VLIDORT User's Guide

2.1.4. Jacobian definitionsAtmospheric Jacobians (also known as weighting functions) are normalized analytic derivativesof the Stokes vector field with respect to any atmospheric property ξ n defined in layer n:∂I(x,μ,φ)Kξ( x,μ,φ)= ξ∂ξ. (2.30)The Fourier series azimuth dependence (c.f. Eq. (2.15)) is also valid:KLM1mmx,μ,φ)= ∑ (2 − δm, 0)C ( φ − φ ) Kξ( x,μ). (2.31)2ξ(0l=mWe use the linearization notation:y∂y= ξ ∂nLp(n)pξ , (2.32)pto indicate the normalized derivative of y n in layer n with respect to variable ξ p in layer p.As noted in section 2.1.3, for the radiation field, input optical properties are {Δ n , ω n , B nl } foreach layer n in a multilayer medium. For Jacobians, we require an additional set of linearizedoptical property inputs { Vn , U n, Z nl} defined with respect to variable ξ n in layer n for which werequire weighting functions. These are:Vn≡ Ln( Δn) ; Un≡ Ln( ωn) ; Znl≡ Ln( Bnl) . (2.33)Δ and itslinearizations{ Vn , U n, Z nl} for a typical atmospheric scenario with molecular and aerosolscattering. One can also define weighting functions with respect to basic optical propertiesthemselves: for example, if ξn= Δn, thenV n≡ Ln( Δn) = Δn .In section 3.2 we give an example of the construction of the input set {n, ωn,Bnl}For surface weighting functions, we need to know how the BRDF matrix operator R in Eq.(2.28) is parameterized. In VLIDORT, we have adopted a 3-kernel BRDF formulation of surfacereflectance similar to the scheme developed in [Spurr, 2003] for LIDORT. In section 2.3, weconfine our attention to the Lambertian case; BRDF implementation is discussed in section 6.3.2.1.5. Solution strategyThe solution strategy has two stages. First, for each layer, we establish discrete ordinate solutionsto the homogeneous RTE (in the absence of sources) and to the RTE with solar source term(section 2.2). Second, we complete the solution by application of boundary conditions and bysource function integration of the RTE in order to establish solutions away from discrete ordinatedirections (section 2.3). In section 2.4, we discuss the pseudo spherical approximation and exactsingle scattering calculations within VLIDORT, and section 2.5 deals with the surface boundarycondition for BRDFs.The complete vector RT solution for a plane-parallel slab was developed by Siewert [Siewert,2000b], and we follow some elements in this formulation. Our description also adheres closely tothe LIDORT treatment, especially concerning this particular integral solution, formulation of theboundary-value problem and linearization methodology.19