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

More documents

Recommendations

Info

88 Chapter 4 4.4 Retrieval of a solar spectrum on a fine spectral grid from the solar measurement In this section we describe the GOME measurements of the Earth radiance and the solar irradiance spectrum by the vectors y ear and y sun , respectively. Both vectors have dimension M and cover the same spectral range 390–400 nm. Each element of these vectors describes the measurement of one particular detector pixel and can be simulated using Eqs. (4.1) and (4.10). Furthermore, the continuous solar spectrum F 0 is described by a vector x of dimension N, which belongs to a discretization of the solar spectrum on a 1 cm −1 wavenumber grid. The forward simulation of GOME earthshine and solar measurements can be described by the linear equations y ear = K ear x + e ear , (4.11) y sun = K sun x + e sun , (4.12) where the M ×N-dimensional kernel matrices K ear and K sun are matrix representations of the corresponding integral kernels k ear and k sun in the Eqs. (4.1) and (4.10). The M-dimensional vectors e ear and e sun contain the measurement error of the Earth radiance and solar irradiance spectrum, respectively. In this section we discuss a retrieval scheme to determine the solar spectrum x from the GOME solar measurements. The retrieved solar spectrum should not be seen as a GOME data product, but as an auxiliary product to enhance the accuracy of GOME Earth radiance simulations, which will be demonstrated at the end of this section. 4.4.1 Inversion scheme The retrieval of the solar spectrum x from the GOME solar irradiance spectrum y sun involves the inversion of Eq. (4.12). This presents an underdetermined problem because the number of measurements M is smaller than the number of parameters N to be retrieved. To gain insight into the inversion of Eq. (4.12) we consider the singular value decomposition (SVD) of the kernel matrix K, viz. K = UΣV T . (4.13) For simplicity, the subscript ‘sun’ is omitted in this subsection. In Eq. (4.13), U is a M × N-matrix with orthonormal columns (u 1 ,...,u M ) spanning the measurement space, V is a N ×N-matrix with orthonormal columns (v 1 ,...,v N ) spanning the state space and Σ is a diagonal matrix containing the singular values (σ 1 ,...,σ N ). The underdetermination of the inversion problem is expressed by the fact that the singular values (σ M+1 ,...,σ N ) are zero. Therefore the estimate of the inversion needs to be written as a truncated sum x est = M∑ n=1 (u T ny) σ n v n . (4.14) The part of the solar spectrum that is described by the vectors v M+1 ,...,v N cannot be retrieved. The vector space spanned by these vectors is referred to as the null-space of the problem, whereas the
Accurate modeling of spectral fine-structure in Earth radiance spectra measured with GOME 89 vectors v 1 ,...,v M describe the row space of the retrieval. The solution (4.14) is also known as the solution of minimum length (see e.g. Menke [1989]) and can be written as x est = K T ( KK T) −1 y , (4.15) which is useful for any numerical implementation. For GOME measurements x est is strongly affected by the measurement error e sun in Eq. (4.12). This is caused by those terms in Eq. (4.14) that belong to small singular values. Here the noise on the measurement y gets amplified and therefore has a significant contribution to x est . However, owing to the spectral smoothing that is part of the forward model of the Earth radiance spectrum in Eq. (4.12), these error contributions are smoothed out and thus have only a minor effect on the corresponding simulations of an Earth radiance spectrum. The minimum length solution in Eq. (4.14) obviously does not contain information of the solar spectrum in the null-space components. Since the null-space components describe the fine spectral structures of the solar spectrum, the retrieved solar spectrum is a smooth version of the highresolution solar spectrum. This smoothing is expressed by the so-called averaging kernel A of the retrieval [Rodgers, 2000]. For the minimum length solution the averaging kernel is given by A = M∑ v i vi T . (4.16) i=1 Thus, the relation between the true solar spectrum and the retrieved spectrum is x = Ax true + e x . (4.17) Here, e x is the error on the retrieved solar spectrum. The averaging kernel A represents the projection of the state vector onto the row space (i.e. the non-null-space) of the inversion problem. On the other hand, (1−A) represents the projection onto the null-space, where 1 is the identity matrix. Figure 4.4 shows the null-space component and the row space component of a solar spectrum retrieved from a simulated GOME solar measurement. Here we used the solar spectrum of Chance and Spurr [1997] as the true high-resolution solar spectrum x true . The figure shows that the null-space part (1−A)x true represents a significant component of the true solar spectrum. The rms of the nullspace component, related to the mean solar spectrum, is 23.4%. By definition, this null-space part is mapped to zero in the forward simulation of the GOME solar irradiance spectrum. 4.4.2 Undersampling error If the solar spectrum is well sampled, the null-space contribution is mapped to zero for any sampling of the slit-averaged spectrum. This is also true for a sampling of the Earth radiance spectrum, which is subject to a similar but slightly stronger spectral smoothing (see Fig. 4.1). However, it is known that the spectral sampling of GOME is too coarse, which results in a model bias for the simulation of the Earth’s radiance spectra of GOME. This error is called the undersampling error of GOME and has to be distinguished from the interpolation error described in Section 4.3.
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 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
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
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
Page 87 and 88: Accurate modeling of spectral fine-
Page 93: Accurate modeling of spectral fine-
Page 103 and 104: 5 Retrieval of cloud properties fro
Page 105 and 106: Retrieval of cloud properties from
Page 125 and 126: References Aben, I., F. Helderman,
Page 127 and 128: References 121 Ellery, A., D. Wynn-
Page 129 and 130: References 123 Lacis, A. A., J. Cho
Page 131 and 132: References 125 Schulz, F., K. Stamn
Page 133 and 134: Summary A spectrum of sunlight that
Page 135 and 136: Summary 129 Earth radiance spectrum
Page 137 and 138: Samenvatting Het door de aardatmosf
Page 139 and 140: Samenvatting 133 met een vector-mod
Page 141: List of publications Peer-reviewed
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?