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

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24 Chapter 2 Figure 2.1: Relevant wavenumber bins, associated with wavenumber shifts ∆ν = ν − ν ′ in , for one, two and three orders of scattering (n max = 1, 2, 3). The displayed lines were calculated using Eq. (2.18) for incoming radiation at wavenumber ν ′ in = 25316 cm−1 (λ = 395 nm) and for standard dry air at a temperature of 240 K. The dotted horizontal line represents an example of a threshold of 10 −5 above which the Raman lines are kept.
A doubling-adding method to include multiple orders of rotational Raman scattering 25 with n > n max are not necessarily present in the optimized grid. This small amount, which ends up wrongly, is distributed to neighboring wavenumber bins which are present in the optimized grid. Fig. 2.1 shows the relevance of the Raman lines versus wavenumber shift for n max =1,2,3. Keeping only the wavenumber bins that satisfy the criterion in Eq. (2.17), we end up with a wavenumber grid which involves considerably less wavenumber bins than the original 1 cm −1 grid. This compressed grid is kept fixed throughout the model atmosphere. In the case of multiple layers, the wavenumber grid of the layer with the most significant wavenumber bins is chosen and is applied to the other layers. With the use of this optimized grid, the numerical implementation of the matrix products of the type in Eq. (2.15) becomes feasible. The optimization of the wavenumber grid has a dramatic impact on the calculation method of the reflection and transmission matrices of the whole model atmosphere. Because the optimization of the wavenumber grid is only valid for one particular wavenumber ν in, ′ the radiative transfer problem needs to be chopped up in pieces. Instead of calculating one big matrix containing all the reflection or transmission properties for all (ν,µ;ν ′ ,µ ′ )-combinations, a separate doubling-adding calculation needs to be performed for each incoming wavenumber ν in. ′ In general, the choice of any finite wavenumber grid automatically leads to loss of radiation at the borders of the spectral range of consideration. We propose to put the light that is scattered outside this spectral range into the corresponding elastic scattering component, such that no radiation is lost. This results in a single scattering albedo matrix ω ij = ω(ν i ;ν j ) which has both Cabannes and Raman scattering contributions on its diagonal elements in cases where the wavenumber shift of Raman scattering exceeds the borders of the spectral range. In this way, for each incoming wavenumber ν j , all probabilities ω(ν i ;ν j ) add up to the total scattering probability ω ray (ν j ). The direct result is that energy is not only conserved for single scattering, but also for higher orders of scattering. The doubling-adding model which takes both multiple scattering and energy conservation into account as described above will be referred to as the DA (Doubling-Adding) model in this paper. The single scattering albedo matrix can be modified in such a way that only one order of Raman scattering is allowed, which is very useful in the comparison with other available algorithms. This single scattering albedo matrix, ω DA1,ij = { ω(ν i ;ν j ) for ν j =ν in ′ , ω cab (ν j )δ ij otherwise , (2.19) is constructed by setting all off-diagonal elements to zero except the ones in the column corresponding to ν ′ in. The diagonal solely contains the scattering probability for elastic Cabannes scattering, ω cab (see Appendix). The version of the doubling-adding model which is initialized with ω DA1,ij will be referred to as DA1. This model properly accounts for multiple elastic Cabannes scattering and includes one order of inelastic Raman scattering; multiple inelastic scattering processes are ignored. 2.3.2 Trade-off between accuracy and computational effort Before modeling of a realistic Earth atmosphere can be initiated, appropriate values for n max and ǫ need to be determined. These parameters govern the balance between accuracy and computational
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 29: 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 48 and 49: 42 Chapter 3 contributions of lower
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
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74 Chapter 3 where S l is the expan
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76 Chapter 3 3.C Appendix: Evaluati
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4 Accurate modeling of spectral fin
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Accurate modeling of spectral fine-
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5 Retrieval of cloud properties fro
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Retrieval of cloud properties from
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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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