Comparison of electromagnetic theory and various approximations for computing the absorption efficiency and single-scattering albedo of hexagonal columns

The applicability of various approximations for computing the absorption efficiency and single-scattering albedo of a randomly oriented hexagonal column is tested versus electromagnetic theory. To calculate the absorption efficiency and single-scattering albedo of the hexagonal column from electromagnetic theory we used a generalization to the separation-of-variables method, which enables continuous calculation of optical properties up to size parameters of 86. We found that the asymptotic absorption efficiency is independent of particle shape, and that, as the size parameter increases, the hexagonal column tends to its asymptotic absorption value more quickly than Mie theory. The asymptotic absorption limit of the hexagonal column is calculated accurately (to within 1%) and rapidly by use of the complex-angular-momentum approximation, indicating that this approximation could be used to calculate the absorption limit of nonspherical particles. The equal-volume sphere best approximates the hexagonal column single-scattering albedo at a strongly absorbing wavelength (e.g., 11.9 mum for an ice particle). However, in the resonance region (e.g., 80 mum for an ice particle) Mie theory fails to approximate the single-scattering albedo of the hexagonal column, but as the size parameter exceeds 10 the error in the sphere approximation reduces to within 2%. At 80-mum wavelength there is a characteristic ripple structure superimposed on the hexagonal column absorption efficiency solutions between size parameters from approximately 1 to 4. The ripple structure is indicative of surface-wave interference and is similar to the sphere but less pronounced on the hexagonal column. We investigated the applicability of ray tracing for calculating the single-scattering albedo at absorbing wavelengths relevant to remote sensing of ice particles in the atmosphere and found it to be within 4% for size parameters between 3 and 42 at 3.7-mum wavelength. At mid-infrared wavelengths (e.g., 8.5 and 11.9 mum) ray tracing is within 5% of electromagnetic theory for size parameters exceeding 10. We also tested the Bryant and Latimer absorption approximation to anomalous diffraction theory by using the separation-of-variables method.     

Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles

The pseudospectral time-domain (PSTD) method is a powerful approach for computing the single-scattering properties of arbitrarily shaped particles with small-to-moderate-sized parameters. In the PSTD method, the spatial derivative approximation based on the spectral method is more accurate than its counterpart based on the finite-difference technique. Additionally, the PSTD method can substantially diminish accumulated errors that increase with the spatial scale and temporal duration of simulation. We report on the application of the PSTD method to the scattering of light by nonspherical ice particles. The applicability of the PSTD method is validated against the Lorenz-Mie theory and the T-matrix method. The phase functions computed from the PSTD method and the Lorenz-Mie theory agree well for size parameters as large as 80. Furthermore, the PSTD code is also applied to the scattering of light by nonspherical ice crystals, namely, hollow hexagonal columns and aggregates, which are frequently observed in cirrus clouds. The phase functions computed from the PSTD method are compared with the counterparts computed from the finite-difference time-domain (FDTD) method for a size parameter of 20 and an incident wavelength of 3.7 microm. The comparisons show good agreement between the two methods.     

Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals

A new geometric-optics model has been developed for the calculation of the single-scattering and polarization properties for arbitrarily oriented hexagonal ice crystals. The model uses the ray-tracing technique to solve the near field on the ice crystal surface, which is then transformed to the far field on the basis of the electromagnetic equivalence theorem. From comparisons with the results computed by the finite-difference time domain method, we show that the novel geometric-optics method can be applied to the computation of the extinction cross section and single-scattering albedo for ice crystals with size parameters along the minimum dimension as small as ~6. Overall agreement has also been obtained for the phase function when size parameters along the minimum dimension are larger than ~20. We demonstrate that the present model converges to the conventional ray-tracing method for large size parameters and produces single-scattering results close to those computed by the finite-difference time domain method for size parameters along the minimum dimension smaller than ~20. The present geometric-optics method can therefore bridge the gap between the conventional ray-tracing and the exact numerical methods that are applicable to large and small size parameters, respectively.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Influence of crystal tilt on solar irradiance of cirrus clouds

The single and multiple scattering and absorption properties of hexagonal ice columns with different degrees of particle orientation are modeled in the solar spectral range by means of a ray-tracing single-scattering code and a Monte Carlo radiative-transfer code. The scattering properties are most sensitive to particle orientation for the solar zenith angles of 50 degrees (asymmetry parameter) and 90 degrees (single-scattering albedo). Provided that the ice columns are horizontally oriented, the usual assumption of random orientation leads to an overestimation (underestimation) of the reflected (transmitted) solar broadband radiation at high Sun elevation and to an underestimation (overestimation) at medium solar zenith angles. The orientation effect is more (less) pronounced in scattering and transmission (absorption) for smaller ice crystals.     

Simulation of infrared scattering from ice aggregates by use of a size-shape distribution of circular ice cylinders

The scalar optical properties (extinction coefficient, mass extinction coefficient, single-scattering albedo, and asymmetry parameter) of a distribution of randomly oriented ice aggregates are simulated generally to well within 4% accuracy by use of a size-shape distribution of randomly oriented circular ice cylinders at wavelengths in the terrestrial window region. The single-scattering properties of the ice aggregates are calculated over the whole size distribution function by the finite-difference time-domain and improved geometric optics methods. The single-scattering properties of the size-shape distribution of circular ice cylinders are calculated by the T-matrix method supplemented by scattering solutions obtained from complex-angular-momentum theory. Moreover, radiative-transfer studies demonstrate that the maximum error in brightness temperature space when the size-shape distribution of circular ice cylinders is used to represent scattering from ice aggregates is only approximately 0.4 K The methodology presented should find wide applicability in remote sensing of ice cloud and parameterization of cirrus cloud scalar optical properties in climate models.     

Geometrical-optics solution to light scattering by droxtal ice crystals

We investigate the phase matrices of droxtals at wavelengths of 0.66 and 11 microm by using an improved geometrical-optics method. An efficient method is developed to specify the incident rays and the corresponding impinging points on the particle surface necessary to initialize the ray-tracing computations. At the 0.66-microm wavelength, the optical properties of droxtals are different from those of hexagonal ice crystals. At the 11-microm wavelength, the phase functions for droxtals are essentially featureless because of strong absorption within the particles, except for ripple structures that are caused by the phase interference of the diffracted wave.     

Simulation of the optical properties of plate aggregates for application to the remote sensing of cirrus clouds

In regions of deep tropical convection, ice particles often undergo aggregation and form complex chains. To investigate the effect of the representation of aggregates on electromagnetic scattering calculations, we developed an algorithm to efficiently specify the geometries of aggregates and to compute some of their geometric parameters, such as the projected area. Based on in situ observations, ice aggregates are defined as clusters of hexagonal plates with a chainlike overall shape, which may have smooth or roughened surfaces. An aggregate representation is developed with 10 ensemble members, each consisting of between 4-12 hexagonal plates. The scattering properties of an individual aggregate ice particle are computed using either the discrete dipole approximation or an improved geometric optics method, depending upon the size parameters. Subsequently, the aggregate properties are averaged over all geometries. The scattering properties of the aggregate representation closely agree with those computed from 1000 different aggregate geometries. As a result, the aggregate representation provides an accurate and computationally efficient way to represent all aggregates occurring within ice clouds. Furthermore, the aggregate representation can be used to study the influence of these complex ice particles on the satellite-based remote sensing of ice clouds. The computed cloud reflectances for aggregates are different from those associated with randomly oriented individual hexagonal plates. When aggregates are neglected, simulated cloud reflectances are generally lower at visible and shortwave-infrared wavelengths, resulting in smaller effective particle sizes but larger optical thicknesses.     

Scattering matrix of infrared radiation by ice finite circular cylinders

Scattering matrix characteristics of polydisperse, randomly oriented, small ice crystals modeled by finite circular cylinders with various ratios of the length to diameter (L/D) ratio are calculated by use of the exact T-matrix approach, with emphasis on the thermal infrared spectral region that extends from the atmospheric short-wave IR window to the far-IR wavelengths to as large as 30 microm. The observed ice crystal size distribution and the well-known power-law distribution are considered. The results of the extensive calculations show that the characteristics of scattering matrix elements of small ice circular cylinders depend strongly on wavelengths and refractive indices, particle size distributions, and the L/D ratios. The applicability of the power-law distribution and particle shapes for light scattering calculations for small ice crystals is discussed. The effects of the effective variance of size distribution on light scattering characteristics are addressed. It seems from the behavior of scattering matrix elements of small ice crystals that the combination of 25 and 3.979 microm has some advantages and potential applications for remote sensing of cirrus and other ice clouds.     

Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths

We investigate the errors associated with the use of circular cylinders as surrogates for hexagonal columns in computing the optical properties of pristine ice crystals at infrared (8-12-microm) wavelengths. The equivalent circular cylinders are specified in terms of volume (V), projected area (A), and volume-to-area ratio that are equal to those of the hexagonal columns. We use the T-matrix method to compute the optical properties of the equivalent circular cylinders. We apply the finite-difference time-domain method to compute the optical properties of hexagonal ice columns smaller than 40 microm. For hexagonal columns larger than 40 microm we employ an improved geometric optics method and a stretched scattering potential technique developed in previous studies to calculate the phase function and the extinction (or absorption) efficiency, respectively. The differences between the results for circular cylinders and hexagonal columns are of the order of a few percent. Thus it is quite reasonable to use a circular cylinder geometry as a surrogate for pristine hexagonal ice columns for scattering calculations at infrared (8-12-microm) wavelengths. Although the pristine ice crystals can be approximated as circular cylinders in scattering calculations at infrared wavelengths, it is shown that optical properties of individual aggregates cannot be well approximated by those of individual finite columns or cylinders.     

Applicability of regular particle shapes in light scattering calculations for atmospheric ice particles

We ascertain the usefulness of simple ice particle geometries for modeling the intensity distribution of light scattering by atmospheric ice particles. To this end, similarities and differences in light scattering by axis-equivalent, regular and distorted hexagonal cylindric, ellipsoidal, and circular cylindric ice particles are reported. All the results pertain to particles with sizes much larger than a wavelength and are based on a geometrical optics approximation. At a nonabsorbing wavelength of 0.55 μm, ellipsoids (circular cylinders) have a much (slightly) larger asymmetry parameter g than regular hexagonal cylinders. However, our computations show that only random distortion of the crystal shape leads to a closer agreement with g values as small as 0.7 as derived from some remote-sensing data analysis. This may suggest that scattering by regular particle shapes is not necessarily representative of real atmospheric ice crystals at nonabsorbing wavelengths. On the other hand, if real ice particles happen to be hexagonal, they may be approximated by circular cylinders at absorbing wavelengths.     

Visible and near infrared reflectances measured from laboratory ice clouds

We present laboratory results of the 0.68 microm visible (VIS) and 1.617 microm near infrared (NIR) reflectances typically used for inferring optical depth and ice crystal size from satellite radiometers, from ice clouds generated in a temperature controlled column cloud chamber. Two types of ice crystals were produced in this experiment: small columns and dendrites with mean maximum dimensions of about 17 and 35 microm. Within experimental uncertainty, the measured reflectances from ice clouds at both wavelengths agree reasonably well with the theoretical results computed from the plane-parallel adding-doubling method for radiative transfer using the measured ice particle morphology. We demonstrate that laboratory scattering and reflectance data for thin ice clouds with optical depths less than 0.4 can be used for validation of the thin cirrus optical depth and ice crystal size that have been routinely retrieved from the satellite VIS-NIR two channel pair.     

Characterization of atmospheric aerosols from infrared measurements: simulations, testing, and applications

An inversion method for the characterization of atmospheric condensed phases from infrared (IR) spectra is described. The method is tested with both synthetic IR spectra and the spectra of particles that flow in a cryogenic flow tube. The method is applied to the IR spectra recorded by the Atmospheric Trace Molecule Spectroscopy instrument carried by the Space Shuttle during three missions in 1992, 1993, and 1994. The volume density and particle size distribution for sulfate aerosol are obtained as a function of altitude. The density and size distribution of ice particles in several cirrus clouds are also retrieved. The probable radius of the ice particles in the high-altitude (10-15-km) cirrus clouds is found to be approximately 6-7 microm.     

In situ particle size measurements using a two-color laser scattering technique

We have developed a technique using light scattered from individual particles in the near-forward direction to measure particle size in the range of 10-200 microm. This technique uses the Mie scattering theory to relate the measured light intensity to particle size based on calibration techniques employing pinholes and water droplets of known size. We have applied a unique two-color optical arrangement to minimize the edge effect which can cause incorrect size measurements for particles that pass through the edge of the laser beam focal volume. In this paper we describe our experimental technique and the results of size measurements obtained with this technique for water droplets and pulverized coal particles.     

Theoretical calculation of uncertainty region based on the general size distribution in the preparation of standard reference particles for particle size measurement

In order to confirm the reliability of particle size measurement technique and to prepare standard reference particles for calibrating particle size measurement devices, experimental and theoretical studies have been conducted about the uncertainty region of particle size measurement for the general particle size distribution. A new theoretical equation to calculate fundamental uncertainty region in the case that the maximum and minimum particle sizes are known, is derived based on Tschebyscheff theory. The uncertainty regions calculated based on the proposed method are applied to poly-disperse particles and a picket-fence distribution composed of two kinds of nearly mono-disperse particles.For the poly-disperse particles, the uncertainty region increases with the increase in particle diameter. For the picket-fence distribution composed of two kinds of nearly mono-disperse particles, the uncertainty region increases around the intermediate particle diameters between the two kinds of particles.Numerical simulation of uncertainty region for the picket-fence distribution has also been carried out. The uncertainty region decreases with the increase in sample size or the decrease in geometric standard deviation.     

Determination of polar stratospheric cloud particle refractive indices by use of in situ optical measurements and T-matrix calculations

A new algorithm to infer structural parameters such as refractive index and asphericity of cloud particles has been developed by use of in situ observations taken by a laser backscattersonde and an optical particle counter during balloon stratospheric flights. All three main particles, liquid, ice, and a no-ice solid (NAT, nitric acid trihydrate) of polar stratospheric clouds, were observed during two winter flights performed from Kiruna, Sweden. The technique is based on use of the T-matrix code developed for aspherical particles to calculate the backscattering coefficient and particle depolarizing properties on the basis of size distribution and concentration measurements. The results of the calculations are compared with observations to estimated refractive indices and particle asphericity. The method has also been used in cases when the liquid and solid phases coexist with comparable influence on the optical behavior of the cloud to estimate refractive indices. The main results prove that the index of refraction for NAT particles is in the range of 1.37-1.45 at 532 nm. Such particles would be slightly prolate spheroids. The calculated refractive indices for liquid and ice particles are 1.51-1.55 and 1.31-1.33, respectively. The results for solid particles confirm previous measurements taken in Antarctica during 1992 and obtained by a comparison of lidar and optical particle counter data.     

Depolarization ratios of single ice particles assuming finite circular cylinders

Results of the depolarization ratio (delta) of single ice particles in fixed orientations are presented to determine whether discrimination between nonspherical ice crystals (causing depolarization) and spherical water droplets (inducing no depolarization) can be made. A T-matrix method is used to compute delta over a range of particle diameters from 0.13 to 4 microm and aspect ratios chi=d/h (d is the diameter and h the height of the particle) from 0.3 to 3, where ice crystals are assumed to have a circular cylindrical shape. The depolarization ratio is primarily dependent on the orientation of the particle. Some orientations return no depolarization, whereas others generate values reaching almost delta=1. Considering the depolarization averaged over all orientations, a dependence of delta with the particle size is observed where values close to 0.25 are reached. No strong influence of the aspect ratio on the depolarization for a given particle size of 2 mum is evident, as values remain in a range between 0.2 and 0.3.     

COP: a data library of optical properties of hexagonal ice crystals

The data library of optical properties of hexagonal ice crystals for radiative modeling, Cirrus Optical Properties (COP), is introduced. It includes phase functions, asymmetry parameters, extinction cross sections, and single scattering albedos. Furthermore, lidar ratios and depolarization are given. The dependence of these parameters on wavelength, particle size, and shape is calculated, and different particle orientations are considered. In addition, a simple FORTRAN code is provided to calculate the corresponding properties of size distributions. Thus the data library is a very flexible tool for determining the optical parameters of ice clouds for climatological purposes and remote sensing. The data library and the FORTRAN code are distributed through electronic mail.     

Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: a comparison of the T-matrix and the finite-difference time-domain methods

We calculated the scattering and absorption properties of randomly oriented hexagonal ice columns using T-matrix theory, employing analytic orientation averaging, and the finite-difference time-domain method, which uses a numerical procedure to simulate random orientation. The total optical properties calculated are the extinction efficiency, absorption efficiency, single-scattering albedo, and the asymmetry parameter. The optical properties are calculated at the wavelengths of 0.66, 8.5, and 12 mum, up to a size parameter of 20 at 0.66 mum and 15 at the two other wavelengths. The phase-matrix elements P11, P12, and P22 are also calculated and compared, up to a size parameter of 20 at 0.66 mum and 15 at 12.0 mum. The scattering and absorption solutions obtained from the two independent electromagnetic methods are compared and contrasted, as well as the central processing unit time and memory load for each size parameter. It is found that the total optical properties calculated by the two methods are well within 3% of each other for all three wavelengths and size parameters. In terms of the phase-matrix elements it is found that there are some differences between the T-matrix and the finite-difference time-domain methods appearing in all three elements. Differences between the two methods for the P11 element are seen particularly at scattering angles from approximately 120 degrees to 180 degrees ; and at the scattering angle of 180 degrees , relative differences are less than 16%. At scattering angles less than 100 degrees , agreement is generally within a few percent. Similar results are also found for the P12 and P22 elements of the phase matrix. The validity of approximating randomly oriented hexagonal ice columns by randomly oriented equal surface area circular cylinders is also investigated in terms of the linear depolarization ratio.     

Ice-crystal absorption: a comparison between theory and implications for remote sensing

The problem of the disagreement between cirrus crystal sizes determined remotely and by in situ measurements is shown to be due to inappropriate application of Mie theory. We retrieved the absorption optical depth at 8.3 and 11.1 mum from 11 tropical anvil cirrus clouds, using data from the High Resolution Infrared Radiation Sounder (HIRS). We related the absorption optical depth ratio between the two wavelengths to crystal size (the size was defined in terms of the crystal median mass dimension) by assuming Mie theory applied to ice spheres and anomalous diffraction theory (ADT) applied to hexagonal columns, hexagonal plates, bullet rosettes, and aggregates (polycrystals). The application of Mie theory to retrievals yielded crystal sizes approximately one third those obtained with ADT. The retrievals of crystal size by use of HIRS data are compared with measurements of habit and crystal size obtained from in situ measurements of tropical anvil cirrus particles. The results of the comparison show that ADT provides the more realistic retrieval. Moreover, we demonstrate that at infrared wavelengths retrieval of crystal size depends on assumed habit. The reason why Mie theory predicts smaller sizes than ADT is shown to result from particle geometry and enhanced absorption owing to the capture of photons from above the edge of the particle (tunneling). The contribution of particle geometry to absorption is three times greater than from tunneling, but this process enhances absorption by a further 35%. The complex angular momentum and T-matrix methods are used to show that the contribution to absorption by tunneling is diminished as the asphericity of spheroidal particles is increased. At an aspect ratio of 6 the contribution to the absorption that is due to tunneling is substantially reduced for oblate particles, whereas for prolate particles the tunneling contribution is reduced by 50% relative to the sphere.