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Ice-crystal absorption: a comparison between theory and implications for remote sensing. Anthony J. Baran, John S. Foot, and David L. Mitchell. The problem of ...
Ice-crystal absorption: a comparison between theory and implications for remote sensing Anthony J. Baran, John S. Foot, and David L. Mitchell

The problem of the disagreement between cirrus crystal sizes determined remotely and by in situ measurements is shown to be due to inappropriate application of Mie theory. We retrieved the absorption optical depth at 8.3 and 11.1 mm from 11 tropical anvil cirrus clouds, using data from the High Resolution Infrared Radiation Sounder ~HIRS!. We related the absorption optical depth ratio between the two wavelengths to crystal size ~the size was defined in terms of the crystal median mass dimension! by assuming Mie theory applied to ice spheres and anomalous diffraction theory ~ADT! applied to hexagonal columns, hexagonal plates, bullet rosettes, and aggregates ~polycrystals!. The application of Mie theory to retrievals yielded crystal sizes approximately one third those obtained with ADT. The retrievals of crystal size by use of HIRS data are compared with measurements of habit and crystal size obtained from in situ measurements of tropical anvil cirrus particles. The results of the comparison show that ADT provides the more realistic retrieval. Moreover, we demonstrate that at infrared wavelengths retrieval of crystal size depends on assumed habit. The reason why Mie theory predicts smaller sizes than ADT is shown to result from particle geometry and enhanced absorption owing to the capture of photons from above the edge of the particle ~tunneling!. The contribution of particle geometry to absorption is three times greater than from tunneling, but this process enhances absorption by a further 35%. The complex angular momentum and T-matrix methods are used to show that the contribution to absorption by tunneling is diminished as the asphericity of spheroidal particles is increased. At an aspect ratio of 6 the contribution to the absorption that is due to tunneling is substantially reduced for oblate particles, whereas for prolate particles the tunneling contribution is reduced by 50% relative to the sphere. © 1998 Optical Society of America OCIS codes: 010.0010, 010.2940, 280.0280, 290.4020.

1. Introduction

To understand the influence of cirrus clouds on the Earth–atmosphere radiation balance it is important to determine and quantify cirrus properties. The cirrus properties to be quantified from the point of view of climate modeling are crystal dimension and shape, optical thickness, ice water path, and altitude.1– 4 Recent observations of high albedos from tropical anvil cirrus clouds5 indicate that these clouds have a major influence on the Earth– atmosphere radiation balance. However, there are few in situ measurements of anvil cirrus clouds be-

A. J. Baran is with the UK Meteorological Office, Satellite Soundings, London Road, Bracknell, Berks RG12, UK. J. S. Foot is with the UK Meteorological Research Flight, Farnborough, Hants GU14 0LX, UK. D. L. Mitchell is with the Desert Research Institute, Reno, Nevada 89506-0220. Received 1 December 1997. 0003-6935y98y122207-09$15.00y0 © 1998 Optical Society of America

cause of their high altitudes, which extend to some 17 km. In situ measurements of ice-crystal size distributions in tropical anvils were recently made as part of the Central Equatorial Pacific Experiment6 ~CEPEX! and the Stratosphere–Troposphere Exchange Project.7 The CEPEX campaign took place in 1993 at latitudes from 20 °S to 2 °N, near 180 °W. The observed temperature near the top of the cloud ranged from 264 to 260 °C. The mean of the maximum crystal dimensions measured during this campaign ranged from ;15 mm near the cloud top to more than 100 mm a few kilometers below the cloud top. Crystal sizes of less than 100 mm were measured with a video ice-particle sampler, as two-dimensional probes do not measure sizes less than ;66 mm.8 The Stratosphere–Troposphere Exchange Project campaign took place during 1987 near Darwin, Australia. Measurements of crystal size were made within ;1 km of the cloud top, with flight temperatures ranging from 278 to 287 °C, and the mean maximum crystal dimensions were 7–13 mm. Most crystal habits dur20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS

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ing the CEPEX were found to be columns and polycrystals ~otherwise known as sideplanes, rosettes, and aggregates thereof !, with little evidence of bullet rosettes. Crystal shapes were not determined during the Stratosphere–Troposphere Exchange Project Campaign. We compare our retrievals of crystal size with the CEPEX measurements because similar cloud-top temperatures were observed in our study and cloud temperature is often used to characterize particle shape and size.9,10 In situ measurements of crystal size are difficult to reconcile with retrieved crystal size based on infrared radiometric measurements made by either aircraftor satellite-based instrumentation. The interpretation of infrared radiometric measurements is usually based on Mie theory, and the reported crystal sizes are less than in situ measured sizes.11–15 For instance, aircraft-based radiometric measurements made at 11.17 and 12.38 mm show that retrieved crystal sizes are approximately one half to one third of the in situ measured crystal sizes.11 Retrievals based on measurements made by the High Resolution Interferometer Sounder12 ~HIS! at 8, 11, and 12 mm also indicated retrieved crystal sizes less than in situ measured sizes, again based on applying Mie theory to calculate absorption coefficients. More recent studies13 demonstrate that, if Mie theory is applied to ice spheres at wavelengths between 8 and 12 mm, the maximum crystal size that can be retrieved is ;30 mm. Similar results based on measurements from the HIS14 also show that for effective radii greater than 30 mm there is little spectral variation between 8 and 12 mm ~size parameter 18; the size parameter is defined as the ratio between particle circumference and wavelength!. Other studies at infrared wavelengths15 also showed that, when Mie theory is applied to radiometric observations, theory and observation cannot be reconciled. In the research reported in all the papers mentioned above, Mie theory was applied to equivalent ice spheres for calculation of the infrared absorption coefficient. Mie theory predicts that photons that have impact parameters ~the perpendicular distance of an incident photon from an axis going through the center of the sphere! greater than the radius of a sphere can be absorbed, in addition to photons that directly collide with the sphere. This process is known as tunneling.16 Tunneling can be interpreted by complex angular-momentum considerations as being responsible for large-angle diffraction and surface waves.16 An important consequence of tunneling is that the Mie absorption efficiency can exceed unity. Complex angular momentum ~CAM! theory can also provide a complete physical description of the Mie resonances and ripple fluctuations.17 It is still common practice to apply Mie theory to calculate the scattering and absorption properties of nonspherical particles. The most common types of crystal shape found in cirrus clouds consist of complex structures, making a numerical approach to scattering and absorption difficult. Inasmuch as nu2208

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merical approaches are computationally demanding, there is a need for approximate alternatives. There are several possible alternatives to Mie theory, such as anomalous diffraction theory18 ~ADT! and geometric ray tracing, or approximating nonspherical particles as spheroids. In ADT the absorption efficiency is completely determined by the geometrical dimensions of the particle and as such cannot exceed unity; in other words, there is no tunneling, and wave resonance phenomena and reflections are ignored. Comparisons19 made between ADT and other approximations indicate that ADT could provide an adequate description of scattering and absorption for nonspherical particles. Another possible alternative to Mie theory is ray tracing. Recent calculations20 show that geometric optics can be applied if the particle size parameters are greater than ;60. This application may be suitable for the treatment of solar radiation incident upon cirrus particles but not for terrestrial radiation. A further approach is to represent nonspherical particles as spheroids. It has been suggested21 that, for size parameters less than 60, radiative properties of cirrus clouds could be described by spheroidal particles. Spheroidal particles have been used in retrievals of cirrus properties by use of a spheroidal– hexagonal model22,23 ~in which the single-scattering properties of prolate spheroids ~aspect ratio, ;4! are combined with hexagonal ice crystals in the region where ray tracing is not applicable!. The spheroidal– hexagonal model was applied to observations22 made by the HIS of the brightness temperature differences between 8.35 and 11.1 mm. Theoretical calculations of the brightness temperature differences based on the spheroid– hexagonal model were found to be in better agreement with the observations than those from the spherical model, further highlighting the inadequacy of the sphere. More-recent calculations24 pointed out that at the wavelengths of 11 and 12 mm the extinction efficiency calculated from Mie theory and from a spheroidal model22 gave similar results in the resonance region ~size parameter, ;5!, the difference being ;5%. This result suggests that spheroidal particles do still support tunneling but to a lesser extent.24 The absorption properties of spheroids found by the T-matrix method25 are investigated further in Section 4. As well as the calculations cited above, there is laboratory evidence that suggests that spheroids also support tunneling.26 In contrast, scattering phase functions from fluffy fly-ash particles27 obtained in microwave experiments suggest that irregularly shaped particles do not support tunneling. Additionally, laboratory measurements of extinction efficiencies of hexagonal column ice crystals28 suggest the applicability of ADT at infrared wavelengths and hence the possible absence of tunneling,29 but the experimental uncertainties are significant. The ability of a particle to support tunneling and surface effects that arise from tunneling depends on surface geometry.16,30 Given that natural ice crystals are particles with sharp edges and no surface curvature,

tunneling on these objects is likely to be destroyed. Assuming spherical or spheroidal shapes in retrievals is likely to be invalid for real cirrus clouds. It is for this reason that we adopt a new radiation scheme, based on ADT,31 for retrieval of crystal size, using HIRS data in Section 7. 2. ADT Determination of Absorption Efficiency and Coefficients

The ADT absorption efficiency32,33 for randomly oriented particles is given by Qabs 5

1 P

S

*F

1 2 exp 2

DG

4pni d1 dP, l

(1)

where P is the projected area of the particle, ni is the imaginary index of refraction, l is the incident wavelength, and d1 is the path length straight through the particle for a given dP. The path length over which absorption occurs is the ratio of the particle volume divided by the projected area, which is called the effective distance de. Thus the absorption efficiency can be written in terms of de as

S

Qabs 5 1 2 exp

D

24pni de . l

(2)

*

`

PQabsn~D!dD.

(3)

0

The size distribution function n~D! is typically represented by an equation of the form n~D! 5 N0 Dn exp~2LD!.

(4)

The size distribution shape is defined by n; for n 5 0, then, n~D! is an exponential size distribution function, and this has been assumed in the rest of the paper. The parameter L defines the size distribution slope, and this is related to the mean maximum # by dimension33 D # 5 1yL. D

tabs 5 babsZ,

(6)

where Z is the geometric depth of the cloud. Thus the absorption optical depth ratio depends on the particle shape and size and on the assumed particle size distribution function. In this paper we report our results in terms of the median mass dimension Dm, as CEPEX crystal sizes are reported in this form. The median mass dimension36 is defined as the particle size that divides the mass of a size distribution into equal parts. The median mass dimension can be found analytically36: Dm 5

b 1 0.67 . L

(7)

To compare the results with the CEPEX, we converted model-predicted mean dimensions to median mass dimensions, using Eq. ~7!, where the b values for various crystal shapes were previously calculated.31 It is noteworthy that median mass dimensions are typically two to three times larger than mean dimensions, depending on the form of the ice-crystal size distribution. 3. Complex Angular-Momentum Theory

Because for convex particles in random orientation P is given by Sy4, where S is the particle surface area,18,34 de can be expressed as ~4V!yS, where V is the volume of the particle. For a given volume a sphere will have the largest de because S is a minimum for the sphere. Typical aspect ratios of hexagonal columns and plates are ;6,35 the aspect ratio being defined as the ratio of the major to the minor axis. It can be shown that these crystals in random orientation have de values that are 0.5– 0.3 those of spheres with equal projected areas. Similar results are obtained for spheroids with the same aspect ratio. The absorption coefficient babs calculated over a size distribution n~D!, where D is the maximum dimension of the particle, is given by babs 5

Given Eq. ~3!, the absorption optical depth is found from

(5)

In CAM the Mie partial-wave terms18 are transformed into the complex angular-momentum plane.16 The purpose of this transformation is to find the major asymptotic contributions to the Mie efficiencies that permit rapid convergence for the absorption and scattering efficiencies while maintaining accuracy.37 Another advantage is that, unlike for the Mie partialwave series, the major physical contributions to the absorption and scattering efficiencies can be understood in terms of edge effects16,18 and geometric optics ~Fresnel reflection, transmission, and multiple internal reflections!. It was previously demonstrated37 that the average CAM absorption efficiency can be represented by three terms: ^Qabs& 5 ^Qabs&F 1 ^Qabs&ae 1 ^Qabs&be.

(8)

The terms on the right-hand side of Eq. ~8! are now briefly described. The first term represents the geometric optics,18 complex angles of refraction, and complex shortcuts through the sphere.37 The other two terms are the above-edge ~ae! and the below-edge ~be! contributions. These terms represent the dynamic effects of surface curvature on diffraction and Fresnel reflection. The large-angle diffraction as previously described in Section 1 is dominated by tunneling, and it is this process that is the major contribution to the above-edge term16 and causes the absorption efficiency to exceed unity. In the calculations that follow, we use real ~nr! and imaginary ~ni ! refractive indices of 1.3 and 0.1, respectively, to calculate Qabs. This choice of real and imaginary refractive index is typical for ice at thermal wavelengths. Figure 1 compares absorption ef20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS

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Fig. 1. Absorption efficiency Qabs plotted against de calculated from CAM and ADT. The geometric optics contribution ~GO! is shown by the crosses, and the edge ~above and below! and the above-edge contributions are shown by the diamonds and the asterisks, respectively. The straight line indicates Qabs 5 1.

ficiencies of a sphere derived from CAM ~including the various contributions! and ADT plotted against the effective distance. The solution given by CAM gives absorption efficiencies roughly 35% greater than those from ADT. Studying the various CAM contributions makes clear that ADT and ^Qabs&F agree closely, particularly for de . 80 mm. The edge terms are dominated by the above-edge term ~i.e., ^Qabs&ae is much greater than ^Qabs&be!. The CAM solution ~which will be almost identical to the Mie solution37! gives Qabs . 1 owing to the contribution of ^Qabs&ae.

Fig. 2. Calculations of absorption efficiency as a function of de from CAM, T matrix ~curves a– c! and ADT. The T-matrix calculations are for ~a! oblate and ~b! prolate spheroids with aspect ratios R of 2 ~curve a!, 4 ~curve b!, and 6 ~curve c!. The straight line indicates Qabs 5 1.

4. Comparison of CAM, T-Matrix Method, and ADT Solutions of Qabs

The T-matrix method25 is used to study the absorption efficiencies of oblate and prolate spheroids, as this can provide some insight into how real ice columns or plates may behave. Figure 2 shows the results of these calculations for randomly oriented oblate and prolate spheroids with aspect ratios R 5 2, 4, 6. Also plotted are the absorption efficiencies calculated from CAM and ADT. As the value of R increases, the absorption efficiency is progressively reduced from values close to those of CAM toward, but greater than, the value predicted from ADT. In the case of the oblate spheroid at R 5 6 the T-matrix calculations becomes close to the ADT value, particularly for de . 80 mm. In the case of the prolate spheroid at R 5 6 the T-matrix solution lies roughly halfway between the CAM and the ADT results. The implication from the results of Figs. 1 and 2 is that for oblate spheroids ~R 5 6! the contribution from the above-edge term becomes insignificant, i.e., the tunneling process is suppressed. For prolate spheroids at the same aspect ratio the implication is that the tunneling contribution is reduced to roughly 50% of the contribution from a sphere with equal de. 2210

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This hypothesis is restated in Fig. 3, which shows again the T-matrix solutions for oblate and prolate spheroids ~R 5 6! and the ADT results. Added to this figure are cross points that show that the T-matrix results can be quite accurately fitted from the CAM contributions. For the oblate spheroid the points are from ^Qabs&F only, and for the prolate spheroid the points are from ^Qabs&F 1 0.5~^Qabs&ae!. We can infer from these results that oblate shapes such as hexagonal plates are unlikely to sustain significant tunneling, and therefore ADT may provide the best estimate of the absorption efficiency of such particles. For prolate shapes such as hexagonal columns tunneling processes are likely to be significantly reduced compared with those in a sphere of equal de. Consideration of the processes that give rise to tunneling16 might further suggest that because tunneling is associated with the interaction between the wave and surface curvature and real ice crystals will not have the same area of curved surface as spheroids, tunneling may become reduced further than 50%. Therefore the absorption efficiency of

Fig. 3. Calculation of absorption efficiency as a function of de for ~a! oblate spheroids and ~b! prolate spheroids, both with an aspect ratio of 6. The crosses represent the result of adding ~a! the geometric optics contribution and ~b! the geometric optics plus 50% of the above-edge contribution from the sphere to ADT.

real ice crystals may well be accurately represented by considerations of ADT. 5. Use of an 8.3-mm Channel

By way of a practical example we show how one can use data from the HIRS 8.3-mm channel to test absorption theory. The imaginary part of the index of refraction for ice has a minimum at 8.3 mm, which is why this channel in conjunction with others is particularly valuable in determining crystal sizes. This fact is illustrated in Fig. 4, which shows predicted emissivity at 8.3- and 11.1-mm HIRS channels under various assumptions with the appropriate refractive index.38 First, Fig. 4~a! shows little sensitivity to particle size when Mie theory is applied to ice spheres. However, when tunneling is removed the measurement becomes more sensitive to large particles, as illustrated in Fig. 4~b!. Figure 4~c! shows a further increase in sensitivity to larger particles when more-realistic crystal shapes are considered. Figure 4~c! suggests a sensitivity to crystal size of as

Fig. 4. Emissivity at 8.3 and 11.1 mm with Mie theory and ADT applied for mean maximum dimensions of 10, 20, 30, 50, 100, and 200 mm. ~a! Mie theory applied to ice spheres ~i.e., including tunneling!, ~b! ADT applied to ice spheres ~excluding tunneling!, ~c! ADT applied to polycrystals.

much as a mean size of 100 mm between the 8.3- and 11.1-mm channels. 6. Radiative Transfer and Retrieval

The monochromatic equation of radiative transfer for the case of a plane-parallel homogeneous, isothermal cloud at thermal wavelengths is m

dI~t, m, u! 5 I~t, m, u! 2 S~t, m, u!. dt

20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS

(9) 2211

For the given boundary conditions of upwelling isotropic radiation and no downwelling radiation from the top of the atmosphere, and assuming no midlevel cloud, the general solution to Eq. ~9! is Il~0, m, u! 5 Ilclear~t, m, u!exp@2~tym!# 1

*

t

exp@2~tym!#Sl~t, m, u!

0

dt , m

(10)

where Il is the upwelling radiances from the cloud, Ilclear is the upwelling radiance from the surface–lower atmosphere, t is the optical depth, and m is the cosine ~from nadir! of the view angle. The term u is the azimuth angle and Sl is the thermal source function, at some vertical optical depth t, within the cloud. In this paper we ignore scattering. The importance of scattering to the brightness temperature difference between 8.3 and 11.1 mm was investigated with a doubling-adding radiative transfer model39 and shown to be negligible over the range of HIRS scan angles of 20°–57° used in this study. Because scattering can be ignored, Sl 5 Bl~Tc! in Eq. ~10!, and the optical depth is the absorption optical depth, tabs. This simplification yields the radiance received at the satellite detector, which is

S DF

Il 5 Ilclear exp 2

S DG

tabs tabs 1 1 2 exp 2 Bl~Tc!, m m

(11)

where the two terms on the right-hand side of Eq. ~11! are the transmission and emission terms. From Eq. ~11! if the clear scene and cloud-top temperature can be determined then tabs can be determined uniquely as

F

tabs 5 2m ln

G

Bl~Tc! 2 Il . Bl~Tc! 2 Ilclear

(12)

We can use Eq. ~12! to determine the absorption optical depth at the HIRS wavelengths at 8.3 and 11.1 mm, and we can use the resulting optical depth ratio ~tabs8.3ytabs11.1! to determine crystal size from Eq. ~3!. We next apply the methodologies outlined in Sections 2 and 7 to retrieve crystal size, assuming Mie theory and ADT, from tropical anvils. 7. Retrieval of Cirrus Crystal Size and Shape from the HIRS 8.3- and 11.1-mm Data A.

Cirrus Case Description

The data presented in this paper were taken from the HIRS instrument on board NOAA-10, a U.S. National Oceanographic and Atmospheric Administration polar-orbiting satellite. Eleven tropical anvils were chosen for this study, which was made from 20 °S to 20 °N latitude during the period 27 January to 10 February 1991. Most of the anvils presented in this study were confined to either the East or the West-Central equatorial Pacific Ocean ~9 cases were from the East and West Pacific, and 2 were from the Indian Ocean!, with no cases located in the central tropical Atlantic Ocean region. 2212

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The anvils were identified against a standard of an 11.1-mm channel brightness temperature; if the temperature was less than 215 K, implying little transmission, the case was deemed most likely to be an anvil. The HIRS 11.1-mm channel is a window channel that is slightly affected by water-vapor emission; the HIRS 8.3-mm channel is, however, a watervapor channel. The 8.3-mm channel receives the majority of its signal from water vapor in the lowest 2 km of the atmosphere and from the surface. For this study the HIRS 8.3-mm channel is effectively a window channel because the tropical cirrus layer is located near the tropopause well above significant amounts of water vapor. B. Estimation of Cloud-Top Temperature and Cloud-Free Brightness Temperature

To solve Eq. ~12! we need to estimate two variables, namely, the cloud-top temperature ~Tc! and the cloud-free upwelling brightness temperature ~Tclear!. To estimate Tc we use an already established method.40 When the brightness temperature difference between the 3.7- and the 11.1-mm channels is zero, the cloud is opaque and the corresponding 11.1-mm brightness temperature is the cloud-top temperature. In this method the range in Tc varied from 276 to 258 °C. Additionally, ship radiosonde balloons were used to confirm the temperatures near the tropopause for some of the cases. These independent temperature profiles confirm our own estimates of the cloud-top temperatures situated near the tropopause. We found cloud-free conditions by using the window channel at 3.7 mm. If this temperature was within 3 °C of the sea surface temperature climatology, the spot was deemed to be free of cloud. The sea surface temperature climatology consists of a monthly 1° resolution for the period 1951–1980 obtained from the Hadley Centre Sea Surface Temperature Archive.41 Having obtained a cloud-free spot, one then uses the cloud-free brightness temperatures as measured by the 8.3- and 11.1-mm channels to determine Tclear. The optical depth for each HIRS spot at 8.3 and 11.1 mm can now be calculated and their ratio obtained to estimate crystal size under the assumption of various crystal habits. In this retrieval we obtain the optical depths around the coldest spot, as we are then confident that the HIRS field of view is completely filled with cloud and that changes in measured Tc are due to changing microphysical conditions rather than to changes in cloud height. C. Crystal Size and Habit Determination: with the CEPEX

Comparison

The Dm values retrieved with Eq. ~7! are shown in Fig. 5 for all the cases, with Dm plotted against the 11.1-mm brightness temperature. The dashed lines in Fig. 5 represent the range ~50 –160 mm! in measured in situ Dm obtained from the CEPEX field studies. This range is taken from the deepest cloud penetration ~;14 km!, which took place on 1 April

the CEPEX, whereas those based on plates and bullet rosettes ~in particular! have a large proportion of high values of retrieved Dm outside the in situ range. The predominant crystal types observed near cloud top during the CEPEX6 were columns and polycrystals, consistent with the results in Fig. 5. Repeating the retrievals given in Fig. 5 but increasing the ADT absorption efficiencies by 35% to represent the tunneling contribution that Mie theory would demand on the basis of an equivalent sphere will reduce all the ADT retrieved sizes. The result of this assumption moves a proportion of the polycrystal and column retrievals below the lower range observed in the CEPEX and would suggest that plates provide the best agreement with the in situ data. Plates, however, were not observed at cloud top during the CEPEX, which therefore is evidence that ADT does provide the best fit to the in situ observations. These satellite-derived measurements adopting ADT are therefore in broad agreement with the in situ measurements made during the CEPEX. 8. Concluding Remarks

Fig. 5. Retrievals of crystal mass dimension ~Dm! plotted against measured brightness temperature, assuming different habits and treatments of absorption. Dashed lines, the CEPEX in situ measurements of mass dimension. Mie theory applied to ice spheres ~triangles!; ADT applied to columns ~squares!, to polycrystals ~diamonds!, to plates ~pluses!, and to bullet rosettes ~crosses!.

1993, and represents the range in size that is likely to be penetrated by the 8.3-mm channel. ~We estimated penetration depth of the 8.3-mm channel by using Beer’s law to determine the exponential attenuation of downwelling radiation.! The range in Fig. 5 is hereafter referred to as the CEPEX range. Comparisons between the retrieved Dm for different habits ~from ADT! and for ice spheres ~from Mie theory! are also shown in Fig. 5. Figure 5 shows that there are significant differences in retrieved Dm values between Mie theory applied to ice spheres and ADT applied to hexagonal columns. Mie theory applied to ice spheres occupies mostly the size range 20 – 40 mm, with all the points lying well below the CEPEX range. It is clear from Fig. 5 that Mie theory applied to ice spheres cannot fit the CEPEX measurements, as explained in Section 4. Retrieved values are almost a factor 3 less than the in situ measurements reported from the CEPEX; this factor is similar to that for other in situ measurements cited in Section 1. By contrast, ADT applied to hexagonal columns leads to retrievals of Dm that occupy mostly the lower half of the CEPEX range. The mean retrieved Dm value for hexagonal columns determined by ADT is 60 mm, compared with 22 mm for Mie theory applied to ice spheres. Retrievals based on polycrystals, plates, and bullet rosettes are also shown in Fig. 5. Results based on polycrystals also lie within the range observed during

We have demonstrated that Mie theory is an inappropriate theory to use in calculating absorption of cirrus particles in the thermal infrared. This conclusion is based both on exact numerical calculations using the T-matrix method applied to spheroids and on comparison of retrievals of crystal size by use of the HIRS 8.3- and 11.1-mm data with representative in situ data. The numerical calculations were based on the spheres’ being deformed into oblate and prolate spheroids so the resulting changes in Qabs could be studied. The calculations showed that for oblate spheroids with an aspect ratio R 5 6 the tunneling contributions become negligible and that differences between ADT and the T-matrix method could be attributed to geometric optics. However, for prolate spheroids ~R 5 6! the geometric optics contribution alone could not explain differences between the T-matrix method and ADT; agreement could be found by inclusion of 50% of the CAM-derived tunneling term. Here we emphasize that ADT does not include tunneling and associated resonance phenomena, and it also ignores reflection. These comparisons demonstrate that for particles with different surface curvatures tunneling can be either negligible or significantly reduced. Because real ice crystals are sharp-edged objects with little or no surface curvature, we consider it reasonable to suggest that tunneling is unlikely to contribute significantly to absorption of such objects. To test this theory we compared available in situ data with retrievals of ice-crystal size based on Mie theory ~applied to ice spheres! and ADT ~applied to different ice-crystal habits! and compared the retrievals with available in situ data. We found that Mie theory applied to ice spheres leads to retrieved sizes that are approximately one third smaller than observed, whereas ADT applied to columns and polycrystals is representative of the CEPEX measurements. These crystal types were the predominant types ob20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS

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served during the CEPEX. It is possible to reach a different conclusion if, for example, we postulate an enhanced absorption that is due to tunneling and this extra contribution added to ADT. This process leads to retrievals based on plates’ providing the best agreement with the in situ sizes, but plates were not observed during the CEPEX. We have demonstrated that the spectral region 8 –12 mm contains information from which microphysical properties of cirrus clouds can be obtained. The results not only are important for remote sensing but also contribute to determination of how the emissivity of cirrus clouds should be calculated in climate models. Support is acknowledged from the UK Department of the Environment for partial funding this research under contract PECDy95. D. Mitchell is funded by the Environmental Sciences Division, U.S. Department of Energy, under grant DE-FG03-94ER61775 as part of the Atmospheric Radiation Measurement Program. Financial support does not constitute an endorsement by these agencies of the views expressed in this paper. We thank M. Mishchenko for the use of his T-matrix code, which A. J. Baran adapted for execution on the Met Office C-90 Cray computer, and P. Francis for radiative transfer calculations. We also thank W. Wiscombe for his CAM codes, which A. J. Baran adapted to the C-90 Cray. Thanks are also due to the reviewer whose challenging comments have greatly improved the scope and content of the paper.

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