Abstract
The remote sensing of atmospheric constituents with limb-viewing satellite instruments or with nadir viewing instruments at large solar zenith angles requires a forward model that simulates the backscattered radiance taking the spherical shape of the Earth atmosphere into account. In addition, many retrieval schemes are based on a linearization of such a forward model. Whenever it is important to take multiple scattering into account (e.g. due to light scattering air molecules, aerosols and clouds) the linearization of the measurement simulation with respect to the parameters to be retrieved is not trivial. Here, the forward-adjoint perturbation theory provides a general method to linearize radiative transfer. In the first part of this review chapter we provide the theoretical background of the linearization approach for a radiative transfer problem in a spherical model atmosphere which is illuminated by a collimated solar beam. Using an operator formulation of radiative transfer allows one to express the linearization approach in a universally valid notation. Depending on the particular formulation of the radiative transfer problem the perturbation of internal sources has to be taken into account in addition. The needed adjoint calculation corresponds to a so-called searchlight problem that requires the use of three-dimensional radiative transfer simulations in general. Subsequently we show how symmetries of the forward radiation field and a proper choice of the radiation sources can be used to simplify the needed adjoint calculations substantially.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C. N. Adams and G. W. Kattawar. Radiative transfer in spherical shell atmospheres, i. Rayleigh scattering. Icarus, 35:139–151, 1978.
D. E. Anderson. The troposphere-stratosphere radiation field at twilight: a spherical model. Planet. Space Sci., 31:1517–1523, 1983.
M. Balluch. A new numerical model to compute photolysis rates and solar heating with anisotropic scattering in spherical geometry. Ann. Geophysicae, 14:80–97, 1996.
G. I. Bell and S. Glasstone. Nuclear Reactor Theory. Van Nostrand Reinholt, New York, 1970.
H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede. Sciamachy: Mission objectives and measurement modes. J. Atmos. Sci., 56:127–150, 1999.
M. Box, S. Gerstl, and C. Simmer. Application of the adjoint formulation to the calculation of atmospheric radiative effects. Beitr. Phys. Atmos., 61:303–311, 1988.
M. Box, S. Gerstl, and C. Simmer. Computation of atmospheric radiative effects via perturbation theory. Beitr. Phys. Atmos., 62:193–199, 1989.
M. A. Box, M. Keevers, and B. H. J. McKellar. On the perturbation series for radiative effects. J. Quant. Spectros. Radiat. Transfer, 39:219–223, 1988.
K. M. Case. Transfer problems and the reciprocity principle. Rev. Mod. Phys., 29:651–663, 1957.
K. M. Case and P. F. Zweifel. Linear Transport Theory. Addison-Wesley, Reading, 1967.
D. G. Collins, W. G. Blaettner, M. B. Wells, and H. G. Horak. Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical shell atmospheres. Appl. Opt., 11:2684–2696, 1972.
A. Dahlback and K. Stamnes. A new spherical model for computing the radiation field available for photolysis and heating at twilight. Planet. Space Sci., 39:671–683, 1991.
A. Doicu, T. Trautmann, F. Schreier, S. Slijkhuis, S. Hilgers, A. von Bargen, M. Hess, and B. Aberle. Radiative transfer models for a spherical shell atmosphere. in preparation.
L. Flynn, J. Hornstein, and E. Hilsenrath. The ozone mapping and profiler suite (omps): The next generation of us ozone monitoring instruments. IEEE IGARSS Proceedings, Anchorage, Alaska, 2004.
O. P. Hasekamp and J. Landgraf. A linearized vector radiative transfer model for atmospheric trace gas retrieval. J. Quant. Spectrosc. Radiat. Transfer, 75:221–238, 2002.
O. P. Hasekamp and J. Landgraf. Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing. J. Geophys. Res., D04203:doi:10.1029/2004JD005260, 2005.
O. P. Hasekamp, J. Landgraf, and R. van Oss. The need of polarization modeling for ozone profile retrieval from backscattered sunlight. J. Geophys. Res., 107:4692, 2002.
B. M. Herman, A. Ben-David, and K. J. Thome. Numerical technique for solving the radiative transfer equation for a spherical shell atmosphere. Appl. Opt., 33:1760–1770, 1994.
J. W. Kaiser and J. P. Burrows. Fast weighting functions for retrievals from limb scattering measurements. J. Quant. Spectrosc. Radiat. Transfer, 77:273–283, 2003.
P. Kunasz and L. H. Auer. Short characteristic integration of radiative transfer problems: formal solution in two-dimensional slabs. J. Quant. Spectrosc. Radiat. Transfer, 39:67–79, 1988.
J. Landgraf and O. P. Hasekamp. Ozone profile retrieval from satellite measurements of nadir backscattered light in the ultraviolet of the solar spectrum. Recent Res. Devel. Geophysics, 4:157–189, 2002.
J. Landgraf, O. P. Hasekamp, and T. Trautmann. Linearization of radiative transfer with respect to surface properties. J. Quant. Spectrosc. Radiat. Transfer, 72:327–339, 2002.
J. Landgraf, O. P. Hasekamp, T. Trautmann, and M. A. Box. A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach. J. Geophys. Res., 106:27291–27306, 2001.
D. J. Lary and M. Balluch. Solar heating rates: the importance of spherical geometry. J. Atmos. Sci., 50:3983–3993, 1993.
J. Lenoble and Z. Sekera. Equation of radiative transfer in a planetary spherical atmosphere. Proc. Nat. Acad. Sci., 47:372–389, USA, 1961.
J. Lewins. Importance. The Adjoint Function. The Physical Basis of Variational and Perturbation Theory in Transport and Diffusion Problems. Pergamon Press, New York, 1965.
R. P. Loughman, E. Griffioen, L. Oikarinen, O. V. Postylyakov, A. Rozanov, D. E. Flittner, and D. F. Rault. Comparison of radiative transfer models for limb-viewing scattered sunlight measurements. J. Geophys. Res., 109:D06303, doi:10.1029/2003JD003854, 2004.
G. Marchuk. Equation for the value of information from weather satellites and formulation of inverse problems. Cosmic Res., 2:394–409, 1964.
G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov. The Monte Carlo Methods in Atmospheric Optics. Springer Verlag, Berlin, 1980.
A. Marshak and A. B. Davis. 3D Radiative Transfer in Cloudy Atmospheres. Springer, New York, 2005.
C. A. McLinden, J. C. McConnell, E. Griffioen, and C. T. McElroy. A vector radiativetransfer model for the odin/osiris project. Can. J. Phys., 80:375–393, 2002.
M. I. Mishchenko, A. A. Lacis, and L. D. Travis. Errors induced by the neglect of polarization in radiance calculations for rayleigh-scattering atmospheres. J. Quant. Spectrosc. Radiat. Transfer, 51:491–510, 1994.
M. I. Mishchenko, L. D. Travis, and A. A. Lacis. Scattering, Absorption and Emission of Light by Small Particles. Cambridge University Press, Cambridge, 2002.
National Oceanic and Atmospheric Administration (NOAA). U.S. Standard Atmosphere. Rep. NOAA-S/T76-1562, Washington, D.C., U.S. Gov. Print. Off., 1976.
L. Oikarinen. Effect of surface albedo variations on uv-visible limb-scattering measurements of the atmosphere. J. Geophys. Res., 107:doi:10.1029/2001JD001492, 2002.
L. Oikarinen, E. Sihvola, and E. Kyrola. Multiple scattering in limb viewing geometry. J. Geophys. Res., 104:31261–31274, 1999.
P. T. Partain, A. K. Heidinger, and G. L. Stephens. High spectral resolution atmospheric radiative transfer: Application of the equivalence theorem. J. Geophys. Res., 105:2163–2177, 2000.
W. F. Payne, E. J. Llewellyn, and J. S. Matsushita. Osiris: An imaging spectrograph for odin. 46th International Astronautical Congress, Oslo, Norway, 1995.
P. Phillips. A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach., 9:84–97, 1962.
O. V. Postylyakov. Linearized vector radiative transfer model mcc++ for a spherical atmosphere. J. Quant. Spectrosc. Radiat. Transfer, 88:297–317, 2004.
C. Rodgers. Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation. Rev. Geophys., 14:609–624, 1976.
A. Rozanov, V. Rozanov, and J. P. Burrows. A numerical radiative transfer model for a spherical planetary atmosphere: combined differential-integral approach involving the picard iterative approximation. J. Quant. Spectrosc. Radiat. Transfer, 69:491–512, 2001.
A. Rozanov, V. Rozanov, and J. P. Burrrows. Combined differential-integral approach for the radiation field computation in a spherical shell atmosphere: non-limb geometry. J. Geophys. Res., 105:22937–43, 2000.
V. V. Rozanov, M. Buchwitz, K. U. Eichmann, R. de Beek, and J. P. Burrows. Sciatran – a new radiative transfer model for geophysical applications in the 240–2400nm spectral region: the pseudo-spherical version. Adv. Space Res., 29:1831–1835, 2002.
V. V. Rozanov, T. Kurosu, and J. P. Burrrows. Retrieval of atmospheric constituents in the uv-visible: A new quasi-analytical approach for the calculation of weighting functions. J. Quant. Spectrosc. Radiat. Transfer, 60:277–99, 1998.
K. K. Sen and S. J. Wilson. Radiative Transfer in Curved Media. World Scientific, Singapore, 1990.
O. I. Smokty. Multiple light scattering in the spherical planetary atmosphere. Pure Appl. Geophys., 72:214–226, 1969.
V. Sobolev. Light Scattering in Planetary Atmospheres. Pergamon Press, Oxford, 1975.
V. V. Sobolev and I. N. Minin. Light scattering in the spherical atmosphere. ISZ (artificial earth satellites), 14, 1962.
F. Spada, M. C. Krol, and P. Stammes. Mcscia: Application of the equivalence theorem in a monte carlo radiative transfer model for spherical shell atmospheres. Atmos. Chem. Phys. Discuss., 6:1199–1248, 2006.
A. Tikhonov. On the solution of incorrectly stated problems and a method of regularization. Dokl. Akad. Nauk SSSR, 151:501–504, 1963.
T. Trautmann and M. A. Box. Green’s function computation in radiative transfer theory. In de Groot, R. A., and Nadrchal, J. (Eds), Physics Computing ’92. World Scientific, Singapore, 1993.
E. A. Ustinov. Inverse problem of the photometry of the solar radiation reflected by an optically thick planetary atmosphere. 2. Numerical aspects and requirements on the observation geometry. Cosmic Res., 29:785–800, 1991.
E. A. Ustinov. Inverse problem of the photometry of the solar radiation reflected by an optically thick planetary atmosphere. mathematical framework and weighting functions of linearized inverse problem. Cosmic Res., 29:519–531, 1991.
E. A. Ustinov. Inverse problem of the photometry of the solar radiation reflected by an optically thick planetary atmosphere. 3. remote sensing of minor gaseous constituents and an atmospheric aerosol. Cosmic Res., 30:170–181, 1992.
R. J. van der A. Improved ozone profile retrieval from combined nadir/limb observations of SCIAMACHY. J. Geophys. Res., 106:14583–94, 2001.
H. H. Walter and J. Landgraf. Towards linearization of atmospheric radiative transfer in spherical geometry. J. Quant. Spectrosc. Radiat. Transfer, 92:175–200, 2005.
H. H. Walter, J. Landgraf, and O. P. Hasekamp. Linearization of a pseudo-spherical vector radiative transfer model. J. Quant. Spectrosc. Radiat. Transfer, 85:251–283, 2004.
H. H. Walter, J. Landgraf, F. Spada, and A. Doicu. Linearization of a radiative transfer model in spherical geometry. J. Geophys. Res., 111:doi:10.1029/2005JD007014, 2006.
Q. Yi, M. A. Box, and T. Trautmann. Higher-order radiative perturbation theory. J. Quant. Spectrosc. Radiat. Transfer, 84:105–114, 2004.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Walter, H.H., Landgraf, J. (2010). Linearization of radiative transfer in spherical geometry: an application of the forward-adjoint perturbation theory. In: Kokhanovsky, A. (eds) Light Scattering Reviews 5. Springer Praxis Books(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10336-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-10336-0_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10335-3
Online ISBN: 978-3-642-10336-0
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)