VLIDORT User's Guide

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L↑↑ ↑k (cot n −cotn −1)[−1] = { [ ] + (cot − cot−1)} n θ θpInLpInInknδnpθnθne− kn− k Hn( 1−knen−kcotθncotθn−1n−1⎡⎢k⎢⎣) Hθnθnn∫en−1θn−kncot θn−1θ∫n−136An( θ ) edθ2sin θkncotkncotθθnAn( θ ) e cotθLdθ+ ∫ dθ2sin θθθn −1p[ A ( θ )] ensin2θkncot θ⎤⎥ . (2.116)⎥⎦The only new quantity here is L [ A ( θ p n)] in the final integral. To evaluate this, we note that theattenuation of the solar beam to a point z with zenith angle in layer n can be written as:⎡NL⎤A n( θ ) = exp⎢−d np( θ ) ε ⎥p . (2.117)⎢⎣⎥∑= p 1 ⎦Here, dnp(θ ) are geometrical distances independent of the optical properties. The total numberof layers in the atmosphere is NL. It follows that:Lp[ An( θ )] = −dnp( θ ) εpAn( θ ) . (2.118)All integrals in the last line of Eq. (2.116) can again be done accurately using trapezium rulesummations. The linearization was checked using finite difference estimates.For most geometrical situations, d ( θ ) = np0 for all layers p > n; this corresponds to points on theline of sight that are illuminated from above. In this case, the attenuation does not depend onextinction coefficients in layers below n, and hence L [ A p n( θ )] = 0 for p > n. However, there aresituations (near the top of the atmosphere for a wide off-nadir viewing path and a high solarzenith angle) in which some points along the line-of-sight are illuminated by direct sunlightcoming from below the horizontal. In this case, the solar path has gone through a tangent heightin the atmosphere, and dnp(θ ) is not necessarily zero for p >n.For a differentiation with respect to other optical properties, the situation is simpler. Definingnow a linearization:Lq qσ ; (2.119)q= σ ∂∂with respect to the scattering coefficient σq in layer q, the only non-vanishing term arising in thelinearization of (2.114) is Lq[Hn] = Hnδnq , so that we have:nkncot θ↑↑ kn(cot θ n −cotθ n −1)−kncot θ n −1Lq[ In−1]= Lq[In] e− δnqknHnedθ2 . (2.120)NOTE. VLIDORT as a single-scatter RT modelθθ∫n −1A(θ ) esin θFollowing user feedback from a number of individuals, VLIDORT (and LIDORT) have nowbeen given a stand-alone single-scatter capability. This is controlled by a single flag, which if set,ensures that the multiple scatter calculation is avoided and the model returns only the singlyscattered radiances and Jacobians. The above single scatter corrections then apply foratmospheric scattered light; and it is only necessary to include for the upwelling field the
transmitted direct-beam reflectance. We also note that an additional delta-M scaling procedurehas been optionally included for the single scatter computation. When the model is running withmultiple scatter included, then this additional scaling can be applied on top of the usual delta-Mscaling applied to the truncated phase function in the diffuse field equations; in this case theNakajima-Tanaka TMS procedure still applies. With VLIDORT in single-scatter-only mode, theoriginal diffuse-field delta-M scaling is not required, though the additional truncation can still beapplied if desired.2.5. Bulk (total column) atmospheric Jacobians.One of the most important applications for VLIDORT has been in forward model simulations forozone profile and total column retrieval. The use of total column as a proxy for the ozone profilewas recognized a number of years ago by scientists at NASA, and column-classified ozoneprofile climatologies were created for the TOMS Version 7 [Wellemayer et al., 1997] and morerecently for Version 8 retrieval algorithms [Bhartia, 2003]. If the profile is represented as a set{ Uj } of partial columns in Dobson Units [DU], then the total column (also in [DU]) isC = ∑U j . For two adjacent TOMS profiles { (1)(2)Ujj} and { Uj } with total columns C (1) andC (2) we define an intermediate profile with column amount C according to:(1)(2)⎛ C − C ⎞(2)⎛ C − C (1)Uj( C)U(2) (1) jU(2) (1) jC C C C⎟ ⎞= ⎜⎟ +⎜. (2.121)⎝ − ⎠ ⎝ − ⎠This defines the profile-column map; it is linear in C. Total column weighting functions arerelated to profile Jacobians by means of chain rule differentiation and the partial derivative:∂Uj( C)U=∂CC(2)j(2)−U− C(1)j(1). (2.122)This map allows us to interpolate smoothly between profile entries in the climatology. In effect,we are drawing on an ensemble of possible profiles of which the climatology is a sample. Othermaps are possible. TOMS Version 8 profiles are specified for 18 latitude bands from pole to pole(10° intervals), and for each month of the year.Suppose now we have a Rayleigh atmosphere with Rayleigh scattering cross-section σ Ray (λ), aircolumn density D p in layer p, ozone partial columns U p , and temperature-dependent ozone crosssections σ O 3p( λ); then the bulk property IOPs are:RayOΔp= σ ( λ)Dp+ σ 3p( λ)Up ;37σ(λ) DRayωp=p;Δp(2.123)Differentiating (2.122) with respect to U p gives the linearized IOP inputs for the profile Jacobian:∂ΔpU =p(σ O 3pλ) ;∂ωpωp∂Δ= −∂U Δ ∂UFinally, we compute the column Jacobian using the chain rule:Kcolnpppp. (2.124)∂I(0, Ω)∂I(0,Ω)∂Up( Ω ) =. (2.125)∂C∂U∂C≡ ∑=1 pp
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 38 and 39: Note the use of the profile-column
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
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NFINELAYERS Integer Number of fine
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6.1.2.4. VLIDORT linearized modifie
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The output file contains (for all 3
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The first call is the baseline calc
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for a 2-parameter Gamma-function si
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Remark. In VLIDORT, the BRDF is a 4
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6.3.3.1. Input and output type stru
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Table C: Type Structure VBRDF_LinSu
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Special note regarding Cox-Munk typ
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squared parameter, so that Jacobian
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6.4. SLEAVE SupplementHere, the sur
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is possible to define Jacobians wit
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etween 0 and 90 degrees.N_USER_OBSG
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6.4.4.2 SLEAVE configuration file c
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