Retrieval of the albedo and phase function from exiting radiances with radiative perturbation theory.

We use radiative perturbation theory to develop a retrieval technique for determining the radiative properties of a scattering medium, such as the Earth's atmosphere, based on measurements of the radiation emerging at either the top or bottom of the medium. In a previous paper [J. Quant. Spectrosc. Radiat. Transfer 54, 695 (1995)] we have shown the capacity of radiative perturbation theory to describe variations in exiting intensity as a linear combination of the parameters that characterize the scattering medium. Here we show that it is possible to set up a matrix relation such that the matrix inversion solves the inverse scattering problem. Using simulated data, we observe that the quality of the solution can be controlled by studying the singular values associated with the kernel matrix, obtaining in this way a stable solution, even in the presence of noise.

Information content of the kernel matrix for the phase function retrieval problem.

We investigate the information content of the radiation measurements to be used in the retrieval of the scattering properties of the atmosphere with the perturbation technique that we previously introduced. Applying this technique to different sets of data, we obtained solutions with varying accuracy. An analysis of these solutions shows that selecting linearly independent data in directions corresponding to small values of the scattering angle increases the number of pieces of information. (This result is in accord with conclusions reached by other researchers, based on a variety of criteria.) This information content should be largely independent of the method or methods employed to perform the inversion procedure.

Investigating biological response in the UVB as a function of ozone variation using perturbation theory.

In order to determine a biological response to ultraviolet radiation, calculations of biologically weighted dose rates are required, which in turn involve the integral over wavelength of an action spectrum multiplied by appropriate surface flux data. To determine a biologically weighted dose rate accurately, a reasonable wavelength resolution is required, involving a full radiative transfer solution to be performed for each wavelength in order to obtain the surface flux information. If biologically weighted dose rates are needed as a function of ozone variation, then the number of radiative transfer solutions quickly makes a large number of ozone variations cumbersome. This paper shows that the perturbation theory developed for atmospheric radiative transfer by Box and co-workers can predict surface fluxes and hence biologically weighted dose rates for a large range of ozone variations very efficiently. The method is then extended to calculate radiation amplification factors. Results for biologically weighted dose rates are presented for a large range of solar zenith angles and ozone loadings using perturbation theory and a full radiative transfer code and show that the perturbation predictions never deviate very far from the radiative transfer solutions.

Information content analysis of aerosol remote-sensing experiments using singular function theory. 2: Scattering measurements.

We employ the singular function theory, which is the natural framework within which to discuss the analysis of first kind Fredholm integral equations, to analyze fully the information available from an aerosol aureole scattering experiment. This information is, of course, of two kinds: first, the number of pieces of information available for a given experimental error level and, second, the type (or location) of this information. To appreciate fully the latter, we apply this theory to the inversion of eleven synthetic data sets. These inversions are compared with those obtained previously from an extinction experiment.

Information content analysis of aerosol remote-sensing experiments using singular function theory. 1: Extinction measurements.

An important early step in any planned remote-sensing experiment is an analysis of the information content of the equations which will ultimately be inverted. In this paper, we employ the singular function theory, which is the natural framework within which to discuss the analysis of first kind Fredholm integral equations. Using this theory, we are able to fully analyze the information available from an aerosol extinction experiment. It is important to appreciate that this information is actually of two forms: first, the number of pieces of information available for a given experimental error level, and second, the type (or location) of this information. To fully appreciate the latter, we apply this theory to the inversion of eleven synthetic data sets, including three multimodal model size distributions.

Experimental validation of the solar aureole technique for determining aerosol size distributions.

This paper describes the inversion procedures by including multiple scattering contributions to almucantar radiance in the solar aureole region and discusses the retrieved size distribution results by comparing them with ground truth measurements. The agreement between the two sets of results is good and provides yet another experimental validation to show that the solar aureole technique is a simple, practical, and accurate method for obtaining the columnar size distribution of atmospheric aerosols. Comparisons between the results obtained by the two approximations--single and multiple scattering--show that the size distribution retrievals obtained by the latter are more accurate than those obtained by the former. Retrieved estimates of surface albedo show that even though surface albedo cannot be accurately determined from solar aureole measurements, these measurements can provide reasonable estimates. A recommendation is made for the use of solar aureole measurements from satellites to retrieve the size distribution and concentration of stratospheric and upper tropospheric aerosols.

Alternate approach to the analysis of solar photometer data.

The standard technique of analyzing solar photometer data to determine atmospheric optical depth and the spectral solar constant is shown to inadvertently weight the data unequally. A new approach is proposed which equally weights all the data. Assuming that the deviations of the data points result from real random variations of optical depth during the period of the measurements, this latter approach is shown to yield more reliable results.

Finite bandwidth and scattered light effects on the radiometric determination of atmospheric turbidity and the solar constant.

Multispectral solar radiometric measurements are routinely performed at a large number of sites, using equipment of varying degrees of sophistication. From the standard Langley plot technique, one may extract the total optical thickness of the atmosphere (and hence the aerosol component) plus the extraterrestrial solar flux. With increasing concern about possible climatic effects of increased turbidity, or changes in the solar constant, it is becoming more important to know the expected accuracy of these results. In this paper, we analytically examine the effects of finite filter bandwidth (in the absence of spectral lines) and find them to be less than one part in a thousand. This is compared with our earlier results on the effects of scattered light, which turns out to be typically an order of magnitude larger.

Finite sun effect on the interpretation of solar aureole.

Although it is usually assumed that solar radiation falls on the earth's atmosphere in the form of plane waves, the finite angular size of the solar disk contradicts this assumption. For most purposes, this finite sun effect on computed or measured radiation quantities is negligible. However, in the region of the solar aureole, which is dominated by aerosol diffraction scattering, measurable effects may be obtained. In this paper, we show that the finite sun effect is related to derivatives of the scattering phase function and that a 1% effect may be obtained close to the sun if enough large particles are present in the atmosphere.

Atmospheric scattering corrections to solar radiometry.

Whenever a solar radiometer is used to measure direct solar radiation, some diffuse sky radiation invariably enters the detector's field of view along with the direct beam. Therefore, the atmospheric optical depth obtained by the use of Bouguer's transmission law (also called Beer-Lambert's law), that is valid only for direct radiation, needs to be corrected by taking account of the scattered radiation. In this paper we shall discuss the correction factors needed to account for the diffuse (i.e., singly and multiply scattered) radiation and the algorithms developed for retrieving aerosol size distribution from such measurements. For a radiometer with a small field of view (half-cone angle < 5 degrees ) and relatively clear skies (optical depths < 0.4), it is shown that the total diffuse contribution represents approximately 1% of the total intensity. It is assumed here that the main contributions to the diffuse radiation within the detector's view cone are due to single scattering by molecules and aerosols and multiple scattering by molecules alone, aerosol multiple scattering contributions being treated as negligibly small. The theory and the numerical results discussed in this paper will be helpful not only in making corrections to the measured optical depth data but also in designing improved solar radiometers.

Retrieval of aerosol size distributions by inversion of simulated aureole data in the presence of multiple scattering.

Inversion of solar almucantar data is a simple and practical method of obtaining aerosol size distributions. In this paper, we have inverted a number of sets of simulated data, using the standard single scattering approximation, to test the errors involved in ignoring multiple scattering. We have also inverted the data using two techniques: one, a modification of the method proposed by Deirmendjian and Sekera; and the other that of McPeters and Green. Inversion results strongly suggest that the accuracy of the retrieved size distribution can be significantly improved by use of our modified Deirmendjian-Sekera approach, in which multiple scattering by molecules is included along with single scattering by molecules and aerosols.

Relationship between two analytic inversion formulae for multispectral extinction data.

Recently Box and McKellar and Fymat have presented analytic inversion formulae for multispectral extinction data in the anomalous diffraction approximation. In this paper, we shall examine the relationship between these two formulae, and in the process we shall derive both the power series expansion and the asymptotic expansion for the extinction in this approximation.

Forwardscattering corrections for optical extinction measurements in aerosol media. 1: Monodispersions.

This paper, Part 1 of two papers, presents a parametric study of the forwardscattering corrections for experimentally measured optical extinction coefficients in homogeneous aerosol media, since some forwardscattered light invariably enters, along with the direct beam, into the finite aperture of the detector. Part 1 treats the case of monodispersions; Part 2, that of polydispersions. Forwardscattering is considered a single-scattering phenomenon, and the corrections are computed by two methods: one, using the exact Mie theory, and the other, the approximate Rayleigh diffraction formula. A parametric study of the dependence of the corrections on the particle size parameter, real and imaginary parts of the complex refractive index, and the half-angle of the detector's view cone has been carried out. The parameter ranges in which the results obtained by the approximate formulation agree well with those obtained by the Mie theory are also investigated. The agreement is especially good for small view cone angles and large particles and improves even more for slightly absorbing aerosol particles. Also discussed is the dependence of these corrections upon the experimental design of the transmission measurement systems.

Forwardscattering corrections for optical extinction measurements in aerosol media. 2: Polydispersions.

This paper, second of two parts, presents a parametric study of the forwardscattering corrections for experimentally measured optical extinction coefficients in polydisperse particulate media, since some forward scattered light invariably enters, along with the direct beam, into the finite aperture of the detector. Forwardscattering corrections are computed by two methods: (1) using the exact Mie theory, and (2) the approximate Rayleigh diffraction formula for spherical particles. A parametric study of the dependence of the corrections on mode radii, real and imaginary parts of the complex refractive index, and half-angle of the detector's view cone has been carried out for three different size distribution functions of the modified Gamma type. In addition, a study has been carried out to investigate the range of these parameters in which the approximate formulation is valid. The agreement is especially good for small-view cone angles and large particles, which improves significantly for slightly absorbing aerosol particles. Also discussed is the dependence of these corrections on the experimental design of the transmissometer systems.

Single and multiple scattering contributions to circumsolar radiation.

Single and multiple scattering contributions to the circumsolar radiation along the almucantar and sun vertical have been computed by a Gauss-Seidel solution to the radiative transfer equation. In the near forward direction, the multiple scattering contributions are significant for optical depths of the order of 0.4. However, the shape of the angular distribution of almucantar radiance up to 10 degrees appears less sensitive to multiple scattering. The results have been compared against an existing radiative transfer code.

Atmospheric turbidity and the circumsolar radiation.

We investigate the possibility of determining the turbidity from the intensity of the circumsolar radiation and find that such a connection can be made only when the size distribution of the aerosol particles is known. However, measurements of both the turbidity and the aureole intensity can give useful information about the size distribution.