The aureoles around stars caused by thin cirrus limit nighttime measurement opportunities for ground-based astronomy, but can provide information on high-altitude ice crystals for climate research. In this paper we attempt to demonstrate quantitatively how this works. Aureole profiles can be followed out to ~0.2° from stars and ~0.5° from Jupiter. Interpretation of diffracted starlight is similar to that for sunlight, but emphasizes larger particles. Stellar diffraction profiles are very distinctive, typically being approximately flat out to a critical angle followed by gradually steepening power-law falloff with slope less steep than −3. Using the relationship between the phase function for diffraction and the average Fourier transform of the projected area of complex ice crystals, we show that defining particle size in terms of average projected area normal to the propagation direction of the starlight leads to a simple, analytic approximation representing large-particle diffraction that is nearly independent of crystal habit. A similar analytic approximation for the diffraction aureole allows it to be separated from the point spread function and the sky background. Multiple scattering is deconvolved using the Hankel transform leading to the diffraction phase function. Application of constrained numerical inversion to the phase function then yields a solution for the particle size distribution in the range between ~50 μm and ~400 μm. Stellar aureole measurements can provide one of the very few, as well as least expensive, methods for retrieving cirrus microphysical properties from ground-based observations.
 Accurate retrieval of the properties of particles in the atmosphere, e.g., ice clouds, is of prime importance to understanding their role in climate change and modeling their effects in global simulations [Comstock et al., 2007]. Knowledge of the impact of aerosols on climate change has improved so much that in their contribution to the “Fourth Assessment Report of the Intergovernmental Panel on Climate Change,” Forster et al.  called AERONET [Holben et al., 1998] a “significant advancement”. However, the impact of cirrus cloud particles is much less certain, because they occur high in the atmosphere and are more difficult to monitor.
DeVore et al.  described a new Sun and Aureole Measurement (SAM) instrument that measures the radiance profile of the solar disk and surrounding aureole (from ~0.6° to 8°) in a narrow spectral band centered on a wavelength of 0.67 μm. Applying the “diffraction approximation” (presented in their paper) to SAM measurements, they were able to retrieve size distributions of cloud particles in the range from 5 to 50 μm. In practice, SAM has been most useful for collecting information on cirrus clouds because of their thinness (i.e., optical depth τ≃0.1 to 3), which is required for the appearance of aureoles.
 In this paper we report what we believe to be a new method for making and interpreting aureole measurements at night. Although the Moon provides an alternative to the Sun, we think that stars provide the best sources, and that planets may have some utility as well. The angular size of the Moon is basically the same as that of the Sun and therefore does not allow for retrieving information on large particles that comes from measurements at small angles. Moreover, both the Moon and planets limit where in the sky measurements can be made, whereas suitable stars are distributed across the entire sky. We have used inexpensive technology to measure starlight diffraction in the angular range from ≲0.03° to ~0.2°, which corresponds to the size range from ~50 μm to 400 μm. Since the climate impact of cirrus is sensitive to its microphysical properties through both particle scattering asymmetry and emissivity/absorptivity [Stephens et al., 1990], it is important to be able to measure the size distributions of large ice crystals such as those we discuss in this work.
 Our approach is based on aureole measurements with a lens and a medium-quality astronomical CCD camera. This remote sensing method has advantages over in situ measurements in that (i) it can readily be carried out on virtually any night when thin cirrus clouds are visible, (ii) it is relatively inexpensive to implement, and (iii) the measurements do not disturb the cloud environment. The aureole approach to monitoring cirrus provides one of the few methods available to retrieve cirrus particle microphysical properties from ground-based observations.
Section 2 describes the various measurements involved in interpreting stellar aureoles, starting with the determination of two important parameters used in aureole profile interpretation, optical depth (section 2.1) and stellar exo-atmospheric irradiance (section 2.2), followed by a discussion of aureole imagery (section 2.3). Examples of star images illustrate the difference between the weak aureoles associated with small particles, e.g., thin water clouds, and the strong ones attributable to the large ice crystals in cirrus.
Section 3 presents various aspects of the theory and modeling of diffraction by ice crystals used to interpret the measurements. The reader is reminded that despite the wide range of crystal habits that need to be considered, the phase function representing diffraction can be calculated simply using the Fourier transform. Moreover, this solution illuminates a simple wavelength scaling relationship for aureole profiles. Examination of phase functions derived for different habits with the same size measure leads to the selection of a size metric based on average projected area as being particularly useful. Phase functions representing diffraction from distributions of particle sizes are discussed next, followed by discussion of the potentially important role of multiple scattering and an analytic approximation to represent aureole profiles involving multiple scattering. Deconvolution of multiple scattering is presented in Appendix A. Next the steps involved in retrieving the particle size distribution from the phase function are presented (section 3.4), either by fitting the data with an analytic form or using constrained numerical inversion. Finally, the effects of the point spread function are shown to be taken into account using an appropriate analytic approximation (section 3.5).
Section 4 illustrates the application of theory and modeling to interpret aureole profiles using an example. The measured aureole profile is separated into three components so that the diffraction profile can be distinguished from the point spread function and the sky background (section 4.1). Next the phase function is deconvolved from the diffraction profile (section 4.2) and then the particle size distribution (PSD) is determined (section 4.4) from it. The section concludes with some more examples of stellar aureoles covering a range of optical depths (section 4.5) and a brief look at an example of an aureole around the planet Jupiter.
Section 5 provides a short summary of our work. This is followed by Appendix A discussing the deconvolution of multiple scattering, where it is approximated as a Poisson process (section A1). The resulting series is solved for single scattering, double scattering, and then for all orders (section A4).
2 Stellar Aureole Measurement
 From our Visidyne measurement site in West Acton, MA (Figure 1), we collected an extensive stellar aureole data set comprising approximately 42 h of imagery from 17 nights between 19 Jun 2011 and 26 Feb 2012 in a narrow 5 nm band around 672 nm. We used multiple exposures to extend the dynamic range. Targets ranged from bright planets such as Jupiter down to magnitude 3 stars. Sky conditions varied from clear to sufficiently cloudy that stars were practically indistinguishable from their aureoles, analogous to the situation with a “fuzzy Sun” [Linskens and Bohren, 1994]. The clouds consisted primarily of cirrus, although water clouds were present at times. The two types of clouds are readily distinguishable from their visual appearance on all-sky imagery as well as from the strength of their aureoles. As with the SAM instrument for solar aureoles, two cameras were used, one to measure particulate optical depth while the second measures the aureole radiance profile at the same time. Supporting data include all-sky imagery, laser cloud altimetry, and both local and synoptic weather data. Figure 2 shows a typical image from the all-sky camera with targets identified through the cirrus cloud.
2.1 Optical Depth
 The central star pixels in the stellar aureole images (discussed in section 2.3) were saturated in order to set the aureole camera exposure times to yield the best signal for the relatively faint aureole. To measure optical depth at the aureole star, a second, coaligned camera collected short-exposure, unsaturated images of the aureole star. A 620−680 nm filter was used to allow short exposures (0.05 to 1 s), even during high optical depth conditions. The line-of-sight transmittance was measured by referencing the measured flux to a clear-sky measurement of the same star. The data were scaled by the frame exposure time, and then the optical depth along the line of sight, τlos, was calculated directly as follows:
where Smea is the measured irradiance and S0 (section 2.2) is a reference measurement on a clear night. Optical depth was measured once per second, and up to five depth values were averaged to reduce scatter due to atmospheric scintillation. Determination of optical depth using Sun or star photometry needs to take into account forward scattering. Since the correction is a strong function of the angular width of the source irradiance measurement [e.g., Shiobara and Asano, 1994; DeVore et al., 2009], the smaller angular width of stars as compared with the Sun significantly reduces the size of the corrections required. Also, to the extent that the data analysis can distinguish between direct and scattered source radiance, the error attributable to forward scattering can be reduced further.
 Some typical optical depth data are shown in Figure 3. The line-of-sight optical depth is plotted in blue, while the start of each 30 s aureole measurement is indicated with a red diamond. In general, the cirrus clouds are not spatially uniform so that rapid changes in τlos occur as the clouds drift through the line-of-sight. The high sample rate is used to identify and then discard aureole measurements that contain significant variation (as indicated by the fluctuations in the blue curve) during the longer aureole integration times (as indicated by the separations between the red diamonds). The primary source of error is atmospheric scintillation, which is seen in this type of optical depth measurement even on clear nights. Analysis of clear night data over 30 s time intervals (the duration of aureole image exposures) returns RMS errors of 0.1 to 0.25 in line-of-sight optical depth. We recommend the use of the conservative error value ±0.125 for data presented here. Temporal variations caused by cloud motion also limit the total exposure time practical for any single star aureole measurement.
2.2 Stellar Exo-Atmospheric Irradiance
 As with the determination of τlos, absolute irradiance values are not necessary for specifying S0. It suffices to define irradiance for a specific camera in terms of the number of photons counted divided by the product of the aperture area of the lens Alens (m2) and the exposure time texp (s), giving units of counts/m2/s. For radiance, one divides the irradiance by the pixel field of view Ωfov (sr), giving units of counts/m2/s/sr.
 We examined 16 unsaturated images containing Capella from 26 Nov 2011, when the sky was very clear. For each image we determined the maximum, Cmax, and average, Cave, number of counts and the corresponding standard deviation, Cstd, in the 21×21 pixel square surrounding Capella. This size square was large enough so that the minimum number of counts at the edges was down by 2 orders of magnitude from the peak. In order to identify those counts associated with the directly transmitted stellar photons, we separated the individual counts into “peak” and “background” bins using the following process. First, we selected a somewhat arbitrary threshold value, Cthr=2Cave, to perform an initial separation. Next we calculated the average, Bave, and standard deviation, Bstd, of the counts in the background bin. Guided by the three-sigma rule [e.g., Kreyszig, 1979], we reset Cthr= min(Bave+3Bstd,Cmax−3Bstd) and redivided the counts into the two bins. Then we calculated the sum of the counts, Cpeak, and the number of pixels, Npeak, in the peak bin and recalculated the average number of counts, Bavg, in the background bin. Then we calculated S0 counts/m2/s/sr as follows:
 For the 16 images analyzed, the average S0=2.2×108counts m−2s−1sr−1 with a standard deviation of 0.1×108 counts m−2s−1sr−1. We selected the maximum of the set, S0=2.5×108counts m−2s−1sr−1, to use under the assumption that it represents the clearest sky conditions.
2.3 Aureole Profiles
 The stellar aureole profiles were collected with a QSI-583wg astronomical camera, interfaced to a Canon 70-200 mm f/# 2.8 lens set to 200 mm focal length. The camera and lens were mounted to a German equatorial mount that located stars and corrected for sidereal motion. The filter was an Astrodon 672 nm filter with 5 nm bandwidth, chosen to minimize light pollution and provide a narrow band for model calculations. The camera was operated for most collects using 4×4 pixel binning to enhance the relatively faint aureoles, resulting in single pixel angular sizes of approximately 22′′. In later collects the binning was reset to 1×1, providing higher angular resolution (5.6′′). Each aureole image was a 30 s camera exposure. After the collection, frames were grouped by optical depth (as measured by the optical depth camera) and stacked to improve signal-to-noise. Images that contained significant cloud spatial gradients were discarded prior to stacking.
 Figure 4 shows a montage of images of aureoles around the star Capella taken through a thin cirrus layer that varied over the course of the measurements. The images are annotated with τlos measured by a second camera. Note the expected correlation between the apparent size and intensity of the aureole and τlos. The dashed yellow circle shows the size of the Moon for reference. The overall brightening of the background represents a combination of the “wings” of the aureoles, scattering of cityshine from the greater Boston area, and detector noise. Figure 5 compares the images of the star Capella as seen through cirrus, clear sky, and a water cloud on a different night. Note that despite the fact that τlos in this case is close to the value that should maximize aureole radiance (see below), the water cloud does not produce a discernable aureole. This is due to the fact that small droplets of water diffract light out to angles larger than our field of regard. (Since the color scalings applied in the two sets of figures differ slightly, one should not place any particular interpretation on the light blue versus black background shading.)
 Figure 6 shows more quantitatively some illustrative aureole profiles about the star Capella as a function of angle. The aureoles span a range of τlos from 0.1 to 2.3. The colored symbols indicate measured radiances, and the solid curves show fits to analytic functions representing the three physical phenomena, the point spread function, stellar diffraction, and sky background, responsible for the shapes of aureole profiles (see section 4 for details). The point spread function results from instrumental scattering and atmospheric turbulence and dominates the aureole profiles at small angles; the resulting radiance is anti-correlated with τlos. The background primarily represents diffusely scattered cityshine and dominates the aureole profiles at large angles; the resulting radiance is positively correlated with τlos. Stellar diffraction gives the aureole its shape at intermediate angles and is the main subject of this paper; the radiance from diffraction peaks for τlos between about 1 and 2.
3 Aureole Theory and Modeling
 Various aspects of the theory and modeling of aureoles are covered in this section to provide a basis for their application in the next section. Despite the wide range of crystal habits that need to be considered, the phase function describing diffraction from individual ice crystals can be calculated simply using the Fourier transform (section 3.1). The solution exhibits a potentially useful wavelength scaling relationship. Airy's analytic solution for a sphere is presented both to provide validation for the implementation of the numerical solution and as a guide for a simpler analytic approximation. Examination of phase functions derived for different habits with the same size measure leads to the selection of a size metric based on the average projected area as being particularly useful. Distributions of particle sizes are discussed next (section 3.2) considering both power-law and exponential analytic forms, which leads to a useful analytic approximation for the phase function for a distribution of particles. Next the diffraction radiance profile is considered (section 3.3) as well as the potentially important role of multiple scattering. Example calculations suggest a useful analytic approximation to represent multiply scattered diffraction profiles, which can be deconvolved using the equations presented in the appendix. Next the steps involved in retrieving the particle size distribution from the phase function are presented (section 3.4), either by fitting the data with an analytic form or using constrained numerical inversion. Finally, the effects of the point spread function are taken into account using an appropriate analytic function (section 3.5).
3.1 Diffraction From Individual Ice Crystals
 Diffraction is the dominant mechanism involved in forming aureoles. The angles involved in stellar aureole measurements tend to be smaller than the size of the Sun and hence smaller than the angles measured in solar aureoles. This difference in angular measurement range has important implications for modeling the particles most affecting the aureole shape. Whereas solar aureole measurements range between ~0.6° and ~8°, and correspond to particle dimensions between ~60 μm and ~5 μm according to the diffraction approximation [DeVore et al., 2009], stellar aureole measurements range between ~0.03° and ~0.2° and correspond to particle dimensions an order of magnitude larger. Ice crystals in the sensitive size range for solar aureoles tend to be compact, e.g., droxtals, and hence are well approximated as spheres. However, the ice crystals relevant to stellar aureole modeling require consideration of more complex shapes [e.g., Baum et al., 2005].
3.1.1 Diffraction Calculation
 Consider diffraction from the eight crystal habits shown in Figure 7. Triangular facets are used to describe the outer surfaces of each crystal. Although not representative of a naturally occurring ice crystal, we include the spherical shape both as an approximation for small ice crystals and to check our numerical calculations of the diffraction patterns against the well-known solution of Airy  for a sphere. We approximate a sphere (Figure 7a) using a surface with 512 triangular facets, generated by starting with an octahedron and sequentially subdividing each triangle into three smaller triangles three times, each time moving the vertices to the surface of the circumscribing sphere. Figure 7b shows a droxtal with maximum volume relative to the circumscribing sphere used to represent small, compact ice crystals [Yang et al., 2003]. While columns, plates, and bullets are observed, aggregations of bullets are more commonly used to represent the larger ice crystals in cirrus. Figures 7c and 7d show a solid hexagonal column and plate with ratios of the width to length or height based on the scaling relations for crystal habits C1e and P1a provided by Heymsfield and Platt . The bullet and bullet rosettes in Figures 7e to 7h use the tip angle and width-to-length scaling relations given by Um and McFarquhar .
Bi et al.  discuss several ways of calculating the scattering of light by particles larger than the wavelength λof the incident radiation. Born and Wolf  relate Fraunhofer diffraction to the 2-dimensional Fourier transform of the projection of the particle on the plane normal to the incident direction of the radiation. Although the line integral formulation of Gordon  may be slightly faster, we have found numerical calculation using the fast Fourier transform [Press et al., 1992] adequate and we can calculate the projected area, σpa, which we have found useful, in the process. Using the small angle approximation [Lenoble, 1985], a phase function Pdiff(θ/λ,φ) representing the angular distribution of diffracted radiation can be written as follows:
where θ is the scattering angle, φ is an azimuthal angle, x and y are cartesian coordinates in the projection plane with their origin within the particle projection, is the 2-dimensional Fourier transform operator, and A(x,y) is the “aperture” function describing the projection of the particle:
We represent the aperture function A(x,y) by a two-dimensional, uniformly spaced grid of points. We set the grid dimensions to be 1024 × 1024 in the examples presented here unless specified otherwise. We set the physical size of the grid in each dimension to the product of a characteristic crystal dimension (e.g., the diameter of a hexagonal plate or the length of a solid column) and the square root of the number of grid points. We project the crystal facets onto a gridded plane oriented normal to the direction of the incident radiation. Each grid point is initialized to zero and set to one if any projected facet covers it.
 Unlike an ordinary phase function, Pdiff is defined only in the forward hemisphere and is meant to represent the diffraction component of the angular distribution of radiation in the near-forward direction. For cases of interest here, i.e., ice crystals large compared with λ and θ≪π/2, diffraction dominates the internal or body scattering component, which can be ignored. However, in order to relate Pdiff(θ,φ) correctly to extinction and optical depth τ, both the diffraction and internal components are included in the extinction cross section σext=2σpa (from the optical theorem, e.g., [Liou, 2002]).
3.1.2 Wavelength Scaling
 From equation ((3)), we see that the amplitude of diffraction phase functions scales inversely with λ2 and that the scattering angle θscales with λ. These observations allow for scaling diffraction phase functions in the near forward direction from one wavelength λ1 to a second wavelength λ2:
The interpretation of equation (5) is simple. For a given size crystal, as the wavelength increases (decreases), the projected area of the crystal becomes smaller (larger) relative to the wavelength and the scattered radiation spreads to larger (smaller) angles. However, since the total diffracted power remains the same, the radiance at any angle decreases (increases) by the square of the wavelength to conserve energy. Although we shall retain λ in a number of equations to emphasize its role, we shall take λ=0.67μm in the calculations and examples in this work.
3.1.3 Airy Solution for a Spherical Particle
 It is useful to consider the analytic solution for a sphere [Airy, 1835]. Expressed as a diffraction phase function, Airy's solution for a sphere, Psph (θ,Dsph), is given by
where χ=πDsph/λ is a non-dimensional scaling parameter, Dsph is the diameter of the sphere, J1 is the Bessel function of the first kind of order 1, and the small angle approximation (sinθ≈θ) has been applied to find the expression on the right.
 Consider the case of light of wavelength λ=0.67μm incident on a spherical particle (Figure 7a) with diameter Dsph=50μm. The dark blue line in Figure 8 shows the Airy solution (equation (6)) for the diffraction phase function. We approximated the sphere using a facetted surface as described in section 3.1.1. However, in this case we subdivided the initial octahedron surface four times so that the resulting spherical surface had 2048 facets. Each facet was projected onto a plane of 2048×2048 points spaced 1.105μm apart. The 1596 points determined to be inside of projected facets were used to generate an aperture function. They represent a projected area of 1948μm2, which is approximately 1% less than the projected cross section of a 50μm diameter true sphere. The red line in the figure shows the phase function generated from application of the Fast Fourier transform [Press et al., 1992] to the aperture function using equation (3). The two curves are remarkably close, but can be distinguished because the minima of the Airy solution are somewhat deeper than those of the numerical one. Although we expect that the limited domain of the numerical Fourier transform is responsible for these differences, they are small compared with the approximations introduced in the following sections.
3.1.4 Analytic Approximation
 It will prove useful to introduce a simple, analytic approximation, Papx(θ,Dsph), to the Airy solution for a sphere in the small angle limit by considering its asymptotic behavior. For small angles, Psph(θ,χ)→χ2/2, and for large angles, Psph(θ,χ)∝(χθ)−3. A simple analytic function Papx(θ,χ) with these asymptotic behaviors is
where parameter ξ is found from the normalization of Papx(θ,χ) using the small angle approximation and taking the upper limit of the integral to infinity:
The green line in Figure 8 shows this analytic approximation. By design, the approximations's plateau at small angles agrees with the Airy and Fourier transform numerical solutions, while at the larger angles, the approximation cuts through their oscillations near the peaks.
3.1.5 Area Diameter
 Figure 9a compares calculations of the diffraction phase functions for the eight crystal habits shown in Figure 7, all with the same maximum size, and averaged over particle orientation. The diffraction phase functions are similar, with plateaus at small scattering angles followed by power-law decreases with slopes of ~−3. The oscillations in the power-law regions tend to average out with distributions of particle sizes and/or shapes. However, the overall amplitudes for the same maximum size vary by a factor of 5 depending upon particle shape. We found that diffraction phase functions for particles with the same volumes exhibited a factor of 3 variability versus crystal habit. For particles with the same ratios of their volumes to their average projected areas, the variability is over an order of magnitude. In other words, none of these measures of particle size proves self-consistent or useful for aureole work. However, the diffraction patterns for particles with the same average projected areas are very similar as can be seen in Figure 9b, prompting us to define the “area diameter” as the diameter of the circle with the same area as that of the average over orientations of the particle projections. The area diameter Da, therefore, appears to be a useful measure of ice crystal size. This should not be surprising since diffraction by a particle is closely related to its projected area.
 Given that phase functions are similar for different crystal habits with the same Da, including spherical particles, and that the phase functions for the latter can be approximated with a simple analytic function, we are led to replace Dsph in the approximation with Da. In terms of the scattering angle θand diameter Da, the analytic approximation in equation (7) becomes
The black line in Figure 9b illustrates how well this approximation fits the diffraction calculations. Therefore, for the remainder of this work, we adopt the form of equation (10) to represent a “universal” phase function for any shape particle with area-diameter Da. The extinction cross section σext is simply twice the projected area:
3.2 Particle Size Distributions
 The simple, analytic approximation Papx(θ,Da) is useful for individual ice crystal habits assuming that they are part of a distribution so that the fine structures of the individual phase functions average out. Next we consider different size distributions of ice crystals and look for ones which are both realistic and have convenient parameterizations.
 Using Papx(θ,Da) the total diffraction phase function P(θ) representing the average over all sizes and habits is given by the following integral over the particle size distribution, PSD:
where the extinction (rather than the scattering) cross section has been used (since absorption is negligible at visible wavelengths), N(Da) is the differential number density of particles along the path through the atmosphere per unit area diameter, and τlos is related to N(Da) as follows:
3.2.1 Power-Law Distribution
 Consider a PSD with the power-law form [e.g., Heymsfield and Platt, 1984]:
where N0 is a normalization constant, −μ is the power-law slope, and the PSD is truncated below Da=Dmin and above Da=Dmax. Substitute equation (14) into equation (13) and use equation (11) to find:
when μ≠3. Figure 10a shows numerical integrations of equation (12) for 2≲μ≲5 with Da extending from Dmin=10μm to Dmax=1000μm using Papx(θ,Da). The asymptotic behaviors of P(θ), constant for small θand proportional to θ−3for large θ, are similar to those of the individual analytic phase function Papx(θ) except that for the steeper PSD slopes, there is a transition region with an intermediate power-law slope between 0 and −3.
 Although an analytic expression can be found for Papx(θ), it involves hypergeometric functions and is not very insightful. However, it is illuminating to apply a different approximation to the single particle phase function by replacing Papx(θ,Da) with what we call the broken-line power-law approximation Pbp(θ,Da):
where θbp is determined from the normalization requirement for Pbp(θ,Da) to be
Figure 8 compares this approximation (the purple curve) with the continuous version (the green curve) from equation (10).
 With the limits of the integral in equation (12) again set to Dmin and Dmax, the analytic result for the total phase function Pbp(θ) naturally divides into three regions depending upon θas follows:
where, importantly,. Substituting this expression into equation (21) gives an equation with the following form for the transition region:
where c1, c2, and c3 are constants that depend upon Dmin, Dmax, and μ. Equations (19) to (22) show a total phase function which is constant for small θ and proportional to θ−3for large θ. If the intermediate transition region of the phase function is modeled with a power-law form, i.e., P(θ)∝θ−ν, then ν≈5−μ. This result is consistent with the relationship between the phase function (aureole) and PSD power-law slopes found using the diffraction approximation [DeVore et al., 2009].
3.2.2 Exponential Distribution
 For later use we note that an exponential form is also sometimes used to model ice crystal PSDs [e.g., Field and Heymsfield, 2003]:
where N0 is a normalization constant, Dchar is a characteristic size, and the PSD is assumed to be truncated below Da=Dmin and above Da=Dmax. Substituting equation (23) into equation (13) and using equation (11) gives
Analytic forms for Papx(θ) are even more complicated than those using the power-law form and are not investigated in this work. However, it is useful to note that since the exponential form tends to fall off very rapidly for Da≫Dchar, frequently little is lost by taking Dmax→∞. As a result, the number of free parameters tends to be one fewer than for the power-law PSD.
3.2.3 Total Phase Function Approximation
 The example phase functions calculated for power-law PSDs presented above (Figure 10a) and the analytic solution using Pbp(θ,Da) (equations (19) and (22)) suggest that an analytic function Papx(θ) representing P(θ) in the first two regions, i.e., for scattering angles before the θ−3 behavior sets in, might be approximated by replacing the θ3term with θνin the denominator of the approximate individual particle phase function Papx(θ,Da):
where P0, θ0, and 0≲ν≲3 are constants to be determined from fitting either calculations or measurement data. In fact, based on our results for the power-law PSD, we might expect ν≃5−μ. Figure 10b compares four of the calculations of P(θ) with fits, Papx(θ). The fits were carried out using the Levenberg-Marquardt method [Press et al., 1992] for values of θ below where the curves appear to start bending over to the θ−3region. The closeness of the fits confirms the potential utility of this functional form.
3.3 Radiance Profiles
 The aureole radiance Lss(θ) resulting from the single scattering of starlight by ice crystals uniformly distributed in a plane-parallel layer depends upon their total phase function P(θ) as follows [e.g., Liou, 2002, pg. 302]:
where the single scattering albedo (the ratio of the scattering to the extinction cross section) has been taken as 1, S0(λ) is the exo-atmospheric irradiance of the star, and the small-angle approximation has been applied. When equation (28) applies, it is a relatively simple matter to solve for P(θ) given measurements of the aureole radiance and τlos, and using a previously determined value of S0(λ). However, numerous authors [e.g., Dave, 1964; Hovenier, 1971; Korkin et al., 2012] have commented on the limited applicability of the single scattering approximation, e.g., to τlos≲0.05. These studies, however, typically considered scattering from small particles at all angles. The situation is distinctly different for scattering from large particles at small angles as discussed next.
3.3.1 Multiple Scattering
 To investigate the limits of the applicability of the single scattering approximation for use in modeling aureole radiance profiles, we used four phase functions representing the full range of scattering angles. Starting with the least forward scattering particles, we looked at Rayleigh scattering, which has a simple, analytic phase function. As an example of aerosols, we used the phase function retrieved by AERONET [Holben et al., 1998] at NASA Goddard Space Flight Center at 18.636 UT on 8 January 2010. We calculated a phase function for altostratus using the PSD shown by Liou  and the Mie code of Bohren and Huffman . For an example of cirrus, we used the phase function calculated by Baum et al.  for a distribution of ice crystal habits and sizes with effective size of 80μm. Figure 11 plots these phase functions on a log-log scale to emphasize the forward scattering direction. This clearly demonstrates the dominance of forward scattering for large particles.
 We digress briefly to describe the radiative transfer (RT) method we used to calculate the scattering of starlight through a uniform, plane-parallel particulate layer. The strongly forward peaked phase functions characteristic of ice crystals in cirrus (Figure 11) tend to cause problems for many RT algorithms. Methods relying on truncating the peak of the phase function at small angles [for example, Joseph et al., 1976; Wiscombe, 1977] work well for hemispheric flux calculations, but are not concerned with details of the angular distribution of radiation and therefore are not appropriate for calculating the profiles of stellar aureole radiance scattered at small angles. We used the successive orders of scattering method [Evans and Marshak, 2005], where the integrals were calculated using a Monte Carlo method [Sobol, 1994]. In order to make the calculations more efficient, we solved the adjoint problem, tracing photons from the sensor back to the source. Figure 12 shows an example of calculations for cirrus with τlos=2.0. The exo-atmospheric stellar irradiance was taken as that of the Sun (S0=0.1475Wcm−2sr−1μm−1) for these calculations. The colored lines labeled “Max Scatterings =” show calculations where the maximum number of scatterings was limited to 1, 2, 3, 4, and 30. (The black curve labeled “Approximation Fit” is discussed later.) The dark blue curve shows the single scattering solution. The fine structure for is numerical noise from the Monte Carlo calculations. This noise decreases somewhat as the maximum number of scatterings increases. The radiance in this region falls off as ≈θ−3. As expected, the aureole broadens somewhat as the maximum number of scatterings increases. For the range of angles shown, the solution appears to have converged using a maximum of three or four scatterings.
 We define a multiple scattering correction factor as the ratio of the aureole radiance calculated using a maximum of 30 scatterings to that calculated for single scattering. Figure 13 shows calculations of the correction factors using the phase functions shown in Figure 11 for τlos=0.1, 0.2, 0.5, 1.0, and 2.0. The Rayleigh scattering case (Figure 13a) shows a significant increase in radiance as the maximum number of scatterings increases and has not converged by four scatterings (not shown). The correction factor is nearly constant with θ, reflecting the shape of the phase function. The aerosol case (Figure 13b) is interesting in that although the phase function is much flatter than that of either altostratus or cirrus, the single scattering solution appears adequate. Unlike the other three cases, which represent conservative scattering, this aerosol case is highly absorptive. The single scattering albedo is only 0.12, which inhibits multiple scattering significantly. The correction factors for the altostatus (Figure 13c) and cirrus (Figure 13d) are close to 1 or nearly constant for θ less than ~1° and ~0.1°, respectively, meaning that the shapes of the aureoles differ only slightly from those of the underlying phase functions.
 Notwithstanding the fact that there is typically relatively little difference between single and multiple scattering in the stellar aureoles from cirrus clouds, it is worthwhile to explore correction algorithms, especially for the higher optical depth cases. And it appears appropriate to try to correct for multiple scattering in these cases, especially at the larger angles. Appendix A presents a technique for deconvolving the effects of multiple scattering.
3.3.2 Aureole Approximation
 To the extent that the shapes of aureoles differ only slightly from those of the underlying phase functions, it is reasonable to assume that an analytic approximation similar to the one found for phase functions could be useful in the near forward direction. Consider the aureole profile approximation Lapx(θ) based on equation (27):
where L0, θ0, and ν are constants to be determined from fitting either model calculations or measurement data.
 As an example, the black line in Figure 12 shows an illustrative fit to the red curve representing the multiple scattering aureole radiance profile for a cirrus cloud with τlos=2.0. The fit for θ≲0.5° is obviously very good.
3.3.3 Phase Function Retrieval
 The phase function P(θ) can be retrieved or deconvolved from a measured diffraction radiance profile Ldif(θ) numerically using the Hankel transform solution described in §A4, specifically equations (A28) and (A1):
where S0 is the exo-atmospheric stellar irradiance and is the Hankel transform (equation (A8)).
 Figure 14 compares the phase function (the blue curve) used to calculate the aureole through a cirrus cloud with τlos=2.0 (shown in Figure 12) with a retrieval (the red curve) using equation (30). The two curves overlap so well that we have divided the latter curve by a factor of 2 so that it can be distinguished in the plot. The deconvolution result exhibits numerical noise for , which some simple smoothing could mitigate. To the best of our knowledge, this deconvolution technique is new, and the effects of measurement errors and noise have yet to be worked out. The result shown in Figure 14 lacks such noise and serves simply to confirm that this technique can work under ideal conditions. The reasonableness of the results presented later in the paper for actual stellar aureole data serve to add confidence in the technique under more realistic conditions. Although RT calculations could be used to check whether a retrieved phase function reproduces the measured aureole profiles, we prefer to perform such checks starting with the PSDs derived from the phase functions and seeing how well the phase functions are reproduced.
3.4 PSD Retrieval
 Given the phase function P(θ) and the optical depth τlos, there are several ways of attempting to retrieve the PSD. We consider two complementary techniques.
3.4.1 Fitting an Analytic Form
 In sections 3.2.1 and 3.2.2, two parametric approximations were shown for representing PSDs. Given τlos, the power-law PSD has three free parameters (μ, Dmin, and Dmax) and the exponential PSD two (Dchar and Dmin, with Dmax set to 1000μm). A χ2 error metric can be defined as follows:
where p is a vector of the free parameters, Pdata(θi) are the values of the phase function derived (deconvolved) from the aureole measurements, wi are suitable weights, e.g., setting is equivalent to assuming 10%errors, and Pmodel(θi,p) are the values of the phase function calculated using the model PSD with parameters p and equations (10) to (13). The Levenberg-Marquardt method [e.g., Press et al., 1992] can be applied to minimize χ2(p) with the required derivatives calculated numerically.
 As an example fit, consider again a model power-law PSD extending from Dmin=10μm to Dmax=1000μm with μ=3.5. The plot in Figure 15a compares the input model PSD (the black curve) used to generate the synthetic aureole with retrievals using analytic models involving the power-law and exponential forms. (The constrained retrievals are discussed below.) Not surprisingly, the retrieval using the power-law form (the red curve) does quite well, while the retrieval using the exponential form (the orange curve) is only close over part of the size range. As with the multiple scattering deconvolution (section 3.3.1) these calculations lack noise and serve simply to confirm that the technique can work under ideal conditions and assuming that the correct underlying form of the PSD is used. Figure 15b compares the phase functions calculated numerically using the retrieved PSDs shown in Figure 15a. The fact that the phase function calculated from the retrieved PSD when an exponential form is used does not match the input PSD in this case reassures us that any arbitrary form cannot be assumed. In principle then, a variety of PSD models could be used and the one producing the best fit to the input phase function selected. However, the next section presents an alternative approach that has worked well for a variety of other physical retrieval problems.
3.4.2 Constrained Numerical Inversion
 To avoid assuming a specific functional form for the PSD as in the previous section, we consider the more direct numerical solution of equation (12). The numerical solution of this Fredholm integral equation for N(Da) as a function of P(θ) can be made easier [King et al., 1978] by replacing the independent variable N(Da) with the product of a slowly varying function, f(Da), and a more rapidly varying one, e.g., :
where β=4 was suggested by the large-particle power-law slopes found by Heymsfield and Platt  and seems to work reasonably well for the current problem. Substituting equation (32) into equation (12) gives
 Following the usual inversion approach [for example, Twomey, 1977], we discretize this equation as follows:
where the Dj form a discrete sequence of m values spanning the range of area diameters from Dmin to Dmax, is suitably defined, e.g., if the values are linearly spaced, and is the average particle density in the size interval about Dj. It is conventional to express equation (34) using matrix notation:
where Ai,j is calculated approximately and analytically by assuming a power-law dependence of the integrand between the end points of each particle size interval.
 In general, A is not square, and hence equation (35) cannot be inverted directly. The least squares solution to this equation, f=(ATA)−1ATg, is problematic because it is under constrained [Twomey, 1977]. Frequently, a practical solution makes use of the addition of an appropriate constraint, e.g., that the solution be smooth in some sense. The constraint takes the form of a second equation that is fit simultaneously. A Lagrange multiplier Λis selected to govern the balance between the least squares errors of the two equations:
To look for a smooth solution, one can constrain (minimize) the first differences by taking H as follows:
or the second differences by taking H as follows:
Press et al.  recommend the following choice for Λ, which balances the two components of the minimization:
where Tr() is the trace of the matrix.
 Figure 15 also shows the results of constrained inversions. Both constraints work reasonably well for the power-law model case, although the second difference did marginally better than the first difference constraint.
3.5 Point Spread Function
 We model stellar aureoles as the sum of the instrumental/atmospheric point spread function (PSF), starlight diffraction, and background, e.g., diffusely illuminated sky. The PSF describes the spreading of the radiation from a point source, such as a star and results from a combination of physical processes, e.g., diffraction by optical elements, saturation of detectors, bleeding of charge on the focal plane, and spreading by atmospheric turbulence. Since the latter process tends to be variable with time and space, we approach modeling the PSF using an empirical parameterization. The Gaussian function G(θ) is frequently used to model PSFs [Racine, 1996]:
where g0 and θg are fitting parameters. The effects of atmospheric turbulence appear to be modeled somewhat better using the Moffat function M(θ) [e.g., Moffat, 1969; Trujillo et al., 2001]:
where m0, θm, and m1 are fitting parameters. However, in this work we have found the Gaussian function preferable, because its functional form is more distinct from that of the aureole approximation (equation (29)). This difference allows the two functional forms to be fit simultaneously without their roles being confused in the process. Therefore, we have adopted the Gaussian functional form for modeling PSFs.
4 Aureole Profile Interpretation
 The process of interpreting aureole profiles begins with separating the three major physical components, the PSF, the stellar diffraction profile, and the background. Next the stellar diffraction profile is deconvolved (for the effects of multiple scattering) to give the phase function, from which the PSD is then derived. The process is illustrated in this section using aureole data for Capella from about 01:10 UT on 17 Dec 2011.
4.1 Diffraction Profile Extraction
 The center of Capella on the image was identified, first, by locating the brightest pixel and then refining the location using the maximum of a quadratic surface fit to the pixels within 0.1° of the brightest point. A set of concentric annuli was defined, and the pixel values within each annulus were averaged to produce the aureole radiance values shown as the black diamonds in Figure 16a.
 Using equation (44) for the PSF, equation (29) for the stellar aureole, and a constant, Lbkg, for the sky background, the sum of these components, Laur(θ), is modeled as follows:
We find the six parameters, g0, θg, L0, θ0, ν, and Lbkg using the Levenberg-Marquardt method to fit the data points. We tried using the formal RMS fluctuations of the pixel values in each annulus to specify the weights used in the fitting process, but this produced error bars that were manifestly smaller than the actual error bars. Consequently, we somewhat arbitrarily adopted 10% error bars for all of the data points to take into account other systematic errors. We found empirically that the specific choice of weighting functions had relatively little effect on the retrieved diffraction pattern of the aureole profile. Figure 16b compares the fit for the stellar aureole with data points after subtracting off the fitted PSF and background values. The plot suggests that the aureole approximation Lapx (θ) (equation (29)) has done a reasonable job of representing the diffraction of starlight by the cirrus cloud particles.
4.2 Phase Function Deconvolution
 Next the phase function of the scatterers is derived from the diffraction radiance profile fit, Lapx (θ), and the exo-atmospheric stellar irradiance, S0, using the deconvolution expression of equation (30). Although they do not need to be calculated explicitly, it may be helpful to show two of the intermediate products in this process. The green curve in Figure 17 shows the multiple scattering probability per unit solid angle, Qms (θ)≡Lapx (θ)/S0 (equation (A1)), adjusted for optical depth by dividing by τlos exp−τlos. The red curve shows the single scattering probability per steradian, Q(θ), deconvolved from Qms(θ) using equation (A27). Note that Q(θ) is both slightly lower and narrower than Qms (θ). This is expected since the effect of multiple scattering is both to increase the total number of photons scattered and to widen their angular distribution. The phase function, the end product of the process, is simply 4πtimes Q(θ) (equation (A12)).
4.3 PSD Solution
 The last step is to retrieve the PSD from the phase function. We use the numerical inversion technique (section 3.4.2), trying both first and second difference constraints, and selecting the one that fits P(θ) better. Figure 18a compares the phase functions derived from the fits (the colored lines) with the phase function (the black curve) scaled from Q(θ) (the red curve in Figure 17). The numerical inversion using the second difference constraint (the green curve) produces a slightly better fit than does the one using the first difference constraint (the red curve).
 Figure 18b compares the PSDs corresponding to the fits in Figure 18a. The matrix used in the inversion solution has dimensions 12×12. Examination of the 12 eigenvalues for the unconstrained problem shows magnitudes ranging from a maximum of 4×107 to a minimum of 5×10−9, indicative of ill-conditioning. By contrast, when the second difference constraint is added in using the Lagrange multiplier as indicated in equation (43), the eigenvalues range from 4×107 to 6×104, a considerable improvement that enables a solution to be found. Consideration of the first few eigenvalues of the unconstrained problem, 4×107, 7×105, 3×104, 2×103, and 9×100, would suggest that, practically speaking, there are perhaps two or maybe three free parameters. Note that although these PSDs correspond well to the phase function deconvolved from the aureole radiance measurement, we cannot say that they are unique, merely reasonable.
4.4 More Examples
 From the data sets we have collected, some 30% of which have been analyzed in detail, we show a few illustrative examples in this subsection. Figure 19 shows more measurement data from the night of 16–17 Dec 2011. The PSF and background magnitudes are anti-correlated as one might expect, the PSF being brightest when τlos is least, while the background (resulting largely from backscattered “cityshine”) is brightest when τlos is greatest. The diffraction profile radiance is maximized for τlos~1 or so, as is consistent from consideration of the single scattering equation (28).
 Figure 20a shows the phase functions deconvolved from the diffraction profile fits in Figure 19. Interestingly, the magnitude of the phase function forward scattering peak seems to be loosely correlated with τlos. Figure 20b shows the PSD solutions derived from the phase functions using constrained numerical inversion. The correlation noted above carries through to the PSDs. We are encouraged by the consistency of the retrievals.
4.4.1 Planetary Aureole Example
 Figure 21a shows an example of aureole measurement data for the planet Jupiter from the early morning of 27 Nov 2011. Note that the aureole profile does not drop down to the background level until about 0.2° in contrast with the case for the star Capella in Figure 16a, offering a potentially useful increase in particle measurement range. Nevertheless, the PSD retrieved using the second difference constraint (Figure 21b) covers roughly the same size range as the retrievals for the star Capella.
 Examination of the first few eigenvalues of the unconstrained problem, 2×105, 7×102, and 5×100, suggests that there are perhaps two free parameters, i.e., no more than in the Capella example shown in section 4.3. This may not be surprising given that the least squares fitting procedure used to extract the diffraction profile employs an analytic approximation (equation (29)) that has three free parameters and yields satisfactory fits.
 In summary, we have demonstrated retrieving and analyzing stellar aureole profiles for the purpose of determining the particle size distributions in ice clouds. (1) We have collected a useful stellar aureole database covering different atmospheric conditions and cloud optical depths. (2) Stellar aureole profiles have clearly been detected and measured through a wide range of cloud conditions. (3) The aureole profiles can be followed out to ~0.2° from stars (~0.5° from Jupiter). (4) The stellar aureoles from cirrus that we have examined have very distinctive profiles, typically being approximately flat out to a critical angle, followed by a steepening power-law decline with a slope less steep than −3. (5) The retrieved phase functions cover the range of sizes from ~50 μm to ~400 μm.
 In the process, we noted that (6) the relation between the diffraction phase functions of complicated ice crystal habits and the Fourier transforms of their projected areas results in a simple wavelength scaling relationship. We compared the phase functions for a variety of crystal habits with the same size metric. We defined the “area diameter” and found that (7) the phase functions for different habits are very similar for the same area diameter and (8) could be approximated using a simple analytic formula. We found a similar analytic approximation to represent possibly multiply scattered diffraction aureole profiles. (9) We utilized this simple aureole approximation along with a Gaussian function for the PSF and a constant for the background to separate the aureole profile data into these components. (10) We developed a technique for deconvolving multiply scattered diffraction profiles to find the phase functions from which (11) we were able to retrieve reasonable PSDs using constrained numerical inversion.
 The next steps are as follows: (a) to develop an instrument for the routine, automatic measurement of thin cirrus microphysical properties, (b) to deploy it at instrumented sites, such as the Atmospheric Radiation Measurement facilities operated by the Department of Energy, and (c) to compare its cirrus microphysical property retrievals with those of other instruments. In this latter category we include specifically (d) independent validation of the retrieved size distributions of cirrus ice crystals by comparison with coincident in situ measurements. In the overall process, a number of instrument improvements are anticipated. For example, we expect that the angular dynamic range of the aureole measurements, which is closely related to the range of particle sizes measured, can be increased by a factor of 2 to 3. Several ways of doing this include the following: (i) reducing the background level by locating the instrument to a darker area, (ii) reducing the PSF width by improving the camera optics, and (iii) using multiple color filters to take advantage of the dependence of diffraction on wavelength (equation (5)). In connection with instrument development, it will be important to investigate the sensitivity of the phase function retrieval to noise in the radiance measurements.
Appendix A: Multiple Scattering Deconvolution
 Small-angle, multiple scattering has been studied extensively for well over half a century, particularly in the context of particle or electron scattering from beams [e.g., Bethe, 1953, and references therein]. Mathematically, the problem involves convolution in the forward direction or deconvolution in the case of determining the single scattering pattern from measurements [e.g., Misell and Burge, 1969; Jones and Misell, 1967]. Assuming cylindrical symmetry in the aureole profiles, we adopt an approach that takes advantage of the Hankel transform [Bracewell, 2000].
 Consider plane-wave radiation from a distant point source, e.g., a star, of irradiance S0 (W/cm2/μm) incident normally on a uniform, plane-parallel, particulate layer, e.g., cirrus. Let L(θ)(W/cm2/sr/μm) be the total radiance (luminance) scattered by the particulate layer at scattering angle θ. The total or multiple scattering probability Qms (θ)(sr−1) per unit solid angle into an annular ring at scattering angle θ is
It is convenient to discretize the scattering angle domain, i.e., replacing continuous functions with vectors. Let Qms[i] be the vector element representing Qms (θi) at scattering angle θi, where 1≤i≤n and covers the range from 0≤θ≤θmax. Then equation (A1) can be written as
 Consider the equation for single scattering in the forward direction from radiation incident normally on a uniform, plane-parallel, particulate layer [e.g., Liou, 2002, pg. 302]. The singly scattered aureole radiance Lss (W/cm2/sr/μm) is
where τis the optical depth measured normal to the layer and p is the phase function normalized to integrate to 4π, Q is the phase function normalized to integrate to 1, and the single scattering albedo is omitted assuming conservative scattering. Or, dividing Lssby S∘, we find that
A1 Poisson Model
 Approximate multiple scattering as a Poisson process [e.g., Ning et al., 1995]. Then Qms is the sum over the probabilities of all orders of scattering:
where ⊗ represents the 2-dimensional convolution operator and Π(n,μ) is the Poisson probability for n scatterings:
and μis the average number of scatterings, which is equal to the optical depth τ. For the particles of interest, the forward scattering radiance forming the aureole is much more intense than scattering at other angles. Consequently, the integral in the convolution operator ⊗ can be approximated by an integral that goes to infinity:
where φ is an azimuthal angle and the small angle approximation, sin θ≈θ, has been employed.
 It is convenient to apply the Hankel transform to equation (A5) so that the convolution operation is replaced by multiplication in the transformed domain using the convolution theorem for Hankel transforms. Let be the Hankel transform operator:
where f(r) is a function to be transformed and J0 is the Bessel function of order 0. The inverse Hankel transform is also . Let represent the Hankel transform of F so that equation (A5) becomes
Given L(θ) and S0 and using equation (A2), this equation allows P(θ) to be calculated.
 The authors acknowledge and thank Rickey Petty and the Atmospheric System Research Program in the Climate and Environmental Sciences Division of the Department of Energy for support through Phase I SBIR grant DE-SC0006325. The authors also thank the anonymous reviewers for their recommendations, particularly their encouragement to take into account the effects of multiple scattering. Finally, the authors thank Brent Holben and his staff for their work in establishing and maintaining AERONET and especially the site at Goddard Space Flight Center.