Retrieval of polar stratospheric cloud microphysical properties from lidar measurements: Dependence on particle shape assumptions


  • J. Reichardt,

    1. Joint Center for Earth Systems Technology, University of Maryland Baltimore County, Baltimore, Maryland, USA
    2. Atmospheric Chemistry and Dynamics Branch, Laboratory for Atmospheres, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
    Search for more papers by this author
  • S. Reichardt,

    1. Joint Center for Earth Systems Technology, University of Maryland Baltimore County, Baltimore, Maryland, USA
    2. Atmospheric Chemistry and Dynamics Branch, Laboratory for Atmospheres, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
    Search for more papers by this author
  • P. Yang,

    1. Department of Atmospheric Sciences, Texas A&M University, College Station, Texas, USA
    Search for more papers by this author
  • T. J. McGee

    1. Atmospheric Chemistry and Dynamics Branch, Laboratory for Atmospheres, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
    Search for more papers by this author


[1] Computational simulations have been performed to deduce the dependence of the microphysical analysis of polar stratospheric cloud (PSC) optical data measured with lidar on the assumptions made about particle shape. In a forward model, PSCs are modeled as crystalline and liquid particles in order to generate synthetic optical properties. The parameterization scheme of the PSC microphysical properties allows for coexistence of up to three different particle types with size-dependent shapes; optical properties of individual crystals, specifically hexagonal and asymmetric polyhedral crystals, are determined using the finite difference time domain (FDTD) method. The set of calculated PSC optical properties is selected based on the wavelengths and measurement capabilities of lidar instruments that are typically used to monitor stratospheric aerosols, such as the Airborne Raman Ozone and Temperature Lidar (AROTEL) and the NASA Langley Differential Absorption Lidar (DIAL) onboard the NASA DC-8 during the Stratospheric Aerosol and Gas Experiment (SAGE) Ozone Loss Validation Experiment (SOLVE) campaign in winter 1999/2000. The retrieval process then inverts these synthetic measurement data using a model approach in which the PSC particles are represented purely by an ensemble of spheroids, and the differences between initial and retrieved microphysical properties are examined. The model simulations show that under the assumption of spheroidal particle shapes, surface area density and volume density of leewave PSCs are systematically smaller by, respectively, ∼10–30% and ∼5–25% than the values found for mixtures of droplets, asymmetric polyhedra, and hexagons.

1. Introduction

[2] Recent studies raise concerns about a positive feedback between the anthropogenically induced climate change and stratospheric ozone depletion [Hartmann et al., 2000]. It is predicted that increasing concentrations of atmospheric carbon dioxide will affect the stratosphere in two ways: First, stratospheric temperatures could decrease [Shindell et al., 1998], and second, the input of water vapor into the stratosphere could increase due to higher temperatures at the tropical tropopause [Kirk-Davidoff et al., 1999]. These effects would result in polar vortices that are more stable, in expanded regions of polar stratospheric cloud (PSC) formation, and in an increase in the number and extent of PSCs [Butchart et al., 2000]. The cumulative effect of these changes could be an increase in ozone depletion. In turn, the reduction of stratospheric ozone concentrations would influence tropospheric greenhouse warming via dynamical, chemical, and radiative feedbacks [Hartmann et al., 2000].

[3] In this context, knowledge of particle sizes and number densities of PSCs is highly important, because they are critical parameters for the modeling of stratospheric chemical processes that lead to the destruction of ozone: Conversion rates of inert reservoir gases into potentially ozone-destroying chemical compounds strongly depend on the phase of the PSC particles and the particle surface area available for heterogeneous processing [Ravishankara and Hanson, 1996]. The size of the PSC particles determines the sedimentation velocity and hence the denitrification and dehydration of the stratosphere.

[4] One approach to the derivation of microphysical properties of stratospheric particles is interpretation of optical data measured in situ with optical particle counters (OPC), or remotely with lidar. Inversion of optical data from PSCs that consist of liquid ternary aerosols (LTA, aqueous solutions of nitric and sulfuric acid) [Mehrtens et al., 1999] or from stratospheric background aerosol [Jäger and Hofmann, 1991; Deshler et al., 1992; Wandinger et al., 1995; Borrmann et al., 2000b] can be accomplished, because the particles are spherical and Mie theory applies.

[5] For PSCs that are formed from solid particles [water ice and nitric acid hydrates (NAH)] an approach other than Mie theory to the analysis of optical data has to be followed, since the appreciable depolarization of the light backscattered from these clouds reveals that the solid PSC particles are predominantly nonspherical. Although in water ice PSCs the ice crystals may reach sizes of up to several tens of a micron [Goodman et al., 1989, 1997], PSC particle sizes can be generally expected to lie in the wavelength range of the observing optical instruments, and therefore optical calculations in the geometrical-optics limit are not applicable in most cases. One rigorous method that has proven to be very successful in calculation of scattering properties of small aspherical particles is the T-matrix technique [Waterman, 1970]. In principle, it allows for the determination of scattering phase matrices of arbitrarily shaped particles. However, for computational reasons, only particle shapes have been considered until very recently that exhibit rotational symmetry, e.g., spheroids and circular cylinders [Waterman, 1970; Mishchenko, 1991; Mishchenko et al., 1996] and Chebyshev particles [Mugnai and Wiscombe, 1989]. Of these principal particle shapes it has become common over the last years to use the spheroidal particle model for the microphysical interpretation of PSC measurements obtained with lidar [Carslaw et al., 1998; Tsias et al., 1999; Wirth et al., 1999; Reichardt et al., 2000; Solve · Theseo 2000 Science Meeting, Abstracts, 2000], with the exception of Liu and Mishchenko [2001] who use circular cylindrical particles also. Without question, the use of the spheroidal model for the interpretation of PSC optical data has been a substantial improvement on the use of the spherical model based on Mie theory. However, it is still an open question whether the spheroidal model describes the optical characteristics of PSC particles properly and how accurately the PSC microphysical properties are retrieved. As pointed out by Reichardt et al. [2000], the spheroid-based retrieval (as likewise any other particle model) relies on the assumption that the optical properties of an ensemble of model particles resemble those of crystalline PSCs with similar particle size distributions and particle number densities. However, there is no convincing observational evidence to support that PSCs are predominantly composed of spheroidal particles. On the contrary, in situ particle sampling, though rare, revealed PSC crystals of irregular, hexagonal, or trigonal shape [Goodman et al., 1989, 1997]. While the effects of particle nonsphericity on OPC measurements of particle microphysical properties have been evaluated [Whitby and Vomela, 1967; Baumgardner et al., 1992; Dye et al., 1992; Borrmann et al., 2000a], a detailed error treatment of lidar data inversion that is based on a specific assumption on the principal particle shape has not been carried out so far.

[6] This paper focuses on the dependence of the retrieval of PSC microphysical properties from lidar measurements on particle shape assumptions. It is the primary motivation of the present effort to investigate this problem by comparison of the results obtained with retrieval algorithms which are based on either a polyhedral or a spheroidal particle model. It is our belief that both spheroidal and polyhedral geometries are too idealized in comparison with the reality. Nevertheless both may serve in providing quantitative information about the specific problem addressed here.

[7] Recently, the finite-difference time-domain (FDTD) technique has become available for calculation of optical properties of small particles of arbitrary shape [Yang and Liou, 1996, 1998; Yang et al., 2000]. The volume of the scattering particle is discretized by using a Cartesian grid mesh which allows construction of particles with corners, edges, and flat crystal faces that may be regarded as representative of crystalline PSCs. Alternatively, T-matrix code has now been developed for a wider variety of particle shapes including star-shaped [Laitinen and Lumme, 1998] and polyhedral particles [Havemann and Baran, 2001; Kahnert et al., 2001]. However, because we wanted to include particles in our study that have no symmetry axis at all, and because computations for these particles are straightforward with the FDTD method, we decided to use the FDTD technique for calculation of the scattering properties of the solid PSC particles.

[8] Our approach to studying the effect of shape assumptions can be described as follows. Based on Mie and FDTD calculations, we generate lidar-relevant optical properties of PSCs, assuming mixtures of droplets, hexagonal crystals, and asymmetric polyhedral crystals. We decided to look at the color ratio, the depolarization ratio, and the lidar ratio, because often these are the PSC optical properties that are measured with lidars. E.g., they have been measured with two of the lidars which were operated on the NASA DC-8 during the Stratospheric Aerosol and Gas Experiment (SAGE) Ozone Loss Validation Experiment (SOLVE) campaign in winter 1999/2000 [Hostetler et al., 2000; McGee et al., 2000; Butler et al., 2001]. Then, assuming spheroidal particle shapes, we apply the retrieval algorithm to the simulated data to extract microphysical information, and finally we compare initial and retrieved PSC microphysical properties. FDTD calculations are computer intensive, with calculation time roughly proportional to the volume of the scattering particle. E.g., determination of the scattering phase matrix at 355 nm for a NAT PSC particle with a maximum dimension of 2 μm and an aspect ratio of 0.5 requires two weeks of computation time on a SGI octane2 workstation (360 MHz). Therefore the maximum size of the particles that can be considered in this study is 2 μm, and for this reason this study is restricted to the analysis of PSCs composed of small solid particles. In particular, this investigation is most applicable to leewave-induced PSCs [Tabazadeh et al., 1997; Carslaw et al., 1998; Tsias et al., 1999]. In section 2 the relation between particle microphysical properties and optical properties that can be measured with lidars is reviewed. Next we describe the microphysical retrieval algorithm that is used for inversion of lidar data (section 3). In section 4 the results of FDTD computations for single crystals are presented, followed by an intercomparison between the polyhedron-based and the spheroid-based retrieval algorithm in section 5. A summary of this study is given in section 6.

2. PSC Microphysical and Optical Properties

[9] The optical and microphysical properties of a cloud are related indirectly. The optical properties at a certain cloud height are averages over all light scattering processes by individual cloud particles at this height (single scattering is assumed). Particle scattering is determined by the scattering phase matrix which in turn depends on the size, shape and composition of the particles. If latter are to be retrieved from lidar measurements, further difficulties stem from the observation geometry. Lidars exclusively sense the backscattering direction of clouds, so that only the information content of the scattering matrices at 180° scattering angle can be exploited. Inversion of lidar measurements of crystalline PSCs is therefore an ill-posed problem which makes derivation of microphysical properties a challenging task.

[10] As previously mentioned, PSC volume density V, surface area density A, and particle size spectrum are of special interest. V is given by

equation image

A is described by a similar equation. In equation (1), N is the number density of the cloud particles, v is the particle volume, and equation image is the normalized size distribution equation image. We choose to parameterize equation image and v in terms of the maximum dimension d of the particles, in a manner similar to that used in the cirrus cloud studies of Auer and Veal [1970] and Heymsfield and Platt [1984].

[11] The complexity of lidar systems used for PSC observations determines the number of PSC optical properties that can be measured. In the following we give a summary of the optical quantities that are accessible to lidar remote sensing in principle. The first two properties, the particle extinction coefficient αpar and the backscatter coefficient βpar, depend on the particle number density of the PSC. The extinction coefficient is given by

equation image

where Qsca is the scattering efficiency of the PSC particle, and G is its geometrical scattering cross section. In addition to the explicit dependence on wavelength and maximum dimension shown in equation (2), Qsca also depends on the index of refraction of the scattering particles, and both Qsca and G are functions of the particles' morphology. In this study we assume that the PSC particles do not absorb light at the laser wavelengths.

[12] The backscatter coefficient is defined by

equation image

pπ denotes the value of the scattering phase function p at the backscattering direction. For PSCs it can be assumed that the cloud particles are randomly oriented (p = p(ϑ), where ϑ is the scattering angle). The scattering phase functions shown in this paper satisfy the normalization condition

equation image

so that p can be interpreted as the probability distribution function of the scattered energy. PSC extinction and backscatter coefficients can be measured with Raman lidars [Ansmann et al., 1992]. Because of their dependence on N, αpar and βpar, and likewise the backscatter ratio (ratio of total to molecular backscattering), cannot be used directly for determination of the particle properties. Preferably, lidar optical properties should be used, if available, which do not depend on the particle number density, and which therefore contain information about the scattering properties of the cloud particles alone. Specifically, these are lidar ratio

equation image

color ratio

equation image

and depolarization ratio

equation image

where βpar and βpar are the backscatter coefficients in the polarization planes orthogonal and parallel, respectively, to the linearly polarized lidar radiation source:

equation image
equation image

Here Δ denotes the depolarization ratio of particles with maximum dimension d. For measurements of color ratios multiwavelength lidars with at least two transmitted wavelengths (kmax = 2) are required, Δpar can be measured with polarization lidars.

3. Retrieval Algorithm

[13] Computations of scattering phase matrices can be carried out for a limited number of particle sizes and shapes only. Consequently, for calculation of PSC optical properties the defining equations for the extinction and backscatter coefficients [equations (2), (3), (8), and (9)] have to be partially discretized. Formally, the approach used here can be described for all coefficients by

equation image

if the functions given in Table 1 are substituted for X and Y. For this study, scattering phase matrices have been calculated for 17 particles (i = 17) with maximum dimensions di between 0.2 and 2.0 μm. Table 2 summarizes di and the lower and upper boundaries of the range of application of Y, dmin(i) and dmax(i). Qsca and G, which are also determined with the optical-property calculations, usually exhibit a smooth dependence on d. So instead of summing over the discrete values of Qsca and G we use interpolated Qsca data and polynomial fits to G for the computation of the integral in equation (10). In equation (10), dmin and dmax are the lower and upper boundaries of equation image.

Table 1. Functions X and Y
FunctionExtinction CoefficientBackscatter Coefficients
Y(λ, d)1pπequation imagepπequation imagepπ
Table 2. Maximum Dimensions di of Particles Used for the Calculations, and Application Ranges of Y(di)
idi, μmdmin(i), μmdmax(i), μm

[14] If different types of particles are assumed to coexist in the PSC, equation (10) has to be rewritten as

equation image


equation image

for all d.

[15] For our investigations we assume that five PSC particle properties, specifically Spar(355 nm), Cpar(355 nm, 532 nm), Cpar(532 nm, 1064 nm), Δpar(532 nm), and Δpar(1064 nm), are available. This data set was measured with, e.g., two of the lidars onboard the NASA-DC8 during the SOLVE mission in winter 1999/2000. These were the four-wavelength Airborne Raman Ozone and Temperature Lidar AROTEL [McGee et al., 2000] that was complemented by a polarization-sensitive aerosol detection receiver [Hostetler et al., 2000], and the DIAL system [Butler et al., 2001]. The five particle optical properties allow for optimization of an equal number of microphysical parameters with the retrieval algorithm.

[16] In this study, we use our model to assess the sensitivity of microphysical retrieval schemes to the assumptions made about particle shapes. In what follows, descriptions of the microphysical model assumptions and of the parameterization are given.

3.1. Model Assumptions

[17] Our model assumptions are based on the following scenario of formation of solid PSC particles. Wind flow over orographic features may, under certain conditions [Dörnbrack et al., 2001], result in wave activity downwind, and in localized temperature minima in the stratosphere. Stratospheric liquid aerosols carried through these cold regions with the wind are cooled rapidly. At temperatures T < ∼195 K the aerosols grow by uptake of H2O and HNO3 to form LTA [Carslaw et al., 1994]. If the air parcel cools to temperatures 3–4 K below the ice equilibrium temperature Tice, water ice embryos form homogeneously within the LTA droplets [Tabazadeh et al., 1997; Carslaw et al., 1998]. NAH particles also nucleate in leewave PSCs, probably heterogeneously on the ice particles (ice-mediated NAH formation) [Koop et al., 1997; Carslaw et al., 1999]. Depending on T, the solid particles grow or evaporate by mass transfer from or to the vapor phase. For T < Tice, ice, NAH and LTA coexist. In the Arctic, this will only be the case in the coldest regions of a leewave temperature perturbation. On temperature increases to T > Tice further downwind, the ice particles evaporate. However, under favorable synoptic conditions NAH particles survive and can be detected several hundreds of kilometers away from their origin [Carslaw et al., 1998; Tsias et al., 1999; Voigt et al., 2000].

3.1.1. Size of the PSC Particles

[18] In situ data suggest a power-law dependence of the number density on particle diameter for PSCs that contain solid particles [Dye et al., 1992]. For this reason we choose for the normalized size spectrum equation image:

equation image

Slope c and the upper boundary dmax of the spectrum are optimized during the retrieval process.

[19] The number density of the solid PSC particles is determined by the nucleation processes. The fraction of the LTA droplets that nucleate ice depends on the cooling rate in the leewave [Tabazadeh et al., 1997]. The fraction of ice particles that serve as NAH nucleation sites depends on the atmospheric conditions and may vary considerably [Carslaw et al., 1998; Wirth et al., 1999]. Calculations by Tabazadeh et al. [1997] and Carslaw et al. [1998] show that high cooling rates will result in large number densities of small solid particles <∼2 μm, and in situ observations support the theoretical values [Voigt et al., 2000]. In our retrieval algorithm we vary dmax accordingly to values up to 2 μm. The lower boundary of the assumed particle distribution is fixed at dmin = 0.175 μm. This approach is not problematic, because scattering of very small particles does not contribute significantly to the lidar signals.

3.1.2. Shape of the PSC Particles

[20] We assume three principal particle shapes: spheres, asymmetric polyhedra, and hexagonal crystals. Pueschel et al. [1992] show photos of irregular submicron solid particles formed in evaporating LTA droplets that impacted the Formvar coating of glass strips. These photos illustrate nicely how PSC particles might look like that nucleated homogeneously in stratospheric LTA. In situ measurements in contrails also demonstrate that the smallest particles are irregular ice crystals [Goodman et al. 1998]. Contrarily, larger ice crystals exhibit well-defined symmetries. 5–50 μm ice crystals sampled in the Antarctic stratosphere [Goodman et al., 1989, 1997] and in a cirrus cloud that formed under conditions similar to those of PSC formation [Heymsfield, 1986] were predominantly hexagonal columns and trigonal plates. In the case of contrails, hexagonal plates have been found to be the prevalent shape [Goodman et al. 1998].

[21] For this reason hexagonal crystals in our model represent PSC particles that acquired crystal symmetry by growing to a certain size. Asymmetric polyhedra are thought to be representative of newly formed irregular ice embryos, which are generally smaller than the hexagonal crystals. We assume the same particle shapes for water ice and NAH, LTA droplets are accounted for by spheres. Lacking detailed information about the partitioning between the various particle types in mixed-phase PSCs, we arbitrarily parameterize the size-dependent fractions fj of the three particle types in our model as

equation image
equation image
equation image

fj are determined by the two parameters d0.5 (maximum dimension at which 50% of the particles are solid hexagonal crystals) and σ (mode widths of the LTA fraction) which are variables of the optimization algorithm. This parameterization allows for PSCs that are formed of any of the three particle types alone, and for mixtures of two or three particle shapes.

[22] Aspect ratio a (ratio of the length of the particle along the symmetry axis to particle widths; columns and prolate spheroids have a > 1, plates and oblate particles have a < 1) of the crystalline PSC particles is assumed to be size dependent as well. Measurements in stratospheric and tropospheric ice clouds confirm that a increases with particle size. Very small ice particles have aspect ratios close to one. We apply these results to the PSC particles by assuming

equation image

with a(d′) = 1.5 (prolate particles) or a(d′) = 0.5 (oblate particles). For d > d′ we set a(d) = a(d′) (note that d′ is a function of m). Slope m is optimized in the retrieval. For the model comparison presented here optical data for aspect ratios of 0.5, 0.75, 1.0, 1.25, and 1.5 have been available. Figure 1 illustrates the size dependence of equation image, f1f3, and a. For this particular example dmax, c, d0.5, σ, and m are 1.8 μm, −4, 1.0 μm, 0.25 μm, and 0.4 μm−1, respectively.

Figure 1.

Parameter set of the retrieval algorithm. The parameter values used for illustration are given in the plots.

3.1.3. Refractive Index of the PSC Particles

[23] Light absorption by PSC particles is not considered. The refractive indices of LTA at the three laser wavelengths are taken from Luo et al. [1996] (T = 191 K). Refractive indices of the asymmetric polyhedral particles and the hexagonal crystals can be chosen independently. For water ice, refractive indices of Warren [1984] are used which have been confirmed in ice PSCs [Adriani et al., 1995]. The refractive index of NAH is a function of the H2O:HNO3 molar ratio [Middlebrook et al., 1994]. If initially water-rich NAH particles rather than NAT nucleated on the ice embryos in the LTA droplets, these particles would have refractive indices smaller than NAT. Over time NAH composition and refractive index would then slowly change to the values of the thermodynamically favored NAT [Marti and Mauersberger, 1993]. In situ data generally confirm the existence of NAT particles in the stratosphere [Fahey et al., 1989; Kawa et al., 1992; Voigt et al., 2000], although exceptions were found [Kawa et al., 1992]. Here we assume that the nitric acid particles are NAT. As refractive indices we take the values measured by Deshler et al. [2000] in a depolarizing Arctic PSC of type I (presumably NAT), although these values are higher than those found in Antarctic PSCs by the same method [Adriani et al., 1995] and seem to be more in agreement with NAH of smaller water content than NAT [Middlebrook et al., 1994]. The data of Deshler et al. [2000] might therefore be regarded as an upper limit to the refractive indices of NAH PSCs.

3.2. Computation of Single-Crystal Phase Matrices

[24] Phase matrices of hexagons and asymmetric polyhedra were calculated with the FDTD code described by Yang and Liou [1996, 1998] and Yang et al. [2000]. Asymmetric polyhedral particles are crystals with seven corners and ten flat crystal faces (ten-faced polyhedra). Details about the realization of these particles are given by Yang et al. [2000], accuracy and errors associated with the FDTD method have been fully discussed by Yang et al. [2000] and by Baran et al. [2001]. Phase functions of spheres are computed with a computer code which is based on the derivation of the Mie theory by van de Hulst [1981]. We smoothed out the Mie resonances of p(di) by using a projection area weighted normal size distribution with center di and an effective variance [Hansen and Travis, 1974] of 0.0067 for the calculations. The integration interval then roughly corresponds with the application ranges of di for all discrete particle sizes. Scattering properties of spheroids (a ≠ 1) used in this sensitivity study were calculated with a T-matrix code which is available at [Mishchenko, 1991]. For the sake of conciseness, optical data of spheroids and spheres were smoothed exactly the same way.

4. FDTD Calculations of Single Crystals

[25] In Figure 2 phase functions of particles with different shapes are compared. Although the general shape of the phase functions is similar at any given d and a, differences between them exist at angles approaching 180° (i.e., the backscattering direction) that are important for lidar observations. At 180° scattering angle hexagonal plates scatter more and hexagonal columns less than the corresponding spheroids. Because the projection areas G of both plates and columns are larger than those of the spheroids, this behavior cannot be explained by differences in G, but must be an effect of the particle shape. Similarly, ten-faced polyhedral crystals with d = 1.6 μm backscatter less than spheroids with d = 1.0 μm, although G values are comparable. From this we conclude that substituting spheroids with equal G for crystals does not reproduce the optical properties on the scale of individual crystals. This has to be kept in mind when retrieval of microphysical properties from lidar data is attempted with a spheroid-based inversion scheme and a parameterization of PSC particle size distributions in terms of surface equivalent radius.

Figure 2.

Scattering phase functions of NAT particles with same maximum dimension d and aspect ratio a, but different shapes. The wavelength is 532 nm.

[26] The size dependence of the lidar-relevant optical properties of single PSC particles is illustrated in Figure 3. Qsca exhibits oscillatory patterns which depend on the volume and the refractive index of the PSC particles. For particle sizes with Qsca values on the rising slope of the first Qsca maximum, lidar ratios are large and highly variable. In contrast, if Qsca is close to the geometrical-optics limiting value of two such as for the NAT hexagons with aspect ratios of 0.5 and 1.5 and with maximum dimensions >∼1.4 μm, lidar ratios are relatively independent of d and show values that are comparable to those found in cirrus clouds where scattering by particles ≫10 μm is dominant (typically, lidar ratios between ∼10 and 30 sr and depolarization ratios between ∼20 and 55% are found in cirrus clouds) [see, e.g., Reichardt et al., 2002]. This finding suggests that the geometrical-optics approximation might be applicable to calculation of the optical properties of NAT hexagons with aspect ratios near one and dimensions >2 μm. Since FDTD computations for this type of particles are especially time consuming, this would be important for future extension of our PSC particle database to larger sizes. Because of the problems associated with the use of the geometrical-optics technique for small particles and in backscattering direction, however, this option has to be studied carefully. Depolarization ratios shown in Figure 3 generally increase with particle maximum dimension (this behavior is characteristic for spheroidal particles as well [Mishchenko and Hovenier, 1995]). Columnar NAT crystals tend to have higher Δ values than planar NAT hexagons, especially for d > 0.7 μm. For most d in the particle size range shown here, Δ of hexagonal columns formed from water ice is significantly smaller than depolarization ratios of NAT crystals.

Figure 3.

Scattering efficiency Qsca, lidar ratio S, and depolarization ratio Δ of single hexagonal crystals as functions of particle maximum dimension for different aspect ratios. The wavelength is 532 nm. S curves of crystals with a = 2.5 have been multiplied by 1/3 to fit into the display range. The geometrical-optics limit of Qsca is shown for comparison (thin dash-dotted line).

[27] In Figure 4 the dependence of the PSC particle optical properties on aspect ratio are presented. For all particle sizes, Qsca, S, and Δ vary smoothly with a. This behavior has been exploited for our retrieval algorithm where optical properties of particles with aspect ratios other than those listed in section 3.1.2 are derived by interpolation. For d = 0.4 μm depolarization ratio increases monotonically with a. In the case of the larger particles, Δ values are minimum for thick to equidimensional plates. Generally, light scattering by columns leads to higher Δ than scattering by plates.

Figure 4.

Scattering efficiency Qsca, lidar ratio S, and depolarization ratio Δ of single hexagonal NAT crystals as functions of aspect ratio for different maximum dimensions. The wavelength is 532 nm.

[28] The wavelength dependence of Qsca, S, and Δ is highlighted in Figure 5. Computational results obtained for a hexagonal NAT plate (a = 0.5) are shown. For all wavelengths low S values correspond with high Δ values, and vice versa. At 1064 nm, plates <1.2 μm do not depolarize, whereas at shorter wavelengths high Δ values are found. This high sensitivity of Δ to particle size and wavelength suggests including lidar measurements of depolarization ratio at 1064 nm, and at 532 or 355 nm in the microphysical retrieval (Mishchenko and Sassen [1998] have come to a similar conclusion for contrail studies). It is interesting to note that the three curves for each of the quantities in Figure 5 are very similar if plotted against the size parameter πd/λ. This behavior is observed for any given aspect ratio. Yet, because of the wavelength dependence of the refractive index, minor differences occur.

Figure 5.

Scattering efficiency Qsca, lidar ratio S, and depolarization ratio Δ of single hexagonal NAT crystals as functions of particle maximum dimension for different wavelengths λ. The aspect ratio is 0.5.

5. Retrieval Intercomparison

[29] In Table 3 the microphysical parameter sets are summarized that are used to investigate the effect of particle shape assumptions on the retrieval of PSC microphysical properties from lidar measurements. The sets have been chosen to specifically address the question how this effect depends on the PSC size distribution, and on the fractional contributions and asphericities of the solid PSC particles. For each parameter set the initial optical properties Spar(355 nm), Cpar(355 nm, 532 nm), Cpar(532 nm, 1064 nm), Δpar(532 nm), and Δpar(1064 nm) are calculated under the assumption that the PSC consists of droplets, and asymmetric ten-faced and hexagonal crystals. The synthetic optical data is then input to the retrieval program and analyzed assuming spheroidal particles. For the retrieval, the microphysical parameters are chosen within reasonable limits and resolution (20–40 values per parameter), and a χ2 search of this parameter space is performed, based on the calculated and input optical data. The statistical weights of the optical properties in the χ2 sum are selected to reflect likely measurement uncertainties: statistical errors of lidar ratio, UV-visible color ratio, visible-infrared color ratio, and both depolarization ratios are 10 sr, 0.2, 0.4, and 0.1, respectively. Because of the generally complex topology of the χ2 manifold, we have to expect not a single minimum, but multiple minima of comparable depths. Therefore, not only the optimum fit but the 20 best solutions are registered. Since we assume both asymmetric and hexagonal particles to be composed of NAT, parameter d0.5 is only relevant for the initial computations with a ternary mixture of particle shapes. It is therefore held constant at the initial value during optimization.

Table 3. Initial Parameter Sets Used for the Retrieval Intercomparison
Setdmax, μmcd0.5, μmσ, μmm, μm−1

[30] In Figure 6 initial microphysical parameters and best fits are summarized for model runs with different size distributions, but same σ, m, and d0.5. Generally, the retrieved values of the asphericities of the spheroids and dmax of the size distribution are larger than the initial values. This is probably an effect of the high statistical weight of the depolarization data: In contrast to equidimensional crystals, spheroids with a = 1 do not depolarize light at 180° scattering angle. In order to match the depolarization ratios of equidimensional to elongated polyhedral crystals (m = 0.4 μm−1), the asphericities of the spheroids have to be maximized, hence the large deviations between initial and retrieved m. Aspherical particles, however, have smaller volumes than compact particles of the same maximum dimension, and this is compensated for by preferably increasing dmax and slightly adjusting slope c of the size distribution. This reasoning may also explain why the width of the LTA mode (σ) is retrieved slightly too small in most cases.

Figure 6.

Initial microphysical parameters (solid symbols) and parameters retrieved with the spheroid-based algorithm (open symbols) for parameter sets 1–3 (top), and 4–6 (bottom). The 20 best fit results are shown for each parameter set, multiples can reduce the number of visible data points to less than 20. For initial parameter d0.5 see Table 3.

[31] The effects of solid-particle asphericity are studied with model runs 7–9 (Figure 7). In all three cases, the size distribution (dmax and c) is well recovered although the spread is quite large if plate-like crystals are assumed (parameter set 7). However, highest fit accuracy is found for prolate spheroids that have large aspect ratios even for small d (m > 1.0 μm−1; e.g., for m = 1.25 μm−1, particles >0.575 μm have aspect ratios of 1.5). In addition in model run 9, the agreement between retrieved and initial σ is poor. In this case, optical properties of PSCs with few small droplets (σ = 0.25 μm) and equidimensional crystals are fitted with particle ensembles that predominantly consist of droplets below and highly aspherical spheroids above d = 1.0 μm. Although initial parameter sets 5 and 9 only differ in m, the retrieved values for σ and m show large differences. This is because lidar data are ambiguous: For a given set of PSC optical properties no unique solution to the inversion problem exists, and consequently any generic inversion algorithm is unstable. One way to address this problem could be introducing additional retrieval constraints based on microphysical considerations (e.g., the retrieved PSC volume density has to lie within reasonable limits). Note, that in runs 7 and 9 the inversion process yields also solutions for spheroids that are highly oblate. This reveals the existence of a second minimum of the χ2 manifold which is comparable in depth to the one found for the solutions with prolate spheroids. The results for model runs 10–12 are illustrated in Figure 7 as well. Parameter sets 10 and 11 represent PSCs with LTA droplets being the dominant particles <1 μm. Parameter set 12 describes the case of a PSC with only two principal particle shapes, it consists of few small droplets and of solid particles which are all equidimensional asymmetric polyhedra (d0.5 = 3 μm). As in all the cases discussed previously, retrieved m does not match the initial values. In all three runs, equation image is retrieved well.

Figure 7.

Same as Figure 6, but for parameter sets 7–9 (top), and 10–12 (bottom).

[32] So far we have studied how the individual parameters of our model parameterization depend on the assumed principal particle shapes. Now we examine the effect the shape assumptions have on PSC bulk properties that are important for stratospheric chemical and microphysical modeling, specifically PSC particle surface area density A and volume density V. In Figure 8 relative differences δA between retrieved and initial surface area density [δA = (ArAi)/Ai] are plotted against corresponding relative differences in volume density. In all cases retrieved A and V are smaller than the initial values (δA and δV < 0). The only exception is model run 12, where some solutions to the inversion problem with positive δV can be found. Generally, absolute values of δV are smaller than those of δA. The important conclusion is that microphysical retrieval algorithms based on spheroidal particles tend to systematically underestimate PSC surface and volume density, if compared to a model where scattering by solid PSC particles is represented by phase matrices of asymmetric polyhedra and hexagonal crystals. In our model simulations of leewave PSCs formed from LTA and NAT particles, the retrieved values for A and V are found to be, respectively, ∼10–30% and ∼5–25% smaller than the initial values, in some cases deviations are even larger. These shape-assumption related uncertainties have to be accounted for in PSC modeling efforts that are based on lidar measurements.

Figure 8.

Relative difference δA between retrieved and initial surface area density versus relative difference δV between retrieved and initial volume density for parameter sets 1–12. The 20 best fit results are shown for each parameter set, multiples can reduce the number of visible data points to less than 20.

6. Summary

[33] An algorithm has been developed for retrieval of microphysical properties from PSC lidar data. The parameterization scheme allows for coexistence of up to three different particle types with size-dependent shapes. The optical properties of solid particles <2 μm have been determined with the FDTD technique, the particles have asymmetric polyhedral or hexagonal shape. In this work, the retrieval algorithm has been applied to investigate the effect of particle shape assumptions on the inversion of lidar data measured in leewave PSCs. The simulations show that the assumption of spheroidal PSC particles results in the retrieval of higher asphericities and, in most cases, of larger maximum particle sizes when compared to model runs with polyhedral particles. Furthermore, leewave PSC surface area density and volume density are systematically smaller by, respectively, ∼10–30% and ∼5–25%.

[34] We plan to extend our FDTD database to larger particles, and to include particles with other geometries and refractive indices. Furthermore we will incorporate geometrical-optics computations of optical properties of particles to which this approximation applies in our microphysical analysis of PSCs. These will likely be large water ice crystals which are found in PSCs of type II.


[35] We would like to thank P. A. Newman and J. F. Gleason for computational support. This work was funded in part by a grant from NASA's Atmospheric Chemistry Modeling and Analysis Program managed by P. L. DeCola. P. Yang's research is supported by the NASA Radiation Sciences Program managed by D. E. Anderson.