Microwave radiometers from space offer limited information regarding cloud vertical structure because the incident wavelength is larger or comparable to the particle size and the scattering and absorption is often in Rayleigh's regime. For nonraining clouds the brightness temperatures are almost linearly related to the vertically integrated cloud liquid water. Thus the retrieval of cloud parameters such as liquid and ice water paths can be based on an approximated radiative transfer scheme. For a single-layered cloud as shown in Figure 4 the solution of the radiative transfer equation including the scattering can be approximated by [Weng et al., 2001]
where μ is the cosine of local zenith (polar) angle, B is the thermal radiance emitted by the cloud with the temperature of T, I0 and I1 are the downwelling and upwelling radiances from the medium above and below the cloud layer, respectively, τ is the optical thickness, and κ and β are related to cloud optical parameters using
where ω, α, and g are cloud single-scattering albedo, similarity parameter, and asymmetric factor, respectively. The coefficients of γ1, γ2, γ3, and γ4 are related to the reflectivity at the boundaries and are discussed in detail by Weng et al. .
3.1. Cloud Liquid Water Algorithm
 Microwave measurements at lower-frequency window channels have been exploited to derive the cloud liquid water of nonraining clouds [Greenwald et al., 1993; Weng and Grody, 1994a; Weng et al., 1997, 2000; Wentz, 1997]. The retrievals of cloud liquid water normally require multiple microwave window channels so that the effects of the absorption of atmospheric gaseous constituents and the emission of the surface can be removed. At low frequencies the atmospheric scattering can be further neglected in equation (2). Thus:
where ε and Ts are the sea surface emissivity and temperature, respectively.
 Since in microwave frequencies the radiance is linearly proportional to temperature, the brightness temperatures are preferred in the algorithm. Using equation (4), the cloud liquid water and total precipitable water can be derived using two AMSU window channels at 23.8 and 31.4 GHz [Weng et al., 2000] by further assuming an isothermal atmosphere. Essentially, cloud liquid water (L) and total precipitable water (V) are derived using
respectively, and where
where κv and κl are the water vapor and cloud liquid water mass absorption coefficients, respectively. Using Rayleigh's approximation, one can express κl in terms of cloud layer temperature (Tl(c)) as
Oxygen optical thickness is parameterized as a function of sea surface temperature through
 Table 3 illustrates some coefficients that can be used for various AMSU channels. In this study, only 23.8 and 31.4 GHz are used for L and V retrievals. The retrievals used in Table 3 were validated against the liquid water obtained from the ground-based radiometer measurements. The RMS error for nonprecipitating cloud liquid is ∼0.05 mm [Grody et al., 2001]. Figure 5 shows a global distribution of cloud liquid water over oceans derived using the AMSU on board NOAA 16. As discussed in section 2.1, the AMSU antenna temperatures are first corrected for the asymmetric bias. The correction scheme is also made separately for each individual satellite. Notice that the AMSU descending measurements during a 24-hour period do not completely cover the globe because of the orbital gaps. It is shown that the algorithm depicts cloud liquid water associated with various systems. The low clouds over oceans off the west coast of South America are detected, although the amount of cloud liquid is small.
Table 3. Parameters Calculated at Four AMSU-A Channels and Used in Total Precipitable Water and Cloud Liquid Water Algorithms
|Parameter||AMSU-A Channel, GHz|
3.2. Cloud Ice Water Path/Particle Size Algorithm
 Microwave remote sensing offers the advantage of being able to measure the ice water path of clouds compared to existing visible and infrared techniques. Visible methods require a number of gross assumptions about ice particle shape, size distribution, and cloud spatial homogeneity to convert from radiance to optical depths to IWP. In addition, thermal infrared techniques require accurate knowledge of cloud temperature. Compared to existing visible and infrared techniques, microwave radiation interacts with ice particles primarily through scattering. The emission and cloud temperature are relatively unimportant. Since ice clouds are above the absorbing part of the atmosphere, they simply modulate the upwelling microwave radiation from below. While the effects of particle shape and size distribution are also important for microwave remote sensing of clouds since they determine the relation between optical depth and IWP, they are more amenable to calculation because the particle sizes are comparable with or smaller than the size of the wavelength. Microwave methods are also complementary to visible and IR methods because the microwave radiation is sensitive to larger ice crystals and to thicker cirrus layers whereas visible/IR radiation is more sensitive to smaller particles and cirrus clouds having lower IWP.
 The retrieval was also recently tested using aircraft millimeter wavelength measurements [Liu and Curry, 1999; Weng and Grody, 2000]. Liu and Curry  presented a method to retrieve IWP using airborne Millimeter-Wave Imaging Radiometer data at 89, 150, and 220 GHz channels. Although the IWP algorithm works well for cirrus clouds in the tropics, an uncertainty arises due to the unknown particle size [Liu and Curry, 1999]. Weng and Grody  proposed an algorithm to derive both IWP and De using dual millimeter wavelength measurements. They found that for a given particle bulk volume density, the brightness temperature at millimeter-wave frequencies can be uniquely related to IWP and De through a two-stream radiative transfer solution. However, the retrievals of cloud ice water path suffer a great amount of uncertainty due to an unknown bulk volume density of ice particles [Weng and Grody, 2000]. Thus it would be a major impact on the ice cloud remote sensing to utilize other sensors, such as visible and infrared radiometers, to identify a possible range of the bulk volume density.
 For a single-layered ice cloud, the radiance at the cloud top can be expressed in a simple form, assuming that cloud optical thickness is very small. The upwelling radiance emanating from an ice cloud is approximated from equation (4) as follows:
where Ω is the scattering parameter and I (τ1, μ) is the upwelling radiance at the cloud base.
 It is clearly seen that microwave radiance is independent of the cloud layer temperature and is directly related to the incident radiation at the cloud base. From a space platform (satellite or aircraft) the upwelling radiance decreases as the scattering parameter increases. A previous study shows that the brightness temperatures calculated using equation (6) have biases of less than a few degrees [Weng and Grody, 2000]. The best accuracy is achieved at the local zenith angles near 54°.
 The variable Ω is important in relation to cloud single-scattering albedo (ω), asymmetry factor (g), and optical thickness (τ) as follows:
For ice particles distributed according to a gamma function, Ω is calculated with Mie theory and is expressed as a function of cloud ice water path (IWP), particle effective diameter (De), and particle bulk volume density (ρi) as follows:
where ΩN is the normalized scattering parameter and is dependent only on the particle effective size parameter being defined as xe = 2π De /λ and m is the complex refractive index. For a smaller De, ΩN at 150 GHz is significantly higher than it is at 89 GHz. However, for a larger De (>1.5 mm), ΩN at both frequencies approaches the same value, indicating that the particle scattering may enter a regime of geometrical optics and become independent of wavelength [Weng and Grody, 2000].
 Using the AMSU-B measurements at 89 and 150 GHz, the scattering parameter ratio is directly related to the particle effective diameter [Zhao and Weng, 2002]:
Note that equation (9) defines a scattering parameter ratio varying between 0 and 1 and allows for a direct determination of De. The ice water path can be derived using
It is evident that IWP and De can be uniquely determined from equations (6–10) for a given constant bulk volume density.
 Notice that the major difficulty in deriving the cloud ice parameters is due to an unknown particle volume density. As shown in equation (10), IWP is directly proportional to the scattering parameter. However, the relationship between ΩN and De is nonlinear and may depend on the particular particle size distribution and bulk volume density. Therefore measurements at two distinct frequencies are normally required to unambiguously determine both IWP and De for a given particle bulk volume density [Evans and Stephens, 1995; Weng and Grody, 2000]. Provided that the bulk volume density of ice particles can be determined independently from other sources, IWP essentially only depends on Ω and De. An error of 30% in the density could result in an error of 25% in IWP [Zhao and Weng, 2002].
 The scattering signature resulting from sea ice and snow particles at higher microwave frequencies is similar to that of the ice particles because the dielectric constants among these scatterers are almost the same. Therefore, for a global application of the IWP and De retrieval algorithm, a procedure is developed to discriminate between the scattering signatures of atmospheres and various surface materials. However, the AMSU alone provides very limited information on surface types due to its lack of polarization measurements. Other data sets such as Advanced Very High Resolution Radiometer infrared data and GDAS surface temperature and surface type data are used in the screening procedure, as well [Zhao and Weng, 2002].
 Surface scattering from snow and sea ice can be largely removed using the measurements at lower AMSU-A frequencies. AMSU-A-derived products of snow cover and sea ice concentration are first used to indicate their presence. The GDAS surface temperature <269 K is used as an additional threshold to identify the scattering signatures of frozen surfaces. The retrieval of atmospheric ice is not performed under these surface conditions. Furthermore, there is no retrieval over high terrains such as the Tibetan Plateau, where the surface temperatures usually are <273 K.
 Deserts also scatter at AMSU 89 and 150 GHz [Weng et al., 2001]. However, scattering from the clouds can be easily separated from scattering from the surface using the satellite infrared measurements and GDAS surface temperatures. If the atmosphere is free from ice clouds, the IR temperature is close to the surface temperature, and therefore the scattering at 89 and 150 GHz must result from the surface. More specifically, for desert scatterers the temperature difference is <10 K and Ω is positive. If the satellite infrared data are not available, AMSU measurements at 183 ± 7 GHz can be used as a substitute because the channel peaks in the lower troposphere and is less affected by the surface. The screening procedure is discussed in detail by Zhao and Weng .
 Figures 6a and 6b display an example of cloud ice water path and effective diameter derived from AMSU for a squall line system over the continental United States. Note that higher amounts of cloud ice water path are associated with larger sizes, which are obvious in northern Mississippi to Alabama where a surface tornado was reported. However, the sizes near 0.5 mm are very common in the system.