Light Scattering Reviews 5

A non-sphericity of a particle plays an important role in light scattering processes, resulting in different scattering and absorption of incident light compared with spherical particles. Among various non-spherical shapes of particles, aggregates of small particles are often applied as model shapes to particles observed actually in nature, such as dust from cometary nuclei, soot aerosols floating in the Earth’s atmosphere, and microbiocells composed of sets of small organic cells.

Solid-liquid suspensions are frequently used in industrial processes. These suspensions usually contain aggregates made up of solid primary particles. Many characterization tools of these suspensions are based on light scattering (Mie theory). However, Mie theory (1908) is not always applicable to practical problems since the scatterer must be a homogeneous sphere. The ordinary particle sizers that use this theory do not make it possible to measure non-spherical particle geometrical characteristics. Extensions of the Mie theory for arbitrary shaped particles or particle aggregates are available nowadays (the T-matrix method, the Generalized Multiparticle Mie (GMM)-solution, etc.). But the computing times of the optical properties via these exact theories do not allow for a real-time analysis. This chapter is therefore dedicated to the search for approximate methods for the estimation of aggregate optical properties, particularly their scattering cross-section.

In the near future, more and more spaceborne or airborne instruments will be able to measure polarized reflectance issued from the atmosphere. To give some examples, currently, the POLarization and Directionality of the Earth’s Reflectance instrument POLDER3/ PARASOL, which is the successor of POLDER2/ADEOS2 and POLDER/ADEOS (Deschamps et al., 1994) measures, since 2005, the polarized signal in the visible spectral range with up to 14 viewing directions. The airborne version of this instrument, called OSIRIS (observing system including polarization in the solar infrared spectrum (Auriol et al., 2008)), is nowadays extended to the near-infrared range and will maybe, in the future, generate a spaceborne version. The Aerosol Polarimetry Sensor (APS), the spaceborne version of the Research Scanning Radiometer (RSP) will be able to measure reflected total and polarized light in visible, near infrared, and short-wave infrared and should be launched in the framework of the Glory mission in 2010 (Mishchenko et al., 2007).

The remote sensing of atmospheric constituents with limb-viewing satellite instruments or with nadir viewing instruments at large solar zenith angles requires a forward model that simulates the backscattered radiance taking the spherical shape of the Earth atmosphere into account. In addition, many retrieval schemes are based on a linearization of such a forward model. Whenever it is important to take multiple scattering into account (e.g. due to light scattering air molecules, aerosols and clouds) the linearization of the measurement simulation with respect to the parameters to be retrieved is not trivial. Here, the forward-adjoint perturbation theory provides a general method to linearize radiative transfer. In the first part of this review chapter we provide the theoretical background of the linearization approach for a radiative transfer problem in a spherical model atmosphere which is illuminated by a collimated solar beam. Using an operator formulation of radiative transfer allows one to express the linearization approach in a universally valid notation. Depending on the particular formulation of the radiative transfer problem the perturbation of internal sources has to be taken into account in addition. The needed adjoint calculation corresponds to a so-called searchlight problem that requires the use of three-dimensional radiative transfer simulations in general. Subsequently we show how symmetries of the forward radiation field and a proper choice of the radiation sources can be used to simplify the needed adjoint calculations substantially.

All the numerical methods of the radiative transfer equation (RTE) solution are based on the replacement of scattering integral by the finite sum. The main problem of such scattering integral representation is the presence of singularities in the radiance angular distribution (RAD): they cannot be included in any quadrature formula in essence (Krylov, 2006). To solve this problem various methods are used by different researchers: Wiscombe (1977), Thomas and Stamnes (2002), Rozanov et al. (2005) and others. The physical model of radiative transfer theory is the ray approximation. In this approximation any break in the boundary conditions spreads into the depth of the medium and generates singularities in the RAD. Scattering in the medium gives, along with the singularities, an anisotropic part in the light field. The difficulties in the calculation of the anisotropic part of the solution led to various truncation methods of the scattering anisotropy that are not completely true: in the general 3-D medium geometry the core of the problem is not the scattering, but the presence of singularities in the RAD. Krylov (2006) showed that the most effective and accurate way of including the singularities in the quadrature formula is their analytical, perhaps approximate, representation and an analytical integration - the method of singularities elimination.

Numerous numerical methods have been developed to solve a plane-parallel radiative transfer problem, including discrete ordinates (Stamnes et al., 1988; Spurr et al., 2001), spherical harmonics (Dave, 1975; Benassi et al., 1984), adding-doubling (Hansen and Hovenier, 1971; Twomey, 1985), successive orders of scattering (Lenoble et al., 2007) etc. An extended reference to numerical methods and publicly available codes can be found in Lenoble (1985), Ricchiazzi et al. (1998), Mayer and Kylling (2005) and Cahalan (2005). In this chapter, we describe the method of spherical harmonics (MSH), in particular its very efficient implementation developed by Karp et al. (1980) and, later, Muldashev et al. (1999). From a numerical standpoint, here are several main components of the spherical harmonics solution: (1) obtaining the system of linear differential equations of MSH, (2) its reduction to the system of linear algebraic equations using singular value decomposition (SVD), (3) use of a system’s matrix symmetry to halve its size for SVD transformation with ~8 times gain in speed, and finally, (4) angular smoothing of the solution for radiance calculations in arbitrary directions. The detail of MSH for the 1-D radiative transfer problem with a uniform surface, and for the 3-D problem with a spatially variable surface, are presented in sections 6.1 and 6.5, respectively. Section 6.2 provides an overview of the 1-D radiative transfer code SHARM (Muldashev et al., 1999; Lyapustin and Wang, 2005) which is one of the most numerically efficient scalar codes. SHARM performs simultaneous monochromatic calculations for multiple sun-view geometries, and allows the user to make multi-wavelength calculations in one run. The code is user-friendly, featuring built-in aerosol models and the most popular models of the bi-directional reflectance factor (BRF) of land and wind-ruffled water surface. Comparisons of SHARM with the benchmark code DISORT showed agreement to 0.02%.

While solving different theoretical and applied problems of planetary atmosphere optics (Sobolev, 1975; van de Hulst, 1980; Yanovitskij, 1997; Liou, 2002; Kokhanovsky, 2004; Hovenier et al., 2004; Garcia et al., 2008), astrophysics (Mihalas, 1978; Dolginov et al., 1995; Leroy, 2000; Nagirner, 2001), scattering media optics and biophysics (Yi et al., 1992; Tuchin, 1997, 2000; Klose, 2009), signal and image transfer (Zege et al., 1991; Gibson et al., 2005), nuclear plant physics (Davison, 1958; Marchuk, 1961; Marchuk and Lebedev, 1971; Sanchez and McCormick, 1982; Ganapol and Kornreich, 1995; Ganapol, 2008), etc., one should investigate the radiative (neutron) transfer process through macroscopically homogeneous and inhomogeneous, absorbing and scattering media (in particular, through turbid media). The study of the features of this process in various real and artificial absorbing and scattering media requires accounting for their shapes, locations of external and internal sources, kinds of functional dependences of local characteristics of the said media on spatial and angular coordinates.

Earth’s climate is known to be sensitive to the light scattering properties of sea ice across the Arctic Basin. For the purpose of understanding its interactions with shortwave radiation, sea ice can be described as a multiply scattering medium whose microstructural properties govern the quantity and quality of light scattering. The ice is populated with numerous inclusions of brine, air, precipitated salt crystals, and impurities embedded within a matrix of pure ice (e.g., see Weeks and Ackley, 1982). The multiple scattering caused by these inclusions in thick, bare sea ice generally causes more than half of the incident solar radiation to be backscattered to the atmosphere (Perovich, 1990). However, the structure of this composite material is complex and so the optical properties of sea ice have proven difficult to quantify and generalize. Since the liquid brine contained in sea ice must remain in freezing equilibrium with the ice, temperature changes can produce significant changes in the size and number distributions of these inclusions which determine the backscattering, absorption, and transmission of shortwave radiation at the frozen surface of the ocean.

Seasonal snow covers large parts of the northern hemisphere annually. It can change the albedo of the surfaces from dark to bright overnight (and back), causing significant climate feedback (Manninnen and Stenberg, 2008; Flanner and Zender, 2006; Pirazzini, 2008; Nolin and Frei, 2001; Roesch et al., 2001). It forms large energy reservoirs which can be exploited by hydro energy power plants, and is the source of big floods when melting. It can significantly impact traffic and construction safety. It changes living and environmental conditions radically, and has major recreational value.

When developing and applying models to light scattering problems, things usually turn very mathematical. This is all in good order, but it may also be a hindrance for a broader audience to gain insight into the overall issues. This chapter aims at discussing a range of light scattering simulation and modeling issues with a minimum of mathematics involved, and with the specific perspective of paper and printing industry applications. Shorter sections of mathematical content are included, but the mathematically interested reader is here pointed to selected references and other chapters in this volume.

Atmosphereless solar-system objects exhibit two ubiquitous light-scattering phenomena at small solar phase angles (sun.object.observer angle α): first, the opposition effect in the intensity of scattered sunlight (e.g., [1]); and, second, the negative degree of linear polarization (I _┴ − I _║)/(I _┴ + I _║). Here I _║ denotes the intensity component parallel to the scattering plane defined by the Sun, the object, and the observer and I _┴ denotes the component perpendicular to that plane [2].