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We investigated the errors associated with the scalar approximation (i.e., an approach that neglects polarization effects) for simulating the intensity of the radiation reflected by an ice cloud. In a case for an optical thickness of τ=1, the relative errors of the scalar approximation are typically between -0.5% and 0.1%. We also investigated the effect of the order of scattering on the degree of linear polarization. It is shown that substantial errors can be introduced by the first-order scattering approximation, and thus, the multiple-scattering effect is essential to an accurate simulation of the polarization configuration of a radiation field. Furthermore, we investigate the effective Mueller matrix pertaining to multiple scattering of light by ice clouds. The effective Mueller matrix is a 4×4 matrix that relates the incident and scattered Stokes parameters. This matrix implicitly contains the effects of all orders of scattering and absorbing events in the entire radiation transfer process. The sensitivity of the (2,1) and (3,1) elements of the effective Mueller matrix to ice crystal shape and size indicates that polarimetric information may be useful for inferring the microphysical properties of ice crystals within ice clouds.
©2006 Optical Society of America
1. Introduction
Despite a substantial amount of efforts made by the atmospheric research community to understand the role of ubiquitous ice clouds in the atmosphere, the optical and radiative properties of these clouds have not been well quantified because of limited knowledge about their macrophysical and microphysical properties [1, 2]. Radiometric measurements under ice cloudy conditions can be used to infer the optical and microphysical properties of ice clouds. For example, the satellite-based measurements of the solar radiation reflected by an ice cloud at a visible band (e.g., 0.65µm) and a near-infrared band (e.g., 1.63µm) can be used to simultaneously retrieve the optical thickness of the cloud and the corresponding effective particle size of ice crystals, as demonstrated by Baum et al. [3] and King et al. [4] using a bi-spectral approach developed originally by Nakajima and King [5] in the case of water clouds consisting of liquid droplets.
At present, remote sensing of atmospheric particulate constituents (e.g., cloud particles and aerosols) is based primarily on the measurements of radiance (I), one of the four Stokes parameters (I, Q, U, V) that are defined to completely specify the intensity and polarization state of a radiation field. The other three Stokes parameters, i.e., Q, U, and V, contain rich information about the interaction between a radiation field and a medium consisting of scattering and absorbing particles. Hansen [6] illustrated that the polarization state of atmospheric radiation under cloudy conditions is more sensitive to cloud microstructure than the corresponding radiation intensity. It has been recognized that polarimetry is a powerful technique for both passive and active remote sensing applications. For example, the merits of utilizing polarimetric measurements in target-detection and in the retrieval of particulate matter (e.g., aerosols) in the atmosphere have been demonstrated by Rakovic and Kattawar [7], Rakovic et al. [8] and Mishchenko and Travis [9]. Polarized backscattering lidar measurements can be used to derive the particle shape, orientation and composition of clouds [10], which are unlikely to be inferred from un-polarized lidar returns. Breon et al. [11] differentiate liquid droplets from ice crystals in clouds using polarization observations acquired from the French Space Agency POLarization and Directionality of the Earth’s Reflectances (POLDER) on board the ADEOS-1/NASDA platform, which provided the polarization ratios at 3 spectral bands (0.443, 0.670 and 0.865 µm) from November 1996 to June 1997. In the very near future, the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) [12] satellite has recently become a member of the NASA A-Train satellite constellation, and the Aerosol Polarimeter Sensor [13, 14] to be launched as a part of the NASA Glory mission are anticipated to measure the polarized reflected radiance. These polarization measurements will provide an unprecedented opportunity to study the optical and microphysical properties of particulate materials (i.e., aerosols, cloud droplets and ice crystals) in the atmosphere.
For a radiative transfer process, the incident and scattered Stokes parameters are related through a 4 by 4 matrix known as the effective Mueller matrix [7] that includes the effects of all orders of multiple scattering and absorption events occurring in the entire physical process from source to receiver. Most current remote sensing techniques are based essentially on radiance measurements and use only the information content in the (1, 1) element of the effective Mueller matrix. However, the (1, 1) effective Mueller matrix element is the least sensitive to small changes in particle morphology (i.e., particles size and shape) and chemical composition. In contrast, other matrix elements, except (1, 4), (4, 1) and (4, 4), are usually quite sensitive to the differences of the single-scattering properties of scatterers. The intent of the present study is to understand the effective Mueller matrix pertaining to the multiple scattering process within an ice cloud consisting of nonspherical ice crystals. This paper proceeds as follows. Section 2 describes the methodology for the simulation of the effective Mueller matrix. Presented in Section 3 are the error analyses regarding the first-order approximation for polarization simulation and the scalar approximation for intensity simulation, which is followed by the examination of the sensitivity of the effective Mueller matrix to the different extent of absorption of ice clouds at a visible and a near-infrared wavelength, and to the sizes and aspect ratios of ice crystals. Finally, the conclusions of this study are given in Section 4.
2. Methodology
2.1 Effective Mueller Matrix
The electric field pertaining to an electromagnetic wave can be decomposed into perpendicular and parallel components (Er and El) with respect to a reference plane containing the propagation direction of the wave. The Stokes parameters (I, Q, U, V) that fully specify the intensity and polarization of this electromagnetic wave are defined [15] as follows:
The interaction of a beam of light and a scattering medium leads to a scattered radiation field whose intensity and polarization configuration are different from the counterparts of the incident radiation beam. If the local meridian planes in a spherical coordinate system are defined as the reference planes for the incident and scattered beams, the incident and scattered Stokes parameters are related through
where the subscripts 0 and s indicate the incident and scattered radiation, respectively; M is the effective Mueller matrix defined in the form of
In general, the effective Mueller matrix is a 4×4 matrix that includes 16 independent elements. From Eqs. (2) and (3), the effective Mueller matrix can be obtained by considering four different polarization configurations for the incident radiation while the incident and observational geometry is kept as the same. In the present numerical computation, we choose these four polarization configurations for the incident light as follows: (1) unpolarized light with (I,Q,U,V)=(1,0,0,0), (2) vertically and linearly polarized light with (I,Q,U,V)=(1,1,0,0), (3) light linearly polarized along 45 degree from the vertical with (I,Q,U,V)=(1,0,1,0), and (4) right-circularly polarized light with (I,Q,U,V)=(1,0,0,1). With these four polarization states for the reflected radiation that can be solved from a vector radiative transfer model, the effective Mueller matrix is given by
where the subscripts indicate the reflected radiation associated with the four aforementioned polarization states of the incident beam.
2.2 Vector Radiative transfer model used in the present study
A number of vector radiative transfer models have been developed for multiple scattering computations that incorporate polarization configuration, including the Second Simulation of the Satellite Signal in the Solar Spectrum (6S) [16], PolRadTran [17], the Monte Carlo method [18], the adding-doubling model [19]. It is worth noting that a scalar-radiative-transfer computational approach that incorporates the effect of polarization has been suggested by Pomraning and McCormick [20].
In this study, a vector radiative transfer model developed by de Haan et al. [21] on the basis of the adding-doubling method is employed to simulate the polarized radiation reflected by an ice cloud layer. The detailed features of this specific radiative transfer computational package have been reported by de Haan et al. [21].
3. Results and discussions
3.1 Single-scattering properties of ice crystals
The morphological habits (or shapes) of ice crystals in the atmosphere are extremely complicated, ranging from hexagons to highly irregular geometries. For simplicity in this sensitivity study, we assume that ice crystals are pristine hexagonal columns with aspect ratios of 2a/L=20µm/75µm and 2a/L=80µm/300µm and hexagonal plates with an aspect ratio of 2a/L=300µm/80µm, where L and a are, respectively, the length and radius of a cylinder that circumscribes a hexagonal ice crystal. The effect of the size distribution of ice particles is not considered here. Furthermore, we assume that ice crystals are randomly oriented. The single-scattering properties of these hexagonal ice crystals are computed at two wavelengths, λ=0.66 µm and λ=2.13 µm, from the geometric optics method [22], a method that is valid in this case because of large size parameters. The complex refractive indices of ice particles at these wavelengths are n=1.3078+i1.66×10-8 and n=1.2674+i5.65×10-4. The single-scattering phase matrix, P, for randomly oriented particles has only six independent non-zero elements in the form of
Figures 1 and 2 show the six independent single-scattering phase matrix elements for randomly oriented hexagonal columns and plates at wavelengths 0.66 µm and 2.13 µm, respectively. The single-scattering properties of hexagonal ice crystals are not discussed here since they have been extensively studied in previous studies (e.g., [23], [24]).
Fig. 1. Six nonzero elements of the single-scattering phase matrix at a wavelength of λ=0.66 µm for randomly oriented hexagonal columns with aspect ratios of 2a/L=20µm/75µm and 2a/L=80µm/300µm and randomly oriented hexagonal plates with an aspect ratio of 2a/L=300µm/80µm. L and a are, respectively, the length and radius of the cylinder circumscribing an ice crystal.
3.2 Vector versus scalar radiative transfer simulations
Many existing radiative transfer models consider only the intensity or radiance of radiation, assuming that the first Stokes parameter is independent of the other three Stokes parameters in radiative transfer simulation. The advantage of this approach (hereafter, referred to as the scalar approximation) is the computational efficiency in numerical simulation. Several previous studies have investigated the errors due to the scalar approximation. Adams and Kattawar [25] showed that the relative errors of the scalar approximation can be larger than 10% for Rayleigh scattering atmospheres. In a case for water clouds consisting of liquid droplets with sizes on the order of or larger than the incident wavelength, the errors of the scalar approximation are less than 1% [6]. In this study, we compare the radiances obtained from the vector and scalar radiative transfer simulations for the radiation reflected by ice clouds at visible and near-infrared wavelengths.
Figure 3 shows the radiance (or intensity) computed by fully considering polarization configuration at two wavelengths, λ=0.66 µm and λ=2.13 µm. Also shown in Fig. 3 are the relative errors of intensity (REOI) computed from the scalar approximation, defined as follows:
where Isca is radiance from the scalar approximation, and Ivec is the counterpart from the corresponding vector radiation transfer simulation. Two incident zenith angles, θ0=0° and 30°, are considered in the simulations. For each panel, the relative azimuth angle between the incident and scattered radiation beams is from 0° to 360°. The viewing zenith angle (θ) from 0° to 90° is along the radial direction, which is uniformly scaled according to cosθ.
Fig. 3. The vector intensity and relative error of intensity (REOI) of hexagonal columns with aspect ratio of 2a/L=80µm/300µm and hexagonal plates with aspect ratio of 2a/L=300µm/80µm for two wavelengths, 0.66 µm and 2.13 µm. The optical thickness of the ice cloud assumed in the computation is τ=1.
At a wavelength of λ=0.66 µm, the variation patterns of the intensity and REOI are symmetrical for θ0=0° because the radiation field is independent of the azimuth angle in this case. For θ0=30°, the maxima of reflected intensities are noticed in the principal plane, i.e., the plane that contains the incident and local zenith directions. It is evident from Fig. 3 that the reflected radiances at λ=0.66 µm are much larger than the cases at λ=2.13 µm because of the absorption of ice crystals at the longer wavelength (the single-scattering albedo in the case of hexagonal columns, for example, is 0.871 at λ=2.13 µm, which, however, is very close to 1 at λ=0.66 µm). Moreover, the reflected intensities in the case of hexagonal plates are slightly larger than the counterparts for hexagonal columns. The values of REOI are between -0.25~0.1% and -0.5~0.1% for λ=0.66 µm and λ=2.13 µm, respectively. Evidently, the scalar approximation for simulating the radiance pertaining to the radiation reflected by ice clouds is quite accurate.
3.3 All orders of scattering versus first-order approximation for polarization
The first order of scattering among all orders of scattering events is the major contribution to the polarization of light reflected by a cloud layer. Some previous studies [10, 18, 26–28] focused on the Mueller matrix with the first-order scattering approximation, which is practically feasible and accurate for an optically thin layer (e.g. in the case of the optical thickness τ<0.1). However, higher order scattering events may be significant if the optical thickness of a cloud is large. Thus, it is necessary to understand the effect of multiple scattering on the polarization configuration of a radiation field.
Fig. 4. The comparisons of the Degree of linear polarization (DLP) from the first-order scattering approximation with that including all orders of scattering for hexagonal columns and plates.
Figure 4 shows the comparison of the degree of linear polarization (DLP) for the first-order and all orders of scattering at a wavelength of λ=0.66 µm when the incident zenith angle (θ0) is 0°. In this case U and V are zero, and the DLP can be defined [29] as follows:
The ice cloud optical thickness for the simulations shown in Fig. 4 is τ=1. For the hexagonal columns (left panel), the first-order result is close to that for the all orders scattering in back-scattering directions. However, substantial errors for the first order approximation are noticed for the scattering angles ranging from 90° to 150°. The overall variation pattern of the DLP in the plate case is quite similar to that in the column case, except that the DLP for the plate is much smaller at large scattering angles (>140°). The similarity of the DLP variations for the plates and columns is fundamentally due to the similarity of the phase matrices for randomly oriented hexagonal columns and plates (see, Fig. 1). Especially, the magnitude of -P12/P11 is larger for ice plates than for ice columns, as evident from Fig.1. This feature is clearly reflected in the results shown in Fig. 4. In reality, the effects due to the particle shape, shown in Fig. 4, would not appear to be especially significant. A real cloud would contain a dispersion of particle shapes and sizes, and for such a situation the impact of particle shape might be largely mitigated because of the effects of polydispersity
3.4 Mueller matrices of hexagonal ice crystal particles
Figure 5 shows the images of the 16 elements of the effective Mueller matrix for ice cloud layers composed of randomly oriented hexagonal column or plate particles at a wavelength of λ=0.66 µm. The cloud layer is assumed to be plane-parallel with an optical thickness of τ=1. All the matrix elements, except M11, are normalized by M11. The effective Mueller matrix images for hexagonal plates differ from those for hexagonal columns, particularly, in the case for M12/M11, M21/M11, M41/M11, M14/M11 and M42/M11. The maximum values for most of the matrix elements for hexagonal plates are significantly larger than those for hexagonal columns.
For passive remote sensing applications based on reflected solar radiation, only the elements M11, M21, M31 and M41 are measurable because natural sunlight is unpolarized, i.e., (I,Q,U,V)=(I,0,0,0). Thus, the first column of the effective Mueller matrix is the most interesting from a perspective of practical applications. Figure 6 shows the first column of the normalized effective Mueller matrix at a wavelength of λ=2.13 µm. Overall, the features of the effective Mueller matrix elements at this wavelength are similar to those shown in Fig. 5 for a wavelength of λ=0.66 µm, except for some slight differences. The element M11 is much smaller at the wavelength λ=2.13 µm, because of the absorption of ice crystals at this wavelength. The left panel of Fig. 6 shows the comparison of the effective Mueller matrix elements between hexagonal columns and plates. Slightly larger values for hexagonal plates are noticed. The right panel of Fig. 6 shows the comparison of the results for hexagonal columns with a size of 2a/L=80µm/300µm and a smaller size of 2a/L=20µm/75µm. The patterns of the images show that the hexagonal column with smaller particle size has larger bidirectional reflectance for all of the four Stokes parameters.
The non-zero M31 and M41 elements shown in Figs. 5 and 6 are due to the effect of multiple scattering, because the (3,1) and (4,1) elements of the corresponding single-scattering phase matrix for randomly oriented ice crystals are zero. However, the values of M31 and M41 are substantially smaller than M11 and M21. Thus, it is quite challenging in practice to use the (3,1) and (4,1) elements to infer the microphysical and optical properties of ice clouds because of small signal-to-noise ratio of the corresponding radiometric measurements.
Fig. 5. Effective Mueller matrices for hexagonal columns with an aspect ratio of 2a/L=80µm/300µm and hexagonal plates with an aspect ratio of 2a/L=300µm/80µm at a wavelength of λ=0.66 µm.
In both Figs. 5 and 6, the element M11 represents the bidirectional reflectance of the cloud layer. The maximum of M11 is noticed in the principal plane with a large viewing zenith angle (θ≈85°). M21 and M31 are associated with linear polarization whereas M41 is associated with circular polarization. The small values of M21/M11, M31/M11 and M41/M11 indicate that the reflected radiation beam of an unpolarized light is slightly polarized. It is quite challenging to reliably measure the value of M41/M11 because the reflected solar radiation is essentially not circularly polarized. The present study indicates that M21/M11 and M31/M11 are sensitive to the wavelength, and particle size and aspect ratio. Therefore, the M21/M11 and M31/M11 elements contain valuable information about the microphysical properties of ice crystals within ice clouds, which can be potentially useful to atmospheric remote sensing.
Fig. 6. Effective Mueller matrix elements M11, M21/M11, M31/M11 and M41/M11 for hexagonal column with aspect ratios of 2a/L=80µm/300µm and 2a/L=20µm/75µm and hexagonal plates with an aspect ratio of 2a/L=300µm/80µm at a wavelength of λ=2.13 µm.
4. Conclusions
A rigorous vector radiative transfer model developed by de Haan et al. [21] is employed to investigate the polarization and the effective Mueller matrix for multiple scattering by cirrus clouds consisting of randomly oriented hexagonal ice crystals at visible and near-infrared wavelengths. The single-scattering properties of ice crystals are computed from the geometrical-optics method.
The radiances computed from the vector radiative transfer simulation and the scalar approximation are compared. In the case for an optical thickness τ=1, the relative errors for the scalar approximation are typically between -0.5% to 0.1%, indicating that the scalar intensity approximation is quite accurate for computing the intensity of a radiation field pertaining to an ice cloudy atmosphere. The degrees of linear polarization (DLP) computed from the first-order scattering approximation are compared with its counterpart that includes all orders of scattering. Substantial errors of DLP computed from the first-order scattering approximation are noticed at the scattering angles from 90° to 150°. Evidently, higher order scattering events are essential to the polarization configuration of reflected radiation.
The effective Mueller matrix elements for hexagonal columns and plates are computed at visible and near-infrared wavelengths. The small values of M21/M11, M31/M11 and M41/M11 indicate slight polarization for the reflected radiation beam of unpolarized light. The sensitivity of the elements M21/M11 and M31/M11 to the wavelength, particle size and aspect ratio may be potentially useful for the retrieval of the microphysical properties of ice crystals on the basis of polarimetric measurements.
Acknowledgments
The authors thank Drs. J. Hovenier and J. F. de Haan for providing their vector adding-doubling radiative transfer code. This research is supported by the National Science Foundation (NSF) CAREER Award research grant (ATM-0239605), and partially by research grants from NASA (NNG04GL24G), and partially supported by the Office of Naval Research under contract N00014-02-1-0478. Regarding his efforts on the study of the optical and radiative properties of ice clouds, Ping Yang specifically acknowledges the support and encouragement of Drs. Donald Anderson and Hal Maring on the NASA Modeling, Analysis and Prediction (MAP) Program and Radiation Sciences Program at NASA Headquarters, respectively. The authors thank the three anonymous reviewers for their constructive comments and suggestions.
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