VLIDORT User's Guide

squared parameter, so that Jacobians for the wind speed can also be determined for this case. It ispossible to calculate higher order contributions, so that R ( Ω,Ω0) = ∑ R Ω Ωs s( ,0) for scatteringorder s, but this is a time-consuming process. For more details see [Jin et al., 2006].In general, one extra order of scattering enhances the BRDFs by 5-15%, depending on thegeometrical configuration; second and higher order contributions are generally at the 1% level orless. We have found that the neglect of multiple glitter reflectances can lead to errors of 1-3% inthe upwelling intensity at the top of the atmosphere, the higher figures being for larger solarzenith angles.In Versions 2.4 and 2.5 of VLIDORT, this multiple-reflected glitter was confined to the directbeam calculation for one extra order of scattering. In version 2.6 of VLIDORT, we havedeveloped a facility for computing multiply-reflected glitter for any order of scattering,applicable to the direct beam BRDF computation as well as the four Fourier component terms.6.3.5. Scalar land surface BRDF kernelsVLIDORT has an implementation of a set of 5 semi-empirical MODIS-type kernels applicable tovegetation canopy [Wanner et al., 1995]; each such kernel must be used in a linear combinationwith a Lambertian kernel. Thus, for example, a Ross-thin BRDF surface type requires acombination of a Ross-thin kernel and a Lambertian kernel:ρtotal( θ , α,φ)= c1 ρRossthin( θ , α,φ)+ c2(6.3.12)Linear factors c 1 and c 2 are interdependent, and are specified in terms of basic quantities of thevegetation canopy. The kernels divide into two groups: those based on volume scatteringempirical models of light reflectance (Ross-thin, Ross-thick), and those based on geometricopticsmodeling (Li-sparse, Li-dense, Roujean). See [Wanner et al., 1995] and [Spurr, 2004] fordetails of the kernel formulae.VLIDORT also has implementations of two other semi-empirical kernels for vegetation cover;these are the Rahman [Rahman et al., 1993] and Hapke [Hapke, 1993] BRDF models. Bothkernels have three nonlinear parameters, and both contain parameterizations of the backscatterhot-spot effect. Here is the Hapke formula:ρhapkeω( μi, μj, φ)=8( μ + μ )⎪⎧⎛ Bh ⎞⎨⎜1+⎟⎪⎩ ⎝ h + tanα⎠i( 2 + cosΘ)j+(1 + 2μ⎪⎫i)(1 + 2μj)−1( )( ) ⎬1+2μ1−ω 1+2μ1−ω ⎪ ⎭ij. (6.3.13)In this equation, the three nonlinear parameters are the single scattering albedo ω, the hotspotamplitude h and the empirical factor B; μ i and μ j are the directional cosines, and Θ is thescattering angle, with α = ½Θ.The important point to note here is that all these kernels are fully differentiable with respect toany of the non-linear parameters defining them. For details of the kernel derivatives, see [Spurr,2004]. It is thus possible to generate analytic weighting functions for a wide range of surfaces in114