A Direct Solution of the Spherical Harmonics Approximation to the Radiative Transfer Equation for an Arbitrary Solar Elevation. Part I: Theory

Dave, J. V.

doi:10.1175/1520-0469(1975)032<0790:adsots>2.0.co;2

Cited by 54 publications

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“…It is seen that errors in the MTF and radiances at (X = +-0) and (X = +-oo) are small for all the expressions of W e , with smallest error for We l = g2/f. The errors were calculated from the relative difference between the current approximation and 'exact' computation codes [Dave, 1975] 3+(1)(1-'n'(1 -LA (X) ) T Table 6 summarizes the expressions for (fi) and (gi). The total number of terms used in (22) in the present paper is 40.…”

Section: The Results Of Calculations Presented Inmentioning

confidence: 77%

Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity

Kaufman¹

1982

J. Geophys. Res.

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A fast yet accurate approximate solution has been developed for radiative transfer in an atmosphere bounded by a surface with nonuniform reflectivity for the purpose of remotely sensing aerosol characteristics (optical thickness and scale height). This solution is derived for the upward radiation at zenith for scattering and absorbing atmospheres and with surface reflectivity in the form of a ‘two‐halves’ field (two halves of field of a different reflectivity such as a sea shore). Any vertical distribution of absorbers and scatterers can be used. The resultant solution is used to develop expressions for the atmospheric line spread function and the atmospheric Modulation Transfer Function. This function can be applied to calculate the upward radiance for any surface reflectivity. The results of calculations are compared with numerical results calculated by the Monte Carlo method. The present results are shown to fit the Monte Carlo calculations almost exactly. This comparison shows that the present fast solution can be used for any surface reflectivity when aerosol optical thickness τA ≤ 1, and from the near UV to the near IR spectrum. The solution is also compared with high resolution satellite data (80 m resolution) obtained from Landsat 1 images of the African coast taken during ground measurements of the aerosol characteristics. The comparison shows that the present approximation can predict the measured radiances over a nonuniform surface reflectivity. It is shown that the horizontal range of the adjacency effects (the range at which the high reflectivity area affects the zenith radiance over the low reflectivity area) are proportionally dependent on the average elevation of the aerosols while independent of their exact vertical profile. In addition, this range is not very sensitive to the aerosol size distribution, optical thickness, and surface reflectance. Thus, this range can be used to retrieve the scale height of aerosols from zenith radiances detected from space.

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Section: The Results Of Calculations Presented Inmentioning

confidence: 77%

Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity

Kaufman¹

1982

J. Geophys. Res.

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“…The boundary conditions at the top and at the bottom of the atmosphere are expressed for the mth azimuthal harmonic in a form given by Marshak, as follows 12,28 :…”

Section: Methods Of Spherical Harmonicsmentioning

confidence: 99%

Radiative transfer code SHARM for atmospheric and terrestrial applications

Lyapustin

2005

Appl. Opt.

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An overview of the publicly available radiative transfer Spherical Harmonics code (SHARM) is presented. SHARM is a rigorous code, as accurate as the Discrete Ordinate Radiative Transfer (DISORT) code, yet faster. It performs simultaneous calculations for different solar zenith angles, view zenith angles, and view azimuths and allows the user to make multiwavelength calculations in one run. The Delta-M method is implemented for calculations with highly anisotropic phase functions. Rayleigh scattering is automatically included as a function of wavelength, surface elevation, and the selected vertical profile of one of the standard atmospheric models. The current version of the SHARM code does not explicitly include atmospheric gaseous absorption, which should be provided by the user. The SHARM code has several built-in models of the bidirectional reflectance of land and wind-ruffled water surfaces that are most widely used in research and satellite data processing. A modification of the SHARM code with the built-in Mie algorithm designed for calculations with spherical aerosols is also described.

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“…Sample calculations showed that the method was about 370 times faster than the iterative method previously used, and also an order of magnitude more accurate. In [77], Dave extended the earlier studies, which considered only the azimuth-independent component of the intensity, to the case in which the azimuth-dependent components are computed. It was established in a most definitive manner that spherical harmonics methods could be used for intensity calculations as well as for the calculations of the flux as carried out in earlier papers [78].…”

Section: Spherical Harmonics Methodsmentioning

confidence: 99%

Computer Modeling in Energy and the Environment

Flatt

1981

IBM J. Res. & Dev.

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Computer modeling has played a signijicant role in the evaluation of possible sources of energy as well as the consequences of the use of energy on the environment. In this paper the contributions of ZBM scientists to improve models in the fields of air pollution, solar energy, plasma physics, coal gasijication, and energy conservation are summarized. These contributions include not only better numerical and programming techniques but improved mathematical models based upon advances in understanding of the physical processes involved.The above outline makes clear the unfortunate necessity to omit a number of IBM studies which have been carried out both in the energy and in the environmental area. Notable among these studies have been those in surface and sub-surface hydrology [l-241. This extensive Copyright 1981 by International Business Machines Corporation. Copying is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Journal reference and IBM copyright notice are included on the first page. The title and abstract may be used without further permission in computer-based and other information-service systems. Permission to republish other excerpts should be obtained from the Editor.

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A Direct Solution of the Spherical Harmonics Approximation to the Radiative Transfer Equation for an Arbitrary Solar Elevation. Part I: Theory

Cited by 54 publications

References 0 publications

Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity

Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity

Radiative transfer code SHARM for atmospheric and terrestrial applications

Computer Modeling in Energy and the Environment

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