Editorial Type:
Article
Article Type:
Research Article

The Polarimetric Radar Scattering Properties of Oriented Aggregates

Robert S. Schrom aNASA Goddard Space Flight Center, Greenbelt, Maryland
bOak Ridge Associated Universities, Oak Ridge, Tennessee

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S. Joseph Munchak aNASA Goddard Space Flight Center, Greenbelt, Maryland

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Ian S. Adams aNASA Goddard Space Flight Center, Greenbelt, Maryland

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Online Publication:
29 Mar 2023
Print Publication:
01 Apr 2023
DOI:
https://doi.org/10.1175/JAS-D-22-0149.1
Page(s):
1145–1161
Open access
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Abstract

The scattering properties of aggregates are studied herein. Early aggregates (<7 monomers) of branched planar crystals and mature aggregates (up to 100 monomers) of columns are randomly generated with varying assumptions about the monomer attachment processes and the orientation behavior during collection. The resulting physical properties of the aggregates correspond well with prior in situ and retrieved sizes and shapes. Assumed azimuthally uniform orientations during collection and monomer pivoting upon attachment resulted in flatter and denser aggregates. The column aggregates had lower density and more spherical shapes than the branched planar crystal aggregates. The scattering properties were calculated using the discrete dipole approximation for a set of orientation angles and transformed to spectral coefficients representing modes of orientation angle variability. The zeroth- and second-order coefficients dominate this variability, with the zeroth-order coefficients representing the scattering properties for randomly oriented particles. The second-order coefficients for backscatter showed differences between horizontal and vertical polarization increasing with density, and these coefficients for specific differential phase increase with both mass and density. Similarly, coefficients for the copolar covariance decreased with density. Rapid changes in the contributions to the radar moments from the second-order coefficients from low to moderate density were observed, likely due to the increasing presence of horizontally aligned monomers in the aggregate structure. Differences in how differential reflectivity and correlation coefficient evolve with the orientation distribution parameters suggest that these measurements, along with specific differential phase and reflectivity, provide complementary information about aggregate sizes, shapes, and orientation distributions.

Schrom’s current affiliation: Earth System Science Interdisciplinary Center, University of Maryland, College Park, College Park, Maryland.

Munchak’s current affiliation: The Tomorrow Companies, Inc., Boston, Massachusetts.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Robert S. Schrom, robert.s.schrom@nasa.gov

Abstract

The scattering properties of aggregates are studied herein. Early aggregates (<7 monomers) of branched planar crystals and mature aggregates (up to 100 monomers) of columns are randomly generated with varying assumptions about the monomer attachment processes and the orientation behavior during collection. The resulting physical properties of the aggregates correspond well with prior in situ and retrieved sizes and shapes. Assumed azimuthally uniform orientations during collection and monomer pivoting upon attachment resulted in flatter and denser aggregates. The column aggregates had lower density and more spherical shapes than the branched planar crystal aggregates. The scattering properties were calculated using the discrete dipole approximation for a set of orientation angles and transformed to spectral coefficients representing modes of orientation angle variability. The zeroth- and second-order coefficients dominate this variability, with the zeroth-order coefficients representing the scattering properties for randomly oriented particles. The second-order coefficients for backscatter showed differences between horizontal and vertical polarization increasing with density, and these coefficients for specific differential phase increase with both mass and density. Similarly, coefficients for the copolar covariance decreased with density. Rapid changes in the contributions to the radar moments from the second-order coefficients from low to moderate density were observed, likely due to the increasing presence of horizontally aligned monomers in the aggregate structure. Differences in how differential reflectivity and correlation coefficient evolve with the orientation distribution parameters suggest that these measurements, along with specific differential phase and reflectivity, provide complementary information about aggregate sizes, shapes, and orientation distributions.

Schrom’s current affiliation: Earth System Science Interdisciplinary Center, University of Maryland, College Park, College Park, Maryland.

Munchak’s current affiliation: The Tomorrow Companies, Inc., Boston, Massachusetts.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Robert S. Schrom, robert.s.schrom@nasa.gov
Keywords: Cloud microphysics; Ice particles; Radars/Radar observations

1. Introduction

Radar measurements that contain polarization information have shown substantial value in better characterizing ice particle properties in the atmosphere. In particular, these measurements provide particle shape information; based on our best theoretical models for shape evolution during particular ice growth processes, this information provides insights into the relative contributions of the growth processes to the observed precipitation. However, the degree of confidence one can have in these insights critically depends on the uncertainty in how ice particle physical and scattering properties evolve during growth.

One important application of polarimetric radar measurements where these uncertainties are evident is the early aggregation of branched planar crystals. As these particles grow from vapor deposition, aggregation occurs when the particles collide during free fall. How the shapes of the early aggregates evolve after the first few collection events is uncertain. For example, Moisseev et al. (2015) hypothesize that the early aggregates are relatively flat with low aspect ratios. These particles may then produce the observed signature of increases in differential reflectivity (ZDR), specific differential phase (Kdp), and reflectivity at horizontal polarization (ZH). However, other researchers hypothesize that aggregation quickly results in more randomly shaped, sparser particles that have near-zero ZDR and produce minimal contributions to Kdp, with the observed polarimetric enhancement attributed mostly to the coexisting pristine ice crystals (e.g., Kennedy and Rutledge 2011; Andrić et al. 2013; Schrom et al. 2015).

Additionally, the orientation behavior of aggregates is important in both understanding the polarimetric radar measurements and modeling ice microphysical processes. In the case of the former, the radiative properties of an ice particle population depend on the distribution of particle orientations in the sensing volume. With respect to ice growth processes, orientation behavior can impact both the collision and collection efficiency of the particles. The evolution of particle shape through both aggregation and riming will also depend on the orientation of the ice particles during collection. Owing to observed positive ZDR and Kdp in many cases of ice particle growth (e.g., Ryzhkov and Zrnić 1998; Kennedy and Rutledge 2011; Matrosov et al. 2017) and simplified hydrodynamic considerations, ice particle orientation distributions are commonly assumed to have mean values corresponding to horizontal orientation (i.e., the angle of the longest axis of the particle with respect to Earth’s surface is 0°) and varying degrees of dispersion about the mean.

The lack of detailed aggregation simulations coupled to detailed simulations of scattering properties has impeded efforts to understand the polarimetric radar signatures of aggregates. Some progress has been made in this area by using idealized shapes to approximate real ice particles; scattering calculations of these idealized shapes (typically spheres and/or spheroids) are highly efficient using the T-matrix method (Waterman 1969) or Rayleigh theory (Rayleigh 1897). However, the resulting scattering properties may poorly approximate the true scattering properties of natural ice particles (e.g., Botta et al. 2013; Schrom and Kumjian 2018), despite these simplified shapes correctly capturing the bulk physical properties of the particles (e.g., maximum dimension, aspect ratio, effective density, and mass). Prior studies have carried out quasi-physical Monte Carlo simulations of aggregation (e.g., Westbrook et al. 2006; Schmitt and Heymsfield 2010; Botta et al. 2010; Kuo et al. 2016; Eriksson et al. 2018), and the interaction of microwave radiation with ice particles is well understood from Maxwell’s equations. However, simulating the polarimetric radar signatures of detailed-shaped aggregates is computationally demanding because of the high spatial resolution needed to capture the particle shapes and the need to perform independent calculations for many particle orientations.1

Some limited studies examining the polarimetric scattering properties of aggregates have been performed. Tyynelä et al. (2011) conduct scattering calculations for aggregates assumed to have fixed horizontal orientations, and their resulting linear depolarization ratios found to be larger than that of comparable spheroids. Subsequently, Tyynelä and Chandrasekar (2014) perform scattering calculations for aggregates with assumed random orientations, and find that both circular depolarization ratio and ZDR are reduced relative to that for pristine particles. Munchak et al. (2022) shows that detailed scattering calculations for a single aggregate of plates produce physically reasonable retrievals of bulk ice properties, with some potential to estimate the degree of horizontal alignment of the aggregate. However, more rigorous analyses of the impacts of particle flutter on these signatures, as well as the natural variability of scattering properties for similar aggregates have yet to be performed.

The main goal of this study is therefore to explore the variability in polarimetric scattering properties of aggregates given certain bulk particle properties and evaluate the potential to efficiently predict their polarimetric scattering properties from these bulk properties. We approach this problem by first simulating aggregation with varying assumptions about the monomer properties, attachment behavior, and their assumed orientation distributions. We use ensembles of these randomly generated aggregates to compute scattering properties for a variety of orientation angles using the Amsterdam discrete dipole approximation (ADDA; Yurkin and Hoekstra 2011) code. The resulting sets of scattering properties are then linked to the physical properties by conducting a spectral analysis of the scattering properties by transforming the scattering calculations from orientation angle space to Legendre-coefficient space. We then find the dominant coefficients for each scattering property and relate them to the physical properties of the particles.

2. Aggregation simulations

a. Description of aggregation procedure

Several prior studies have generated aggregates for the purposes of understanding how physical properties of these particles evolve and for populating scattering databases of ice particles (e.g., Westbrook et al. 2006; Schmitt and Heymsfield 2010; Xie et al. 2011; Botta et al. 2013; Leinonen et al. 2013; Nowell et al. 2013; Kuo et al. 2016; Brath et al. 2020). The common strategy for generating these particles is to select monomers (i.e., single pristine ice crystals) and randomly attach them to a growing aggregate or an initial monomer at the onset of aggregation. The attachment points of the monomer with the aggregate are then calculated as the closest nonintersecting point between an aggregate and a monomer along some direction. Generally, the relative angles between the monomer and the aggregate are randomly selected, as is the direction that one particle moves toward the other.

Assuming purely random orientations of monomers and aggregates when generating these synthetic aggregates will bias the physical properties and the corresponding scattering properties, especially given the evidence of preferred horizontal orientation from polarimetric radar measurements of ice precipitation. Additionally, we hypothesize that monomer motions that occur after impact such as pivoting and slipping will alter the aggregate structure. Owing to the complexity of these motions, they have not been considered in prior simulations of aggregation, and we will focus only on the effect of monomer pivoting during aggregation. To simulate pivoting (illustrated in Fig. 1), we assume that the particle pivots about its initial attachment point p1 and that p1 is the closest distance between the monomer and the aggregate along the specified attachment direction (assumed to be along the negative z axis; Fig. 1a). Once p1 is found, the monomer will then rotate about this point with a rotation axis r1 defined by the cross product of the vector from p1 to the monomer center of mass and the negative z axis (Fig. 1b). This rotation continues until a second attachment point p2 is made between the monomer and aggregate. A second pivoting then occurs about the rotation axis r2 = p2 − p1 (Fig. 1c). As with the first pivot, this rotation continues until a third attachment point between the monomer and aggregate is found, and the monomer is then incorporated into the aggregate.

Fig. 1.
Fig. 1.

An illustration of the pivoting procedure we use herein. (a) The monomer falls toward the aggregate along the direction v. (b) The first attachment point is at point p1, and the first pivot occurs at p1 with rotations about the axis r1 in the direction indicated by the purple curved arrow. The labeled point c indicates the monomer center of mass. (c) The second attachment point resulting from the first pivot is labeled as p2. The second pivot occurs about the line connecting p2 to p1 with rotations about the axis r2 in the direction indicated by the gold curved arrow.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

We use the Euler angles α, β, and γ to specify particle orientation herein; an illustration of how these angles define the particle orientation is shown in Fig. 2. The orientation of the aggregate during collection will depend on the coupling between the surrounding air motion and the particle, requiring intensive fluid dynamic calculations to accurately model this system. Additionally, observations of aggregate orientation in natural clouds are limited. Owing to this lack of knowledge, we use simplified assumptions about the aggregate orientation behavior to better understand how this element of aggregation impacts the evolution of the bulk physical properties. We first assume that the two largest principal axes of an aggregate’s inertia momentum tensor are oriented horizontally (i.e., along Earth’s surface). We then rotate the aggregate randomly prior to each monomer collection event under two different assumptions:

  1. uniform random distributions of α, cosβ, and γ [referred to herein as full-Euler random (ER)] and

  2. uniform random distributions of α, with β = γ = 0 [referred to herein as azimuthally random (AR)].

Fig. 2.
Fig. 2.

A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

The AR assumption has been observed in laboratory studies of aggregates in free fall (e.g., Westbrook and Sephton 2017; McCorquodale and Westbrook 2021); the ER assumption provides a useful baseline to evaluate the impact of preferred orientations on the aggregate properties. For each of these aggregate orientation assumptions, we perform two sets of aggregation simulations: one set with monomer pivoting and one set without monomer pivoting. In the case of no pivoting, the monomers are simply incorporated into the structure upon their first contact with the aggregate. Additionally, we perform separate aggregation simulations for branched planar crystal monomers and for solid hexagonal column monomers, since aggregates of both types of these pristine habits exist in nature. We also aim to test how their physical and scattering properties differ. For the branched planar crystal aggregates, we focus on the early formation stage of these aggregates and generate particles sequentially from two to six monomers. The complex nature of the branched planar crystal shapes makes the aggregation simulations relatively inefficient. The relative simplicity of columns allows us to aggregate a much greater number of them efficiently, and therefore, we construct each column aggregate by collecting a random number of column monomers from 10 to 100.

The branched planar crystal shapes are generated randomly using the method of Schrom and Kumjian (2019) with a Gaussian distribution of a-axis lengths of mean 1 mm and standard deviation of 0.3 mm and restricted to a-axis lengths of 0.3–1.5 mm. The aspect ratio ϕ of the branched planar crystals is determined from the power-law relation (as in Schrom and Kumjian 2019)
ϕ=(0.001a)d,
where a is the a-axis length (in mm), d = 0.45, and 0.001 mm is the length of the hexagonal core where branched growth initiates. For the column monomers, the c-axis lengths are randomly sampled between 0.4 and 1.2 mm, and the a-axis length is determined from the empirical relation for hexagonal column dimensions from Auer and Veal (1970). For the branched planar crystal aggregates, we therefore have four sets of synthetic aggregates: ER pivoting, AR pivoting, ER nonpivoting, and AR nonpivoting. For the column aggregates, we have two sets of synthetic aggregates: ER nonpivoting and AR nonpivoting. We choose not to include sets with pivoting for the column aggregates owing to the more simplified shapes of the column monomers and larger number of monomers within these sets of aggregates; these factors will tend to reduce the impact of pivoting on the particle properties. Indeed, simulated ER and AR column aggregates with pivoting (not shown) have similar distributions of maximum dimension, density, and aspect ratio compared to the respective ER and AR column aggregates without pivoting. A summary of the generated aggregates with different orientation assumptions, pivoting assumptions, and habits are shown in Table 1.
Table 1

Description of the assumptions and properties used to simulate the different sets of aggregates. The abbreviations used in the aggregate experiment names are defined as follows: ER corresponds to full-Euler-random orientations, AR corresponds to azimuthally random orientations, BPA corresponds to branched planar crystal aggregates, CA corresponds to column aggregates, NP corresponds to no pivoting, and P corresponds to pivoting. The aggregation monomer increment refers to whether an aggregate is saved sequentially after each monomer collection event (as in the case of branched planar crystals) or an aggregate is saved after a random number of monomers are collected.
Table 1

b. Simulated physical properties

Randomly selected examples of the simulated aggregates are plotted in Fig. 3. For each synthetic aggregation experiment, there is a fair amount of variability in the structure of the resulting ice particles. Generally, the ER nonpivoting branched planar aggregates have more chaotically oriented monomers than the ER pivoting, AR pivoting, and AR nonpivoting branched planar aggregates. Both sets of column aggregates have more sparse and more spherical shapes than the branched planar aggregates.

Fig. 3.
Fig. 3.

Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ER-BPA-P aggregate farthest to the right).

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

Aside from mass, many particle physical properties have arbitrary definitions. As such, we define the physical properties in ways that are most relevant to the distribution of mass of the particle, since the mass distribution is most closely tied to the near-field interactions of dipoles within a particle that generate the observed polarimetric signals (e.g., Lu et al. 2014). We use the convex hull of points associated with an aggregate to calculate the particle volume and maximum dimension; the latter is the maximum distance between two convex hull points. We then determine the effective density of the particle as the ratio of the particle mass to the convex hull volume. This method for calculating effective density will produce higher values compared to using circumscribing idealized shapes such as ellipsoids or spheres, since the convex hull more tightly encompasses the particle. However, this estimate of density is a more direct measure of the relative sparseness of the aggregate internal structure.

Recent work (Jiang et al. 2019; Dunnavan et al. 2019) has shown that ellipsoids can characterize the physical properties of aggregates relatively well, where two aspect ratios are determined by fitting an ellipsoid to the aggregate. We determine these two aspect ratios by first calculating the three eigenvalues of the covariance matrix of aggregate dipole locations. Using the covariance matrix allows for aspect ratios to depend more strongly on the distribution of mass throughout the structure compared to using an ellipsoid fit over the exterior. Two aspect ratios are found from the ratios of the largest eigenvalue to the two smaller eigenvalues, with the largest aspect ratio denoted by ϕ1 and the smallest aspect ratio denoted by ϕ2 (analogous to ellipsoid aspect ratios b/a and c/a, respectively). Using the aggregate maximum dimension dmax, we define characteristic particle dimensions a, b, and c with
a=dmax2,
b=aϕ1,
c=aϕ2.
An illustration of these dimensions is shown in Fig. 4.
Fig. 4.
Fig. 4.

Side and top view perspectives of an aggregate and a corresponding ellipsoid. The dimensions of the ellipsoid correspond to the characteristic dimensions of the particle (a, b, and c).

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

The simulated particles have mass and maximum dimension values that generally fall within empirical relations derived from in situ measurements (e.g., Locatelli and Hobbs 1974; Mitchell et al. 1990) shown in Fig. 5a, with a range in mass up to 0.5 mg at a given maximum dimension. The measurements of maximum dimension from these studies have uncertainties associated with the impact of the particle with the collection surface, measurements taken at a single viewing angle, and the small sample sizes, limiting our ability to rigorously compare them to the aggregates generated herein. The two-column aggregate sets reach larger sizes and masses compared to the branched planar crystal aggregates. The nonpivoting ER branched planar crystals generally have the lowest mass relative to maximum dimension.

Fig. 5.
Fig. 5.

Scatterplots of (a) mass vs maximum dimension, (b) effective density vs mass, and (c) the two fitted ellipsoid aspect ratios for the synthetic aggregates described herein. The contours in (a) encompass 75% of each aggregate set; in (b) and (c), the dashed and solid contours encompass 75% and 25% of the particles, respectively. The empirical relations in (a) labeled SP Agg. and Col. Agg. correspond to aggregates of side planes and aggregates of columns, respectively.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

There are more substantial differences between the aggregation sets with respect to effective density. Both column aggregate sets have low densities, generally <50 kg m−3 (Fig. 5b). The branched planar crystal aggregates have larger effective densities relative to the column aggregates, and show a general decrease in effective density with respect to mass. On average, the AR branched planar aggregates with pivoting have the highest densities with ∼25% of the values between 125 and 275 kg m−3. Of the branched planar crystal aggregates, the ER branched planar aggregate without pivoting set has the lowest densities on average with values mostly <100 kg m−3 (Fig. 5b).

There are also noticeable differences in the relations between the two aspect ratios ϕ1 and ϕ2 among the aggregate sets (Fig. 5c). The column aggregate sets have ϕ1 and ϕ2 values that are generally closer to 1 than the branched planar aggregates, indicating that these particles are more spherical. The ER nonpivoting branched planar aggregates have ϕ1 values relatively close to ϕ2 values, indicating that these particles are closer to prolate shapes than oblate shapes. These particles occupy a similar location in dual-aspect-ratio space as the retrieved data from Jiang et al. (2019). The aggregate set with the lowest aspect ratios is the AR pivoting branched planar aggregates, with ϕ2 values mostly between 0.1 and 0.4. Values of ϕ1 for these particles are >0.5, indicating shapes that are more similar to oblate spheroids than prolate spheroids.

3. Scattering calculations

a. Simplified assumptions for the particle orientation distribution

Particle fall behavior is typically parameterized in terms of orientation angle distributions (e.g., Ryzhkov et al. 2011), and we use the Euler angles herein to describe such distributions. Observational evidence as well as laboratory studies with idealized analog particles (e.g., Westbrook and Sephton 2017) suggest that aggregates have a tendency to rotate about a vertical axis (i.e., rotations of α). However, there is still a great deal of uncertainty regarding the conditions where this mode of rotation dominates the orientation behavior during free fall. Unsteady oscillations for particles with large Reynolds numbers may lead to more tumbling over chaotic rotation axes (McCorquodale and Westbrook 2021). Additionally, turbulent air motion in the atmosphere is also likely to perturb the rotation axis. We make the simplifying assumption that α and γ are uniformly distributed. This assumption allows us to characterize the particle orientation distribution as a function of β, the angle of tilt of the particle from its z axis in the base or laboratory reference frame (Fig. 2).

In particular, we characterize the distribution of β angles with beta distributions of the form [Abramowitz and Stegun 1972, Eq. (26.5.1), p. 944]
F(υ;a,b)=1B(a,b)υa−1(1−υ)b,
where a and b are the orientation distribution parameters, B(a, b) is the beta function, and υ ranges from 0 to 1 and relates to β by
β=cos⁡−1(1−2υ).
The advantages of using this distribution compared to the common Gaussian canting angle distribution (e.g., Ryzhkov 2001) are that the domain of this distribution is identical to the domain of β, it has a simple form that can be analytically integrated for scattering properties defined with a series of Legendre polynomials (appendix B), and it has the ability to capture both sharply peaked (i.e., near-horizontal alignment) and uniform distributions. We fix a = 1 and allow b ≥ 1, resulting in orientation distributions where the mode is at β = 0 (i.e., horizontal orientation). Values of b = 1 correspond to random uniform orientation distributions; as b increases, the dispersion of orientations decreases (i.e., the amount of fluttering or canting decreases). Examples of various orientation distributions with different b values are shown in Fig. 6.
Fig. 6.
Fig. 6.

Beta distributions for υ=(1−cos⁡β)/2 plotted with respect to β.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

b. Spectral representation of the radar moment orientation variability

Understanding how the radiative properties of a particle are linked to its physical properties is a fundamental problem in forward simulating radar measurements. When performing scattering calculations with ADDA for many orientations, the variability of the radar moments with orientation is unknown a priori and needs to be characterized using discrete variables. Our approach in characterizing the orientation variability of the radar moments is to transform these functions from orientation angle space to spectral space. As shown in appendix A, the radar moments can be transformed from functions of the Euler angles to a series of Wigner D-matrix coefficients. Furthermore, applying the assumption of uniformly distributed α and γ angles for the particles allows for the variability in β of the radar moments to be expressed in spectral space by a series of Legendre polynomial coefficients as (appendix A):
f(υ)=∑l=0∞ClPl(1−2υ)=∑l=0∞ClPl′(υ),
where f(υ) is the radar moment as a function of υ, Cl are its expansion coefficients, and Pl′(υ)=Pl(1−2υ) are Legendre polynomials of order l in terms of υ. Each Cl is the contribution of Pl′(υ) (Fig. 7) to the radar moment f(υ). As such, we seek to find the coefficients that contribute most to the variability of each radar moment with β and relate them to the physical properties.
Fig. 7.
Fig. 7.

Legendre polynomials up to order l = 6.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

The direct dependence of the radar moments on particle orientation cannot be observed with current remote sensing technology. Instead, an orientation-averaged property 〈f〉 is observed; 〈f〉 depends on the orientation distribution F(α, β, γ) and the radiative property as a function of the Euler angles f(α, β, γ) with (Mishchenko and Yurkin 2017)
⟨f⟩=∫02π∫0π∫02πF(α,β,γ)f(α,β,γ)sin⁡(β)dαdβdγ.
Applying our simplified particle orientation distribution assumptions stated in the previous section, we have
⟨f⟩=∫01f(υ)F(υ;a,b)dυ,
and from appendix B,
⟨f⟩=∑l=0∞wl(a,b)Cl,
where wl(a, b) are the orientation distribution weights for the spectral coefficients. Because the zeroth-order Legendre polynomial is 1, w0 = 1 for all values of b. Figure 8 shows these weights for different values of b. For b = 1, the l > 1 weights are all zero, deviating from zero as b increases. The increasing magnitude of these weights with b indicates that as the orientation distribution becomes more peaked near β = 0, the higher-order coefficients contribute more to the orientation-averaged scattering properties.
Fig. 8.
Fig. 8.

Orientation distribution weights for the Legendre series coefficients as functions of b. The weights are normalized by a factor of 1/2l+1 for visualization purposes.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

Despite the simplified forms of the radar moments for uniformly distributed α and γ, calculations for completely asymmetric particle shapes must still be conducted on a set of unique values for all three Euler angles. The chosen set of Euler angle triplets determines the maximum number of spectral coefficients that can be resolved, as shown in detail by McEwen et al. (2015).

c. ADDA simulation configuration

We use a general configuration for our scattering simulations that is the same for each synthetic aggregate set. This configuration includes the particle orientation angles, the frequencies of the radiation, the scattering angles, and the direction of incident radiation (set fixed along the x axis of the base reference frame). The particle orientation angles are set according to Gauss–Legendre quadrature nodes for α and β (as in Wieczorek and Meschede 2018), and seven equally spaced nodes between 0° and 360° are used for γ. This angular grid allows for relatively efficient projection onto spherical harmonics using the Python SHTOOLS wrapper (Wieczorek and Meschede 2018), though more efficient quadrature methods are known for the spherical nodes (e.g., Lebedev quadrature). Our choice of orientation angles corresponds to representations of spherical harmonics to order 6. We use the Python SHTOOLs wrapper to calculate the Cl values from the calculations performed on this set of orientation angles for each particle and radar moment. The one-dimensional orientation angle grids for α, β, and γ that form the three dimension grid of orientation nodes (resulting in 588 independent calculations) are listed in Table 2. We perform the calculations at Ku and Ka band; the precise wavelengths are given in Table 2.

Table 2

Specifications for the ADDA scattering calculations used herein.
Table 2

The additional necessary input to ADDA are dimensionless indices for the x, y, and z locations of the dipoles defining the aggregate shape. The length of the dipole is then needed to incorporate the correct physical size into the calculations. We use 15.2 μm for the dipole size to be consistent with the spacing of points used in generating the aggregation simulations and to satisfy the conditions where DDA is valid.

4. Results

a. Analysis of the scattering moment coefficients

Based on the conceptual model of individual elements of a particle scattering with different phases relative to each other, with attendant near-field effects on this scattering, it is clear that how the scattering properties vary with respect to β depends on the wavelength of radiation, the particle size, and particle shape. Thus, the series of Legendre coefficients for a set of scattering properties with respect to β contain the shape and size information relevant for particle scattering at a particular frequency. Similarly, the size relative to the wavelength and the shape determine how many coefficients are needed to accurately characterize how the scattering properties change with respect to β. We can find the dominant terms of the series for each radar scattering property by constructing power spectra from the coefficients using Parseval’s theorem for spherical harmonics (Wieczorek and Meschede 2018),
Il=(Cl)2,
where Il is the power of coefficient Cl of order l. Each Il contributes a fraction of the total power of the spectrum and therefore can be interpreted as the relative importance of Cl in characterizing how the radar moment varies with β. The scattering properties we consider herein are the squared horizontally polarized scattering amplitude |Shh|2, the squared vertically polarized scattering amplitude |Svv|2, specific differential phase Kdp, the real part of the horizontal–vertical scaled covariance Re(Shh*Svv), and the imaginary part of the horizontal–vertical scaled covariance Im(Shh*Svv), and we denote the corresponding spectral coefficients as Chhl, Cvvl, Cdpl, Crel, and Ciml, respectively.

The mean spectra for each scattering property over each particle set are shown in Fig. 9. In general, the spectra have peaks in the even coefficients and troughs in the odd coefficients indicating that the symmetric Legendre polynomials relative to β = 90° dominate the scattering behavior with respect to orientation (Fig. 7). As such, we only discuss the even coefficients herein. This symmetry is a consequence of averaging the scattering properties over α and γ, effectively making the particles appear rotationally symmetric. For the horizontal and vertical power coefficients, the zeroth-order coefficients are dominant, followed by the second-order and fourth-order coefficients (Fig. 9). For the Kdp coefficients, the second-order coefficient is dominant, with the zeroth-order and fourth-order coefficients substantially (>70 dB) lower in magnitude. The Crel have similar power spectra to the Chhl and Cvvl coefficients; the Ciml have similar power spectra to the Cdpl coefficients (Fig. 9). The sixth-order coefficients for all the variables are further reduced relative to the fourth-order coefficients.

Fig. 9.
Fig. 9.

Power spectra for (a) Chh at Ku band, (b) Chh at Ka band, (c) Cvv at Ku band, (d) Cvv at Ka band, (e) Cdp at Ku band, (f) Cdp at Ka band, (g) Cre at Ku band, (h) Cre at Ka band, (i) Cim at Ku band, and (j) Cim at Ka band. The colors indicate the aggregation experiment set and the lines are the mean spectra for each aggregation set. The dashed black lines are provided as a visual aid for the relative magnitude of the spectral coefficients.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

Generally, the behavior of these coefficient spectra are similar between Ku and Ka bands, with the fourth-order and sixth-order coefficients decreasing slower with order at Ka band compared to Ku band. The increasing contribution of the higher-order coefficients at Ka band results from increased non-Rayleigh scattering behavior. For the Cdpl coefficients, the power spectra are similar between Ku and Ka bands.

b. Relations between scattering properties and physical properties

The dependence of scattering properties on the exact shape of ice particles leads to a fundamental uncertainty in the relations between scattering properties and a set of physical properties (e.g., Schrom and Kumjian 2018), and a corresponding uncertainty in retrieving physical properties from remote sensing measurements. However, the scattering calculations we present show that the scattering property coefficients have clear relations to the physical properties, especially mass and effective density. Since P0′(υ)=1 (Fig. 7), the zeroth-order coefficients correspond to the mean values of the radar moment over β (i.e., those for uniform random orientation distributions). Because the orientation-averaged radar moments are weighted sums of the spectral coefficients, the higher-order coefficients enhance or reduce 〈f〉 relative to C0.

Figure 10 shows the zeroth-order and second-order coefficients for the horizontal and vertical powers at Ku and Ka band. Chh0 is highly correlated to the mass at Ku band, increasing approximately with the square of mass as predicted by Rayleigh theory, since Chh0 is proportional to the backscatter cross section for uniform random orientations (Fig. 10a). To reduce the impact of different particle masses on the higher-order coefficients, we examine the ratio of Chh2 to Chh0. Positive (negative) values of this ratio indicate backscatter cross sections for aggregates with preferred horizontally aligned orientations that are larger (smaller) than for aggregates with uniform random orientations. Ratios of Chh2 to Chh0 increase with effective density, with the largest values > 0.1. For the lowest density values (<100 kg m−3; associated with the column aggregates), the values range from −0.05 to 0.07 (Fig. 10b). Additionally, the ratio of Chh2 to Chh0 is negatively correlated with the ratio of Cvv2 to Cvv0, indicating that as the density of the particles increase, the second-order coefficients contribute to enhancements in backscatter at horizontal polarization and reductions in backscatter at vertical polarization (Fig. 10c). This behavior implies that ZDR is generally positive and increases with density for aggregates that have orientation distributions with preferred horizontal alignment.

Fig. 10.
Fig. 10.

Scatterplots of (a) Chh0 at Ku band vs mass, (b) Chh2/Chh0 at Ku band vs effective density, (c) Cvv2/Cvv0 at Ku band vs effective density, (d) Chh0 at Ka band vs mass, (e) Chh2/Chh0 at Ka band vs effective density, and (f) Cvv2/Cvv0 at Ka band vs effective density.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

At Ka band, there is more variability in the scattered power at horizontal and vertical polarization. The Chh0 of the particles with the smallest mass also increase with mass squared. However, the Chh0 increases at a slower rate relative to that at Ku band for mass > 0.1 mg, indicative of non-Rayleigh scattering behavior. For the ratio of Chh2 to Chh0, there is a general positive trend with increasing effective density. However, there is a wider range of values for a given density relative to Ku band and the majority of values are negative (Fig. 10e). There is a clearer trend in the ratio of Cvv2 to Cvv0 at Ka band, despite a large degree of variability; this ratio becomes more negative as density increases (Fig. 10f). The combined effect of the reduction in backscattered power at vertical polarization and neutral to slight reduction in backscattered power at horizontal polarization leads to a general net positive ZDR for aggregates with preferred horizontal orientation (not shown).

At both Ku and Ka band, the Kdp spectral coefficients depend on both mass (Fig. 11) and density, with the latter implied by the density variation between aggregate sets (Fig. 5b). As shown in Fig. 9, the contribution of Cdp0 to the total Kdp orientation function is minimal relative to Cdp2. The relatively small values of Cdp0 are a result of the mean Kdp being near 0° km−1, corresponding to uniform random particle orientations. The higher-density particles such as the pivoting branched planar aggregates and nonpivoting, azimuthally random branched planar aggregates have the largest Cdp2 values for a given mass, with the coefficient magnitudes increasing with mass (Fig. 11b). The coefficients at Ku- and Ka-band scale almost exactly according to the ratio of their wavelengths (Figs. 11c,d), suggesting that forward scattering of the particles at these frequencies is well-approximated by Rayleigh theory (contrary to backscatter).

Fig. 11.
Fig. 11.

Scatterplots of (a) Cdp0 at Ku band vs mass, (b) Cdp2 at Ku band vs mass, (c) Cdp0 at Ka band vs Cdp0 at Ku band, and (d) Cdp2 at Ka band vs Cdp2 at Ku band.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

We also consider the covariance between backscattered horizontally and vertically copolarized radiation. This quantity is primarily used in the estimation of the copolar correlation coefficient ρhv defined as
ρhv=|⟨Shh*Svv⟩|⟨|Shh|2⟩⟨|Svv|2⟩.
Based on the spectral analysis shown in Fig. 9, Ccov0 and Ccov2 dominate the backscatter (where Ccovl=Crel+iCiml), and therefore, the orientation-averaged covariance is well approximated as
⟨Shh*Svv⟩∼w0Ccov0+w2Ccov2,
where w0 = 1 and w2 is determined by the orientation distribution (as in appendix B). From this approximation, ρhv can be written
ρhv∼|Ccov0|2+w22|Ccov0|2+2w2Re(Ccov0Ccov2)⟨|Shh|2⟩⟨|Svv|2⟩.
Since |Ccov0| is much greater than |Ccov2|, ρhv can additionally be approximated as
ρhv∼|Ccov0|2+2w2Re(Ccov0Ccov2)⟨|Shh|2⟩⟨|Svv|2⟩.
Using this expression for ρhv allows us to use only real components for the covariance as well as to scale the coefficients within the order of magnitude of the ρhv values by |Chh0|2. Scaling these components allows for us to examine how the relevant terms of the covariance contribute to ρhv individually. For b = 1, only the zeroth-order coefficient term associated with random particle orientation contributes to the covariance. As b increases, the contribution from the second term of the covariance contributes more to the total covariance and therefore to ρhv.

The two components of the covariance scaled by |Chh0|2 are plotted for Ku and Ka band in Fig. 12. At Ku band, both the ratio of |Ccov0|2 to |Chh0|2 and the ratio of 2Re(Ccov0Ccov2) to |Chh0|2 decrease with density. At Ka band, the ratio of |Ccov0|2 to |Chh0|2 behaves similarly to that at Ku band (Fig. 12c). The ratio of 2Re(Ccov0Ccov2) to |Chh0|2 shows no trend with density for density > 100 kg m−3, with mostly negative values between −0.5 and −0.2.

Fig. 12.
Fig. 12.

Scatterplots of (a) |Ccov0|2/|Chh0|2 at Ku band vs effective density, (b) 2Re(Ccov0Ccov2)/|Chh0|2 at Ku band vs effective density, (c) |Ccov0|2/|Chh0|2 at Ka band vs effective density, and (d) 2Re(Ccov0Ccov2)/|Chh0|2 at Ka band vs effective density.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

The scattering properties of the Monte Carlo aggregates generated herein show minimal dependence on the particle aspect ratio, similar to the results of Ekelund and Eriksson (2019). One reason aspect ratio is less important is that it may be erroneously influenced by elements of the ice structure that have limited impact on the scattering properties. In particular, aggregates composed of only a few monomers have highly nonuniform distributions of mass within the bulk shape (e.g., convex hull, ellipsoidal fits) implied by the aspect ratios, and the aspect ratio provides negligible information about the monomer orientations within the aggregate or the monomer aspect ratios. For aggregates with a larger number of monomers (e.g., the column aggregates generated herein), the mass distribution becomes more homogeneous. However, these particles generally have relatively small polarimetric enhancements (cf. Figs. 10–12), making it difficult to assess whether the bulk aspect ratios are important for the scattering properties of aggregates with larger numbers of monomers.

The simulated polarimetric radar observables averaged over each aggregation set have varying dependencies on the assumed orientation dispersion parameter (b). We show these relations in Figs. 13 and 14, where we first calculate the orientation-averaged polarimetric radar variables over a range of individual b values for each aggregate in the set, and then average the polarimetric radar variables over each set. At Ku band, the average ZDR and Kdp per mass have nearly identical relative increases with increasing b, with the highest ZDR values of 2.5–3.5 dB at b = 100 found for both sets of branched planar aggregates with pivoting and the set of AR branched planar aggregates without pivoting (Fig. 13). The ER and nonpivoting branched planar aggregates have smaller ZDR values at b = 100 of ∼1 dB that are similar to the ZDR of the AR column aggregates. The ER column aggregates have the smallest maximum ZDR values with ZDR at b = 100 of <0.5 dB (Fig. 13). Values of the mean orientation-averaged ρhv minimize near b = 4 with values between 1 and 0.97, increases to >0.995 at b = 100. As expected, the aggregate sets with the smallest ρhv minima tend to have the largest ZDR maxima. In terms of the standard deviation of the polarimetric radar observables for each aggregation set, the values increase with b, with the ER column aggregates having values less than half that of the other aggregate sets (Fig. 13d).

Fig. 13.
Fig. 13.

Orientation-averaged Ku-band (a) mean of ZDR, (b) mean of ρhv, (c) mean of Kdp, (d) standard deviation of ZDR, (e) standard deviation of ρhv, and (f) standard deviation of Kdp. The line color indicates the aggregation experiment as labeled in (f).

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

Fig. 14.
Fig. 14.

As in Fig. 13, but for Ka band.

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0149.1

The relations between these orientation-averaged scattering properties and b are similar at Ka band to those shown at Ku band. The maximum values of ZDR (found for the pivoting branched planar aggregates and nonpivoting AR branched planar aggregates) are ∼0.5 dB higher at Ka band (Fig. 14a). The shapes of the mean orientation-averaged ρhv curves have somewhat more variability at Ka band than those at Ku band, with the ρhv minima occurring at a wider range of b values (Fig. 14b). Values of the mean Kdp per mass are proportionally higher at Ka band compared to Ku band, owing to the shorter wavelength (Fig. 14c). The values of ρhv at b = 100 are also lower at Ka band than those at Ku, suggesting that some small flutter can still produce measurable ρhv reductions at this wavelength, especially for the ER branched planar aggregates. The variability in the orientation-averaged ρhv across the branched planar aggregate sets is much larger at high b values compared to the variability at Ku band at these b values. Both sets of ER branched planar aggregates have their highest standard deviations of ρhv at b = 100, suggesting certain particles may have much lower ρhv when assumed to flutter lightly (Fig. 14e).

5. Discussion and conclusions

The shapes, orientation behavior, and resulting scattering properties of aggregates are complex. The simplified, Monte Carlo aggregation experiments presented herein show a wide range of bulk physical properties are possible from initial assumptions about the monomer properties, their attachment behavior, and the fall behavior of the aggregate. The synthetic aggregates assumed to fall with uniform azimuthal rotations about their vertically aligned axes tend to have flatter shapes with denser structures than the synthetic aggregates assumed to fall with completely random orientations. Additionally, the inclusion of monomer pivoting upon attachment produces a similar effect of flatter and denser structures. The nonpivoting, completely random branched planar aggregates have the sparsest structures. The column aggregates show comparatively lower-density structures with more isotropically distributed mass (i.e., less flat) than the branched planar aggregates. The degree of horizontal orientation of the monomers is related to density because consistently more aligned particles pack together more efficiently in space compared to more randomly aligned monomers.

We find that the polarimetric scattering properties of the aggregates depend most strongly on mass and effective density (defined based on the convex hull volume of the particles). Our spectral analysis of the scattering coefficients shows that for the zeroth-order coefficient that corresponds to random uniform particle orientation, backscatter is highly correlated to mass and increases monotonically with mass at both Ku band and Ka band. Increases in effective density between 50 and 150 kg m−3 are associated with rapid increases in the second-order backscatter coefficient at horizontal polarization and rapid decreases in the second-order backscatter coefficient at vertical polarization. Similarly, the components of the copolar covariance at these densities decrease rapidly with density. The changes in these coefficients with density are likely due to the increasing inhomogeneity of the aggregate internal structure of partially aligned monomers.

For orientation distributions with increasing horizontal alignment, the second-order coefficients enhance the orientation-averaged backscatter at horizontal polarization, and reduce the orientation-averaged backscatter at vertical polarization, resulting in increases in ZDR. These relations with respect to effective density are likely a reflection of how the degree of monomer horizontal alignment and packing within an aggregate tend to increase with effective density, especially when considering the effective density in terms of the convex hull of the aggregate. With increasing horizontal alignment of the monomers, the near-field interactions of the dipoles enhance the internal electric fields at horizontal polarization and reduce the internal electric fields at vertical orientation. ZDR and Kdp increases monotonically with the orientation b parameter, while ρhv has minima between b values of 2–5. These unique behaviors suggest the potential for these variables to provide complementary information about aggregate shapes and orientation distributions.

Our results suggest that conventional measures of aspect ratio have indirect relations to the polarimetric scattering properties. This lack of importance of aspect ratio is likely due to the polarization effects being influenced by the monomer sizes, orientations relative to the polarization state of the incident radiation, and proximity to each other, and this information is not captured by aspect ratio. Individual ice crystals and rimed particles are more likely to show strong scattering property dependencies on aspect ratios, since the mass within these particles is more uniformly distributed and the particle symmetries reduce the ambiguity in defining the aspect ratio.

Our numerical experiments shown herein are relatively simplified, and therefore, the true hydrodynamics of falling aggregates may vary substantially to the assumptions we make herein. For example, the fall behavior of the aggregates is likely to change as increasing numbers of monomers become incorporated into the structure, changing the mass and projected area (e.g., Heymsfield and Westbrook 2010). These more complex natural orientation distributions may lead to polarimetric scattering properties that cannot be reproduced with our more simplified assumptions. Additionally, monomers may slip as well as pivot during their attachment. This slipping effect may further increase the density of the aggregate structures as the monomers become more closely packed.

Finally, natural aggregates are typically composed of a variety of ice crystal habits, partially rimed particles and other aggregates; our synthetic aggregates have only single habits and only include pristine-aggregate collection. However, it is possible that aggregates composed of mixed habits may have physical and scattering properties that are similar to those simulated herein, with the possible exception to aggregates composed of irregular ice particles. For these aggregates of irregular ice particles, it is possible that the mass is more homogeneously distributed compared to aggregates of branched planar crystals and aggregates of columns, and bulk properties such as aspect ratio may be more important than characterizing the monomer properties.

To best evaluate model simulations of ice clouds, it is important to accurately simulate the scattering properties associated with the simulations. We have shown that the effective density of the simulated aggregates depend substantially on the monomer habits, attachment assumptions of the monomers, and orientation of the aggregates as they fall. However, the relations between the scattering properties and mass and density suggest the potential for the scattering properties to be predicted without explicit knowledge of the detailed structure of the monomers within an aggregate. Such predictions could be used in a forward model such as in Schrom and Kumjian (2019) to efficiently predict probabilities of scattering properties given a set of physical properties such as mass and effective density. These probabilities may then be used in a forward simulation to characterize the mean and uncertainty of model-simulated radar measurements to more quantitatively use radar measurements to inform microphysical model parameterizations.

1

Some solvers that allow for the calculation of scattering properties at different orientations efficiently after an intensive initial computation do exist (e.g., Mackowski 2002).

Acknowledgments.

This work is supported by the NASA GPM Ground Validation program and the NASA Postdoctoral Program, administered by Oak Ridge Associated Universities. We thank the reviewers for their thorough comments that substantially improved this manuscript.

Data availability statement.

The ADDA simulations for the aggregate experiments and the corresponding geometric property files are stored in an online repository (available at https://github.com/rskschrom/aggregate_scattering).

APPENDIX A

Representation of Scattering Properties for Azimuthally Uniform, Fluttering Particles

a. General representation

The scattering properties are square integrable over the Euler angles, and therefore, f(α, β, γ) can be expressed with an expansion of Wigner D-matrix elements as (McEwen et al. 2015)
f(α,β,γ)=∑l=0∞2l+18π2∑m=−ll∑n=−llCmnlDmnl*(α,β,γ),
with coefficients Cmnl, and the asterisk indicates the complex conjugate. The Wigner D-matrix elements are the orthogonal basis for functions of the Euler angles, analogous to spherical harmonics being the orthogonal basis for functions on a sphere (Shen et al. 2018).

b. Representation with uniform γ

Averaging f(α, β, γ) over γ and using
Dmnl(α,β,γ)=e−imαdmnl(β)e−inγ,
where dmnl is the “small” d-matrix element, gives
⟨f(α,β)⟩γ=∑l=0∞2l+18π2∑m=−llCm0ldm0l(β)eimα.
The d-matrix elements with n = 0 are related to the spherical harmonics Yml(α,β) with (McEwen 2011)
eimαdm0l(β)=(l−m)!(l−m)!Plm(cos⁡β)eimα=4π2l+1Ylm(α,β),
where Plm(cos⁡β) are the associated Legendre polynomials, and therefore, ⟨f(α,β)⟩γ can be written as an expansion in spherical harmonics as
⟨f(α,β)⟩γ=∑l=0∞∑m=−llClmYlm(α,β),
with
Clm=2l+116π3Cm0l.

c. Representation with uniform α and γ

Averaging Eq. (A5) over α gives
⟨f(β)⟩α,γ=∑l=0∞ClPl(cos⁡β),
where Pl is the ordinary Legendre polynomial of order l, and Cl are the expansion coefficients.

APPENDIX B

Analytical Orientation Averaging of Scattering Properties Expressed with Beta Distributions

The nth moments In(a, b) of orientation distribution
F(υ;a,b)=1B(a,b)υa−1(1−υ)b,
where υ=(1−cos⁡β)/2, are
In(a,b)=∫01υnF(υ;a,b)dυ=B(a+n,b)B(a,b).
A Legendre polynomial of order l can be expressed in terms of υ as
Pl′(υ)=Pl(1−2υ)=∑k=0lpkυk,
where pk are the polynomial coefficients that result after substituting x = 1 − 2υ into the conventional Legendre polynomials Pl(x). The scattering properties expressed by a Legendre series expansion
f(υ)=∑l=0∞ClPl′(υ)
can be integrated over the orientation distribution F(υ; a, b) by
⟨f⟩=∑l=0∞wl(a,b)Cl,
where wl(a, b) are the orientation distribution coefficient weights defined by
wl(a,b)=∑k=0lpkIk(a,b).

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  • Tyynelä, J., and V. Chandrasekar, 2014: Characterizing falling snow using multifrequency dual-polarization measurements. J. Geophys. Res. Atmos., 119, 8268–8283, https://doi.org/10.1002/2013JD021369.

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  • Tyynelä, J., J. Leinonen, D. Moisseev, and T. Nousiainen, 2011: Radar backscattering from snowflakes: Comparison of fractal, aggregate, and soft spheroid models. J. Atmos. Oceanic Technol., 28, 1365–1372, https://doi.org/10.1175/JTECH-D-11-00004.1.

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  • Waterman, P. C., 1969: Scattering by dielectric obstacles. Alta Freq., 38, 348–352.

  • Westbrook, C. D., and E. K. Sephton, 2017: Using 3-D-printed analogues to investigate the fall speeds and orientations of complex ice particles. Geophys. Res. Lett., 44, 7994–8001, https://doi.org/10.1002/2017GL074130.

    • Search Google Scholar
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  • Westbrook, C. D., R. C. Ball, and P. R. Field, 2006: Radar scattering by aggregate snowflakes. Quart. J. Roy. Meteor. Soc., 132, 897–914, https://doi.org/10.1256/qj.05.82.

    • Search Google Scholar
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  • Wieczorek, M. A., and M. Meschede, 2018: SHTools: Tools for working with spherical harmonics. Geochem. Geophys. Geosyst., 19, 2574–2592, https://doi.org/10.1029/2018GC007529.

    • Search Google Scholar
    • Export Citation

  • Xie, Y., P. Yang, G. W. Kattawar, B. A. Baum, and Y. Hu, 2011: Simulation of the optical properties of plate aggregates for application to the remote sensing of cirrus clouds. Appl. Opt., 50, 1065–1081, https://doi.org/10.1364/AO.50.001065.

    • Search Google Scholar
    • Export Citation

  • Yurkin, M. A., and A. G. Hoekstra, 2011: The discrete-dipole-approximation code ADDA: Capabilities and known limitations. J. Quant. Spectrosc. Radiat. Transfer, 112, 2234–2247, https://doi.org/10.1016/j.jqsrt.2011.01.031.

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    • Search Google Scholar
    • Export Citation

  • Tyynelä, J., J. Leinonen, D. Moisseev, and T. Nousiainen, 2011: Radar backscattering from snowflakes: Comparison of fractal, aggregate, and soft spheroid models. J. Atmos. Oceanic Technol., 28, 1365–1372, https://doi.org/10.1175/JTECH-D-11-00004.1.

    • Search Google Scholar
    • Export Citation

  • Waterman, P. C., 1969: Scattering by dielectric obstacles. Alta Freq., 38, 348–352.

  • Westbrook, C. D., and E. K. Sephton, 2017: Using 3-D-printed analogues to investigate the fall speeds and orientations of complex ice particles. Geophys. Res. Lett., 44, 7994–8001, https://doi.org/10.1002/2017GL074130.

    • Search Google Scholar
    • Export Citation

  • Westbrook, C. D., R. C. Ball, and P. R. Field, 2006: Radar scattering by aggregate snowflakes. Quart. J. Roy. Meteor. Soc., 132, 897–914, https://doi.org/10.1256/qj.05.82.

    • Search Google Scholar
    • Export Citation

  • Wieczorek, M. A., and M. Meschede, 2018: SHTools: Tools for working with spherical harmonics. Geochem. Geophys. Geosyst., 19, 2574–2592, https://doi.org/10.1029/2018GC007529.

    • Search Google Scholar
    • Export Citation

  • Xie, Y., P. Yang, G. W. Kattawar, B. A. Baum, and Y. Hu, 2011: Simulation of the optical properties of plate aggregates for application to the remote sensing of cirrus clouds. Appl. Opt., 50, 1065–1081, https://doi.org/10.1364/AO.50.001065.

    • Search Google Scholar
    • Export Citation

  • Yurkin, M. A., and A. G. Hoekstra, 2011: The discrete-dipole-approximation code ADDA: Capabilities and known limitations. J. Quant. Spectrosc. Radiat. Transfer, 112, 2234–2247, https://doi.org/10.1016/j.jqsrt.2011.01.031.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    An illustration of the pivoting procedure we use herein. (a) The monomer falls toward the aggregate along the direction v. (b) The first attachment point is at point p1, and the first pivot occurs at p1 with rotations about the axis r1 in the direction indicated by the purple curved arrow. The labeled point c indicates the monomer center of mass. (c) The second attachment point resulting from the first pivot is labeled as p2. The second pivot occurs about the line connecting p2 to p1 with rotations about the axis r2 in the direction indicated by the gold curved arrow.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ER-BPA-P aggregate farthest to the right).

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Side and top view perspectives of an aggregate and a corresponding ellipsoid. The dimensions of the ellipsoid correspond to the characteristic dimensions of the particle (a, b, and c).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Scatterplots of (a) mass vs maximum dimension, (b) effective density vs mass, and (c) the two fitted ellipsoid aspect ratios for the synthetic aggregates described herein. The contours in (a) encompass 75% of each aggregate set; in (b) and (c), the dashed and solid contours encompass 75% and 25% of the particles, respectively. The empirical relations in (a) labeled SP Agg. and Col. Agg. correspond to aggregates of side planes and aggregates of columns, respectively.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Beta distributions for υ=(1−cos⁡β)/2 plotted with respect to β.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Legendre polynomials up to order l = 6.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Orientation distribution weights for the Legendre series coefficients as functions of b. The weights are normalized by a factor of 1/2l+1 for visualization purposes.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Power spectra for (a) Chh at Ku band, (b) Chh at Ka band, (c) Cvv at Ku band, (d) Cvv at Ka band, (e) Cdp at Ku band, (f) Cdp at Ka band, (g) Cre at Ku band, (h) Cre at Ka band, (i) Cim at Ku band, and (j) Cim at Ka band. The colors indicate the aggregation experiment set and the lines are the mean spectra for each aggregation set. The dashed black lines are provided as a visual aid for the relative magnitude of the spectral coefficients.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Scatterplots of (a) Chh0 at Ku band vs mass, (b) Chh2/Chh0 at Ku band vs effective density, (c) Cvv2/Cvv0 at Ku band vs effective density, (d) Chh0 at Ka band vs mass, (e) Chh2/Chh0 at Ka band vs effective density, and (f) Cvv2/Cvv0 at Ka band vs effective density.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Scatterplots of (a) Cdp0 at Ku band vs mass, (b) Cdp2 at Ku band vs mass, (c) Cdp0 at Ka band vs Cdp0 at Ku band, and (d) Cdp2 at Ka band vs Cdp2 at Ku band.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Scatterplots of (a) |Ccov0|2/|Chh0|2 at Ku band vs effective density, (b) 2Re(Ccov0Ccov2)/|Chh0|2 at Ku band vs effective density, (c) |Ccov0|2/|Chh0|2 at Ka band vs effective density, and (d) 2Re(Ccov0Ccov2)/|Chh0|2 at Ka band vs effective density.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Orientation-averaged Ku-band (a) mean of ZDR, (b) mean of ρhv, (c) mean of Kdp, (d) standard deviation of ZDR, (e) standard deviation of ρhv, and (f) standard deviation of Kdp. The line color indicates the aggregation experiment as labeled in (f).

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    As in Fig. 13, but for Ka band.

Fig. 1.

An illustration of the pivoting procedure we use herein. (a) The monomer falls toward the aggregate along the direction v. (b) The first attachment point is at point p1, and the first pivot occurs at p1 with rotations about the axis r1 in the direction indicated by the purple curved arrow. The labeled point c indicates the monomer center of mass. (c) The second attachment point resulting from the first pivot is labeled as p2. The second pivot occurs about the line connecting p2 to p1 with rotations about the axis r2 in the direction indicated by the gold curved arrow.
Fig. 2.

A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.
Fig. 3.

Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ER-BPA-P aggregate farthest to the right).