In order to learn more about the Martian polar caps, it is important to compare and contrast the behavior of both frozen H2O and CO2 in different parts of the electromagnetic spectrum. Because relatively little attention has been given, thus far, to observing the seasonal Martian polar caps in the thermal microwave part of the spectrum, in this experiment, a 35-GHz handheld radiometer was used to measure the microwave emission and scattering from layers of manufactured CO2 (dry ice). Compared to natural snow crystals, results for the dry ice layers exhibit lower microwave brightness temperatures for similar thicknesses, regardless of the incidence angle of the radiometer. For example, at 50° H (horizontal polarization) and with a covering 18 cm of dry ice, the brightness temperature was 76 K. When the total thickness of the dry ice was 27 cm, the brightness temperature was 86 K. The lower brightness temperatures are due to a combination of the lower physical temperature and the larger crystal sizes of the commercial CO2 crystals compared to the snow crystals. While little is known about the CO2 and water snowpacks on Mars, it is likely that the particles are in close contact with one another as is the case for ice sheets on Earth – the grains are interconnected. This would qualify as a dense media. In densely packed media, the particles do not scatter independently; rather, they interact with other particles. Dense media calculations compare very favorably with the observed values from the handheld radiometer. The calculated versus observed TBs are within 10% for each case with the exception of the 26 cm layer thickness and the 0.8 mm particle size (15%) and for the 14 cm layer thickness and the 2.0 mm particle size (16%). Thus, it appears that dense media transfer modeling (DMRT) will be useful for modeling the flow of energy emerging from frozen CO2 deposits. Because the dry ice used in this experiment was manufactured in the shape of cylindrical pellets, an effort was made to see what effect, if any, the shape of the crystal, for different particle sizes, has on microwave scattering. Results from a discrete dipole scattering (DDSCAT) model show that differently shaped crystals, having the same effective radius of a sphere, give very similar cross section/efficiencies.
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 The microwave region contributes little to the total radiation budget of Earth or Mars, compared to the ultraviolet, visible and infrared wavelengths. However, because ice crystals appreciably scatter and absorb (depending on the crystal size) upwelling microwave radiation emanating from the Earth at frequencies above about 10 GHz, microwave radiometry offers the potential to assess the thickness and the extent of the Martian seasonal caps using remote sensing techniques. In terms of remotely sensing the Martian seasonal and permanent ice caps, relatively little attention has been thus far given to observing the thermal microwave part of the spectrum. An advantage of using this approach is that microwaves are indifferent to daylight and darkness. Therefore, the thickness and extent of the caps can be estimated even during the polar night period. [Foster et al., 1998]. This 1998 paper laid the groundwork for the present study.
 Although much of what is known about the composition and structure of the Martian polar caps is a result of laboratory work and modeling, in the microwave region of the spectrum, there is a need to conduct basic experiments related to how microwaves are scattered and or absorbed by accumulations of CO2 crystals having various sizes. A problem, of course, with experimental measurements, is how to make them under conditions, which are analogous to the conditions expected on Mars. Otherwise, the results may not fully explain what is observable on Mars. Nonetheless, initial experiments with preliminary findings are useful for helping to design further experiments and to validate modeling results.
 The purpose of this paper is to measure the passive microwave brightness temperatures of frozen CO2 (dry ice) at 35 GHz (∼0.8 cm), using a handheld radiometer. The CO2 brightness temperatures will be modeled using dense media radiative transfer theory and compared to the radiometric observations. A discrete dipole scattering model will be employed to assess what affect the shape of the CO2 crystal, for a range of particle sizes, has on microwave scattering. Unfortunately, there are no direct (in situ) Viking Lander measurements of the Martian polar caps, and there are few orbital measurements (either from the Mariner, Mars, Phobos or even Pathfinder missions), that can be used as a standard of reference for comparison with the laboratory measurements of CO2 crystals and the modeling results on CO2 extinction efficiency described in this paper. However, the measured response from dry ice can be compared with the modeled results to assess whether or not the model can be used to accurately gage the extinction of CO2 and H2O crystals having different sizes.
2. Passive Microwave Radiometry
 At millimeter wavelengths or microwave frequencies, the incident energy passes into and through snow layers. Therefore, the grain size and penetration depth become important considerations in the scattering equations. This makes the microwave portion of the spectrum well suited for exploring crystal or grain structure in snowpacks and even deposits of CO2.
 Microwave emission from a snow (or CO2 layer) over a ground medium consists of contributions from the snow itself and from the underlying ground. Both contributions are governed by the transmission and reflection properties of the air-snow and snow-ground boundaries and by the absorption/emission and scattering properties of the snow layers. If the snowpack or CO2 deposit is thick (greater than penetration depth of the wavelength), then it may be treated as a semi-infinite medium and contributions from the ground will not be as important [Chang et al., 1976].
 As an electromagnetic wave emitted from the underlying ground propagates through a snowpack or CO2 deposits, it is scattered by the randomly spaced particles in all directions. Consequently, when the wave emerges at the snow/air interface, its amplitude has been attenuated, and thus the brightness temperature is low. Dry snow absorbs very little microwave energy, and therefore it contributes very little in the form of self-emission [Ulaby and Stiles, 1980; Foster et al., 1984]. For snowpacks on Earth, snow crystals are effective scatterers of microwave energy for frequencies greater than about 10 GHz. The snow crystals redistribute part of the cold sky radiation, which reduces the upwelling radiation measured with a radiometer [Schmugge, 1980]. The deeper the snow, the more snow crystals are available to scatter the upwelling microwave energy, and thus brightness temperatures are lower.
 Some success has been achieved in developing algorithms that utilize one or more microwave frequencies to derive snowpack thickness. For instance, the difference in brightness temperature between the 37 GHz channel and the 18 GHz channels on the Scanning Multi-channel Microwave Radiometer (SMMR) instrument, on board the Nimbus–7 satellite, has been used to derive snow depth from a uniform snow field.
where SD is snow depth in centimeters and C is a coefficient related to the average crystal size.
 The path length of the radiating energy is related to how closely packed the particles are to each other (density), and is therefore important in terms of a mediums ability to scatter radiation. While field measurements of snow density are routinely made, this parameter cannot be extracted using remote sensing technology. A representative value of 300 kg m−3 is typically assumed in developing algorithms for midlatitude snowpacks in midwinter [Foster et al., 1998]. This value varies in response to the snow water equivalent of the snow, and thus it can change (increase) even if the depth of the snowpack remains fairly constant. Very little information is available about possible H2O snow densities on Mars.
 For CO2 snowpacks, a density value of between 1060 and 1070 kg m−3 has been estimated by a number of authors, including Yamada and Person  and Tsujimoto et al. . Presumably, CO2 snow can also be fluffy in nature (much lower density), but no representative values are known. Keiffer  states that the seasonal winter polar caps of Mars are mainly CO2 grains. G. B. Hansen (personal communication) believes that in some areas of the seasonal caps, CO2 grain sizes on the order of 1 cm (radius) or larger may be likely, and thus the density could be about 1560 kg m −3, near the bulk density of CO2.
 Large snow crystals are especially effective scatters of microwave energy [Hall et al., 1986; Armstrong et al., 1993]. Foster et al.  have shown that for snow, the shape of the crystal is insignificant (for crystals that are smaller than the wavelength), in comparison to the size of the crystal and the spacing between the crystals, in scattering the microwave radiation emanating from the ground and passing through the snowpack. In equation (1), “C” will be smaller with a larger crystal size. For example, if the average crystal radius is 0.3 mm, C is 1.59, and if the radius is 0.5 mm, C is 0.39 [Foster et al., 1998].
 For H2O ice, the complex index of refraction has been measured by Chang et al.  and Warren  to be approximately 1.78 for the real part and 0.0024 for the imaginary part at 35 GHz and for temperatures of about −10°C. The emissivity of the snow and the underlying frozen soil is very close to 0.98 at a frequency of 35 GHz [Ulaby et al., 1981; Foster et al., 1984].
 Only a few measurements are available of either the dielectric or the refractive index for frozen CO2, and none near 35 GHZ. Because no Debye relaxation absorption is expected in the microwave region, the imaginary index should be very low since CO2 is not a polar molecule. At 1000 GHz, Hansen  gets a real value of 1.444 and an imaginary value of 0.0048 for solid CO2. At lower frequencies, Simpson et al.  obtained a dielectric constant of 2.25 for solid CO2 in the frequency range between 2.2 and 12 GHz, for a density of 1400 kg m −3 and for temperatures between 113 and 183 K. Note that this density is between the densities given for CO2 earlier in this section. Estimating an uncertainty of about 10% in their value for the dielectric constant, gives a refractive index of 1.5 (+ or − 0.1). The loss tangent, represented by the imaginary part of the refractive index, is listed as <0.004 throughout this same frequency range. For temperatures greater than 77 K, Warren  showed that, away from the absorptive bands, the refractive index varies only from 1.40 at 1 μm to 1.44 at microwave frequencies. Based on the above, for this investigation, we have selected a value of 1.44 for the real and 0.005 for the imaginary part of the refractive index.
 The emissivity of CO2, between 6 and 2,000 microns, varies from between 0.7 to 0.95, using plausible estimates of grain size and contamination (G. B. Hansen, personal communication). Hansen  argues that the reason the observed emissivities are low (in comparison to water ice and dust, for instance) for many areas of the Martian seasonal caps is attributable to the differences in CO2 grain sizes as well as the unusual optical properties of CO2 ice – determined by the complex index of refraction. However, as the CO2 grain size increases, the emissivity also increases and is similar to that for water-ice mixtures.
 Thirty-five gigahertz may not necessarily be the optimum frequency to be used for deriving thicknesses of solid CO2. However, since passive microwave radiation at 35 GHz has been successfully used to derive snowpack thickness, with snow crystals that typically range in size between 100 microns and 0.5 cm, and since it is likely that the CO2 grains on Mars are near this size range, between 10 microns and 1.0 cm, 35 GHz is a reasonable starting point for conducting these experiments.
 An out-of-doors site was deemed necessary for this experiment because the numerous thermal emission sources in an indoor cold laboratory (walls, tables, etc.) would contaminate the microwave measurements. Outside radiation sources could possibly also affect the radiometer readings, such as a heavy rainfall or over hanging trees; however, to ensure that this would not be the case, measurements were made in the open and on days when no precipitation was falling. In the 35-GHz region of the spectrum, the atmosphereic radiation contributes only minutely to TB – very little emission from water vapor. Readings made from the radiometer of the sky taken during the experiments were on the order of 40 K.
 Even if there is negligible scattering, there will still be some extraneous reflections. For example, there will be a small amount of reflected atmospheric emission from the surface/air boundary as well as reflected emission from the atmosphere that penetrates the snow and dry ice where it is then reflected backward off of the aluminum sheet (purpose of aluminum described later in this section). Furthermore, as mentioned above, dry ice is a weak emitter.
 To conduct these experiments, approximately 400 lbs of dry ice pellets were required – purchased from a commercial supplier. This type of commercially available frozen CO2 is produced by compressing and then rapidly expanding CO2 gas. Liquid CO2 is allowed to expand by reducing its pressure to sea level atmospheric pressure (∼1013 mb). This spontaneously converts the liquid to both a gas and a solid. If the expansion occurs in a cold chamber, the snow, which represents approximately 40% of the liquid conversion, can be compacted to conform to the chamber shape and size. The most common forms of manufactured dry ice are pellets and solid blocks.
 A 1 m by 2 m plate of aluminum sheet metal was positioned on the ground so that microwave emissions from the underlying soil layers would be minimized, and a 35-GHz handheld radiometer was then used to make measurements of the CO2 pellets (deposited on the plate). Only a 35-GHz radiometer was available for this study. Measurement units are in volts, which are converted to brightness temperatures by simply multiplying recorded volts on the voltimeter by 50. The radiometer measures only passive microwave radiation. The bandwidth is 600 MHz, and the temperature sensitivity is 1.5 K. The beam width is approximately 10 degrees. At a height above the surface of 1 m, the standard distance used in these experiments, the footprint size is about 0.025 m2. The incidence angle was controlled by attaching an inclinometer to the radiometer. By rotating the instrument 90 degrees, both horizontal and vertical polarizations can be obtained.
 The experiments were conducted in Beltsville, Maryland, in February 1999 and in February 2000. Patches of wet snow were on the ground during the 2000 experiment, but there was no snow at all in 1999, therefore dry ice measurements were not compared with natural snow measurements. For each of these experiments, passive microwave radiation emanating from within layers of manufactured CO2 pellets was measured with a 35-GHz handheld radiometer. Both large (0.8–1.0 cm in length) and small (0.4–0.6 cm in length) cylindrical-shaped dry ice pellets (Figures 1a and 1b), at a temperature of 197 K (−76 degrees C), were measured. Nonabsorbing foam, having negligible emissivity, was positioned around the sides of the plate in order to keep the dry ice in place and to assure that the incremental deposits were level (Figure 2). The passive beam did not directly intercept the foam. The radiometer was actually handheld rather than mounted on a tripod for this experiment since it proved difficult to correctly position the tripod over the foam panels.
 Thirty-five-gigahertz measurements of the plate were made through the dry ice deposits in the following way. Layers of dry ice were built up and measurements were repeated for the increasing CO2 pack. First, 7 cm of large CO2 pellets were poured onto the sheet metal plate, then an additional 7 cm were added, and finally, 12 cm were added on top of the 14 cm base. Measurements were made each time the thickness of the deposit was increased. The same process was repeated for the smaller grain pellets. Additionally, in February of 2000, measurements were taken of a 25 kg (27 cm × 27 cm × 27 cm) solid cube of CO2, which was cut in half and then remeasured. After the above described series of measurements were made, the CO2 pellets were then placed on top of a melting snowpack (the temperature of the snow was 0°C), and measurements were made with the radiometer. As a final part of this experiment, soil particles were spread on top of the dry ice, and once again, microwave measurements were made with the radiometer.
 As mentioned previously, a radiometer will measure the weakly emitted radiation from snow crystals as well as sky reflection. If there is no scattering from the media being measured, then the brightness temperature (TB) is equal to the emissivity multiplied by the physical temperature of the object. Emissivity can be calculated once the dielectric constant is known. With scattering, as is the case of a media such as snow, the TB is reduced because of the cold sky contributions (reflection) to the snow. The sky contributions need to be accounted for when determining the TB of the snow or carbon dioxide pellets.
4. Results and Discussion
 The 1999 and 2000 experiments demonstrate the effects that different dry ice pellet sizes and varying pellet thicknesses have on the 35-GHz TB (Figure 3). The larger dry ice pellets (triangles on Figure 3) typically have slightly higher TBs than do the smaller-sized pellets (circles on Figure 3). For instance, for a thickness of 8 cm, the 0.8 cm diameter pellets have a TB of approximately 105, whereas the 0.4 cm diameter pellets have a TB of about 95. The size of the crystal affects scattering considerably more than it does absorption, at a frequency of 35 GHz, for both CO2 and H2O crystals. This is classic Mie Theory – scattering increases as the particle size approaches the size of the wavelength. Note that on Figure 3, because of the aluminum plate at the bottom of the dry ice deposits, there is an inverse relationship between thickness and scattering.
 Because the aluminum plate blocks emission from the ground, and the cold sky is reflected from the plate, smaller thicknesses of dry ice will actually have lower brightness temperatures. Thus, it is observed that the TB of the dry ice gradually increases with increasing thickness. For instance, from 7 cm to 27 cm (Figure 3 – 1999, small dry ice pellets) the TB increases by 35 K, and in 2000 the TB for similarly sized pellets increased by about 40 K. The low temperature of the dry ice is largely responsible for the low TBs. Volume scattering by the dry ice crystals would not result in such low TBs, regardless of the thickness.
 When the smaller and larger pellets were added together to produce an accumulation of 37 cm, the resulting TB was actually lower than that measured for either the smaller or larger pellets having a thickness of only 26 cm. Why this is the case is not known for sure, but it may be due to a combination of the large particles and increased thickness of the deposit acting in unison to cause the brightness temperatures to saturate. At this thickness, the 35-GHz sensor is no longer sufficiently sensitive to volume scattering, and the deposit behaves more like a dense media (see sections 4.1 and 4.3).
 For the solid block of dry ice, there was no increase in scattering when the thickness of the block was doubled. The term solid may be misleading. Actually, the individual CO2 grains were compressed to form a packed powder. At a thickness of 10 cm, the brightness temperature was comparable to that of the grains (Figure 3). The thicker dry ice cube actually showed a slight decrease in scattering, perhaps because the compressed ice was emitting more radiation than it was scattering.
 Finally, in our admittedly rather feeble attempt to simulate a sullied Martian snowpack, a thin layer of soil particles (approximately 2 mm thick) was scattered on top of the heap of CO2 crystals. In the microwave portion of the spectrum, dry soil has a high emissivity and refractive index compared to that of snow. The volume of snow crystals (whether composed of H2O or CO2), and the resulting scattering of microwave energy, overwhelms the emission from the thin layer of soil particles added to the top of the snowpack.
4.1. Radiation Transfer Modeling
 Radiative transfer theory describes the transport of electromagnetic energy of a specific intensity through a dielectric medium. Snow is but one such medium. The microwave radiation emitted by a covering of H2O or CO2 snow is dependent upon the physical temperature, crystal characteristics and the density of the snow. A basic relationship between these properties and the emitted radiation can be derived by using the radiative transfer approach [Chandrasekhar, 1960; England, 1975; Chang et al., 1976; Kong et al., 1979]. The lack of precise information about crystal size, shape and the snowpack density is compensated for by using averages for these parameters, based on field and laboratory observations. For computational purposes, assumptions are made that the averages are representative of conditions encountered throughout the snowpack. These quantities are then used as input to radiative transfer equations to solve the energy transfer through the snow covering [Foster et al., 1998].
 While little is known about the CO2 and water snowpacks on Mars, it is likely that the particles are in close contact with one another as is the case for ice sheets on Earth – the grains are interconnected. This would qualify as a dense media. In densely packed media, the particles do not scatter independently; rather, they interact with other particles. When the dielectric contrast between the scatterers and the background is small, a weak scattering assumption can be applied to solve the multiple scattering problem in a dense medium [West, 1994]. However, for nontenuous scatterers, this assumption is no longer valid, and dense medium radiative transfer theory (DMRT) must be used.
 Volume scattering models fall into two groups, incoherent and coherent methods. In incoherent volume scattering theories, such as radiative transfer, the effect of the phases of the fields scattered between neighboring particles can be ignored. So, single scattering properties of constituent particles are used to determine the volume scattering. This type of approach is often applied to nondense media such as newly fallen snow. However, when the density of the medium increases, so does the complexity of volume scattering analysis. As the scattering density increases, multiple scattering between particles is more important, and the coherent approach becomes especially complex [Siqueira, 1996].
 When calculating wave propagation and scattering in dense media, the correlation between particles has to be taken into account [Tsang and Ishimaru, 1985]. For dense media radiative wave theory, phase coherence between scatters is accounted for by using the analytic quasi-crystaline approximation. This approximation accounts for phase coherence between scatters on the microscopic scale, and radiative transfer is used to model the behavior of energy transfer on the macroscopic scale. Using this approach, calculations have been made of the radiation transfer in and through CO2 deposits having different-sized particles (effective radius [spherical particles] is used in the DMRT calculations). The modeling results are then compared to observations made from a 35-GHz radiometer.
 For this investigation, the wavelength chosen is 8500 μm (0.85 cm), corresponding to a frequency of 35 GHz. It has been demonstrated [Chang et al., 1987] that for a snowpack less than a meter in depth, more information about the snow water equivalent and thickness can be derived when using a frequency of about 35 GHz than when using higher or lower frequencies.
 To derive the modeled brightness temperatures, the following formula was used:
where TB is the brightness temperature, e is the emissivity, T is the physical temperature and sky is the cold sky reflection.
 Using the large 0.8 cm pellets and for a 7 cm layer, for example, the formula would be 90 = 196e + 30 (1-e), where 90 K is the TB resulting from volume scattering, 196 K is the physical temperature of dry ice, 30 is the TB of the aluminum plate. For this case, the modeled TB is 71 K.
Table 1 shows values for observed versus calculated microwave brightness for CO2 crystals having different radii and thicknesses. A dense media radiative transfer (DMRT) model, based on quasi-crystalline approximation (QCA), with a sticky particle assumption was used to calculate brightness temperatures for the given parameters [Tsang et al., 2000]. This DMRT model is applicable to moderate-sized particles (particle sizes comparable to the wavelength). For this study, the stickiness parameter is assumed to be 0.1. The reflectivity of the bottom interface with the aluminum plate is given as 1.0. Using the sticky particle assumption, particles are allowed to adhere together to form clusters, which better describes what is observed in snowpacks. DMRT calculations compare very favorably with the observed values for two particles sizes and the three thicknesses listed. The observed versus calculated is within 10% for each case with the exception of the 26 cm layer thickness and the 0.8 mm particle size (15%) and for the 14 cm layer thickness and the 2.0 mm particle size (16%). The shallowest layer (7 cm) gives the best results – less than about 5% difference for each particle size. Thus, it appears that DMRT can be useful for modeling the flow of energy emerging from frozen CO2 deposits.
Table 1. Brightness Temperatures From Dense Media Model Calculations Compared With Observations
 The dry ice crystal shape, formed in laboratory cold chambers, is typically pseudo-octahedral, where two four-sided pyramids share a common base. Specifically, this type of crystal is known as a tetragonal ditetragonal bipyramid, and they have been observed for CO2 crystals of sizes up to 1 mm in diameter. Because the dry ice used in this experiment was manufactured in the shape of cylindrical pellets, an effort was made to see what effect, if any, the shape of the crystal for different particle sizes, has on microwave scattering.
 A particle scattering cross section model was used to assess whether the shape of different-sized dry ice pellets affects their scattering properties. The model calculates the scattering cross section for a variety of particle shapes, including cylinders, spheres and hexagons. Crystals were modeled having an effective radius (radii of a sphere of equal volume) of 500, 1,000, 5,000, and 10,000 μm. The discrete dipole scattering (DDSCAT) model employed here is a Fortran program that calculates scattering and absorption coefficients of electromagnetic radiation by arbitrary targets using the discrete dipole approximation (DDA). With this approximation, the targets are replaced by an array of point dipoles. The electromagnetic scattering problem for the arrays is then solved, essentially exactly [Draine, 1988] (updated by Draine and Flatau  and Foster et al. ).
 According to B. Draine (personal communication), DDSCAT can be used for any isotropic material. Even if the material is anisotropic, it can be used providing that certain dielectric tensor conditions are satisfied. For best results, the dielectric constant should not be too large (< ∼ 4). DDSCAT is a versatile program and has been used to address scattering from materials such as snow, ammonia or interstellar dust [Draine, 1988; West et al., 1989]. The program code incorporates Fast Fourier Transform methods [Goodman et al., 1991].
 Again, the wavelength chosen is 8500 μm (0.85 cm), and a value of 1.44 was used for the real part and 0.005 was used for the imaginary part of the refractive index of frozen CO2,. Three different target orientations with calculations for two incident polarizations states are available with this model. Here, randomly oriented dipoles are specified. Scattering intensities are computed for two scattering planes at intervals of 30 degrees in the scattering angle theta; phi = 0 for the x-y plane, and phi = 90 for the x-z plane. The true thickness of a deposit is not required for emission boundary conditions; scattering or absorption results from the array of point dipoles [Foster et al., 1998].
Table 2 gives values for extinction, absorption and scattering efficiency of frozen carbon dioxide crystals for differently shaped and sized particles using a frequency of 35 GHz and a refractive index of 1.44 + 0.0005. For comparison purposes, Table 3 gives extinction, absorption and scattering efficiency values, as modeled for H20 crystals. The scattering values are all several times greater than absorption for the range of H2O particles listed in Table 3. For the CO2, the scattering values for the smallest particles (500 μm) are close to those for absorption, but for the 1,000 μm size particles, scattering is several times larger than absorption. Note that for the largest particles (5,000 and 10,000 μm), scattering is an order of magnitude greater than absorption.
Table 2. Extinction Efficiency, Absorption, and Scattering for Different Shaped Carbon Dioxide Crystalsa
Size Parameter (α) = 2πr/λ
The refractive index is 1.44+0.005 i, and the wavelength is 0.85 cm (35 GHz).
Table 3. Extinction Efficiency, Absorption, and Scattering for Different Shaped Water (Snow) Crystalsa
Size Parameter (α) = 2πr/λ
The refractive index is 1.78+0.0024 i, and the wavelength is 0.85 cm (35 GHz).
 The definitions of scattering and absorption efficiency are given below. If a(eff) is the effective radius of the particle, then
where Csca is the scattering cross section,
where Cabs is the absorption cross section,
where a(eff)=(3V/4π)1/3 (radius of a sphere of equal volume).
 From these DDSCAT model calculations (Tables 2 and 3) it can be seen that differently shaped crystals, having the same effective radius of a sphere, give very similar cross section/efficiencies. However, for the CO2 crystals, for sizes 1,000 μm or smaller, tetrahedron shapes scatter more than cylinders, while for sizes larger than 1,000 μm, the reverse is true. For water crystals, the tetrahedral shapes also scatter slightly more than cylinders.
 Referring to Tables 2 and 3 and Figure 4, notice that extinction decreases with crystal size for the largest crystals (10,000 μm). According to these tables, extinction, absorption and scattering each decrease from 5,000 μm to 10,000 μm. When the particle size is greater than the wavelength (8,500 μm), extinction no longer increases but rather oscillates [Ulaby et al., 1981]. Calculations of the attenuation cross sections of large ice and water spheres have shown that the normalized attenuation cross section increases up to a size parameter (α) of 1, and then from there decreases to a size parameter of 5 [Atlas and Wexler, 1963; Battan et al., 1970].
 While it is hoped that the measurements described in this paper provide some insight to possible microwave observation of the seasonal water and CO2 caps on Mars, in order to truly comprehend properties of the caps, more information is needed along several fronts. For instance, the radiative transfer and dense media models can be improved if they are constrained by observations. In addition, what are the actual size and shape of the CO2 crystals on Mars? Surely, they are not in the shape of manufactured pellets. This information is critical in determining the thickness of the deposits. Also, a better knowledge of the dielectric constant of CO2 at microwave frequencies and improved radiative transfer algorithms would give us more confidence in estimating brightness temperatures. Knowing more about the quantity and quality (grain size) of dust and other contaminants that falls on the caps could also be important regarding the radiative transfer of energy from within and through the dry ice layer or layers.
 In order to ensure that measurements of CO2 ice and water snow are more meaningful, a number of factors must be considered when conducting future experiments. The radiometer system must be reliably calibrated and if at all possible, a second radiometer used to confirm the measurements. For optimum results, a stable atmosphere is needed, such as a cold, clear night. Measurement of the physical temperature versus depth and of the structure versus depth is also desirable concomitant with the radiometric measurements.
 Other uncertainties also need to be addressed in the coming years, such as better measurements for the optical constants of dust, and does CO2 condensation vary locally? Because dry ice (CO2) quickly sublimates at the temperatures which we were working (above 0 degrees C), we were not able to repeat the radiometric measurements in a reliable manner. Furthermore, our 35-GHz radiometer has an analog rather than a digital readout, which results in an inexact reading of the observed values. Therefore, error limits could not be given on Table 1, and error bars could not be drawn about the observations on Figures 3 and 4. To remedy this, in the future, additional measurements will be made, using larger reserves of dry ice, so that error bars can be generated. Although some of the considerations from the above paragraphs are beyond the scope of this paper, they need to be addressed eventually before a greater understanding can be realized of the thermal emission observations of the Martian polar caps.
5. Conclusions and Future Plans
 In this study it was found that compared to natural snow crystals, the dry ice crystals exhibited lower brightness temperatures. This is attributed to both the greater scattering of the larger pellets of CO2, which are about an order in magnitude larger than the largest snow crystals, and the colder physical temperatures of the dry ice. For instance, a brightness temperature of only 86 K was measured over the aluminum plate when the thickness of the dry ice was 27 cm, for an incidence angle of 50 degrees (horizontal polarization).
 Dense media calculations compare very favorably with the observed values from the handheld radiometer. The observed versus calculated TBs are within 10% for each case with the exception of the 26 cm layer thickness and the 0.8 mm particle size (15%) and for the 14 cm layer thickness and the 2.0 mm particle size (16%). Thus, it appears that DMRT will be useful for modeling the flow of energy emerging from frozen CO2 deposits.
 From the DDSCAT model calculations, it can be seen that differently shaped crystals, having the same effective radius of a sphere, give very similar cross section/efficiencies. However, for the CO2 crystals, for sizes 1,000 μm or smaller, tetrahedron shapes scatter slightly more than cylinders, while for sizes larger than 1,000 μm, the reverse is true. For water crystals, the tetrahedron shapes also scatter minutely more than do cylinders [Foster et al., 1998].
 Eventually, we would like to be able to construct an artificial CO2 snowpack having larger crystals at the bottom and smaller crystals at the surface. It would be useful to measure crystals of various sizes with the 35-GHz handheld radiometer and, in addition, to use another radiometer tuned to a higher (85 GHz) frequency.
 The authors would like to thank Andrew Klein and Eric Erbe for assisting with the experiment and Marla Moore for her comments. Also, Gary Hansen was especially helpful in making many useful suggestions.