Corresponding author: I. S. Adams, Remote Sensing Division, Naval Research Laboratory, Code 7223, Bldg. 2/215E, 4555 Overlook. Ave. SW, Washington, D.C. 20375, USA. (firstname.lastname@example.org)
 The capability of current and future sensors to make accurate measurements of polarized microwave radiation allows for the investigation of particle shape and orientation. Additionally, the dichroic properties of media consisting of particles with preferential alignment will alter the polarization state of radiation emitted and reflected by the surface below a cloud boundary. Therefore, a deep understanding of the influence of particle orientation and shape upon radiation are required for remote sensing of both cloud and surface properties. In this study, we compute the scattering properties of three horizontally aligned snow crystals: two dendrites and an hexagonal plate. Additionally, we create two approximations using cylindrical plates. One uses a plate with a diameter equal to the maximum dimension of the respective snow crystal, utilizing an effective dielectric model which assumes the disk to be a matrix of ice with air inclusions. The other approximation uses a cylindrical plate of equal mass, with a radius chosen to conserve mass. To simplify the analysis, all particles have equal thickness. The results show a strong polarization response, particularly in the Q element of the Stokes vector. This polarization response is captured well by the two approximations. While the approximations are applicable for certain cases, there are discrepancies between the scattering properties of the idealized snow crystals and the two cylindrical plate models that may limit the generality of the approximations. Further radiative transfer studies are required to test the full applicability of the crystal approximations.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Given advances in current and future space-borne passive remote sensing missions, a detailed understanding of the fully polarimetric scattering properties of precipitating particles is important. In particular, the polarized response of preferentially aligned particles will both provide information about cloud characteristics and augment the radiation originating from below the cloud layers. The Global Precipitation Measurement (GPM) Microwave Imager (GMI) is a suite of dual-polarized (vertical and horizontal) microwave and millimeter wave radiometers with channels at 10.65, 18.7, 36.5, 89.0, and 165.5 GHz, and additional vertically polarized channels at 23.8 and 183.31 GHz [Newell et al., 2007]. Recently, Skofronick-Jackson and Johnson have shown that high-frequency observations exhibit improved sensitivity to ice and snow hydrometeors over lower frequency measurements. The inclusion on GMI of dual-polarization capabilities at 165.5 GHz necessitates modeling and analysis of the scattering properties of oriented snow particles at high frequencies. Such information is useful not only in sensing of precipitation, but also in the sensing of surface and other atmospheric phenomena, in particular passive sensing of near ocean-surface vector winds [Gaiser et al., 2004]. WindSat utilizes fully polarimetric observations of brightness temperature (Tb) at 10.7, 18.7, and 37.0 GHz, along with dual-polarized measurements at 6.8 and 23.8 GHz, to infer surface ocean vector winds [Bettenhausen et al., 2006]. For such applications, the ability to sense surface vector winds through precipitation requires polarimetric modeling of precipitating ice in the microwave regime [Adams et al., 2008].
 The aim of this paper is to model a number of idealized snow crystals in order to investigate the single-scattering and bulk scattering properties of snow crystals formed due to a-axis growth. The scattering properties of the various dendrite approximations are computed using DDSCAT. We highlight the complexities of handling horizontally aligned particles, and we observe the modifications to the Stokes vector by oriented snow crystals. We assess the suitability of approximating each of the snow crystals with cylindrical plates. By comparing the angular structure of the phase matrix for each of the particles and the approximations, we probe the influence of the small-scale structure of snow crystals on electromagnetic scattering.
2. Properties of Snow Crystals
2.1. Dendrite Shapes
 While the six-fold (hexagonal) symmetry of snow crystals is a consequence of the lattice structure of ice, the macroscopic growth characteristics of these crystals are determined by temperature and supersaturation. The ambient air temperature actuates either c-axis (columnar) or a-axis (platelike) crystal growth. Moreover, supersaturation influences the size and complexity of the crystal, e.g., the growth of star-shaped dendrites most commonly associated with snow requires high supersaturation.Noel et al.  use lidar measurements to estimate the populations of platelike, columnlike, and irregular crystals under various temperature, humidity, and airflow conditions. Platelike crystals were observed primarily at temperatures above −20°C. Due to aggregation, pristine snow crystals are more prevalent in the upper levels of cirrus clouds, as these crystals will fall and aggregate with other crystals [Schmitt and Heymsfield, 2010]. For this experiment we will employ three idealized snow crystals that represent dendrite growth in the −10 to −20°C temperature range [Libbrecht, 2005]. One is commonly used in the literature [O'Brien and Goedecke, 1988], which we will term a stellar dendrite. The other is formed using six offset ellipses rotated about a central axis of rotation. This crystal is identified as a simple star. We also look at hexagonal plates. The three idealized snow crystals are displayed in Figure 1. The reason for using particles that are representative of snow crystals in only a quasi-realistic sense is to impose a set of constraints that allows us to directly compare the scattering properties of the various distributions and densities of ice within the platelike structure of the considered crystals. Thus, all particles have a set thickness of 70μm Additionally, scattering calculations are performed at fixed size intervals, where the maximum dimension of the crystal Dmax ranges from 0.5 mm to 10 mm at a spacing of 0.5 mm. Defining the particle over this size range does result in artificially large particles with unrealistic aspect ratios at the large end; however, this allows us to determine the radiative significance of thin, extended bodies. Such particles sizes and dimensions are also in line with previous studies [Liu, 2004]. Although we expect that hexagonal plates will populate the small end of the size distribution and stellar dendrites will exist primarily at larger sizes, we compare all particles at all sizes in order to observe size- and shape-dependent features of the calculated scattering properties.
2.2. Particle Orientation
 Using SSM/I data and radiative transfer simulations, Prigent et al.  determined that polarization differences in some observations of precipitation at 37 and 89 GHz could only be explained by preferential alignment of nonspherical hydrometeors. Also, depolarization signatures in radar [Matrosov et al., 2005] and lidar [Noel and Sassen, 2005; Noel and Chepfer, 2010; Zhou et al., 2012] indicate that, under some conditions, the maximum dimensions of particles will tend to align perpendicularly to airflow. Through modeling, Klett  demonstrated that thermal and turbulent fluctuations, in combination with the size of the particle, explain the distribution of hydrometeor tilt angles, such that turbulence can disrupt preferential orientation. Larger particles may remain in a preferential orientation, but will wobble while falling, and this effect has been observed in radar data [Matrosov et al., 2005], where the standard deviation of the flutter was approximately 9°. Examinations of cirrus clouds using lidar [Noel and Chepfer, 2010; Zhou et al., 2012] suggest that aligned crystals are more prevalent in warm or mixed-phase clouds, and the −10 to −30°C temperature range in which these crystals are found favors platelike morphologies. For this study we will only consider particles with strict horizontal orientation; however, insection 5, we will address the implications of snow crystal canting.
3. Scattering by Axially Aligned Particles
 To compute the single-scattering properties of snow crystals, we employ version 7.1 of the publicly available DDSCAT code [Draine, 1988; Draine and Flatau, 1994; B. T. Draine and P. J. Flatau, User guide for the discrete dipole approximation code DDSCAT 7.1, 2010, available at http://arxiv.org/abs/1002.1505]. DDSCAT offers the flexibility of computing the scattering, extinction, and emission characteristics of particles with arbitrary geometries, orientations, and refractive indices using the discrete dipole approximation (DDA). The DDA is a common method for computing the scattering properties of snow [Liu, 2004; Kim, 2006; Petty and Huang, 2010], though the examples in the literature assume complete random orientation for all particles. While aspects of the DDSCAT code favor handling random orientations, the software provides all the necessary information needed to compute the scattering properties for particles with preferential orientation.
3.1. Amplitude Scattering Matrix
Figure 2 shows the local reference frame within which we define the scattering properties of a particle. The amplitude scattering matrix S relates the incident polarized electric field to the scattered polarized electric field [Van de Hulst, 1957; Bohren and Huffman, 2004; Mishchenko et al., 2002]. Each author uses a slightly different notation, with slightly different conventions for the orientation of the electric fields. We will use the notation of Mishchenko et al.  throughout this paper:
where Sij are the elements of the amplitude scattering matrix, and
The electric field E is specified in spherical coordinates, where is the radial component of the electric field, lies within the plane of incidence, defined by and the z axis, and completes the right-handed coordinate system such that . The component of the electric field may also be considered the vertically polarized or parallel component, while refers to the horizontal, or perpendicular, polarization. Given that the propagating wave is transverse, in the far field, the radial component is 0; therefore, we use to denote unit vector describing the direction of propagation, r is the radial distance from the particle, k is the wave number of the propagating wave. The superscripts “inc” and “sca” correspond to the incident and scattered waves, respectively.
 In order to compute the amplitude scattering matrix, we use DDSCAT to compute the scattering matrix fml, described by Draine , for the prescribed particle orientations and scattering directions:
The scattering matrix is similar in form to the amplitude scattering matrix, with only a few small details contrasting the two equations. The left sides of the respective equations appear to be equivalent. DDSCAT allows the user to choose between the particle or the laboratory reference frames when defining the orientation of the scattered radiation. Choosing the particle reference frame ensures that our scattered directions are consistent, leaving any differences to the respective right sides of the equations. While we may choose to orient the incident electric fields in an arbitrary configuration, any that we choose will be referenced to the laboratory frame. Since the orientations of the particle and the incident electric field may be configured such that
the conversion between the scattering matrix and the amplitude scattering matrix requires a single rotation from the laboratory frame to the particle frame, along with scaling by the wave number k. Thus, for a horizontally aligned particle that is either rotationally symmetric or rotationally averaged,
Mishchenko et al.  gives a complete and general treatment for rotating particle reference frames, while Xie and Miao  treat rotations of the output of DDSCAT specifically. From the amplitude scattering matrix, the phase and extinction matrices, Z and K, respectively, as well as the emission (absorption) vector, Ke, can be derived. The formulae for these properties and a description of their significance is given in Appendix A.
3.2. Horizontal Orientation
 The scattering properties of particles with arbitrary orientation are unwieldy. Choosing a proper orientation scheme can simplify the problem by imposing symmetries. In the case of horizontally aligned particles, with either rotational symmetry or random azimuthal orientation, the extinction matrix reduces to block diagonal form having three independent elements. That is, the extinction between I and Q is decoupled from the extinction between U and V—definitions for I, Q, U, and V are given in equation (A1) [Mishchenko et al., 2002]:
For rotationally symmetric particles where the axis of symmetry is aligned in the zenith direction, there are no preferred azimuthal directions. That is, any rotation about the axis of symmetry produces the same geometric configuration. Therefore, the phase matrix depends on both the incident and scattered zenith angles, but it depends only upon the difference between the incident and scattered azimuth angles. Additionally, the extinction matrix and emission vector exhibit no azimuthal variation. This logic may be extended to an ensemble of particles, such as the idealized snow crystals, with uniform, random orientation about the spin axis. Finally, only the first two elements of the emission vector are non-zero.
4.1. Single-Scattering Calculations
 We use DDSCAT to compute the scattering properties for our idealized crystals at three frequencies used for passive microwave imagery: 37 GHz, 89 GHz and 165 GHz. The refractive indices are computed using Mätzler  with the three chosen frequencies for a temperature of −13.15°C. Furthermore, we developed two approximations to our dendrite models consisting of cylindrical plates. The rotational symmetry of cylinders eliminates the DDSCAT computations required for azimuthal averaging, drastically reducing the runtime of DDSCAT. One approximation uses an approach similar to the fluffy spheres, in which effective dielectric constants are calculated for plates of equal maximum dimensions. The plates are considered to be matrices of ice with air inclusions. The mixing formula is given by Shivola , where v = 0.85 [Petty and Huang, 2010]. The ratio of the air inclusions to the ice matrix is such that the mass of the idealized snow crystal is conserved. The other approximation comprises cylindrical plates of equal thickness with diameters calculated to conserve the masses of the idealized snow crystals. The grid spacings for all particles and approximations are 14 μm, making the columnar dimensions 5 grid points. The basal dimensions scale with Dmax.
 The scattering parameters are computed in increments of 5°, between 0° and 90° for incident zenith angles, θinc, and between 0° and 180° for scattered zenith θsca and scattered azimuth ϕsca. Since all particles and approximations are either azimuthally averaged or rotationally symmetric, ϕsca is referenced to an incident azimuth angle ϕinc = 0; therefore, Δϕ = ϕsca. Due to the exhaustive volume of data needed to describe the scattering properties of horizontally aligned particles, we focus our discussion of single-scattering calculations to those computed at 165 GHz, with a few examples at 37 and 89 GHz.
Figure 3 shows elements Zi1 of the phase matrix for i = 1,2,3,4 for the idealized stellar dendrite and the accompanying approximations for Dmax = 2 mm The array of images shows the angular contribution of radiation scattered into the line of sight for a zenith angle of 55°, which is typical for conically scanning radiometers. As only the elements of the first column are shown, the images show the angular contributions to the Stokes vector from unpolarized incident radiation. This is not an unreasonable simplification given that I is about seven times larger than Qfor a cloud-free atmosphere over the ocean at 37 GHz 55° from zenith. At higher frequencies, optical depths, and near surface wind speeds,Q becomes smaller relative to I. Over land, measurements of Q are typically much smaller than the Q of ocean observations.
 Looking at the resulting intensity Z11 in the first row, scattering concentrates broadly about Δϕ = 0, with the intensity peaking in the forward scattering direction, as expected. The equal mass approximation obviously overestimates scattering. The effective permittivity model fits more closely, but still overestimates the scattering. For Z21, we see a strong negative peak at 180∘ − θsca, indicating that the horizontally aligned particle interacts strongly with the Eϕ component of the electric field. Z31 is less than an order of magnitude smaller than Z21; however, any contributions to the U component of the Stokes vector require asymmetries in the distribution of the incident radiation due to the antisymmetry of S12 and S21, and thus Z31, about Δϕ = 0°. Also, S21 and S12 are zero for Δϕ = 0°. Z41 is an order of magnitude smaller than Z31, but has the same properties with regards to symmetry.
Figure 4 conveys the same information presented in Figure 3, but for a stellar dendrite with Dmax = 8 mm. Scattering is considerably more focused toward the forward scattering direction, and we see the well defined lobes described by Van de Hulst . The equal mass plate grossly overestimates scattering, as with the smaller particle size. The effective permittivity approximation underestimates scattering, though the scattering is more focused. The two approximations present a ridge structure that is far less defined in the phase matrix of the idealized stellar dendrite. The simple star, not shown, also exhibits a muted, almost non-existent, set of ridges much like the stellar dendrite; whereas, the hexagonal plate inFigure 5 appears more like the plate approximations with a strong set of ridges. The ridge structure becomes more exaggerated at larger values of Dmax for platelike morphologies and approximations, but never materializes strongly for the more sparsely distributed dendrites. Also note, in Z41the small, notch-like, local minima betweenθinc = 30° and θinc = 60°, which is also observed, to a lesser extent, in Z31. This feature is only observed for the two idealized dendrites, not for the approximations nor for the hexagonal plates. These morphology-dependent effects are the result of differences in diffraction among the various particles and approximations [Van de Hulst, 1957; Kim, 2006]. The path lengths through the dendrites — particularly at higher zenith angles, where the direction of propagation is more closely aligned with the basal plane of the crystal — differ for the particles and approximations, as well as for the azimuthal orientations of the idealized crystals. Diffraction-dependent features are apparent at 89 GHz and 165 GHz, but not at 37 GHz, and are more prevalent at larger particle sizes.
 To further investigate the scattering contribution, we perform the integration of the phase matrix from equation (A2), normalizing by the incident radiation, to determine the relative scattering contribution:
Since the integrand of equation (A2) has been normalized by the incident radiation field, this scattering contribution represents a uniform distribution of incident radiation. Based on scattering symmetries and the impact of these symmetries on the amplitude scattering matrix [Van de Hulst, 1957; Mishchenko et al., 2002], as well as the structure of the Stokes phase matrix, i.e. equation (A4),
Figure 6 shows the scattering contribution from I of the incident radiation and Figure 7 shows the scattering contribution from Q of the incident radiation for all idealized particles and approximations at all frequencies plotted versus Dmax. As expected, we observe that the intensity of the scattered radiation increases with particle size, and this trend holds across all the frequencies considered.
 As σ21norm <= 0 in Figure 6, and since I is, by definition, always positive, the scattering contribution of I results in contributions to Q that are negative. The interpretation of Figure 7 is a bit more nuanced, as the incident linear polarization difference Q may be positive or negative. Given the sign of σ12norm, if Q is positive, then there is additional enhancement of the scattered I, but if Q is negative, then the contribution from Q to Icounteracts the self-contribution ofI, which is given by σ12norm. Finally, the self-contribution ofQ, given by σ22norm, retains the same sign as the incident Q. For θsca = 55° the scattering contribution results in an enhancement to the radiance, and, since component I of the incident radiation is larger than the magnitude of Q, the scattered Q will tend to be negative, even when the incident Q is positive.
 Comparing the scattering contributions at the different frequencies, scattering at 89 GHz exhibits an order of magnitude increase over 37 GHz, while the scattering contribution at 165 GHz is roughly three times that of 89 GHz. Among the approximations, the equal mass plate approximates the hexagonal plate quite well, with respect to scattering, while overestimating the stellar dendrite at all frequencies. The effective permittivity model slightly underestimates scattering for the stellar dendrite at all but the smallest particle sizes. Neither approximation works well for the simple star, though the equal mass plate appears to be a slightly better fit.
 Since extinction is only meaningful for forward propagating waves, θ = θinc = θsca. Looking at the angular dependence of extinction in Figure 8, the hexagonal plate is best approximated by the equal mass plate while the effective permittivity model works best for the stellar dendrite. Neither approximation, again, quite fits the simple star. The extinction matrix is handled such that when K12< 0, self-extinction ofQ retains the sign of Q, the contribution from I makes Q more positive, and the contribution from Q enhances I. Any dichroic effects between U and V depend on the signs of the Stokes elements as well as the sign of the extinction matrix element K34 in equation (6). For zenith angle θ= 0°, there is only self-extinction, i.e., there is no dichroism. As the zenith angle increases, so does the cross-talk betweenI and Q and between U and V.We neglect to show the relationship between extinction and particle size, as the results echo those of the scattering component: self-extinction monotonically increases with both particle size and frequency, with the polarizing components monotonically becoming more negative. This behavior applies to all frequencies. The angular dependence is a function of the size parameter, i.e., the ratio between the particle size and the wavelength of the radiation.Figure 9 shows K11 at 37 GHz for the same 2 mm particle as considered in Figure 8. At this smaller size parameter, the extinction decreases monotonically with increasing θ. Also, the effective permittivity approximation slightly overestimates extinction for the stellar dendrite, while only underestimating extinction a small amount for the simple star. Contrast this to the extinction for Dmax = 8 mm at 165 GHz shown in Figure 10, where the extinction for the simple star is best approximated by an equal mass plate.
 For emission, the zenith angle θ is the same angle that is prescribed for extinction. The plots of the emission vector normalized by the Planck function, given in Figures 11 and 12, show that small differences in the scattering and extinction properties between the idealized particles and the approximations lead to larger relative discrepancies in emission. Also, while the scattering and extinction for the hexagonal plates dwarf those properties of the other two snow crystals, at large particle sizes, emission by either the simple star or the stellar dendrite approaches and even exceeds that of the hexagonal plate. Emission at 37 GHz and 89 GHz is negligible.
4.2. Scattering Medium
 To investigate the scattering properties of an ice cloud, we integrate the single-scattering calculations over a particle size distribution using a two-moment snow size distribution representative of a midlatitude stratiform cloud [Field et al., 2007]. Since the masses of each of the idealized particles are directly proportional to Dmax2, ice water content is proportional to the second moment of the distribution, and other moments of the distribution may be calculated from the second moment using a temperature dependent relationship. The ice water content of the cloud is 0.15 g m−3. While each hypothetical cloud in this analysis is composed of a single habit of ice crystal, realistic clouds comprise a range of snowflake morphologies, the structures of which are determinedby the local environmental conditions governing growth and aggregation. The application of the particle size distribution here is to highlight the bulk scattering effects of any one of the chosen morphologies and to make comparisons among the idealized models and the simplifications. A more realistic implementation would populate small crystal sizes with hexagonal plates, intermediate crystal sizes with the simples stars, and large sizes with the stellar dendrites, and would also include other crystal geometries and snow aggregates.
 As with the single-scattering properties fromsection 4.1, we look at the integrated scattering contribution from I and Q in Figures 13 and 14, except that we observe the dependence on θsca. At small zenith angles, there is little contribution to Q; however, the modification to I from Q is the largest. At higher zenith angles, the scattered light is increasingly polarized, favoring negative Q, while the modification of I due to Q reduces.
 Typically, particles of smaller sizes comprise a considerably higher proportion of the particle population than larger particles, thus providing more weight to the scattering properties of the smaller particles. This is evident in the 37-GHz data shown inFigure 13 , where the effective permittivity model overestimates scattering of the stellar dendrite. Conversely, the effective permittivity model underestimates scattering for the simple star, underlining the sensitivity of this approximation to particle density and choice of particle size distribution. While the scattering properties of the hypothetical cloud are strongly weighted by smaller particles, we observe the influence of larger particles by examining the extinction, as shown in Figure 15. For the medium, extinction by the simple star is not represented well by either approximation. Figures 8 and 10 elucidate this, showing a better fit for the effective permittivity model for small sizes and a closer representation by the equal mass plate for larger sizes. Emission, given in Figure 16, is not negligible at 165 GHz. Here we see a poor fit between the hexagonal plate and the equal mass plate, as was obvious in the single-scattering properties. Based on the analysis of the single-scattering properties, the emission calculations for the snowing medium offer no unexpected results.
 While the consideration of strict horizontal alignment of ice particles presents an extreme case, this analysis gives an upper bound to any polarization effects. The results at 37 GHz show that while the contributions of snow crystals may not be ignored, these are small. Scattering and extinction at 89 GHz and 165 GHz, along with emission at 165 GHz are significant. As has been observed in other studies, simple particle approximations fail to properly capture all the features of the scattering properties over the full parameter space. The equal mass plate approximation consistently overestimates scattering, much like the solid sphere approximation overestimates the scattering for randomly oriented particles considered at similar frequencies [Liu, 2004]. The effective permittivity approximation may overestimate or underestimate depending on particle size and density, but it approximates the stellar dendrite much more closely than the equal mass plate. The behavior of the effective permittivity approximation is troubling, since assumptions of particle density and particle size distribution will affect the accuracy of the approximation.
 Due to the extended nature of the plates and dendrites, we observe some shape-dependent features in the scattering properties of the idealized crystals and the approximations. As highlighted insection 4, there are features of the angular weighting of incident radiation for the phase matrix, specifically for the resulting U and Vcomponents of the scattered radiation, that do not match between the plate and dendrite geometries. In general, the equal mass plate is the best analog for the hexagonal plate, and this seems obvious due to the similarities in particle geometries. Likewise, the effective-permittivity-modeled plate most closely represents the stellar dendrite. While the distribution of ice about the basal plane of the stellar dendrite is not random, this sparse distribution is geometrically more similar to a random distribution than the more discrete branching of the simple star or the uniform structure of the hexagonal plate. Particle shape effects must be considered at 89 GHz and 165 GHz, as the size of the small-scale features of the crystals are on the order of, or larger than, the wavelengths. The differences in particle morphology have little influence at 37 GHz due to the longer wavelength.
 In practice, the effective permittivity plates are appropriate at 37 GHz, and the effects of pristine ice crystals will be small. For higher frequencies, determining the full applicability and generality of the approximations requires radiative transfer calculations over a range of ice water contents and measurement geometries for various particle size distributions. While radiative transfer calculations have not been included in the analysis, the results may be put into the context of space-based measurements of precipitation and ocean vector winds. Due to extinction of the large radiative background signals of land and ocean, the observedQ for a cloud with horizontally aligned particles would be positive compared to the Q emanating from a cloud of randomly oriented scatterers. Contributions from scattering and emission would counteract the positive polarization difference. In terms of horizontally and vertically polarized radiometric measurements, if Q—the analog to polarization difference—is positive, then the measured vertical polarization will be larger than the horizontal. For the other Stokes parameters, a small amount of cross-talk due to extinction could corrupt the measuredU and V, though based on the magnitudes of K34 and U and V emanating from an ocean surface, this effect should be small. Multiple scattering will only affect U and V when there are large horizontal inhomogeneities [Adams et al., 2008]. The dichroic effects of aligned particles will be muted by a number of factors. As stated previously, turbulence and thermal gradients induce particle flutter, and a distribution of canting angles reduces the measured polarization difference [Xie and Miao, 2011]. Absorption by water vapor and supercooled liquid droplets will also mask the presence of aligned particles due to the attenuation of Q, and the polarization difference due to precipitation must be separated from that of the background, e.g., the ocean surface. For an optically thick cloud, the observation of positive Q at high frequencies will suggest the presence horizontally aligned ice crystals. Since passive ocean surface vector wind retrievals only utilize frequencies at or below 37 GHz, the effects of pristine snow crystals will be dominated by the presence of liquid precipitation and large aggregates.
Appendix A:: Scattering, Extinction, and Emission
 The single-scattering properties of a particle encompass the scattering, extinction, and emission of electromagnetic radiation. These properties may be calculated directly from the elements of the amplitude scattering matrix described inequation (1). In order to characterize the full polarization state of a propagating plane wave, we describe radiation using the Stokes vector, the elements of which are defined to be [Mishchenko et al., 2002]
where ϵ is the permittivity of the medium in which the wave propagates, and μ is the permeability of the medium. Note that I is proportional to the sum of vertically and horizontally polarized radiation, while Q is proportional to vertical minus horizontal.
 To characterize the change in the Stokes specific intensity vector I at position r along some direction of propagation for some angular frequency ω through a medium comprising randomly distributed hydrometeors, we enlist the vector radiative transfer equation:
where ds is a path length measured along , and , , and are the extinction matrix, phase matrix, and emission vector, respectively, which have been averaged over the various particle sizes and types considered. The phase and extinction matrices transform the Stokes vector of the incident wave into that of the scattered wave, while the emission vector specifies the emission components for each element of the Stokes vector.
A1. Phase Matrix
 The phase matrix Z gives the transformation between the scattered Stokes vector and the incident Stokes vector for :
The elements of the amplitude scattering matrix define the phase matrix:
A2. Extinction Matrix
 Extinction is the loss due to both absorption and scattering. For the special case of , i.e., exact forward scattering, the 4 × 4 extinction matrix K describes the attenuation of the forward propagating wave due to scattering by a particle. The resultant Stokes vector over a small surface element ΔS perpendicular to , due to extinction, is
where O(r−2) describes the component of the elastically scattered spherical wave, proportional to , propagating in the direction of . As with the phase matrix, the elements of the amplitude scattering matrix define the extinction matrix:
A3. Emission Vector
 As all absorbing bodies also emit radiation, the Stokes emission vector characterizes angular distribution of the intensity and polarization state of the emitted energy. The ith element of Ke is a function of the ith elements of the first columns of the extinction and phase matrices:
The physical temperature of the particle is denoted by T and is the direction of emission, originating inside the particle, and Ib(T, ω) is the Planck function. For details on the derivation of equation (A7), see Mishchenko et al. .