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Harshvardhan and Michael D. King

Abstract

Computational results have been obtained for the spherical albedo, global transmission, and global absorption of plane-parallel layers composed of cloud droplets. These computations, obtained using the doubling method for the entire range of single scattering albedos (0 ≤ ω 0 ≤ 1) and for optical depths between 0.1 and 100, are compared with corresponding results obtained using selected multiple scattering approximations. Both the relative and absolute accuracies of asymptotic theory for thick layers, three diffuse two-stream approximations, and two integrated two-stream approximations are presented as a function of optical thickness and single scattering albedo for a scattering phase function representative of cloud droplets at visible wavelengths. The spherical albedo and global absorption computed using asymptotic theory are found to be accurate to better than 5% for all values of the single scattering albedo, provided the optical thickness exceeds about 2. The diffuse two-stream approximations have relative accuracies that are much worse than 5% for the spherical albedo over most of the parameter space, yet are accurate to within 5% in the global absorption when the absorption is significant. The integrated delta-Eddington scheme appears to be the most suitable model over the entire range of variables, generally producing relative errors of less than 5% in both the spherical albedo and global absorption.

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Michael D. King

Abstract

A hemispherical radiometer has been used to obtain spectrally narrow-band measurements of the downward hemispheric diffuse and total (global) flux densities at varying solar zenith angles on 14 days over Tucson. Data are presented which illustrate the effects of temporally varying atmospheric conditions as well as clear stable conditions on the ratio of the diffuse to direct solar radiation at the earth's surface. The ground albedo and the effective imaginary term of the complex refractive index of atmospheric particulates are derived from the diffuse-direct ratio measurements on seven clear stable days at two wavelengths using the statistical procedure described by King and Herman (1979). Results indicate that the downwelling diffuse radiation field in the mid-visible region in Tucson can be adequately described by Mie scattering theory if the ground albedo is 0.279±0.100 and the index of absorption is 0.0306±0.0082.

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Michael D. King

Abstract

A method is presented for determining the scaled optical thickness of clouds from reflected solar radiation measurements. The procedure compares measurements of the reflection function with asymptotic expressions for the reflection function of optically thick layers. Analytic formulas are derived which explicitly show the dependence of the reflection and transmission functions of nonabsorbing atmospheres, on cloud optical thickness (τc), ground albedo (Ag) and asymmetry factor (g). For nonconservative atmospheres, the dependence of the reflection function on single scattering albedo (ω0) and asymmetry factor are contained implicitly in the asymptotic functions and constants. These asymptotic expressions for both conservative and nonconservative atmosphere are shown to be valid when the scaled optical thickness (1−gc≥??1.45, corresponding to clouds of optical thickness τc≥??9. By utilizing the asymptotic expression for the reflection function of an optically thick, conservatively scattering atmosphere. a simple expression is obtained relating the measured reflection function to scaled optical thickness. This expression shows that the ground albedo produces a constant bias in the derived optical thickness, regardless of the value of the measured reflection function.

High-resolution images of the reflection function of clouds have been obtained with a multichannel scanning radiometer operated from a high-altitude aircraft. An image of the reflection function of clouds obtained from a stratiform cloud system in central Oklahoma is analyzed using two different phase functions. Results show that details of the single scattering phase function have an impact on the derived optical thickness, although the dominant influence is the cloud asymmetry factor which appears explicitly in the analysis.

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Michael D. King and Harshvardhan

Abstract

Computational results have been obtained for the plane albedo, total transmission and fractional absorption of plane-parallel atmospheres composed of cloud droplets. These computations, which were obtained using the doubling method, are compared with comparable results obtained using selected radiative transfer approximations. Both the relative and absolute accuracies of asymptotic theory for thick layers and delta-Eddington, Meador–Weaver and Coakley–Chýlek approximations are compared as a function of optical thickness, solar zenith angle and single scattering albedo. Asymptotic theory is found to be accurate to within 5% for all optical thickness greater than about 6. On the other hand, the Coakley–Chýlek approximation is accurate to within 5% for thin atmospheres having optical thickness less than about 0.2 for most values of the solar zenith angle. Though the accuracies of delta-Eddington and Meador-Weaver approximations are less easily summarized it can generally be concluded that the delta-Eddington approximation is the most accurate for conservative scattering when the solar zenith angle is small, while the Meador–Weaver approximation is the most accurate for nonconservative scattering (ω0 ≤ 0.9). Selected results from the Eddington approximation are presented to illustrate the effect of delta function scaling in the delta-Eddington approximation. In addition, selected results from the single scattering approximation and asymptotic theory are presented in order to help explain the strengths and limitations of the various approximations.

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Michael D. King

Abstract

A method is presented for determining the single scattering albedo of clouds at selected wavelengths in the visible and near-infrared wavelength regions. The procedure compares measurements of the ratio of the zenith to nadir propagating intensifies deep within a cloud layer with radiative transfer computations of the same. Analytic formulas are derived which explicitly show the dependence of the internal intensity ratio on ground albedo, optical depth, single scattering albedo and cloud asymmetry factor. The single scattering albedo and cloud asymmetry factor enter the solution in such a way that a similarity relationship exists between these two parameters. As a result, the accuracy with which the single scattering albedo can be determined is dictated by the accuracy with which the asymmetry factor can be estimated. A method of observation is described whereby aircraft measurements of the zenith and nadir propagating intensifies can be used to determine the similarity parameter as a function of wavelength. Since the fractional absorption of a cloud depends on the similarity parameter and not on the single scattering albedo and asymmetry factor separately, this poses no severe limitation to the method. An accurate knowledge of the ground albedo and total optical thickness of a cloud are unnecessary for a solution, provided one associates the wavelength for which the intensity ratio is a maximum with conservative scattering. Under this internal calibration approach, uncertainties in the ground albedo are very nearly compensated by uncertainties in the cloud optical thickness.

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Michael D. King

Abstract

The sensitivity of constrained linear inversions to the selection of the Lagrange multiplier is demonstrated for the can of inferring columnar aerosol size distributions from spectral aerosol optical depth measurements. Since negative values of the aerosol size distribution constitute an unphysical solution, the Lagrange multiplier is varied within a restricted range until a range of values is reached for which all elements of the solution vector are positive. In addition to the constraint that the solution vector be positive, it is necessary for the final solution to be a smooth function and to satisfy the original integral equation to within the noise level of the measurements. An iterative method is presented whereby an initial estimate of the size distribution is modified until the final solution satisfies both the positivity constraint and the requirement that the regression fit to the data using the inverted size distribution he consistent with the measurement errors. A formula for calculating the variances and covariances in the inversion solution is derived and applied to optical depth measurements obtained at the University of Arizona and at Goddard Space Flight Center. In the former case an estimate of the measurement errors is available and thus the inversion formula and error analysis explicitly includes the magnitudes of the measurement variances. In the latter case the measurement errors are not known and the analysis assumes the errors in the measurements are equal and uncorrelated. Results of the error analysis show that the variances in the solution vector are large for radii where the information content of the measurements is small.

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Teruyuki Nakajima and Michael D. King

Abstract

A method is presented for determining the optical thickness and effective particle radius of stratiform cloud layers from reflected solar radiation measurements. A detailed study is presented which shows that the cloud optical thickness (τ c) and effective particle radius (re) of water clouds can be determined solely from reflection function measurements at 0.75 and 2.16 μm, provided τ c ≳ 4 and re ≳ 6 μm. For optically thin clouds the retrieval becomes ambiguous, resulting in two possible solutions for the effective radius and optical thickness. Adding a third channel near 1.65 μm does not improve the situation noticeably, whereas the addition of a channel near 3.70 μm reduces the ambiguity in deriving the effective radius.

The effective radius determined by the above procedure corresponds to the droplet radius at some optical depth within the cloud layer. For clouds having τ c ≳ 8, the effective radius determined using the 0.75 and 2.16 μm channels can be regarded as 85%–95% of the radius at cloud top, which corresponds in turn to an optical depth 20%–40% of the total optical thickness of the cloud layer.

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Michael D. King and Robert J. Curran

Abstract

The flux density measured at satellite altitude with a fixed field of view radiometer differs from the true flux density reflected by the earth-atmosphere system within the field of view of the radiometer. This difference is due to angular response characteristics of the radiometer, solid angle effects due to geometry, and angular reflectance effects of the earth-atmosphere system. All of these effects lead to uncertainties in the interpretation of instantaneous earth radiation budget measurements. The differences between the true flux density and the measured flux density are shown to be significant when the field of view of the radiometer is, large and when the atmosphere has a nonuniform, or spatially dependent, reflectance (albedo). A simulation experiment is described whereby the scene within the field of view of a nadir looking sensor is divided into a large number of equal area elements, each of which reflects radiation with one of two different reflectance models (corresponding to cloud-free and cloudy areas). The conditional mean values of the measured flux density, given values of the true flux density, are shown to differ significantly from the conditional means of the inverse problem, that of finding the mean value of the true flux density given a value for the measured flux density. The differences between the true flux density and the measured flux density are examined as a function of satellite altitude, field of view of the radiometer and solar zenith angle (including the effects of a terminator within the field of view) for both Lambertian and non-Lambertian reflectance models.

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Michael D. King and Benjamin M. Herman

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A statistical technique is developed for inferring the optimum values, of the ground albedo and the effective imaginary term of the complex refractive index of atmospheric particulates. The procedure compares measurements of the ratio of the hemispheric diffuse to directly transmitted solar flux density at the earth's surface with radiative transfer computations of the same as suggested by Herman et al. (1975). A detailed study is presented which shows the extent to which the ratio of diffuse to direct solar radiation is sensitive to many of the radiative transfer parameters. Results indicate that the optical depth and size distribution of atmospheric aerosol particles are the two parameters which uniquely specify the radiation field to the point where ground albedo and index of absorption can be inferred. Varying the real part of the complex refractive index of atmospheric particulates as well as their vertical distribution is found to have a negligible effect on the diffuse-direct ratio. The statistical procedure utilizes a semi-analytic gradient search method from least-squares theory and includes a detailed error analysis.

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Michael D. King and Dale M. Byrne

Abstract

A solar radiometer has been used to monitor solar irradiance at eight discrete wavelengths. From these monochromatic measurements at varying zenith angles the total optical depth has been deduced by a computerized curve-fitting method. A unique technique will be described whereby the ozone absorption optical depths, and hence total ozone content of the atmosphere, can be inferred directly from the spectral variation of total optical depth. This procedure permits a systematic determination of total ozone content on a daily basis when other measurements are not available. Using the ozone absorption optical depths determined in this manner, the values of aerosol optical depth may be obtained more accurately by subtracting the molecular scattering and estimated ozone absorption contributions from the total optical depth.

A technique is also described for estimating the absorption optical depths at wavelengths where additional molecular absorption other than ozone occurs. Results are presented as 1) daily values of total ozone content and 2) molecular absorption optical depths due to water vapor and oxygen at two of the radiometer wavelengths. The total ozone content exhibits the characteristic seasonal cycle with peak values in April.

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