Skip to main content
SpringerLink
Log in

LIDORT and VLIDORT: Linearized pseudo-spherical scalar and vector discrete ordinate radiative transfer models for use in remote sensing retrieval problems

  • Chapter
Book cover Light Scattering Reviews 3

Part of the book series: Springer Praxis Books ((ENVIRONSCI))

Abstract

The modern treatment of the radiative transfer equation (RTE) in plane-parallel media dates back to the pioneering work by Ambartsumian and Chandrasekhar in the 1940s (Chandrasekhar, 1960;Ambartsumian, 1961). Using a formulation in terms of the Stokes vector for polarized light, Chandrasekhar was able to solve completely the polarization problem for an atmosphere with Rayleigh scattering, and benchmark calculations from the 1950s are still appropriate today (Coulson et al., 1960). The scalar (intensity-only) and vector (with polarization) radiative transfer equations in one vertical dimension may be solved in a number of ways. These include the doubling-adding method, the discrete ordinates approach, the successive orders of scattering method, Gauss-Seidel iteration, and (not least) the Monte Carlo approach. For a review of solution methods, see for example (Lenoble, 1985). Most solution methods for scalar and vector RTEs divide into two camps: the doubling/adding approach and the discrete ordinate method. For descriptions of the former, see for example (Hansen and Travis, 1974;de Haan et al., 1987;Hovenier et al., 2004). The well-known scalar DISORT discrete ordinate model was developed in the 1980s and released for general use in plane-parallel multi-layer multiple scattering media (Stamnes et al., 1988a); this was extended to the vector model VDISORT in the 1990s (Schulz and Stamnes, 2000).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ambartsumian, V. A., 1961: Scientific Papers, vol. 1, Armenian Academy of Sciences, Erevan.

    Google Scholar 

  • Anderson, E., Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, 1995: LAPACK User’s Guide, 2nd edition, Society for Industrial and Applied Mathematics, Philadephia.

    Google Scholar 

  • Barichello, L., R. Garcia, C. Siewert, 2000: Particular solutions for the discrete-ordinates method. J. Quant. Spectrosc. Radiat. Transfer, 64, 219–226.

    Article  CAS  Google Scholar 

  • Bovensmann, H., J. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. Rozanov, K. Chance, and A. Goede, 1999: SCIAMACHY: Mission objectives and measurement modes. J. Atmos. Sci., 56, 127–150.

    Article  Google Scholar 

  • Callies J., E. Corpaccioli, M. Eisinger, A. Hahne, and A. Lefebvre, 2000: GOME-2-Metop’s second generation sensor for operational ozone monitoring, ESA Bulletin, 102.

    Google Scholar 

  • Caudill T.R., D.E. Flittner, B.M. Herman, O. Torres, R.D. McPeters, 1997: Evaluation of the pseudo-spherical approximation for backscattered ultraviolet radiances and ozone retrieval. J. Geophys. Res., 102, 3881–3890.

    Article  CAS  Google Scholar 

  • Chami M., R. Santer, E. Dilligeard, 2001: Radiative transfer model for the computation of radiance and polarization in an ocean-atmosphere system: polarization properties of suspended matter for remote sensing. Applied Optics, 40, 2398–2416.

    Article  CAS  Google Scholar 

  • Chandrasekhar, S., 1960: Radiative Transfer. Dover Publications, New York.

    Google Scholar 

  • Christi M.J., and G.L. Stephens, 2004: Retrieving profiles of atmospheric CO2 in clear sky and in the presence of thin cloud using spectroscopy from the near and thermal infrared: a preliminary case study. J. Geophys. Res., 109, D04316, doi: 10.1029/2003JD004058.

    Article  CAS  Google Scholar 

  • Coulson K., J. Dave, and D. Sekera, 1960: Tables related to radiation emerging from planetary atmosphere with Rayleigh scattering. University of California Press, Berkeley.

    Google Scholar 

  • Crisp D., R.M. Atlas, F-M. Breon, L.R. Brown, J.P. Burrows, P. Ciais, B.J. Connor, S.C. Doney, I.Y. Fung, D.J. Jacob, C.E. Miller, D. O’Brien, S. Pawson, J.T. Randerson, P. Rayner, R.J. Salawitch, S.P. Sander, B. Sen, G.L. Stephens, P.P. Tans, G.C. Toon, P.O. Wennberg, S.C. Wofsy, Y.L. Yung, Z. Kuang, B. Chudasama, G. Sprague, B. Weiss, R. Pollock, D. Kenyon, and S. Schroll, 2004: The Orbiting Carbon Observatory (OCO) mission. Adv. Space Res., 34, 700.

    Article  CAS  Google Scholar 

  • Cox, C., W. Munk, 1954a: Statistics of the sea surface derived from sun glitter. J. Mar. Res., 13, 198–227.

    Google Scholar 

  • Cox, C., W. Munk, 1954b: Measurement of the roughness of the sea surface from photographs of the sun’s glitter. J. Opt. Soc. Am., 44, 838–850.

    Google Scholar 

  • Dahlback A., and K. Stamnes, 1991: A new spherical model for computing the radiation field available for photolysis and heating at twilight. Planet. Space Sci., 39, 671.

    Article  Google Scholar 

  • de Rooij, W.A., and C.C.A.H. van der Stap, 1984: Expansion of Mie scattering matrices in generalized spherical functions. Astron. Astrophys., 131, 237–248.

    Google Scholar 

  • de Haan, J.F., P.B. Bosma, and J.W. Hovenier. 1987: The adding method for multiple scattering of polarized light. Astron. Astrophys., 183, 371–391.

    Google Scholar 

  • Deuze, J.L., P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, and D. Tanre, 2000: Estimate of the aerosol properties over the ocean with POLDER. J. Geophys. Res., 105, 15329.

    Article  CAS  Google Scholar 

  • ESA, 1995: GOME Global Ozone Monitoring Experiment Users Manual, ed. F. Bednarz. ESA SP-1182.

    Google Scholar 

  • Garcia, R.D.M., and C.E. Siewert, 1986: A generalized spherical harmonics solution for radiative transfer models that include polarization effects. J. Quant. Spectrosc. Radiat. Transfer, 36, 401–423.

    Article  CAS  Google Scholar 

  • Garcia, R.D.M, and C.E. Siewert, 1989: The Fn method for radiative transfer models that include polarization. J. Quant. Spectrosc. Radiat. Transfer, 41, 117–145.

    Article  CAS  Google Scholar 

  • Grainger, R.G., J. Lucas, G.E. Thomas, and G.B.L. Ewen, 2004: Calculation of Mie Derivatives, Applied Optics, 43(28), 5386–5393.

    Article  Google Scholar 

  • Hansen, J.E., and L.D. Travis, 1974: Light scattering in planetary atmospheres. Space Sci. Rev., 16, 527–610.

    Article  Google Scholar 

  • Hapke, B., 1993: Theory of Reflectance and Emittance Spectroscopy. Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Hasekamp, O.P., and J. Landgraf, 2001: Ozone profile retrieval from backscattered ultraviolet radiances: the inverse problem solved by regularization. J. Geophys. Res., 106,D8, 8077–88.

    Article  CAS  Google Scholar 

  • Hasekamp, O.P., and J. Landgraf, 2002: A linearized vector radiative transfer model for atmospheric trace gas retrieval. J. Quant. Spectrosc. Radiat. Transfer, 75, 221–238.

    Article  CAS  Google Scholar 

  • Hasekamp, O.P., and J. Landgraf, 2005a: Retrieval of aerosol properties over the ocean from multispectral single-viewing-angle measurements of intensity and polarization: Retrieval approach, information content, and sensitivity study. J. Geophys. Res., 110, D20207, doi:10.1029/2005JD006212.

    Article  Google Scholar 

  • Hasekamp, O.P., and J. Landgraf, 2005b Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing. J. Geophys. Res., 110, D04203, doi:10.1029/2004JD005260.

    Article  Google Scholar 

  • Hasekamp, O., J. Landgraf, and R. van Oss, 2002: The need of polarization monitoring for ozone profile retrieval from backscattered sunlight. J. Geophys. Res., 107, 4692.

    Article  CAS  Google Scholar 

  • Hoogen, R., V.V. Rozanov, and J.P. Burrows, 1999: Ozone profiles from GOME satellite data: Algorithm description and first validation, J. Geophys. Res., 104, 8263–8280.

    Article  CAS  Google Scholar 

  • Hovenier, J.W., C. van der Mee, and H. Domke, 2004: Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods. Kluwer, Dordrecht.

    Google Scholar 

  • Jiang, Y., X. Jiang, R-L. Shia, S.P. Sander, and Y.L. Yung, 2003: Polarization study of the O2 A-band and its application to the retrieval of O2 column abundance. EOS Trans. Am. Geophys. Union, 84, 255.

    Google Scholar 

  • Lacis, A., J. Chowdhary, M. Mishchenko, and B. Cairns, 1998: Modeling errors in diffuse sky radiance: vector vs. scalar treatment. Geophys. Res. Lett., 25, 135–138.

    Article  Google Scholar 

  • Landgraf, J., O. Hasekamp, T. Trautmann, and M. Box, 2001: A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach. J. Geophys. Res., 106, 27291–27306.

    Article  CAS  Google Scholar 

  • Landgraf, J., O. Hasekamp, and T. Trautmann, 2002: Linearization of radiative transfer with respect to surface properties. J. Quant. Spectrosc. Radiat. Transfer, 72, 327–339.

    Article  CAS  Google Scholar 

  • Lenoble, J. (Ed.), 1985: Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. Deepak Publishing, Hampton, VA.

    Google Scholar 

  • Levelt P.F., G.H.J. van den Oord, M.R. Dobber, A. Mälkki, J. de Vries, P. Stammes, J.O.V. Lundell, and H. Saari, 2006: The Ozone Monitoring Instrument, IEEE Transact. Geoscience Remote Sensing, 44, 1093–1101.

    Article  Google Scholar 

  • Li, W., K. Stamnes, R. Spurr, and J. Stamnes, 2007: Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach. II. Case Study for Santa Barbara Channel.

    Google Scholar 

  • Liu, X., K. Chance, C.E. Sioris, R.J.D. Spurr, T.P. Kurosu, R.V. Martin, and M.J. Newchurch, 2005: Ozone profile and tropospheric ozone retrievals from the global ozone monitoring experiment: Algorithm description and validation. J. Geophys. Res., 110, D20307, doi:10.10.29/2005JD006240.

    Article  Google Scholar 

  • Mackowski, D.W., and M.I. Mishchenko, 1996: Calculation of the T matrix and the scattering matrix for ensembles of spheres. J. Opt. Soc. Am. A, 13, 2266–2278.

    Article  Google Scholar 

  • Mishchenko, M.I., 2002: Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics. Applied Optics, 41, 7114–7135.

    Google Scholar 

  • Mishchenko, M.I., and L.D. Travis, 1997: Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight. J. Geophys. Res., 102, 16989.

    Article  Google Scholar 

  • Mishchenko, M., A. Lacis, and L. Travis, 1994: Errors induced by the neglect of polarization in radiance calculations for Rayleigh scattering atmospheres. J. Quant. Spectrosc. Radiat. Transfer, 51, 491–510.

    Article  Google Scholar 

  • Mishchenko, M.I., B. Cairns, J.E. Hansen, L.D. Travis, R. Burg, Y.J. Kaufman, J.V. Martins, and E.P. Shettle, 2004: Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements. J. Quant. Spectrosc. Radiat. Transfer, 88, 149–161.

    Article  CAS  Google Scholar 

  • Mishchenko, M.I., L.D. Travis, and A.A. Lacis, 2006: Scattering, Absorption and Emission of Light by small particles. Cambridge University Press, Cambridge, U.K.

    Google Scholar 

  • Munro, R., R. Siddans, W. J. Reburn, and B. J. Kerridge, 1998: Direct measurements of tropospheric ozone from space, Nature, 392, 168–171.

    Article  CAS  Google Scholar 

  • Nakajima, and,T. M. Tanaka, 1998: Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation. J. Quant. Spectrosc. Radiat. Transfer, 40, 51–69.

    Article  Google Scholar 

  • Natraj, V., and R. J. D. Spurr, 2007: A linearized approximately spherical two orders of scattering model to account for polarization in vertically inhomogeneous scattering-absorbing media, J. Quant. Spectrosc. Radiat. Transfer, in press.

    Google Scholar 

  • Natraj, V., R. Spurr, H. Boesch, Y. Jiang, and Y.L. Yung, 2007: Evaluation of errors from neglecting polarization in the forward modeling of O2 a band measurements from space, with relevance to the CO2 column retrieval from polarization-sensitive instruments. J. Quant. Spectrosc. Radiat. Transfer, 103, 245–259.

    Article  CAS  Google Scholar 

  • Quirantes, A., 2005: A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers. J. Quant. Spectrosc. Radiat. Transfer, 92, 373–381.

    Google Scholar 

  • Rahman, H., B. Pinty, and M. Verstrate, 1993: Coupled surface-atmospheric reflectance (CSAR) model. 2. Semi-empirical surface model usable with NOAA advanced very high resolution radiometer data. J. Geophys. Res., 98, 20791.

    Article  Google Scholar 

  • Rodgers, C.D., 2000: Inverse Methods for Atmospheric Sounding: Theory and Practice, World Scientific Publishing, Singapore.

    Google Scholar 

  • Rozanov A.V., V.V. Rozanov, and J.P. Burrows, 2000: Combined differential-integral approach for the radiation field computation in a spherical shell atmosphere: Nonlimb geometry. J. Geophys. Res., 105, 22937–22942.

    Article  Google Scholar 

  • Rozanov, V.V., and A.A. Kokhanovsky, 2006: The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications. Atmos. Res., 79, 241–265.

    Article  Google Scholar 

  • Rozanov, V, T. Kurosu, and J. Burrows, 1998: 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, 277–299.

    Article  CAS  Google Scholar 

  • Sancer, M., 1969: Shadow-corrected electromagnetic scattering from a randomly-rough ocean surface. IEEE Trans. Antennas Propag., AP-17, 557–585.

    Google Scholar 

  • Schulz, F.M., and K. Stamnes, 2000: Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model. J. Quant. Spectrosc. Radiat. Transfer, 65, 609–620.

    Article  CAS  Google Scholar 

  • Schutgens, N., and P. Stammes, 2003: A novel approach to the polarization correction of spaceborne spectrometers. J. Geophys. Res., 108, 4229. doi:10.1029/2002JD002736.

    Article  Google Scholar 

  • Siewert, C.E., 1982: On the phase matrix basic to the scattering of polarized light. Astron. Astrophys., 109, 195–200.

    Google Scholar 

  • Siewert, C.E., 2000a: A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer. J. Quant. Spectrosc. Radiat. Transfer, 64, 109–130.

    Google Scholar 

  • Siewert, C.E., 2000b: A discrete-ordinates solution for radiative transfer models that include polarization effects. J. Quant. Spectrosc. Radiat. Transfer, 64, 227–254.

    Google Scholar 

  • Spurr, R., 2002: Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment. J. Quant. Spectrosc. Radiat. Transfer, 75, 129–175.

    Google Scholar 

  • Spurr, R.J.D., 2003: LIDORT V2PLUS: A comprehensive radiative transfer package for UV/VIS/NIR nadir remote sensing; a general quasi-analytic solution. Proc. S.P.I.E. International Symposium, Remote Sensing 2003, Barcelona, Spain, September.

    Google Scholar 

  • Spurr, R.J.D., 2004: A new approach to the retrieval of surface properties from earth-shine measurements. J. Quant. Spectrosc. Radiat. Transfer, 83, 15–46.

    Google Scholar 

  • Spurr, R. J. D., 2006: VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media. J. Quant. Spectrosc. Radiat. Transfer, 102(2), 316–342, doi: 10.1016/j/jqsrt.2006.05.005.

    Google Scholar 

  • Spurr, R., and M. J. Christi, 2006: Linearization of the interaction principle: Analytic Jacobians in the radiant model. J. Quant. Spectrosc. Radiat. Transfer, 103(3), 431–446, doi: 10.1016/j.jqsrt.2006.05.001.

    Article  CAS  Google Scholar 

  • Spurr, R., T. Kurosu, and K. Chance, 2001: A linearized discrete ordinate radiative transfer model for atmospheric remote sensing retrieval. J. Quant. Spectrosc. Radiat. Transfer, 68, 689–735.

    Article  CAS  Google Scholar 

  • Spurr, R.J.D., K. Stamnes, H. Eide, W. Li, K. Zhang, and J. Stamnes, 2007: Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach. I. Analytic Jacobians from the linearized CAO-DISORT model. J. Quant. Spectrosc. Radiative Transfer, 104, 428–449.

    Article  CAS  Google Scholar 

  • Sromovsky, L.A., 2005: Effects of Rayleigh-scattering polarization on reflected intensity: a fast and accurate approximation method for atmospheres with aerosols. Icarus, 173, 284.

    Google Scholar 

  • Stam, D.M., J.F. de Haan, J.W. Hovenier, P. Stammes, 1999: Degree of linear polarization of light emerging from the cloudless atmosphere in the oxygen A band. J. Geophys. Res., 104, 16843.

    Article  CAS  Google Scholar 

  • Stamnes, K., and P. Conklin, 1984: A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres. J. Quant. Spectrosc. Radiat. Transfer, 31, 273.

    Article  Google Scholar 

  • Stamnes, K., S.-C. Tsay, W. Wiscombe, and K. Jayaweera, 1988a: Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media. Applied Optics, 27, 2502–2509.

    CAS  Google Scholar 

  • Stamnes K., S-C. Tsay, and T. Nakajima, 1988b: Computation of eigenvalues and eigenvectors for discrete ordinate and matrix operator method radiative transfer, J. Quant. Spectrosc. Radiat. Transfer, 39, 415–419.

    Article  CAS  Google Scholar 

  • Stammes, P., J.F. de Haan, and J.W. Hovenier, 1989: The polarized internal radiation field of a planetary atmosphere. Astron. Astrophys, 225, 239–259.

    Google Scholar 

  • Stamnes K., S-C. Tsay, W. Wiscombe, and I. Laszlo, 2000: DISORT: A general purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting media. Documentation of Methodology Report, climate. gsfc.nasa.gov/wiscombe/Multiple-Scatt/.

    Google Scholar 

  • Thomas, G. E., and K. Stamnes, 1999: Radiative Transfer in the Atmosphere and Ocean. Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Ustinov, E.A., 2001: Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. J. Quant. Spectrosc. Radiat. Transfer, 68, 195–211.

    Google Scholar 

  • Ustinov, E.A., 2005: Atmospheric weighting functions and surface partial derivatives for remote sensing of scattering planetary atmospheres in thermal spectral region: General adjoint approach. J. Quant. Spectrosc. Radiat. Transfer, 92, 351–371.

    Google Scholar 

  • Van Oss R.F., R.H.M. Voors, and R.J.D. Spurr, 2001: Ozone Profile Algorithm, OMI Algorithm Theoretical Basis Document. Volume II, OMI Ozone products (Bhartia. P.K., ed.), ATBD-OMI-02, Version 1.0, September.

    Google Scholar 

  • Van Oss, R.F., and R.J.D. Spurr, 2002: Fast and accurate 4 and 6 stream linearized discrete ordinate radiative transfer models for ozone profile retrieval. J. Quant. Spectrosc. Radiat. Transfer, 75, 177–220.

    Article  Google Scholar 

  • Vestrucci, M., and C.E. Siewert, 1984: A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light. J. Quant. Spectrosc. Radiat. Transfer, 31, 177–183.

    Article  Google Scholar 

  • Wanner, W., X. Li, and A. Strahler, 1985: On the derivation of kernels for kernel-driven models of bidirectional reflectance. J. Geophys. Res., 100, 21077.

    Article  Google Scholar 

  • Wauben, W.M.F., and J.W. Hovenier, 1992: Polarized radiation of an atmosphere containing randomly-oriented spheroids. J. Quant. Spectrosc. Radiat. Transfer, 47, 491–500.

    Article  Google Scholar 

  • Wiscombe, W., 1971: The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions. J. Atmos. Sci., 34, 1408–1422.

    Google Scholar 

  • Yin, J., T. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, 2006: Analytic solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface. Applied Optics, 45, 7433–7455.

    Google Scholar 

  • Zhao, D., and Y. Toba, 2003: A spectral approach for determining altimeter wind speed model functions. J. Ocean., 59, 235–244.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RT SOLUTIONS Inc., 9 Channing Street, Cambridge, MA, 02138, USA

    Robert Spurr

Authors
  1. Robert Spurr
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Environmental Physics, University of Bremen, Bremen, Germany

    Alexander A. Kokhanovsky

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Praxis Publishing Ltd, Chichester, UK

About this chapter

Cite this chapter

Spurr, R. (2008). LIDORT and VLIDORT: Linearized pseudo-spherical scalar and vector discrete ordinate radiative transfer models for use in remote sensing retrieval problems. In: Kokhanovsky, A.A. (eds) Light Scattering Reviews 3. Springer Praxis Books. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48546-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation