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At a wavelength corresponding to negligible electromagnetic absorption by ice, rigorous numerical simulations based on solving Maxwell’s equations show a backscattering peak associated with the phase function of randomly oriented hexagonal ice crystals. The backscattering peak, which has important implications to the interpretation of lidar observations, exists in the cases of smooth regular, smooth irregular, and roughened hexagonal ice crystals. The backscattering peak width is inversely proportional to the size parameter. The theoretical prediction of the backscattering peak is consistent with observations.
© 2015 Optical Society of America
1. Introduction
Lidar technology has been widely used to study ice clouds [1–7]. From either ground-based or satellite-based lidars, the microphysical and optical properties of ice clouds can be inferred from lidar backscattering signals associated with these clouds. As the backscattering properties of ice clouds are fundamental to lidar retrievals, it is necessary to enhance our knowledge about the phase function of ice clouds near 180° scattering angle. In particular, it is critical to determine whether or not a backscattering peak exists for ice cloud particles [8].
The backscattering properties of ice cloud crystals at wavelengths corresponding to negligible electromagnetic absorption by ice have been studied with numerical simulations. The studies are mainly based on geometric optical methods, because the ice crystal size parameter (πD/λ, where D is characteristic dimension and λ is wavelength) is too large for the application of rigorous numerical scattering models [9].
Ice clouds are often assumed to be primarily formed of smooth regular hexagonal columns/plates and aggregates of pristine elements. The assumed particle morphologies lead to a backscattering peak associated with the scattering phase function, due to the corner-reflection effect related to smooth regular hexagonal crystals [10–12]. In particular, Borovoi et al. [10] suggest that the corner reflection dominates backscattering for randomly oriented smooth hexagonal plates and columns. Furthermore, their study also suggests that Fraunhofer diffraction associated with backscattered plane-parallel beams leads to a backscattering peak with a width that is proportional to λ /L with L indicating the characteristic minimum dimension of an ice crystal (i.e., the thickness of a plate or the cross-section diameter of a column).
The studies of the optical properties of ice crystals and relevant downstream applications have been thoroughly reviewed by Baran [8,14], Yang et al. [9], Liou and Takano [13], and Yang and Liou [15]. The single-scattering properties calculated based on pristine regular hexagonal ice crystals with smooth surface are not consistent with observations, particularly, from remote sensing perspective, whereas those from roughened crystals [16–18] show much closer agreement. It should be pointed out that, like the case of roughened ice crystals, application of inhomogeneous ice crystals containing air bubbles to theoretical light scattering and radiative transfer simulations also improves the consistency between theory and observation [19, 20]. We call special attention to the fact that the surface roughness of ice crystals suppresses the corner-reflection effect; as a result, geometric optics methods suggest no pronounced backscattering peak for a cloud formed of roughened ice crystals [21–23].
However, both smooth-particle models and rough-particle models in conjunction with the application of the geometric optics method fail to replicate the backscattering phase function derived from the Cloud Aerosol Lidar with Orthogonal Polarization (CALIPSO) observations (see Figs. 8-10 of [22]). Without directly solving Maxwell’s equations, geometric optics methods localize electromagnetic waves in terms of rays, and, the effects of the phase interference among rays are usually excluded for practical computations. Higher-order terms in the multipole expansion of the scattered field, which make significant contributions to backscatter, cannot be considered in geometric optics methods [24]; therefore, the backscattering properties calculated from geometric optics methods may be quite inaccurate.
We investigated the backscattering phase function of ice crystals simulated with the pseudo-spectral time domain method (PSTD) [25,26] and the invariant imbedding T-matrix method (II-TM) [27,28]. Both the PSTD and II-TM numerically solve Maxwell’s equations, and fully considers the effects of electromagnetic wave interference and polarization characteristics.
2. Study of the backscattering peak with numerical simulations
Figure 1 shows the normalized phase function of hexagonal ice crystals, calculated with both the PSTD and II-TM, at a visible wavelength (λ = 0.5 µm) with an index of refraction of 1.31. The imaginary part is set to be zero so that the result is applicable to all wavelengths corresponding to negligible absorption by ice. Figure 1(a) shows the phase function of hexagonal ice crystals calculated from the PSTD. The black line shows the phase function of randomly oriented smooth hexagonal columns, with a size parameter of 50 and an aspect ratio of 1 (i.e., hexagon’s width equals height) following Liu et al. [23]. A general increasing trend of backscattering with small oscillations is found between 170° and 180°, plus a narrow peak between 178° and 180° (Fig. 1b). The increasing trend is caused by the corner-reflection effect [10–12] and by geometric optical rays that undergo high orders of internal reflections. The narrow peak associated with a pristine hexagonal ice crystal has been explained in terms of Fraunhofer diffraction pattern of plane-parallel corner-reflection beams [10], but cannot be explained by the conventional ray-tracing technique that does not consider interference. Figures 1(c) and 1(d) show similar results based on the II-TM for an ice crystal size parameter of 50 and an aspect ratio of 1. Both the narrow peak and the increasing trend are seen in the II-TM simulations [28], but the magnitude of the increasing trend is more pronounced in the II-TM result than in the PSTD counterpart. The differences between the PSTD and II-TM results are understandable. In the PSTD solution, the phase function of an ensemble of randomly oriented ice crystals is averaged over a finite number of orientations because of limited computational resource, leading to errors. On the contrary, the II- TM calculates the scattering properties of randomly oriented crystals in an analytical form, assuring that no uncertainty can occur during the orientation-averaging process.
Fig. 1 Hexagonal ice crystal backscattering peaks. (a) Phase function of hexagonal ice crystals, calculated from the PSTD. Black line for smooth regular hexagons, red line for roughened regular hexagons, and blue line for smooth irregular hexagons. (b) Ratio of the phase function value to the backscattering counterpart in the scattering angle region of 175°-180°, calculated from the PSTD. (c-d) similar to (a-b) except for the II-TM results.
Because ice crystals with rough surfaces show much closer agreement with observations, we also show the phase function of hexagonal ice crystals with the degree of surface roughness specified in terms of σ = 0.3 in Fig. 1 (note, the physical meaning of parameter σ can be found in previous studies [29, 30]). Rough surfaces do not produce specular reflection, thus, the aforesaid increasing trend induced by the corner-reflection effect is less significant in this case [Figs. 1(a)-1(c)]. However, the narrow backscattering peak exists, and the backscattering phase function value at 180° is 1.5-2 times larger than the counterpart at 175° [Figs. 1(b)-1(d)].
Considering that, in reality, ice crystals may deviate from regular hexagonal ice crystals, we also show the phase function of irregular smooth hexagonal ice crystals [30]. Random perturbations specified with σ = 0.3 were added to the ice particle face orientations, and the relevant phase function is computed with the PSTD [26]. Furthermore, the scattering phase functions of 20 smooth surfaced irregular hexagonal particles are averaged to obtain the ensemble mean. The results are shown in Figs. 1(a) and 1(b) with blue solid lines. The backscattering properties of irregular hexagonal ice crystals are similar to those of roughened hexagonal ice crystals, including the narrow backscattering peak.
In addition to particle shape and surface roughness, particle size is important for backscattering properties, and we analyze the magnitude and shape of the backscattering peak by varying particle size. To quantify the backscattering peak, we define an amplification factor given by ζ = P(θ)/ Pflat(θ), where P(θ) is the actual phase function value at scattering angle θ, and Pflat(θ) is the corresponding phase function value assuming that there is no backscattering peak between 175° and 180°. In this study, the Pflat(θ) of roughened hexagonal ice crystals is calculated from the phase function between 150° and 175° using linear extrapolation, and the peak width is calculated as the half-height width of the ζ peak. The ζ peaks associated with roughened hexagonal ice crystals for size parameters 50 and 100 are shown in Figs. 2(a) and 2(b), respectively. The maximum values of ζ in Figs. 2(a) and 2(b) are similar, but the half widths of the ζ peaks are different in these two cases, indicating that the half-height width of the backscattering peak depends on the size parameter.
Fig. 2 Width of the backscattering peak is inversely proportional to the particle size. (a) Amplification factor of roughened hexagons with a size parameter of 50. (b) Amplification factor of roughened hexagons with a size parameter of 100. (c) Half-height width of the backscattering peak as a function of the size parameter. The black line denotes the best fit to the discrete data. (d) Backscattering phase function versus the size parameter.
Figure 2(c) shows the half-height width of the backscattering peak as a function of the size parameter. The width of the backscattering peak is inversely proportional to the size parameter, i. e., the backscattering peak width associated with large particles is narrower than that for small particles. Figure 2(d) suggests that the magnitude of backscattering peak is not sensitive to particle size.
The present results suggest that the backscattering peak exists in the cases of smooth regular, smooth irregular, and roughened hexagonal ice crystals, and the width of the backscattering peak is inversely proportional to the size parameter. In the case of smooth hexagonal ice crystals, the present finding is consistent with that reported by Borovoi et al. [10] who explained the physical phenomenon using the concept of Fraunhofer diffraction of corner-reflection beams emerging in the backscattering direction. Based on these results, the aforesaid backscattering peak is expected to be noticed from observation. Note that hexagonal model is used for qualitative analysis, and the habits of ice crystals in the real world are much more complex.
3. Analysis of observations
Measured ice crystal phase functions from previous studies reported in the literature are plotted in Fig. 3. The blue dots are in situ polar nephelometer (PN) measurements of approximately 4000 ice cloud samples from Jourdan et al. [17], where the error bars denote their standard deviation. We normalized the phase functions measured by PN so that it is independent of particle number concentration. The red dots are polar nephelometer measurements of laboratory generated ice cloud crystals from Barkey and Liou [31]. The measured phase functions are consistent with the phase function of roughened hexagons calculated from the PSTD and II-TM, and are also consistent with the results from the improved geometric optical method (IGOM [32]) simulations [22] and satellite retrievals [33]. However, the polar nephelometer cannot measure the backscattering phase function of ice crystals, because the distance between the laser beam receiver/transmitter and the particle needs to be sufficiently far in order that the backscattering phase function can be measured.
Fig. 3 Normalized phase function of ice crystals derived from observations. Dots are off-backscattering phase function from polar nephelometers, circles are backscattering phase function derived from previous retrievals, and solid lines represent phase function of roughened and irregular hexagons.
In our case, we derive the backscattering phase function of ice cloud crystals from the lidar ratio. A lidar instrument emits a linearly polarized laser beam into the cloud and receives the backscattering laser signals. The distance between the lidar and the cloud (~10 km for a ground-based lidar, and 700 km for CALIPSO) is much larger than the distance between the laser transmitter and receiver, allowing effective derivation of the backscattering phase function. The lidar ratio is defined as the ratio of the extinction coefficient to the backscatter coefficient, and equals 4π/P11(180°) at a wavelength corresponding to negligible absorption, where P11(180°) is the normalized backscattering phase function. Chen et al. [34] estimated the lidar ratio to be 29 ± 12 (sr) using ground-based lidar, implying a backscattering phase function of 0.31-0.74 at a wavelength of 532 nm. Josset et al. [35] estimated the lidar ratio to be 33 ± 5 (sr) based on CALIPSO, implying a backscattering phase function value between 0.33 and 0.45. These results are also consistent with Raman lidar measurements [36]. It should be pointed out that observed lidar ratio strongly depends on ice crystal size, particle shape, and cloud temperature [37]. In the process of retrieving the lidar ratio, the multiple scattering effects are considered, and the lidar ratio is an average attribute of numerous ice crystals. Note that the results from Jourdan et al. [17] are measured at the 804 nm wavelength, the results from Barkey and Liou [31] are measured at 670 nm, and the lidar wavelength considered in this study is 532 nm. Because the refraction indices of ice at the three wavelengths have similar values and ice crystals are large compared to a visible-light wavelength, the phase functions are quite close. Therefore, a peak is likely to exist at 180° with an amplification of 1.5-2. Uncertainties are introduced by observational, retrieval, and statistical errors, and the phase function of ice cloud particles depends on particle size, shape and surface roughness, but the mean backscattering phase function is significantly (95% confidence level) larger than the phase function around 160°, indicating that the backscattering peak suggested by numerical models is plausible.
An ice cloud crystal backscattering peak generated from single scattering is important for lidar observations, and we expect that radiative transfer models can better simulate the lidar backscatter of ice clouds when the backscattering peak is considered. Therefore, we simulate the lidar signals related to ice clouds with and without the backscattering peak, and compare the results with observations.
We calculate the phase function of ice crystals using the IGOM, and adjust the backscattering phase function with an empirical equation. We use the IGOM because the size parameter of ice crystals could be larger than 1000 [22], and the applicable range of the PSTD and II-TM methods does not include large ice crystals. In the simulation, the wavelength is 0.532 μm, refractive index is 1.31 + 1.5x10−9i, the surface roughness coefficient is 0.3, the aspect ratio is 1. A gamma size distribution n(D) = N0Dexp(−4D/70) is assumed, where N0 is a constant that normalizes the size distribution, and D is the characteristic dimension with units in μm. The phase function for hexagonal ice columns simulated by the IGOM is shown in Fig. 4(a). The phase function simulated by the IGOM is consistent with the observations (Fig. 3), except for the backscatter. We adjusted the backscatter with the empirical equation:
in which we set the constant R to be 0.7 and δc = 2πD(π-θ)/λ. The equation is empirically derived to fit the simulated backscattering peak data in Fig. 2. The lidar signals of opaque ice cloud layer (typically optical depth>3.5) observed by CALIPSO are simulated using the Monte Carlo radiative transfer model developed by Hu et al. [38]. In the Monte Carlo simulations, the optical depth of cloud layer is set to be 4, and the cloud extinction coefficient varies from 0.5 to 4 km−1.Fig. 4 Effect of backscattering peak on lidar observations. (a) Phase function of ice cloud crystals calculated from the IGOM (blue line), and the adjusted phase function using an empirical equation for backscatterer (red line). (b) Lidar multiple scattering coefficient, simulated with a Monte Carlo radiative transfer model. Blue points are simulated using the unadjusted phase function, and red points are simulated using the adjusted phase function. Shaded area denotes CALIPSO retrieval intervals from Josset et al. [35]. (c) Lidar layer-integrated attenuated backscatter of opaque clouds simulated with the Monte Carlo radiative transfer model. Shaded area denotes standard deviation interval of CALIPSO observations during 2009-2010.
Figure 4(b) shows the lidar multiple scattering coefficient for CALIPSO, defined as the ratio of apparent experimental visible extinction coefficient to true visible extinction coefficient. If the backscattering peak is not considered (blue line), the multiple scattering coefficient is 0.35-0.5, but after adjusting the backscatter (red line), the multiple scattering coefficient increases to 0.45 −0.65, which is within the range (shaded area) retrieved by Josset et al. [35]. Figure 4(c) shows the simulated CALIPSO layer-integrated attenuated backscatter of opaque clouds. The layer-integrated attenuated backscatter simulated with the adjusted phase function is within the range of the observations (shaded area); therefore, simulations with the inclusion of the backscattering peak are more consistent with CALIPSO observations.
4. Discussion and conclusions
Numerical simulations directly solving Maxwell’s equations show a narrow backscattering peak in the phase function of not only randomly oriented smooth regular hexagonal but also irregular smooth and roughened hexagonal ice crystals. Our numerical results indicate that the width of the backscattering peak is inversely proportional to the size parameter. The backscattering peak is consistent with observations, and has an important implication in interpreting lidar observations associated with ice clouds. After adjusting the roughened ice crystal phase function with a backscattering peak, the lidar layer-integrated attenuated backscatter simulated by a Monte Carlo radiative transfer model is consistent with observations.
At present, the physical mechanism of the backscattering peak for roughened ice cloud crystals is not clear, but the backscattering peak is likely to be associated with wave interferences. In particular, the empirical fitting to the backscattering peak in terms of Eq. (1) illustrates the similarity between the backscattering peak and Fraunhofer diffraction. In the case of smooth particles, Borovoi et al. [10] attributed the backscattering peak to the Fraunhofer diffraction of collimated beams emerging in the backscattering direction. In the case of roughened particles, the cross sections of collimated beams are very small, leading to wide-spread diffraction patterns. Furthermore, the magnitude of backscattering peak does not increase significantly with particle size [Fig. 2(d)]. Thus, the backscattering peak is unlikely to be caused by a set of plane-parallel backscattering beams in the near-field in the case of an ice crystal with roughened surface.
One possible explanation is that the backscattering peak of roughened ice cloud crystals is a result of coherent backscattering. When radiation propagates through a medium with a large number of scattering events, constructive phase interferences of waves propagating along the same light-scattering paths but in opposite directions arise when observing in the backscattering direction, resulting in a backscattering peak [39–42]. In the case of a large ice crystal, the reflected/refracted light waves are a superposition of waves associated with various locations on the ice surface, and the backscattering waves may be enhanced by constructive interferences of waves propagating along the same light-scattering paths but in opposite directions.
Coherent backscattering may also be generated by multiple scattering between ice cloud crystals, but the effect is insignificant for lidar observations. The critical off-backscatter angle α = π-θ necessary to observe the coherent backscatter generated by the multiple scattering of ice crystals is [43]:
where λ is wavelength and l* is the mean free path. The typical extinction coefficient of ice cloud is 1/km, the mean free path of visible light in the ice cloud medium is ~103m, and the wavelength λ is ~m. Therefore, αcritical is ~ rad, which is much smaller than the off-backscatter view angle of most lidars and renders the coherent backscatter generated by inter-particle multiple scattering insignificant in lidar observations.Acknowledgments:
The authors are grateful to Dr. O. Jourdan for sharing the PN data set. The authors are also thankful to Dr. L. Bi and Dr. C. Liu for helping the numerical simulations. Chen Zhou acknowledges support from NASA Earth and Space Science Fellowship Program (NNX12AN57H). Ping Yang acknowledges support by a NSF grant (AGS-1338440) and a NASA grant (NNX11AK37G) and partially by the endowment funds (TAMU 512231-10000) related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University.
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