Introduction
The purpose of optical models of snow and clouds is to quantify the measurable reflectance and transmittance as functions of wavelength and medium properties, especially particle size. The “optical” grain-size derived from measured optical data is the direct inversion of such models. If different models give different results we have the problem of ambiguity. The present paper shows that this problem exists, and proposes a way to solve it. Modeling the reflectance and transmittance of strongly scattering media is a delicate task. Different approaches using slightly different assumptions, even within the same model, lead to different results, making comparisons difficult. A major problem arises from the fact that the shape and size of scattering particles are often not known, and even if they are known cannot easily be described. As a result, a certain degree of arbitrariness in the choice of these quantities evolves in the different models. It would be highly desirable to have a simple, analytic multiple-scattering model available as a standard for comparison and to better understand the physics involved. Such a model can give detailed insight into the spectral behavior of scattering and emitting media, even if not all the properties are realistic. The ice-lamella system presented here represents such a model. It is purely one-dimensional, with all lamellae being parallel, all waves being plane waves and all wave interactions being limited to two directions (incident and reflected waves at the same linear polarization).
In contrast to the optical industry where controlled glass-lamellae systems play important roles in adjusting light and heat transmittance and reflectance, in the present case, the lamellae are assumed to be freely arranged and slightly variable in thickness, leading to a kind of average behavior. Free arrangement means that the expected ice-volume fraction v is independent of position, which also means that a fraction v of particle surfaces are in contact. This property assures, on average, incoherent interactions between different lamellae even at high values of v, a rather surprising result found by Reference MätzlerMätzler (1997, Reference Mätzler1998a). The model gives a surprisingly realistic spectral description of ice clouds and snowpacks from microwave to the ultraviolet (UV) of homogeneous situations. As well as the transmittance and reflectance, the model gives the emittance through Kirchhoff’s law. The model is described for vertical incidence in section 2, and the extension to oblique incidence is straightforward. In section 3 the model results are presented, and comparisons are made with other snow-reflectance models, the emphasis being placed on comparison of effective grain-sizes and on the spectral behavior.
2. The Model
2.1. Model geometry
Let us assume a model for a purely one-dimensional scattering medium to represent either a snowpack or an ice cloud, i. e. a pack of thin, horizontal ice sheets of mean thickness d, packed to a volume fraction v (0 < v < 1), so that on average there are N lamellae per meter depth (Fig. 1). The lamellae are assumed to be freely arranged, a concept introduced by Reference MätzlerMätzler (1997).
Fig. 1. Pack of freely arranged ice lamellae, representing the ice-lamella system, a model of a cloud or of a snowpack. Freely arranged lamellae can be in direct contact (second from bottom) thus increasing the average thickness.
Expressing N by v and d we obtain
The main advantage of this geometry is that the reflectance can be computed simply and exactly, in the sense that only two spatial directions are involved in the scattering process, a forward and a backward ray. In this case the integro-differential radiative transfer equation decays into a pair of first-order differential equations (see Equation (6)) identical to the ones used in the two-flux model for which exact solutions are known. Furthermore, certain icy scatterers are indeed horizontally aligned (ice lenses, horizontally aligned snow layers, plate-like ice crystals in cirrus clouds in non-turbulent condition), and thus are favorably approximated. Since volume scattering of spherical or other ice grains is also well approximated by two-stream theories (Reference Meador and WeaverMeador and Weaver, 1980), their results can be related and compared to the present simple model.
2.2. Reflection of a single ice lamella
For a dielectric lamella with negligible losses, the reflectivity is given by the Airy equation (Reference Born and WolfBorn and Wolf, 1980)
where r 1 is the Fresnel reflectivity of the air–ice interface and P is the one-way phase through the lamella. For vertical incidence these quantities are given by
and
where n = n′ + i·n″ is the complex refractive index of ice and k is the vacuum wavenumber.
2.3. Average reflectivity of a lamella
Since d is assumed to be slightly different for different lamellae, the phase terms in Equation (2) are smeared out when averaged over many lamellae, except for very small values of P. For the same reason, coherent superpositions of reflections at different lamellae disappear due to the variable distance between them, i.e. the free arrangement of the lamellae. Noting that r 1 ≪ 1, the denominator of Equation (2) can be approximated by 1. The average lamella reflectivity r av can then be written as
For a small phase, Equation (5) gives the coherent reflectivity of the lamella through the first maximum of the sine function (at one-quarter wavelength), and it provides a continuous transition from the coherent to the incoherent situation at larger thickness (or larger phase).
2.4. Two-flux scattering coefficient
Scattering consists of reflections of incident radiation at the ice lamellae. Since all lamellae are horizontally aligned, the reflections from vertically incident radiation result in radiation propagating in the opposite direction. Freely arranged scatterers interact incoherently (Reference MätzlerMätzler, 1998a), so a radiative-transfer treatment can be used to describe the scattering of the total layer. In this one-dimensional geometry, the so-called two-flux or two-stream model (e.g. Reference IshimaruIshimaru, 1978) is exact. In this model, the radiative transfer of the up- and downwelling intensities, I 1(z) and I 2(z), respectively, can be described by the following pair of equations (emission being omitted here, but included later), assuming an upward-directed z axis:
where the minus sign in the first term on the right applies for I 1 and the plus sign for I 2. For constant coefficients these equations are solved analytically, leading to the results given below. For N lamellae per meter depth the scattering coefficient γ s is given by γ s = Nr av. However, due to the free arrangement, this quantity is reduced by the probability (1 – v) of two adjacent lamellae being in contact, so γ s is given by
The contacting lamellae have an increased total thickness, so we can introduce the average thickness as d av = d/(1 − v). This quantity probably represents a measured mean thickness more closely than d. Thus Equation (7) can also be expressed by
2.5. Two-flux absorption coefficient
So far, dielectric losses have been ignored. Small losses can be included by the absorption coefficients of ice γ a,ice and air γ a,air. The resulting absorption coefficient γ a of the lamella model is
In the case of high dielectric losses, absorption happens locally within the topmost lamella, leading to (Equation 9′) below.
2.6. Reflectance, transmittance and emittance of the slab
Let us neglect atmospheric absorption, and let us assume that the slab of height h (thus consisting of N · h lamellae) is situated above a non-reflecting background. Then the reflectivity (or reflectance), r, and the transmissivity (transmittance), t, of the slab are determined by the two-flux model, defining the reflectivity r 0 for infinite thickness, the transmission function t 0 and the damping coefficient (eigenvalue) γ 2 of the model (e.g. Reference MätzlerMätzler, 1987):
where the model parameters are given by
An exception to Equation (9) must be made when the ice lamella becomes opaque; then only the reflections at lamellae near the top of the pack will contribute. A useful approximation for the pack reflectivity and transmissivity is obtained by including incoherent reflections at the three topmost air–ice interfaces and neglecting multiple scattering:
Finally, the emissivity (or emittance) e of the slab is obtained from Kirchhoff’s law, stating that e is equal to the absorptivity a:
This equation follows from energy conservation. The emisivity of the whole (slab and background) system is given by e 0 = 1 – r = e +t.
3. Discussion
For thin packs, i.e. small h (≪ 1/γ s), the slab reflectivity simplifies to r ≅γ s h. On the other hand, for a sufficiently deep pack, the reflectivity is only a function of the ratio x = γ s/γ a:
The often used Approximation (13) is approached for large values of x (e.g. for snow at visible wavelengths). Inserting x from Equations (7′) and (8) for γ a,air = 0 and for large P we obtain
and thus from Approximation (13) we find
a well-known approximation for the reflectance of a strongly volume-scattering object (e.g. Reference BohrenBohren, 1987). Here, the constant K is given by
; thus for lossless ice we obtain from Equation (3):
Inserting the refractive index of ice in the visible range (n′ = 1.33), we obtain r 1 = 0.020 and K = 7.06. According to an early snowpack model of spherical ice particles (Reference Bohren and BarkstromBohren and Barkstrom, 1974), the reflectance of a deep snowpack can be written as
where D BB is the sphere diameter of the Bohren and Barkstrom model. This result agrees with Approximation (15) by choosing
Another comparison can be made using the snow model of Reference Wiscombe and WarrenWiscombe and Warren (1980). Reflectivities of thick snow-packs computed with this model for different grain diameters D ww were taken from Reference MarshallMarshall (1989) and from Reference Sergent, Pougatch, Sudul and BourdellesSergent and others (1993) at a wavelength of 1 μm, and the results are shown by the data on the upper curve of Figure 2. The curve represents Equation (12) for x = 4.915 mm/D ww. Comparing this result with Equation (14) for r 1 = 0.0171 (n′ = 1.301) and γ a,ice = 0.024 mm−1, we find x = 1.42/d av, so agreement is achieved with the lamella model if
The lower curve in Figure 2 represents the model of Reference De Haan, Bosma and HovenierDe Haan and others (1987), with x = 2.69 mm/D DH. Agreement with the lamella model is achieved if the De Haan grain diameter D DH is given by
Comparison of Equations (18a–c) indicates that different snow-reflectance models lead to slightly different results (all within about a factor of 2) with respect to the grain diameter. This is why a reference standard could help to identify differences between models and ultimately improve the modeling work. The above comparison also means that the present simple model fits well within the snow-reflectance models. On average, the lamella thickness is about equal to the grain radius.
Fig. 2. Decrease of the reflectance r 0 of pure snow (wavelength 1 μm) with increasing grain diameter D of spherical ice grains. Data points (diamonds) along the upper curve are computed using the model of Reference Wiscombe and WarrenWiscombe and Warren (1980); the curve represents Equation (13) with x = 4.915 mm/D. The lower curve represents Equation (13) with x = 2.69 mm/D. The data points were computed by Reference Sergent, Leroux, Pougatch and GuiradoSergent and others (1998), using the model of Reference De Haan, Bosma and HovenierDe Haan and others (1987).
Another comparison of the lamella-pack model was made with data of the NASA Adavanced Spaceborne Thermal Emission and Reflection Radiometer (NASA-ASTER spectral library (http://asterweb.jpl.nasa.gov). The results are shown in Figure 3 for three different effective grain-sizes D AS = 0.024, 0.082 and 0.178 mm. The lamella thickness, d, was assumed to be equal to D AS, and a constant ice-volume fraction v = 0.1 was assumed. The spectral ice data were taken from Reference WarrenWarren (1984). According to the description of the ASTER library, the ASTER spectra were modeled based on broad-band measurements (2–14 μm) made by Reference Salisbury, D’Aria and WaldSalisbury and others (1994) at the Johns Hopkins University Infrared Laboratory, Baltimore, MD. The model quoted is Reference WaldWald’s (1994). The agreement between the spectra of the lamella pack and the ASTER snowpack is excellent. Especially at wavelengths of < 1.4 μm, the ASTER data and the lamella-pack model give almost identical results. From this coincidence it can be concluded that the ASTER grain-size corresponds to the original lamella thickness d for v = 0.1. Expressing the comparison by d av we have
Fig. 3. Reflectance of thick snowpacks vs wavelength between 0. 2 and 2.8 μm for three effective grain-sizes: 0.024 mm (uppermost pair of curves) 0.082 mm (middle pair) and 0. 178 mm (lowest pair). The smoother curves represent the lamella-pack model with the lamella thickness equal to the grain-size for v = 0.1; the noisier curves represent data from the ASTER spectral library.
Unfortunately, in the ASTER library the “grain-size” D AS is not clearly defined. After discussion with the authors of the database it seems probable that “ASTER grain-size” means radius. In this case the agreement with the results of Equations (18a–c) and Figure 2 is much better. In fact, Reference WaldWald (1994) used “radius” and “grain-size” as synonyms.
From updated spectral information on the complex refractive index of pure-water ice (Reference WarrenWarren, 1984; Reference Mätzler, Schmitt, De Bergh and FestouMätzler, 1998b), microwave to UV reflectivity and transmissivity spectra were computed for given packs. Examples of reflectance and transmittance data of two 10 cm thick snowpacks at a temperature of 266 K are shown in Figure 4a and b. Figure 4c shows the spectra of an ice cloud (shortest wavelength is 200 nm). The corresponding reflectivity spectra for infinite thickness are also shown. The computations are based on Equations (9), (9′) and (10). At lower frequencies where the phase P is small, we obtain a reflectivity which increases with increasing k 2 d, i.e. with frequency, so the emissivity decreases, as is observed for dry snow in the microwave range. In Figure 4a and b the reflectivity in the 2–100 GHz range is compared with the results of the recent Microwave Emission Model of Layered Snowpacks (MEMLS; Reference Wiesmann, Mätzler and WeiseWiesmann and others, 1998, Reference Wiesmann and MätzlerWiesmann and Mätzler, 1999;) for the same thickness, density and temperature, and for correlation lengths p MEMLS of the isotropic heterogeneity fitted to the present data. It is found that p MEMLS is significantly larger than d, and its influence on the scattering coefficient is stronger than for d. Indeed, three-dimensional Rayleigh scattering increases with k 4(p MEMLS)3, whereas in the one-dimensional geometry, scattering increases with k 2 d. The shape of the MEMLS spectra (Fig. 4a and b) is also slightly steeper than in the present model. Thus there is a functional difference between scattering in one-and three-dimensional heterogeneity at microwave frequencies, whereas at optical frequencies both types of heterogeneity produce coincident spectra and coincident grain-size dependence.
Fig. 4. (a, b) Radio to UV spectra of transmissivity t and reflectivity r of a thin snowpack consisting of a 10 cm deep ice-lamella pack with d = 0.05 mm, v = 0.1 (a) and d = 0.02 mm, v = 0.1 (b). Also shown is the reflectivity r 0 of the same snow, but at infinite thickness. The data points labeled with crosses are MEMLS results of r for the same snow density, thickness and temperature (266 K), but with p MEMLS = 0.2 mm (a) and p MEMLS = 0.12 mm (b). (c) Radio to UV spectra of transmissivity t and reflectivity r of an ice cloud consisting of a 100 m deep ice-lamella pack with d = 3 μm, v = 2 × 10−6. Also shown is the reflectivity r 0 of the same cloud, but at infinite thickness. Absorption in moist air was neglected.
A comparison between the two correlation lengths d and p MEMLS follows from geometrical considerations, referring to the specific surface s = S/V of a granular medium where S is the total surface of particles within volume V. In the three-dimensional case, the equation of Reference Debye, Anderson and BrumbergerDebye and others (1957) applies:
whereas in the one-dimensional case of Figure 1, s is given by
Comparing Equations (19) and (20) gives
In view of Equation (21) the discrepancy in Figure 4a and b between p MEMLS and d is not too severe.
For the general behavior of the model as seen in Figure 4a–c, the following can be noted: There is a broad maximum of the infinite reflectivity r 0 in the 10–1000 GHz range. This maximum decreases with decreasing d, whereas the maxima increase at short wavelengths. This property is intrinsic to volume scattering when the wavelength changes from larger to smaller than the characteristic size of the scatterers. The reason why the maximum of r 0 is so flat over the 10–1000 GHz range is the common behavior of γ a and γ s in this frequency range, both increasing with the square of frequency. The behavior is different at <10 GHz where γ a converges to a frequency-independent value, leading to an increase of r 0 with frequency squared.
The transmissivity shows a high-frequency cut-off near 100 GHz for the 10 cm snowpacks and near 1000 GHz for the ice cloud. The difference is mainly due to the different water-equivalent depth, decreasing from 10 mm for the snowpack to 0.2 mm for the cloud. The transition from Equation (9) to Equation (9′), from transparent to opaque lamellae, takes place deep within the cut-off region, at frequencies of >4000 GHz, with a return to Equation (9) at a wavelength of < 3 μm (f > 105 GHz). At the transition point, the reflectivity for Equation (9′) is larger than for Equation (9), leading to visible jumps in the spectra. The transition from incoherent to coherent lamella reflections, as expressed by P = 3π/4 in Equation (7), occurs in the decreasing part of r, near 1000 GHz in Figure 4a and near 3000 GHz in Figure 4b (see the slight change in slope).
4. Conclusions
A simple, physical, multiple-scattering model was presented for describing the reflectance and the transmittance (Equations (9) and (10)) over a very large frequency range in a volume-scattering medium, such as snowpacks and clouds, consisting of ice and air. The one-dimensional geometry consists of a slab of freely arranged, horizontally aligned ice lamellae of a given original thickness d. Due to occasional contacts between adjacent lamellae, the average lamella thickness d av is slightly larger than d. Either one of these parameters describes the structure, together with the ice-volume fraction v. The shortwave reflectance spectra up to a wavelength of 2.8 μm coincide with snow spectra modeled for spherical ice grains using Mie theory for grain radii about equal to d av. The decrease of the reflectance with increasing grain-size is the same in both types of model. A certain discrepancy between different snow-reflectance models was observed (Equations 18a–d).
Concerning the microwave range, there is no general agreement between the ice-lamella model and scattering in a three-dimensional heterogeneity. Nevertheless the present one-dimensional geometry gives an approximate agreement with reflectivities computed with a snow-emission model if the correlation length p MEMLS of the three-dimensional medium is properly adjusted to d. By using the information available on the complex refractive index of ice, very broad-band spectra for snowpacks and clouds can be constructed from the formulae presented here. Due to the simplicity of the ice-lamella pack, this model could be used as a reference in the development, validation and improvement of more elaborate models.
Acknowledgements
The author would like to thank T. Grenfell and S. Warren for providing the ice refractive-index data from Reference WarrenWarren (1984) and for an updated version. Thanks also go to the Jet Propulsion Laboratory, California Institute of Technology, for providing the ASTER data and to A. Wald and J. Salisbury for detailed discussion of aspects of their papers.
