Rotational Raman scattering in the Earth's atmosphere ... - SRON

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42 Chapter 3 contributions of lower order. In this work, we present a vector radiative transfer model, where inelastic Raman scattering is taken into account using the radiative transfer perturbation theory approach. Similar to the work of Vountas et al. [1998], the vector radiative transfer equation is separated in an unperturbed radiative transfer problem describing radiative transfer in a Rayleigh scattering atmosphere, and a perturbation term for additional Raman scattering (Section 3.2). In Section 3.3, the effect of this perturbation on the measurement simulation is described by a classical perturbation series using the Green’s function formalism. In this approach the n-th order perturbation term can be interpreted as a correction to the simulated measurement for n orders of Raman scattering. To first order the numerical implementation of the perturbation can be eased significantly by the adjoint formulation of radiative transfer, which is also discussed in Section 3.3. Section 3.4 shows numerical simulations for the Stokes parameters I, Q and U of the reflected sunlight. Here the effect of one order of Raman scattering on the polarization components is considered in detail. Furthermore, the accuracy of both the single scattering approximation and the scalar approach of radiative transfer is studied for the simulation of Ring structures in the reflectance spectra. Finally, in Section 3.5 we investigate the suitability of the single scattering and scalar radiative transfer approximation for the simulation of Ring structures in polarization sensitive measurements of GOME within the error margins of the instrument. Here, we focus on both the spectral range 290–313 nm, which contains information on the vertical ozone distribution in the probed atmosphere, and the filling-in of the Ca II Fraunhofer lines, which can be used to retrieve cloud properties. 3.2 The radiative transfer problem 3.2.1 The radiative transfer equation including both elastic Cabannes scattering and inelastic Raman scattering The radiance and state of polarization of light can be described by an intensity vector I, which has the Stokes parameters I, Q, U, and V as its components (see e.g. Chandrasekhar [1960], Hovenier and van der Mee [1983]) I = [I,Q,U,V ] T . (3.1) Here, the superscript T indicates the transposed vector. The intensity vector is defined with respect to a certain reference plane, which in this paper is given by the local meridian plane. In general, the intensity vector is a function of altitude z, of wavelength λ and of direction Ω = (µ,ϕ), where ϕ is the azimuthal angle measured clockwise when looking downward and µ is the cosine of the zenith angle (µ < 0 for downward directions and µ > 0 for upward directions). The domain of these variables define the so-called phase space of the radiative transfer problem. In the following, we consider a plane-parallel, macroscopically isotropic atmosphere bounded from below by a Lambertian reflecting surface. For such an atmosphere the equation of transfer for
A vector radiative transfer model using the perturbation theory approach 43 polarized light is given in its forward formulation by ˆLI = S, (3.2) with the transport operator [Bell and Glasstone, 1970, Marchuk, 1964, Box et al., 1989, Hasekamp and Landgraf , 2001] ∫ ∞ ∫ [ { [µ ∂ ˆL = d˜λ d˜Ω δ(λ−˜λ) 0 4π ∂z + β ext(z,λ) ] δ(Ω− ˜Ω)E − A(˜λ) } ] π δ(z) Θ(µ) |µ| Θ(−˜µ) |˜µ| − J(z, ˜Ω, ˜λ|z,Ω,λ) ◦ . (3.3) Here d˜Ω=d˜µd˜ϕ, E is the 4 × 4 unity matrix, Θ represents the Heaviside step function, and δ is the Dirac-delta with δ(Ω− ˜Ω) = δ(µ− ˜µ)δ(ϕ− ˜ϕ). β ext is the extinction coefficient and the symbol ’◦’ indicates that ˆL is an integral operator acting on a function to its right. The first term of the radiative transfer equation describes the extinction of light for a wavelength λ in a direction Ω and the second term represents the isotropic reflection on a Lambertian surface, where the matrix A(˜λ) is given by [ ] A(˜λ) = diag A(˜λ), 0, 0, 0 (3.4) with Lambertian albedo A(˜λ). Finally, J(z, ˜Ω, ˜λ|z,Ω,λ) defines the source function at height z for scattered light from direction ˜Ω to Ω with a change in wavelength from ˜λ to λ. To simplify matters we assume a model atmosphere consisting of N 2 , O 2 and Ar molecules only. For Cabannes and rotational Raman scattering by an ensemble of randomly oriented molecules J is described by J(z, ˜Ω, ˜λ|z,Ω,λ) = ∑ { χ N δ(λ−˜λ) βcab scat,N (z, ˜λ) Z cab N (˜λ, 4π ˜Ω,Ω) N= N 2 ,O 2 ,Ar + βram scat,N (z, ˜λ,λ) Z ram (˜Ω,Ω) 4π } . (3.5) Here βscat,N cab and βram scat,N represent the scattering coefficients of N 2, O 2 , and Ar for Cabannes and rotational Raman scattering, respectively, with corresponding scattering phase matrices Z cab N and Zram . Here, χ N is the volume mixing ratio which for standard dry air is 0.7809, 0.2095, 0.0093 for N 2 , O 2 , Ar, respectively. Notice that Ar as an inert gas has an isotropic polarizability and thus rotational Raman scattering on Ar does not exist, βscat,Ar ram = 0. The elastic feature of Cabannes scattering is represented by the Dirac-delta δ(λ−˜λ), whereas the change in wavelength from ˜λ to λ due to inelastic Raman scattering is described by the scattering coefficient βscat,N ram (z, ˜λ,λ). The scattering source function can be written in a more compact form, viz. J(z, ˜Ω, ˜λ|z,Ω,λ) = δ(λ−˜λ) βcab scat(z, ˜λ) Z cab (˜λ, 4π (3.6) + βram scat(z, ˜λ,λ) Z ram (˜Ω,Ω) , 4π (3.7)
Page 1 and 2: VRIJE UNIVERSITEIT Rotational Raman
Page 5 and 6: Contents 1 Introduction 1 1.1 Obser
Page 7 and 8: 1 Introduction 1.1 Observing skylig
Page 9 and 10: Introduction 3 Extracting the wealt
Page 11 and 12: Introduction 5 and Spurr, 1997, Vou
Page 13 and 14: Introduction 7 energy: E rot (v,J)
Page 15 and 16: Introduction 9 Figure 1.4: Probabil
Page 17 and 18: Introduction 11 scattered radiance
Page 19 and 20: Introduction 13 approximation in wh
Page 21 and 22: Introduction 15 scattering have bee
Page 23 and 24: 2 A doubling-adding method to inclu
Page 25 and 26: A doubling-adding method to include
Page 43: A doubling-adding method to include
Page 46 and 47: 40 Chapter 3 nm with a spectral res
Page 50 and 51: 44 Chapter 3 if we use the effectiv
Page 52 and 53: 46 Chapter 3 This in turn results i
Page 54 and 55: 48 Chapter 3 This may be shown in a
Page 56 and 57: 50 Chapter 3 The forward and adjoin
Page 58 and 59: 52 Chapter 3 Here, ∆ = diag [1, 1
Page 60 and 61: 54 Chapter 3 Figure 3.3: Relative R
Page 62 and 63: 56 Chapter 3 biases the simulation
Page 64 and 65: 58 Chapter 3 3.4.2 Approximation me
Page 66 and 67: 60 Chapter 3 where I ram,app and I
Page 68 and 69: 62 Chapter 3 3.5 Simulation of pola
Page 70 and 71: 64 Chapter 3 Figure 3.10: Relative
Page 72 and 73: 66 Chapter 3 dependence on the vert
Page 74 and 75: 68 Chapter 3 Figure 3.13: Same as F
Page 76 and 77: 70 Chapter 3 3.6 Summary A vector r
Page 78 and 79: 72 Chapter 3 where λ is the wavele
Page 80 and 81: 74 Chapter 3 where S l is the expan
Page 82 and 83: 76 Chapter 3 3.C Appendix: Evaluati
Page 85 and 86: 4 Accurate modeling of spectral fin
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Accurate modeling of spectral fine-
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Accurate modeling of spectral fine-
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5 Retrieval of cloud properties fro
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References Aben, I., F. Helderman,
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References 121 Ellery, A., D. Wynn-
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References 123 Lacis, A. A., J. Cho
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References 125 Schulz, F., K. Stamn
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Summary A spectrum of sunlight that
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Summary 129 Earth radiance spectrum
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Samenvatting Het door de aardatmosf
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Samenvatting 133 met een vector-mod
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