A new approach to retrieve cloud base height of marine boundary layer clouds
J. M. Li,
Key Laboratory for Semi-Arid Climate Change of the Ministry of Education, College of Atmospheric Sciences, Lanzhou University, Lanzhou, China
Corresponding author: J. M. Li, Key Laboratory for Semi-Arid Climate Change of the Ministry of Education, College of Atmospheric Sciences, Lanzhou University, Lanzhou, Gansu 730000, China. (firstname.lastname@example.org)
 A novel approach for estimating marine boundary layer cloud base height (CBH) is proposed based on calculated boundary layer lapse rates, collocated cloud top height (CTH), cloud top, and ocean surface temperatures from the A-Train satellite constellation. The method takes advantage of the assumption that decreases of temperature within and below water clouds may follow the different constant apparent lapse rates in the same region, respectively. The CBHs derived from the new method compare well with the coincident CBH product from the active sensors of Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) and CloudSat. The correlation coefficient, the mean difference, and the standard deviation are 0.79 (0.54), 0.02 km (0.03 km), and ±0.35 km (±0.54 km), respectively, when CTH is derived from CALIPSO data (or Moderate Resolution Imaging Spectroradiometer retrieval). Besides the relatively small bias, the most important advantage of this method compared to previous CBH retrieval techniques is that it is independent of boundary layer cloud types, optical thickness, and illumination.
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 Clouds may significantly affect the radiation budget and water cycles of the Earth by altering their various properties. Besides the cloud microphysical properties, the cloud base height (CBH), which is one of the important cloud macrophysical structures, is of particular significance for the infrared radiation at the surface. CBH is also extremely important for air safety, airborne military activities, and surveillance operations [Vislocky and Fritsch, 1997]. Thus, an accurate method for estimating CBH may further improve our understanding for the impact of clouds on the Earth's radiation budget and provide useful information for improvement of aviation safety.
 Ground-based active instruments, such as lidar or radar, can help obtain relatively accurate CBH locally by receiving the backscattered return signal [Wang and Sassen, 2001]. However, they cannot provide CBH information on large scales, especially over the ocean. The advantage of satellites is that high-resolution, two-dimensional distributions of the microphysical and macrophysical properties of clouds may be retrieved on a global scale [Huang et al., 2006]. However, the retrieval of CBH is considerably complex and not directly available from passive satellite observations. Until now, several efforts for retrieving CBH from satellite instruments have been proposed and improved. For example, previous approaches obtained mainly the cloud geometrical thickness (CGT) by applying both satellite-derived cloud optical depth and cloud top temperature, and some additional empirical assumptions that connect cloud optical depth and cloud top temperature with CGT [Smith et al., 1993], and then subtract the CGT from the cloud top height (CTH) to further derive CBH. Hutchison  presented a technique for CGT estimation based on cloud optical thickness, cloud effective radius, and the assumption that liquid water content is constant throughout the vertical extent of the cloud. Similar approaches also were used in optically thin marine stratocumulus and the early convective water clouds [Bennartz, 2007; MeerkÖtter and Zinner, 2007]. Besides the CBH techniques based on visible and infrared cloud parameters, Wilheit and Hutchison  also proposed a method to retrieve the CBH by combining passive microwave brightness temperature and infrared cloud top temperature. However, these methods work mainly for special cloud types (e.g., relatively thin, stratiform, or convective clouds) and are limited to daytime application. However, retrieval of cloud properties also during nighttime is of particular significance for operational and climate change applications.
 In this investigation, we propose a simple approach for retrieving the CBH of marine boundary layer clouds using calculated boundary layer lapse rates, collocated CTH, cloud top, and ocean surface temperature information from the A-Train satellite constellation. A similar method to derive the CTH has been presented. (Sun-Mack, S., et al., Regional apparent boundary layer lapse rates determined from CALIPSO and MODIS data for cloud height determination, submitted to Journal of Applied Meteorology and Climatology, 2013). Here we further improve this methodology and apply it to estimate CBH. An obvious advantage of the method compared to previous CBH retrieval techniques is that it is independent of boundary layer cloud types, optical thickness, and illumination. A detailed presentation of the novel method is provided in section 2. Major results and discussions are given in sections 3 and 4.
2 Data and Method
2.1 Basic Idea
 Generally speaking, the mean ocean boundary layer lapse rate Γa can be expressed as
where Tct and Tsfc are the cloud top and sea surface temperatures, respectively. Zct is the CTH, and Ζsfc denotes the surface elevation. It is worth noting that in this study, the lapse rate is based on the sea surface and cloud top temperatures instead of the air temperatures over the ocean surface and at the cloud top in the standard definition of the lapse rate, respectively. However, the cloud top and surface temperatures can be different from the actual air temperatures at cloud top and above the surface. Thus, we introduce the same term proposed by Sun-Mack et al. (submitted manuscript, 2013) and denote the computed value (left part in equation (1)) as the apparent lapse rate (Γa) in order to distinguish it from the standard definition of the lapse rate. Hereinafter, the apparent lapse rate will be called the lapse rate for convenience. In a previous study, Minnis et al.  used a fixed lapse rate of 7.1 K km−1 between the sea surface and cloud top temperatures to estimate the marine boundary layer depth over the NE Pacific. In more recent work by Sun-Mack et al. (submitted manuscript, 2013), they pointed out that the derived lapse rates over the snow-free ocean range from 5 to 9 K km−1, with mean values of about 6.9 and 7.2 K km−1 during daytime and nighttime, respectively. These results clearly showed that the lapse rates take on a wide range of values at individual locations.
 In fact, the temperature difference (Τsfc − Τct) between the ocean surface and the cloud top in equation (1) may be further divided into two parts. One part is the temperature variation within the cloud, and the other part is the difference below cloud base. That is,
where Γaw and Γab are the apparent lapse rates within and below the water cloud, respectively. In addition, Zcb and Zct represent cloud base and top heights derived from active satellite sensors, respectively. Sea surface elevations Zsfc are set equal to zero in this study. In order to retrieve the cloud base height, equation (2) can be transformed as a theoretical multivariable linear regression model as follows:
where both the coefficients a1 = 1/Γab and a2 = (−Γaw)/Γab are expressed as a function of the lapse rates Γaw and Γab. ΔT is the temperature difference between the ocean surface and the cloud top obtained from satellite data sets. In addition, H is the cloud thickness. For a geographical grid box (such as 5° × 5°) with n marine cloud observational samples, our hypothesis here is as follows: the lapse rates Γaw and Γab are approximated by different constants within the same grid; thus, these cloud samples all satisfy equation (3), that is,
 Thus, by using the instantaneous satellite observations of Zcb, ΔT, and H for each cloud sample, the optimum estimates of coefficients a1 and a2 in each grid box, that is, and , can be given by
where XT is the transpose of the matrix X. Finally, the regression equation about estimated CBH () is
 The regression coefficients and equation are easily inferred from the above model by using historical satellite data sets. If we denote μ as the bias between real (Zcb) and estimated CBH (), then equation (8) can be written as:
 Finally, the final expression can be written mathematically as follows:
 Here the bias μ may be considered as a correction factor. Its values may be estimated roughly based on historical satellite observation (Zcb) and retrieval (), and furthermore, they are grouped using different ΔT and Zct. Eventually, one bias matrix is created in each grid box. The missing values in each bias matrix are filled by interpolating surrounding values or replacing them with μ under the same ΔT and Zct condition in a neighboring grid box. After obtaining all regression coefficients and bias matrices in each 5° × 5° grid box, the CBH of water cloud over ocean in the tested month is estimated finally according to equation (10).
2.2 Satellite Data
 This study uses the collocated Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) cloud product and CloudSat 2B-GEOPROF-Lidar data set. In addition, only those clouds with CTH below 4 km over oceans between 70°N and 70°S are investigated in the study.
 The following MODIS level-2 collection 5.1 cloud parameters (MYD06) [Platnick et al., 2003] are used in this study: cloud top temperature, cloud multilayer flag (CMLF), surface temperature, cloud optical thickness (COT), and cloud fraction. The spatial resolution is 1 × 1 km for CMLF and COT and 5 × 5 km for other parameters.
 The CloudSat 2B-GEOPROF-Lidar data set, which combines both the CloudSat and Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) data streams [Mace et al., 2009], is used to identify single-layer water clouds and avoid contamination of optically thin ice clouds in the water cloud top temperature retrieval. In addition, it also provides the essential information of cloud top and base heights for the regression model.
2.3 Retrieval Procedure
 Our procedure for retrieving CBH from satellite data can be summarized by the following steps.
 First, the main criterion for the selection of cloud samples in each grid (5° × 5°) is the existence of a single-layer cloud field (over 90% cloudy). By using collocated MODIS level-2 collection 5.1 cloud products (MYD06) and 2B-GEOPROF-Lidar data set of CloudSat, the single water clouds can be picked out exactly by limiting the cloud layer number in the 2B-GEOPROF-Lidar data set and the cloud fraction from MYD06 products.
 Second, for each cloud sample, cloud top temperature Tct and surface temperature Tsfc are derived from MYD06, while cloud top and base heights (Zct/Zcb) are obtained from the 2B-GEOPROF-Lidar data set. It is worth noting that the parameter Flagbase/Flagtop in the 2B-GEOPROF-Lidar data set can tell us if the cloud base and top are observed by either CALIPSO or CloudSat or both. In view of the larger vertical resolution of CALIPSO (only 30 m below 8 km) than CloudSat, only those samples whose cloud top heights are just identified by CALIPSO (that is, Flagtop = 2) are selected for use in the study. However, cloud base information from either CloudSat or CALIPSO depends on whether the cloud is opaque or transparent for CALIPSO.
 Third, the optimum estimates of the regression coefficients a1 and a2 (that is, , ) and bias (μ) matrix in each 5° × 5° grid box are inferred according to equations (2)–(9). The time window of satellite data sets that is used to derive the regression coefficients and bias matrix is from January 2008 to June 2008.
 Finally, the CBHs in other months (July and October of 2009) are tested and estimated according to equation (10) according to the above steps.
 Since marine boundary layer clouds frequently exceed the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) detection limit of single scattering optical depth (ετ < 3, where ε is the multiple scattering factor, and τ is the optical depth) [Li et al., 2011], the lidar signal can be completely attenuated within a penetration depth of about several hundred meters for most boundary layer clouds. When CALIPSO fails to detect the CBHs of optically thick clouds, the CBHs from CloudSat are used. However, due to the reflection from the surface and the 1 km pulse length of the Cloud Profiling Radar (CPR), sensitivities of the CPR in the lowest 1 km near the surface are reduced. This strong surface signal could be a large source of uncertainty when clouds have the base below 1 km. To further verify the quality of CALIPSO/CloudSat data in this study, a simple empirical analysis is first performed to see if there are any artifacts due to CALIPSO/CloudSat base height limitations. A previous study has shown that cloud thickness is correlated with the square root of cloud optical depth [Minnis et al., 1992], implying that there should be a somewhat monotonically increasing relationship between cloud thickness and optical depth. However, if the cloud base height from either CALIPSO or CloudSat is in error, then there should be a tailing off of cloud thickness even as optical depth increases. By analyzing this relationship for clouds with top height below and above 1.5 km, we confirmed that the relationship is monotonically increasing when optical depth is below 50, regardless of whether CBH is obtained from CloudSat or CALIPSO. Because the important multiple scattering effect in the analysis of the CALIOP signal may help reduce the attenuation and enhance the detectability of CALIOP, the detection limit of optical depth of CALIOP can possibly reach 16 when the depolarization ratio is as high as 0.4. However, there are relatively few cloud samples with a depolarization ratio as high as 0.4 in this investigation. Those cloud samples, whose cloud base height may be observed by CALIPSO and whose optical depth is beyond the detection limit of CALIOP, might be problematic and may need to be excluded. Statistical results show that these cloud samples account for just 7% of the total samples. This issue may be caused by uncertainties in matching the different satellite data sets and in-cloud optical depths derived from MODIS. In addition, we just use those samples with cloud top above 0.1 km in order to further minimize the impact of the surface on CALIPSO height detection. Therefore, the cloud base information used in this study from CALIPSO and CloudSat is still reliable. In summary, in all cloud samples, about 15.6% of CBH information is provided by CloudSat. Among them, about 74.3% of CloudSat's samples have tops below 1 km.
 Figure 1 depicts the mean geographic distributions of apparent lapse rate within (Γaw) and below (Γab) marine boundary clouds during daytime and nighttime during January 2008 to June 2008, which are calculated by using the optimum estimates of coefficients a1 and a2 in each grid. Several obvious features are seen. First, regional distributions of apparent lapse rate are obvious. The distributions of lapse rates at daytime are similar to those at nighttime. Larger differences are mostly located in the middle and high latitudes. Second, for Γaw, its values range from −3 K km−1 to 6 K km−1. Moreover, the extent of lapse for temperature is degressive from the tropics to high latitudes. Thermal inversions tend to occur more frequently over the oceans at high latitudes, especially in storm tracks during nighttime. However, for the Γab, its minimum values are primarily located in the tropics with a value of about 6 K km−1. From 30° poleward, the Γab value increases with latitude, and the maximum values even reach 18 K km−1, especially during nighttime. However, it is worth noticing that the statistical results of both Γaw and Γab in this study are representative of water clouds (cloud top < 4 km) and cannot represent the whole troposphere. In addition, since seasonal variations of apparent lapse rates are relatively small, only mean geographic distributions of apparent lapse rates from January 2008 to June 2008 are given. For the bias μ, the global statistical results show that its peak value and the corresponding occurrence frequency are about −0.11 km and 1.5%, respectively. However, most of its regional mean values range from −0.15 to 0.15 km (figure not shown).
 Cloud top height (CTH) is an important input parameter for the new method. However, CTH information from active satellite sensors is not always available due to limited spatial and temporal resolutions. Recently, Sun-Mack et al. (submitted manuscript, 2013) derived the CTH of marine boundary layer clouds by using calculated regional lapse rates and matched cloud top/surface temperatures from MODIS. They pointed out that the new regional lapse rates used in their study should provide more accurate low cloud top heights than currently available techniques, such as cloud top parameterization [Zuidema et al., 2009], MODIS Collection-6 zonal lapse rates [Baum et al., 2012], and GEOS-5 temperature profiles. Thus, to expand our CBH retrieval algorithm to long-term passive satellite cloud products, we also retrieve rough estimates of CTH from MODIS data based on the method proposed by Sun-Mack et al. (submitted manuscript, 2013) and further apply the retrieved CTHs with the new method to derive CBHs.
 As an example, Figure 2a shows the satellite (CALIPSO\CloudSat) observed and retrieved CTHs/CBHs of a boundary layer water cloud scene for 27 July 2009 (daytime). The subplot in the upper right corner of Figure 2a shows the location of this cloud scene (green thick line), which follows CALIPSO's track. The square symbols represent the CTHs from CALIPSO observations (blue color) and MODIS retrievals (black color). The hexagram symbols represent the CBHs from CALIPSO/CloudSat observations (green color), only MODIS retrievals (red color), and joint retrievals from MODIS and CALIPSO (blue color), respectively. Here the only difference between MODIS CBHs retrievals and joint CBH retrievals from MODIS and CALIPSO is CTHs from MODIS retrievals and CALIPSO observations, respectively. The mean value of water cloud geometry thickness in Figure 2a is approximately 0.5 km, and the cloud optical depth also is small (<8). The statistical results indicate that the mean value of the CTH difference between the CALIPSO observations and the MODIS retrievals is about −0.38 ± 0.04 km; however, the mean CBH differences are about −0.42 ± 0.07 km and −0.07 ± 0.08 km for only MODIS retrievals and joint retrievals from MODIS and CALIPSO, respectively. A similar case is shown in Figure 2b but for thicker water clouds at 17 July 2009 (nighttime).
 Finally, we select 244,628 single-layer water cloud samples for July and October 2009 during daytime and nighttime and compare matched CALIPSO/CloudSat CBH data with retrieval results (see Figure 3). In Figure 3a (bigger plot), the x axis is for joint CBHs retrievals from MODIS and CALIPSO, and the y axis is for GEOPROF-observed (CALIPSO/CloudSat) CBH. The color values represent the sample number. In addition, the black dots are mean values, and the horizontal thin black lines are the error bars. The interval size of the x and y axes for the bigger plot is 0.03 km. It is very clear that CBHs derived by the novel method agree reasonably well with those obtained from the measurements of space-based radar or lidar. The correlation coefficient, the mean CBH difference, and the standard deviation are 0.79, 0.02 km, and ±0.35 km, respectively. In addition, the subplot in the lower right corner of Figure 3a shows the probability density distributions of CBH differences for the absolute difference (blue line) and the relative difference (black line). The interval sizes of relative and absolute differences (x axis) are 1% and 0.03 km, respectively. Here we define the relative difference as and the absolute difference as . Their peak values are −0.11 and 0.03 km, and the occurrence frequencies are 1.48% and 6%, respectively. Similar with Figure 3a, Figure 3b shows a comparison of CBHs between only MODIS retrievals and GEOPROF observations. The relatively low correlation coefficient (see Figure 3b) clearly indicates that the accuracy of CTH is very important for determining the CBH by using the new method. In fact, the cloud top information is also an essential parameter to other CBH retrieval techniques, such as the optical method used in a previous study [MeerkÖtter and Zinner, 2007]. The optical method needs the cloud optical depth and the effective radius as inputs to retrieve the cloud thickness under the adiabatic assumption condition, and then subtracts the cloud thickness from the CTH to derive the CBH. To further assess whether the new technique has any potential improvement compared to previous algorithms, the optical method was also applied to identical experimental data sets during daytime. Compared with coincident 2B-GEOPROF-Lidar data sets, the mean CBH difference, the standard deviation, and the correlation coefficient for our technique (optical method) are about 0.014 km (−0.13 km), ±0.34 km (±0.43 km), and 0.8 (0.78), respectively, during the daytime when CTH information is obtained from CALIPSO. However, when CTHs are just retrieved from MODIS, the corresponding mean CBH difference, standard deviation, and correlation coefficient for our technique (optical method) are about 0.02 km (−0.1 km), ±0.58 km (±0.63 km), and 0.54 (0.53), respectively. The statistical results indicate that the use of the apparent lapse rate method can slightly reduce the errors of CBH retrievals compared to the optical method. In addition, a comparison between the observed and retrieved regional mean values of CBH also shows that the new method still can work for the regional means (figure not shown).
 However, besides CTH, the explanation of the differences between satellite observation and retrieval from several error sources in this study still should be taken into account. These differences are most likely explained by the uncertainties in the derived cloud top temperature Tct and the surface temperature Tsfc from MODIS observations. On average, if the errors in both cloud top and surface temperatures from MODIS are about 1 K, these uncertainties in Tct and Tsfc amount to errors of ~4% for the CBH estimation. The other error sources (such as broken clouds and thin cirrus) are already eliminated by limiting the sample selection. Thus, these uncertainties in CBH estimation from the novel method are still acceptable.
 A novel method for estimating the CBH of marine water clouds (cloud top < 4 km) is proposed based on collocated near-simultaneous measurements of active and passive remote sensors. Its utility has been demonstrated by comparisons of its retrieval results with the CBH data from space lidar and radar. Besides the relatively small bias, the most important advantage of this method compared to previous CBH retrieval techniques (that is, optical method) is that it is independent of boundary layer cloud types, optical thickness, and illumination. Thus, it may serve as a valuable supplement to currently available passive techniques.
 Actually, the apparent lapse rate method can be applied to all satellite instruments that are capable of providing cloud top temperature and surface temperature. Furthermore, any improvements of CTH retrieval algorithms based on passive sensors (such as MODIS) would help to expand this method to long-term passive satellite cloud products. In future studies, the method will be further refined and tested by focusing on extending the number of test scenes with respect to season and region (such as ice-covered ocean or land).
 This research was financially jointly supported by the National Basic Research Program of China under 2013CB955802, the National Science Foundation of China under grant 41205015, and the Fundamental Research Funds for the Central Universities (lzujbky-2013-105). We also would like to thank the MODIS, CALIPSO, and CloudSat science teams for providing excellent and accessible data products.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.