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We consider the scattering of a plane electromagnetic wave obliquely incident on a plane-parallel layer of discrete random medium with non-scattering boundaries. We solve the Lax integral equation for the conditional configuration-averaged exciting field coefficients by assuming a special-form solution, that is, by representing the conditional configuration-averaged exciting field coefficients as a linear combination of the coefficients corresponding to an up-going and a down-going wave. This solution representation is supposed to be valid within the whole domain occupied by the particles, even in the close proximity of the boundaries. By balancing the waves with different propagation directions and wavenumbers we derive two homogeneous systems of equations corresponding to the generalized Lorenz–Lorentz law and two inhomogeneous systems of equations corresponding to the generalized Ewald–Oseen extinction theorem. It is shown that (i) the two homogeneous systems of equations of the generalized Lorenz–Lorentz law reduce to a single homogeneous system of equations corresponding to a semi-infinite discrete random medium at normal incidence; (ii) the dispersion equation is direction and polarization independent; and (iii) the two inhomogeneous systems of equations of the generalized Ewald–Oseen extinction theorem can be reduced to two scalar equations by means of the addition theorem for vector spherical harmonics. It is also shown that the same dispersion equation can be obtained without assuming a special-form solution representation in the proximity of the boundaries.

A new method for modeling the radiative transfer in inhomogeneous three-dimensional media illuminated by a Gaussian beam is described. This approach, called the Spectral Spherical Harmonics Discrete Ordinate Method (SSHDOM), uses the Fourier expansion method to transform the three-dimensional radiative transfer into an one-dimensional equation in the spectral domain, and the Spherical Harmonics Discrete Ordinate Method (SHDOM) for its solution. Specifically, (i) the source function is represented in the spectral domain through a spherical harmonic expansion, (ii) the spectral one-dimensional radiative transfer equation is integrated along discrete ordinates through a spatial grid, and (iii) the solution method is based on the Picard iteration. Both SSHDOM and SHDOM algorithms are implemented in a common computer code.

In this paper, the vector radiative transfer equation is derived by means of the vector integral Foldy equations describing the electromagnetic scattering by a group of particles. By assuming that in a discrete random medium the positions of the particles are statistically independent and by applying the Twersky approximation to the order-of-scattering expansion of the total field, we derive the Dyson equation for the coherent field and the ladder approximated Bethe–Salpeter equation for the dyadic correlation function. Then, under the far-field assumption for sparsely distributed particles, the Dyson equation is reduced to the Foldy integral equation for the coherent field, while the iterated solution of the Bethe–Salpeter equation ultimately yields the vector radiative transfer equation.

In this paper we apply the Green function formalism and the Wigner transform method to derive the radiative transfer equation for a layer of sparse discrete random medium bounded by scattering rough interfaces. The radiative transfer equation is formulated with respect to the Wigner transform of the dyadic correlation function for the electromagnetic field. The results are obtained under the effective field and on-shell approximations, and the meaning of these approximations is discussed in detail.

In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum representation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.

In remote sensing of planetary bodies, the development of analysis techniques that lead to quantitative interpretations of datasets has relatively been deficient compared to the wealth of acquired data, especially in the case of regoliths with particle sizes on the order of the probing wavelength. Radiative transfer theory has often been applied to the study of densely packed particulate media like planetary regoliths, but with difficulty; here we continue to improve theoretical modeling of spectra of densely packed particulate media. We use the superposition T-matrix method to compute the scattering properties of an elementary volume entering the radiative transfer equation by modeling it as a cluster of particles and thereby capture the near-field effects important for dense packing. Then, these scattering parameters are modified with the static structure factor correction to suppress the irrelevant far-field diffraction peak rendered by the T-matrix procedure. Using the corrected single­ scattering parameters, reflectance (and emissivity) is computed via the invariant-imbedding solution to the scalar radiative transfer equation. We modeled the emissivity spectrum of the 3.3 μm particle size fraction of enstatite, representing a common regolith component, in the mid-infrared (~5 – 50 μm). The use of the static structure factor correction coupled with the superposition T-matrix method produced better agreement with the corresponding laboratory spectrum than the sole use of the T-matrix method, particularly for volume scattering wavelengths (transparency features). This work demonstrates the importance of proper treatment of the packing effects when modeling semi-infinite densely packed particulate media using finite, cluster-based light scattering models.

We use the volume integral equation formulation to consider frequency-domain electromagnetic scattering of a damped inhomogeneous plane wave by a particle immersed in an absorbing medium. We show that if absorption in the host medium is sufficiently weak and the particle size parameter is sufficiently small, then (i) the resulting formalism (including the far-field and radiative-transfer regimes) is largely the same as in the case of a nonabsorbing host medium, and (ii) one can bypass explicit use of sophisticated general solvers of the Maxwell equations applicable to inhomogeneous-wave illumination. These results offer dramatic simplifications for solving the scattering problem in a wide range of practical applications involving absorbing host media.

A vector radiative transfer equation with an additional source term typical of dense media is obtained. The analysis includes (i) the derivation of an integral equation for the correlation matrix of the exciting field coefficients accounting for the correlation between the particles, (ii) the derivation of an integral representation for the specific coherency dyadic in terms of this matrix, and (iii) the simplification of the integral equation for the correlation matrix and of the integral representation for the specific coherency dyadic by employing a series of approximations which are characteristic of sparse media.

The computation of the coherent field in the case of a plane electromagnetic wave obliquely incident on a discrete random layer with non-scattering boundaries is addressed. For dense media, the analysis is based on a special-form solution for the conditional configuration-averaged exciting field coefficients, and is restricted to the computation of the so-called zeroth-order fields without a special treatment of the boundary regions. In this setting, we calculate the coherent fields reflected and transmitted by the layer, and the coherent field inside the layer. We found that these fields are analytically equivalent to plane electromagnetic waves, and investigated the fulfillment of the boundary conditions for the electric fields at the layer interfaces. The results are then particularized to the cases of normal incidence and a semi-infinite discrete random medium. For sparsely distributed particles, we present a self-consistent derivation of the coherent field and discuss the Twersky and Foldy approximations.

We analyze the possibilities of orbital photopolarimetric measurements to study properties of aerosols in the Earth's atmosphere. As an example, we consider the case when such measurements are performed within a narrow spectral channel centered at 1.378 µm that allows to retrieve microphysical characteristics of stratospheric aerosols separately from those of tropospheric aerosols. We consider the case of stratospheric aerosols caused by volcanic eruption, and adopt the model of the stratosphere in the form of a homogeneous plane-parallel layer composed of polydisperse spherical particles. We use numerically exact solutions of the vector radiative transfer equation to theoretically simulate measurements carried out at various numbers of scattering angles, including: (i) radiance measurements alone; (ii) polarization measurements alone; and (iii) radiance and polarization measurements together. The results of computations show that the simultaneous use of radiance and polarization measurements at a sufficiently large number of scattering angles enables one to retrieve the optical thickness, effective radius, and refractive index of aerosols with adequate accuracy. We demonstrate how the accuracy of the derived values of the optical parameters of aerosols depends on the accuracy of measurements of the intensity and polarization of the reflected light, optical thickness of aerosol layer itself, effective radius of aerosols, width of the particle size distribution, and number of viewing angles.

We use the recently generalized version of the multi-sphere superposition T-matrix method (STMM) to compute the scattering and absorption properties of microscopic water droplets contaminated by black carbon. The soot material is assumed to be randomly distributed throughout the droplet interior in the form of numerous small spherical inclusions. Our numerically-exact STMM results are compared with approximate ones obtained using the Maxwell-Garnett effective-medium approximation (MGA) and the Monte Carlo ray-tracing approximation (MCRTA). We show that the popular MGA can be used to calculate the droplet optical cross sections, single-scattering albedo, and asymmetry parameter provided that the soot inclusions are quasi-uniformly distributed throughout the droplet interior, but can fail in computations of the elements of the scattering matrix depending on the volume fraction of soot inclusions. The integral radiative characteristics computed with the MCRTA can deviate more significantly from their exact STMM counterparts, while accurate MCRTA computations of the phase function require droplet size parameters substantially exceeding 60.

In this paper we present the vector radiative transfer theory for a discrete random medium illuminated by a Gaussian beam. The analysis is based on a plane wave spectrum representation for a Gaussian beam and uses an approach developed previously for a discrete random medium illuminated by a plane electromagnetic wave. Specifically, we establish an integral representation for the coherent field, define an approximate coherent field that satisfies the differential equation fulfilled by the coherent field corresponding to a plane electromagnetic wave and matches the Gaussian beam at the interface of the particulate medium, and finally, derive the vector radiative transfer equation. For weakly focused Gaussian beams, the resulting equation is the traditional radiative transfer equation.

For a macroscopically plane-parallel discrete random medium, the boundary conditions for the specific coherency dyadic at a rough interface are derived. The derivation is based on a modification of the Twersky approximation for a scattering system consisting of a group of particles and the rough surface, and reduces to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave propagating in a discrete random medium with non-scattering boundaries. In a matrix-form setting, the boundary conditions for the specific coherency dyadic imply the boundary conditions for specific intensity column vectors which in turn, yield the expressions for the reflection and transmission matrices. The derived expressions are shown to be identical to those obtained by applying a phenomenological approach based on a facet model to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave.

A new research algorithm is presented here as the second part of a two-part study to retrieve aerosol microphysical properties from the multispectral and multiangular photopolarimetric measurements taken by Aerosol Robotic Network's (AERONET's) new-generation Sun photometer. The algorithm uses an advanced UNified and Linearized Vector Radiative Transfer Model and incorporates a statistical optimization approach.While the new algorithmhas heritage from AERONET operational inversion algorithm in constraining a priori and retrieval smoothness, it has two new features. First, the new algorithmretrieves the effective radius, effective variance, and total volume of aerosols associated with a continuous bimodal particle size distribution (PSD) function, while the AERONET operational algorithm retrieves aerosol volume over 22 size bins. Second, our algorithm retrieves complex refractive indices for both fine and coarsemodes,while the AERONET operational algorithm assumes a size-independent aerosol refractive index. Mode-resolved refractive indices can improve the estimate of the single-scattering albedo (SSA) for each aerosol mode and thus facilitate the validation of satellite products and chemistry transport models. We applied the algorithm to a suite of real cases over Beijing_RADI site and found that our retrievals are overall consistent with AERONET operational inversions but can offer mode-resolved refractive index and SSA with acceptable accuracy for the aerosol composed by spherical particles. Along with the retrieval using both radiance and polarization, we also performed radiance-only retrieval to demonstrate the improvements by adding polarization in the inversion. Contrast analysis indicates that with polarization, retrieval error can be reduced by over 50% in PSD parameters, 10-30% in the refractive index, and 10-40% in SSA, which is consistent with theoretical analysis presented in the companion paper of this two-part study.

We present a numerical testbed for remote sensing of aerosols, together with a demonstration for evaluating retrieval synergy from a geostationary satellite constellation. The testbed combines inverse (optimal-estimation) software with a forward model containing linearized code for computing particle scattering (for both spherical and non-spherical particles), a kernel-based (land and ocean) surface bi-directional reflectance facility, and a linearized radiative transfer model for polarized radiance. Calculation of gas absorption spectra uses the HITRAN (HIgh-resolution TRANsmission molecular absorption) database of spectroscopic line parameters and other trace species cross-sections. The outputs of the testbed include not only the Stokes 4-vector elements and their sensitivities (Jacobians) with respect to the aerosol single scattering and physical parameters (such as size and shape parameters, refractive index, and plume height), but also DFS (Degree of Freedom for Signal) values for retrieval of these parameters. This testbed can be used as a tool to provide an objective assessment of aerosol information content that can be retrieved for any constellation of (planned or real) satellite sensors and for any combination of algorithm design factors (in terms of wavelengths, viewing angles, radiance and/or polarization to be measured or used). We summarize the components of the testbed, including the derivation and validation of analytical formulae for Jacobian calculations. Benchmark calculations from the forward model are documented. In the context of NASA's Decadal Survey Mission GEOCAPE (GEOstationary Coastal and Air Pollution Events), we demonstrate the use of the testbed to conduct a feasibility study of using polarization measurements in and around the O2 A band for the retrieval of aerosol height information from space, as well as an to assess potential improvement in the retrieval of aerosol fine and coarse mode aerosol optical depth (AOD) through the synergic use of two future geostationary satellites, GOES-R (Geostationary Operational Environmental Satellite R-series) and TEMPO (Tropospheric Emissions: Monitoring of Pollution). Strong synergy between GEOS-R and TEMPO are found especially in their characterization of surface bi-directional reflectance, and thereby, can potentially improve the AOD retrieval to the accuracy required by GEO-CAPE.

This Essay traces the centuries-long history of the phenomenological disciplines of directional radiometry and radiative transfer in turbid media, discusses their fundamental weaknesses, and outlines the convoluted process of their conversion into legitimate branches of physical optics.

The problem of backscattering of light by a discrete random medium illuminated by an obliquely incident plane electromagnetic wave is considered. The analysis is performed in a linear-polarization basis and includes (i) a complete derivation of the cross reflection matrix for a layer with densely and sparsely distributed particles, (ii) the design of an approximate method for computing the ladder and cross reflection matrices in the case of a semi-infinite medium with a sparse distribution of particles, (iii) the derivation of the relations between the elements of the ladder and cross reflection matrices in the exact backscattering direction for dense and sparse media, and (iv) the development of practical algorithms for solving the underlying integral equations by the method of Picard iterations and the discrete ordinate method. Simulation results for particles with large size parameters are also presented.

We study multiple scattering of light by particles embedded in an absorbing host medium using a recently developed single-scattering and vector radiative-transfer methodology directly based on the Maxwell equations. The first-principles results are compared with those rendered by the conventional heuristic approach according to which the single-scattering properties of particles can be computed by assuming that the host medium is nonabsorbing. Our analysis shows that the conventional approach yields very accurate results in the case of aerosol and cloud particles suspended in an absorbing gaseous atmosphere. In the case of air bubbles in water, the traditional approach can cause large relative errors in reflectance, but only when strong absorption in the host medium makes the resulting reflectance very small. The corresponding polarization errors are substantially smaller.

Increasing our understanding of the Earth-atmosphere system has been a scientific and political priority for the last few decades. This system not only touches on environmental science, but it has applicability to our broader understanding of planetary atmospheres in general. While this issue focuses primarily on electromagnetics, other fundamental fields of science, including fluid and thermodynamics play major roles. In recent years, significant research efforts have led to advances in the fields of radiative transfer and electromagnetic scattering from irregularly shaped particles. Recently, several workshops and small conferences have taken place to promote the fusion of these efforts. Late in 2013, for instance, two such meetings took place. The Optical Characterization of Atmospheric Aerosols (OCAA) meeting took place in Smolenice, Slovakia to promote a better understanding of microphysical properties of aerosol particles, and the characterization of such atmospheric particles using optical techniques. A complementary conference was organized in Nagoya, Japan, the 3rd International Symposium on Atmospheric Light Scattering and Remote Sensing (ISALSaRS), whose goal is to fuse the advances achieved in particle characterization with remote-sensing techniques. While the focus of these meetings is slightly different, they represent the same aspects of this rapidly growing field. This Topical Issue is the first of two parts. Within this issue we analyze different aspects of the problem of atmospheric characterization and present a broad overview of the topical area. Research includes theory and experiment, ranging from fundamental microphysical properties of individual aerosol particles to broad characterizations of atmospheric properties. Since this is an active field, we also have encouraged the submission of ideas for new methodologies that may represent the future of the field.

We report a novel method for detecting cloud particles in the atmosphere. Solar radiation backscattered from clouds is studied with both satellite data and a radiative transfer model. A distinct feature is found in the angle of linear polarization of solar radiation that is backscattered from clouds. The dominant backscattered electric field from the clear-sky Earth-atmosphere system is nearly parallel to the Earth surface. However, when clouds are present, this electric field can rotate significantly away from the parallel direction. Model results demonstrate that this polarization feature can be used to detect super-thin cirrus clouds having an optical depth of only ~0.06 and super-thin liquid water clouds having an optical depth of only ~0.01. Such clouds are too thin to be sensed using any current passive satellite instruments.

This paper summarizes the scientific legacy of Peter C. Waterman (1928–2012) who introduced concepts and theoretical techniques that have had a major impact on the fields of scattering by particles and particle groups, optical particle characterization, radiative transfer, and remote sensing. A biographical sketch is also included.

The exact multiple sphere superposition method is used to calculate the coherent and incoherent contributions to the ensemble-averaged electric field amplitude and Poynting vector in systems of randomly positioned nonabsorbing spherical particles. The target systems consist of cylindrical volumes, with radius several times larger than length, containing spheres with positional configurations generated by a Monte Carlo sampling method. Spatially dependent values for coherent electric field amplitude, coherent energy flux, and diffuse energy flux, are calculated by averaging of exact local field and flux values over multiple configurations and over spatially independent directions for fixed target geometry, sphere properties, and sphere volume fraction. Our results reveal exponential attenuation of the coherent field and the coherent energy flux inside the particulate layer and thereby further corroborate the general methodology of the microphysical radiative transfer theory. An effective medium model based on plane wave transmission and reflection by a plane layer is used to model the dependence of the coherent electric field on particle packing density. The effective attenuation coefficient of the random medium, computed from the direct simulations, is found to agree closely with effective medium theories and with measurements. In addition, the simulation results reveal the presence of a counter-propagating component to the coherent field, which arises due to the internal reflection of the main coherent field component by the target boundary. The characteristics of the diffuse flux are compared to, and found to be consistent with, a model based on the diffusion approximation of the radiative transfer theory.

First-principle analysis of the functional design of a well-collimated radiometer (WCR) reveals that in general, this instrument does not record the instantaneous directional flow of electromagnetic energy. Only in special cases can a sequence of measurements with a WCR yield the magnitude and direction of the local time-averaged Poynting vector. Our analysis demonstrates that it is imperative to clearly formulate the physical nature of the actual measurement afforded by a directional radiometer rather than presume desirable measurement capabilities. Only then can the directional radiometer be considered a legitimate part of physically based remote sensing and radiation-budget applications. We also emphasize the need for a better understanding of the nature of measurements with panoramic radiometers.

A data library is developed containing the scattering, absorption, and polarization properties of ice particles in the spectral range from 0.2 to 100 μm. The properties are computed based on a combination of the Amsterdam discrete dipole approximation (ADDA), the T-matrix method, and the improved geometric optics method (IGOM). The electromagnetic edge effect is incorporated into the extinction and absorption efficiencies computed from the IGOM. A full set of single-scattering properties is provided by considering three-dimensional random orientations for 11 ice crystal habits: droxtals, prolate spheroids, oblate spheroids, solid and hollow columns, compact aggregates composed of eight solid columns, hexagonal plates, small spatial aggregates composed of 5 plates, large spatial aggregates composed of 10 plates, and solid and hollow bullet rosettes. The maximum dimension of each habit ranges from 2 to 10 000 μm in 189 discrete sizes. For each ice crystal habit, three surface roughness conditions (i.e., smooth, moderately roughened, and severely roughened) are considered to account for the surface texture of large particles in the IGOM applicable domain. The data library contains the extinction efficiency, single-scattering albedo, asymmetry parameter, six independent nonzero elements of the phase matrix (P11, P12, P22, P33, P43, and P44), particle projected area, and particle volume to provide the basic single-scattering properties for remote sensing applications and radiative transfer simulations involving ice clouds. Furthermore, a comparison of satellite observations and theoretical simulations for the polarization characteristics of ice clouds demonstrates that ice cloud optical models assuming severely roughened ice crystals significantly outperform their counterparts assuming smooth ice crystals.

In this paper we have analyzed circumstances under which a rigid particulate sample can behave optically as a true discrete random medium consisting of particles randomly moving relative to each other during measurement. To this end, we applied the numerically exact superposition T-matrix method to model far-field scattering characteristics of fully ordered and quasi-randomly arranged rigid multiparticle groups in fixed and random orientations. We have shown that, in and of itself, averaging optical observables over movements of a rigid sample as a whole is insufficient unless it is combined with a quasi-random arrangement of the constituent particles in the sample. Otherwise, certain scattering effects typical of discrete random media (including some manifestations of coherent backscattering) may not be accurately replicated.

The main objective of this Report is to formulate the general theoretical framework of electromagnetic scattering by discrete random media rooted in the Maxwell-Lorentz electromagnetics and discuss its immediate analytical and numerical consequences. Starting from the microscopic Maxwell-Lorentz equations, we trace the development of the first-principles formalism enabling accurate calculations of monochromatic and quasi-monochromatic scattering by static and randomly varying multiparticle groups. We illustrate how this general framework can be coupled with state-of-the-art computer solvers of the Maxwell equations and applied to direct modeling of electromagnetic scattering by representative random multi-particle groups with arbitrary packing densities. This first-principles modeling yields general physical insights unavailable with phenomenological approaches. We discuss how the first-order-scattering approximation, the radiative transfer theory, and the theory of weak localization of electromagnetic waves can be derived as immediate corollaries of the Maxwell equations for very specific and well-defined kinds of particulate medium. These recent developments confirm the mesoscopic origin of the radiative transfer, weak localization, and effective-medium regimes and help evaluate the numerical accuracy of widely used approximate modeling methodologies., Review paper

Current problems of the theory of multiple scattering of electromagnetic waves by discrete random media are reviewed, with an emphasis on densely packed media. All equations presented are based on the rigorous theory of electromagnetic scattering by an arbitrary system of non-spherical particles. The main relations are derived in the circular-polarization basis. By applying methods of statistical electromagnetics to a discrete random medium in the form of a plane–parallel layer, we transform these relations into equations describing the average (coherent) field and equations for the sums of ladder and cyclical diagrams in the framework of the quasi-crystalline approximation. The equation for the average field yields analytical expressions for the generalized Lorentz–Lorenz law and the generalized Ewald–Oseen extinction theorem, which are traditionally used for the calculation of the effective refractive index. By assuming that the particles are in the far-field zones of each other, we transform all equations asymptotically into the well-known equations for sparse media. Specifically, the equation for the sum of the ladder diagrams is reduced to the classical vector radiative transfer equation. We present a simple approximate solution of the equation describing the weak localization (WL) effect (i.e., the sum of cyclical diagrams) and validate it by using experimental and numerically exact theoretical data. Examples of the characteristics of WL as functions of the physical properties of a particulate medium are given. The applicability of the interference concept of WL to densely packed media is discussed using results of numerically exact computer solutions of the macroscopic Maxwell equations for large ensembles of spherical particles. These results show that theoretical predictions for spare media composed of non-absorbing or weakly absorbing particles are reasonably accurate if the particle packing density is less than ∼30%. However, a further increase of the packing density and/or absorption may cause optical effects not predicted by the low-density theory and caused by near-field effects. The origin of the near-filed effects is discussed in detail.

Direct computer simulations of electromagnetic scattering by discrete random media have become an active area of research. In this progress review, we summarize and analyze our main results obtained by means of numerically exact computer solutions of the macroscopic Maxwell equations. We consider finite scattering volumes with size parameters in the range, composed of varying numbers of randomly distributed particles with different refractive indices. The main objective of our analysis is to examine whether all backscattering effects predicted by the low-density theory of coherent backscattering (CB) also take place in the case of densely packed media. Based on our extensive numerical data we arrive at the following conclusions: (i) all backscattering effects predicted by the asymptotic theory of CB can also take place in the case of densely packed media; (ii) in the case of very large particle packing density, scattering characteristics of discrete random media can exhibit behavior not predicted by the low-density theories of CB and radiative transfer; (iii) increasing the absorptivity of the constituent particles can either enhance or suppress typical manifestations of CB depending on the particle packing density and the real part of the refractive index. Our numerical data strongly suggest that spectacular backscattering effects identified in laboratory experiments and observed for a class of high-albedo Solar System objects are caused by CB.

Measurements with directional radiometers and calculations based on the radiative transfer equation (RTE) have been at the very heart of weather and climate modeling and terrestrial remote sensing. The quantification of the energy budget of the Earth's climate system requires exquisite measurements and computations of the incoming and outgoing electromagnetic energy, while global characterization of climate system's components relies heavily on theoretical inversions of observational data obtained with various passive and active instruments. The same basic problems involving electromagnetic energy transport and its use for diagnostic and characterization purposes are encountered in numerous other areas of science, biomedicine, and engineering. Yet both the discipline of directional radiometry and the radiative transfer theory (RTT) have traditionally been based on phenomenological concepts many of which turn out to be profound misconceptions. Contrary to the widespread belief, a collimated radiometer does not, in general, measure the flow of electromagnetic energy along its optical axis, while the specific intensity does not quantify the amount of electromagnetic energy transported in a given direction. The recently developed microphysical approach to radiative transfer and directional radiometry is explicitly based on the Maxwell equations and clarifies the physical nature of measurements with collimated radiometers and the actual content of the RTE. It reveals that the specific intensity has no fundamental physical meaning besides being a mathematical solution of the RTE, while the RTE itself is nothing more than an intermediate auxiliary equation. Only under special circumstances detailed in this review can the solution of the RTE be used to compute the time-averaged local Poynting vector as well as be measured by a collimated radiometer. These firmly established facts make the combination of the RTE and a collimated radiometer useful in a well-defined range of applications. However, outside the domain of validity of the RTT the practical usefulness of measurements with collimated radiometers remains uncertain, while the theoretical modeling of these measurements and the solution of the energy-budget problem require a more sophisticated approach than solving the RTE.

This note serves as a short introduction to the reprint of our article "T-matrix computations of light scattering by nonspherical particles: a review" (JQSRT 1996; 55:535:75). We first discuss the motivation for writing that article and explain its historical context. This is followed by a short overview of more recent developments.

Although the note by Hapke and Nelson has virtually no relevance to our original publication, it contains a number of statements that are misleading and/or wrong. We, therefore, use this opportunity to dispel several profound misconceptions that continue to hinder the progress in remote sensing of planetary surfaces.

Several spectacular backscattering effects observed for particulate planetary surfaces have been interpreted in terms of the effect of weak localization (WL) of electromagnetic waves. However, the interference concept of WL explicitly relies on the notion of phase of an electromagnetic wave and is strictly applicable only when particles forming the surface are widely separated. Therefore, one needs a definitive quantitative proof of the WL nature of specific optical effects observed for densely packed particulate media. We use numerically exact computer solutions of the Maxwell equations to simulate electromagnetic scattering by realistic models consisting of large numbers of randomly positioned, densely packed particles. By increasing the particle packing density from zero to ~40%, we track the onset and evolution of the full suite of backscattering optical effects predicted by the low-density theory of WL, including the brightness and polarization opposition effects (BOE and POE). We find that all manifestations of WL, except the circular polarization ratio and POE, are remarkably immune to packing-density effects. Even POE can survive packing densities typical of planetary regolith surfaces. Our numerical data coupled with the results of unique observations at near-backscattering geometries demonstrate that the BOE and POE detected simultaneously for high-albedo solar system objects are caused by the effect of WL.

We analyze ground-based radar polarimetric observations of Saturn's rings at a wavelength of 12.6 cm by employing the model of a vertically and horizontally plane-parallel homogeneous slab composed of clumpy particles in the form of fractal aggregates of small ice monomers. Our model takes full account of the effects of polarization, multiple scattering, and coherent backscattering. Using efficient superposition T-matrix and vector radiative transfer codes, we perform computations of the backscattering circular polarization ratio for fractal aggregates generated with a cluster–cluster aggregation model and having the following characteristics: monomer refractive index m=1.78+i0.003; monomer packing density p=0.2; fractal dimensions Df=2.5 and 3; and overall fractal radii R in the range 4⩽R⩽10 cm. In order to obtain physically realistic values of single-scattering properties of the aggregates we perform averaging over an ensemble of clusters generated for the same values of fractal parameters but having different geometrical configurations of the monomers. We conclude that in the framework of the above morphological model of Saturn's rings and the specific cluster–cluster aggregation procedure, it may be problematic to obtain a satisfactory and realistic agreement between theoretical computations and the observed values of the radar circular polarization ratio.

We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, Df, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers NS, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when Dfo2 but increases rapidly when Df42. However, a systematic reduction in light absorption with Df is observed for clusters with sufficiently large NS, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.

This paper continues a systematic theoretical analysis of electromagnetic scattering by a group of arbitrarily sized, shaped, and oriented particles embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The previously developed microphysical approach is used to derive the generalized form of the radiative transfer equation (RTE) applicable to a large group of sparsely, randomly, and uniformly distributed particles. The derivation of the RTE directly from the macroscopic Maxwell equations yields unambiguous and definitive analytical expressions for the participating quantities and thereby fully resolves the lasting controversy caused by the conflicting outcomes of several phenomenological approaches.

We analyse the results of computations of the intensity and degree of linear polarization of diffusely reflected sunlight for the centre of a planetary disc in the phase-angle range 0 ◦

The optical and radiative properties of dust particles in solar and thermal infrared regions are investigated. Dust particles are assumed to be spheres and spheroids for a comparison aimed at understanding the nonsphericity effect of these particles on the radiation at the top of a dusty atmosphere. The classical Lorenz–Mie theory is employed to compute the optical properties of spherical dust particles. To compute the single-scattering properties of spheroidal dust particles, a combination of the T-matrix method and an approximate method is used in the present study. In the approximate method, applicable to large particles, the geometric optics method is applied to the computation of the scattering phase matrix. A combination of the solution from the geometric optics method and the contribution of the so-called edge effect is used to compute the extinction efficiency of a spheroidal particle whose absorption efficiency is computed by adding the so-called above- and below-edge effect (a term from the well-known complex angular momentum theory) to the geometric optics result. Numerical results show that the results from the T-matrix method and the present approximate approach converge at a size parameter of 50 for computing the integrated scattering properties (i.e., the extinction efficiency, single-scattering albedo, and asymmetry factor). Additionally, the phase functions computed from the two methods are quite similar for size parameters larger than 40 although some considerable differences may still be noticed for other phase matrix elements. Furthermore, the effect of surface roughness on the single-scattering properties of spheroidal particles is discussed. The present radiative transfer simulations illustrate the nonsphericity effect of dust particles is significant at short wavelengths, however, not at the thermal infrared wavelengths.

For a sparse, disordered, plane-parallel particulate medium, we analyze quantitatively the effects of particle microphysical properties on the values of the linear and circular backscattering polarization ratios. Using numerically exact T-matrix and vector radiative-transfer codes, we performed computations for the following models: (1) a semi-infinite homogeneous layer composed of randomly oriented, polydisperse oblate spheroids with the real part of the refractive index equal to 1.2, 1.4 and 1.6 and the imaginary part of the refractive index equal to 0 and 0.01; (2) a semi-infinite homogeneous layer composed of randomly oriented, polydisperse, oblate circular cylinders with the refractive index 1.4; (3) finite homogeneous layers of various optical thickness composed of randomly oriented, polydisperse, oblate spheroids with the refractive index 1.55. Our computations demonstrate that the values of the polarization ratios depend substantially on particle shape, real and imaginary parts of the particle refractive index, particle size relative to the wavelength, illumination geometry and optical thickness.

We use the superposition T-matrix method to compute the scattering matrix elements and optical cross-sections for a variety of complex soot and soot-containing aggregate particles in random orientation at a visible wavelength 0.628 μm. It is shown that random variations in the geometrical configuration of monomers in a soot cluster for fixed fractal dimension and prefactor, monomer size, and number of monomers have a rather weak effect on scattering and absorption, at least in the visible part of the spectrum. Thus, the electromagnetic scattering and absorption characteristics of a single cluster realization are sufficient to represent the mean values obtained by averaging over many realizations of the “equivalent” clusters generated for the same fractal parameters. However, the results for the soot clusters differ fundamentally from those calculated for the volume-equivalent soot sphere and for the corresponding external mixture of soot monomers, assuming that there are no electromagnetic interactions between the monomers. We also compute and analyze the scattering and absorption properties of aerosols formed by semi-external aggregation of larger ammonium sulfate, silica, or dust particles with soot clusters as well as semi-external aggregates consisting of several components with different sizes and refractive indices. Depending on its chemical composition and size, the larger particle that is in touch with a soot cluster can strongly influence, or even dominate, the overall optical characteristics of the aggregate. Aggregation can result in stronger extinction, absorption, and scattering cross-sections relative to those computed for the corresponding external mixture. Possibly owing to mutual shadowing, the optical cross-sections of multi-component aggregates are smaller than those of their externally mixed counterparts, but by no more than ∼20%. Implications of our study for analyses of remote sensing observations and atmospheric radiation balance computations are discussed.

The NASA Glory mission is intended to facilitate and improve upon long-term monitoring of two key forcings influencing global climate. One of the mission's principal objectives is to determine the global distribution of detailed aerosol and cloud properties with unprecedented accuracy, thereby facilitating the quantification of the aerosol direct and indirect radiative forcings. The other is to continue the 28-yr record of satellite-based measurements of total solar irradiance from which the effect of solar variability on the Earth's climate is quantified. These objectives will be met by flying two state-of-the-art science instruments on an Earth-orbiting platform. Based on a proven technique demonstrated with an aircraft-based prototype, the Aerosol Polarimetry Sensor (APS) will collect accurate multiangle photopolarimetric measurements of the Earth along the satellite ground track within a wide spectral range extending from the visible to the shortwave infrared. The Total Irradiance Monitor (TIM) is an improved version of an instrument currently flying on the Solar Radiation and Climate Experiment (SORCE) and will provide accurate and precise measurements of spectrally integrated sunlight illuminating the Earth. Because Glory is expected to fly as part of the A-Train constellation of Earth-orbiting spacecraft, the APS data will also be used to improve retrievals of aerosol climate forcing parameters and global aerosol assessments with other A-Train instruments. In this paper, we detail the scientific rationale and objectives of the Glory mission, explain how these scientific objectives dictate the specific measurement strategy, describe how the measurement strategy will be implemented by the APS and TIM, and briefly outline the overall structure of the mission. It is expected that the Glory results will be used extensively by members of the climate, solar, atmospheric, oceanic, and environmental research communities as well as in education and outreach activities.

We employ the numerically exact superposition T-matrix method to perform extensive computations of elec nottromagnetic scattering by a volume of discrete random medium densely filled with increasingly absorbing as well as non-absorbing particles. Our numerical data demonstrate that increasing absorption diminishes and nearly extinguishes certain optical effects such as depolarization and coherent backscattering and increases the angular width of coherent backscattering patterns. This result corroborates the multiple-scattering origin of such effects and further demonstrates the heuristic value of the concept of multiple scattering even in application to densely packed particulate media.

This tutorial paper provides a general overview of the hierarchy of problems involving electromagnetic scattering by particles and clarifies the place of the radiative transfer theory and the theory of coherent backscattering in the context of classical electromagnetics. The self-consistent microphysical approach to radiative transfer is compared with the traditional phenomenological treatment.

We tested the applicability of three rigorous radiative transfer computational approaches, namely, the discrete ordinates radiative transfer (DISORT) method, the adding–doubling approach, and an efficient computational technique based on Ambartsumian's nonlinear integral equation for computing the bidirectional reflectance of a semi-infinite layer. It was found that each of these three models, in a combination with the truncation of the forward peak of the bulk scattering phase functions of ice particles, can be used to simulate the bidirectional reflectance of a semi-infinite snow layer with appropriate accuracy. Furthermore, we investigate the sensitivity of the bidirectional reflectance of a homogeneous and optically infinite snow layer to ice crystal habit and effective particle size. It is shown that the bidirectional reflectance is not sensitive to the particle effective size in the visible spectrum. The sensitivity of the bidirectional reflectance in the near-infrared spectrum to the particle effective size increases with the increase of the incident wavelength. The sensitivity of the bidirectional reflectance to the effective particle size and shape is attributed fundamentally to the sensitivity of the single-scattering properties to particle size and shape. For a specific ice crystal habit, the truncated phase function used in the radiative transfer computations is not sensitive to particle effective size. Thus, the single-scattering albedo is primarily responsible for the sensitivity of the bidirectional reflectance to particle size, particularly, at a near-infrared wavelength.

The superposition T -matrix method is used to compute the scattering of unpolarized light by semi-external aerosol mixtures in the form of polydisperse, randomly oriented two-particle clusters with touching components. The results are compared with those for composition-equivalent external aerosol mixtures, in which the components are widely separated and scatter light in isolation from each other. It is concluded that aggregation is likely to have a relatively weak effect on scattering and radiative properties of two-component tropospheric aerosols and can be replaced by the much simpler external-mixture model in remote sensing studies and atmospheric radiation balance computations.

Starting from first principles, we present a detailed analysis of the concept of single scattering of light by a small volume element filled with sparsely and randomly positioned particles. We first derive the formulas of the far-field single-scattering approximation, which treats the volume element as a single scatterer, and discuss its range of applicability, using for illustration exact T-matrix results for randomly oriented two-sphere clusters. Our second approach is to treat the volume element as a small cloud of particles and apply the so-called first-order-scattering approximation. We demonstrate that although the two approaches are based on somewhat different sets of assumptions, they give essentially the same result for the electromagnetic response of a sufficiently distant polarization-sensitive detector.

Sunlight illuminating the Earth's atmosphere is scattered by gas molecules and suspended particles, giving rise to blue skies, white clouds, and optical displays such as rainbows and halos. By scattering and absorbing the shortwave solar radiation and the longwave radiation emitted by the underlying surface, cloud and aerosol particles strongly affect the radiation budget of the terrestrial climate system. As a consequence of the dependence of scattering characteristics on particle size, morphology, and composition, scattered light can be remarkably rich in information on particle properties and thus provides a sensitive tool for remote retrievals of macro- and microphysical parameters of clouds and aerosols.