The difference between cloud-top altitude Ztop and infrared effective radiating height Zeff for optically thick ice clouds is examined using April 2007 data taken by the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) and the Moderate-Resolution Imaging Spectroradiometer (MODIS). For even days, the difference ΔZ between CALIPSO Ztop and MODIS Zeff is 1.58 ± 1.26 km. The linear fit between Ztop and Zeff, applied to odd-day data, yields a difference of 0.03 ± 1.21 km and can be used to estimate Ztop from any infrared-based Zeff for thick ice clouds. Random errors appear to be due primarily to variations in cloud ice-water content (IWC). Radiative transfer calculations show that ΔZ corresponds to an optical depth of ∼1, which based on observed ice-particle sizes yields an average cloud-top IWC of ∼0.015 gm−3, a value consistent with in situ measurements. The analysis indicates potential for deriving cloud-top IWC using dual-satellite data.
 Infrared (IR) atmospheric window (∼11 μm) radiances are routinely used to estimate cloud-top heights from passive satellite sensors [e.g., Minnis et al., 1995; Rossow and Schiffer, 1999]. The cloud effective radiating temperature Teff is estimated from the observed 11-μm brightness temperature T11 and matched to local temperature soundings to find the cloud-top height. Although it is recognized that Teff corresponds to some level below the tops of optically thin clouds, it is commonly assumed that optically thick clouds have sharp boundaries. The latter are generally treated as blackbodies with T11, after correcting for atmospheric absorption and cloud particle scattering, assumed to be equivalent to the cloud-top temperature. Recently, Sherwood et al.  demonstrated, however, that even deep convective clouds do not have such sharply defined boundaries in the IR spectrum. They found that cloud-top heights derived from the eighth Geostationary Operational Environmental Satellite (GOES-8) were 1–2 km below the convective cloud tops detected by lidar data collected over Florida. Those and other results require new approaches to interpret the infrared brightness temperatures of optically thick clouds. Measurements from active sensors combined with passive infrared radiances are needed to address this outstanding problem.
 Until recently, active remote sensing of optically thick clouds has been extremely limited. Ground-based radars and lidars profile the atmosphere continuously, but observe at only one location. They are also unlikely to detect optically thick ice cloud tops because lidars can only penetrate to optical depths of less than about 3 into the cloud and cloud radars often have no returns from smaller ice crystals common at the tops of such clouds. Airborne active sensors sample larger areas during field campaigns and can outline the tops of the clouds, but they collect data for only a few days during a given experiment. With the 2006 launch of the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite into orbit behind the Aqua satellite in the A-Train, coincident and nearly simultaneous global lidar and infrared radiance measurements are now available. This study uses the measurements from CALIPSO and the Aqua Moderate-Resolution Imaging Spectroradiometer (MODIS) to develop a new method to estimate the physical top of optically thick ice clouds from passive IR imager data.
2. Data and Methodology
 Like Aqua, CALIPSO follows a Sun-synchronous orbit with an approximately 1330-LT equatorial crossing time ∼90 s behind Aqua. Because CALIPSO is offset by 7–18° east of Aqua, the Aqua sensors typically observe the CALIPSO ground track at viewing zenith angles VZA of 9–19°. The primary CALIPSO instrument is the Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP), which has 532 and 1064-nm channels for profiling clouds and aerosols [Winker et al., 2007]. The CALIOP, with footprints nominally 70-m wide and sampled every 330 m, allows the global characterization of cloud vertical structure at resolutions up to 30 m. The CALIPSO data used here are the April 2007 Version 1.21 1/3 km cloud height products [Vaughan et al., 2004].
 Cloud properties derived from 1-km Aqua MODIS radiances using the Clouds and the Earth's Radiant Energy System (CERES) project cloud retrieval algorithms [Minnis et al., 2006] were matched with CALIOP data [see Sun-Mack et al., 2007]. The CERES cloud properties were determined from MODIS radiances using updated versions of the daytime Visible Infrared Solar-Infrared Split Window Technique (VISST) and the nighttime Solar-infrared Infrared Split-window Technique (SIST) [Minnis et al., 1995]. The products include cloud temperature, height, thermodynamic phase, optical depth, effective ice crystal diameter De, and other cloud properties.
 The VISST/SIST first determines Teff, which corresponds to the cloud effective height Zeff, located somewhere within the cloud [e.g., Minnis et al., 1990]. Above 500 hPa, Zeff is determined by matching Teff to a local atmospheric temperature sounding. For optically thin ice clouds, an empirical correction is applied to estimate the true cloud-top temperature Ttop based on cloud emissivity [Minnis et al., 1990]. The true cloud-top altitude Ztop for those clouds is the lowest level in the sounding corresponding to Ttop. Optically thick clouds are assumed to have sharp boundaries and, therefore, most IR radiation reaching the satellite sensor is emitted by the uppermost part of the cloud. In these cases, VISST and SIST assume that Teff is equivalent to Ttop and Ztop = Zeff. VISST accounts for the effects of infrared scattering so, for these clouds, Teff is slightly greater than T11.
 Matched VISST and CALIPSO data from every even day during April 2007 were selected to develop a relationship between the effective and true cloud-top heights of optically thick ice clouds. Clouds with effective emittance exceeding 0.98 (visible optical depth τ > 8) are considered to be optically thick. This definition includes a wide variety of clouds including thick cirrus and convective cloud anvils and cores. Polar clouds (latitudes >60°) were excluded to avoid mischaracterizing them over ice and snow. The method is tested using the odd-day April 2007 MODIS-CALIPSO non-polar matched data.
3. Cloud-Top Height Correction
Figure 1 shows CALIOP backscatter intensity profiles (Figure 1a) and scene classifications for a 1-h segment of a 27 April 2007 CALIPSO orbit. It began in darkness over North America, crossed the Pacific and Antarctica into daylight, and ended in the Indian Ocean. The scene classifications (Figure 1b), which show cloud and aerosol locations, are overlaid with black dots corresponding to Ztop from CERES-MODIS for optically thick, single-layer ice clouds. These are evident as the gray areas underneath the clouds. The absence of black dots indicates that the cloud is liquid water, multilayered, or optically thin cirrus. Generally, Ztop is 1–2 km below the CALIPSO top ZtopCAL.
 The cloud-top height pairs for all even days during April 2007 are plotted in Figure 2 as density scatter plots with linear regression fits. In Figure 2a, the average difference between the 15,367 ZtopCAL and their Zeff pairs increases slightly with increasing altitude. The mean difference, Zeff − ZtopCAL, is −1.58 ± 1.26 km. The linear regression fit plotted over the data,
yields a squared linear correlation coefficient R2 = 0.89. According to equation (1), the difference ΔZ between Ztop and Zeff rises from ∼1.25 km for Zeff = 5 km to more than 2 km for Zeff > 14 km.
 Applying equation (1) to Zeff in Figure 1b yields the new values in Figure 1c that are generally very close to the corresponding ZtopCAL. Figure 2b compares the 15,170 values of ZtopCAL and Ztop computed with equation (1) for all April 2007 odd-day data. For ZtopCAL > 3 km, the data are centered along the line of agreement, while lower cloud heights are overestimated. The correction yields a mean difference of −0.03 ± 1.21 km and R2 = 0.91. This empirical correction effectively eliminates the bias and slightly reduces the random error in the estimated Ztop. The correction is robust in that it applies well to two independent datasets.
 For Zeff < 3 km, the data are centered on the line of agreement in Figure 2a indicating no correction is needed. The correction results in unphysical values at those altitudes and should not be applied. This overestimation is due to uncertainties in the atmospheric profile of temperature in the lower layers [e.g., Dong et al., 2008] or to misclassification of supercooled-liquid water or mixed-phase clouds as ice clouds by the CERES-MODIS Aqua algorithm. The basic assumption that the correction is for ice clouds would be violated for those and other low-level clouds. The tops of water clouds are unlikely to be more than a few hundred meters above Zeff [e.g., Dong et al., 2008]. For low clouds, a better estimate of Zeff and a more accurate phase classification are needed before applying a correction to obtain Ztop. That effort is beyond the scope of this paper.
 To minimize the impact of low-altitude temperature and phase uncertainties, the regression was performed using the even-day data (13,046 samples) only for ice clouds with effective pressures, peff < 500 hPa, yielding
 Applying equation (2) to odd-day clouds having peff < 500 hPa yields an average difference of −0.08 ± 1.15 km, a value nearly equal to the mean difference of −0.13 ± 1.14 km that would be obtained by applying equation (1) to the same odd-day dataset. If equation (2) is used to estimate Ztop for all of the odd-day data, the mean difference is 0.07 ± 1.24 km. The results are essentially the same for both fits. The 500-hPa cutoff effectively precludes the introduction of any new low-cloud height biases.
 The instantaneous differences can mainly be attributed to uncertainties in the temperature profiles used to convert temperature to altitude, data spatial mismatches, VZA dependencies, and variations in cloud microphysics. The small portions of the satellite pixel sampled by the narrow lidar footprint can cause some significant differences if cloud height varies within the pixel. Errors in the temperature profiles can move Zeff up or down. For example, some Zeff values between 6 and 14 km in Figure 2a are below their ZtopCAL counterparts and account for ∼1 km of the range in ΔZ. It could also account for some of the extreme overestimates. This type of error will occur some of the time since the temperature profiles are based on numerical weather analysis assimilation of temporally and spatially sparse observations. The VZA has little impact here.
 To examine the impact of cloud microphysics on ΔZ, radiative transfer calculations were performed by applying the Discrete Ordinates Radiative Transfer (DISORT [Stamnes et al., 1988] method to an example case. For a given layer, the thickness can be expressed as
where Δτi is the visible optical depth for cloud layer i, the visible extinction efficiency Q has a value of ∼2 [e.g., Minnis et al., 1998], IWCi is the layer ice water content, the density of ice is δ = 0.9 g cm−3, and Dei is the effective diameter of the ice crystals in the cloud layer.
 The DISORT calculations assumed an 8-km thick cloud extending to 13 km in a tropical atmosphere. The cloud was divided into 198 layers with Δzi decreasing from less than 110 m at the base to 40 m at the top. The bottom-layer optical depth was specified at 12 to ensure that the cloud is optically thick. Teff was determined for a range of IWC and three values of De using the same mean IWC but with three IWC profiles: IWC decreasing linearly from the layer above the base to cloud top, uniform IWC, and IWC increasing from the layer above the base to the top. Zeff was determined from Teff and the simulated cloud-top height correction is ΔZ = 13 km –Zeff. The optical depth (IWC) of the layer above Zeff, the top layer, is the sum of τi (IWCi) above Zeff. Figure 3 shows the results for both uniform and decreasing-with-height IWC. Assuming that ∼1.5 km of the range in ΔZ (Figure 2a) is due to inaccurate temperature profiles, the observed range is ∼4.5 km. That extreme value of ΔZ could occur for De = 180 μm and IWC = 0.01 gm−3 (Figure 3c) or for smaller values of IWC and De (Figure 3b), but is unlikely for very small particles (Figure 3a). The average bias at Zeff = 14 km (Figure 2a) is 2.1 km, a value that can be explained, at VZA = 14°, with uniform or decreasing IWC = 0.014 gm−3 and De = 80 μm (Figure 3b), or with smaller or larger values of IWC and De. For a given value of IWC, ΔZ in Figures 3a–3c is similar for both uniform and decreasing IWC profiles, except that, for a given ΔZ, the IWC is slightly smaller for the decreasing case. For the increasing-with-height case (not shown), ΔZ rapidly approaches zero with increasing IWC for all particle sizes.
 The decreasing-with-height IWC profile is probably most realistic, however, for simplicity, only the uniform IWC case results are considered in the following calculations. Although its value at 5 km is 62 μm, the CERES-MODIS observed mean De varies almost linearly from 55 μm at Zeff = 6 km to 76 μm at 12.6 km, then down to 64 μm at 15 km (not shown). At 14 km, De ∼ 68 μm, requiring IWC to be ∼ 0.011 gm−3. At Zeff = 9 km, ΔZ = 1.6 km and De = 68 μm, requiring IWC = 0.019 gm−3. Since the optical depth corresponding to ΔZ is relatively constant (Figure 3d and other IWC cases), IWC can be estimated at each altitude using the proportional relationship
where ΔZ is determined from equation (1), De is the mean at Zeff, and the proportionality constant k was determined from equation (4) to be 0.000334 gm−3, using the estimate of IWC for Zeff = 14 km and De = 68 μm. Values of uniform IWC were estimated for Zeff = 5–15 km and fitted using a third order polynomial regression to obtain
where Zeff is in km. The R2 equals 0.77 indicating that the average IWC is strongly dependent on cloud height. This fit does not apply below 5 km. While the mean IWC varies between 0.01 and 0.02 gm−3, it is somewhat sensitive to the IWC vertical profile and much larger or smaller values of IWC could result from any individual CERES-MODIS/CALIPSO data pair.
 The behavior of (5) is not surprising given that IWC has been observed to decrease with decreasing cloud temperature. (Teff was not used as the independent variable here because the height differences were more highly correlated with Zeff than with Teff.) Heymsfield and Platt  reported that the mean IWC in cirrus clouds varied from 0.027 gm−3 at T = −25°C to 0.001 gm−3 at −58°C and that IWC variability at a given temperature was typically an order of magnitude or greater. Wang and Sassen  found IWC ranging from 0.017 to 0.001 gm−3 between −20 and −70°C for comparable clouds. Garrett et al.  observed IWCs up to 0.3 gm−3 in a thick anvil cloud, while smaller values, ranging from 0.0001 to 0.02 gm−3, were observed by McFarquhar and Heymsfield  in the top 2 km of three tropical anvils. The mean IWC values estimated here for the top portions of thick ice clouds are well within the range of observations. The variation in the observed IWCs can also explain much of the random error seen in Figure 2b.
Figures 3e and 3f show that the top-layer τ, constant at ∼1.15 for De = 80 and 180 μm, increases to 1.5 for De = 10 μm (Figure 3d) and to larger values when IWC < 0.01 gm−3. The corresponding values for the decreasing IWC case are 0.9 and 1.2 for De = 80 and 10 μm, respectively, and slightly greater for the increasing IWC case. The values of τ for the larger particles are close to that used by Sherwood et al.  to estimate where Zeff should be relative to the lidar-observed top for convective anvils. The difference is mostly due to scattering. Based on the lidar-derived optical depths, Sherwood et al.  concluded that the large values of ΔZ, similar to those in Figure 2a, did not correspond to τ = 1, but to τ ≥ 10. Given the above analysis and the observed range of IWC, it appears that an average value of 2 km for ΔZ is quite reasonable and corresponds to τ ∼ 1 for the size of ice crystals retrieved with VISST. For the matched CALIPSO-CERES data used here, the mean height where the CALIPSO beam was fully attenuated is 1.3 km below Zeff, a value much greater than the 150 m calculated for the airborne lidar used in the Sherwood et al.  analysis. It is not clear why that earlier study produced such different results from the current analysis, but may be due to assumptions used in the τ retrievals from the airborne lidar or differences in power between it and the CALIOP. Nevertheless, the current results are consistent with the expected values of IWC near cloud top.
 While the small range (9°–19°) in VZA for this study precludes development of an empirical correction for VZA dependence, the plots in Figure 3 suggest a simple cosine variation of ΔZ with VZA. Thus, calculating Ztop′ using either equations (1) or (2), the VZA-corrected estimate of Ztop is
where ΔZ = (Ztop′ − Zeff) cos(VZA). Validating equation (6) will require a comprehensive combined imager-lidar dataset having a wide range of VZAs.
5. Concluding Remarks
 The effective cloud radiating height, Zeff, may be adequate for radiative transfer calculations in climate or weather models, but the physical boundaries of a cloud are needed to determine where condensates form and persist. The upper boundary is inadequately represented by Zeff for ice clouds. A simple parameterization was developed that uses Zeff to provide, on average, an unbiased estimate of Ztop for optically thick ice clouds. It complements other parameterizations used to estimate Ztop for optically thin cirrus. The random errors in Ztop determined with the new parameterization are consistent with the variations in IWC observed near cloud top in previous in situ measurements. Reducing the instantaneous uncertainty in Ztop may be possible using combinations of different spectral channels or dual-angle views, but the reduction will be limited by the accuracy of the temperature profile. When applied, the parameterization estimate of Ztop should have some level above the tropopause as an upper limit to minimize unrealistic results. If the observed cloud penetrates into the stratosphere, however, Zeff and, hence, Ztop can be underestimated because the VISST selects Zeff as lowest the altitude where Teff is found in the sounding. Additionally, the correction should not be applied to low-level clouds. Although this correction for Ztop is a function of Zeff determined from the 11-μm brightness temperature, it is probably applicable to Zeff determined using other techniques such as CO2 slicing. Although only 1 month of CALIPSO data was used here, the results appear robust. Testing with data from other seasons is required to confirm that contention and data from other satellites, that are not near the CALIPSO ground track, would be needed to verify the formulation for off-nadir angles. With an expanded dataset, it may also be possible to refine the correction in terms of cloud type (e.g., cirrus, anvil, or convective core).
 This research was supported by NASA through the NASA Energy and Water Cycle Studies Program and the CALIPSO and CERES projects.