VLIDORT User's Guide

more algebra (chain rule differentiation, this time not normalizing), we find the followingderivatives:∂Δ= 1∂ ω − ω∂ ; =1ll Ray l∂σΔ ;∂ β β − β=,; (3.6a)∂σNσ+ σσ RayRayRayaerRay∂Δ∂α Gas1= 1;∂ΔNf∂e = ;∂Δ∂e2∂Δ∂z∂Δ∂z21= N= 0 ;= 0;( 1−f );∂ω∂αGasω= −Δ∂ω Nf ( z 1− ω)=∂eΔ1∂ω N( 1−f )( a2− ω)=∂eΔ;2∂ω∂z =21∂ω=∂zNfeΔ1∂βl; = 0∂α; (3.6b);N( 1−f ) eΔ2;;Gasβ=∂efzNσ( β − β )∂ l 1 l , aer1aerβ (1 − f ) z=∂eNσ+ σaerlRay; (3.6c)( β − β )∂ l2 l , aer2β=∂zfeNσ+ σRay( β − β )∂ l 1 l , aer1aerβ (1 − f ) e=∂zNσ+ σaerlRay( β − β )∂ l2 l , aer2+ σRayl; (3.6d); (3.6e)l. (3.6f)3.1.3. Additional atmospheric inputsVLIDORT is a pseudo-spherical model dealing with the attenuation of the solar beam in acurved atmosphere, and it therefore requires some geometrical information. The user needs tosupply the earth’s radius R earth and a height grid {z n } where n = 0, 1, ... NLAYERS (the totalnumber of layers); heights must be specified at layer boundaries with z 0 being the top of theatmosphere. This information is sufficient if the atmosphere is non-refracting.If the atmosphere is refracting, it is necessary to specify pressure and temperature fields {p n } and{t n }, also defined at layer boundaries. The refractive geometry calculation inside VLIDORT isbased on the Born-Wolf approximation for refractive index n(z) as a function of height:n( z)= 1+α0p(z) / t(z). Factor α 0 depends slightly on wavelength, and this must be specified bythe user if refractive bending of the solar beams is desired. To a very good approximation it isequal to 0.000288 multiplied by the air density at standard temperature and pressure. VLIDORThas an internal fine-layering structure to deal with repeated application of Snell’s law. In thisregard, the user must specify the number of fine layers to be used for each coarse layer (10 isusually sufficient).3.1.4. Surface property inputsThe kernel-based BRDF treatment has now been separated from the main VLIDORT code.Calculation of the BRDF kernels and Fourier components of the BRDF is now performed in adedicated BRDF supplement. Thus, VLIDORT now receives total BRDFs and their Fouriercomponents (and if required, the surface-property linearizations of these quantities), withoutknowledge of the individual kernels used to construct these quantities. A brief description of theavailable BRDF kernels and their inputs are given here. For a fuller treatment, consult the BRDFsupplement appendix (section 6.3).41