Understanding Errors in Cloud Liquid Water Path Retrievals Derived from CloudSat Path-Integrated Attenuation

Matthew Lebsock aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Hanii Takahashi aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
bJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Richard Roy aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Marcin J. Kurowski aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Lazaros Oreopoulos cEarth Sciences Division, NASA Goddard Space Flight Center, Greenbelt, Maryland

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Abstract

An algorithm that derives the nonprecipitating cloud liquid water path Wcld from CloudSat using a surface reference technique (SRT) is presented. The uncertainty characteristics of the SRT are evaluated. It is demonstrated that an accurate analytical formulation for the pixel-scale precision can be derived. The average precision of the SRT is estimated to be 34 g m−2 at the individual pixel scale; however, precision systematically decreases from around 30 to 40 g m−2 as cloud fraction varies from 0% to 100%. The retrievals of clear-sky Wcld have a mean bias of 0.9 g m−2. Output from a large-eddy simulation coupled to a radar simulator shows that an additional bias of −8% may result from nonuniformity within the footprint of cloudy pixels. The retrieval yield for the SRT, measured relative to all warm clouds over ocean between 60°N and 60°S latitude is 43%. The SRT Wcld is compared with one estimate of Wcld from the Moderate Resolution Imaging Spectroradiometer (MODIS) using an adiabatic cloud profile and an effective radius derived from 3.7-μm reflectance. A strong correlation between the mean MODIS Wcld and SRT Wcld is found across diverse cloud regimes, but with biases in the mean Wcld that are cloud-regime dependent. Overall, the mean bias of the SRT relative to MODIS is −13.1 g m−2. Systematic underestimates of Wcld by the SRT resulting from nonuniform beamfilling cannot be ruled out as an explanation for the retrieval bias.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew Lebsock, matthew.d.lebsock@jpl.nasa.gov

Abstract

An algorithm that derives the nonprecipitating cloud liquid water path Wcld from CloudSat using a surface reference technique (SRT) is presented. The uncertainty characteristics of the SRT are evaluated. It is demonstrated that an accurate analytical formulation for the pixel-scale precision can be derived. The average precision of the SRT is estimated to be 34 g m−2 at the individual pixel scale; however, precision systematically decreases from around 30 to 40 g m−2 as cloud fraction varies from 0% to 100%. The retrievals of clear-sky Wcld have a mean bias of 0.9 g m−2. Output from a large-eddy simulation coupled to a radar simulator shows that an additional bias of −8% may result from nonuniformity within the footprint of cloudy pixels. The retrieval yield for the SRT, measured relative to all warm clouds over ocean between 60°N and 60°S latitude is 43%. The SRT Wcld is compared with one estimate of Wcld from the Moderate Resolution Imaging Spectroradiometer (MODIS) using an adiabatic cloud profile and an effective radius derived from 3.7-μm reflectance. A strong correlation between the mean MODIS Wcld and SRT Wcld is found across diverse cloud regimes, but with biases in the mean Wcld that are cloud-regime dependent. Overall, the mean bias of the SRT relative to MODIS is −13.1 g m−2. Systematic underestimates of Wcld by the SRT resulting from nonuniform beamfilling cannot be ruled out as an explanation for the retrieval bias.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew Lebsock, matthew.d.lebsock@jpl.nasa.gov

1. Introduction

The cloud liquid water path Wcld is an essential climate variable according to the World Meteorological Organization’s Global Climate Observing System (Bojinski et al. 2014). However, large uncertainty remains in observations of Wcld. There are two common methods for observing Wcld. The first method relates a microwave brightness temperature enhancement caused by condensed liquid water over radiometrically cold ocean surfaces (Wilheit and Chang 1980). The second method uses a bispectral visible and near-infrared method to derive a visible optical thickness and cloud-top effective radius (Nakajima and King 1990) that can be further combined to provide Wcld. A number of studies using coincident observations from the A-Train (Stephens et al. 2002) have demonstrated large differences in retrieved Wcld between these two methods (Bennartz 2007; Greenwald et al. 2007, 2018; Greenwald 2009; Seethala and Horváth 2010; Lebsock and Su 2014). While these studies have been useful in highlighting specific sources of bias, significant uncertainty remains in both retrieval methods. There is a need to identify and characterize independent remote sensing methods to constrain Wcld to resolve these lingering discrepancies around this essential climate variable.

Previously, Lebsock et al. (2011) used measurements of the CloudSat path-integrated attenuation (PIA) to derive Wcld for shallow clouds over the ocean. They used a surface reference technique (SRT) to derive the PIA based on the difference in the surface echo between cloudy pixels and nearby clear-sky pixels. An important assumption of that work was that, although the random errors are relatively large for the CloudSat SRT technique, in the absence of precipitation the derived Wcld is nearly bias free. The SRT method was subsequently used to study the aerosol effect on precipitation formation in warm clouds (Suzuki et al. 2013), to quantify the Moderate Resolution Imaging Spectroradiometer (MODIS) cloud water mass deficit due to missed cloud detection and failed low cloud retrievals over ocean (Lebsock and Su 2014) and to evaluate the CloudSat release-05 Radar-Visible Optical Depth Cloud Water Content Product (Leinonen et al. 2016b).

In a later study, Lebsock and Suzuki (2016) used output from a large-eddy simulation (LES) coupled to a radar model to evaluate the precision and bias error characteristics of a spaceborne radar SRT retrieval for the total water path Wtotal, which includes precipitation. They estimated that precision in the derived Wtotal is the larger of either 20 g m−2 or 30%. They further found that biases from systematic differences in the clear-sky attenuation between cloudy and clear pixels tend to bias total water path retrievals high by 5–10 g m−2 because cloudy pixels have more water vapor than nearby clear-sky pixels. Most striking, they found that nonuniform beamfilling (NUBF), or inhomogeneity within the radar footprint, could result in underestimates of up to 50% in the derived total water path.

The purpose of this paper is to use actual CloudSat data to revisit the uncertainty characteristics of the nonprecipitating Wcld retrievals from the CloudSat SRT. In so doing we determine that the LES-based results of Lebsock and Suzuki (2016) significantly overestimated the precision of the SRT retrieval. We present a modified uncertainty model and demonstrate that it is able to accurately reproduce the observed variability of clear-sky retrievals. We further confirm that the bias in the SRT retrieval method is usually small in the case of nonprecipitating clouds by analyzing both actual CloudSat data and analyzing results form an LES coupled to a CloudSat simulator. Then, SRT retrievals are compared with Wcld retrievals from MODIS. These comparisons, conducted by cloud regime, are used to demonstrate how the SRT-retrieved Wcld and its associated uncertainty estimate might be used in the future to understand the bias and precision in the collocated MODIS retrievals.

2. Data and methods

a. Data products

CloudSat, release version 5, data over ocean from 2006 to 2010 and from 60°N to 60°S latitude are used throughout this study. The CloudSat GEOPROF product provides a measurement of the normalized surface cross section Σ0 and radar reflectivity factor Z as described in Tanelli et al. (2008). The GEOPROF product also includes a cloud mask (Marchand et al. 2008), however, CloudSat is not sensitive enough to detect many shallow clouds. For this reason, a combined radar–lidar hydrometeor mask called GEOPROF lidar has also been developed (Mace and Zhang 2014). This product includes a variable that lists the layer-top heights of up to five distinct cloud layers, which we use as a more precise cloud/clear discriminator than can be provided from the radar only. We also use the 2-m temperature from the ECMWF auxiliary (ECMWF-AUX) product, which is a weather analysis interpolated in space and time to the CloudSat footprint (Partain and Cronk 2017).

Cloud properties from MODIS are also used: specifically, the collection 6 MODIS cloud products from Aqua subset to a 15 pixel (±5 km) swath centered on the CloudSat ground track called MAC06S0 (Savtchenko et al. 2008). These files contain the same data fields as their full-swath parent product MYD06 (Platnick et al. 2017). Specifically, we use the visible cloud optical depth τvis and effective radius derived from the 3.7-μm channel re. An adiabatic cloud liquid water path (Bennartz 2007) is calculated as W = (5/9)ρlτvisre, where ρl is the density of liquid water.

The 1° × 1° cloud regimes derived using a clustering algorithm applied to MODIS collection-6 level-3 daily cloud-top-pressure and cloud-optical-depth joint histograms (Oreopoulos et al. 2016) are used to place results in the context of specific cloud regimes (CR). This dataset assigns each 1° × 1° region into 1 of 12 CRs, which have distinct geographical distributions, cloud morphology, radiative effects, and precipitation characteristics (Oreopoulos et al. 2016; Leinonen et al. 2016a). There are 12 unique cloud regimes in the MODIS dataset listed in Table 1 along with a brief description. The reader is referred to the above references for more detailed information on the CRs. We use the CR’s to understand and quantify what (if any) cloud-type or regime dependence exists in the SRT and MODIS Wcld data.

Table 1

Brief description of each of the 12 MODIS cloud regimes. See Oreopoulos et al. (2016) for more detail.

Table 1

b. Retrieval method

The cloud liquid water path Wcld is derived from an estimate of the two-way radar path-integrated attenuation (in decibel units) due to cloud water:
PIAcld=10log10(e2τcld),
where PIAcld does not include gaseous absorption and τcld is the cloud optical depth at the CloudSat frequency of 94.05 GHz. Assuming a homogenously filled radar footprint, we estimate the liquid water path as
Wcld=PIAcld2κ(T)=α(T)PIAcld,
where κ(T) is the temperature-dependent mass absorption coefficient. For cloud droplets that are much smaller than the radar wavelength (i.e., Rayleigh limit), this coefficient is much larger than the scattering coefficient and is given by (Doviak and Zrnic 1993)
κ(T)=10log10(e)6πλρlIm[m(T)21m(T)2+2],
where m is the complex refractive index of liquid water, ρl is its density, and λ is the radar wavelength. This small-particle approximation begins to lose applicability in the presence of drizzle, so we are cautious to only explore nonprecipitating clouds in this work.
Solving Eq. (2) for Wcld requires an estimate of the PIAcld, which is calculated here using a clear-sky surface reference technique:
PIAcld=Σ0,clr̂Σ0,cld,
where Σ0 is the normalized surface cross section in decibel units, Σ0,cld is the observed surface cross section for the cloudy pixel of interest, and Σ0,clr̂ is an estimate of the clear-sky cross section. For this estimate, we use the mean Σ0,clr within a window of ±50 pixels around the pixel of interest. If there are no clear-sky pixels within the ±50-pixel window, then Σ0,clr̂ is not estimated and no retrieval is performed. The SRT assumes that the gaseous attenuation and the ocean surface cross section are not systematically different between clear and cloudy pixels. This is not generally true because cloudy pixels tend to have larger water vapor paths than nearby clear pixels; however, Lebsock and Suzuki (2016) show that the resultant bias in Wcld is only on the order of 5–10 g m−2. The calculated PIAcld can be negative and corresponds to a negative Wcld. We note that while negative retrievals are obviously unphysical, it is important to retain them when aggregating data so as not to introduce bias into composite results.
The uncertainty in the derived Wcld is expressed using standard uncertainty propagation as
εWcld=α2(εΣ0,cld2+εΣ0,clr2)+PIAcld2(αT)2εT2,
where εΣ0,clr is the sample standard deviation in the estimate of Σ0,clr̂, εΣ0,cld is the uncertainty in the measured Σ0 of the cloudy pixel for which the retrieval is being performed, and εT is the uncertainty in the cloud temperature. Note that this formulation is slightly different than Lebsock and Suzuki (2016), who did not consider uncertainty in cloud temperature.

Uncertainty in temperature results in a small contribution to the total uncertainty because of the relative insensitivity of α to temperature. Figure 1 shows the inverse two-way extinction quantity α plotted as a function of temperature. Local variations in this parameter are between 0.24% and 1.28% K−1 over the range of temperatures shown. Throughout this work, we approximate the cloud temperature using the ECMWF 2-m temperature, the CALIPSO cloud height, and a fixed 7.5 K km−1 lapse rate (Zuidema et al. 2009) to define α. We attempted to use MODIS cloud-top temperatures; however, we found that the cloud-top temperature data were missing for many small cumulus clouds. In contrast, this indirect method for estimating cloud temperature is available for every pixel. Zuidema et al. (2009) find that the temperature lapse rate varies between 6.9 and 7.6 K km−1 in subtropical marine boundary layers. Assuming typical cloud heights vary between 0.5 and 2.5 km, we estimate the potential range of error in Wcld due to incorrect cloud temperature to be 0.08%–2.24%.

Fig. 1.
Fig. 1.

The two-way inverse extinction coefficient α(T) for liquid water plotted as a function of temperature at 94.05 GHz.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Uncertainty in the estimation of PIAcld dominates the total uncertainty budget for Wcld meaning emission temperature uncertainty can generally be ignored. In this case, estimating uncertainty in the retrieved Wcld amounts to estimating εΣ0,cld and εΣ0,clr. In the absence of strongly attenuating deep convection, the surface is always bright and therefore in a high signal-to-noise ratio (SNR) regime for CloudSat. In this SNR regime it can be shown that uncertainty in the measured Σ0,cld (in decibel units) is a function of only the speckle error, which results from interference between randomly phased reflected electric fields from the randomly distributed scatterers within the target volume, and is given by (Doviak and Zrnic 1993)
εΣ0,cld=10log10(e)Npulses,
where Npulses is the number of consecutive uncorrelated radar pulses that are averaged in the radar reflectivity measurement. For CloudSat this radar parameter varies between 577 and 679 (Tanelli et al. 2008). Taking 577 pulses as a conservative estimate gives εΣ0,cld = 0.18 dB, which we will keep fixed throughout this analysis. The expected value of the standard deviation of the Σ0,clr would be exactly 0.18 if there were no natural variability in the observed surface echo. However, we estimate in section 3a that the square root of the mean variance in Σ0,clr is 0.87 dB. As a result, the uncertainty in the derived Wcld is not dominated by the instrumental measurement precision but rather by the natural mesoscale variability of the clear-sky water vapor and near surface winds.

We note that the distribution of Σ0,clr occasionally contains outliers indicating a highly nonnormal distribution, which can significantly affect the sample mean and variance. To evaluate the impact of these cases, we perform a test of normality (Shapiro and Wilk 1965) for each distribution of Σ0,clr associated with individual retrievals. The Shapiro–Wilk test returns a p value indicating the likelihood that the sample distribution is normally distributed. A p value < 0.05 indicates that the null hypothesis of a normal distribution can be rejected with 95% confidence. We emphasize that this statistical test cannot confirm that the distribution is in fact normally distributed.

c. Data filtering

The retrieval method is only applied to a limited set of pixels following a filtering procedure. Some of these filters are necessary while others have an arbitrary element. The filters applied are explicitly described below:

  1. Surface type—Retrievals are limited to ocean surfaces only. The navigation_land_sea_flag variable = 2 in GEOPROF is used to identify ocean pixels.

  2. Precipitation—The retrieval is limited to nonprecipitating clouds. The Precip-Column variable precip_flag = 0 is used to identify nonprecipitating clouds. Precip_flag = 0 corresponds to an attenuation-corrected reflectivity in the lowest bin exceeding −15 dBZ (Haynes et al. 2009). This is a conservative criterion that excludes clouds that are suspected of containing drizzle.

  3. Cloud temperature—The retrieval is limited to clouds with a cloud-top temperature (CTT) > 273.15 K, as calculated indirectly from the 2-m air temperature, the cloud-top height (CTH) from CALIPSO, and fixed temperature lapse rate of 7.5 K km−1.

  4. Cloud height—The retrieval is limited to clouds with a CTH ≤ 5 km indicated in the GEOPROF lidar cloud_layer_heights.

  5. Cloud identification—The GEOPROF lidar cloud_layer_heights variable = 0 to indicate clear sky. Note that there is a possibility that a CloudSat footprint contains partial cloud cover that is undetected by the CALIPSO lidar because the lidar footprint does not fill the radar footprint area.

  6. Missing data—On extremely rare occasions, Σ0 is missing, in which case no retrieval is performed.

3. Results

a. Evaluation of errors using clear sky

To begin we evaluate the retrieval formalism by performing the cloud water path retrieval on clear-sky pixels, as indicated by the combined radar–lidar cloud mask. For these pixels the correct answer is known to be 0 g m−2. If the formalism in section 2b is correct, it would be expected that after averaging N pixels the mean retrieved Wcld would be 0, and following from the central limit theorem the distribution of retrieved Wcld would be normal with standard deviation following from the law of total variance given by
εWcld¯=i=1NεWcld2(i)N.
Note that because these are clear-sky pixels, there is no uncertainty in the cloud temperature.

This clear-sky experiment provides us with an opportunity to confirm our understanding of the measurement uncertainty model. To ensure that the sampling is commensurate with the retrievals performed on cloudy pixels we require that at least one pixel in the window contain a cloud with CTT ≥ 273.15 K and CTH ≤ 5 km. For these clear-sky retrievals we assume a cloud temperature of 280 K for the purposes of translating uncertainties in PIA to a retrieved Wcld. Results would scale according to Fig. 1 for different cloud temperatures.

Table 2 and Fig. 2 show a summary assessment of the retrieval mean bias and the ability of the analytical uncertainty estimate to reproduce the observed variability. The table shows separate rows for all pixels and for subsets of data in which pixels with p value ≤ 0.05 have been removed. The removed pixels are those for which the null hypothesis that the distribution of Σ0,clr is normally distributed can be rejected with 95% confidence. A final row shows a subset in which pixels with p value ≤ 0.05 and surface wind speed < 4 m s−1 have been removed. First, the results demonstrate that, as expected, the mean bias is negligibly small. Second, the observed variability and analytical error estimates are significantly reduced when the p value ≤ 0.05 pixels are removed from the population and further reduced when wind speeds < 4 m s−1 are removed. Third, the results show that the analytical uncertainty estimate is systematically too large and indicates that outliers are biasing the variance in the estimate of Σ0,clr. Furthermore, Fig. 2 clearly shows a sharply peaked, nonnormal distribution of retrieved Wcld and significant overestimate of uncertainty when the p value ≤ 0.05 and low-wind-speed pixels are included. This is an unexpected result for such a large sample size and is the likely result of occasional outliers causing the distributions of Σ0 to behave pathologically. We are drawing from real-world sample distributions whose exact properties cannot be determined; however, it can easily be shown that drawing from analytical distributions such as the Cauchy distribution, which does not have a defined finite mean and variance, results in Eq. (7) consistently underestimating variance. Removing the low-p-value and low-wind-speed pixels reduces the uncertainty and results in a more normal distribution; however, there remains a slight overestimate of the observed variability by the analytical model. Figure 3 demonstrates that the extent of the uncertainty overestimation is a function of p value. The uncertainty overestimate systematically decreases with increasing p value and significantly increases for p values < 0.05. Even for p value = 1 there is a small residual overestimate of the analytical model because the statistical test cannot confirm the null hypothesis that the distribution is in fact normally distributed. From these results we define an empirical uncertainty adjustment term to account for nonnormality:
Δε=7.89+1.77ln(p value),
which is only valid for p value > 0.05. The adjustments would become significantly larger for the smallest p values; however, we will exclude these pixels from the remainder of the analysis. The final column in Table 2 shows the adjusted uncertainty is indeed in much better agreement with the observed standard deviation than the unadjusted estimates.
Fig. 2.
Fig. 2.

(a) The histogram of the derived clear-sky water paths along with the standard deviation (STD) of the observed distribution and the analytical error estimate (Error). (b) The same, filtered for p value > 0.05 and wind speed > 4 m s−1 and also including the adjusted error (Adj Error) from Eq. (8).

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Fig. 3.
Fig. 3.

(left) The observed STD minus the unadjusted analytical error estimate as a function of the Shapiro–Wilk p value and (right) a histogram of the p values. Note that p value = 0 is not included in either panel. For p value = 0 cases, the difference between the observed STD and the analytical error estimate is −105.6 g m−2, which is substantially larger than for nonzero p values < 0.05.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Table 2

Summary of clear-sky retrievals of Wcld. The uncertainty estimate is derived from Eqs. (5) and (7), and the adjusted uncertainty estimate includes the p-value correction shown in Eq. (8).

Table 2

The value of Σ0,clr has a strong dependence on wind speed and a much smaller dependence on the sea surface temperature (SST). The dependance on wind speed is particularly strong for wind speeds less than about 4 m s−1 (Tanelli et al. 2008; Haynes et al. 2009), whereas the magnitude of Σ0,clr is relatively stable at higher wind speeds. There are larger natural variations in the individual values from which we estimate Σ0,clr̂  in low-wind-speed environments, and as a result we expect errors in estimating Σ0,clr̂ to be substantially larger in low-wind-speed regimes. Figure 4 demonstrates that this is indeed the case. Both the standard deviation of the clear-sky retrievals and the analytical error estimates have a strong nonlinear increase below wind speeds of 4–5 m s−1. Figure 4 also shows that despite the increasing error the mean bias is still relatively close to 0 albeit with a small negative bias at low wind speeds. We conclude that we are still able to adequately quantify the error in low-wind-speed environments; however, that error is large and for the remainder of the paper we will remove pixels in which wind speed is below 4 m s−1. While these pixels correspond to a small fraction of our global sample, they are concentrated regionally in the western Pacific Ocean warm pool and parts of the intertropical convergence zone, where larger errors would be frequently expected when using the SRT retrieval.

Fig. 4.
Fig. 4.

(top) The observed STD of retrieved clear-sky Wcld (blue) and the adjusted error estimate (red) as a function of near surface wind speed for all clear-sky pixels with p value > 0.05. (bottom) The retrieved mean Wcld.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Figure 5 shows a more detailed evaluation of the retrieval uncertainties in terms of cloud regimes. Figures 5a and 5b show the retrieval mean, standard deviation, error, and adjusted error estimate as a function of cloud fraction and MODIS cloud regime, respectively. Most important, the retrieved mean is generally near 0 g m−2 regardless of cloud regime or cloud fraction, indicating a relatively bias-free estimate of PIAcld across a diversity of cloud conditions. Again, we see that the analytical error estimates consistently overestimate the observed variability; however, the adjusted error estimate does a credible job of bringing the uncertainty into agreement with the observed standard deviation across regimes even though the adjustment term has no knowledge of the cloud regime or cloud fraction. Last, the measurement standard deviation shows a clear dependence on the cloud regimes, which is well tracked by the adjusted error estimates. The mean adjusted error estimated is 34 g m−2 (Table 2), and Fig. 5 shows that after filtering by p value and wind speed, this value is fairly representative across the diversity of cloud regimes.

Fig. 5.
Fig. 5.

Results sorted according to (a) the lidar cloud faction or (b) the MODIS CR. For (a) and (b), the top plot shows the observed STD, analytical error estimate, and adjusted error (Adj Error). The bottom plot shows the mean retrieved Wcld. The top plot can be interpreted as random error, and the bottom plot can be interpreted as measurement bias.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

We conclude from the evaluation of clear-sky Wcld that random retrieval errors originating from application of the SRT are well understood. There is negligible bias in estimating Σ0,clr̂ across a diversity of cloud regimes. The random uncertainty in PIAcld translate into typical uncertainties in Wcld of 34 g m−2 after filtering for pixels of the Σ0,clr distribution with p value > 0.05 and wind speeds > 4 m s−1. This precision is well characterized using an analytical framework after applying an adjustment factor to account for the nonnormal distribution of the observed Σ0,clr.

b. NUBF

Thus far, these clear-sky retrievals neglect a potentially important source of error, that is the effect of NUBF of the field of view in the cloudy pixels. The SRT retrieval is an inversion of the observed attenuation that is an exponential function of τ,cld [Eq. (1)]. Because the exponential function is convex everywhere, Jensen’s inequality states that the retrieved optical depth must be less than or equal to the true mean optical depth (τcld,retτcld¯). Since τ,cld maps linearly into Wcld, we see that NUBF always results in an underestimate of the true cloud water path.

We use an LES of the Barbados Oceanographic and Meteorological Experiment (BOMEX; Siebesma et al. 2003) field experiment to evaluate the expected biases due to NUBF on the retrievals. BOMEX was a nonprecipitating shallow marine cumulus field experiment where clouds very similar to those that are the ideal targets of the SRT retrievals were observed. The LES model of Matheou and Chung (2014) is used for the simulation with the initial/boundary conditions and forcings described in Siebesma et al. (2003). The LES is run with horizontal and vertical resolution of 20 m on a domain size of 12.8 × 12.8 km2. The vertical profile of the atmosphere above the LES top boundary (located at 3 km) is filled in from Modern-Era Retrospective Analysis for Research and Applications, version 2 (Gelaro et al. 2017), data appropriate for the region and weather regime. We couple a radar simulator to six different snapshots of the BOMEX LES as described in Roy et al. (2021). The simulator assumes a constant unattenuated radar surface cross section, but it does include resolved subpixel variability in both cloud liquid and water vapor. To evaluate the NUBF, the Σ0 and Wcld fields are averaged over the radar’s two-way beam pattern using an idealized Gaussian CloudSat-like antenna pattern. We then normalize the simulated Σ0 such that the mean is 0 dB for clear-sky pixels separately at the model and radar resolution. After accounting for CloudSat’s along-track integration by performing a time-average of the antenna pattern, the resulting footprint has size 1.4 km (cross track) by 1.7 km (along track), where the widths are defined using the 6-dB point of the two-way propagation pattern (Tanelli et al. 2008). The LES domain was divided into an integer number of parallel along track segments that are separated by the cross-track beamwidth of 1.4 km to ensure footprint independence. Each along-track pixel is separated by the along-track width of 1.7 km again to maintain statistical independence of each pixel. The end result is a reduction in the number of CloudSat-resolution pixels relative to the native model resolution by an approximate factor of 502. Last, an estimate of PIAcld was made by subtracting the individual Σ0 from the mean Σ0,clr.

Figure 6 (left panel) summarizes the influence of NUBF on the retrieved Wcld for BOMEX. The figure shows the relationship between Wcld and PIA at both the model resolution and the CloudSat resolution. The relationship is linear at the model resolution with the scatter due to water vapor absorption. We fit a line to this relationship with intercept forced through 0 and find a slope of α = 105 g m−2 dB−1. We then use the CloudSat-resolution simulated PIAcld to invert this linear fit to derive the Wcld providing simulated retrievals. The mean bias of these synthetic retrievals is −8%, although it is clear in Fig. 6 that the biases can be significantly larger as Wcld increases. We reemphasize that the negative bias results because the relationship between PIAcld and Wcld is not linear at the CloudSat resolution. To provide a heuristic understanding of the relationship at the radar resolution we approximate the NUBF by a pixel that is filled with a homogenous cloud element with subpixel cloud fraction fcld and optical depth τcld and an area of homogenous clear sky with area 1 − fcld and τcld = 0. To further simplify the model, we assume that there are no subpixel variations in the gaseous attenuation. The observed PIA averaged over the radar pixel in decibel units can then be parameterized in terms of fcld as
PIAcld¯=10log10(fcldexp{[ln(10)Wcld¯10αfcld]}+(1fcld)).
The right panel of Fig. 6 shows the fractional bias of a partially filled footprint according to Eq. (9) as a function of fcld and Wcld¯. Three key points can be gleaned from Eq. (9) and Fig. 6. First the NUBF bias increases as fcld decreases. Second, regardless of the cloud fraction the NUBF bias tends to 0 as Wcld¯ goes to 0. Third, for a given cloud fraction the bias increases monotonically as the Wcld¯ increases. While a more realistic description of NUBF changes the magnitude of these effects, it does not alter these three general conclusions.
Fig. 6.
Fig. 6.

(left) The relationship between Wcld and the apparent PIA from the BOMEX LES experiment. Here the y axis shown as PIA is the difference of the individual simulated Σ0 from the mean of the clear-sky pixels so that it includes the effect of water vapor and cloud liquid water variability. The relationship is shown at both the model native resolution (20 m2) and the CloudSat radar footprint resolution (1700 × 1400 m2). The linear fit at the model resolution is PIA = Wcld/105. (right) The CloudSat-resolution fit, based on Eq. (8) with fcld = 0.32.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

We find the value of fcld that minimizes the mean square deviations of residuals of the CloudSat-resolution data, giving fcld = 0.32. This should be contrasted with the actual cloud fraction in the BOMEX simulations is 0.21. In this sense it is important to interpret fcld as an effective cloud fraction that encompasses all of the effects of NUBF as opposed to an actual cloud fraction. Nevertheless, the approximation of Eq. (9) provides a possible approach to mitigate NUBF biases in the SRT by using high-resolution imagery from the MODIS bands 1 and 2 that resolve visible and near-infrared radiances at 250 m to estimate radar subfootprint cloud fraction.

We comment that the effects of NUBF can be significantly larger when using the SRT to infer the total water path (including precipitation), in which case optical depths are substantially larger and the α term itself is spatially variable and introduces considerable additional inhomogeneity within the radar beam. Lebsock and Suzuki (2016) showed that when estimating the total water path in shallow convective precipitating cloud field this bias could be as large as −50%.

c. Retrieval yield

The retrieval for Wcld presented here is powerful in that it is extremely simple and has a fairly well-understood error characterization, including minimal sources of mean bias. However, the retrieval can only be performed reliably on nonprecipitating, marine, liquid-phase clouds, for which a clear-sky surface cross section reference can be established. We further limit the retrieval to pixels in which p value > 0.05 and wind speed is > 4 m s−1. From 2006 to 2010 we find 80 257 183 cloudy pixels with a CTT ≥ 273.15 K and CTH ≤ 5 km between 60°N and 60°S over ocean. Of these, 13.8% are flagged as precipitating (including drizzle). We are unable to estimate Σ0,clr̂ for 25.2% of pixels, and 16.3% have a p value ≤ 0.05. Last, 12.9% have wind speed < 4 m s−1. These conditions combine to limit retrieval yield to 43.1% between 60°N and 60°S latitudes. Note that this is the retrieval yield defined relative to only single layer clouds liquid clouds. There is a notable geographical pattern to the retrieval yield that is also evident in the yield binned by MODIS cloud regimes (Fig. 7). The retrieval yield is found to be maximized for the MODIS CRs 1 and 10–12, which correspond to shallow subtropical shallow cumulus convection (CR10–11), disorganized cloud fields, with low cloud fraction (CR12), and tropical cirrus (CR1). Note that, while the retrieval yield is fairly high for CR1, the actual number of retrievals in this regime is relatively small because single layer liquid clouds are rarely observed because of the preponderance of high cloud. There is a significant reduction in the retrieval yield for the other CRs that can be attributed primarily to the inability to consistently estimate Σ0,clr̂ relative to CRs 10–12. Precipitation occurrence, which is a significant factor in reducing retrieval yield, is relatively constant across the cloud regimes.

Fig. 7.
Fig. 7.

(top) Map of the retrieval yield, (middle) the retrieval yield binned by the MODIS cloud regime, and (bottom) the frequency of various factors decreasing retrieval yield binned by MODIS cloud regime.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

d. Cloud retrievals

The geographical distribution of the retrieved mean Wcld is shown in Fig. 8. We emphasize that this Wcld is conditioned on low cloud occurrence for which a retrieval is performed and is not suitable for comparison with all-sky datasets or model outputs. The geography of the distribution is fairly straightforward, with relatively constant values between 20 and 30 g m−2 over much of the midlatitude oceans and the eastern margins of the subtropical ocean basins. Each of these areas is characterized by significant coverage of stratocumulus clouds. In contrast, cumulus-dominated geographies such as the western Pacific warm pool and the Indian Ocean have smaller mean Wcld near 10 g m−2. We note that these low values of Wcld in cumulus regimes are a direct result of the large numbers of small lidar-detected cumulus that go undetected by CloudSat and often do not have MODIS cloud retrievals, which highlights the unique role for the SRT retrieval to fill the gaps in the other retrieval methods.

Fig. 8.
Fig. 8.

The map of the conditional Wcld for the SRT.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Next, we compare the SRT retrievals with MODIS Wcld derived according to the adiabatic assumption and the 3.7-μm re. Figure 9 shows good correlation between the retrieved mean SRT Wcld binned by MODIS Wcld. However, there is a significant bias between the two estimates. MODIS has a mean Wcld of 43.1 g m−2, and the SRT has a mean Wcld of 30.0 g m−2. Also shown in Fig. 9 is a range of possible biases for the SRT retrieval from two sources: 1) NUBF from the parameterization in Eq. (8) and Fig. 6, which always biases the SRT negatively (low), and 2) systematic water vapor attenuation bias of 10 g m−2, which may bias the retrieval positively (high) as estimated from Lebsock and Suzuki (2016). Over much of the range of MODIS Wcld, it appears that NUBF could explain some of the MODIS–SRT bias, but with the caveat that this parameterization is based on a single LES case and may not be generalizable.

Fig. 9.
Fig. 9.

(left) A scatterplot of the mean SRT binned by MODIS Wcld. The gray error bars show the STD of the SRT retrievals within each MODIS bin. The red error bars show a range of potential bias based on NUBF using the parameterization in Fig. 6 and Eq. (9) and water vapor bias assumed to be 10 g m−2 from Lebsock and Suzuki (2016). (right) The mean difference between SRT lidar and MODIS Wcld as a function of MODIS Wcld.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Some of the bias between MODIS and the SRT can be explained by NUBF causing an underestimate in the latter. Applying the NUBF correction (e.g., Fig. 6) to every pixel increases the mean SRT Wcld to 35.3 g m−2, which is in better agreement with MODIS. However, there is a great deal of uncertainty in the applicability of this parameterization, which is based on a single LES, to global data, as stated earlier. Furthermore, we expect that water vapor attenuation bias in the SRT cannot explain the bias because it should only further decrease the SRT and thus increase the discrepancy.

Of course, MODIS LWP is also subject to bias, particularly due to possible overestimates in re that would result in overestimates of Wcld. Recently, unresolved variability has been shown to account for a 1–2 μm overestimate of the MODIS re (Zhang et al. 2016; Werner et al. 2018). Three-dimensional radiative effects have been estimated using Multiangle Imaging SpectroRadiometer (MISR) optical depths near cloud-bow scattering angles to suggest a zonal mean bias in re derived from the 3.7-μm channel of 2–7 μm (Liang et al. 2015). Unresolved variability and three-dimensional effects also influence the retrieved optical depth. Depending on the scattering geometry and cloud morphonology these biases can be either positive or negative (e.g., Várnai and Marshak 2002); for the subtropical cumulus cloud regimes that are overrepresented in this dataset, the optical depth bias is likely negative because of large solar zenith angles. The combined effect of potential re and τ bias on the MODIS Wcld is difficult to assess.

Figure 10 shows the scatterplot of the mean SRT against MODIS categorized by MODIS CRs and Table 3 shows the mean Wcld for each CR. The geographical distributions of MODIS CRs are shown in Oreopoulos et al. (2016), and a brief description of each was provided here in Table 1. In general, there is correlation between the SRT and MODIS retrievals regardless of regime, however, there are clearly regime dependent biases. The only MODIS CRs in which SRT tends to be larger than MODIS is CR2, which contains the strongest convective storms, and has a very small number of SRT retrievals. For all other CRs, MODIS exceeds the SRT. This is particularly true for CRs 8–12, which correspond to stratocumulus, cumulus, and broken cloud regimes. These CRs also dominate the retrieval yield (Fig. 7) and therefore largely drive the overall SRT–MODIS bias. It is plausible that these regimes contain a disproportionate amount of small cumulus that only fill part of the CloudSat footprint and would therefore be especially susceptible to NUBF-induced underestimates for the SRT.

Fig. 10.
Fig. 10.

Scatterplot of the mean SRT lidar binned by MODIS Wcld for each of the 12 MODIS cloud regimes. The gray error bars show the STD of the SRT retrievals within each MODIS bin.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

Table 3

CloudSat Wcld and MODIS Wcld (g m−2) averaged over different cloud regime.

Table 3

The regime dependent biases in Wcld suggest that there may be geographic patterns of bias as well. Figure 11 shows that there are indeed indications of coherent geographical patterns in the differences between the data. MODIS is uniformly larger than SRT almost everywhere with the exception of the western Pacific warm pool and small regions off the east coast of Central America and North Africa. These are regions where deeper liquid-phase cumulus convection exists than is found in the subtropical trade winds and are home to the CR 2, which is the regime in which the SRT mean exceeds the MODIS mean.

Fig. 11.
Fig. 11.

The mean CloudSat SRT − MODIS Wcld. Only common pixels are used in this plot.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0235.1

4. Summary and discussion

This paper quantifies the error characteristics and retrieval yield of cloud liquid water path retrievals derived from estimates of the CloudSat path-integrated attenuation using the SRT. The advantage of this technique is that it permits estimation of the PIA with minimal bias and well characterized random uncertainty that can be reduced through averaging. A semianalytical estimate of the retrieval precision that includes an empirical adjustment for the effects of nonnormally distributed clear-sky radar surface cross section enables the accurate estimation of the precision across a diversity of cloud regimes. The precision in the SRT Wcld is found to be 34 g m−2 after filtering pixels that have low wind speeds and statically identifiable nonnormal distributions of clear-sky surface cross section. This result represents the random component of the error inherent in the SRT method. In addition, we demonstrate using a single LES, that for nonprecipitating liquid-phase marine cumulus clouds, the SRT, as applied to the CloudSat footprint, has potential bias from NUBF that is on the order of −8% when averaged over many clouds. Future work will determine better how this bias may vary across cloud regimes. In particular, there is a need to reproduce the NUBF analysis using an LES with a larger domain and longer integration times such as that by Bretherton and Blossey (2017) that tends to produce aggregated shallow convection with larger Wcld that is more representative of the satellite data relative to the idealized small-domain BOMEX LES.

A disadvantage of the SRT retrieval is that it cannot be performed under certain conditions. It cannot be applied on precipitating pixels because precipitation-sized drops do not exhibit a linear relationship between liquid mass and absorption. It also cannot be performed for pixels that are in areas of complete cloud cover because the technique requires clear-sky pixels to estimate the clear-sky surface echo. Also, empirical evidence suggests that pixels for which the surface reference is highly nonnormally distributed or are associated with very low near surface winds speeds have unacceptably large uncertainty and should be excluded. As a result, the retrieval yield is 43.1% over oceans between 60°N and 60°S.

Because retrieval yield is maximized for nonprecipitating low cloud fraction regimes the SRT is best suited to examine Wcld in fair weather cumulus regimes over oceans. This is likely to prove to be a useful reference point against which we can evaluate Wcld derived from shortwave reflectance measurements like those from MODIS. These regimes are known to be highly challenging for MODIS retrievals due to three-dimensional radiative transfer effects (Zhang et al. 2012; Várnai and Marshak 2002), cloud detection (Zhou et al. 2015), and algorithm failures (Cho et al. 2015).

While the focus of this paper is on understanding the errors and yield of the SRT method, we perform an initial comparison with MODIS Wcld derived from the 3.7-μm effective radius and an adiabatic assumption We find that MODIS overestimates relative to SRT by 13.1 g m−2, a value that we deem plausible on the basis of a simple model of NUBF, which always acts to bias the SRT low. Future research is needed to understand the sources of these biases, including the three-dimensional radiative transfer artifacts in the MODIS data and a more comprehensive evaluation of NUBF on the SRT retrieval. Use of high-resolution channels on MODIS may prove particularly useful in this regard.

While this study is limited to nonprecipitating shallow clouds, the general principles can be extended to other situations. Of particular interest, one could imagine using this method to constrain the mean Wcld in shallow warm clouds beneath thin cirrus or of supercooled liquid or mixed-phase clouds at middle and high latitudes. Many of the conclusions with regard to uncertainty also apply to precipitating clouds and the use of PIA to constrain the total (cloud + precipitation) liquid water path. Furthermore, CloudSat produces a 94-GHz radiometric brightness temperature (Dobrowalski and Tanelli 2019) that can be used in a similar fashion as the PIA-SRT to constrain the Wcld (Berry et al. 2020). The aspects of the SRT uncertainty presented here will apply in a similar but quantitatively different way to those retrievals as well.

Acknowledgments.

This work was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and was funded by the CloudSat mission. CloudSat and MODIS data used in this study were downloaded from the CloudSat Data Processing Center (http://www.cloudsat.cira.colostate.edu/). Author Oreopoulos gratefully acknowledges funding by NASA’s CloudSat Science Team program.

Data availability statement.

The data product shown in this paper will be released as a new data product in CloudSat, release 06, available online (http://www.cloudsat.cira.colostate.edu/).

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Save
  • Bennartz, R., 2007: Global assessment of marine boundary layer cloud droplet number concentration from satellite. J. Geophys. Res. Atmos., 112, D02201, https://doi.org/10.1029/2006JD007547.

    • Search Google Scholar
    • Export Citation
  • Berry, E., G. G. Mace, and A. Gettelman, 2020: Using A-Train observations to evaluate east Pacific cloud occurrence and radiativeeffects in the community atmosphere model. J. Climate, 33, 61876203, https://doi.org/10.1175/JCLI-D-19-0870.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bojinski, S., M. Verstraete, T. C. Peterson, C. Richter, A. Simmons, and M. Zemp, 2014: The concept of essential climate variables in support of climate research, applications, and policy. Bull. Amer. Meteor. Soc., 95, 14311443, https://doi.org/10.1175/BAMS-D-13-00047.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bretherton, C. S., and P. N. Blossey, 2017: Understanding mesoscale aggregation of shallow cumulus convection using large‐eddy simulation. J. Adv. Model. Earth Syst., 9, 27982821, https://doi.org/10.1002/2017MS000981.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cho, H.-M., and Coauthors, 2015: Frequency and causes of failed MODIS cloud property retrievals for liquid phase clouds over global oceans. J. Geophys. Res. Atmos., 120, 41324154, https://doi.org/10.1002/2015JD023161.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dobrowalski, G., and S. Tanelli, 2019: Level 2B-TB94 process description and interface control document. California Institute of Technology Jet Propulsion Laboratory Doc., 9 pp., https://www.cloudsat.cira.colostate.edu/cloudsat-static/info/dl/2b-tb94/2B-TB94_PDICD.P1_R05.rev0_.pdf.

    • Search Google Scholar
    • Export Citation
  • Doviak, R., and D. Zrnic, 1993: Doppler Radar and Weather Observations. Academic Press, 562 pp.

  • Gelaro, R., and Coauthors, 2017: The Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2). J. Climate, 30, 54195454, https://doi.org/10.1175/JCLI-D-16-0758.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Greenwald, T. J., 2009: A 2 year comparison of AMSR-E and MODIS cloud liquid water path observations. Geophys. Res. Lett., 36, L20805, https://doi.org/10.1029/2009GL040394.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Greenwald, T. J., T. S. L’Ecuyer, and S. A. Christopher, 2007: Evaluating specific error characteristics of microwave-derived cloud liquid water products. Geophys. Res. Lett., 34, L22807, https://doi.org/10.1029/2007GL031180.

    • Crossref
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  • Fig. 1.

    The two-way inverse extinction coefficient α(T) for liquid water plotted as a function of temperature at 94.05 GHz.

  • Fig. 2.

    (a) The histogram of the derived clear-sky water paths along with the standard deviation (STD) of the observed distribution and the analytical error estimate (Error). (b) The same, filtered for p value > 0.05 and wind speed > 4 m s−1 and also including the adjusted error (Adj Error) from Eq. (8).

  • Fig. 3.

    (left) The observed STD minus the unadjusted analytical error estimate as a function of the Shapiro–Wilk p value and (right) a histogram of the p values. Note that p value = 0 is not included in either panel. For p value = 0 cases, the difference between the observed STD and the analytical error estimate is −105.6 g m−2, which is substantially larger than for nonzero p values < 0.05.

  • Fig. 4.

    (top) The observed STD of retrieved clear-sky Wcld (blue) and the adjusted error estimate (red) as a function of near surface wind speed for all clear-sky pixels with p value > 0.05. (bottom) The retrieved mean Wcld.

  • Fig. 5.

    Results sorted according to (a) the lidar cloud faction or (b) the MODIS CR. For (a) and (b), the top plot shows the observed STD, analytical error estimate, and adjusted error (Adj Error). The bottom plot shows the mean retrieved Wcld. The top plot can be interpreted as random error, and the bottom plot can be interpreted as measurement bias.

  • Fig. 6.

    (left) The relationship between Wcld and the apparent PIA from the BOMEX LES experiment. Here the y axis shown as PIA is the difference of the individual simulated Σ0 from the mean of the clear-sky pixels so that it includes the effect of water vapor and cloud liquid water variability. The relationship is shown at both the model native resolution (20 m2) and the CloudSat radar footprint resolution (1700 × 1400 m2). The linear fit at the model resolution is PIA = Wcld/105. (right) The CloudSat-resolution fit, based on Eq. (8) with fcld = 0.32.

  • Fig. 7.

    (top) Map of the retrieval yield, (middle) the retrieval yield binned by the MODIS cloud regime, and (bottom) the frequency of various factors decreasing retrieval yield binned by MODIS cloud regime.

  • Fig. 8.

    The map of the conditional Wcld for the SRT.

  • Fig. 9.

    (left) A scatterplot of the mean SRT binned by MODIS Wcld. The gray error bars show the STD of the SRT retrievals within each MODIS bin. The red error bars show a range of potential bias based on NUBF using the parameterization in Fig. 6 and Eq. (9) and water vapor bias assumed to be 10 g m−2 from Lebsock and Suzuki (2016). (right) The mean difference between SRT lidar and MODIS Wcld as a function of MODIS Wcld.

  • Fig. 10.

    Scatterplot of the mean SRT lidar binned by MODIS Wcld for each of the 12 MODIS cloud regimes. The gray error bars show the STD of the SRT retrievals within each MODIS bin.

  • Fig. 11.

    The mean CloudSat SRT − MODIS Wcld. Only common pixels are used in this plot.

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