In the first stage of the Atmospheric Infrared Sounder/Advanced Microwave Sounding Unit-A/Humidity Sounder for Brazil (AIRS/AMSU/HSB) geophysical retrieval algorithm, cloud liquid-water profiles are retrieved from the microwave measurements. These profiles are used in the forward radiative-transfer calculation for microwave channels in subsequent retrieval steps. The retrieval of cloud liquid water is based on a model for moisture condensation in which the relative humidity for onset of clouds is a retrieved parameter, which makes the retrieval more robust with respect to errors in the retrieved temperature profile and horizontal inhomogeneities of the moisture field within an instrument footprint. Retrieved cloud liquid is compared with ground-based radiometric measurements, and the vertical distribution is compared to relative humidity profiles from dedicated radiosondes launched underneath the Aqua satellite.
 Three instruments on the Aqua satellite, the Atmospheric Infrared Sounder (AIRS), the Advanced Microwave Sounding Unit-A (AMSU-A), and the Humidity Sounder for Brazil (HSB), are used co-operatively to retrieve parameters of the atmosphere and surface [see Aumann et al., 2003; Lambrigtsen, 2003; Lambrigtsen and Calheiros, 2003]. In the first stage of the retrieval process, cloud liquid-water profiles, microwave surface emissivity, and preliminary estimates of the temperature and water-vapor profiles are derived from the microwave channels on AMSU-A and HSB. The cloud-liquid profiles and surface emissivity (along with updated temperature and water-vapor profiles) are then used in the microwave forward calculation during subsequent stages of the physical retrieval algorithm in which AIRS infrared measurements are added. The algorithm is described as “physical” because it seeks to minimize a cost function involving the difference between measured radiances and those calculated from the retrieved state of the atmosphere and surface. There are many precedents for retrieval of integrated cloud liquid water over the oceans from satellite-based microwave instruments, but to calculate brightness temperatures of sounding channels, the vertical profile of cloud liquid is needed, rather than just the integrated amount. Because the HSB has channels in the opaque water line at 183 GHz, the retrieval algorithm produces a cloud-liquid profile over both land and water surfaces. The microwave retrieval and forward algorithms have been described previously [Rosenkranz, 2001, 2003] but this paper documents version 4.0, which contains some significant improvements in the liquid-water retrieval algorithm, such as allowing the relative humidity at which liquid water begins to form to be a variable.
 Some issues related to estimation of cloud-liquid profiles have also been discussed by Wilheit and Hutchison , who did a sensitivity study for retrieval of cloud-base altitude using similar microwave channels. In a simulation with a uniform cloud layer, they found a potential for vertical resolution of ∼1 km, which in the lower troposphere would be equivalent to ∼100 hPa in pressure.
 The HSB operated from June 2002 until 5 February 2003, when it was powered off because of excessive current in the scan motor. Although spacecraft operators have made several attempts to restart HSB, none have yet succeeded. This paper is concerned with results obtained for the period when HSB data were available. The AMSU-A has channels that are sensitive to total cloud liquid water over the ocean, but it alone cannot provide information on the vertical distribution of moisture.
2. Moisture and Condensation Model
 The microwave algorithm uses AMSU-A channels 4–14 (53–57 GHz) for estimation of temperature, and the remaining microwave channels (24, 31, 50, 89, 150, 183 ±1, ±3, and ±7 GHz) to estimate atmospheric moisture and surface brightness. Although the window channels from 24 to 89 GHz are sensitive to the differences in absorption spectra of vapor and liquid integral amounts, the four HSB weighting functions at 150 to 183 GHz each sense different levels in the atmosphere, where opacity may be contributed by water-vapor line and continuum absorption, or (in addition) liquid water. Hence the radiometric measurements alone are not sufficient to separately estimate profiles of vapor and liquid. Wilheit  pointed out that the physics of water-vapor condensation add some a priori information or constraints that can be used to retrieve the vertical distribution of liquid; according to this concept, liquid water is placed at the altitudes where the measurements force relative humidity into saturation. However, this constraint is subject to uncertainties in the retrieval. Although the water-vapor profile is saturated within the cloudy part of the field of view, the clouds may not be spatially resolved; furthermore, the saturation point depends on temperature, and the retrieved temperature profile contains some error. Therefore, in the AIRS/AMSU/HSB algorithm, the relative humidity at which liquid water begins to form, denoted HL, is retrieved as one parameter of the atmospheric state, along with the relative humidity profile.
 The condensation model links both the water vapor and cloud liquid water mixing-ratio profiles to a single retrieved profile H and a parameter HL (i.e., H depends on the pressure level, HL does not). H and HL are measured in percent of saturation vapor. We first define the functions
The derivative of brightness temperature with respect to H is used in the retrieval algorithm, and we note that equation (1) has a continuous first derivative everywhere. The choice of a value for c, which determines the curvature of the bend in the ramp function, is somewhat arbitrary, but the discussion in the next section will show that its importance is minor. The liquid-water mixing ratio averaged over an AMSU-A footprint is calculated as
where c1 is a coefficient equivalent to a liquid/air mass mixing ratio of 10−5 per percent. The average water-vapor mixing ratio in the footprint is calculated as
where ρS is the saturation vapor mixing ratio. Thus equation (4) implies that the value of ρV/ρS lies between zero and HL/100. The saturation vapor density ρS is computed from the temperature profile, using a formula adapted from Liebe  for saturation vapor pressure pS in hPa:
Supercooled liquid water clouds are allowed, hence saturation is calculated with respect to liquid water (by extrapolation) when the temperature is below 273 K.
 As illustrated in Figure 1, for HL = 100 (solid curves), H is equal to relative humidity when it is within the approximate range 10–90%; near HL, H changes from a water-vapor variable to liquid-water, and values of H ≫ HL increase liquid water while the vapor remains fixed. The rounded transitions of slope improve the convergence properties of the retrieval solution. Because both H and HL are variables, the retrieval can find values of liquid water that satisfy the brightness temperatures of window and water-vapor channels even with some error in the temperature profile.
3. Moisture/Surface Retrieval Solution
 The H profile, HL, and four surface emissivity parameters To, T1, T2 and pρ are concatenated into a vector P. The surface model is described by Rosenkranz and Barnet . The cost function to be minimized is
In the above, Pest is the estimate of P, Po is its a priori mean value, and SP is its covariance matrix with respect to Po. is a vector containing the measured antenna temperatures of AMSU channels 1, 2, 3, 15, and the four HSB channels, and Se is their noise covariance matrix, assumed diagonal. TB is a brightness temperature vector computed at each iteration from the current estimated values of temperature, moisture, and surface brightness, using the transmittance model [Rosenkranz, 2003], and Θ′ is a “tuning” correction for antenna sidelobes and possible systematic forward-model errors. Sf is a diagonal covariance matrix which approximately represents errors in TB resulting from errors in the temperature profile retrieval and tuning. Superscript t indicates the transpose. The solution to equation (6) is obtained by Newtonian iteration with damping as described by Rosenkranz . The matrices Se and Sf have fairly small elements, so the minimum of equation (6) generally occurs where the vector TB + Θ′ is close to the observations. Since TB is computed using the vapor and liquid mixing ratios, the role of H (in P) is only to introduce the a priori statistics and constraints into the algorithm. The constant coefficients in equations (2) and (3) determine the scaling relation between H and the liquid water profile, but do not have crucial importance for the retrieved values of liquid, because the latter must produce a TB that minimizes equation (6).
 The influence of the a priori statistics, Po and SP, on the retrieval is a function of the degree to which the brightness temperatures are sensitive to variations in each parameter. In general, the statistics help to regularize the solution, preventing it from producing profiles with highly unlikely features such as oscillations that correspond to small variations in brightness temperatures; such features would make the first term of equation (6) large. On the other hand, features that have a strong relation to the brightness temperatures (an example would be the integral of cloud liquid, at least over a water surface) are tightly constrained by the measurements, because errors in those large vertical-scale features would make the second term of equation (6) large.
 The a priori relative humidity is obtained from climatological (location-dependent) databases of temperature and vapor mixing ratio, but limited to <90%. Hence the initial cloud liquid-water profile always has very small values. The covariance matrix of relative humidity was calculated from the TIGR profile ensemble [Chedin et al., 1985]. For HL and the surface-related elements of P, it is necessary to postulate statistics based on physical considerations and previously observed ranges of variation. In order not to allow more degrees of freedom in the model than the measurements will support, these statistics are defined differently depending on the type of surface over which the measurements were taken. Over non-frozen water and land surfaces, HL was assumed to have a mean of 100 and an a priori variance of 64, i.e., (8%)2. This variance of HL is roughly consistent with Sf and with the interpretation of HL as being necessary to compensate for errors in the saturation water vapor. Over cryosphere surfaces (sea ice, snow covered land, glacier), the retrieval needs greater freedom to fit the surface brightness spectrum, so the variance of HL is set to zero, which fixes HL at a value of 100.
 In pre-launch simulations done at the Jet Propulsion Laboratory, using clouds from an NCEP forecast model, RMS errors for integrated liquid-water content were 0.045 mm over water surfaces, and 0.146 mm over land surfaces. The reason for the difference is that the higher emissivity of a land surface reduces the cloud signal in the more transparent AMSU channels. Frozen surfaces such as ice and snow present even greater difficulties for retrieval owing to variability of their emissivity spectra with respect to time and other contingent conditions.
4. Comparisons With Other Radiometers and Radiosonde Measurements
 Integrated liquid water is also measured over oceans by the AMSR-E instrument on Aqua, using an algorithm similar to the one described by Wentz and Spencer . To compare AMSR-E retrievals (version 3) with those of AMSU/HSB, both were put on a 0.5° latitude/longitude grid for the day of 6 September 2002. The RMS difference was 0.049 mm over 151,865 grid points selected to exclude rain (as indicated by AMSR-E).
Figure 2 compares the integral of cloud liquid water from the AMSU/HSB algorithm with measurements by ground-based radiometers at the Nauru Island (Tropical Western Pacific) site of the Atmospheric Radiation Measurement program (ARM-TWP) (see Ackerman and Stokes  or http://www.arm.gov/sites/). The RMS error is consistent with the simulations. Note that the ARM algorithm can produce negative liquid-water content, which gives an indication of its noise level. The ground-based radiometer's retrieval accuracy is not limited by uncertainty in the background emissivity, as in the case of a satellite-based retrieval. The retrieval on 20 January 2003 (marked in Figure 2) is an outlier. Examination of this case showed several areas of heavy clouds in nearby satellite footprints. The field of view of the ground-based radiometer is very small compared to that of the satellite instruments, so horizontal inhomogeneity of the clouds may have contributed to the large error on this day.
 Because Figure 2 is a comparison of retrievals from two radiometric instruments, it does not exclude the possibility of both retrievals being in error, for example through the theory of absorption by clouds. However, English  has validated cloud liquid water retrievals from an airborne microwave radiometer by means of in situ measurements with a hot-wire probe.
 Special validation radiosondes were launched at the ARM sites during Aqua overpasses and “best estimate” profiles have been produced from soundings launched just before and after each overpass [Tobin et al., 2005]. Although radiosondes do not measure liquid water, their relative humidity profiles can be used to deduce the altitude of a cloud. It was assumed that a cloud was present at pressure levels where the relative humidity was >99%. The mean cloud pressure was calculated for each of those soundings, and is compared in Figure 3 to the first moment of the vertical distribution of liquid water with pressure, retrieved from the AMSU/HSB data. The correlation coefficient between the two measures of cloud altitude is 0.63, which indicates some degree of skill in locating clouds vertically.
Figures 4 and 5are similar comparisons for the Southern Great Plains (ARM-SGP) site. It is evident in Figure 4 that some clouds over land (even with liquid water >0.15 mm) are missed by the AMSU/HSB algorithm, a problem seen also in the simulations. Figure 5 shows no significant correlation, although that may in part be due to the smaller range of cloud altitudes at this site than at TWP.
 In the operational algorithm, profiles with integrated liquid water >0.5 mm are flagged and not used in the later retrieval stages, except to derive cloud parameters. For the purpose of validating the liquid water retrieval, these high-liquid cases are included here, as long as they satisfy the other criteria for acceptance.
 The microwave transmittance model used in the retrieval algorithm does not include any treatment of scattering in the atmosphere, and it therefore usually fails to provide a satisfactory fit to observations that contain heavy rain. Nevertheless, there are some satellite footprints for which both rain, using the algorithm of Chen and Staelin , and liquid water are retrieved, which implies that some of the retrieved liquid water represents rain. Presumably, these are footprints for which the raindrop-size distribution gives rise predominantly to absorption rather than scattering. An example from a frontal system over the North Atlantic in the 23° to 45° latitude range is shown in Figure 6, where the peak value, over altitude, of liquid-cloud density is plotted against the rain rate at AMSU-A resolution. It appears that for those footprints in which rain was detected, the rain rate and liquid density are linearly related. The liquid-water absorption model is based on the Rayleigh approximation, which restricts its accuracy to small droplets (in general, diameter <100 μm). Therefore the retrieved values of liquid density are probably inaccurate for the case of rain. The rain-rate algorithm is based on a statistical relationship to radar measurements, and is not restricted with respect to drop size. Thus from Figure 6 one should only infer the existence of a quantitative relation, without taking the slope of the regression line as being well-calibrated.
 This examination of results from the AMSU/HSB algorithm for liquid-water retrieval showed that it has some ability to estimate the vertical distribution as well as the integrated amount over a water surface. Retrieval of integrated cloud liquid water over a land surface was also demonstrated, although only in a fraction of cases. Further work is needed to characterize the situations that cause failure of cloud detection, and to enlarge the comparison dataset with a greater range of cloud altitudes. Although HSB is no longer in operation, the algorithm could be applied to data from present and future NOAA satellites, which carry similar instruments. Possibilities for improvement of the algorithm by incorporation of infrared measurements, as suggested for example by Wilheit and Hutchison  or Shao and Liu  will also be explored.
 This work was supported by NASA under contracts NAS5-31376 and NNG04HZ51C. The author thanks D. H. Staelin for comments on the manuscript.