Numerical accuracy of “equivalent” spherical approximations for computing ensemble-averaged scattering properties of fractal soot aggregates

Numerical accuracy is quantitatively assessed in conjunction with the application of four “equivalent” spherical approximations in the computation of the optical properties of small aggregate soot particles. The approximations are based on equal volume, equal surface area, the radius of gyration, and a collection of independent spheres with the same volume and the same volume-to-projected area ratio as the original nonspherical particle. A diffusion-limited cluster-cluster aggregation algorithm is used to specify the geometries of soot particles. Furthermore, the Generalized Multi-particle Mie (GMM) method is utilized to compute the single-scattering properties of individual soot aggregate particles assumed to be randomly oriented in space. The ensemble-averaged single-scattering properties of the particles are obtained by accounting for the probability distribution functions (PDF) of the number of monomers per aggregate at two wavelengths, 0.628 and 1.1 μm. It is shown that all of the aforementioned equivalent-spherical approximations lead to large errors in the computation of the phase function.     

Interior radiances in optically deep absorbing media—III. Scattering from haze L

The interior radiances are calculated within an optically deep absorbing medium scattering according to the Haze L phase function. The dependence on the solar zenith angle, the single scattering albedo, and the optical depth within the medium is calculated by the matrix operator method. The development of the asymptotic angular distribution of the radiance in the diffusion region is illustrated through a number of examples; it depends only on the single scattering albedo and on the phase function for single scattering. The exact values of the radiance in the diffusion region are compared with values calculated from the approximate equations proposed by Van de Hulst. The variation of the radiance near the lower boundary of an optically thick medium is illustrated with examples. The attenuation length is calculated for various single scattering albedos and compared with the corresponding values for Rayleigh scattering. The ratio of the upward to the downward flux is found to be remarkably constant within the medium. The heating rate is calculated and found to have a maximum value at an optical depth of two within a Haze L layer when the sun is at the zenith. The location of this maximum moves toward the top of the haze layer as the solar zenith angle increases and also as the single scattering albedo decreases. When the single scattering albedo is less than 0·8, the downward flux is so small within the diffusion region that experimental measurements are probably not possible.     

A three-parameter analytic phase function for multiple scattering calculations

A very simple procedure has been developed to fit the first three moments of an actual phase function with a three parameter analytic phase function. The exact Legendre Polynomial decomposition of this function is known which makes it quite suitable for multiple scattering calculations. The use of this function can be expected to yield excellent flux values at all depths within a medium. Since it is capable of reproducing the glory, it can be used in synthetic spectra computations from planetary atmospheres. Accurate asymptotic radiance values can also be achieved as long as the single scattering albedo ω₀ ?0.9.     

Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium: errata

In our recent Letter,(1) several typographical errors were present. On p. 487, in Fig. 2, the equations for the following Mueller matrix elements should read as S(14) = (RO - LO), S(22) = (HH + VV) - (HV + VH), S(23) = (PH + MV) - (PV + MH), S(24) = (RH + LV) - (RV + LH), S(32) = (HP + VM) - (HM + VP), S(33) = (PP + MM) - (PM + MP), S(34) = (RP + LM) - (RM + LP), S(41) = (OR + OL), S(42) = (HR + VL) - (HL + VR), S(43) = (PR + ML) - (PL + MR), and S(44) = (RR + LL) - (RL + LR). Also on p. 487, in the left-hand column, line 10 from the top should read as follows: mfp? = 1/[mua + mus(1 - g)], was 0.957 cm.     

A fast infrared radiative transfer model based on the adding-doubling method for hyperspectral remote-sensing applications

A fast infrared radiative transfer (RT) model is developed on the basis of the adding-doubling principle, hereafter referred to as FIRTM-AD, to facilitate the forward RT simulations involved in hyperspectral remote-sensing applications under cloudy-sky conditions. A pre-computed look-up table (LUT) of the bidirectional reflection and transmission functions and emissivities of ice clouds in conjunction with efficient interpolation schemes is used in FIRTM-AD to alleviate the computational burden of the doubling process. FIRTM-AD is applicable to a variety of cloud conditions, including vertically inhomogeneous or multilayered clouds. In particular, this RT model is suitable for the computation of high-spectral-resolution radiance and brightness temperature (BT) spectra at both the top-of-atmosphere and surface, and thus is useful for satellite and ground-based hyperspectral sensors. In terms of computer CPU time, FIRTM-AD is approximately 100-250 times faster than the well-known discrete-ordinate (DISORT) RT model for the same conditions. The errors of FIRTM-AD, specified as root-mean-square (RMS) BT differences with respect to their DISORT counterparts, are generally smaller than 0.1 K.     

Interior radiances in optically deep absorbing media—II Rayleigh scattering

The interior radiances are calculated within an optically deep absorbing medium scattering according to the Rayleigh phase function. The accuracy of the matrix operator method is improved by many orders of magnitude through the use of accurate starting values obtained by the Runge-Kutta method rather than from the single scattering approximation. The radiance and flux are given for a range of solar zenith angles and for single scattering albedos of 1, 0.99, 0.9, 0.5 and 0.1. The development of the asymptotic angular distribution of the radiance is illustrated. It is shown that this asymptotic distribution is probably physically unobservable when ω₀ < 0.8, since the flux is less than 10^-8 of its original value at the beginning of the asymptotic region. The ratio of the upward to downward flux is calculated and is shown to be remarkably constant within the medium except very close to the boundaries. The heating rate within the medium is found to be very nearly proportional to the downward flux, except near the boundaries. When the single scattering albedo is small, a number of examples illustrate the significant contribution of the direct solar flux to the total flux even at great optical depths within the medium. The total downward flux decreases exponentially with optical depth away from boundaries when the single scattering albedo is greater than or equal to 0.9; when it is less than or equal to 0.5 only an approximate exponential fit can be obtained within the region accessible to experimental observation.     

Solutions of the equation of transfer for a medium bounded by a perfect specular reflector in terms of those for a perfect absorber

In a recent paper, Castiet al.(su1) showed that the solution to the equation of radiative transfer for a homogeneous, plane-parallel atmosphere bounded below by a perfect specular reflector could be expressed in terms of the solutions for the same atmosphere bounded from below by a perfect absorber. They only considered the case of isotropic atmospheric scattering and hence there was no azimuthal dependence. A simple proof of this relation is given here. Furthermore, it is shown to be valid in the more general case of anisotropic scattering in an inhomogeneous atmosphere.     

Comment on the transmission matrix for a dielectric interface

In a recent paper the reflection and transmission matrices for a dielectric interface based on Fresnel formulas are derived [Garcia RDM. Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium. J Quant Spectrosc Radiat Transfer 2012;113:306–17]. Although the final formulas appear to be correct, we found that there are some significant conceptual and logical flaws in the derivation. Here we explain that the misunderstanding is due to the different physical significances of the Stokes parameters for the coherent and diffuse radiation field and that the so-called transmission factor directly originates from the physical definition of the Stokes parameters. We also clarify a few incorrect interpretations in the aforementioned paper about previously published works.     

Energy transfer between laser filaments in liquid methanol

We demonstrate energy exchange between two filament-forming femtosecond laser beams in liquid methanol. Our results are consistent with those of previous works documenting coupling between filaments in air; in addition, we identify an unreported phenomenon in which the direction of energy exchange oscillates at increments in the relative pulse delay equal to an optical period (2.6 fs). Energy transfer from one filament to another may be used in remote sensing and spectroscopic applications utilizing femtosecond laser filaments in water and air.