VLIDORT User's Guide

βl,aer(1)(2)fz1e1βl+ ( 1−f ) z2e2βl= . (3.2c)σaerThe quantity σ aer is the combined scattering coefficient and e aer the combined extinctioncoefficient. In Eq. (3.2c) we have given the combined expression for just one of the Greekconstants; the other 5 are constructed in a similar fashion. Thus, the quantity β l,Aer is the l-thcoefficient in the Legendre polynomial expansion of the phase function. Here, e 1 , z 1 and arethe extinction coefficient, single scatter albedo and Legendre expansion coefficient for aerosoltype 1; similar definitions apply to aerosol type 2.3.1.2. Linearized optical property inputsFor the linearized inputs with respect to a parameter ξ for which we require weighting functions,we define normalized quantities:ξ ∂Δξ ∂ωξ ∂Blφ ξ=Δ ∂ξ; ϕ ξ=ω ∂ξ; Ψl , ξ=B ∂ξ. (3.3)lThese may be established by differentiating the definitions in Eq. (3.1). We give one examplehere: if there is a single absorbing gas (ozone, for example), with C the partial column of tracegas in any given layer, and σ gas the absorption coefficient, then we have Δ = Cσ gas+ δRay+ τaerin the above equations. For trace gas profile Jacobians, we require the derivatives in Eq. (3.3) asinputs, taken with respect to C. These are:C ∂ΔCσgasφC≡ = ;Δ ∂CΔC ∂ωCσgas ξ ∂BlϕC≡ = − ; Ψl, ξ≡ = 0ω ∂CΔB ∂ξ. (3.4)Jacobian parameters may be elements of the retrieval state vector, or they may be sensitivityparameters which are not retrieved but will be sources of error in the retrieval. As anotherexample (keeping to the notation used for the above bi-modal aerosol model), we will assumethat the retrieval parameters are the total aerosol density N and the bimodal ratio f. All otherquantities in the above definitions are sensitivity parameters.For the retrieval Jacobians (with respect to N and f) the relevant inputs are found by partialdifferentiation of the definitions in Eq. (3.1). After some algebra, one finds (we have justconsidered one for the Greek-matrix elements for simplicity):∂Δ∂ΔN = N∂N∂N∂ωN =∂NβN =∂NNσaeraerNσNσ= Δ− ωΔaerΔ aer∂ l aer l , aer∂Δ∂Δaer; f = f = fN( e 1− e2) ; (3.5a)∂f∂f;( β − β )aer+ σRayl;∂ωf =∂fffNl[( z e − z e ) − ω( e −2)]1 1 2 2 1eaerΔ( β − β )∂ β fN(z e − z el1 1 2 2)l , aer∂f=Nσ+ σRaylβ(1)l; (3.5b). (3.5c)For sensitivity Jacobians, the quantities σ Ray , α gas , e 1 , z 1 , e 2 and z 2 are all bulk property modelparameters that are potentially sources of error. [We can also consider the phase functionquantities γ Ray , γ 1 and γ 2 as sensitivity parameters, but the results are not shown here]. After a lot40