1. Introduction

Globally, cirrus clouds can permanently cover more than 30% of the earth's surface (Wylie et al. 1994). With such a spatial and temporal coverage it is well known that cirrus clouds are an important component of the earth–atmosphere radiation balance (e.g., Stephens and Webster 1981; Liou and Takano 1994; Donner et al. 1997; Kristjansson et al. 2000). Quantifying the radiative impact of cirrus on cloud–climate feedback within general circulation models is difficult due to their natural variability in terms of ice crystal size and shape (Heymsfield and McFarquhar 2002). In the paper by Kristjansson et al. (2000) it is shown that ice crystal size and shape are both as important as each other in determining the cloud longwave radiative flux at top-of-the-atmosphere. In order to improve cirrus parameterization within global circulation models so that uncertainties in longwave radiative fluxes can be reduced, it is important to obtain global information on ice crystal size and shape.

At infrared wavelengths retrieval of effective crystal size is accomplished by using dual and trispectral techniques based in the terrestrial atmospheric window region of 8–12 μm (Ackerman et al. 1990; Strabala et al. 1994; Ackerman et al. 1995; Baran et al. 1998; Stubenrauch et al. 1999). Remote sensing techniques based in this wavelength region are able to retrieve ice crystal effective size due to the variation in the complex refractive index of ice between 8 and 12 μm (Ackerman et al. 1990). However, the retrievals of ice crystal effective size using wavelengths between 8 and 12 μm in some of the above papers were based on Mie theory applied to ice spheres, which limited their retrievals of ice crystal effective size to less than 30 μm (Strabala et al. 1994; Ackerman et al. 1995). These two authors also noted that retrievals of ice crystal effective size based on Mie theory were less than in situ measured sizes. It is also reported in Spinhirne and Hart (1990) that retrievals based on ice spheres were approximately one-half to one-third of the in situ measured crystal sizes. In a more recent paper, Fu and Sun (2001) show by assuming randomly oriented hexagonal ice columns that there is no sensitivity to retrieval of ice crystal size for effective sizes greater than 25 μm at two wavelengths in the terrestrial window region.

There are two principal reasons as to why this underprediction should occur by assuming ice spheres. The first is described in terms of the “effective distance” or photon pathlength d_e, which is expressed as the volume-to-area ratio of the particle, a concept first introduced by Bryant and Latimer (1969). For a given volume the sphere has the least surface area, it therefore follows that for all convex particles of a given volume the sphere will have the greatest d_e and so will exhibit the largest absorption as discussed in the paper by Mitchell and Arnott (1994). For randomly oriented convex particles the averaged projected area is given by one-fourth of the surface area as found by Vouk (1948). The second reason as to why absorption is greater for spheres than nonspherical ice crystals is discussed in Mitchell et al. (1996) and Baran et al. (1998) in terms of “above-edge” contributions being larger for ice spheres than nonspherical ice crystals. This second reason is now more fully discussed.

Mie theory predicts that photons that have impact parameters (i.e., the perpendicular distance of an incident photon from an axis going through the center of the sphere) greater than the radius of the sphere can be absorbed, in addition to photons that directly impact the sphere. This process is known as “tunneling” or the above-edge contribution and is a conceptual picture provided by applying asymptotic methods through analytic continuation of the Mie partial wave series to the complex angular momentum (CAM) plane as described by Nussenzveig (1988). Importantly, by considering plane wave incidence on a hard sphere it was shown by Nussenzveig (1988) that the above-edge contribution is responsible for large-angle diffraction and surface waves. Applying complex angular momentum considerations to the problem of absorption is described in the paper by Nussenzveig and Wiscombe (1980), and the accuracy of the CAM approximation is shown to be within 5% of Mie theory for the absorption efficiency (i.e., the absorption cross section divided by the projected area of the particle) at a size parameter (product of particle radius and wavenumber; here, wavenumber is 2π/λ, where λ is the incident wavelength) of 10. The relative CAM errors decrease for increasing size parameter and real refractive index (Nussenzveig and Wiscombe 1980). The CAM errors (measured relative to Mie theory) are further investigated in this paper at the absorbing wavelengths of 8.55 and 11.0 μm for the absorption coefficient integrated over representative measured cirrus size distribution functions provided by Fu (1996). In the paper by Mitchell et al. (1996) it is speculated that the above-edge contribution to absorption is unlikely to exist for natural ice particles due to these particles having sharp edges. In the paper by Baran et al. (1998) it is argued that the above-edge contribution should diminish relative to the sphere as the particle becomes more nonspherical. If this argument is correct, then the consequence of the spherical assumption would be to underpredict the in situ particle effective crystal size due to the above-edge contribution being more significant for ice spheres than for nonspherical ice crystals. The impact of the above-edge contribution is to increase absorption by the ice sphere thereby requiring a smaller effective crystal size to account for significant measured brightness temperature differences between the 8.5- and 11.0-μm channels (see section 5).

In order to show that the above argument is a plausible physical reason Baran et al. (1998) used upwelling radiometric measurements from the high-resolution infrared sounder (HIRS) instrument at the wavelengths of 8.3 and 11.1 μm to retrieve crystal median mass dimension (D_m) from 11 tropical anvils. The retrieved D_m was compared with in situ measurements of D_m obtained during the Central Equatorial Pacific Experiment [CEPEX; see McFarquhar and Heymsfield (1996) for details of the CEPEX campaign]. The retrievals of D_m assumed ice spheres and a variety of nonspherical ice crystals. The ice sphere absorption coefficient was calculated using Mie theory, and the absorption coefficients of the various nonspherical ice crystals were calculated using the modified anomalous diffraction approximation (ADA) due to Mitchell et al. (1996). Essentially, this modified ADA includes contributions from geometric optics in the form of refraction and internal reflection but ignores above-edge contributions and external reflection. Baran et al. (1998) found that by applying the modified ADA to calculate the absorption coefficient of the various nonspherical ice crystals led to better agreement between the retrieved D_m and in situ D_m than retrievals based on the ice sphere. The retrievals of D_m based on the ice sphere were found to be about one-third of the in situ D_m. This finding led to the speculation by Baran et al. (1998) that the physical reason as to why the ice sphere underpredicted D_m was due to the above-edge contribution since this contribution was neglected for the various nonspherical ice crystals, which gave much better agreement with the in situ measurements.

Based on more rigorous electromagnetic theory, Fu et al. (1998) estimated that the above-edge contribution to the absorption efficiency of the randomly oriented hexagonal ice column was diminished relative to the ice sphere. Baran and Havemann (1999) found a similar result based on electromagnetic theory and CAM, and they estimated that the above-edge contribution was reduced to about two-thirds in the resonance region (i.e., size parameters approximately in the range 1–10). Moreover, Baran et al. (2001) has estimated that the ice aggregate above-edge contribution is diminished relative to the ice sphere above-edge contribution by about one-half in the resonance region. These more recent findings based on rigorous electromagnetic theory are more consistent with Baran et al. (1998) who argued that as the particle becomes more nonspherical the above-edge contribution should be diminished relative to the sphere.

In this paper a further study is made of the impact of differing absorption processes (i.e., refraction, reflection, and above-edge contributions) on the retrieval of ice crystal effective size under the assumption of different ice crystal habits. Unlike previous studies, more rigorous theory is applied in order to calculate the absorption coefficients of the various ice crystal shapes. The study is divided into the following sections. Section 2 describes the CAM approximation adopted for calculating the geometric optics and above edge absorption coefficients of the ice sphere and comparisons are made against Mie theory for the absorption coefficient integrated over representative cirrus size distribution functions. Section 3 describes the methodology of calculating the absorption coefficients of the various nonspherical ice crystals. Section 4 describes the cirrus case study and the methodology of retrieving ice crystal effective size. Section 5 compares and contrasts the retrieval of ice crystal effective size under various assumptions of ice crystal shape and absorption processes. Section 6 contains discussion and conclusions.

2. The complex angular momentum approximation applied to absorbing spheres

The behavior of 〈Q_abs〉, 〈Q_go〉, and 〈Q_ae〉 as a function of maximum dimension are shown in Figs. 1 and 2 for the ice sphere at the wavelengths of 8.55 and 11.0 μm, respectively. The complex refractive indices for ice used in the calculations are taken from Warren (1984). The figures show that by far the greatest contribution to 〈Q_abs〉 is 〈Q_go〉 but that 〈Q_ae〉 is not insignificant at maximum dimensions around 35 μm. The total contribution to 〈Q_abs〉 by 〈Q_ae〉 for maximum dimensions around 35 μm is about 20% and 11% at 8.55 and 11.0 μm, respectively. The dependence of 〈Q_ae〉 on crystal size and complex refractive index has previously been discussed in Baran and Havemann (1999). For the two wavelengths considered here, the complex refractive index N of ice is N = 1.29 + 0.039i and 1.09 + 0.25i, where i is the imaginary index, at 8.55 and 11.0 μm, respectively. As a consequence of the real part of N being greater at 8.55 than 11.0 μm, the 〈Q_ae〉 contribution to 〈Q_abs〉 is therefore more significant at 8.55 than at 11.0 μm. Note from the figures that for maximum dimensions greater than 1000 μm 〈Q_abs〉 ∼ 〈Q_go〉 as 〈Q_ae〉 becomes insignificant due to geometric optics becoming dominant at such large maximum dimensions.

The CAM solution of the absorption coefficient β_abs at 8.55 and 11.0 μm is now tested against solutions for β_abs using Mie theory. In this section β_abs is defined as

View Expanded

where Q_abs(Cam,Mie) is the absorption efficiency calculated from either CAM or Mie theory, P(D) is the projected area as a function of maximum sphere dimension D, and n(D) is the measured size distribution function taken from Fu (1996). The effective radius R_e of the size distribution is defined as

R_eD_e(4)

where D_e = 3/2 ∫ V(D)n(D)dD/∫ P(D)n(D)dD, the term V(D) is the geometric volume of the sphere.

Figure 3 shows ε plotted as a function of R_e. The figure shows that as R_e increases, ε decreases, which is not a surprising result since CAM is an asymptotic approximation. The relative percentage error in CAM, as shown by Fig. 3 at 8.55 μm, is less than 3% for all R_e. However, at 11.0 μm, the relative percentage error in CAM for the smallest R_e of 4.0 μm is 11%, but by an R_e greater than about 12 μm ε is less than 10% and is less than 5% by an R_e of 100 μm. The absolute error is shown in Fig. 4 where Abs is plotted against R_e for both wavelengths. For R_e < 10 μm, Abs is about 1.0 at 11.0 μm, but for R_e > 10 μm Abs is generally much less than 0.1 at 8.55 and 11.0 μm, respectively. The absolute and relative percentage error in CAM at 8.55 and 11.0 μm is not significant and is accurate enough to be a useful asymptotic approximation for the purposes of this paper. However, in general it should be noted that the CAM approximation as described in Nussenzveig and Wiscombe (1980) does not resolve individual resonance contributions (these resonance contributions appear as a “ripple” superimposed on the Mie efficiencies) on the CAM efficiencies since only a “coarse-grained” average is considered. Therefore, it should be noted that when comparing CAM efficiency solutions with Mie theory, large errors in CAM could result due to the absence of the ripple on the CAM efficiencies, this being particularly true for nonabsorbing or weakly absorbing spheres (see, e.g., Grandy 2000).

3. Methodology for calculating the absorption coefficients of nonspherical particles

The absorption coefficient β_abs of the hexagonal ice column and ice aggregate in this section is calculated from

View Expanded

where Q_{abs(FDTD,IGO)} is the absorption efficiency calculated from either finite difference time domain (FDTD) or improved geometric optics (IGO), P(D) is the projected area as a function of maximum crystal dimension D, and n(D) are the measured size distribution functions taken from Fu (1996), which, in the midlatitudes, have similar shapes to the distributions used by Francis et al. (1999).

The ice aggregate is a mathematical realization of irregular ice crystals that occur in cirrus clouds, and it consists of a collection of eight smooth hexagonal ice column elements, with the aspect ratio of each element varying between 1.39 and 2.25. The aspect ratio of the aggregate is close to unity [Yang and Liou (1998); a figure of the ice aggregate model can be seen in Fig. 2 of the same paper]. The hexagonal ice column and ice aggregate absorption efficiencies are calculated using the FDTD method using a grid resolution of λ/30 for crystal maximum dimensions up to 40 μm (Yang et al. 2000a). For maximum dimensions greater than 40 μm and up to 3500 μm, the absorption efficiency for both ice crystal habits is calculated using the improved geometric optics method due to Yang and Liou (1996). The aspect ratio [e.g., the ratio of the crystal length (L) to twice the radius (a)] of the hexagonal ice column varies with maximum dimension as given in Yang et al. (2000b), which is described by

View Expanded

The IGO method employs the ray tracing procedure to compute the field on the particle surface, which is then transformed to the far field via the electromagnetic equivalence principle. This approach allows the computation of the absorption efficiency down to much smaller maximum dimensions than the conventional geometric optics approach. In order to match absorption efficiency solutions between the FDTD and IGO methods, corresponding Mie absorption efficiency solutions are scaled to ensure that absorption solutions between FDTD and IGO are continuous and that the correct absorption asymptote is reached. In the papers by Fu et al. (1998) and Baran and Havemann (2000) it is shown that the absorption asymptote is independent of ice crystal shape, and absorption in the asymptotic region can be computed accurately using either Mie theory or CAM, respectively. This approximate matching procedure between the FDTD and IGO methods follows the composite method of Fu et al. (1998). It is shown in Fu et al. (1998) that the relative error in the composite method is generally less than about 3%.

Figure 5 shows solutions obtained for Q_abs plotted against maximum dimension using Eq. (10) at the wavelength of 8.55 μm assuming the hexagonal ice column. Also shown in the figure are the Mie (filled triangles) and FDTD (filled circles) Q_abs solutions, which are shown for comparison. Figure 6 shows the same as Fig. 5 but for the hexagonal ice column at 11.0 μm. Figures 5 and 6 show that there are no discontinuities between FDTD and IGO absorption solutions by applying the composite method.

Figure 7 shows the same as Fig. 5 but for the ice aggregate at 8.55 μm, the figure shows that IGO overpredicts absorption for 4 μm < D < 60 μm. The Mie Q_abs solutions closely follow those predicted by FDTD for maximum dimensions less than about 30 μm. However, Fig. 7 shows different behavior than Fig. 5 in that the IGO solutions merge with the composite solutions at maximum dimensions greater than about 50 μm. In this case from Eq. (11) Q_abs = Q_abs(IGO) since c = 0 and d = 1 for maximum dimensions greater than about 50 μm, physically this implies that the tunneling effect is negligible for ice aggregates at this wavelength and the IGO contribution is completely dominant. This behavior in Q_abs for the ice aggregate is very different to the ice sphere shown in Fig. 1 and to the hexagonal ice column shown in Fig. 5. In the paper by Baran and Havemann (1999), Fig. 11b of that paper shows Q_abs plotted as a function of size parameter assuming a randomly oriented hexagonal ice column of aspect ratio 6, where the size parameter is defined in terms of the volume-to-projected area ratio. In the paper by Baran et al. (2001), it is shown that the above-edge contribution does depend on the assumed aspect ratio and as the aspect ratio of the hexagonal ice column increases the above-edge contribution decreases, this approximates the behavior of the above-edge contribution to more irregular ice crystals. From Fig. 11b in Baran and Havemann (1999), at a size parameter of about 16, Q_abs ∼ 0.93 using an electromagnetic approximation based on the separation of variable method (SVM) due to Rother et al. (1999). A size parameter of 16 transformed to the ice aggregate corresponds to a maximum dimension of about 200 μm and from Fig. 7 this gives Q_abs ∼ 0.90 using the composite curve. The relative difference between the SVM and composite method is about 3.2%, which is the expected accuracy of the composite method according to Fu et al. (1998). This level of accuracy is sufficient for the purposes of this paper.

Figure 8 shows the same as Fig. 7 but for the aggregate at 11.0 μm; note that at 11.0 μm, the difference between the composite and IGO methods is smaller for the aggregate [Q_ae(ns) ∼ 0.09 from Fig. 8 at D ∼ 70 μm using Eq. (12) below] than for the hexagonal ice column [Q_ae(ns) ∼ 0.20 from Fig. 6 at D ∼ 50 μm]. Figures 7 and 8 show that there are no discontinuities between FDTD and IGO absorption solutions by applying the composite method.

The impact of particle geometry and absorption process on the retrieval of ice crystal effective size at the wavelengths of 8.55 and 11.0 μm is discussed in section 5, but the aircraft measurements are now discussed.

4. Cirrus case and aircraft radiometric measurements at 8.55 and 11.0 μm

For the level run below the cirrus the average measured emissivity at 8.55 μm was 0.434, and at 11.0 μm it was 0.532. Figure 9 shows the ratio of emissivities plotted against the emissivity measured at 11.0 μm together with the sensitivity of the measurements to ice crystal effective size. The figure shows that the ratio of the emissivities lie between about 0.67 to 0.87 with the average being 0.82. The average emissivity is significantly less than unity, implying that there is likely to be useful information on crystal size contained in these measurements. The measured emissivities lie in the r_e range between about 20 and 36 μm with the figure showing no evidence of saturation in terms of crystal size. In the paper by Fu and Sun (2001), it was concluded that retrieval of crystal effective size greater than 25 μm was not possible. However, it should be noted that this conclusion was based on the assumption of the hexagonal ice column, whereas in this paper Fig. 9 is based on the ice aggregate. As an interesting aside, Fig. 9 indicates that there is significant sensitivity in the emissivity ratio between the r_e = 34 and 36 μm curves. The size distributions used for these two curves do predict a similar r_e if the aggregate is assumed, yet the corresponding radiative properties would seem to indicate a larger difference in r_e. Physically, this means that r_e may not uniquely describe the radiative properties of cirrus cloud, as discussed in Mitchell (2002). However, this apparent discrepancy does not change any of the analysis or conclusions presented in this paper. The methodology adopted for the retrieval of crystal effective size is now briefly described.

a. Methodology of retrieval

In this paper the zero scattering approximation is assumed since the study is concerned with absorption processes. Moreover, Francis et al. (1999) have previously shown that the neglect of scattering at the wavelengths of 8.55 and 11.0 μm is likely to result in retrieval errors of crystal effective size of only 4 μm at most. The aircraft measures downwelling radiance at cloud base, I^↓(z_b, θ), which, under the zero scattering approximation, is given by

View Expanded

where z_t represents the cloud top and z_b the cloud base; θ is the viewing angle; τ_abs is the absorption optical thickness, which is given by the absorption coefficients described in section 3; and the Planck blackbody emission at the mean cloud emitting temperature 〈T〉 is given by B(〈T〉).

The value of 〈T〉 used in the radiative transfer calculations was 233.6 K. The term I^↓(z_t, θ) given in Eq. (14) was determined by moderate-resolution transmittance (MODTRAN) using a local radiosonde profile. Given I^↓(z_b, θ), I^↓(z_t, θ) and 〈T〉, the cloud emissivity at 8.55 and 11.0 μm can be found by inverting Eq. (14). The crystal effective size can then be retrieved by relating r_e to the emissivities found at 8.55 and 11.0 μm. The results of retrieving r_e under various assumptions is discussed in section 5.

5. Retrieval of ice crystal effective size under various assumptions

6. Discussion and conclusions

The geometric optics and above-edge contributions to the absorption efficiency of ice spheres and nonspherical ice crystals have been studied using the CAM approximation and a composite method based on the FDTD method and an improvement to geometric optics, respectively. The importance of the geometric optics and above-edge contributions were studied for different particle shapes, which were the ice sphere, randomly oriented hexagonal ice column, and randomly oriented ice aggregate. The study was centered on the important terrestrial window wavelengths of 8.55 and 11.0 μm, which are often used for remote sensing of cirrus cloud. It was found that although the geometric optics component was the most important contribution to the total absorption efficiency of the sphere and nonspherical ice crystals (i.e., generally greater than 80% in the resonance region), the above-edge contribution was not insignificant. In order to study the impact of the absorption processes on the retrieval of ice crystal effective size, aircraft radiometric measurements of downwelling radiance from cirrus cloud were utilized at the wavelengths of 8.55 and 11.0 μm.

The impact of the above-edge contribution on the retrieval of the ice sphere effective size was shown to be significant. The retrieved r_e using the geometric optics contribution only was found to be 30.4 ± 14.2 μm; however, with the above-edge contribution included, the retrieved r_e reduced to 15.5 ± 7.2 μm. The impact of the above-edge contribution reduced the retrieved r_e by approximately half the value obtained by assuming the geometric optics component only. Moreover, the retrieved r_e with the above-edge contribution included was found to be about 2 to 3 times smaller than the estimated in situ ice crystal effective size. This finding is typical of what is found when comparison is made between in situ measurements of r_e and retrievals of ice crystal effective size assuming the ice sphere. The physical reason for this discrepancy between remote sensing retrievals of r_e based on ice spheres and in situ measurements is due to the above-edge contribution. It was speculated in Mitchell et al. (1996) and argued in Baran et al. (1998) that the physical reason as to why ice spheres underpredicted in situ measurements of ice crystal effective size is due to the above-edge contribution; the findings of this paper appear to support those arguments.

The impact of the above-edge contribution on retrievals of r_e assuming the hexagonal ice column and ice aggregate were far less dramatic than for the ice sphere. For the case of the hexagonal ice column the difference in mean r_e between retrievals based on the geometric optics component only and with the above-edge contribution included was found to be 4.0 μm. The same retrieval process was applied to the ice aggregate and the difference in mean r_e was found to be 1.2 μm. The impact of the above-edge contribution on the retrievals of ice crystal effective size was systematically diminished as the ice crystal complexity increased and the crystal symmetry decreased. This finding is consistent with the arguments contained in Baran et al. (1998) that the importance of above-edge contributions should be diminished as ice crystal nonsphericity increases. However, in general, above-edge contributions should not be ignored in the computation of ice crystal single-scattering properties in the resonance region since it has a complex dependence on crystal geometry, crystal size, and complex refractive index; solutions based on electromagnetic theory should always be preferred in that region.

The retrievals of r_e based on the ice aggregate were found to be in better agreement with the lower end of the estimated cirrus cloud in situ measurements of r_e than retrievals based on the hexagonal ice column.