VLIDORT User's Guide

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Note the use of the profile-column mapping derivatives ∂ U p∂ C . Merely adding up the partialcolumn weighting functions is equivalent to assuming that the response of the TOA field tovariations in total ozone is the same for all layers – the profile shape remains the same. Equation(2.125) is the correct formula to account for shape variation.It is perfectly possible to set up VLIDORT to deliver a set of profile Jacobians; a columnweighting function would then be created externally from the sum in Eq. (2.125). This is notvery efficient, since for a 13-layer atmosphere, it requires us to calculate 13 separate profileweighting functions and then sum them. However, a facility was introduced in the scalarLIDORT code Version 2.5 to have LIDORT calculate the column Jacobian directly; in effect, thesummation in Eq. (2.125) is done internally. This is a much more efficient procedure. In this casethe linearized IOP inputs are expressed in terms of the profile-column mapping derivatives:∂Δp∂C= σO 3p( λ∂Up) ;∂C∂ωp∂Cωp= − σΔpO 3p( λ∂Up) . (2.126)∂CThis feature has been retained in all LIDORT Version 3 codes, and the single-scatter corrections(outgoing and nadir), surface treatments and performance enhancements (in particular thelinearization of the reduced BVP problem) have been upgraded to ensure that the columndifferentiation is done internally inside LIDORT if the requisite flag is turned on for columnlinearization. The model will generate either profile or column Jacobians for atmosphericquantities.38
3. The numerical VLIDORT model3.1. Preparation of inputs3.1.1. Basic optical property inputsIn this section, we give a brief introduction to the input requirements for VLIDORT, in particularthe determination of optical property inputs (including linearized quantities). It is already clearthat for a Stokes vector computation using VLIDORT, we require the input set {Δ n , ω n , B nl } foreach layer n, where Δ n is the total optical thickness, ω n the total single scatter albedo, and B nl theset of Greek matrices specifying the total scattering law. The form for B nl is given in Eq. (2.24)in terms of the six Greek constants {α l , β l , γ l , δ l , ε l , ζ l } which must be specified for each momentl of a Legendre function expansion in terms of the cosine of the scattering angle. The values β lare the traditional phase function expansion coefficients, the ones that appear as inputs to thescalar version; they are normalized to 4π.As an example, we consider a medium with Rayleigh scattering by air molecules, some trace gasabsorption, and scattering and extinction by aerosols. Dropping the layer index, if the layerRayleigh scattering optical depth is δ Ray and trace gas absorption optical thickness α gas , with theaerosol extinction and scattering optical depths τ aer and δ aer respectively, then the total opticalproperty inputs are given by:Δ = αgas+ δRay+ τaer ;δ aer+ δ Rayω = ;ΔδB ,+ δ B ,Ray l Ray aer l aerBl=δRay+ δ. (3.1)aerThe Greek matrix coefficients for Rayleigh scattering are given by the following table.α l β l γ l δ l ε l ζ ll=0 0 1 0 0 0 0l=1 0 0 0 3(1− 2ρ)0 02 + ρl=2 6(1 − ρ)(1 − ρ)6(1− ρ)0 0 02 + ρ 2 + ρ−2 + ρFor zero depolarization ratios, the only surviving Greek constants are: β 0 = 1.0, β 2 = 0.5, α 2 =3.0, γ 2 = −√6/2 and δ 1 = 1.5. Aerosol quantities must in general be derived from a suitableparticle scattering model (Mie calculations, T-matrix methods, etc.).We consider a 2-parameter bimodal aerosol optical model with the following combined opticalproperty definitions in terms of the total aerosol number density N and the fractional weighting fbetween the two aerosol modes:Δ aer= Ne aer≡ N[ fe1 + (1 − f ) e2]; (3.2a)aerfz1 e1+ ( 1−f ) z2e2ωaer≡eaere; (3.2b)aer= σ39
Page 1: User’s GuideVLIDORTVersion 2.6Rob
Page 5 and 6: Table of Contents1H1. Introduction
Page 7 and 8: 1. Introduction to VLIDORT1.1. Hist
Page 9 and 10: Table 1.1 Major features of LIDORT
Page 11 and 12: In 2006, R. Spurr was invited to co
Page 13: corrections, and sphericity correct
Page 16 and 17: Matrix Π relates scattering and in
Page 18 and 19: m⎛ P⎞l( μ)0 0 0⎜⎟mmm ⎜ 0
Page 20 and 21: In the following sections, we suppr
Page 22 and 23: of the single scatter albedo ω and
Page 24 and 25: Here T n−1 is the solar beam tran
Page 26 and 27: ~ + ~ ~ (1)~ ~ + ~ ~ (2)~ ~ − ~ 1
Page 28 and 29: The solution proceeds first by the
Page 30 and 31: Linearizations. Derivatives of all
Page 32 and 33: For the plane-parallel case, we hav
Page 34 and 35: One of the features of the above ou
Page 36 and 37: L↑↑ ↑k (cot n −cotn −1)[
Page 40 and 41: βl,aer(1)(2)fz1e1βl+ ( 1−f ) z2
Page 42 and 43: For BRDF input, it is necessary for
Page 44 and 45: streams were used in the half space
Page 46 and 47: anded tri-diagonal matrix A contain
Page 48 and 49: in place to aid with the LU-decompo
Page 50 and 51: In earlier versions of LIDORT and V
Page 53 and 54: 4. The VLIDORT 2.6 package4.1. Over
Page 55 and 56: (discrete ordinates), so that dimen
Page 57 and 58: Table 4.2 Summary of VLIDORT I/O Ty
Page 59 and 60: Table 4.3. Module files in VLIDORT
Page 61 and 62: Finally, modules vlidort_ls_correct
Page 63 and 64: end program main_VLIDORT4.3.2. Conf
Page 65 and 66: $(VLID_DEF_PATH)/vlidort_sup_brdf_d
Page 67 and 68: Finally, the command to build the d
Page 69 and 70: Here, “s” indicates you want to
Page 71 and 72: The main difference between “vlid
Page 73 and 74: to both VLIDORT and the given VSLEA
Page 75 and 76: STATUS_INPUTREAD is equal to 4 (VLI
Page 77 and 78: 5. ReferencesAnderson, E., Z. Bai,
Page 79 and 80: Mishchenko, M.I., and L.D. Travis,
Page 81: Stamnes K., S-C. Tsay, W. Wiscombe,
Page 84 and 85: Table A2: Type Structure VLIDORT_Fi
Page 86 and 87: DO_WRITE_FOURIER Logical (I) Flag f
Page 88 and 89:
USER_LEVELS (o) Real*8 (IO) Array o
Page 90 and 91:
angle s, Stokes parameter S, and di
Page 92 and 93:
DO_SLEAVE_WFS Logical (IO) Flag for
Page 94 and 95:
6.1.1.8. VLIDORT linearized outputs
Page 96 and 97:
NFINELAYERS Integer Number of fine
Page 98 and 99:
6.1.2.4. VLIDORT linearized modifie
Page 100 and 101:
The output file contains (for all 3
Page 102 and 103:
The first call is the baseline calc
Page 104 and 105:
for a 2-parameter Gamma-function si
Page 106 and 107:
Remark. In VLIDORT, the BRDF is a 4
Page 108 and 109:
6.3.3.1. Input and output type stru
Page 110 and 111:
Table C: Type Structure VBRDF_LinSu
Page 112 and 113:
Special note regarding Cox-Munk typ
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squared parameter, so that Jacobian
Page 116 and 117:
6.4. SLEAVE SupplementHere, the sur
Page 118 and 119:
is possible to define Jacobians wit
Page 120 and 121:
etween 0 and 90 degrees.N_USER_OBSG
Page 122:
6.4.4.2 SLEAVE configuration file c
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VLIDORT User's Guide

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