Scattering parameter computation - Approximations

When dealing with radiative transfer calculations in cloudy atmospheres, interaction pro-cesses between the radiation and gas molecules and between radiation and hydrometeors have to be considered. The inuence of gases is described by their absorption coecients.

The eect of hydrometeors is included through their single-scattering properties, namely the absorption vector, the scattering matrix, and the extinction matrix. The intensity of the interaction between radiation and hydrometeors depends on various parameters like size, phase, orientation, shape, and refractive index of the particles, as well as on the considered wavelength.

As long as spheres are considered, the single-scattering properties can be accurately cal-culated by the Lorenz-Mie theory (Mie, 1908), since even frequencies in the submillimeter range together with particles with dimension around 1 cm do not result in size parameters larger than 60 (see Eq. 2.29). For sucient small particles, the single-scattering calculations can be performed by the Rayleigh approximations (e.g. Bohren and Clothiaux, 2006).

To achieve this goal for non-spherical particles, two approaches have been used so far.

One is to assume a known particle shape and therefore to describe it as accurately as possible by approximations like the T-matrix method (Mishchenko et al., 1996) or the discrete dipole approximation (DDA; Draine and Flatau, 2000). The second one is based on the Lorenz-Mie theory for the calculation of the single-scattering properties of spherical particles. Thereby, the problem of scattering at non-spherical particles is approximated

through modied diameters, and therefore densities, and dielectric properties (e.g. Liu and Curry, 2000; Evans et al., 2002; Bennartz and Petty, 2001; Kim et al., 2007).

In the following subsection some of the mostly common used methods to calculate the single-scattering properties in microwave region will be described.

Lorenz-Mie theory

When homogeneous spheres (or coated spherical objects) are considered the single-scattering properties in the microwave range can be described analytically by a series of spherical harmonics. The expansion coecients to this series are named Mie-coecients and only dependent on the complex index of refraction of the sphere n and the dimensionless size-parameter χ as described in Eq. 2.29. Details of the derivation of the function and the coecients describing the scattering properties can be found in Bohren and Clothiaux (2006) or Petty (2006).

Fig. 2.4 shows examples of the single-scattering properties in terms of eciencies (see Eqs. 2.38-2.40) and single-scattering albedo calculated with the Lorenz-Mie theory for typ-ical particles. The particles and the corresponding size parameters are described in Fig. 2.3 (water droplet, ice crystal, rain drop, and graupel/snow particle) for frequencies between 1 and 1000 GHz. In general, independent of particle type the absorption and scattering eciencies increase with increasing frequency due to the size parameter dependence, and the larger the particles the stronger their inuence on the radiation. For cloud droplets, absorption is the dominant process throughout the frequency range. Below 20 GHz, the interaction of rain drops with the radiation is due to absorption. Above 20 GHz absorp-tion eciency equals scattering eciency resulting in a single-scattering albedo of 0.5.

Rain drops have in general a much greater extinction eciency than frozen particles below 50 GHz and a much lower single-scattering albedo. This is the reason why remote sensing technics working at lower frequencies are essentially sensitive to water emission. At higher frequencies (above 50 GHz), graupel/snow extinction becomes comparable to rain extinc-tion. Graupel and snow particles are strong scatterers (single-scattering albedo is close to 1 above 10 GHz). Together with rain drops, they dominate the signal in the microwave frequency range. Since ice crystals are small with a very small imaginary part of the refrac-tive index, there is almost no absorption present at frequencies below 300 GHz and their eect is negligible compared to other hydrometeors. Above 300 GHz, their inuence gets noticeable through increased scattering eciency.

In Fig. 2.5 the dependency of the single-scattering properties to particle dimension at window frequencies in the millimeter-wavelength range is shown. The results were again obtained with Lorenz-Mie theory and reect that small particles are almost transparent, especially for the solid and mixing (graupel/snow) phase at lower frequencies. Larger graupel or snow particles (d > 2 mm) have a single-scattering albedo close to 1, whereas liquid particles converge to 0.5 as the size parameter gets larger. At this relatively low frequencies, ice crystals are hardly detectable.

Figure 2.4: Mie eciencies calculated with Lorenz-Mie theory for frequencies between 1 and 1000 GHz for typical particles: cloud droplet (d = 10 µm at 10^◦C, top left), rain drop (1000 µm at 10^◦C, top right, ice crystal (60 µm at -20^◦C, middle left), and graupel/snow particle (2000 µm at -5^◦C, middle right). The bottom row shows comparisons of the extinction eciencies (left) and the single-scattering albedo (right).

Figure 2.5: Mie eciencies and single-scattering albedo calculated with Lorenz-Mie theory for dierent frequencies (50, 89, 150, and 220 GHz) and dierent particle types/dielectric properties (see sec. 2.7): water droplets (top), ice crystals (middle), and graupel/snow particles as a mixing between pure ice and air (bottom).

T-matrix

The scattering properties for non-spherical, rotationally symmetric particles with an ad-ditional symmetry to their plane of rotation can be described by the T-matrix method (Mishchenko et al., 1996). For the calculation of the single-scattering properties, the Ex-tended Boundary Condition Method (EBCM; Waterman, 1971) is applied. For spheres, the formulations used in the T-matrix method reduce to the one describing the Lorenz-Mie theory.

Calculations with the T-matrix method show the importance of considering the shape of frozen hydrometeors and the Chebychev shaped (Czekala et al., 1999) falling rain drops in the scattering calculations. It can be seen that there is a signicant impact on the intensity and the polarization signal of microwave radiation in the atmosphere.

Discrete dipole approximation

The Discrete Dipole Approximation (DDA) is an approximation of a continuous target by a nite mesh of polarizable elements/dipoles. This method for describing the scat-tering properties of arbitrarily shaped particles is widely used in microwave radiometry (see Evans and Stephens, 1995a; Liu, 2004) and astrophysics. It was rst developed by Purcell and Pennypacker (1973). Draine and Flatau (2000) reviewed the method and compiled well-documented code to this approximation that is freely available under http://arxiv.org/abs/astro-ph/0008151v4.

By using the DDA it is possible to calculate the electro-magnetic scattering and ab-sorption of arbitrarily shaped particles, as long as the distance between the single dipoles is small compared to the considered wavelength. The method is not well suited for parti-cles with very large complex refractive indices since these require much narrower distance between the dipoles thereby requiring much more memory size. The technique was used by Evans and Stephens (1995a) to investigate the impact of particle shape and orientation on the scattering properties of dierent ice habits by computing the single-scattering prop-erties at dierent frequencies. They found that the extinction increases dramatically with frequency and that the dierence in extinction coecients can be up to a factor of three between the various shapes (smallest for rosettes and highest for solid columns). Since oriented plates polarize most, they result in the highest polarization dierence followed by rosettes and columns. The highest scattering ratio (a measure for the forward scattering) can be found for solid columns with decreasing value for thinner particles.

The authors of the DDA code (Draine and Flatau, 2000) note that accurate calcula-tions of electro-magnetic scattering are possible for size-parameters less than 15 and when the complex refractive index n of the scattering medium is not larger compared to unity (|n − 1| < 1, Draine and Flatau, 2000). Within the ESA (European Space Agency) funded study Development of Radiative Transfer model for the frequency range between 200 and 1000 GHz (Sreerekha et al., 2005) validation calculations for the DDA have been per-formed. It was found that single-scattering properties calculated with the DDA code and Lorenz-Mie theory for spherical particles and frequencies, so that |n| kd < 0.5, with k as

Figure 2.6: Scattering and absorption eciencies as a function of the size parameter. The relative errors are given with respect to the Lorenz-Mie reference calculations. The right diagrams were calculated for a higher resolution in size parameter range than in the left diagrams. However, the left diagrams show accuracies up to size parameters of ∼ 15.

the wave number and d as the dipole separation, agree well (see Fig. 2.6). Similar results have been achieved by Liu (2004). The accuracy of DDA calculations depends strongly on the number of dipoles N: the larger the number, the more accurate the calculated single-scattering properties. On the other hand, the computation time scales with the number of dipoles, so that an optimal number needs to be found. For a given number of dipoles N, the accuracy of the DDA approximations depends mainly on the refractive index: smaller refractive indices can be described more accurately.

Parameterizations

The Lorenz-Mie calculations for the single-scattering properties are much more time e-cient than methods describing the particle shapes more exactly. Therefore it is worth to seek for parameterizations of the particle properties that allow the calculations of the single-scattering properties by the Lorenz-Mie formalism, even though the particles are non-spherical. This means that the shape is assumed as a sphere and the non-spherical character is included in the density or dielectric property assumptions. Various approaches to parameterizations of the properties of non-spherical particles exist (e.g. Liu and Curry,

2000; Evans et al., 2002; Bennartz and Petty, 2001; Kim et al., 2007), whereof the one by Liu (2004) will be presented in this subsection.

Liu - equal-mass sphere approximation To describe the single-scattering properties of non-spherical ice and snow particles, Liu (2004) approximates the non-sphericity by assuming equivalent spheres. These spheres have the same mass as the approximated non-spherical particles and a radius and density between a 'solid-sphere' and a 'soft-sphere' particle description. Their dielectric properties are described by the Maxwell-Garnett mixing formula (Eq. 2.56). The diameter of the equivalent mass-spheres is described with a quantity named softness parameter (SP) dened as

SP = D − D₀

D_max− D₀, (2.41)

where D, D0, and Dmaxare the diameters of the equivalent sphere, the solid-sphere, and the soft-sphere, respectively. Hereby, solid means an equal-mass sphere with the density 0.916 gm⁻³ of solid ice. The equal-mass soft-sphere is described by a spherical particle with diameter being equal to the maximum dimension of the non-spherical particle and reduced density. This means that the particle is inated by keeping the mass constant, so that the density decreases. The SP can be derived by nding a particle diameter so that its single-scattering properties after Lorenz-Mie theory t to the one calculated with the DDA approximation for an more exact description of the particle. The SP varies with the particle shape, the frequency, and the single-scattering property. Values for two dierent habits implemented in the applied RT model are shown in Tab. 2.1.

In the microphysical scheme of Méso-NH the cloud ice particles are assumed to be solid-spheres with a density of 0.916 gm⁻³. Therefore, the assumed diameter corresponds to a solid-sphere diameter. For particles with a solid-sphere diameter D0 > 45 µm, a relation for the maximum diameter Dmax is given by Liu and Curry (2000) in terms of the solid-sphere diameter D0 as

Dmax = 2.71D^0.737₀ . (2.42)

Smaller particles are assumed to produce no signicant scattering signal in the consid-ered microwave region. The diameter D of the approximated particle is given by Eq. 2.41 and its density can be deduced from the equivalent mass assumption through

ρ = D0

D · 0.916. (2.43)

For snowakes, a generalized γ-distribution function with respect to maximum diameter D_max is assumed and discretized into appropriate bins. The density of each single particle varies with its size and is dened by

Frequency (GHz) Rosettes Snowakes σ_sca σ_sca

85.5 0.54 0.33

150.0 0.36 0.27

220.0 0.26 0.22

Table 2.1: Softness parameters (SP) for the calculation of the equal-sphere diameter to derive the scattering cross section σsca. The SP are averaged over dierent shapes of the corresponding particles.

ρ(D_max) = 6m

πD_max³ (2.44)

with the mass m as

m = aD_max^b−3. (2.45)

For snowakes a is 0.02 and b is 19.6. The solid-sphere diameter for these particles can be calculated by assuming equal masses. According to Eqs. 5a and 5b of Liu (2004), the diameter of the solid-sphere for these snowakes is given by

D = D_max(ρ_max/ρ_e)^(1/3). (2.46) Thereby, the eective density ρe is dened as the mass of the particle divided by the volume of a circumscribing sphere. With the values for SP in Tab. 2.1 and Eq. 2.41, the equal-sphere diameter for snowakes can be derived and the single-scattering properties calculated by the Lorenz-Mie theory.