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- Author or Editor: Gilbert N. Plass x

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## Abstract

The calculation of atmospheric transmission functions for infrared spectral lines having a pressure-broadened line shape is extended to include the case where the fractional concentration of the radiating gas varies with height. The radiation exchange between two atmospheric layers can now be calculated under a wide variety of conditions. A recent formulation of the radiation problem by Plass is used to show that the conversion factor for changing parallel-beam radiation to diffuse radiation varies between one and two, depending on the particular frequency interval and optical thickness considered. The value of the conversion factor is two for a frequency interval involving only very weak lines; the value is 1.78 when the centers of the lines are black, if the lines do not overlap appreciably; the value approaches unity as the lines overlap more and more. The use of an average value for the conversion factor can lead to errors of 17 per cent in the calculation of radiation exchange. Since the angular integrations necessary to obtain the diffuse radiation are often considerably easier to perform when the correct line shape is used than when its frequency dependence is neglected, there appears to be no reason to introduce an additional uncertainty in radiation calculations by the use of an average value for the conversion factor to diffuse radiation. It is also shown that the radiation exchange for a strong line can be obtained for the varying pressure conditions in the atmosphere merely by a change of variables in the result for the transmission function of a slab under constant pressure conditions.

## Abstract

The calculation of atmospheric transmission functions for infrared spectral lines having a pressure-broadened line shape is extended to include the case where the fractional concentration of the radiating gas varies with height. The radiation exchange between two atmospheric layers can now be calculated under a wide variety of conditions. A recent formulation of the radiation problem by Plass is used to show that the conversion factor for changing parallel-beam radiation to diffuse radiation varies between one and two, depending on the particular frequency interval and optical thickness considered. The value of the conversion factor is two for a frequency interval involving only very weak lines; the value is 1.78 when the centers of the lines are black, if the lines do not overlap appreciably; the value approaches unity as the lines overlap more and more. The use of an average value for the conversion factor can lead to errors of 17 per cent in the calculation of radiation exchange. Since the angular integrations necessary to obtain the diffuse radiation are often considerably easier to perform when the correct line shape is used than when its frequency dependence is neglected, there appears to be no reason to introduce an additional uncertainty in radiation calculations by the use of an average value for the conversion factor to diffuse radiation. It is also shown that the radiation exchange for a strong line can be obtained for the varying pressure conditions in the atmosphere merely by a change of variables in the result for the transmission function of a slab under constant pressure conditions.

## Abstract

The radiance and polarization of the earth's atmosphere with haze and clouds has been calculated for a realistic model which includes Rayleigh scattering from the atmospheric molecules, Mie scattering from aerosols which are distributed with height according to the Elterman model, and Mie scattering from water droplets in clouds. Two different size distributions are used for the water droplets: one for the haze C model and the other from the nimbostratus model. Multiple scattering from the atmosphere and planetary surface is included to all orders by a Monte Carlo technique. The exact scattering matrix for the aerosol haze C and nimbostratus models is calculated from the Mie theory. Solar radiation is assumed to be unpolarized and incident from the zenith. The radiance and polarization are calculated at Î» = 0.4 and 0.7 Î¼ for a model of a cloud in the real atmosphere and the results are compared to those for the cloud alone and for the atmosphere alone. Relatively large polarization values are obtained except for viewing angles near the zenith or nadir or for photons transmitted through optically thick clouds. The polarization is typically larger at Î» = 0.4 Î¼ than at Î» = 0.7 Î¼, because of the greater contribution from Rayleigh scattering at the former wavelength.

## Abstract

The radiance and polarization of the earth's atmosphere with haze and clouds has been calculated for a realistic model which includes Rayleigh scattering from the atmospheric molecules, Mie scattering from aerosols which are distributed with height according to the Elterman model, and Mie scattering from water droplets in clouds. Two different size distributions are used for the water droplets: one for the haze C model and the other from the nimbostratus model. Multiple scattering from the atmosphere and planetary surface is included to all orders by a Monte Carlo technique. The exact scattering matrix for the aerosol haze C and nimbostratus models is calculated from the Mie theory. Solar radiation is assumed to be unpolarized and incident from the zenith. The radiance and polarization are calculated at Î» = 0.4 and 0.7 Î¼ for a model of a cloud in the real atmosphere and the results are compared to those for the cloud alone and for the atmosphere alone. Relatively large polarization values are obtained except for viewing angles near the zenith or nadir or for photons transmitted through optically thick clouds. The polarization is typically larger at Î» = 0.4 Î¼ than at Î» = 0.7 Î¼, because of the greater contribution from Rayleigh scattering at the former wavelength.

## Abstract

The equations for radiative transfer are integrated exactly for a band of spectral lines which do not overlap and for an Elsasser band. The result of the two-fold integration of the absorption over frequency and over the atmospheric path can be expressed in terms of the Legendre functions. Here it is assumed that the mixing ratio is constant, that the line intensity is independent of temperature, and that the Lorentz line shape is valid. Asymptotic forms of the Legendre functions are used to obtain the solutions to the following problems from these exact results. The regions of validity of the single-line and strong-line approximations are precisely stated. It is shown that the strongest line in the band absorbs more radiation than any other line in an atmospheric layer, when the overlap of the lines can be neglected. For an Elsasser band, an expression is derived for the line strength that gives the maximum absorption in an atmospheric layer for radiation emitted either by a black body at another level or by another atmospheric layer.

## Abstract

The equations for radiative transfer are integrated exactly for a band of spectral lines which do not overlap and for an Elsasser band. The result of the two-fold integration of the absorption over frequency and over the atmospheric path can be expressed in terms of the Legendre functions. Here it is assumed that the mixing ratio is constant, that the line intensity is independent of temperature, and that the Lorentz line shape is valid. Asymptotic forms of the Legendre functions are used to obtain the solutions to the following problems from these exact results. The regions of validity of the single-line and strong-line approximations are precisely stated. It is shown that the strongest line in the band absorbs more radiation than any other line in an atmospheric layer, when the overlap of the lines can be neglected. For an Elsasser band, an expression is derived for the line strength that gives the maximum absorption in an atmospheric layer for radiation emitted either by a black body at another level or by another atmospheric layer.

## Abstract

The model of the stratosphere proposed by Strong and Plass assumed that the spectral lines had the Lorentz shape. Actually the center of the line shifts to a lower frequency by an amount proportional to the pressure, and the shape of the line becomes asymmetrical at large distances from the center. The effect of the line shift on atmospheric heat-transfer is shown to be small and can be neglected except for very accurate calculations. On the other hand, for the strong spectral lines that play the dominant role in determining atmospheric heat-transfer, the difference between the results calculated using the Lorentz and asymmetrical line-shapes is large. A discussion is given of the effect of the asymmetry on the actual bands of interest in the atmosphere. The difficulties of making exact theoretical calculations of atmospheric heat-transfer are also discussed.

## Abstract

The model of the stratosphere proposed by Strong and Plass assumed that the spectral lines had the Lorentz shape. Actually the center of the line shifts to a lower frequency by an amount proportional to the pressure, and the shape of the line becomes asymmetrical at large distances from the center. The effect of the line shift on atmospheric heat-transfer is shown to be small and can be neglected except for very accurate calculations. On the other hand, for the strong spectral lines that play the dominant role in determining atmospheric heat-transfer, the difference between the results calculated using the Lorentz and asymmetrical line-shapes is large. A discussion is given of the effect of the asymmetry on the actual bands of interest in the atmosphere. The difficulties of making exact theoretical calculations of atmospheric heat-transfer are also discussed.

## Abstract

The upward and downward flux at various levels in the atmosphere and ocean is calculated by a Monte Carlo method which includes all orders of multiple scattering. A realistic model of the atmosphere-ocean system is used. In the atmosphere, both Rayleigh scattering by the molecules and Mie scattering by the aerosols as well as molecular and aerosol absorption are included in the model. Similarly, in the ocean, both Rayleigh scattering by the water molecules and Mie scattering by the hydrosols as well as absorption by the water molecules and hydrosols are considered. Separate single-scattering functions are calculated from the Mie theory for the aerosols and the hydrosols with an appropriate and different size distribution in each case. The scattering angles are determined from the appropriate scattering function including the strong forward-scattering peak when there is aerosol or hydrosol scattering. Both the reflected and refracted rays, as well as the rays that undergo total internal reflection, are followed at the ocean surface, which is assumed smooth. The ocean floor is represented by a Lambert surface. The upward flux as measured either just above the ocean surface or at the top of the atmosphere shows a significant dependence on the turbidity of the ocean water. The upward and downward flux is calculated at various wavelengths from 0.40 to 0.65 Î¼ and for three ocean models: clear, medium turbid, and turbid. The dependence of the flux on the albedo of the ocean floor is presented. The upward as well as the downward flux is larger just below the ocean surface than just above it at those wavelengths with relatively little absorption by water molecules and under those conditions when there is relatively little scattering from the hydrosols. The ratio of the upward to downward flux in the ocean at points away from boundaries is of the order of 0.1 in transparent regions of the spectrum for a clear ocean, but decreases rapidly as either the turbidity or wavelength increases.

## Abstract

The upward and downward flux at various levels in the atmosphere and ocean is calculated by a Monte Carlo method which includes all orders of multiple scattering. A realistic model of the atmosphere-ocean system is used. In the atmosphere, both Rayleigh scattering by the molecules and Mie scattering by the aerosols as well as molecular and aerosol absorption are included in the model. Similarly, in the ocean, both Rayleigh scattering by the water molecules and Mie scattering by the hydrosols as well as absorption by the water molecules and hydrosols are considered. Separate single-scattering functions are calculated from the Mie theory for the aerosols and the hydrosols with an appropriate and different size distribution in each case. The scattering angles are determined from the appropriate scattering function including the strong forward-scattering peak when there is aerosol or hydrosol scattering. Both the reflected and refracted rays, as well as the rays that undergo total internal reflection, are followed at the ocean surface, which is assumed smooth. The ocean floor is represented by a Lambert surface. The upward flux as measured either just above the ocean surface or at the top of the atmosphere shows a significant dependence on the turbidity of the ocean water. The upward and downward flux is calculated at various wavelengths from 0.40 to 0.65 Î¼ and for three ocean models: clear, medium turbid, and turbid. The dependence of the flux on the albedo of the ocean floor is presented. The upward as well as the downward flux is larger just below the ocean surface than just above it at those wavelengths with relatively little absorption by water molecules and under those conditions when there is relatively little scattering from the hydrosols. The ratio of the upward to downward flux in the ocean at points away from boundaries is of the order of 0.1 in transparent regions of the spectrum for a clear ocean, but decreases rapidly as either the turbidity or wavelength increases.

## Abstract

The upward and downward radiance is calculated for a realistic model of the atmosphere-ocean system by a Monte Carlo method. All known processes are taken into account which affect the solar photons, including scattering and absorption by atmospheric and oceanic molecules, and by aerosols and hydrosols, as well as reflection and refraction at the ocean surface. The scattering angles are chosen from distributions calculated from Mie theory for the aerosols and hydrosols and thus take account of the strong forward-scattering peak. Typical radiance values are presented at six wavelengths from 0.40 to 0.65 Î¼, for three different solar angles, and for three different models of the ocean with various amounts of turbidity. The minimum value of the upward radiance just above the ocean surface as a function of the nadir angle of observation increases 640% from the turbid to the clear ocean model. Even at the top of the atmosphere the increase is 40%. Thus, detectors in either airplanes or satellites should be able to obtain important information about the turbidity of the ocean. Other features shown in the results include the development with depth of the downward radiance both within and without the allowed cone into which radiation may enter the ocean from the sun and sky, the development of the asymptotic form for the downward radiance with depth, and the dependence of the radiance at various depths upon the turbidity of the ocean as well as the wavelength of the radiation.

## Abstract

The upward and downward radiance is calculated for a realistic model of the atmosphere-ocean system by a Monte Carlo method. All known processes are taken into account which affect the solar photons, including scattering and absorption by atmospheric and oceanic molecules, and by aerosols and hydrosols, as well as reflection and refraction at the ocean surface. The scattering angles are chosen from distributions calculated from Mie theory for the aerosols and hydrosols and thus take account of the strong forward-scattering peak. Typical radiance values are presented at six wavelengths from 0.40 to 0.65 Î¼, for three different solar angles, and for three different models of the ocean with various amounts of turbidity. The minimum value of the upward radiance just above the ocean surface as a function of the nadir angle of observation increases 640% from the turbid to the clear ocean model. Even at the top of the atmosphere the increase is 40%. Thus, detectors in either airplanes or satellites should be able to obtain important information about the turbidity of the ocean. Other features shown in the results include the development with depth of the downward radiance both within and without the allowed cone into which radiation may enter the ocean from the sun and sky, the development of the asymptotic form for the downward radiance with depth, and the dependence of the radiance at various depths upon the turbidity of the ocean as well as the wavelength of the radiation.

## Abstract

The radiance and color of the twilight sky are calculated for single scattered radiation with the use of spherical symmetric models of the earth's atmosphere. Spherical geometry is used throughout the calculations with no plane parallel approximations. Refraction effects are taken into account through fine subdivision of the atmosphere into spherical shells of fixed index of refraction. Snell's law of refraction is used to calculate a new direction of travel each time that a photon traverses the interface between layers. Five different models of the atmosphere were used: a pure molecular scattering atmosphere; molecular atmosphere plus ozone absorption; and three models with aerosol concentrations of one, three and ten times normal together with molecular scattering and ozone absorption. The results of the calculations are shown for various observation positions and local viewing angles in the solar plane for wavelengths in the range of 0.40 to 0.75 Âµm.

## Abstract

The radiance and color of the twilight sky are calculated for single scattered radiation with the use of spherical symmetric models of the earth's atmosphere. Spherical geometry is used throughout the calculations with no plane parallel approximations. Refraction effects are taken into account through fine subdivision of the atmosphere into spherical shells of fixed index of refraction. Snell's law of refraction is used to calculate a new direction of travel each time that a photon traverses the interface between layers. Five different models of the atmosphere were used: a pure molecular scattering atmosphere; molecular atmosphere plus ozone absorption; and three models with aerosol concentrations of one, three and ten times normal together with molecular scattering and ozone absorption. The results of the calculations are shown for various observation positions and local viewing angles in the solar plane for wavelengths in the range of 0.40 to 0.75 Âµm.

## Abstract

The degree and direction of polarization, the ellipticity, and the radiance of the radiation at various levels in the atmosphere and ocean are calculated by a Monte Carlo method which includes all orders of multiple scattering. Both Rayleigh scattering by the molecules and Mie scattering by the aerosols, as well as molecular and aerosol absorption, are included in the model of the atmosphere. Similarly, in the ocean, both Rayleigh scattering by the water molecules and Mie scattering by the hydrosols, as well as absorption by the water molecules and hydrosols, are considered. Separate single-scattering phase matrices are calculated from Mie theory for the aerosols and hydrosols. Both the reflected and refracted rays, as well as the rays that undergo total internal reflection, are followed from the ocean surface which is assumed to be flat. The degree and direction of polarization, the ellipticity, the radiance and the flux are given as functions of the turbidity of the ocean, the solar zenith angle, and the wavelength of the radiation.

## Abstract

The degree and direction of polarization, the ellipticity, and the radiance of the radiation at various levels in the atmosphere and ocean are calculated by a Monte Carlo method which includes all orders of multiple scattering. Both Rayleigh scattering by the molecules and Mie scattering by the aerosols, as well as molecular and aerosol absorption, are included in the model of the atmosphere. Similarly, in the ocean, both Rayleigh scattering by the water molecules and Mie scattering by the hydrosols, as well as absorption by the water molecules and hydrosols, are considered. Separate single-scattering phase matrices are calculated from Mie theory for the aerosols and hydrosols. Both the reflected and refracted rays, as well as the rays that undergo total internal reflection, are followed from the ocean surface which is assumed to be flat. The degree and direction of polarization, the ellipticity, the radiance and the flux are given as functions of the turbidity of the ocean, the solar zenith angle, and the wavelength of the radiation.