Earth-Science Reviews 64 (2004) 189–241 www.elsevier.com/locate/earscirev
Optical properties of terrestrial clouds
A. Kokhanovsky*
Institute of Environmental Physics, University of Bremen, Otto Hahn Allee 1, D-28334 Bremen, Germany Institute of Physics, 70 Skarina Avenue, Minsk 220072, Belarus Received 18 January 2002; accepted 16 April 2003
Abstract
The aim of this review is to consider optical characteristics of terrestrial clouds. Both single and multiple light-scattering properties of water clouds are studied. The numerous results discussed can be used for solutions to both inverse and direct problems of cloud optics. An introduction to main microphysical characteristics of water and ice clouds, including the effective size, the liquid water content and path, the concentration of particles, is given. The refractive index of water and ice, which is of importance for optical waves propagation in clouds, is also considered. We consider extinction and absorption coefficients, phase functions, and asymmetry parameters of water and ice clouds and their relationships with the size, shape, and concentration of water droplets and ice crystals. Simple analytical results for absorption, reflection, and transmission characteristics of cloudy media are given. They are valid for optically thick clouds with optical thickness larger than 5. We have reviewed studies of polarization of solar light reflected from clouds. Image transfer problems have been considered as well. Modern methods of passive remote sensing of water and ice clouds are discussed. The review is finished by the discussion of optical properties of inhomogeneous clouds, which is a main topic of modern cloud optics research. D 2003 Elsevier B.V. All rights reserved.
Keywords: Radiative transfer; Light scattering and propagation; Water and ice clouds
1. Introduction serve as a blanket to protect the Earth’s surface against cooling at night. This is due to the fact that the Water, ice, and mixed clouds are major regulators of maximum of the terrestrial emission is in the far infra- solar fluxes in the terrestrial atmosphere (Kondratyev red, where water droplets are highly absorbing. and Binenko, 1984; Liou, 1992, 2002). They reflect a Clouds, mists, and fogs are very common. They large portion of incoming visible radiation back to reduce the visibility of objects in the atmosphere and outer space. The light energy absorbed by water drop- limit capabilities of atmospheric vision, remote sens- lets and ice crystals leads to heating of atmospheric ing, and detection systems (Zege et al., 1991). Thus, it layers. Another important role of cloudy media is to is of a great importance to understand the peculiarities of light interaction with cloudy media, which can be considered as a huge ‘‘colloid’’, composed of liquid * Institute of Environmental Physics, University of Bremen, and solid water particles, dispersed in the air. It should P.O. Box 330440. Otto Hahn Allee 1, D-28334 Bremen, Germany. Tel.: +49-421-218-4475; fax: +49-421-218-4555. be pointed out that water droplets and air around them E-mail address: [email protected] can be contaminated by various fine particles (e.g., (A. Kokhanovsky). soot and dust particles). These aerosol particles influ-
0012-8252/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0012-8252(03)00042-4 190 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 ence both light-scattering and absorption properties of 2. Microphysical characteristics cloudy media, making the studies of light propagation in cloudy media even more difficult (Menon et al., 2.1. Water clouds 2002). The very existence of clouds is due to fine- aerosol particles, the so-called cloud nuclei (Twomey, 2.1.1. Particle-size distributions 1977), with typical radii between 5 and 200 nm Water clouds consist of small liquid droplets that (Mason, 1975). generally have the spherical shape although particles Studies of optical and microphysical properties of of other shapes can exist due to different external cloudy media have a long and fruitful history. The influences. For instance, the deformation of large results of these investigations have been summarized in particles due to the gravitation force is of importance numerous books and papers (Kondratyev and Binenko, for raining clouds with particles having radii 1 mm 1984; Zege et al., 1991; Liou, 1992, 2002; Kokhanov- and larger (Macke and Grossklaus, 1998). The aver- sky, 2001a). The main properties of cloudy media are age radius of droplets in nonraining water clouds is well understood now, but unsolved problems remain. usually around 10 Am and the approximation of The most important of these include the accounting for spherical particles works quite well. Natural clouds the three-dimensional shape of clouds (Evans, 1998; with droplets of uniform size throughout never occur Mackeetal.,1999;Schreier,2001;Schreierand due to the variability of physical properties of air both Macke, 2001), and their inhomogenity in horizontal in space and time domains (Twomey, 1977). Thus, and vertical directions (Cahalan et al., 1994, 2001; one can consider a radius of a droplet, a, as a random Platnick, 2000, 2001). The influence of aerosols inside value, which is characterized by the distribution cloudy media, which can be one of explanations of the function f(a). This function is normalized by the anomalous absorption paradox (Danielson et al., 1969; following condition:
Rozenberg et al., 1978; Stephens and Tsay, 1990), also Z l should be clarified. f ðaÞda ¼ 1: ð2:1Þ Another current issue is the characterization of 0 optical properties of ice clouds, which have extremely complex microstructure (Liou, 1992, 2002; Liou and The integral Takano, 1994; Macke et al., 1996; Yang and Liou, Z 1998; Kokhanovsky and Macke, 1999; Yang et al., a2 FðaÞ¼ f ðaÞda ð2:2Þ 2000, 2001) and appear almost with the same fre- a1 quency as water clouds. One rarely finds two identical crystals in ice clouds because preferential shapes vary gives the fraction of particles with radii between a1 with temperature and pressure (Mason, 1975), which and a2 in a unit volume of a cloud. The distribution vary across cloudy media (Yang et al., 2001).It function f(a) can be represented as a histogram, should be stressed that optical properties of a single graphically or in a tabular form (Ayvazian, 1991). particle and, as a result, optical properties of a cloud However, it is the most common to use an analytical as a whole are highly influenced by particles’ shapes form of this function, involving only two or three (Mishchenko et al., 1995, 1996, 1999, 2000). parameters (Deirmendjian, 1969). Of course, this is a A complete review across all fields of cloud optics, great simplification of real situations occurring in which is in a state of a rapid transition, would be natural clouds, but most optical characteristics of a difficult within a single book, not to mention in a single cloud as a whole practically do not depend on the fine article. Thus, I will concentrate mostly on analytical structures of particle-size distributions (PSD) f(a) results, which can be used for cloud optical properties (Hansen and Travis, 1974). McGraw et al. (1998) studies. The formulae presented can be applied for found that the local optical properties of polydisper- rapid estimations of light fluxes in cloudy atmosphere sions can be modelled with high accuracy by just the and will assist understanding the information content of first six moments of the particle-size distribution, and transmitted and reflection functions of cloudy media in the use of certain combinations of moments can respect to microphysical characteristics of clouds. reduce the number of parameters even further. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 191
In most cases, the function f(a) can be represented Cðl þ 4Þ hWi¼ w0; ð2:13Þ by the gamma distribution (Deirmendjian, 1969): l3Cðl þ 1Þ
l Àl a f ðaÞ¼Na e a0 ; ð2:3Þ where where 3 4pa0 2 v0 ¼ ; s0 ¼ 4pa ; w0 ¼ qv0 ð2:14Þ llþ1 3 0 N ¼ lþ1 ð2:4Þ Cðl þ 1Þa0 are corresponding parameters for monodispersed par- ticles with radii a . One can obtain in the case of the is the normalization constant and C(l +1) is the 0 most often employed cloud model with a0 =4Am and Gamma function. It follows that the first derivative l = 6 (Cloud C.1 model of Deirmendjian, 1969): f V(a0) = 0 and the second derivative f U(a0) < 0 (see Eq. (2.3)). Thus, the function (Eq. (2.3)) has the maximum 7 14 7 hVi¼ v0; hRi¼ s0; hWi¼ w0; ð2:15Þ at a = a0. Note that Eq. (2.4) follows from Eq. (2.1) 3 9 3 and the definition of the Gamma function: c À 16 3 c À 12 2 Z l where v0 2.7 Â 10 m , s0 2 Â 10 m , c À 10 CðlÞ¼ xlÀ1eÀxdx: ð2:5Þ w0 2.7 Â 10 g. Although parameters (Eq. 0 (2.14)) are small, the very large numbers of cloud One can see that the parameter l characterizes the droplets (typically, 100 particles in cm3)create width of the particle-size distribution f(a), being important factors for atmospheric processes (see next smaller for wider distributions. Moments section). Z l Eq. (2.3) allows to characterize the cloud droplet n n ha i¼ a f ðaÞda ð2:6Þ distribution only by two parameters: a0 and l. 0 However, it should be remembered that neither is of the distribution (Eq. (2.3)) are calculated from the constant and can vary inside the body of a cloud. following simple analytical equation Thus, their values depend on the averaging scale, with large averaging scales producing more broad a n Cðl þ n þ 1Þ hani¼ 0 : ð2:7Þ particle-size distributions with smaller values of l. l Cðl þ 1Þ The value of l = 2 was found to be rather repre- Eq. (2.7) can be used to find the average volume of sentative (Khrgian and Mazin, 1952) and is advised particles to be used in low-resolution cloud satellite retrieval Z l algorithms. The parameter l =6 (Deirmendjian, 4p hVi¼ a3f ðaÞda; ð2:8Þ 1969), used in the derivation of Eq. (2.15), is 3 0 typical only for small averaging scales (Fomin the average surface area and Mazin, 1998). General features of the droplet Z l spectra in water clouds were studied in a great hRi¼4p a2f ðaÞda; ð2:9Þ detail by, e.g., Warner (1973). 0 Parameters a0 and l are defined in terms of the specific unimodal cloud droplet distribution (Eq. and the average mass of droplets (2.3)). It is of an advantage to characterize cloud hWi¼qhVi; ð2:10Þ particle-size distributions by their moments.
3 Moments can be retrieved from optical measure- where q = 1 g/cm is the density of water. It follows ments without reference to specific distribution laws that (McGraw et al., 1998). Cðl þ 4Þ Hansen and Travis (1974) found that the effective hVi¼ v ; ð2:11Þ l3Cðl þ 1Þ 0 radius
Cðl þ 3Þ ha3i hRi¼ s ; ð2:12Þ a ¼ ð2:16Þ l2Cðl þ 1Þ 0 ef ha2i 192 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 is one of the most important parameters of any values of D may indicate the presence of the second particle-size distribution. It is proportional to the mode in the range of large particles (Ayvazian, 1991). average volume/surface ratio of droplets. The param- Eq. (2.19) and the results for a0 and l just reported eter (Eq. (2.16)) can be also defined for nonspherical lead to the effective radius, aef, of water droplets being particles as we will see later. The coefficient of in the range from 5 to 50 Am, depending on the cloud variance (CV) of the particle-size distribution type. Near-global survey of the value of aef using A V V A s satellite data shows that typically 5 m aef 15 m D ¼ ð2:17Þ (Han et al., 1994). We see that water clouds with hai aef >15 Am are rare. This can be used to discriminate where satellite pixels with ice crystals even at wavelengths sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ice and water absorption coefficients are almost Z l 2 equal. Such a possibility of discrimination is due to s ¼ ða ÀhaiÞ f ðaÞda; ð2:18Þ much larger (e.g., in 5–10 times) effective sizes of ice 0 crystals as compared to droplets. The large size of ice is also of importance, especially for narrow droplet crystals will reduce the reflection function in near distributions. The value of s is called the standard infrared considerably as compared to droplets. This deviation. The CV, which is equal to the ratio of the reduction can be easily detected. Note that clouds with standard deviation to the mean radius hai, is often aef >15 Am are often raining (Masunaga et al., 2002). expressed in percent. Pinsky and Khain (2002) showed that the threshold of It follows for the PSD (Eq. (2.3)): the occurrence of drizzle is around aef =15 Am. Then strong collisions of droplets start. 3 1 aef ¼ a0 1 þ ; D ¼ pffiffiffiffiffiffiffiffiffiffiffi ð2:19Þ Some authors prefer to use the representation of the l 1 þ l particle-size distribution by the following analytical form (Ayvazian, 1991): and, therefore, 0 1 2 a 2 ln 1 1 À D 1 B am C l ¼ À 1; a0 ¼ aef : ð2:20Þ f ðaÞ¼pffiffiffiffiffiffi exp@À A; ð2:21Þ D2 1 þ 2D2 2pra 2r2
For instance, we obtain at l =3: aef =2a0, D = 0.5, which is called the lognormal distribution. The rela- s = hai/2. The effective radius, a , is always larger ef tions between the values of aef, hai, D and the than the mode radius, a0. parameters of the gamma and the lognormal par- Eq. (2.20) gives the meaning of the parameter l in ticle-size distributions are presented in Table 1. The the PSD (Eq. (2.3)). In situ measurements show that value of Def in this table represents the effective the value of a0 often varies from 4 to 20 Am (Mason, variance defined as 1975) and la[2,8] in most of cases. It should be Z l 2 2 pointed out that clouds with smaller droplets are not ða À aef Þ a f ðaÞda stable due to the coagulation and condensation pro- 0 Z Def ¼ l : cesses (Twomey, 1977). Particles with a large radius 2 2 aef a f ðaÞda cannot reside in atmosphere for a long time due to the 0 gravitation force. Thus, several physical processes This value is often used instead of the coefficient of lead to the existence of the most frequent mode radius variance, D, (Hansen and Travis, 1974) because of a range. One can obtain from Eq. (2.19) and inequality special importance attached to the value of the effec- 2 V l V 8 that the value of Da[0.3,0.6]. Thus, it tive radius of droplets, aef, in a cloudy medium as follows that the standard deviation of the radius of compared to the average radius, hai, of droplets. For particles in water droplets is from 30% to 60% of an instance, light extinction in clouds is governed mostly average radius in most cases. Smaller and larger by values of aef and liquid water content independ- values of D do occur, but values of D smaller than ently on the particle-size distribution (Kokhanovsky, 0.1 were never observed (Twomey, 1977). Larger 2001a). A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 193
Table 1 Droplet-size distributions n f(a) B hai aef DDef ha i lþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Àl a l a0 Cð1 þ l þ nÞ Gamma distribution Bale a0 a (1 + (1/l)) a (1 + (3/l)) ð1=l þ 1Þ 1/(l +3) lþ1 0 0 l C 1 l !a0 Cðl þ 1Þ ð þ Þ 2 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ln 1 2 2 p 2 2 2 am ffiffiffiffiffiffi 0.5r 2.5r r2 r n n r Lognormal distribution exp À p ame ame e À 1 e À 1 a e 2 a 2r2 2pr m
2.1.2. The concentration of droplets where l = z2 À z1 is the geometrical thickness of a The number concentration of particles, N, in addi- cloud. If Cw = const, tion to size and shape of particles, is of importance for the optical waves propagation, scattering, and extinc- w ¼ Cwl: ð2:23Þ tion in cloudy media. The concentration of droplets depends on the concentration CN of atmospheric The geometrical thickness of clouds varies, depending condensation nuclei. The value of CN is smaller over on the cloud type (Landolt-Bernstein, 1988). It usually oceans than over continents; thus, the concentration of is in the range 500–1000 m for stratocumulus clouds. droplets in marine clouds is on average smaller than Near-global data obtained by Han et al. (1994) from over continents. Generally, the smaller concentration satellite measurements show that the liquid water path of droplets over oceans means that they can grow is typically in the range 50–150 g/m2. The annual larger, producing clouds with larger droplets over mean is equal to 86 g/m2 (Han et al., 1994). oceans, which is confirmed by the analysis of the Cloud systems can easily cover area S c 103 km2 satellite optical imagery as well (Han et al., 1994). (Kondratyev and Binenko, 1984); thus, the total This influences the occurrence and the rate of precip- amount of water W = wS (for idealized clouds having itation. the same liquid water path for the whole cloudy area) Swensmark and Friis-Christensen (1997) and stored in such a water cloud system is equal approx- Marsh and Swensmark (2000) have speculated that imately to 108 kg, if we assume that w = 100 g/m2, cosmic ray ionization could influence the production which is a typical value (Han et al., 1994). This under- of condensation nucleii and, therefore, cloud proper- lines the importance of clouds both for climate prob- ties. This is of importance for climate change prob- lems and human activity (e.g., crops production, etc.). lems as well (Swensmark, 1998). The dimensionless volumetric concentration of 2.2. Ice clouds droplets Cv = NhVi and the liquid water content (LWC) Cw = qCv or (see Eq. (2.10)) Cw = NhWi are Microphysical properties of ice clouds cannot be often used in cloud studies as well. characterized by a single particle-size distribution The typical variability of N, Cv, and Cw in water curve as is the case for liquid clouds even if one clouds is presented in Table 2 (Mason, 1975; Fomin considers relatively small volumes of a cloudy and Mazin, 1998). Numbers in Table 2 are represen- medium. This is due to extremely complex shapes tative, but it should be remembered that these values of ice particles in crystalline clouds. Major shapes of can change in broader range in real situations. The ice crystals are plates, columns, needles, dendrites, liquid water content is not constant throughout a stellars, and bullets (Mason, 1975; Volkovitsky et al., cloud, but has larger values near the top of a cloud 1984; Takano and Liou, 1989, 1995; Liou, 1992, (Feigelson, 1981; Paul, 2000; Yum and Hudson, 2002; Yang et al., 2001). Combinations of bullets 2001). The liquid water path (LWP), w, is defined as Table 2 Typical range of values N, C , and C in water clouds Z v w z2 À 3 3 N (cm ) Cv Cw (g/m ) w ¼ CwðzÞdz; ð2:22Þ 20–1000 10À 7 –10À 6 0.01–1 z1 194 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 and needles are also common. The Magano–Lee It should be pointed out, however, that the whole classification of natural crystals includes 80 shapes concept of a single-distribution function breaks down (Magano and Lee, 1966), ranging from the elementary for ice clouds. Recall that the microphysical properties needle (the classification index N1a) to the irregular of most water clouds (as far as their optical properties germ (the classification index G6). are concerned) can be characterized by just three The concentration of crystals, N, varies with numbers (e.g., the effective radius of droplets, the height, often in the range 50–50,000 crystals per coefficient of variance of the droplet distribution cubic meter. The ice water content function f(a), and the concentration of droplets) (Deir- mendjian, 1969). C ¼ NhWi; ð2:24Þ i By contrast, at least 80 multidimensional particle- where hWi is average mass of crystals, usually in the size distributions are needed if one would like to use range 10À 4 –10À 1 g/m3. Thus, the average mass of a the classification of crystals developed by Magano crystal is in the range 2 Â 10À 3–2Â 10À 9 g. Crystals and Lee (1966). This calls for the introduction of a have a bulk density, q, less than that of the bulk ice new way for particle characterization in the case of (q = 0.3–0.9 g/cm3) due to the presence of impurities complex particulate systems such as ice clouds. The and bubbles inside ice particles (Landolt-Bernstein, same problem arises in the optics of mineral aerosol, 1988). The size of crystals is usually characterized by blown from the Earth’s surface to atmosphere (Volten their maximal dimension, D, which is related to the et al., 2001; Mishchenko et al., 2002). effective size (Yang et al., 2000). D is usually in the One of the possible solutions of the problem lies in range 0.1–6 mm for single crystals and 1–15 mm for the characterization of ice crystals in an elementary snow crystal aggregates. Smaller crystals (e.g., with volume of a cloudy medium by the function maximal sizes around 20 Am) also are present in ice XN XM ! ! ! ! clouds (Yang et al., 2001). The mode D0 of size f ða; b Þ¼ crfrða Þþ cifiðb Þ; ð2:25Þ distribution curves f(D) depends on the shape of r¼1 i¼1 crystals, with characteristic values of D being 0.5 0 where f ða!Þ is the size distribution of particles of mm for plates and columns, 1 mm for needles, r a regular shape (e.g., hexagonal plates or columns), sheaths, and stellars, and 2 mm for dendrites. Distri- ! f ð bÞ is the statistical distribution of particles with bution curves can be modelled by gamma distribu- i random surfaces or the so-called irregularly shaped tions with half-widths of distributions, D , being 1/2 particles. Note that instead of single variables (e.g., larger for larger values of D. Frequently, D c D¯ , 1/2 the radius of particles a), we need to introduce vector where D¯ is the average size of crystals. ! parameters a! and b in the case of crystalline clouds. Simple shapes of ice crystals (e.g., hexagonal In particular, we have for the components of the two- prisms) can be characterized by two dimensions: the pffiffiffi dimensional vector parameter, a!, with coordinates a , length of the prism, L, and the diameter, D ¼ a 3, 1 a in the case of an idealized cloud with hexagonal where a is the side of a hexagonal cross section. Even 2 columns: a1 = D, a2 = L. in this most simple case, two-dimensional distribution ! Functions f ð bÞ present statistical distributions of functions, f(D,L), should be used. Note that f(D,L) can i some statistical characteristics of particles (e.g., aver- be approximately reduced to one-dimensional func- age radii, correlation lengths, etc.). Values of c and c tions, f(D), due to the existence of empirical relation- i r give concentrations of different crystal habits. Clearly, ships between the length of crystals and their diameter ! the simplest case is to consider the function f ð!a; bÞ as in natural clouds (Auer and Veal, 1970). For instance, a sum of two functions: for hexagonal columns with the semiwidth, ~, and the length, L, (Mitchell andffiffiffi Arnott, 1994), ~ = 0.35L at ! ! ! ! p f ða; bÞ¼c1f1ð aÞþc2f2ð bÞ; ð2:26Þ L < 100 Am and ~ = 3.48 L otherwise. Pruppacher and ! Klett (1978) give the following empirical relationship where function f1ð aÞ represents particles of a regular for plates: L = 2.4483~0.474 at ~a[5 Am, 1500 Am]. shape (say, hexagonal cylinders) and the function ! Similar relationships can be obtained for other shapes f2ð bÞ represents statistical parameters of a single par- (Mitchell and Arnott, 1994). ticle of irregular shape. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 195
This irregularly shaped particle can be presented, the refractive index of aerosol in visible is larger than e.g., as a fractal (Macke and Tzschichholz, 1992; that for ice. Macke et al., 1996; Kokhanovsky, 2003a). It should ! be pointed out that the function f2ð bÞ in this case 2.3. Refractive index of liquid water and ice represents ‘‘fictive’’ particles (Kokhanovsky, 2001a) which do not exist in a cloud at all. However, ice The complex refractive index of particles suspended cloud optical characteristics, calculated using f2 in the atmosphere is another important parameter in ! ð bÞ, indeed represent quite well the optical character- atmospheric optics studies (Liou, 1992). This is due to istics of particles with extremely diverse shapes the fact that not only the size, shape, and concentration (Macke et al., 1996; Kokhanovsky, 2003a). of particles influence the light propagation in atmos- A similar approach to optical characterization of phere. The chemical composition, thermodynamic irregularly shaped particles was developed by Pelto- phase, and temperature of particles are of importance niemi et al. (1989), Peltoniemi (1993), and Muinonen as well. The refractive index of water droplets and ice et al. (1996). It is based on Monte-Carlo calculations crystals varies with the temperature and has been of light scattering by a large particle with a rough tabulated by many authors (see, e.g., Hale and Querry, surface. The model of spheres with rough surfaces 1973 for liquid water and Warren, 1984 for ice). was successfully applied to the optical characteriza- Spectral dependencies of real and imaginary parts tion of irregularly shaped aerosol particles (Volten et of the complex refractive index of water and ice are al., 2001). This model presumably can be extended for presented in Figs. 1 and 2. The differences in light the case of ice clouds as well. For this, the parameters absorption by liquid and solid water are considerable, of the irregularity of the so-called fictive particle a fact that can be used for the retrieval of the cloud (Kokhanovsky, 2001a) should be changed accord- thermodynamic phase (liquid, ice, or mixed-phase ingly. This is due to different morphology of ice clouds) from ground (Dvoryashin, 2002) and satellite crystals (Magano and Lee, 1966) as compared to (Knap et al., 2002) measurements. In particular, we mineral aerosol (Okada et al., 2001). Also note that see that the absorption band around the wavelength,
Fig. 1. Real part of the refractive index of water and ice. 196 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 k = 1.55 Am, for ice is shifted to larger wavelengths as liquid water absorption can also influence the cloud compared to liquid water. This shift in the absorption droplets’ size retrievals. band position can be easily detected with the use of modern spectrometers (see, e.g., Bovensmann et al., 1999). 3. Local optical characteristics The real part of the refractive indices of water, nw, and ice, ni, do not vary considerably in the visible and 3.1. Water clouds near-infrared regions of electromagnetic spectra. Gen- erally, ni(k) Fig. 2. Imaginary part of the refractive index of water and ice. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 197 horizontally infinite homogeneous plane-parallel Huffman, 1983; Kokhanovsky, 2003b). It is known layer, which is, of course, a great simplification that (Shifrin, 1951) of a real cloud field (Cahalan et al., 1994, 2001). Cext ¼ R=2 ð3:4Þ The value l=1/rext is called the photon-free path length and gives the average distance between in a good approximation for water droplets in the photon-scattering events in a cloud. Knowing the visible range of the electromagnetic spectrum. Here R average number of scattering events, n¯, it is easy to is the surface area of a droplet. From Eqs. (3.3) and find the average distance, L = n¯l, which photon (3.4)~ travels in the medium before it escapes out of the scattering layer. We have for the average total time, NhRi rext ¼ : ð3:5Þ which photon spends in a cloud: t = L/c, where c is 2 the group light speed. Thus, the extinction coefficient depends on the prod- The value of l is often in the range 10–200 m for uct of the number concentration of particles and their water clouds; thus, the value of rext is in the range average surface area. By use of 0.005–0.1 mÀ 1, depending on the cloud type, and C perhaps even larger for dense fogs and mists. The N ¼ v ; ð3:6Þ hVi extinction coefficient, rext, determines the meteorolog- ical range of visibility (Zege et al., 1991; Liou, 1992), where hVi is the average volume of particles (see Eq. which is defined as Sm = 3.91/rext. In particular, we (2.8)) and C is the volumetric concentration of À1 v have at rext = 0.1 m : Sm = 39.1 m. In a cloudless particles, we get from Eq. (3.5) atmosphere, this number is approximately 1000 times larger. 3Cv rext ¼ ; ð3:7Þ A more advanced understanding of the image 2aef reduction by cloudy media can be achieved on the base where we introduced the effective radius of the image transfer theory (Zege et al., 1991; Zege and Kokhanovsky, 1994, 1995; Barun, 1995, 2000; 3hVi aef ¼ : ð3:8Þ Zuev et al., 1997; Katsev et al., 1998). This theory hRi treats a cloud as a high-frequency spatial filter. The The extinction coefficient decreases with the size of cloud greatly reduces high spatial frequencies of particles at C = const. observed objects, causing fine features of objects to v For constant C , the value of r can be expressed be lost due to multiple light scattering and photon v ext via the liquid water content, C , which is often direction randomization process. Clouds with larger w measured in cloudy media: particles have ‘‘better’’ optical transfer functions (higher values) as compared to clouds with very fine 3Cw rext ¼ ; ð3:9Þ droplets, a fact that can be used for cloud microphysical 2qaef parameters monitoring purposes (Zege and Kokhanov- where q = 1 g/cm3 is the density of water. It follows 3 sky, 1992; Zuev et al., 1997). for typical values aef =6 Am and Cw = 0.4 g/m that À 1 The extinction coefficient varies with height in a rext = 0.1 m . This is rather dense cloud with the cloud (see Eq. (3.2)) and, for a given value of the ratio I0/I = e at the distance 10 m for light incident vertical coordinate, z, is given by perpendicular to a cloud layer (see Eq. (3.1)). We should underline two peculiarities of water Z l clouds, which follow from Eq. (3.9). First of all, the rext ¼ N f ðaÞCextda; ð3:3Þ 0 extinction does not depend on the wavelength, k, and, second, it depends only on the ratio of liquid water where N is the number concentration of particles. The content to the effective radius, aef. The first point leads value of the extinction cross section, Cext, is obtained to the conclusion that clouds do not change the spectral from Mie theory valid for spherical droplets (Shifrin, composition of a direct light beam in visible. Thus, 1951; Van de Hulst, 1957; Kerker, 1969; Bohren and spectral transmittance methods of particulate media 198 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 microstructure determination (Shifrin and Tonna, be performed with Mie theory although a simple cor- 1993) cannot be applied in cloud optics. Another im- rection to Eq. (3.9) allows for its use in the near-infrared portant property, which follows from Eq. (3.9), is inde- region of the electromagnetic spectrum (Kokhanovsky pendence of the value of rext on the type of particle-size and Zege, 1995, 1997a,b; Kokhanovsky, 2001a): distribution. Even the width of the PSD is of no im- () portance at a given value of the effective radius of 3Cw t r 1 ; 3:11 ext ¼ þ 2=3 ð Þ particles. This is the reason why the satellite methods of 2qaef ðkaef Þ cloud microstructure determination (Arking and Childs, 1985; Nakajima and King, 1990; Han et al., where k =2p/k and t is the parameter which only 1994) are concerned mostly with the retrievals of aef slightly depends on the dispersion of the particle-size values and the liquid water path (see Eqs. (2.23) and distribution. It follows that for most typical widths of (3.9)) particle-size distributions in clouds t = 1.1 (Kokhanov- 2 sky and Zege, 1997b). We get from Eq. (3.11) that the w ¼ qa s ð3:10Þ 3 ef extinction coefficient of cloudy media slightly in- creases with the wavelength in the visible and near and not the droplet-size distribution itself. infrared. The spectral dependence of rext for the The type of particle-size distribution in clouds is gamma PSD (Eq. (2.3)) at aef =4Am, l = 6 is presen- not possible to find from extinction measurements in ted in Fig. 3 for the spectral range 0.2–5 Am. The the visible. On the other hand, there is an advantage in data were obtained with simple Eq. (3.11) and Mie 3 that one can calculate the extinction coefficient of theory, assuming that Cw = 0.1 g/m . The value of t cloudy media even with limited information on stat- was equal to 1.1. The accuracy of Eq. (3.11) istical properties of droplets and crystals. This is decreases with wavelength/size ratio, but the error is especially important for ice clouds as we will see later. smaller than 5% at k <2 Am and aef >4 Am. Smaller Eq. (3.9) is valid only in the visible range of the values of aef are of a rare occurrence in natural clouds electromagnetic spectrum and for comparatively large (Han et al., 1994). Eq. (3.9) gives a constant value of droplets. Calculations for longer wavelengths should 0.375 mÀ 1 in the case, presented in Fig. 3. We see Fig. 3. Extinction coefficient of a cloudy medium with water droplets, characterized by the PSD (Eq. (2.3)) at aef =4Am, l = 6. The value of Cw is 0.1 g/m3. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 199 ! ! that Eq. (3.11) has a high accuracy and gives the where E0 is the electric field of an incident wave, E is correct spectral dependence the extinction coefficient the electric field inside a particle, k =2p/k is the wave (at least up to the wavelength equal to 1.5 Am). number, m = n À iv is the refractive index, V is the The accuracy of Eq. (3.11) at larger wavelengths volume of a droplet or crystal. The first coarse can be increased if one accounts for the dependence of approximation is to assume that the value of the extinction cross section on the refractive index of particles, which could be done, for instance, in the ! ! !* ! ! ! ! Eðr ÞE ðr ÞcBðnÞE0ðr ÞE0*ðr Þð3:15Þ framework of the Van de Hulst approximation (Acker- man and Stephens, 1987; Kokhanovsky, 2001a). due to the weakness of absorption. Thus, it follows in Parameterizations developed by Mitchell (2000) and this case Harrington and Olsson (2001) can be used even out- C ¼ BðnÞaV; ð3:16Þ side the geometrical optics limit. abs Clearly, the accuracy of Eq. (3.11) increases with where a =4pv/k and we assumed that a particle is aef. Thus, the case presented in Fig. 3 provides the uniform (n and v do not depend on the radius vector r!). maximal error of the approximation (3.11) for terres- Clearly, we have that B ! 1asn ! 1 for arbitrarily trial clouds having aef >4 Am. Typically, errors are shaped particles (Kokhanovsky, 2001a). It follows even smaller. from Eqs. (3.13) and (3.16): 3.1.2. Absorption coefficient rabs ¼ BðnÞaNhVið3:17Þ Clouds both scatter and absorb incident radiation. or The probability of photon absorption, b, is defined by the following equation: rabs ¼ BðnÞaCv: ð3:18Þ rabs b ¼ ; ð3:12Þ We have calculated the value B = rabs/aCv for the PSD r ext (Eq. (2.3)) with l = 6 and aef =4 Am with Mie theory where rabs is the absorption coefficient and rext is the and presented it in Fig. 4 as the function of the wave- extinction coefficient (see, e.g., Eq. (3.11)). The value length. One can see that the value of B almost does not of b is close to zero in the visible and near infrared, depend on the wavelength. This means that also there is which allows to obtain simple equations for reflection only weak dependence of B on n(k), v(k), and xef = kaef and transmission functions of cloudy media in terms at k V1.8 Am.Forlargervaluesofk, Eq. (3.15) of series b j/2, where j is an integer number (Rozen- becomes invalid due to higher absorption of light by berg, 1962, 1967; Van de Hulst, 1980; Minin, 1988; droplets or crystals. We have from Fig. 4: B c 5/3. Kokhanovsky, 2002b). The accuracy of approximation (3.18) at B = 5/3 is The absorption coefficient is defined as: better than 5% below k = 1.8 Am for values of aef in Z l the range 4–6 Am. For larger particles, the accuracy rabs ¼ N Cabs f ðaÞda; ð3:13Þ reduces due to the neccesity to account for terms 0 proportional to a2 and higher. Then Eq. (3.18) should where Cabs is the absorption cross section (Shifrin, be modified (see, e.g., Kokhanovsky and Macke, 1951). It follows for the scattering coefficient, rsca = 1997; Kokhanovsky, 2001a). rext À rabs. The absorption cross section can be cal- Eq. (3.18) can be written as culated from Mie theory for spherical particles. Another way to calculate the absorption cross section, BðnÞaC r ¼ w ; ð3:19Þ which is valid for nonspherical particles as well, is to abs q use the exact integral (Kokhanovsky, 2001a; Markel, where C is the liquid water content and q is the density 2002): w Z of water. One can see that the spectral variation of the 2k ! ! absorption coefficient of liquid clouds approximately ! v ! ! * ! 3!; Cabs ¼ ! nðr Þ ðr ÞEðr ÞE ðr Þd r ð3:14Þ coincides with the spectral dependence of the absorp- E V 0 tion coefficient of liquid water itself (see Fig. 2) at least 200 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 4. Spectral dependence B(k), obtained for the same conditions as in Fig. 3. up to 1.8 Am. The same applies to ice clouds, namely, The parameter of considerable importance in the rabs f 4pvi(k)/k, where vi(k) is the imaginary part of radiative transfer problems in clouds is not the absorp- the refractive index of ice. This feature can be used for tion coefficient itself, but the ratio of the absorption the cloud thermodynamic phase determination by and extinction coefficients. It follows from Eqs. (3.11) remote-sensing techniques (see Fig. 2). Note that B in and (3.19) for this ratio Eq. (3.19) depends on the shape of particles (Kokha- b B˜ aa ; 3:21 novsky and Macke, 1997). ¼ ef ð Þ À 2/3 À1 It should be pointed out that water droplets can where B˜ =(2/3)B(n)(1 + txef ) is close to 1. The collect absorbing particles from surrounding air. Liquid value of b is obtained from the measurement of the aerosol particles penetrate into droplets. They dissolve reflection function of clouds in the infrared (Naka- and change the value of a in Eq. (3.18). Soot particles jima and King, 1990). Thus, Eq. (3.21) can be can form a discontinuous layer on the surface of a used to find the effective radius of droplets droplet. All these effects will produce an enhancement (aef c b/a), which corresponds to a given value of the absorption coefficient as compared to simple Eq. of b. The uncertainty in the value of a (see Eq. (3.18) (Prishivalko et al., 1984; Kokhanovsky, 1989; (3.20)) can introduce additional errors in the Asano et al., 2001). Note that the effects of aerosol retrieved values of the effective radius of droplets particles (e.g., soot) can be modelled by inserting into (Asano et al., 2001). Eq. (3.18) the value of Let us estimate this error, neglecting the term À 2/3 txef for the sake of simplicity. Then it follows: XN a ¼ a þ c a ; ð3:20Þ 3b w i i a ¼ ð3:22Þ i¼1 ef 2Ba where aw and ai are absorption coefficients of water and and, therefore, (daef/aef)=(À da/a). Aerosol particles impurities, respectively, and ci is the relative concen- trapped in droplets make the value of a larger. Thus, tration of impurities. The problem of black carbon the effective radius, aef, of polluted clouds (as influence on the light absorption in cloud was studied obtained from Eq. (3.22) without account for pollu- in detail by Chylek et al. (1996) and Liu et al. (2002). tants) appears to be smaller as compared to values A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 201 obtained from in situ measurements (e.g., using laser (3.11) and (3.12). The effective radii of particles were diffractometers (Kokhanovsky, 2001a)). This was equal to 4 and 16 Am. The accuracy of the approx- found also experimentally (Nakajima et al., 1991). imation is better than 10% at wavelengths smaller To correct this effect, one should use the value of a than 2 Am (see Fig. 6). They are typically smaller than (Eq. (3.20)) in Eq. (3.22). The problem, however, is 5% at k V 1.8 Am. complicated by the fact that values of absorption coefficients and concentrations of impurities are not 3.1.3. Phase function known a priori. On the other hand, one can formu- Until now, we have considered only the extinction late the inverse problem in such a way that they can and the absorption characteristics of cloudy media. be derived from measured spectral reflectances However, clouds not only attenuate propagated sig- simultaneously with the value of the effective radius nals and absorb light but also scatter incident light in of droplets (Zege et al., 1998c). all directions. Generally, the probability of photon Eq. (3.18) is limited to the case of weak absorption. scattering by a droplet depends on the size of the It can be generalized to account for larger absorption droplet and the energy of the incident photon (or the of light by water droplets. We found, parameterizing light wavelength). Mie theory results, that Now we should characterize the probability of photon scattering in a given direction, specified by rabs ¼ BaCv½1 À aaef : ð3:23Þ the scattering angle, h (h = 0 corresponds to the for- ward scattering). It is known that the intensity of light Eq. (3.23) transforms to Eq. (3.18) for small values of scattered by water droplets is much larger in the absorption (aaef ! 0). forward hemisphere than it is in the backward direc- In conclusion, we present the spectral dependence tions (Deirmendjian, 1969; Kokhanovsky et al., of the probability of photon absorption in Fig. 5. Data 1998a). were obtained using Mie theory and approximate Let us introduce the probability dP of light solution, given by Eq. (3.23) with account for Eqs. scattering in the direction, specified by the vector Fig. 5. Probability of photon absorption, obtained for the same conditions as in Fig. 3. Results for the effective radius, aef =16 Am are also shown. 202 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 6. Error of the geometrical optics approximation for the probability of photon absorption, obtained from data presented in Fig. 5 for effective radii 4 and 16 Am. Data for the effective radius 6 Am are also shown. ! X inside the solid angle dX. This probability will be depend on the angle (see Eq. (3.25)). Note that the proportional to the value of dX/4p.Namely,we value of x(X) is sometimes also called the phase have: function or the scattering indicatrix (Ishimaru, 1978; Van de Hulst, 1980). dX dP ¼ xðXÞ ; ð3:24Þ Light scattering by water droplets is azimuthally 4p symmetric due to their spherical shape, thus where x(X) is the coefficient of proportionality. It Z 1 p follows from Eq. (3.24): pðÞh sinhdh ¼ 1: ð3:27Þ Z 2 0 dX xðXÞ ¼ P: ð3:25Þ The phase function p(h) (or the scattering indicatrix) 4p 4p can be calculated from Mie theory (Shifrin, 1951).It The value of P represents the probability of photon does not depend on the concentration of particles, by survival in the scattering process. It is equal to 1 if the definition, but on their refractive index and size. there is no absorption of light by a particle and it is Functions p(h), obtained from Mie theory, for differ- smaller than 1 if some photons are absorbed by a ent particle-size distributions are presented in Fig. 7 at particle. Clearly, the probability of photon survival, P, k = 0.65 Am. Main features of phase functions of water is equal to the single-scattering albedo (Chandrase- clouds in visible are (see Fig. 7): khar, 1950): sharp forward–backward asymmetry; r w ¼ sca : ð3:26Þ weak dependence on the size of droplets in the 0 r ext range of scattering angles 20–60j; It should be pointed out that b = rabs/rext is a small enhanced scattering near rainbow hr (approximately parameter for water clouds (see Fig. 5) in the visible 138j for water droplets) and glory hg (180j) scat- and P u w0 =1À b c 1inthiscase.Thevalue tering angles (Tricker, 1970; Greenler, 1980; Kon- p(X)=x(X)/P is called the phase function. It is equal nen, 1985; Spinhirne and Nakajima, 1994); to one if the probability of light scattering does not strong forward peak. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 203 Fig. 7. Phase functions of water clouds obtained from Mie theory for the same conditions as in Fig. 3. Data for aef equal to 6 and 16 Am are also given for the comparison. Amplitudes of peaks at angles h =0, hr, p can be coming of this equation is that it does not account for used as indicators of the size of droplets (Van de the influence of the size of droplets on the phase Hulst, 1957; Shifrin and Tonna, 1993; Kokhanovsky function (see Fig. 7). However, it can be useful for and Zege, 1997b). general modelling of single and multiple light scatter- Approximate equations for the phase functions of ing in cloudy media (Zege et al., 1993, 1995; Katsev water clouds, derived using the geometrical optics et al., 1998). concept, have been presented by many authors (e.g., The solution of the radiative transfer equation, see Shifrin, 1951; Van de Hulst, 1957; Nussenzveig, which describes the radiative transfer through cloudy 1992; Zege et al., 1993; Kokhanovsky and Zege, media, is simplified if one uses the expansion of the 1997b; Grandy, 2000). A detailed comparison of the phase function in series: geometrical optics and Mie theory was performed by Xl Liou and Hansen (1971), who conclude that the ray pðhÞ¼ asPsðcoshÞ: ð3:28Þ optics results have a high accuracy for large droplets s¼0 with size parameters larger than 400. For some angular regions, errors are small even for compara- Here Ps(cos h) is the Legendre polynomial. However, tively small droplets (see, e.g., the region 20–60j in note that in practice, the number of coefficients, as, Fig. 7). Geometrical optics equations are not very could be taken to be finite. It is approximately equal to simple even for spherical droplets (Kokhanovsky, 2kd, where d is the average diameter of particles and k 2001a). is the wave number (Kokhanovsky, 1997). The coef- Thus, we found, parameterizing Mie theory results, that the phase function of a cloud can be presented Table 3 Parameters b , b , and h approximately by the following equation: i i i ibi bi hi X5 2 1 1744.0 1200.0 0.0 pðhÞ¼QexpðÀChÞþ biexp½Àbiðh À hiÞ ; 2 0.17 75.0 2.5 i¼1 3 0.30 4826.0 p where Q = 17.7, C = 3.9, h is given in radians and 4 0.20 50.0 p 5 0.15 1.0 p constants bi, bi, hi are given in Table 3. The short- 204 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 ficients, as, for cloudy media calculated with Mie extinction coefficient) the coefficient of photon dif- À 1 theory at k = 0.65 Am for the PSD (Eq. (2.3)) with fusion, D=[3rext(1 À g)] , in a cloudy medium. l = 6 are presented in Fig. 8. Approximate equation We stress that it is the value of integral (Eq. for as, based on the combination of the geometrical (3.30)) (and not the phase function itself), which optics and the Fraunhofer diffraction, were obtained determines a number of important radiative charac- by Kokhanovsky (1997, 2001a). They can be used to teristics of clouds (Zege et al., 1991). For instance, avoid long numerical calculations with Mie theory as we have for the value of the total cloud reflectance xef ! l. or the spherical albedo in the case of a semiinfinite We emphasize that both geometrical optics results cloud (Zegep etffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi al., 1991; Kokhanovsky, 2001a): (Kokhanovsky, 1997) and data presented in Fig. 8 rl ¼ expðÀ4 b=3ð1 À gÞÞ . Thus, the asymmetry show that the number smax, which corresponds to parameter needs a special attention. We will consider max the largest coefficient as , is equal approximately this parameter in the next section in more detail. max max to 4xef/5. Also we have approximately s as =2. It follows from Eq. (3.28) using the orthogonality 3.1.4. Asymmetry parameter of Legandre polynomials Let us represent the phase function p(h) of a large Z particle (x l,2x (n 1) l)as(Kokhanov- p ef ! ef À ! 2s þ 1 sky, 2001a) as ¼ pðhÞPsðcoshÞsinhdh: ð3:29Þ 2 0 D D G G In particular, we have a0 = 1 (see Eq. (3.27)) and r p ðhÞþr p ðhÞ pðhÞ¼ sca sca : ð3:31Þ a1 =3g, where D G rsca þ rsca Z 1 p g ¼ pðhÞsinhcoshdh ð3:30Þ D 2 Here p (h) is the phase function associated with the 0 Fraunhofer diffraction of light by a droplet; pG(h)is is the so-called asymmetry parameter, giving the the phase function associated with rays, which pene- average cosine of the photon-scattering angle. Note trate the particle, reflect, and refract inside the droplet; G D that the value of g determines (together with the rsca and rsca are scattering coefficients associated with Fig. 8. Coefficients as obtained for the same conditions as in Fig. 3. Data for aef equal to 6 Am are also shown. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 205 these processes. Note that Eq. (3.31) corresponds to of the electromagnetic spectrum (see Fig. 1). We will the Van de Hulst’s (1957) localization principle. It assume that n = 1.333. Then it follows that gG = holds only approximately and ignores the possible 0.7686 (Kokhanovsky, 2001a) and g = 0.8843 accord- interference of waves (Glautshing and Chen, 1981) ing to Eq. (3.32), where we accounted for the equality D G that originate due to separate diffraction and geo- rsca = rsca, which also holds in the visible region of metrical optics scattering processes. the electromagnetic spectrum, where absorption can It follows from Eqs. (3.30) and (3.31) be neglected (Van de Hulst, 1957).Thevalueof asymmetry parameter g of water clouds, in practice, rD gD þ rG gG weakly depends on the size of particles and wave- g sca sca ; 3:32 ¼ D G ð Þ length. This dependence can be approximated by the rsca þ rsca following equation (Kokhanovsky, 2001a): where C Z p g ¼ g0 À ; ð3:35Þ 1 x2=3 gD ¼ pDðhÞsinhcoshdh; ð3:33Þ ef 2 0 Z where C is the constant, which does not depend 1 p on the size of droplets (but possibly depends on gG ¼ pGðhÞsinhcoshdh: ð3:34Þ their shape) and g0 u g(xef ! l). The integral for 2 0 g0 is given by Van de Hulst (1957). It depends One can obtain (Van de Hulst, 1957; Kokhanovsky, only on the refractive index of a particle and equal 2001a,b) that gD c 1. Note that the function pG(h) approximately to 0.8843 at n = 1.333, as stated does not depend on the size of large nonabsorbing above. 2/3 particles (Van de Hulst, 1957; Kokhanovsky and Results of calculations of the value C=(g0Àg)xef Nakajima, 1998).Thus,gG depends only on the with Mie theory are presented in Fig. 9 for the refractive index, n, of water in this case. The value particle-size distribution (Eq. (2.3)) at l =6, aef =4 of n does not change considerably in the visible region Am, and k V 1 Am. It follows that C c 0.5. Thus, we Fig. 9. Spectral dependence C(k) obtained for the same conditions as in Fig. 3. 206 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 have from Eq. (3.35) the value of G =1À g, which is total reflectance. Smaller values of hh2i mean that often called the co-asymmetry parameter: photons penetrate to larger depths in cloudy media before their escape back to the free space, where the 1 source of radiation is located. This also makes the G ¼ 0:12 þ : ð3:36Þ 2x2=3 total path lengths of photons in cloudy media larger, ef which in turn produces the larger total absorption of 2 It is the value of G and not g itself which plays an radiation inside clouds with larger values of hh i. important role in the radiative transport in cloudy The total fraction of radiation absorbed by a semi- media. For instance, values of g equal to 0.75 (typical infinite cloud is given by the following expression (Kokhanovsky, 2001a): for ice clouds) and 0.85 (typical for water clouds) are ()sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi comparatively close to each other, but corresponding b values of G (0.25 and 0.15, respectively) differ. This is A ¼ 1 À exp À4 3 1 g a noteworthy feature for comparing cloudy media ð À Þ with different values of asymmetry parameters. or It is important to understand the physical meaning sffiffiffiffiffiffiffiffiffiffiffi of the parameter G. For this, we use the expansion 2b 2 A ¼ 4 cosh =1À (h /2) + ... in Eq. (3.30).R Then it follows 3hh2i 2 2 1 p 2 that G = hh i/2, where hh i¼2 0 h pðhÞsinhdh is the averaged square of the scattering angle. We neglect as b ! 0. Clearly, the ratio high-order terms in the expansion of cosh which is A possible due to a peaked character of the phase n¯ ¼ ; b function in Eq. (3.30) (see Fig. 7). The average value of the squared scattering angle, can be used as an estimation of the average number of therefore, governs the radiative transport in cloudy photon-scatteringqffiffiffiffiffiffiffiffiffiffi events in a cloud layer. In particular, media. In particular, semiinfinite clouds with close we get n¯ ! 4 2 as b ! 0. Smaller values of b 3bhh2i values of ratios b/hh2i also have close values of the and hh2i give larger values of n¯ as one might expect. Fig. 10. Asymmetry parameter obtained for the same conditions as in Fig. 3. Results for the effective radius aef =6 Am are also shown. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 207 Fig. 11. Error of the geometrical optics approximation for the asymmetry parameter obtained from data presented in Fig. 10 for effective radii 4 and 6 Am. Eq. (3.36) can be used only for nonabsorbing It was shown (Kokhanovsky et al., 1998a) that it holds channels (e.g., in visible). However, it can be modi- approximately for water clouds: fied in a full analogy with Eq. (3.23) to account for z A 1 light absorption at k 1 m (Kokhanovsky, 2001a): F 0:03 : ¼ þ 2=3 1 5xef G 0:12 eaa : 3:37 ¼ þ 2=3 À ef ð Þ 2xef It suggests that approximately 97% of light is scattered by a local volume of a cloudy medium in We obtained fitting Eq. (3.37) to Mie theory results the range of angles smaller than 90j. The backscatter- that e = 0.15. Comparisons of calculations with simple ing signal is, therefore, quite low. It increases, how- Eq. (3.37) and Mie theory are presented in Fig. 10. ever, due to multiple-scattering processes that take They were performed for the particle-size distribution place inside a cloud (Bissonnete et al., 1995; Zege et A A (Eq. (2.3)) at l =6, aef =4 m, and aef =6 m for al., 1995, 1998a,b). wavelengths, k V 2.4 Am. It follows that the accuracy of Eq. (3.37) (see Fig. 11) is better than 5% at aef =6 3.2. Ice clouds Am in visible and near infrared for k V 2.3 Am. It is A also better than 5% for droplets with aef =4 m for 3.2.1. Extinction coefficient V A values of k 2.0 m. Local optical characteristics of ice and mixed Another important characteristic of the phase func- clouds cannot be calculated so easily as for water tion of a cloudy medium is the probability of light clouds. In particular, the Mie theory (Shifrin, 1951) scattering in the backward hemisphere: cannot be used due to the complex shape and internal Z structure of ice crystals (Takano and Liou, 1989; 1 p F ¼ pðhÞsinhcoshdh: Macke, 1993, 1994; Macke et al., 1996, 1998; Mitch- 2 p=2 ell and Arnott, 1994; Yang and Liou, 1998; Yang et 208 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 al., 2000, 2001). Main results obtained in the optics of will provide us with only a coarse approximation. It crystalline clouds were summarized by Volkovitsky et follows from Eq. (3.42) that al. (1984) and Liou (1992, 2002). 3Cv Ice crystals are usually much larger than the wave- aef ¼ : ð3:44Þ length of the incident radiation. Thus, the extinction 2rext cross section, C , does not depend on the wavelength ext Eq. (3.44) suggests a method for indirect determi- and the refractive index of particles (Shifrin, 1951; nation of the effective radius of particles in complex Van de Hulst, 1957): crystalline cloudy media. Cext ¼ 2s; ð3:38Þ 3.2.2. Absorption coefficient where s is the cross section of a particle, projected on The absorption cross section of a single crystal at the plane perpendicular to the incidence direction. It small absorption (aaef ! 0) can be found in a full follows for N identical crystals in a fixed orientation anology with Eq. (3.16): that Cabs ¼ BaV; ð3:45Þ rext ¼ 2Ns; ð3:39Þ where V is the volume of a crystal, a =4pv/k and B is where N is number concentration of particles and rext is the coefficient proportionality, which depends on the the extinction coefficient. However, crystals differ by shape and the real part of the refractive index of their shape, size, and orientation; thus, the extinction particles (Kokhanovsky and Macke, 1997), but not coefficient can be calculated as on their size (see Eq. (3.14)). For an ensemble of crystals of identical shapes: rext ¼ 2Ns¯; ð3:40Þ rabs ¼ BaCv ð3:46Þ where s¯ is the average cross section of particles. This and for N distinct shapes of crystals in a cloud: equation can be written as XN s¯ rabs ¼ a BjCvj: ð3:47Þ rext ¼ 2Cv ; ð3:41Þ j¼1 V¯ Values of Bj do not depend on the imaginary part of where Cv is the volumetric concentration of crystals the refractive index and the size of crystals, but only ¯ ¯ and V is the average volume of crystals (Cv = NV). It on the real part of the refractive index, n, of crystals follows for convex crystals of the same shape in and their shapes. The most probable values of B= ¯ random orientation (Van de Hulst, 1957) that R=4s¯, (1/C )SN B C in crystalline clouds can be found where R¯ is the average surface area of particles. There- v j =1 j vj from experimental measurements of ratios rabs/Cva in fore, we get: ice clouds of different microstructures. The probability of photon absorption b = rabs/rext 3Cv can be found from Eqs. (3.42) and (3.47): rext ¼ ; ð3:42Þ 2aef b ¼ Naaef ; ð3:48Þ where where 3V¯ aef ¼ ð3:43Þ XN R¯ 2 N ¼ BjCvj: ð3:49Þ 3Cv is the effective radius of particles. This result is similar j¼1 to Eq. (3.7). Only a portion of ice crystals (e.g., Using the model of fractal particles (Macke, 1994), hexagonal plates and columns) are convex. For this one can obtain that N c 1.8. Due to the complexity of and other reasons, one should expect that Eq. (3.42) crystal shapes in ice clouds, the accurate value of N A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 209 cannot be found theoretically. Thus, measurements of which in many cases (see, e.g., Eqs. (3.21) and (3.48)) this parameter are needed. do can be modelled by a collection of spherical It is readily apparent from Eq. (3.48) that the particles (Grenfell and Warren, 1999). single-scattering albedo The phase function of hexagonal cylindrical crys- tals outside the diffraction region, presented in Fig. w0 ¼ 1 À Naae ð3:50Þ 12, was obtained using the Monte-Carlo ray tracing approach (Macke, 1994) for identical hexagonal ice for crystalline media. Note that the formulae presented cylinders of length 0.5 mm in random orientation at here are valid only as aaef ! 0. Generalization of wavelength 0.5 Am. The side of the cross section was these equations for arbitrary light absorption by drop- taken to be equal to 0.08 mm. Dominant features of lets was given by Kokhanovsky and Macke (1997). the phase functions of this type are enhanced scatter- Most recent parameterizations were developed by ing at angles of 22j and 46j, corresponding to the so- Yang et al. (2000) and McFarquhar et al. (2002). called halos. Greenler (1980) states that he observes 22j halos in Wisconsin on 70 to 80 days of a typical 3.2.3. Phase function year. The second halo at 46j is of a rare occurrence in The phase function of ice clouds in the visible is natural conditions due to the presence of crystals of the average over ensemble of phase functions of other shapes, which obscure halo phenomena. Both crystals with different shapes (Takano and Liou, rainbow and glory (Tricker, 1970; Greenler, 1980; 1995). It depends on the size of crystals in the Konnen, 1985) are absent for ice clouds. small-angle scattering region and primarily on shape The backward cross section of hexagonal ice and structure of crystals at larger scattering angles. crystals of arbitrary orientation is calculated for visi- The phase function of ice clouds cannot be modelled ble light by means of a ray-tracing code by Borovoi et (Liou and Takano, 1994) by spherical polydisper- al. (2000). It is shown that backscattering of the tilted sions. This is in contrast with integral light-scattering crystals is caused by a corner reflectorlike effect. A and absorption characteristics of crystalline clouds, very large peak of backscattering is found for a tilt Fig. 12. Phase functions of hexagonal ice cylinders with the aspect ratio(length/size of the side of the hexagonal cross section) of 5.88 and fractal particles in random orientation at the wavelength 0.5 Am. Only the geometrical optics contribution of both phase functions is shown. 210 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 angle of 32.5j between the principal particle axis and al. (1999)). In particular, the phase function of hori- the incidence direction. This peak is caused by multi- zontally oriented crystals (e.g., plates) depends on ple total internal reflections for part of the rays that are directions of light incidence and observation sepa- incident upon the skewed rectangular faces. rately and not on a single parameter (scattering angle) Realistic phase functions of a crystalline cloud that as for spherical or randomly oriented particles. Such account also for diffraction of light at small angles and clouds belong to a broad class of the so-called size/shape distribution of crystals are tabulated by anisotropic media. The radiative transfer in ice clouds Takano and Liou (1989) and Liou (1992) and are with horizontally oriented crystals was studied by characterized by an asymmetry parameter equal to Liou (2002). This is a major topic of modern cloud 0.75. optics research. Note that clouds with horizontally Calculation of phase functions of crystals of differ- oriented crystals are characterized by dichroism and ent shapes can be avoided by introducing the single birefringence (Van de Hulst, 1957). fictive particle with the phase function similar to the phase function of an ensemble (Kokhanovsky, 2001a, 3.2.4. Asymmetry parameter 2003a,b,c). Particles with random stochastic surfaces Asymmetry parameters of crystals depend on the were studied by Peltoniemi (1993) and Muinonen et shape, but not on their size in the visible (Macke et al., al. (1996). The model of a fractal particle was intro- 1996; Kokhanovsky and Nakajima, 1998), due to duced by Macke and Tzschichholz (1992), Macke large size of ice crystals. Size-dependent wave cor- (1994), and Macke et al. (1996). Depending on the rections (see, e.g., Baran and Havemann, 1999), such parameters of roughness, these models can quite well as given by the second term in Eq. (3.36), can be describe the phase functions of ice crystals (Macke, neglected for crystals as k ! 0. 1994; Kokhanovsky, 2003a,b,c) and the mineral frac- The real part of the refractive index of ice crystals tion of the atmospheric aerosol (Volten et al., 2001; is also important, but it varies only slightly in the Kokhanovsky, 2003a,b,c). visible (see Fig. 1); this dependence can be neglected. Phase function of a fractal particle, which may be a The asymmetry parameter for ice clouds will be the better description of complex shapes in crystalline average value over an ensemble of particles of differ- clouds, is presented (along with that of a randomly ent shapes. Macke et al. (1998) found from extensive oriented hexagonal cylinder) in Fig. 12. Data were numerical calculations that the asymmetry parameter, obtained using the Monte-Carlo ray-tracing code g, is in the range 0.79–0.85 for columns, 0.83–0.94 (Macke, 1994; Macke et al., 1996). The fractal phase for plates, and 0.74 for polycrystals, represented by a function is characterized almost the same value of the fictive fractal particle in the visible. It is in the range asymmetry parameter ( g c 0.74) as the phase func- 0.83–0.87 for spherical droplets. We see, therefore, tion of randomly oriented hexagonal crystals given in that the value of g for cloudy media with ice crystals the same figure. should be somewhere between 0.74 and 0.87. Larger Linear combinations of functions presented in Figs. values of g indicate that ice clouds are composed of 7 and 12 with weights depending on the concentration plates only, which is never the case in natural con- of spherical particles, hexagonal cylinders and irreg- ditions. We see that the co-asymmetry parameter, ularly shaped particles in the cloud can be used to G=1À g, changes from 0.13 for water clouds to describe natural clouds. Note that other regular 0.26 for irregularly shaped particles. Actual values shapes, such as plates, can also contribute to the total of g for a given cloud can be found only from direct phase function of an elementary volume of a cloudy measurements in natural ice clouds such as were medium. Measurements of phase functions of crystal- performed by Garett et al. (2001), who found that line clouds were presented by Volkovitsky et al. the value of g is in the range 0.73–0.76, depending on (1984), Gayet (1998), Saunders et al. (1998), Barkey the cloud area under investigation. They obtained et al. (1999), Baran et al. (2001) and Jourdan et al. values of g from measurements of the phase function (2003). inside the ice cloud in the angle range 10–175j.In Horizontal orientation of crystals may complicate one case, the value of g reached 0.81, but sublimation the phase function in up to 40% of clouds (Chepfer et was noted, rendering it to be a typical case. The A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 211 asymmetry parameter of ice clouds does not vary assume constant values for them. Measurements by considerably for visible light and, on average, is equal Garett et al. (2001) suggest that Gi =0.26 and to 0.745, a value close to that for a fractal fictive Gw = 0.13. Then we have from Eq. (3.51): particle and to the value for hexagonal crystals, which G give the phase function, presented in Fig. 12. ¼ À 2 ð3:52Þ 0:13 We suggest that an asymmetry parameter value of 0.74 should be used for theoretical modelling of light or approximately =(4hh2i)À2. The correlation propagation in crystal clouds. The corresponding between values of G and was confirmed experimen- value of G is equal to 0.26, which does not depend tally (Garett et al., 2001). Therefore, measurements of G on the size of crystals because they are large as (or, alternatively, hh2i) in mixed clouds can yield the compared to the wavelength in the visible range of value of a liquid fraction, . the electromagnetic spectrum. G does not depend on It is evident (Kokhanovsky and Macke, 1997; the shape of crystals because for a given cloud, we Kokhanovsky, 2001a) that in the near-infrared range have a statistical and very broad distribution of of the electromagnetic spectrum the larger size of shapes, which produce in the end a saturated value crystals generally leads to larger asymmetry parame- of the asymmetry parameter for a completely chaotic ter. Detailed calculations of optical characteristics of scattering. This scattering can be modelled by a single ice crystals in the infrared for different shapes of fractal fictive particle (Macke, 1994; Kokhanovsky, crystals were performed by Zhang and Xu (1995) 2003a). It should be stressed, however, that the and Macke et al. (1998). Parameterization schemes for calculation of g involves also averaging over scatter- local optical characteristics of ice clouds were devel- ing angle (see Eq. (3.30)). Thus, the model of chaotic oped by Ebert and Curry (1992), Fu and Liou (1993), scattering is more appropriate for the value of g than Fu (1996), Kokhanovsky and Macke (1997), Yang et for the phase function itself. The phase function al. (2000), Harrington and Olsson (2001), and McFar- should be modelled as a combination of light scatter- quhar et al. (2002). Selected parameterizations of ing by regular and irregularly shaped particles with cloud local optical characteristics given in this review different weights, as discussed above (see, e.g., Liou, are summarized in Table 4. 2002). Again, such a model can be established only from measurements performed in natural clouds. Now we consider briefly the case of mixed ice– 4. Radiative characteristics water clouds for which the value of G is G ¼ Gi þ Gw; ð3:51Þ 4.1. The visible range where Gi and Gw are co-asymmetry parameters of ice Let us consider now the global optical character- and water particles, respectively, and x is the fraction istics of clouds such as their reflection and trans- of water droplets in a mixed cloud, defined as the ratio mission functions (Van de Hulst, 1980). They can be of number of droplets to a total number of particles in measured remotely by airborne, satellite and ground- a mixed cloud. Gi and Gw do not vary significantly in based radiometers and spectrometers (Kondratyev and mixed clouds; thus, in the first approximation, we can Binenko, 1984; Liou, 2002). The aim of this section is Table 4 Integral local optical characteristics of clouds in visible and near infrared LOC Water cloud Ice cloud Range of validity rabs (5/3)a(1 À aaef)Cv BaCv aaef ! 0 À 2/3 rext (3/2aef)Cv(1 + 1.1xef ) (3/2aef)Cv xef ! l,2xef (n À 1) ! l À 2/3 1 À g 0.12 + 0.5xef À 0.15aaef 0.26 xef ! l,2xef (n À 1) ! l, aaef ! 0 Here a =4pv/k and xef =2paef/k. The parameter B depends on the shape of crystals in a cloud (Kokhanovsky and Macke, 1997). It follows for fractals: BÂ1.8. 212 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 to introduce simple analytical formulae, which can be effective radius in Fig. 13 were 6 and 16 Am, which used to relate global optical characteristics of clouds cover the typical range for natural water clouds. to their local optical characteristics discussed in the R(#0,#,u) can be smaller and larger than 1, depending previous section. This will also mean that we obtain on the incidence angle, which implies that depending direct relationships between observable quantities, on particular viewing geometries, clouds can be even such as cloud reflection and transmission functions, more reflective than the ideally white Lambertian with microphysical parameters of clouds (e.g., the size surface. This is mostly due to peculiarities of the and refractive index of particles, see Table 4). phase function of cloudy media [e.g., in the back- We start from the study of the reflection function of scattering (# c #0,u c p) region]. It follows from a cloudy medium under an assumption that the cloud Fig. 13 that in the range of solar angles 30–60j and can be represented as a homogeneous plane-parallel nadir observation, the reflection function of a semi- layer. The absorption of light by particles is neglected infinite water cloud is almost equal to the reflection in this section. This is a standard assumption in the function of an ideally white Lambertian reflector. It visible range of the electromagnetic spectrum. How- differs from 1 not more than by 10% for these geo- ever, in some cases the absorption of light by water metries (Zege et al., 1991). 0 clouds in visible cannot be neglected. This is due to The reflection function Rl(#0,#,u) can be repre- various sources of pollution (Melnikova and Mikhai- sented by the following simple approximate equation lov, 1994, 2000; Asano et al., 2002; Vasilyev and (Kokhanovsky, 2002a): Melnikova, 2002). 0 b1 þ b2cos#cos#0 þ pðhÞ The reflection function of a cloud R(#0,#,u)is Rlð#0;#;uÞ¼ ð4:5Þ defined as the ratio of reflected light intensity 4ðcos# þ cos#0Þ I(# ,#,u) for the case of a cloud to that of an ideal 0 where h = arccos(À cos#cos#0 + sin#sin#0cosu) is the Lambertian ideally white reflector scattering angle, p(h) is the phase function of a cloudy medium, and b1 and b2 are constants. Eq. (4.5) obeys Ið#0;#;uÞ 0 0 Rð#0;#;uÞ¼ ; ð4:1Þ to the reciprocity principle [Rl(# ,#,u)=Rl(#,# ,u), I*ð# Þ 0 0 0 see, e.g., Zege et al., 1991]. where We found that b1 = 1.48, b2 = 7.76 for nadir obser- vations (Kokhanovsky, 2002a). Comparison of the I*ð#0Þ¼Fcos#0 ð4:2Þ approximation and exact data in Fig. 13 shows that the accuracy of Eq. (4.5) is better than 2% at #0 <85j, is the intensity of light reflected from the ideally white # =0j. Constants b1 and b2 for other viewing geo- Lambertian reflector, pF is the solar flux on the area metries can be found using parametrizations of results perpendicular to the direction of incidence, #0 is the obtained from the exact radiative transfer codes (see, solar angle, # is the observation angle, and u is the e.g., Mishchenko et al., 1999). Approximations for the relative azimuth between solar and observation direc- 0 function Rl(#0,#,u)at# p 0j were also obtained by tions (Minin, 1988). Also, we will use l = Acos#A, Melnikova et al. (2000a,b) and Kokhanovsky (2003c). l0 = cos#0. For the Lambertian ideally white reflector Eq. (4.5) can be used to find the reflection function from Eq. (4.1), R u 1, which, by definition, does not of a finite cloud R(l,l0,u,s) via the following formula depend on the viewing and observation geometry. (Germogenova, 1963; Van de Hulst, 1980, Minin, Although clouds are white when looking from 1988): space, their reflection function R(#0,#,u) is not equal 0 to one, but depends on the viewing and observation Rðl; l0; u; sÞ¼Rlðl; l0; uÞÀtðsÞK0ðlÞK0ðl0Þ; geometry. Results of calculations of the reflection ð4:6Þ function of an idealized semiinfinite nonabsorbing where 0 1 water cloud Rl (#0,#,u) at the wavelength, k = 0.65 t ¼ ð4:7Þ Am and the nadir observation are presented in Fig. 13. 0:75sð1 À gÞþa Calculations were performed for the gamma particle- is the global transmittance of a cloud, g is the asym- size distribution, given by Eq. (2.3) at l = 6. Values of metry parameter studied above in detail, and K0(l)is A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 213 0 Fig. 13. Reflection function of an idealized semiinfinite nonabsorbing cloud Rl(0,#0,0) obtained from the exact radiative transfer code (Mishchenko et al., 1999) and approximation (4.5) at the wavelength k = 0.65 Am and the effective radii of droplets aef = 6 and 16 Am. It is assumed that particles in a cloud are characterized by the gamma particle-size distribution (Eq. (2.3)) with the parameter l =6. the escape function (Van de Hulst, 1980). Eqs. (4.6) where Pl is the Legendre polynomial and, according to and (4.7) are valid only in the absence of light Eq. (4.10), is presented in Fig. 14. We see that K0(l) absorption in a cloud. The escape function is defined almost does not depend on g at l z 0.2 (# <78j). This via the solution of the characteristic integral equation is the case even for g = 0 and for the Mie-type phase (Van de Hulst, 1980). Note that the parameter a in Eq. functions (Kokhanovsky, 2001a).Thereissome (4.7) is defined as follows (Sobolev, 1972): dependence of the escape function on the microstruc- Z ture of the cloud at the range of observation angles 1 2 # =80–90j, but the cloud-top inhomogeneity plays a a ¼ 3 K0ðlÞl dl: ð4:8Þ 0 role at such grazing observation angles, preventing use of the idealized plane-parallel layer approximation. The escape function, K0(l), only weakly depends on The variability of K0(l)atl = 0.2–1.0 for different the cloud microstructure and can be presented by the values of g = 0.75–0.9 is well inside the 2% sensitiv- following equation (Zege et al., 1991; Kokhanovsky, ity range. This coincides with the error of Eq. (4.9) at 2001a): l z 0.2, being smaller than 2%. 3 Note that function (4.9) also describes the angular K ðlÞ¼ ð1 þ 2lÞ: ð4:9Þ 0 7 distribution of solar light transmitted by a cloud (Zege et al., 1991; Kokhanovsky, 2001a). The function K0(l0) calculated with exact radiative The substitution of Eq. (4.9) into Eq. (4.8) yields: transfer code with g equal to 0.75, 0.85, and 0.9 in the 15 case of Heney–Greenstein phase function a ¼ c1:07 ð4:11Þ Xl 14 l pðhÞ¼ ð2l þ 1Þg PlðcoshÞ; ð4:10Þ independently of cloud microstructure. The value of a, l¼0 numerically calculated by King (1987), assuming the 214 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 14. Escape function, calculated with exact radiative transfer code for the Heyney–Greenstein phase function at g = 0.75, 0.8 and 0.9 (Yanovitskij, 1997) and with approximation given by Eq. (4.9). fair weather cumulus cloud model, is given approx- 15, prepared from data given by Trishchenko et al., imately by 1.07 in agreement with our estimation. 2001). Kokhanovsky and Rozanov (2003) found that King (1987) used Mie theory to find the phase the error of the modified reflection function at the function of a cloudy medium for the calculation of nadir observation is smaller than 5% for solar angles K0(l0) and a via Eq. (4.8). Yanovitskij (1997) found smaller than 74j. the same value of a for Heney–Greenstein phase Eq. (4.6) is readily modified to account for the functions with asymmetry parameters in the range Lambertian light reflection from the underlying sur- 0.0–0.9. This supports the approximation of a fixed face (Sobolev, 1972): value of a in Eq. (4.7), given by Eq. (4.11), inde- r TðlÞTðl Þ pendently of cloud microphysical parameters. Rˆ ðl; l ; u; sÞ¼Rðl; l ; u; sÞþ s 0 ; 0 0 1 À r r The error of Eq. (4.6) is smaller than 1% at s z 10 s (King, 1987). It rapidly increases with s. Kokhanov- ð4:12Þ sky and Rozanov (2003) proposed to modify Eq. (4.6) ˆ to diminish the error of this equation for smaller s.In where R is the reflection function of a Lambertian particular, they proposed to use instead of t in Eq. surface-cloud system, rs is the spherical albedo of the (4.6): Lambertian surface, which may depend on the wave- length, R u Rˆ (rs = 0), 2 2 t* ¼ t Àt˜; t˜ ¼ð4:86À13:08ll0 þ12:76l l0Þ=s: TðlÞ¼tK0ðlÞð4:13Þ is the diffuse transmittance of a cloud layer (Sobolev, This modification allows to increase the accuracy 1972) and r =1À t is the spherical albedo of a non- of approximate formulae considered for smaller s absorbing cloud. (down to s = 5). This range of optical thicknesses is We have neglected the direct solar light term (Liou, that most frequently observed in water clouds both 2002) in Eq. (4.12) due to a large thickness of clouds with satellite and ground-based techniques (see Fig. under consideration. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 215 Fig. 15. Frequency of registration of different optical thicknesses of cloudy media obtained from satellite and ground measurements as presented by Trishchenko et al. (2001). Finally, substituting Eqs. (4.6) and (4.13) into de Hulst, 1980; King, 1987; Nakajima and King, Eq. (4.12), it follows for the reflection function of 1992; Vasilyev and Melnikova, 2002): a Lambertian surface-nonabsorbing cloud system (Kokhanovsky et al., 2003): Rðl; l0; u; sÞ¼Rlðl; l0; uÞ À2cs 0 mle Rˆ ðl; l ; u; sÞ¼Rlðl; l ; uÞ À KðlÞKðl Þ; ð4:15Þ 0 0 1 À l2eÀ2cs 0 tð1 À rsÞ À K0ðlÞK0ðl0Þ: where c is the so-called diffusion exponent, K(l) and 1 À rsð1 À tÞ Rl are the escape function and the reflection function ð4:14Þ of an absorbing semiinfinite medium with the same local optical characteristics as a finite layer under This formula can be used as a basis for the semi- study, respectively. Eq. (4.15) accounts for the influ- analytical cloud retrieval algorithm (Kokhanovsky et ence of light absorption on the reflection function of ˆ u 0 al., 2003). Note that R(l,l0,u,s) Rl(l,l0,u)at clouds. The reflection function decreases if additional rs =1. absorbers are present in cloud droplets. Constants l and m are defined by the following integrals (Van de 4.2. The near-infrared range Hulst, 1980): Z Unfortunately, the relatively simple Eq. (4.6) can- 1 not be used to calculate the reflection function of a l ¼ 2 i2ðgÞgdg; ð4:16Þ finite cloud in the near-infrared region of the electro- À1 Z magnetic spectrum due to the presence of absorption 1 bands of liquid water (see Fig. 2). Alternatively, the m ¼ 2 KðgÞiðÀgÞgdg; ð4:17Þ following formula applies (Germogenova, 1963; Van 0 216 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 i(g) being the angular distribution of light in deep the similarity parameter in the radiative transfer text- layers of a cloud, defined as the so-called asymptotic books (Van de Hulst, 1980). It is a useful parameter, regime (Sobolev, 1972). describing the optical properties of clouds. We see that Functions Rl(l,l0,u), K(l), and constants m and l y ! 4s as x0 ! 1. Substituting y into Eqs. (4.18)– have the following asymptotic forms when light (4.21) yields: absorption by droplets is relatively small (x0 ! 1) 0 (Van de Hulst, 1980): Rlðl; l0; uÞ¼Rlðl; l0; uÞÀyK0ðlÞK0ðl0Þ; ð4:22Þ 0 Rlðl; l ; uÞ¼R ðl; l ; uÞ 0 l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 ay KðlÞ¼K0ðlÞ 1 À ; ð4:23Þ 1 À x0 2 À 4 K0ðlÞK0ðl0Þ; ð4:18Þ 3ð1 À gÞ m ¼ 2y; ð4:24Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! and 1 À x0 l ¼ 1 À ay: ð4:25Þ KðlÞ¼K ðlÞ 1 À 2a ; ð4:19Þ 0 3ð1 À gÞ Eqs. (4.22)–(4.25) were derived assuming that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ! 1. Alternatively, the right-hand sides of Eqs. (4.22)–(4.25) give us the first terms of the expansion 1 À x0 m ¼ 8 ; ð4:20Þ of corresponding functions with respect to y (Minin, 3ð1ÀgÞ 1988). The accuracy of these equations decreases with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y quite rapidly. The next terms of the expansions (f y2) are given by Minin (1988) and Melnikova et 1 À x l ¼ 1 À 4a 0 ; ð4:21Þ al. (2000b). Still higher terms are not known or quite 3 1 g ð À Þ complex (Minin, 1988), but it has been shown that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the following equations account approximately for and c ! 3ð1 À x0Þð1 À gÞ as the single-scattering higher-order terms (Rozenberg, 1962; Zege et al., 0 albedo, x0!1. Thus, at x0=1, Rl=Rl, K(l)=K0(l), 1991; Kokhanovsky et al., 1998b, 2003): m = l = 0, and Eq. (4.15) transforms into Eq. (4.6). 0 Rlðl; l ; uÞ¼R exp½Àyuðl; l ; uÞ ; ð4:26Þ Note that Eqs. (4.18)–(4.21) follow from the 0 l 0 asymptotic analysis of the radiative transfer equation mKðlÞKðl0Þ¼½1 À expðÀ2yÞ K0ðlÞK0ðl0Þ; (Van de Hulst, 1980). Integration of Eq. (4.18) with ð4:27Þ qrespectffiffiffiffiffiffiffiffiffiffiffi to all angles yields rl =1À y, where y ¼ 4 1Àx0 l ¼ expðÀayÞ; ð4:28Þ 3ð1ÀgÞ and rl is the spherical albedo of a weakly absorbing semiinfinite cloud (Van de Hulst, 1980). where a viewing function u is defined by Thus, the parameter y can be interpreted as a fraction K0ðlÞK0ðl Þ of photons absorbed in a weakly absorbing semi- uðl; l ; uÞ¼ 0 ð4:29Þ 0 R0 ðl; l ; uÞ infinite cloud. It depends both on the single-scattering l 0 albedo and the asymmetry parameter. Larger values of and does not depend on x0 and s. The viewing g imply that photon scattering increases at small function has only a weak dependence on the micro- angles. Thus, the photon path length before its escape physical characteristics of clouds (e.g., the droplet-size from the medium is also increased. As a consequence, distribution). This follows from the low sensitivity of this results in increased light absorption in the 0 the functions Rl(l,l0,u) and K0(l) in Eq. (4.29) to the medium. Clouds having larger values of g, therefore, microphysical characteristics of clouds (see Figs. 13 absorb more light. Media having different values of and 14). The physical sense of the function (4.29) has x0 and g, but the same values ofqyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, have the same been discussed by Kokhanovsky (2002b). The accu- 1Àx values of rl. The parameter s ¼ 0 is called racy of Eq. (4.26) was studied by Kokhanovsky 3ð1Àx0gÞ A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 217 (2002c). It is better than 10% at y V 0.8 for the nadir Eq. (4.30) can be used to determine the optical observation and solar angles in the range 0–75j. thickness and effective radius from spectral reflection Eqs. (4.26)–(4.28) avoid the solution of integral function measurements over extended cloud fields equations (Van de Hulst, 1980; Yanovitskij, 1997) for (Kokhanovsky and Zege, 1996; Kokhanovsky et al., the determination of functions Rl(l,l0,u) and K(l)in 2002). Eq. (4.15). They transform into exact asymptotic Surface reflection is accounted for by Eq. (4.12). results (Eqs. (4.22)–(4.25)) as y ! 0. Substitution of Eq. (4.30) into Eq. (4.12) yields: Substituting Eqs. (4.27) and (4.28) into Eq. (4.15), we have: ˆ Rðl; l0; u; sÞ¼Rlðl; l0; u; sÞ ÀxÀy ! Rðl; l ; u; sÞ¼Rlðl; l ; uÞÀte K ðlÞK ðl Þ; 0 0 0 0 0 trs À expðÀx À yÞÀ Tðl; l0Þ; ð4:30Þ 1 À rsr where x = cs and the global transmittance, t, is given ð4:33Þ by (Kokhanovsky et al., 2003): where sinhy t ¼ ð4:31Þ sinhðay þ xÞ Tðl; l0Þ¼tK0ðlÞK0ðl0Þð4:34Þ Eq. (4.31) yields Eq. (4.7) at x = 1. The value of 0 is the transmittance function of a cloud, r is the total exp(À x) describes light attenuation in deep layers of a reflectance of a cloud, t is given by Eq. (4.31), and Rl cloud (Van de Hulst, 1980). is given by Eq. (4.32). It is assumed that Eq. (4.30) was first proposed by Rozenberg (1962, tK(l)K(l ) c tK (l)K (l ). This assumption is valid 1967), but he assumed that a = 1 in Eq. (4.31), which 0 0 0 0 for single-scattering albedos close to one. is not consistent with the exact asymptotic result given by Eq. (4.7). 4.3. Total reflectance The range of applicability of Eq. (4.26) with respect to higher values of y can be readily extended Let us find the approximate solution for the total using the following correction (Kokhanovsky, 2002c): reflectance r in Eq. (4.33). Clearly, the value of r p 0 1 À t. This is due to light absorption in a cloudy Rlðl; l0; uÞ¼Rlðl; l0; uÞ medium.  exp½Àyð1 À cyÞuðl; l ; uÞ : ð4:32Þ 0 The total reflectance or the spherical albedo r is where c = 0.05. The value of c was obtained by the defined by (Sobolev, 1972): parameterization of calculations with exact radiative Z Z transfer code, developed by Mishchenko et al., 1999. 1 2p p Kokhanovsky (2002c) found that the error of Eq. r ¼ du d#sin2# 2pZ 0 0 (4.32) is smaller than 5% at the nadir observation, p the solar angle smaller than 75j and y V 1.18.  d#0sin2#0Rð#0;#;u; sÞ: ð4:35Þ Detailed studies of accuracy of Eq. (4.30) were per- 0 formed by Kokhanovsky et al. (1998b). In particular, they found that the error of Eq. (4.30) is smaller than Let us also introduce the plane albedo z Z Z 15% at s 10 and x0 > 0.95. Kokhanovsky and Roza- 2p p nov (2003) proposed to substitute t exp( x) in Eq. 1 À Rð#0Þ¼ du d#sin2#Rð#0;#;u; sÞ (4.30) by ˆt = t exp(À x) À ˜t, where ˜t has been introduced 2p 0 0 in Section 4.1. The accuracy of the modified equation is and the diffuse transmittance given in Fig. 16a–d (Kokhanovsky and Rozanov, 2003). It is better than 6% at solar angles smaller than Z Z 1 2p p 70j at nadir observation, if s z 5, k V 1.5 Am for the Tð# Þ¼ du d#sin2#Tð# ;#;u; sÞ: 0 2p 0 Deirmendjian’s Cloud C.1 model with aef =6Am. 0 0 218 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 16. (a) Dependence of the reflection function of a plane-parallel homogeneous cloud layer with water droplets on the solar zenith angle at nadir observation for different values of the cloud optical thickness at the effective radius of droplets equal to 6 Am, the wavelength equal to 1.55 Am (exact data - symbols, approximation - lines). The droplet-size distribution is given by Eq. (2.3) at l=6 (Kokhanovsky and Rozanov, 2003). (b) The same as in Fig. 16a but for various effective radii and s=5, 100 (Kokhanovsky and Rozanov, 2003). (c) The error of approximation obtained using data in Fig. 16a (Kokhanovsky and Rozanov, 2003). (d) The error of approximation for various values of y=0.12 (1), 0.27 (2), 0.37 (3), 0.83 (4), and 1.2 (5) and s=5 (Kokhanovsky and Rozanov, 2003). For the case of idealized semiinfinite nonabsorbing and clouds (Sobolev, 1972) and as a result of the con- Z Z 2p p=2 servation of energy law, 1 0 du d#sin2#Rlð#0;#;uÞ¼1; ð4:37Þ 2p 0 0 Z Z 1 2p p=2 i.e., all photons injected into a cloudy medium are du d#0sin2#0 reflected back in outer space after an infinite travel 2p 0 0 0 Z time. Here Rl(# ,#,u) is the reflection function of a p=2 0 d#sin2#R0 # ;#;u 1 4:36 semiinfinite nonabsorbing cloud. The reflection func-  lð 0 Þ¼ ð Þ 0 0 tion Rl(#0,#,u) of a cloudy medium only weakly A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 219 depends on its microstructure (see Fig. 13); by defi- The accuracy of Eq. (4.42) is studied in Fig. 17a,b.We nition, it does not depend on either the optical thick- conclude that the accuracy of this formula is better ness, s = rext L, or the single-scattering albedo, x0 = than 5% at x0>0.93 for crystalline clouds. For water rsca/rext. Here rext is the extinction coefficient and clouds, the accuracy of this formula is better than 5% rsca is the scattering coefficient of a cloudy layer of at x0>0.97. the geometrical thickness L. Combining Eqs. (4.31), (4.38) and (4.42), we have It follows from Eqs. (4.30) and (4.35) for absorb- for the total reflectance of a cloud layer: ing clouds: sinhðx þ ayÞexpðÀyÞÀsinhðyÞexpðÀx À yÞ r ¼ : r ¼ rl À texpðÀx À yÞ; ð4:38Þ sinhðx þ ayÞ ð4:45Þ where Z Z 1 2p p=2 Substitution of Eq. (4.45) into Eq. (4.33) enables rl ¼ du d#sin2# 2p the reflection function of a cloud over underlying Z 0 0 p=2 surface with the spherical albedo rs to be calcu-  d#0sin2#0Rlð#0;#;uÞð4:39Þ lated. Such calculations were performed and pre- 0 sented in a graphical form by Kokhanovsky et al. (1998b). and we take the normalization condition (Van de Total light absorptance in a cloud layer is given by Hulst, 1980) Z a = 1 À r À t, where r is calculated by Eq. (4.42) and t 1 by Eq. (4.31). As a result, we have the following 2 dllK0ðlÞ¼1 ð4:40Þ 0 analytical equation for the total light absorbtion inside a plane-parallel cloud having a finite thickness: into account. The approximate formula (4.10) for sinhðx þ ayÞ½1 À expðÀyÞ À sinhðyÞ½1 À expðÀx À yÞ K0(l) obeys the exact integral relation (4.40). The a ¼ constant rl represents the total reflectance of a semi- sinhðx þ ayÞ infinite layer. According to the definition (Eq. (4.36)), ð4:46Þ rl =1 at x0 = 1. Eq. (4.39) is not readily analytically It follows as a u 1: integrated at arbitrary values of x0, but it follows from Eqs. (4.18) and (4.39) as x ! 1 (see also the dis- 0 sinhðxÞ sinhðyÞ cussion in the previous section) that r ¼ ; t ¼ ; sinhðx þ yÞ sinhðx þ yÞ rl ¼ 1 À y: ð4:41Þ sinhðx þ yÞÀsinhðxÞÀsinhðyÞ a ¼ ; ð4:47Þ sinhðx þ yÞ At larger values of y using the same substitution as was used in the derivation of Eq. (4.28) from Eq. which yields the well-known formulae presented else- (4.25), the integral (Eq. (4.39)) gives approximately where (see, e.g., Zege et al., 1991). Overall, the global radiative characteristics of rl ¼ expðÀyÞð4:42Þ cloudy media are well described by only two param- 1/a eters, which are called the generalized radiative trans- or (see Eq. (4.28)) rl = l . Similar result for the fer parameters (Zege et al., 1991): plane albedo reads (Kokhanovsky, 2001a): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À x0 Rlðl0Þ¼exp½ÀyK0ðl0Þ : ð4:43Þ x ¼ s 3ð1 À x Þð1 À gÞ; y ¼ 4 : 0 3ð1 À gÞ It easy to show that the diffuse transmittance is given ð4:48Þ by the following equation (Kokhanovsky, 2001a): The parameter x describes the attenuation of a light Tðl0Þ¼tK0ðl0Þ: ð4:44Þ field in deep layers of semiinfinite weakly absorbing 220 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 17. (a) Cloud spherical albedo calculated with the exact radiative theory (Yanovitskij, 1997) (symbols) and according to approximation (4.42) (lines) at g = 0.75, 0.8 and various x0. In exact calculations, Heney–Greenstein phase function was assumed. (b) Errors of Eq. (4.42) as obtained from data in Fig. 17a. media. For the light intensity in deep layers, I(l,s)= ative characteristics of optically thick cloud layers are w(l)exp( À x), where the angular distribution of light determined by parameters x and y, which govern field w(l) does not depend on the optical depth, s.We asymptotic regime and light reflection for semiinfinite see from Eq. (4.42) that y = ln(1/rl). Thus, the radi- cloudy media. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 221 Table 5 solar light, namely, the preferential direction of oscil- Radiative characteristics of cloudy media lation of the electric vector in a light beam (Kokha- Radiative Equation novsky, 2003b). Generally speaking, both reflected and characteristic transmitted light is elliptically polarized. Reflection R(l0,l,u)=Rl(l,l0,u) À t exp(À x À y) The polarization arises mainly due to single scat- function K0(l)K0(l0) tering of light by a water droplet, an aerosol particle or Plane albedo R(l )=Rl(l ) À texp(À x À y)K (l ) 0 0 0 0 an ice crystal. Molecular scattering is another source Spherical albedo r = rl À texp(À x À y) Transmission T(l0,l,u)=tK0(l0)K0(l) of the polarized light in the terrestrial atmosphere. function Multiple scattering, which is of particular importance Diffuse T(l0)=tK0(l0) for clouds, leads to randomization both photons transmittance directions and polarization states. This decreases light Global t = sinhy/sinh(ay + x) transmittance polarization due to increase in the optical thickness of 0 a cloud. Another effect of multiple scattering is the Here Rl(l0,l,u)=Rl(l0,l,u)exp[ À yu(l0,l,u)], u(l0,l,u)=K0 0 introduction of ellipticity in scattered light beams. It (l0)K0(l)/Rl(l0,l,u), Rl(l0)=exp[À yK0(l0)], K0(l0)= 0 (3/7) (1 + 2l0), rl = exp( À y), a = 1.07, and Rl(l0,l,u) is the should be pointed out that the solar light scattered by reflectionpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function of a semiinfiniteqffiffiffiffiffiffiffiffiffiffiffi nonabsorbing cloud medium, spherical water droplets becomes partially linearly x s 3 1 x 1 g , y 4 1Àx0 . ¼ ð À 0Þð À Þ ¼ 3ð1ÀgÞ polarized. This is due to the fact that the ellipticity of singly scattered light is equal to zero for spherical This concludes the consideration of reflection, scatterers if the incident light is unpolarized (Van de transmission, and absorption characteristics of cloudy Hulst, 1957). media. For the sake of convenience, we present main Studies of polarization characteristics of solar light equations of this section in Table 5. They can be used transmitted and reflected by cloudy media have a long for a rapid estimation of radiative characteristics of history (Hovenier, 1971; Hansen, 1971a,b; Hansen water clouds, giving that their microphysical or local and Hovenier, 1974; Gehrels, 1974; Coffeen, 1979; optical parameters are known. Also they give a con- Coulson, 1988; Knibbe, 1997; Kokhanovsky, 2003b). venient formulation and starting point for a semi- However, the real burst of research in this area was analytical solution of the inverse problem, namely, given by a launch of the Polarization and Direction- for finding s and aef from spectral measurements of ality of Earth Reflectances (POLDER) instrument on reflected light (Kokhanovsky et al., 2003). board the Japanese ADEOS-I satellite (Deschamps et For instance, measurements of light fluxes above al., 1994; Chepfer et al., 2001). The POLDER was and below clouds were performed using pyranometers able to transmit to ground stations a huge amount of by Asano et al. (2002).Theyusedlook-uptable information about polarization characteristics of light approach to retrieve cloud properties. Taking into reflected from cloudy media, aerosols, and underlying account that the cloud system studied was optically surfaces at several wavelengths. Specifically, the first ! thick, more simple solutions given here can be equally three components of the Stokes vector SrðI; Q; U; VÞ applied, leading in the decrease in the computational (Chandrasekhar, 1950) have been measured for wave- burden. lengths, k, equal to 443, 670, and 865 nm. There is no doubt that even more advanced polarimeters (see, e.g., Cairns et al., 1999) with more wide spectral 5. Polarization characteristics coverage will appear on board different satellites in future, which makes further theoretical studies of 5.1. Introduction polarization characteristics of cloudy media extremely important. This is due to potential possibilities for Cloudy media do not only absorb and scatter ra- global retrievals of cloud microphysical parameters diation. They also can be considered as major sources (Breon and Goloub, 1998), the shape of particles (along with aerosols, gases, and underlying surfaces) of (Deschamps et al., 1994) and the optical thickness polarized light in the terrestrial atmosphere. Clouds of clouds (Kokhanovsky, 2000) based on polarization introduce a new property to incoming unpolarized measurements. 222 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Of course, similar information can be obtained the framework of the modified asymptotic theory. from the reflected intensity measurements. However, Formulae obtained are rather simple and allow for it could well appear that the degree of polarization semianalytical solution of the inverse problem can be used as a source of additional information (Kokhanovsky and Weichert, 2002).Theycanbe about cloud particle-size distributions close to the top applied to water clouds due to weak absorption of of a clouds (Breon and Goloub, 1998). Indeed, the light by water droplets in a broad spectral range (at high proportion of photons scattered from a thin least up to 2.2 Am, see Fig. 2). upper layer of a cloud in creating light polarization is quite understandable. Multiply scattered light 5.2. Reflection and transmission matrices fluxes from deep layers are hardly polarized at all. Radiative characteristics, on the other hand, represent Let us introduce the reflection Rˆ (l,l0,u) and the cloud as a whole. Thus, the effective radius transmission Tˆ(l,l0,u) matrices of cloudy media. derived from radiative measurements is an average Reflection and transmission matrices depend on the of large ensembles, possibly very different particle- optical thickness, s, of a cloud, microstructure of a size distributions, present in different parts of cloudy cloud, cosine l0 of the solar angle #0, cosine l of media. the viewing angle #, and the relative azimuth u = It has been emphasized by Hansen (1971a) that the w À w0, where w0 and w are azimuths of the Sun fact that the polarization data refers to cloud tops may and observer, respectively. The importance of these be not a drawback, but an advantage. Indeed, it is matrices is that they can be used for the calcu- ! easier to interpret observations which do not refer to lation of the Stokes vector of reflected S ðI; Q; U; ! r some average values over the entire cloud depth. VÞ and transmitted StðIV; QV; UV; VVÞ light under The polarization characteristics of cloudy media arbitrary illumination of a cloudy layer. Namely, can be studied applying numerical codes (Hansen, it follows for plane-parallel cloudy layers (Hove- 1971a; Liou, 1992) based on the vector radiative nier, 1971): Z Z transfer equation solution. However, one can also use 1 2p ! 1 the fact that cloud fields are optically thick in most Srðl; uÞ¼ dlVlV duVRˆ ðl; l;Vu À uVÞ cases (Kondratyev and Binenko, 1984). This allows p 0 0 ! application of vector asymptotic analytical relations  S0ðlV; uVÞ; ð5:1Þ (Domke, 1978) derived for optically thick disperse Z Z 1 2p media with arbitrary phase functions and absorption. ! 1 S ðl; uÞ¼ dlVlV duVTˆðl; lV; u À uVÞ These solutions help clarify physical mechanisms t p 0 0 and main features behind the polarization change ! ! s  S ðlV; uVÞþS ðl; uÞexp À ; due to the increase of the size of droplets or the 0 0 l thickness of a cloud. Analytical solutions also pro- 0 vide an important tool for the simplification of ð5:2Þ the inverse problem. They can be used, e.g., in ! where S ðl; uÞ represents the Stokes vector of the studies of the information content of polarimetric 0 incident light beam. The second term in Eq. (5.2) measurements. is small for most of cloudy media (sH1) and can The drawback of well-known vector asymptotic be neglected. theory for optically thick layers (Domke, 1978) is In a good approximation, the Sun is a source of related to the fact that the main equations of this unpolarized light incident from only one direction (l , theory depend on different auxiliary functions and 0 u ). Then it follows: parameters. Calculations of these functions and 0 parameters are rather complex due to their gene- ! 1 ! S ðl; uÞ¼ dðl À l Þdðw À w Þp F; ð5:3Þ ral applicability in terms of the single-scattering 0 l 0 0 albedo. 0 ! The special case of weakly absorbing light scatter- where vector F is normalized so that the first element ! ing media was considered by Kokhanovsky (2001a) in of pF (namely, pF1) is the net flux of solar beam per A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 223 unit area of a cloud layer. Eqs. (5.1) and (5.2) simplify is the degree of linear polarization and if one accounts for Eq. (5.3): R P ¼ 41 ð5:10Þ ! ˆ ! c Srðl; uÞ¼ Rðl; l0; w À w0ÞF; ð5:4Þ R11 ! ! is the degree of circular polarization. One can show S l; u Tˆ l; l ; w w F; 5:5 tð Þ¼ ð 0 À 0Þ ð Þ (Van de Hulst, 1980) that for particular viewing geo- l l u p where only first element of the column vector metries (e.g., 0 =1, =1, =0, ) the values of R31 and R are equal to zero and light is partially linearly 0 1 41 polarized. The degree of circular polarization, Pc,is F1 B C equal to zero in this case. B C B C It is common to use also a different definition of B F2 C ! B C the degree of polarization, namely, F ¼ B C ð5:6Þ B C B F3 C Q @ A P ¼À ð5:11Þ I F4 for linearly polarized light beams. It follows from Eq. differs from zero for unpolarized solar radiation. Thus, (5.11) at the nadir viewing geometry (R31 = R41 =0) only elements Rj1, Tj1 ( j = 1, 2, 3, 4) are relevant to that studies of solar light propagation, scattering and R polarization in terrestrial atmosphere. Other elements P ¼À 21 : ð5:12Þ are of interest only in the case of polarized (e.g., laser) R11 incident beams. Knowledge of matrices Rˆ and Tˆ is of primary We will use this definition of the degree of polar- importance for understanding radiative and polariza- ization in this review. The second component of the tion characteristics of light fluxes, which emerge from Stokes vector, Q, is proportional to the difference a cloud body. They play a role of well-known Mueller Il À Ir, where values of Il and Ir are intensities of light matrices (Shurcliff, 1962; Tynes et al., 2001) for polarized parallel and perpendicular to a meridional multiple-scattering cloudy media. Note that Eq. (5.4) plane, which contains the normal to a layer and a can be written in scalar form for unpolarized illumi- viewing direction. Thus, light polarized perpendicular nation: to the meridional plane is characterized by the positive polarization. Note that the sign of polarization degree I ¼ R11F1; Q ¼ R21F1; U ¼ R31F1; V ¼ R41F1 strongly depends on the observation geometry. For ð5:7Þ instance, it is positive in the main rainbow region (Van de Hulst, 1957). It could be both positive and negative in the glory-scattering region. and similarly to the transmitted light. Thus, the The degree of polarization of singly scattered intensity of the reflected light is determined by the light for solar illumination is given in Fig. 18a,b. element R of the reflection matrix. This is the only 11 They were obtained using Mie theory for polydis- element of major concern of most cloud optics studies persions of water droplets at the wavelength, k, equal up to date. The degree of polarization is given by to 443 nm. The particle-size distribution was given qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by Eq. (2.3). 2 2 P ¼ Pl þ Pc ; ð5:8Þ Polarized light scattering by hexagonal crystals were studied by Labonnote et al. (2001) and Liou (2002), the first paper being devoted to a special case where of inhomogeneous crystals. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi It should be pointed out that the Stokes vector 2 2 R21 þ R31 elements I, Q, U, V (see Eq. (5.7)) can be used to Pl ¼ ð5:9Þ R11 determine the polarization ellipse parameters such as 224 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Fig. 18. (a) Degree of polarization of light singly scattered by a polydispersion of spherical water droplets characterized by the gamma particle- size distribution (Eq. (2.3)) with the half-width parameter l = 6 and the effective radius aef = 4, 6, 8, 10, and 16 Am as the function of the scattering angle at k = 443 nm. (b) Same as in Fig. 1a, but at aef =6Am and different values of the half-width parameter l = 2, 4, 6, 8, 10, and 12. the ellipticity and the angle of preferential oscilla- Thus, reflection and transmission matrices tions of the electric vector in a light beam (Shurcliff, allow determination of both intensity and polar- 1962). ization states of reflected and transmitted light A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 225 beams under arbitrary illumination of a cloud Eqs. (5.13) and (5.14), derived by Kokhanovsky layer. (2001a) for small probabilities of photon absorption, are much simpler than initial asymptotic formulae 5.3. Modified asymptotic theory obtained by Domke (1978). They allow calculation of the reflection and the transmission matrices of thick 5.3.1. General equations weakly absorbing disperse media by simple means if The optical thickness of water clouds is usually the solution of the problem for a semiinfinite medium large. Some mixed and crystalline clouds are also with the same phase matrix as for a finite layer under extended in the vertical direction for many hundred consideration is available (Wauben, 1992; Van de meters and could be optically thick (Uchiyama et al., Hulst, 1980). This reduction of a problem to the case 1999). Thus, asymptotic vector theory (Domke, 1978; of a semiinfinite nonabsorbing medium is of impor- Van de Hulst, 1980) is applicable to cloudy media. tance for radiative transfer theory, in general, and to Using this theory, the azimuthally averaged reflection cloud optics, in particular. matrix Rˆ (l,l0,s) and transmission matrix Tˆ(l,l0,s)of a homogeneous plane-parallel weakly absorbing tur- 5.3.2. Nonabsorbing media bid layer of a large optical thickness, s, can be written It follows from Eqs. (5.12)–(5.14) for the degree in the following form as y ! 0 (Kokhanovsky, of polarization at the nadir observation in the case of 2001a): nonabsorbing clouds (Kokhanovsky, 2001a) that: P0 ðl Þ ˆ ˆ ˆ l 0 Rðl; l0; sÞ¼Rlðl; l0ÞÀTðl; l0; sÞexpðÀx À yÞ; Pðl0Þ¼ ; ð5:17Þ 1 À tuðl0Þ ð5:13Þ where ˆ ! !T Tðl; l0; sÞ¼t K0ðlÞK0 ðl0Þ; ð5:14Þ K0ð1ÞK0ðl0Þ uðl0Þ¼ 0 ð5:18Þ Rlð1; l0Þ where x and y are given by Eq. (4.46), t is given by Eq. (4.31) and and t is given by Eq. (4.7). Our calculations show that Z Z !we have roughly u c 1 and, therefore (see Eq. 5.17), 1 2p c 0 ! 3 1 2 0 ! P(l0) Pl(l0)/r, where we used r + t = 1 for non- K0ðlÞ¼ l þ d11 duRˆ lð1; l; uÞ e; 4 p 0 0 absorbing clouds. It means that brighter clouds have ð5:15Þ smaller values of the degree of polarization and vice versa. The degree of polarization for a semiinfinite where nonabsorbing cloud at the nadir observation 0 1 0 1 0 Rl21ð1; l0Þ @ A Plðl Þ¼À ð5:19Þ e!¼ : ð5:16Þ 0 0 Rl11ð1; l0Þ 0 depends on the solar angle and the effective radius of 0 0 Rˆ l(l,l0) is the azimuthally averaged reflection particles. Here Rl11(1,l0) and Rl21(1,l0) are corre- matrix for a semiinfinite medium with the same local sponding elements of the reflection matrix (see Eq. optical properties as a finite medium under study. The (5.4)) of a nonabsorbing semiinfinite medium. Note ! vector KT ðl Þ in Eq. (5.15) is transpose to the vector that Eq. (5.17) was derived accounting for the equality ! 0 ! Kðl0Þ. The vector K0ðlÞ also occurs in the so-called (Wauben, 1992) K02(1) = 0. Kokhanovsky and Wei- Milne problem (Minin, 1988; Wauben, 1992).It chert (2002) proposed the parameterization of this describes the angular dependence and the polarization function in terms of aef at the rainbow-scattering characteristics of light emerging from semiinfinite region. nonabsorbing turbid layers with sources of radiation Thus, Eqs. (5.17) reduces the calculation of the located deep inside the medium. degree of polarization of light reflected by a cloud in 226 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 the visible to the case of an idealized situation of a polarization states and propagation directions of pho- nonabsorbing semiinfinite media. Calculations with tons in highly scattering nonabsorbing semiinfinite Eq. (5.17) and exact vector radiative transfer equation layers irrespective of local optical properties of the are given in Fig. 19 at the rainbow-scattering geom- medium where they propagate. 0 etry. The value of Pl(l0) in Eq. (5.17) was obtained from the exact vector radiative transfer calculations. 5.3.3. Weakly absorbing media We see that accuracy of Eq. (5.17) is high. Note also Eq. (5.17) is easily generalized to account for weak the decrease of the degree of polarization with cloud absorption of light in a cloudy medium. Namely, it optical thickness. This can be used for the cloud follows from Eqs. (5.12) and (5.13) (Kokhanovsky, optical thickness determination at the rainbow-scatter- 2001b): ing geometry (Kokhanovsky, 2000). In more general case (see Eqs. (5.13) and (5.14)), Plðl0Þ ! 0 Pðl0Þ¼ ; ð5:20Þ the functions K0ðlÞ and Rˆ l(l,l0,u) for a nonabsorb- 1 À u*ðl0ÞtexpðÀx À yÞ ing semiinfinite light-scattering medium should be found numerically. They depend on the phase matrix where the global transmittance, t, is given by Eq. of a random medium in question. However, by defi- (4.31) and nition they do not depend on the single-scattering albedo, x0, and the optical thickness, s.Another K0ð1ÞK0ðl0Þ u*ðl0Þ¼ : ð5:21Þ interesting feature of these functions is their weak Rlð1; l0Þ dependence on the cloud microphysical parameters (size, shape, and chemical composition of particles) of Values of Pl(l0) and Rl(1,l0) represent the degree a medium under study in the broad range of angular of polarization and reflection function of a semiinfin- parameters. This is due to the randomization of both ite weakly absorbing cloud at the nadir viewing Fig. 19. Dependence of the degree of polarization on the solar angle for cloud optical thickness 8, 15, and 30 at k = 0.65 Am, #0 =37j, aef =6 Am, l = 6 (see Eq. (2.3)) according to the exact radiative transfer calculations (symbols) and Eq. (5.17) (lines). A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 227 geometry. They can expressed via the correspondent where Z l Z l functions for nonabsorbing media. In particular, Eq. ! t! !V Ài t r¯ V !V; : (4.32) can be used to find Rl(1,l0). Similar equation Ið Þ¼ Iðr Þe dr ð6 3Þ Àl Àl holds for Pl(l ) (Kokhanovsky, 2002c): 0 Z l Z l ! ! Ài!t !r V ! I0ðt Þ¼ I0ðr VÞe dr V; ð6:4Þ 0 Àl Àl Plðl0Þ¼Plðl0Þexp½yuð1 À 0:05yÞ : ð5:22Þ Z l Z l !! Sðt!Þ¼ Sðr!VÞeÀi t r Vd!r V; ð6:5Þ l l Eqs. (4, 32), (5.20)–(5.22) reduce the calculation for a À À weakly absorbing finite layer to a more simple case of and t! is the space frequency. One of the main a semiinfinite nonabsorbing medium. The accuracy of problems of the image transfer theory is to determine these equations has been studied by Kokhanovsky the value of the Fourier transform of the PSF, namely, (2001a, 2002c). the optical transfer function (OTF) Sð!tÞ. After that the Note that the dependence of the degree of polar- value of Ið!tÞ can be determined from Eq. (6.2) and for ization of reflected light on the size of droplets has the light intensity in the image plane, it follows from been studied by Hansen (1971a) and Kokhanovsky Eq. (6.3): (2003b). Z l Z l 1 !! !V ! i t r ! Iðr Þ¼ 2 Iðt Þe dt: ð6:6Þ 4p Àl Àl 6. Image transfer The optical transfer function depends on the prop- 6.1. Introduction erties of media between an object and an image (Wells, 1961; Volnistova and Drofa, 1986; Zege et Let us consider now pecularities of image transfer al., 1991). Different methods of calculation of this in cloudy media. From vision theory, a cloud is a function and the modulation transfer function (MTF) high-frequency filter. The transmission of high-fre- ASðt!ÞA Mðt!Þ¼ quency signals (in time and space domains) is reduced ASð0ÞA considerably in comparison with low-frequency sig- nals as a result of light scattering. Any diffused source are described by Ishimaru (1978) and Zege et al. of light can be considered as a superposion of point (1991). light sources. Thus, in linear optical systems the ! image of such an object with intensity I0ðr VÞ is a 6.2. Optical transfer function linear superposition of images of point sources (Zege et al., 1991): We will consider here the only case of the image transfer through a cloud with large particles at normal Z l Z l incidence of a light beam. In this case, the OTF is a ! ! ! ! ! ! Iðr Þ¼ dr VI0ðr VÞSðr V; r Þdr V; ð6:1Þ real function. It does not depend on the azimuth for Àl Àl randomly oriented particles. It was obtained (Zege et al., 1991) that where the point spread function (PSF) Sðr!V; !r Þ Sðt*Þ¼exp½Às þ x0sgðt Þ ; ð6:7Þ describes the process of the transformation of the ! object intensity I0ðr VÞ in the initial plane to the image where t*=tL is the dimensionless frequency, L is the ! intensity Iðr Þ in the image plane. The point spread geometrical thickness of a layer, t ¼ At!A and Z function is a main notion of the image transfer theory. 1 Eq. (6.1) has a more simple form in a frequency gðt*Þ¼ Pðt*mÞdm; domain: 0 Z 1 l ! ! ! Pðt*mÞ¼ pðhÞJ0ðht*mÞhdh: ð6:8Þ Iðt Þ¼Sðt ÞI0ðt Þ; ð6:2Þ 2 0 228 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 Table 6 Here J0(ht*m) is the Bessel function. The optical transfer function can be obtained from Eqs. (6.7) Phase functions p(h) and their Fourier–Bessel transforms p(r)( is the normalization constant, x is the size parameter) (Zege et al., and (6.8) if one has information on the optical thick- 1991; Kokhanovsky, 2001a) ness of a cloud (V 7, in the approximations (6.7) and p(h) p(r) (6.8)), the single-scattering albedo, x0, and the phase function p(h). (2 exp(À h))/u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 For media with the complex microstructure (e.g., þ r crystal clouds), the crux of the difficulty is that the 2 2exp(À h) 3/( 2 + r2)3/2 phase function is not known a priori. In this case, (2/ 2)exp exp(À ( 2r2)/2) different models to describe this function can be used (À h2/2 2) 8 &' ÀÁ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ (see Table 6). For instance, if one supposes that the <> 2 2 arccos r À r 1 À r ; rV2x phase function is described by the Gaussian function: (4J 2(hx))/h2 p 2x 2x 2x ! 1 :> 2 h2 0; r > 2x pðhÞ¼ exp À ; ð6:9Þ b2 2b2 and, therefore, where b is constant, which is inversely proportional to ()"# ! the effective radius of droplets, and h is the scattering ðÞbt* 2 Sðt*Þ¼exp Às 1 À x 1 À ; angle, it follows that 0 6 "# "# 2 ðbmt*Þ ðbt*Þ2s Pðt*mÞ¼exp À ð6:10Þ Mðt*Þ¼exp À : ð6:15Þ 2 6 and, therefore, In particular, we have at t*=0: rffiffiffiffi ! x s p bt* Sð0Þ¼exp½ Àsð1 À x0Þ ; ð6:16Þ SðtÞ¼exp Às þ 0 erf pffiffiffi ; ð6:11Þ bt* 2 2 which means that S(0) differs from one only due to the pffiffiffi where erfðbt*= 2Þ is the probability integral, given light absorption process. Note that the value of b is by inversely proportional to the average size of particles. Z Thus, one can see that the loss of contrast is higher for x x 1 t2 thicker clouds with larger single-scattering albedos ffiffiffi ffiffiffiffiffiffi À 2 erf p ¼ p e dt: ð6:12Þ and smaller droplets or crystals. 2 2p Àx The dependence of the OTF on the size and refractive index of droplets was studied in detail by One canpffiffiffi obtain at large values of frequency, t*, erf ðbt*= 2Þ¼1 and, therefore, Zege and Kokhanovsky (1994), Zuev et al. (1997), and Kokhanovsky (2001a) using the geometrical optical Sðt*Þ¼expðÀsÞ: ð6:13Þ approximation for the phase function p(h). This al- lowed them to obtain analytical relationships between Thus, the OTF is determined by the nonscattered values of OTF and cloud microphysical parameters. H light alone at t* 1. It follows directly from Eqs. Zege and Kokhanovsky (1992) used these relation- (6.7) and (6.8) as well. ships for the optical particle sizing problem solution. Note that at large optical thickness, the number of Comparisons of calculations the OTF with Eq. nonscattered photons is low. Thus, the value of the (6.7) and Monte-Carlo calculations (Drofa and Usa- OTF is small and the contrast of the image is low. It chev, 1980) were performed by Zege and Kokhanov- follows at small values of the frequency, t*, that sky (1994). It was found that the accuracy of rffiffiffiffi"# 2 approximate Eq. (6.7) for cloudy media with an bt* 2 ðbt*Þ erf pffiffiffi ¼ bt* 1 À ð6:14Þ effective radius of water droplets 6 Am and s =1 is p 6 2 better than 5% in the visible. The accuracy decreases A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 229 0 with optical thickness, s, which should be less than The analytical results for functions Rl(l,l0,u) and 5–7 to apply Eq. (6.7). K0(l) have been presented above. Thus, the global transmittance, t, and correspondingly the total reflec- tance, r =1À t, can be obtained from Eqs. (7.1) and 7. Satellite remote sensing of cloudy media (7.2) and knowledge of the surface albedo rs and the measured value Rˆ (l,l0,u,s). 7.1. Optical thickness For such a retrieval, it is not necessary to know the optical thickness of clouds and the average size of Equations presented in previous sections can be droplets. This is an extremely important point for used for a rapid estimations of the radiative, polar- climate studies, where the global and temporally ization and image characteristics of cloudy media. averaged value of the cloud reflectance, r =1À t,is They also can be used to check the accuracy of new an important parameter. Usually, r < 0.8 for natural algorithms using the fact that the numerical solution of water clouds in the visible (Danielson et al., 1969), the radiative transfer equation (Thomas and Stamnes, which implies that clouds with optical thicknesses 1999) and results presented above for optically thick larger than 70–100 appear not very often (Trishchenko layers should converge as s ! l and b ! 0. et al., 2001). The reduced reflectance in visible can be Kokhanovsky and Macke (1999) used approxima- also related to aerosol absorption in clouds (Melnikova tions, presented in Table 5, to study the influence of and Mikhailov, 1994, 2000) and to the inhomogeneity the shape of particles on the radiative transfer in and finite size of clouds (Stephens and Tsay, 1990). clouds. They found that clouds with nonspherical Using Eq. (7.1), we obtain t u K at rs = 0, and t =0 particles are more reflective (larger values of reflec- and r = 1 at rs = 1. This shows that all photons incident tance) as compared to clouds with spherical droplets on optically thick nonabsorbing clouds over surfaces with the same value of volume to particle surface area with rs = 1 survive and return back to outer space. ratio. The opposite is true for transmittance. Similar They yield no information about actual cloud thick- results were obtained by Takano and Liou (1989). ness. This explains why the retrieval of cloud param- The most important area of the application of eters over bright surfaces (e.g., snow and ice) hardly asymptotical solutions lies in the field of remote sens- can be performed in visible (Platnick et al., 2001). The ing (Rozenberg, 1967; Rozenberg et al., 1978; King, information on the global transmittance, t, can be used 1981, 1987; Kokhanovsky and Zege, 1996; Kokha- to find the scaled optical thickness (Rozenberg et al., novsky, 2000, 2001a; Kokhanovsky et al., 2003). 1978; King, 1987), given by In particular, this approach avoids or reduces (if thin clouds are also under consideration) the need for s* ¼ sð1 À gÞ: ð7:3Þ precalculation and storage of the so-called look-up tables, which is a standard technique in passive cloud It follows from Eqs. (7.1) and (7.2) that remote sensing (Arking and Childs, 1985; Rossow et 4 s* ¼ ðtÀ1 À aÞ; ð7:4Þ al., 1989; Nakajima and King, 1990). 3 We have for the global transmittance from Eq. (4.14) after simple algebraic calculations: where t is given by Eq. (7.1). Again, the value of s* can be obtained although there is no information ð1 À r ÞK t ¼ s ; ð7:1Þ about the size of droplets and the actual optical 1 À rsð1 þ KÞ thickness of clouds. Eq. (7.4) can be used for the retrieval of s* from where the function K is introduced. It is given by the measurement Rˆ (l,l0,u,s) at a single wavelength (King, 1987). Eq. (7.3) is used for the derivation of 0 the optical thickness, s, (see Eq. (7.3)) if the value of g Rlðl; l ; uÞÀRˆ ðl; l ; u; sÞ KuKðl; l ; uÞ¼ 0 0 : is known (Rossow, 1989) (approximately 0.74 for ice 0 K ðlÞK ðl Þ 0 0 0 clouds as previously discussed). However, for warm ð7:2Þ clouds, the asymmetry parameter g depends on the 230 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 size of droplets even for nonabsorbing channels (see tion on the cloud reflection function, assuming that the Eq. (3.35)). Often, the dependence g(aef) is neglected surface is Lambertian with albedo rs, is easily taken and it is assumed that aef =10 Am for water clouds into account, leading to (see Eqs. (4.14) and (4.33)): (Rossow and Schiffer, 1999). Then it follows from Eq. 0 t1ðaef ; wÞ½1 À rs1 (3.35) that g = 0.86 at k = 0.65 Am and aef =10 Am. Rˆ 1ðaef ; wÞ¼Rl À Utilizing Eqs. (7.3) and (7.4), this value of g can be 1 À rs1½1 À t1ðaef ; wÞ used for a crude estimation of the optical thickness of  K0ðlÞK0ðl0Þ; ð7:5Þ liquid clouds. Errors can be introduced if one assumes the fixed a Rˆ ða ; wÞ¼R0 expðÀyða Þð1 À cyða ÞÞuÞ 2 ef l& ef ef priori defined value of g. It follows from Eq. (3.35) at exp x a ; w y a k = 0.65 Am that g = 0.84–0.87 at aef =4–20Am. From À ½À ð ef ÞÀ ð ef Þ Eq. (7.3), we have s = Is*, I u (1 À g)À 1 c 6.3–7.6 ' t ða ; wÞr and sa[9.4,11.5] at s* = 1.5, depending on the value of À 2 ef s2 1 À rs2r2ðaef ; wÞ g used. The assumption that aef =10Am yields g = 0.86 and I = 7.2, s = 10.7.  t2ðaef ; wÞK0ðlÞK0ðl0Þ: ð7:6Þ This leads to a relative error of 7–14% in the retrieved optical thickness (i.e., a range of possible The subscripts ‘‘1’’ and ‘‘2’’ refer to wavelengths, k1 values from s = 9.4 to s = 11.5 instead of s = 10.7). and k2, in visible and near-infrared channels, respec- This uncertainty in the optical thickness can be tively. The values of rs1 and rs2 give us the surface removed if measurements in the near-infrared region albedos in visible and near-infrared respectively. The of the electromagnetic spectrum are performed, ena- explicit dependence of functions involved on the bling the size of droplets and, therefore, the asymme- parameters aef and w to be retrieved is introduced in try parameter g to be estimated. For this, however, we brackets. The liquid water path w is preferred to the should be sure that we have a liquid and not ice or a optical thickness in retrieval procedures due to the mixed-phase cloud. Another uncertainty arises due to independence of w on the wavelength. The optical the possible contamination of clouds by absorbing thickness is uniquely defined if aef and w are known. aerosols (Asano et al., 2001, 2002). Then Eq. (7.1) is Eqs. (7.5) and (7.6) form a nonlinear system of not valid. two algebraic equations having two unknowns (aef and w), which can be solved by standard methods 7.2. Size of droplets and programs. In particular, we can find the value of w from Eq. (7.5) analytically (Kokhanovsky et al., As was specified above for correct estimation of 2003). Substitution of this result in Eq. (7.6) gives us optical thickness of clouds from space, we need to a single transcendent equation for the effective radius know the effective radius of droplets. The size of of droplets determination (Kokhanovsky et al., droplets can be found if the reflection function in 2003). The accuracy of this semianalytical retrieval near-infrared region spectrum is measured simultane- algorithm has been studied by Kokhanovsky et al. ously (Nakajima and King, 1990; Kokhanovsky and (2003). Zege, 1996). This is due to the fact that the reflection function in the infrared strongly depends on the 7.3. Thermodynamic state of clouds probability of photon absorption by droplets. This probability is proportional to the effective radius of The discrimination of liquid water from ice clouds droplets as was discussed previously (see Eq. (3.21)). is of importance for many applications including The influence of absorption and scattering of light flight safety and Earth climate studies. The size and by molecules and aerosol particles on the measured shape of particles in warm and ice clouds are different. value R(l,l0,u,s) is often neglected in the cloud This influences the energy transmitted and reflected retrieval algorithms. However, correction can be easily by a cloud. taken into account if needed (Wang and King, 1997; This discrimination can be performed, taking into Goloub et al., 2000). The influence of surface reflec- account the difference in angular or spectral distribu- A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 231 tion of reflected light. We present results of calcula- polarization at the rainbow geometry is studied tion of the reflection function of cloudy media with (Goloub et al., 2000; Kokhanovsky, 2003b). liquid and frozen water droplets in Fig. 20. It follows from this figure that minima in the reflection function 7.4. Cloud-top height and cloud fraction of ice clouds (e.g., near 1.5 and 2.0 Am) are moved to larger wavelengths as compared to the case of liquid Another important characteristic of a cloud is its droplets. Of course, this is due to the difference in height. It can be retrieved using data from space-borne spectral behaviour of the imaginary parts of the lidars (Winker and Trepte, 1998). Passive measure- complex refractive index of liquid water and ice (see ments also can be used. For instance, Yamomoto and Fig. 2). Minima for liquid water also moved to larger Wark (1961) proposed the use of the oxygen A band, wavelengths as compared to the absorption bands of centred at 0.761 Am. The physical basis of this water vapour. These different positions of minima can method depends on a deep minimum around 0.761 be easily registered with modern spectrometers (see, Am due to oxygen absorption. This minimum is not e.g., Dvoryashin, 2002; Knap et al., 2002). shown in Fig. 20 because only scattering and absorp- Another possibility is to consider different angular tion of light by cloud particles was accounted for in behaviour of the reflection function for ice and water the calculation for this figure. The depth of the clouds at specific scattering geometries (e.g., rainbow, absorption line will depend on the cloud height. glory and halo scattering). In particular, the reflection Photons can hardly penetrate thick clouds and be function of water clouds, as distinct from ice clouds, absorbed by the oxygen in the air column below the has a maximum near the rainbow-scattering angle, cloud. This will increase the value of the reflection which also can be easily detected. This feature function at 0.761 Am for the case of clouds at high becomes even more pronounced if the degree of altitudes. The depth of the absorption line is larger for Fig. 20. Spectral dependence of the reflection function of cloudy media for the nadir observation and the solar angle equal to 60j. Clouds are composed of water or ice spherical particles with effective radius 6 Am. It is assumed that particles in a cloud are characterized by the gamma particle-size distribution (Eq. (2.3)) with the parameter l = 6. The geometrical thickness of cloud is equal to 500 m. The liquid water path is 100 g/m2, which gives the optical thickness of 27 at the wavelength 0.55 Am. The reflection of light from surface and the scattering and absorption of light by aerosols and gases are neglected. 232 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 low clouds. Practical application of the method, how- function of a fractal fictive scatterer. Mie theory ever, is not so simple (Kuze and Chance, 1994; should not be used in this case. Koelemeijer et al., 2001). First of all, the depth of For thinner clouds, one should build the precalcu- line also depends on the oxygen absorption cross lated table of reflection functions, which should be section, which varies with temperature and pressure. compared with experimental data to establish both the Thus, one should use a priori assumptions on the optical thickness of clouds and the size/shape of temperature and pressure variation with height in the droplets (Masuda et al., 2002). This is not an easy Earth’s atmosphere. The generally unknown surface problem. In particular, the model of ice spheres cannot albedo and cloud geometrical thickness also can be used for this purpose (Mishchenko et al., 1995, influence the retrieval accuracy. Other possible sour- 1996; Chepfer et al., 1998; Rolland et al., 2000; ces of errors are described in detail by Kuze and Doutriaux-Bucher et al., 2000). Chance (1994) and Koelemeijer et al. (2001). Yang et al. (2001) considered the influence of The largest complication arises for pixels that are vertical variability of ice crystals habits and crystal only partially covered by clouds. Ignoring them will dimensions on the satellite ice cloud retrieval algo- lead to a big reduction of data; hence, to overcome rithms. They accounted for the fact that more complex this problem, Koelemeijer et al. (2001) proposed an shapes and larger sizes of crystals are more typical for algorithm that retrieves simultaneously effective the base of the cloud as compared to its top, where cloud-top height/pressure and cloud fraction, assum- crystals are smaller and more rounded. The authors ing that cloud is a Lambertian surface. Such an state that the vertical distribution of optical character- assumption leads to underestimation of cloud-top istics of clouds can be neglected if visible channels are heights as compared to in situ measurements. used, but the vertical inhomogeneity should be fully Global information on cloud fraction/cover, Q,is accounted for if one is interested in the average size of of a considerable importance by itself (Batey et al., particles retrievals (e.g., from the measurements of the 2000). Usually, the value of Q varies in the range cloud reflection function at 2.11 Am moderate reso- 0.55–0.85, depending on the exact region under lution imaging spectrometer channel (King et al., study. Globally, clouds cover around 60% of the 1992)). Information on the vertical structure of a atmosphere. This once more underlines the impor- crystalline cloud is not known a priori, which compli- tance of clouds in the radiation balance and atmos- cates the retrieval procedure. We see, therefore, that the phere heating rates studies. creation of a suitable look-up table for reflection func- Palle and Butler (2002) state that the global cloud tions of crystalline clouds is not at all a trivial problem. cover increased during the past century. They argue Note that ice clouds are often studied using against a dominating role of the solar activity (via ground-based systems. This allows us to derive the galactic cosmic rays) on the cloud formation. cloud optical thickness and the effective radius of particles. For this, measurements of direct light and 7.5. Remote sensing of crystalline clouds light scattered at small angles usually are performed. The important problem here is to account for multiple Remote sensing of optical thicknesses and effective light scattering using the radiative transfer equation size of droplets in crystalline clouds is complicated by (Thomas and Stamnes, 1999). For layers with optical low optical thickness (usually smaller than 5), the thickness smaller than approximately 5, a so-called high spatial and temporal variability of cloud proper- small-angle approximation (Zege et al., 1991) can be ties, nonspherical shape of particles and their possible used. The intensity of the diffused light, Id, is given by horizontal orientation. If the thickness of a crystalline a simple integral in this case (Zege et al., 1991; cloud is high, then the optical thickness can be derived Kokhanovsky, 2001a): in the same way as was discussed above, assuming an asymmetry parameter of 0.74 (see Eqs. (7.1)–(7.4)). Z l I The problem is to find the corresponding reflection I ðs;#Þ¼ 0 ½CðsÞÀexpðÀsÞ J ðr#Þrdr; d 2p 0 function of a semiinfinite medium (see Eq. (7.4)). One 0 possibility is to calculate it beforehand using the phase ð7:7Þ A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 233 where (see Table 6) Then the reflection function of a vertically inhomoge- neous optically thick cloud can be presented by Eq. CðsÞ¼exp½Àð1 À x0pðrÞÞs ð7:8Þ (4.6) as in the case of a homogeneous layer. However, the meaning of parameters in this equation is different is the normalized mutual coherence function for in an inhomogeneous case. That is, s is the optical electric fields separated by the distance r/k for a plane 0 thickness of a vertically inhomogeneous cloud, Rl is incident wave, where k =2p/k; I is the intensity of 0 the reflection function of a semiinfinite cloud with the incident light and it was assumed that a cloud is same vertical distribution of optical characteristics as illuminated along the normal. The function a finite cloud, and g is the average asymmetry Z 1 l parameter. The value of g does not change consid- pðrÞ¼ pðhÞJ ðhrÞhdh ð7:9Þ l 2 0 erably with droplet size (see Eq. (3.35) as xef ! ); 0 thus, it can be assumed to be a priori defined value is the Fourier–Bessel transform of the phase function (say, e.g., 0.86; Rossow and Schiffer, 1999). Then p(h). The angle # gives the angle that measured accounting also for the weak sensitivity of the reflec- 0 transmitted light beam makes with the normal to a tion function Rlto the size of droplets (see Fig. 13), cloud layer. Most frequently used functions p(h) and we state that Eq. (4.6) can be also applied to the their transforms p(r) are given in Table 6. Function derivation of optical thickness of vertically inhomo- (7.8) can be used to study the loss of coherence of geneous clouds. This suggests that one cannot retrieve optical waves in cloudy media (Zege and Kokhanov- the vertical distribution of extinction coefficient in a sky, 1995). cloud from the reflection function measurements in In conclusion, we note that both ice and water the visible. Lidar returns, however, can provide this clouds are often studied with lidar techniques (Platt, information (Zege et al., 1998a,b). 1978a,b; Bissonnete et al., 1995; Sassen, 1991, 2001; The situation in the near-infrared is not so simple Winker and Trepte, 1998; Mishchenko and Sassen, because the derived size of droplets depends on the 1998; Zege et al., 1998a,b; Winker, 1999; McGill et average photon penetration depth, which is of course a al., 2002). function of wavelength (Platnick, 2000). It means that the derived effective size of droplets is a function of the wavelength for vertically inhomogeneous clouds. 8. Inhomogeneous clouds Generally, the absorption increases with wavelength; hence, penetration depth decreases with wavelength. 8.1. Vertical inhomogeneity Thus, we arrive to the conclusion that reflection functions at larger wavelength (e.g., 2.11 Am) will Vertical and horizontal inhomogeneity of clouds give us values of the droplets radii closer to the cloud can be dealt with in the framework of the Monte- tops and reflection functions at smaller wavelengths Carlo methods of the radiative transfer equation (e.g., 1.6 Am) give us the radius of droplets deep solutions (Schreier and Macke, 2001). This is of inside clouds. These different values of radii are considerable importance for crystalline media, as was considered as a shortcoming of the retrieval method discussed in the previous section. Unfortunately, for vertically inhomogeneous clouds. On the other Monte-Carlo methods are extremely slow and cannot hand, this opens a new possibility to study the vertical be applied in the operational satellite cloud retrieval distribution of droplets’ sizes, analysing spectral algorithms. reflectances of clouds, obtained both from airborne Accounting for the vertical inhomogeneity, how- and satellite platforms. ever, can be easily done in the framework of the In conclusion, we note that radii of droplets theory of optically thick layers as discussed previ- increase with height (Brenguier et al., 2000). It means ously. Corresponding equations are presented by that values of radii retrieved at 1.6 Am should be Sobolev (1972) and Yanovitskij (1997). For the sake smaller than those obtained from the channel at 2.11 of simplicity, we consider here only the case of clouds Am. This is also observed experimentally (Platnick et in the visible, where light absorption can be neglected. al., 2001). 234 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 8.2. Horizontal inhomogeneity Usually, the influence of horizontal inhomogenity of clouds on their radiative characteristics is studied in Real cloud fields are horizontally inhomogeneous the framework of the independent column (or pixel) (Titov, 1998; Marshak et al., 1995, 1998) and have approximation (Cahalan et al., 1994). This approxi- complex shapes, which complicates theoretical mod- mation neglects the horizontal photon transport elling (Rogovtsov, 1991, 1999). This also produces between adjacent columns. Then one can obtain for unphysical angular dependence of cloud optical the reflection function of a pixel in the framework of thickness for different solar angles (Loeb and this approximation (Pinkus and Klein, 2000): Davies, 1996; Loeb and Coakley, 1998) if the model Z l of the horizontally inhomogeneous layer is used in R¯ ¼ RðsÞf ðsÞds; ð8:1Þ the retrieval procedure for inhomogeneous cloud 0 fields. Fouilloux et al. (2000) found that derived where f(s) is the optical thickness distribution function cloud optical thicknesses and effective radii for for a given pixel and R(s) is the reflection function for inhomogeneous clouds depend on the averaging a horizontally homogeneous cloud with a given opti- scale. Therefore, they state that comparisons between cal thickness, s. Cahalan et al. (1994), Fu et al. (2000), aircraft measurements and satellite observations are and Schreier and Macke (2001) found that this not valid for heterogeneous clouds (which is the case approximation has a high accuracy for the domain- for most clouds). This puts the validation of cloud averaged radiative fluxes. satellite products with airborne radiometers in ques- It is known (Cahalan et al., 1994) that R¯ < R(s¯), tion. where s¯ is the average optical thickness, which means Approximations for horizontal photon transport that the values of s¯ obtained from measurements over within real-world clouds were developed by Platnick horizontally inhomogeneous clouds in the assumption (2001). In particular, he derived analytic approxima- of a horizontally homogeneous plane-parallel clouds tions for the root-mean-square horizontal displace- are underestimated. Thus, correction of the optical ment of reflected and transmitted photons relative to thickness obtained by an empirical factor is needed the incident cloud-top location. (Rossow and Schiffer, 1999). Fig. 21. The coefficient of variance of the cloud optical thickness spatial distribution (Barker et al., 1996). A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 235 It is difficult to apply Eq. (8.1) to the radiative tions. In reality, most clouds have very complicated transfer problem analysis due to the necessity to shapes and structures. Accounting for these features perform calculations R(s) many times for a given will be a main topic of cloud optics research for years pixel. The problem can be greatly simplified (Kokha- to come. The progress in respect to the description of novsky, 2003d) if one uses approximate analytical local optical characteristics of crystalline clouds with solutions, which are valid for optically thick clouds particles of complex shapes is expected due to large (see e.g., Section 4), in combination with analytical efforts currently undertaken in this direction. forms for the function f(s), which is usually given by Other topics of high priority, which will be devel- the gamma (Barker et al., 1996) or the lognormal oping in a future, are studies of clouds with lidars (Nakajima et al., 1991) distributions. The values of (e.g., space-based). This will allow provide informa- the coefficient of variance of this distribution obtained tion on cloud geometrical parameters with a high from experimental measurements over a number of spatial resolution. Depolarization and polarization cloud scenes are given in Fig. 21, which was prepared techniques will also gain further momentum in cloud using data of Barker et al. (1996). We see that the optics studies. coefficient of variance is in the range 20–80%. All of these developments will allow us to charac- The lower limit of integration in Eq. (8.1) should terize cloud properties and their temporal and spatial be 10 to allow for the use of Eq. (4.14) in integral variability on a scale that was not possible in the past. (8.1). The integral in the range from 0 to 10 can be In turn, the information obtained will make it possible estimated using the exact radiative transfer equation.¯ to understand and describe the influence of clouds on This contribution is small at large values of s and can climate and its change, which remains a major unre- be neglected. solved problem of a modern climatology. 9. Summary Acknowledgements The goal of this review was to consider various The author would like to thank A. Macke, K. approaches to calculate local and global optical Masuda and M. Mishchenko for providing their characteristics of cloudy media. Main tools reviewed computer codes, which made this research possible. are the geometrical optics approximation and the Also, I acknowledge with many thanks important theory of optically thick turbid layers. Both theories discussions on the cloud optics with J.P. Burrows, M. can be substituted by exact solutions (the Mie theory King, T. Nakajima, S. Platnick, V.V. Rozanov, W. von for local optical characteristics and the radiative Hoyningen-Huene, and E. P. Zege. This work was transfer theory for the radiative characteristics) in funded by the BMBF via GSF/PT-UKF (07UFE12/8) the case of homogeneous clouds with spherical par- and by the German Space Agency (50EE0027 and ticles. 50EE9901). However, they appear to be very important in bringing the forward propagation models closer to References the reality (e.g., account for nonspherical shape of crystals, effects of cloud inhomogeneity). Approxima- Ackerman, S.A., Stephens, G.L., 1987. The absorption of solar tions developed allow consideration of cases that are radiation by cloud droplets: an application of anomalous diffrac- tion theory. J. Atmos. Sci. 44, 1574–1588. difficult or impossible to handle with exact techni- Arking, A., Childs, J.D., 1985. Retrieval of clouds cover parameters ques. Another important feature of the approximate from multispectral satellite images. J. Appl. Meteorol. 24, methods is the possibility to simplify inverse prob- 323–333. lems of cloud optics. Asano, S., Uchiyama, A., Yamazaki, A., Tanizono, M., 2001. Ef- The cloud optics studies are far from being com- fects of aerosols on retrieval of the microphysical properties of water clouds from the airborne solar spectral reflectance meas- pleted. Most results of this review are valid only for urements. Proc. SPIE 4150, 208–216. homogeneous plane parallel clouds, which is a good Asano, S., Uchiyama, A., Yamazaki, A., Gayet, J.-F., Tanizono, M., starting point for cloud optical properties investiga- 2002. Two case studies of winter continental-type water and 236 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 mixed-phase stratocumuli over the sea: 2. Absorption of solar Chepfer, H., Brogniez, G., Fouquart, Y., 1998. Cirrus clouds micro- radiation. J. Geophys. Res. 107 (10.1029/2001JD001108). physical properties deduced from POLDER observations. Auer, A.H., Veal, D.L., 1970. The dimension of ice crystals in J. Quant. Spectrosc. Radiat. Transfer 60, 375–390. natural clouds. J. Atmos. Sci. 27, 919–926. Chepfer, H., Brogniez, G., Goloub, P., Bre´on, F.M., Flamant, P.H., Ayvazian, G.M., 1991. Propagation of Millimeter and Submillim- 1999. Observations of horizontally oriented ice crystals in cirrus eter Waves in Clouds. Gidometeoizdat, Leningrad. clouds with POLDER-1/ADEOS-1. J. Quant. Spectrosc. Radiat. Baran, A.J., Havemann, S., 1999. Rapid computation of the optical Transfer 63, 521–543. properties of hexagonal columns using complex angular mo- Chepfer, H., Goloub, P., Riedi, J., De Haan, J.F., Hovenier, J.W., mentum theory. J. Quant. Spectrosc. Radiat. Transfer 63, Flamant, P.H., 2001. Ice crystal shapes in cirrus clouds 499–519. derived from POLDER/ADEOS-1. J. Geophys. Res., D 106, Baran, A.J., Francis, P.N., Labonnote, L.-C., Doutriaux-Boucher, 7955–7966. M., 2001. A scattering phase function for ice cloud: tests of Chylek, P., Lesins, G.B., Videen, G., Wong, J.G.D., Pinnick, R.G., applicability using aircraft and satellite multi-angle multi-wave- Ngo, D., Klett, J.D., 1996. Black carbon and absorption of length radiance measurements of cirrus. Q. J. R. Meteorol. Soc. solar radiation by clouds. J. Geophys. Res. 1001 (D18), 127, 2395–2416. 23365–23371. Barker, H.W., et al., 1996. A parameterization for computing grid- Coffeen, D.L., 1979. Polarization and scattering characteristics in the averaged solar fluxes for inhomogeneous marine boundary layer atmospheres of Earth, Venus, and Jupiter. J. Opt. Soc. Am. 69, clouds: II. Validation using satellite data. J. Atmos. Sci. 53, 1051–1064. 2304–2316. Coulson, K.L., 1988. Polarization and Intensity of Light in Atmos- Barkey, B., Liou, K.N., Gellerman, W., Sokolsky, P., 1999. An phere. Deepak, Hampton. analog light scattering experiment of hexagomal icelike par- Danielson, R.E., Moore, D.R., van de Hulst, H.C., 1969. The trans- ticles: Part II. Experimental and theoretical results. J. Atmos. fer of visible radiation through clouds. J. Atmos. Sci. 26, Sci. 56, 613–625. 1078–1087. Barun, V.V., 1995. Visual perception of retroreflective objects Deirmendjian, A., 1969. Electromagnetic Scattering on Spherical through light scattering media. Proc. SPIE 2410, 470–479. Polydispersions. Elsevier, Amsterdam. Barun, V.V., 2000. Influence of cloud aerosol microstructure on the Deschamps, P.-Y., Breon, F.-M., Leroy, M., Podaire, A., Bricaud, backscattering signal from the object shadow area. Izv. Rus. A., Buriez, J.-C., Seze, G., 1994. The POLDER mission: in- Acad. Nauk, Atmos. Ocean. Phys. 36, 258–265. strument characteristics and scientific objectives. IEEE Trans. Batey, M., Harshvardhan, Green, R., 2000. Geometrically effective Geosci. Remote Sens. 32, 598–614. cloud fraction for solar radiation. Atmos. Res. 55, 115–129. Domke, H., 1978. Linear Fredholm integral equations for radiative Bissonnete, L.B., et al., 1995. Lidar multiple scattering from clouds. transfer problems in finite plane-parallel media: II. Embedding Appl. Phys., B Lasers Opt., B 60, 355–362. in a semi-infinite medium. Astron. Nachr. 299, 95–102. Bohren, C.F., Huffman, D.R., 1983. Absorption and Scattering of Doutriaux-Bucher, M., Buroez, J.-C., Brogniez, G., Labonnote, Light by Small Particles. Wiley, NY. L.-C., 2000. Sensitivity of retrieved POLDER directional cloud Borovoi, A., Grishin, I., Naats, E., Oppel, U., 2000. Backscattering optical thickness to various ice particles models. Geophys. peak of hexagonal ice columns and plates. Opt. Lett. 25, Res. Lett. 27, 113–116. 1388–1392. Drofa, A.S., Usachev, A.L., 1980. On vision in a cloudy medium. Bovensmann, H., Burrows, J.P., Buchwitz, M., Frerick, J., Noel, S., Izv. Acad. Nauk SSSR, Fiz. Atmos. Okeana 16, 933–938. Rozanov, V.V., 1999. SCIAMACHY: mission objectives and Dvoryashin, S.V., 2002. Remote determination of the ratio between measurement modes. J. Atmos. Sci. 56, 127–150. the coefficients of water and ice absorption in clouds in the Brenguier, J.-L., Pawlowska, H., Schu¨ller, L., Preusker, R., Fischer, 2.15–2.35 Am spectral range. Izv. Atmos. Ocean. Phys. 38, J., Fouquart, Y., 2000. Radiative properties of boundary layer 523–528. clouds: droplet effective radius versus number concentration. Ebert, E.E., Curry, J.A., 1992. A parametrization of ice cloud opti- J. Atmos. Sci. 57, 803–821. cal properties for climate studies. J. Geophys. Res. 97, Breon, F.-M., Goloub, P., 1998. Cloud droplet effective radius from 3831–3836. spaceborne polarization measurements. Geophys. Res. Lett. 25, Evans, F.K., 1998. The spherical harmonics discrete ordinate meth- 1879–1883. od for three-dimensional atmospheric radiative transfer. J. At- Cahalan, R.F., Ridgway, W., Wiscombe, W.J., Bell, T.L., 1994. The mos. Sci. 55, 429–466. albedo of fractal stratocumulus clouds. J. Atmos. Sci. 51, Feigelson, E.M. (Ed.), 1981. Radiation in a Cloudy Atmosphere. 2434–2455. Gidrometeoizdat, Leningrad. 280 pp. Cahalan, R.F., Oreopoulos, L., Wen, G., Marshak, A., Tsay, S.-C., Fomin, B.A., Mazin, I.P., 1998. Model for an investigation of radi- DeFelice, T., 2001. Cloud characterization and clear-sky correc- ative transfer in cloudy atmosphere. Atmos. Res. 47–48, tion from Landsat-7. Remote Sens. Environ. 78, 83–98. 127–153. Cairns, B., et al., 1999. The research scanning polarimeter: calibra- Fouilloux, A., Gayet, J.-F., Kriebel, K.-T., 2000. Determination of tion and ground-based measurements. Proc. SPIE 3754, cloud microphysical properties from AVHRR images: compar- 186–196. isons of three approaches. Atmos. Res. 55, 65–83. Chandrasekhar, S., 1950. Radiative Transfer. Oxford Press, Oxford. Fu, Q., 1996. An accurate parameterization of the solar radiative A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 237 properties of cirrus clouds for climate models. J. Climate 9, physical properties obtained from airborne measurements. 2058–2082. J. Geophys. Res. 108 (doi:10.1029/2002/D002723). Fu, Q., Liou, K.N., 1993. Parametrization of the radiative properties Katsev, I.L., Zege, E.P., Prikhach, A.S., Polonsky, I.N., 1998. Effi- of cirrus clouds. J. Atmos. Sci. 50, 2008–2025. cient technique to determine backscattered light power for var- Fu, Q., et al., 2000. Cloud geometry effects on atmospheric solar ious atmospheric and oceanic sounding and imaging systems. absorption. J. Atmos. Sci. 57, 1156–1168. J. Opt. Soc. Am., A 14, 1338–1346. Garett, T.J., 2001. Shortwave, single-scattering properties of arctic Kerker, M., 1969. The Scattering of Light and Other Electromag- ice clouds. J. Geophys. Res. 106 (D14), 15155–15172. netic Radiation. Academic Press, NY. Gayet, J.-F., 1998. In situ measurements of the scattering phase Khrgian, A.H., Mazin, I.P., 1952. On droplet size distribution in function of stratocumulus, contrails and cirrus. Geophys. Res. clouds. Pap. Cent. Aerol. Obs., N 7, 56–61 (Moscow). Lett. 25, 971–974. King, M.D., 1981. A method for determining the single scattering Gehrels, T. (Ed.), 1974. Planets, Stars and Nebulae Studied with albedo of clouds through observation of the internal scattered Photopolarimetry. University of Arizona Press, Tuscon. radiation field. J. Atmos. Sci. 38, 2031–2044. Germogenova, T.A., 1963. Some formulas to solve the transfer King, M.D., 1987. Determination of the scaled optical thickness of equation in the plane layer problem. In: Stepanov, B.I. (Ed.), clouds from reflected solar radiation measurements. J. Atmos. Spectroscopy of Scattering Media. AN BSSR, Minsk, Minsk, Sci. 44, 1734–1751. pp. 36–41. King, M.D., et al., 1992. Remote sensing of cloud, aerosol, and Glautshing, W.J., Chen, S.-H., 1981. Light scattering from water water vapour properties from the moderate resolution imaging droplets in the geometrical optics approximation. Appl. Opt. 20, spectrometer (MODIS). IEEE Trans. Geosci. Remote Sens. 30, 2499–2509. 2–27. Goloub, P., Herman, M., Chepfer, H., Riedi, J., Brogniez, G., Cou- Knap, W.H., et al., 2002. Cloud thermodynamic-phase determina- vert, P., Seze, G., 2000. Cloud thermodynamical phase classi- tion from near-infrared spectra of reflected light. J. Atmos. Sci. fication from the POLDER spaceborne instrument. J. Geophys. 59, 83–96. Res. D 105, 14747–14759. Knibbe, W.J.J., 1997. Analysis of Satellite Polarization Observa- Grandy Jr., W.T. 2000. Scattering of Light from Large Spheres tions of Cloudy Atmospheres. PhD thesis, Free University of Cambridge Univ. Press, NY. Amsterdam, Amsterdam, Netherlands. Greenler, R., 1980. Rainbows, Halos, and Glories. Cambridge Univ. Koelemeijer, R.B., Stammes, P., Hovenier, J.W., de Haan, J.F., Press, Cambridge. 2001. A fast method for retrieval of cloud parameters using Grenfell, T.C., Warren, S.G., 1999. Representation of a nonspher- oxygen A band measurements from the global ozone monitoring ical ice particle by a collection of independent spheres for scat- experiments. J. Geophys. Res. 106 (D4), 3475–3490. tering and absorption radiation. J. Geophys. Res. D 104, Kokhanovsky, A.A., 1989. The geometrical optics approximation 31697–31709. for the absorption cross section of a layered sphere. Atmos. Opt. Hale, G.M., Querry, M.R., 1973. Optical contacts of water in 2, 908–912. the 200-nm to 200-Am wavelength region. Appl. Opt. 12, Kokhanovsky, A.A., 1997. Expansion of a phase function of large 555–563. particles on Legandre polynomials. Izv. Rus. Acad. Sci., Atmos. Han, Q., et al., 1994. Near global survey of effective droplet Ocean. Phys. 33, 692–696. radii in liquid water clouds using ISCCP data. J. Climate 7, Kokhanovsky, A.A., 2000. Determination of the effective radius of 465–497. drops in water clouds from polarization measurements. Phys. Hansen, J.E., 1971a. Multiple light scattering of polarized light in Chem. Earth, Part B 5–6, 471–474. planetary atmospheres: Part II. Sunlight reflected by terrestrial Kokhanovsky, A.A., 2001a. Light Scattering Media Optics: Prob- water clouds. J. Atmos. Sci. 28, 1400–1426. lems and Solutions. Praxis-Springer, Chichester. Hansen, J.E., 1971b. Circular polarization of sunlight reflected by Kokhanovsky, A.A., 2001b. Reflection and transmission of polar- clouds. J. Atmos. Sci. 28, 1515–1516. ized light by optically thick weakly absorbing random media. Hansen, J.E., Hovenier, J., 1974. Interpretation of the polarization J. Opt. Soc. Am. A 18, 883–887. of Venus. J. Atmos. Sci. 31, 1137–1160. Kokhanovsky, A.A., 2002a. A simple approximate formula for the Hansen, J.E., Travis, L.D., 1974. Light scattering in planetary at- reflection function of a homogeneous semi-infinite turbid me- mospheres. Space Sci. Rev. 16, 527–610. dium. JOSA, A 19, 957–960. Harrington, J.Y., Olsson, P.Q., 2001. A method for the parametriza- Kokhanovsky, A.A., 2002b. Statistical properties of a photon gas in tion of cloud optical properties in bulk and bin microphysical random media. Phys. Rev., E 66 (037601). models. Implications for arctic cloudy boundary layers. Atmos. Kokhanovsky, A.A., 2002c. Reflection and polarization of light by Res. 57, 51–80. semi-infinite turbid media: simple approximations. J. Colloid Hovenier, J.W., 1971. Multiple scattering of polarized light in plan- Interface Sci. 251, 429–431. etary atmospheres. Astron. Astrophys. 13, 7–29. Kokhanovsky, A.A., 2003a. Optical properties of irregularly shaped Ishimaru, A., 1978. Wave Propagation and Scattering in Random particles. J. Phys. D 36, 915–923. Media. Academic Press, New York. Kokhanovsky, A.A., 2003b. Polarization Optics of Random Media. Jourdan, O., Oshchepkov, S., Gayek, J.F., Scherbakov, V., Tsaka, Springer-Praxis, Berlin. H., 2003. Statistical analysis of cloud light scattering and micro- Kokhanovsky, A.A., 2003c. Reflection of light from nonabsorbing 238 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 semi-infinite cloudy media: a simple approximation. J. Quant. tionships in science and technology. In: Fischer, G. (Ed.), Group Spectrosc. Radiat. Transfer 81, 125–140. V: Geophysics and Space Research. V.5: Meteorology. Subvo- Kokhanovsky, A.A., 2003d. The influence of horizontal inhomoge- lume b. Physical and Chemical Properties of the Air. Springer- neity on radioactive characteristics of clouds: an asymptotic case Verlag, Berlin. 570 pp. study. IEEE Trans. Geosci. Remote Sens. 41, 61–70. Liou, K.N., 1992. Radiation and Cloud Processes in the Atmos- Kokhanovsky, A.A., Macke, A., 1997. Integral light scattering and phere. Oxford Univ. Press, Oxford. absorption characteristics of large nonspherical particles. Appl. Liou, K.N., 2002. Introduction to Atmospheric Radiation. Academ- Opt. 36, 8785–8790. ic Press, NY. Kokhanovsky, A.A., Macke, A., 1999. The dependence of the ra- Liou, K.N., Hansen, J., 1971. Intensity and polarization for single diative characteristics of optically thick media on the shape of scattering by polydisperse spheres: a comparison of ray optics particles. J. Quant. Spectrosc. Radiat. Transfer 63, 393–407. and Mie theory. J. Atmos. Sci. 28, 995–1004. Kokhanovsky, A.A., Nakajima, T.Y., 1998. The dependence of Liou, K.N., Takano, Y., 1994. Light scattering by nonspherical phase functions of large transparent particles on their refractive particles: remote sensing and climate implications. Atmos. index and shape. J. Phys. D 31, 1329–1335. Res. 31, 271–298. Kokhanovsky, A.A., Rozanov, V.V., 2003. The reflection function Liu, L., Mishchenko, M.I., Menon, S., Macke, A., Lacis, A.A., of optically thick weakly absorbing turbid layers: a simple ap- 2002. The effect of black carbon on scattering and absorption proximation. J. Quant. Spectrosc. Radiat. Transfer 77, 165–175. of solar radiation by cloud droplets. J. Quant. Spectrosc. Radiat. Kokhanovsky, A.A., Weichert, R., 2002. The determination of the Transfer 74 (2), 195–204. droplet effective size and optical depth of cloudy media from Loeb, N.G., Coakley Jr., J.A. 1998. Inference of marine stratus polarimetric measurements. Appl. Opt. 41, 3650–3658. cloud optical depths from satellite measurements: does 1D Kokhanovsky, A.A., Zege, E.P., 1995. Local optical parameters of theory apply? J. Atmos. Sci. 11, 215–233. spherical polydispersions: simple approximations. Appl. Opt. Loeb, N.G., Davies, R., 1996. Observational evidence of plane 34, 5513–5519. parallel model biases: apparent dependence of cloud optical Kokhanovsky, A.A., Zege, E.P., 1996. The determination of the depth on solar zenith angle. J. Geophys. Res. 101, 1621–1634. effective radius of drops and liquid water path of water clouds Macke, A., 1993. Scattering of light by polyhedral ice crystals. from satellite measurements. Earth Res. Space 2, 33–44. Appl. Opt. 32, 2780–2788. Kokhanovsky, A.A., Zege, E.P., 1997a. Optical properties of aero- Macke, A., 1994. Modellierung der Optischen Eigenschaften von sol particles: a review of approximate analytical solutions. Cirruswolken. PhD thesis, University of Hamburg. J. Aerosol Sci. 28, 1–21. Macke, A., Grossklaus, M., 1998. Light scattering by nonspherical Kokhanovsky, A.A., Zege, E.P., 1997b. Physical parametrization of raindrops: implications for lidar remote sensing of rain rates. local optical characteristics of cloudy media. Izv. RAN, Fiz. J. Quant. Spectrosc. Radiat. Transfer 60, 355–363. Atmos. Okeana 33, 209–218. Macke, A., Tzschichholz, F., 1992. Scattering of light by fractal Kokhanovsky, A.A., Babenko, V.A., Barun, V.V., 1998a. On particles: a qualitative estimate exemplary for two-dimensional asymptotic values of light fluxes scattered by large spherical triadic Koch island. Physica A 191, 159–170. particles between two angles. J. Phys. D 31, 1817–1822. Macke, A., Mueller, J., Raschke, E., 1996. Scattering properties of Kokhanovsky, A.A., Nakajima, T., Zege, E.P., 1998b. Physically- atmospheric ice crystals. J. Atmos. Sci. 53, 2813–2825. based parametrizations of the shortwave radiative characteristics Macke, A., Francis, P., McFarquhar, G.M., Kinne, S., 1998. The of weakly absorbing optically thick media: application to liquid role of ice particle shapes and size distributions in the single water clouds. Appl. Opt. 37, 9750–9757. scattering properties of Cirrus clouds. J. Atmos. Sci. 55, Kokhanovsky, A.A., et al., 2003. A semi-analytical cloud retrieval 2874–2883. algorithm using backscattering radiation in 04.–2.4 Am spectral Macke, A., Mitchell, D.L., Bremen, L.V., 1999. Monte Carlo radi- range. J. Geophys. Res. D 108, 12 (10.1029/2001JD001543). ative transfer calculations for inhomogeneous mixed phase Kondratyev, K.Ya., Binenko, V.I., 1984. Impact of Cloudiness on clouds. Phys. Chem. Earth, Part B 24, 237–241. Radiation and Climate. Gidrometeoizdat, Leningrad. Magano, C., Lee, C.V., 1966. Meteorological classification of Konnen, G.P., 1985. Polarized Light in Nature. Cambridge Univ. natural snow crystals. J. Fac. Sci., Hokkaido Univ. 7, Press, Cambridge. 321–362. Kuo, L., Labrie, D., Chylek, P., 1993. Refractive indices of water Markel, V.A., 2002. The effects of averaging on the enhancement and ice in the 0.65- to 2.5-Am spectral range. Appl. Opt. 32, factor for absorption of light by carbon particles in micro- 3531–3540. droplets of water. J. Quant. Spectrosc. Radiat. Transfer 72, Kuze, A., Chance, K.V., 1994. Analysis of cloud top height and 765–774. cloud coverage from satellites using the O2 A and B bands. Marsh, N.D., Swensmark, H., 2000. Cosmic rays, clouds and cli- J. Geophys. Res. 99, 14481–14491. mate. Phys. Rev. Lett. 85, 5004–5007. Labonnote, L.C., Brogniez, G., Buriez, J.-C., Doutriaux-Boucher, Marshak, A., et al., 1995. Radiative smoothing in fractal clouds. M., Gayet, J.-F., Macke, A., 2001. Polarized light scattering by J. Geophys. Res., D 100, 26247–26261. inhomogeneous hexagonal crystals. J. Geophys. Res., D 106, Marshak, A., Davis, A., Wiscombe, W., Cahalan, R., 1998. Radia- 12139–12153. tive effects of sub-mean free path liquid water variability in Landolt-Bernstein, L., 1988. Numerical data and functional rela- stratoform clouds. J. Geophys. Res., D 103, 19557–19567. A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 239 Mason, B.J., 1975. Clouds, Rain and Rainmaking. Cambridge Scattering by Nonspherical Particles: Theory, Measurements, Univ. Press, Cambridge. and Applications. Academic Press, New York. Masuda, K., Ishimoto, H., Takashima, T., 2002. Retrieval of cirrus Mishchenko, M.I., Travis, L.D., Lacis, A.A., 2002. Scattering, Ab- optical thickness and ice-shape information using total and po- sorption, and Emission of Light by Small Particles. Cambridge larized reflectance from satellite measurements. JQSRT 75, Univ. Press, Cambridge. 39–51. Mitchell, D.L., 2000. Parametrization of the Mie extinction and Masunaga, H., Nakajima, T.Y., Nakajima, T., Kachi, M., Suzuki, absorption coefficients for water clouds. J. Atmos. Sci. 57, K., 2002. Physical properties of maritime low clouds as re- 1311–1326. trieved by combined use of Tropical Rainfall Measuring Mis- Mitchell, D.L., Arnott, W.P., 1994. A model predicting the evolu- sion (TRMM) Microwave Imager and Visible/Infrared Scanner: tion of ice particle spectra and radiative properties of cirrus 2. Climatology of warm clouds and rain. J. Geophys. Res. 107 clouds: II. Dependence of absorption and extinction on ice crys- (10.1029/2001JD001269). tal morphology. J. Atmos. Sci. 51, 817–832. McFarquhar, G.M., Yang, P., Macke, A., Baran, A.J., 2002. A new Muinonen, K., Nousiainen, T., Fast, P., Lumme, K., Peltoniemi, J.I., parameterization of single scattering solar radiative properties 1996. Light scattering by Gaussian random particles: ray optics for tropical anvils using observed ice crystal size and shape approximation. J. Quant. Spectrosc. Radiat. Transfer 55, distributions. J. Atmos. Sci. 59, 2458–2478. 577–601. McGill, M.J., Hlavka, D., Hart, W., Scott, V.S., Spinhirne, J., Nakajima, T., King, M.D., 1990. Determination of the optical thick- Schmid, B., 2002. The cloud physics lidar: instrument de- ness and effective particle radius of clouds from reflected solar scription and initial measurement results. Appl. Opt. 41, radiation measurements. Part 1. Theory. J. Atmos. Sci. 47, 3725–3733. 1878–1893. McGraw, R., Nemesure, S., Schwartz, S.E., 1998. Properties and Nakajima, T., King, M.D., 1992. Asymptotic theory for optically evolution of aerosols with size distributions having identical thick layers: application to the discrete ordinates method. Appl. moments. J. Aerosol Sci. 29, 761–772. Opt. 31, 7669–7683. Melnikova, I.N., Mikhailov, V.V., 1994. Spectral scattering and Nakajima, T., King, M.D., Spinhirne, J.D., Radke, L.F., 1991. De- absorption coefficients in strati derived from aircraft measure- termination of the optical thickness and effective particle radius ments. J. Atmos. Sci. 51, 925–931. of clouds from reflected solar radiation measurements. Part II. Melnikova, I.N., Mikhailov, V.V., 2000. Vertical profile of spectral Marine stratocumulus observations. J. Atmos. Sci. 48, optical parameters of stratus clouds from airborne radiative 728–750. measurements. J. Geophys. Res. 105 (D), 23255–23272. Nussenzveig, H.M., 1992. Diffraction Effects in Semiclassical Scat- Melnikova, I.N., Dlugach, Zh.K., Nakajima, T., Kawamoto, K., tering. Cambridge Univ. Press, London. 2000a. Calculation of the reflection function of an optically Okada, K., Heintzenberg, J., Kai, K., Qin, Y., 2001. Shape of at- thick scattering layer for a Heney–Greenstein phase function. mospheric mineral particles collected in three Chinese arid-re- Appl. Opt. 39, 4195–4204. gions. Geophys. Res. Lett. 28, 3123–3126. Melnikova, I.N., Domnin, P.I., Mikhailov, V.V., Radionov, V.F., Palle, E., Butler, C.J., 2002. The proposed connection between 2000b. Optical characteristics of clouds derived from measure- clouds and cosmic rays: cloud behaviour during the past 50– ments of reflected or transmitted solar radiation. J. Atmos. Sci. 120 years. J. Atmos. Sol.-Terr. Phys. 64, 327–337. 57, 2135–2143. Paul, S.K., 2000. Cloud drop spectra at different levels and with Menon, S., Saxena, V.K., Durkee, P., Wenny, B.N., Nielsen, K., respect to cloud thickness and rain. Atmos. Res. 52, 303–314. 2002. Role of sulfate aerosol in modifying the cloud albedo: a Peltoniemi, J.I., 1993. Light scattering in planetary regolits and closure experiment. Atmos. Res. 61, 169–187. cloudy atmospheres. PhD thesis, University of Helsinki. Minin, I.N., 1988. Radiative Transfer Theory in Planetary Atmos- Peltoniemi, J.I., Lumme, K., Muinonen, K., Irvine, W.M., 1989. pheres. Nauka, Moscow. Scattering of light by stochastically rough particles. Appl. Opt. Mishchenko, M.I., Sassen, K., 1998. Depolarization of lidar returns 28, 4088–4095. by small ice crystals. Geophys. Res. Lett. 25, 309–312. Pinkus, R., Klein, S.A., 2000. Unresolved spatial variability and Mishchenko, M.I., et al., 1995. Effect of particle nonsphericity on microphysical process rates in large-scale models. J. Geophys. bidirectional reflectance of Cirrus clouds. Proc. of the 1995 Res. 105 (D22), 27059–27065. ARM Science Meeting, San Diego, CA, March 19–23. Pinsky, M.B., Khain, A.P., 2002. Effects of in-cloud nucleation and Mishchenko, M.I., Rossow, W.B., Macke, A., Lacis, A.A., 1996. turbulence on droplet spectrum formation in cumulus clouds. Sensitivity of cirrus cloud albedo, bidirectional reflectance and Q. J. R. Meteorol. Soc. 128, 501–534. optical thickness retrieval accuracy to ice particle shape. J. Geo- Platt, C.M.R., 1978a. Lidar backscattering from horizontally ori- phys. Res. D 101, 16973–16985. ented ice crystals. J. Appl. Meteorol. 17, 482–488. Mishchenko, M.I., Dlugach, J.M., Yanovitskij, E.G., Zakharova, Platt, C.M.R., 1978b. Some microphysical properties of an ice N.T., 1999. Bidirectional reflectance of flat, optically thick par- cloud from lidar observations of horizontally oriented crystals. ticulate layers: an efficient radiative transfer solution and appli- J. Appl. Meteorol. 17, 1220–1224. cations to snow and soil surfaces. J. Quant. Spectrosc. Radiat. Platnick, S., 2000. Vertical photon transport in cloud remote sensing Transfer 63, 409–432. problems. J. Geophys. Res. 105 (D18), 22919–22935. Mishchenko, M.I., Hovenier, J.W., Travis, L.D. (Eds.), 2000. Light Platnick, S., 2001. Approximations for horizontal photon transport 240 A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 in cloud remote sensing problems. J. Quant. Spectrosc. Radiat. Shifrin, K.S., Tonna, G., 1993. Inverse problems related to light Transfer 68, 75–99. scattering in the atmosphere and ocean. Adv. Geophys. 34, Platnick, S., Li, J.Y., King, M.D., Gerber, H., Hobbs, P.V., 2001. A 175–252. solar reflectance method for retrieving the optical thickness and Shurcliff, W.A., 1962. Polarized Light. Harvard Univ. Press, Cam- droplet size of liquid water clouds over snow and ice surfaces. bridge. J. Geophys. Res. 106 (D14), 15185–15199. Sobolev, V.V., 1972. Light Scattering in Planetary Atmospheres. Prishivalko, A.P., Babenko, V.A., Kuzmin, V.N., 1984. Scattering Nauka, Moscow. and Absorption of Light by Inhomogeneous and Anisotropic Spinhirne, J.D., Nakajima, T., 1994. Glory of clouds in the near Spherical Particles. Nauka i Tekhnika, Minsk. infrared. Appl. Opt. 33, 4652–4662. Pruppacher, H.P., Klett, J.D., 1978. Microphysics of Clouds and Stephens, G.L., Tsay, S.-C., 1990. On the cloud absorption anom- Precipitation. D. Reidel Publ., Dordrecht. aly. Q. J. R. Meteorol. Soc. 116, 671–704. Rogovtsov, N.N., 1991. Radiative transfer in scattering media of Swensmark, H., 1998. Influence of cosmic rays on Earth’s climate. different configurations. In: Ivanov, A.P. (Ed.), Scattering and Phys. Rev. Lett. 81, 5027–5030. Absorption of Light in Natural and Artificial Dispersed Media. Swensmark, H., Friis-Christensen, E., 1997. Variations of cosmic Nauka I Tekhnika, Minsk, pp. 58–81. ray flux and global cloud coverage. A missing link in solar- Rogovtsov, N.N., 1999. Properties and Principles of Invariance climate relationships. J. Atmos. Sol.-Terr. Phys. 59, 1225–1232. Belarussian Polytechnical Academy, Minsk. Takano, Y., Liou, K.-N., 1989. Solar radiative transfer in cirrus Rolland, P., Liou, K.N., King, M.D., Tsay, S.C., McFarquhar, G.M., clouds: 1. Single scattering and optical properties of hexagonal 2000. Remote sensing of optical and microphysical properties of ice crystals. J. Atmos. Sci. 46, 3–19. cirrus clouds using moderate-resolution imaging spectroradiom- Takano, Y., Liou, K.-N., 1995. Solar radiative transfer in cirrus eter channels: methodology and sensitivity to physical assump- clouds: 1. Light scattering by irregular ice crystals. J. Atmos. tions. J. Geophys. Res., D 105, 11721–11738. Sci. 452, 818–837. Rossow, W.B., 1989. Measuring cloud properties from space: a Thomas, G., Stamnes, K., 1999. Radiative Transfer in Atmosphere review. J. Climate 2, 419–458. and Ocean. Cambridge Univ. Press, Cambridge. Rossow, W.B., Schiffer, R.A., 1999. Advances in understanding Titov, G.A., 1998. Radiative horizontal transport and absorption in clouds from ISCCP. Bull. Am. Meteorol. Soc. 80, 2261–2287. stratocumulus clouds. J. Atmos. Sci. 55, 2549–2560. Rossow, W.B., Garder, L.C., Lacis, A.A., 1989. Global, seasonal Tricker, R.A.R., 1970. Introduction to Meteorological Optics. cloud variations from satellite radiance measurements: Part I. Elsevier, London. Sensitivity of analysis. J. Climate 2, 419–458. Trishchenko, A.P., Li, Z., Chang, F.-L., Barker, H., 2001. Cloud Rozenberg, G.V., 1962. Optical characteristics of thick weakly ab- optical depth and TOA fluxes: comparison between satellite and sorbing scattering layers. Dokl. AN SSSR 145, 775–777. surface retrievals from multiple platforms. Geophys. Res. Lett. Rozenberg, G.V., 1967. Physical foundations of light scattering 28, 979–982. media spectroscopy. Sov. Phys., Usp. 91, 569–608. Twomey, S., 1977. Atmospheric Aerosols. Elsevier, London. Rozenberg, G.V., Malkevitch, M.S., Malkova, V.S., Syachinov, V.I., Tynes, H.H., Kattawar, G.W., Zege, E.P., Katsev, I.L., Prikhach, 1978. The determination of optical characteristics of clouds A.S., Chaikovskaya, L.I., 2001. Mueller matrix for multiple from measurements of the reflected solar radiation using data light scattering in turbid media: comparison of two independ- from the Sputnik ‘‘KOSMOS-320’’. Izv. Acad. Sci. USSR, Fiz. ent methods. Appl. Opt. 40, 400–412. Atmos. Okeana 10, 14–24. Uchiyama, A., Asano, S., Shiobara, M., Fukabori, M., 1999. Sassen, K., 1991. The polarization lidar technique for cloud re- Ground-based Cirrus observations: I. Observation system and search. Bull. Am. Meteorol. Soc. 78, 1885–1903. results of frontal Cirrostratus Clouds on June 22 and 30, 1989. Sassen, K., 2001. Cloud and aerosol research capabilities at FARS. J. Meteorol. Soc. Jpn. 77, 522–551. Bull. Am. Meteorol. Soc., 82, 1119–1138, 1885–1903. Van de Hulst, H.C., 1957. Light Scattering by Small Particles. Saunders, C., Rimmer, J., Jonas, P., Arathoon, J., Liu, C., 1998. Wiley, NY. Preliminary laboratory studies of the optical scattering proper- Van de Hulst, H.C., 1980. Multiple Light Scattering: Tables, For- ties of the crystal clouds. Ann. Geophys. 16, 618–627. mulas and Applications. Academic Press, New York. Schreier, R., 2001. Solarer Stralungstransport in der inhomogenen Vasilyev, A.V., Melnikova, I.N., 2002. Short-Wave Solar Radiation Atmosphare. PhD thesis, Institute fur Meereskunde, Kiel. in the Earth Atmosphere, Calculation. Observation. Interpreta- Schreier, R., Macke, A., 2001. On the accuracy of the independent tion. St. Petersburg State University, St. Petersburg. column approximation in calculating the downward fluxes in the Volkovitsky, O.A., Pavlova, L.N., Petrushin, A.G., 1984. Optical UVA, UVB, and PAR spectral regions. J. Geophys. Res. 106 Properties of the Ice Clouds. Gidrometeoizdat, Leningrad. (D13), 14301–14312. Volnistova, L.P., Drofa, A.S., 1986. Quality of image transfer Schuller, L., Armbruster, W., Fischer, J., 2000. Retrieval of cloud through light scattering media. Opt. Spectrosc. 61, 116–121. optical and microphysical properties from multispectral radian- Volten, H., Mun˜oz, O., Rol, E., de Haan, J.F., Vassen, W., Hovenier, ces. Atmos. Res. 55, 35–45. J.W., Muinonen, K., Nousiainen, T., 2001. Scattering matrices Shifrin, K.S., 1951. Scattering of Light in a Turbid Media Gostekh- of mineral aerosol particles at 441.6 and 632.8 nm. J. Geophys. teorizdat, Moscow English translation: NASA Tech. Trans. TT Res. 106 (D15), 17375–17401. F-447, 1968, NASA, Washinghton, DC. Wang, M., King, M.D., 1997. Correction of Rayleigh scattering A. Kokhanovsky / Earth-Science Reviews 64 (2004) 189–241 241 effects in cloud optical thickness retrievals. J. Geophys. Res. under multiple light scattering in a medium. Opt. Spectrosc. 72, 102 (D22), 25915–25926. 220–226. Warner, J., 1973. The microstructure of cumulus clouds: Part IV. Zege, E.P., Kokhanovsky, A.A., 1994. Analytical solution for opti- The effect on the droplet spectrum of mixing between cloud and cal transfer function of a light scattering medium with large environment. J. Atmos. Sci. 30, 256–261. particles. Appl. Opt. 33, 6547–6554. Warren, S.G., 1984. Optical constants of ice from the ultraviolet to Zege, E.P., Kokhanovsky, A.A., 1995. Influence of parameters of the microwave. Appl. Opt. 23, 1206–1225. coarse aerosols on properties of light fields and optical transfer Wauben, W.M.F., 1992. Multiple Scattering of Polarized Radia- functions: analytical solutions. Izv. Acad. Nauk, Atmos. Ocean. tion in Planetary Atmospheres. PhD thesis, Free University of Phys. 30, 777–783. Amsterdam. Zege, E.P., Ivanov, A.P., Katsev, I.L., 1991. Image Transfer Wells, W.H., 1961. Loss of resolution in water as a result of multi- Through a Scattering Medium. Springer-Verlag, New York. ple small-angle scattering. J. Opt. Soc. Am. 59, 686–691. Zege, E.P., Katsev, I.L., Polonsky, I.N., 1993. Multicomponent ap- Winker, D.M., 1999. Global observations of aerosols and clouds proach to light propagation in clouds and mists. Appl. Opt. 32, from combined lidar and passive instruments to improve radia- 2803–2812. tion budget and climate studies. Proc. AMS 10th Conference on Zege, E.P., Katsev, I.L., Polonsky, I.N., 1995. Analytical solution to Atmospheric Radiation, 290–293. lidar return signals from clouds with regard to multiple scatter- Winker, D.M., Trepte, C.R., 1998. Laminar cirrus observed near ing. J. Appl. Phys. B 60, 345–353. the tropical tropopause by LITE. Geophys. Res. Lett. 25, Zege, E.P., Katsev, I.L., Polonsky, I.N., 1998a. Effects of multiple 3351–3354. scattering in laser sounding of a stratified scattering medium: 1. Yamomoto, G., Wark, D.Q., 1961. Discussion of the letter by R.A. General theory. Izv. Atmos. Ocean. Phys. 34, 36–40. Hanel, ‘Determination of cloud altitude from a satellite’. J. Geo- Zege, E.P., Katsev, I.L., Polonsky, I.N., 1998b. Effects of multiple phys. Res. 66, 3596. scattering in laser sounding of a stratified scattering medium: 2. Yang, P., Liou, K.N., 1998. Single-scattering properties of complex Peculiarities of sounding of the atmosphere from space. Izv. ice crystals in terrestrial atmosphere. Contrib. Atmos. Phys. 71, Atmos. Ocean. Phys. 34, 227–234. 223–248. Zege, E.P., Kokhanovsky, A., Katsev, I.L., Polonsky, I.N., Pri- Yang, P., Liou, K.N., Wyser, K., Mitchell, D., 2000. Parameteriza- khach, A.S., 1998. The retrieval of the effective radius of snow tion of the scattering and absorption parameters of individual ice grains and control of snow pollution with GLI data. In: Mis- crystals. J. Geophys. Res. 105, 4699–4718. hchenko, M.I., Travis, L.D., Hovenier, J.W. (Eds.), Conference Yang, P., et al., 2001. Sensitivity of cirrus bidirectional reflectance on Light Scattering by Nonspherical Particles: Theory, Meas- to vertical inhomogenity of ice crystal habits and size distribu- urements, and Applications, Amer. Meteor. Soc., Boston, MA, tions for two moderate-resolution imaging spectroradiometer pp. 288–290. (MODIS) bands. J. Geophys. Res. 106 (D15), 17267–17291. Zhang, J., Xu, L., 1995. Light scattering by absorbing hexagonal Yanovitskij, E.G., 1997. Light Scattering in Inhomogeneous At- ice crystals in cirrus clouds. Appl. Opt. 34, 5867–5874. mospheres. Springer-Verlag, NY. Zuev, V.E., Belov, V.V., Veretennikov, V.V., 1997. Linear Systems Yum, S.S., Hudson, J.G., 2001. Microphysical relationships in Theory in Optics of Disperse Media. Sibirian Brunch of the warm clouds. Atmos. Res. 57, 81–104. Russian Academy of Sciences Publishing, Tomsk. Zege, E.P., Kokhanovsky, A.A., 1992. On sizing of big particles