this effect was introduced for satellite geometries in [Spurr, 2003].Correction involves an exact single scatter calculation along the line of sight from ground toTOA: in this case, Eq. (2.102) is still valid, but now the geometry is changing from layer tolayer. The same correction has been adopted for VLIDORT.