Cloud response to Earth’s changing climate is one of the largest sources of uncertainty among global climate model (GCM) projections. Two of the largest sources of uncertainty are the spread in equilibrium climate sensitivity (ECS) and uncertainty in radiative forcing due to uncertainty in the aerosol indirect effect. Satellite instruments with sufficient accuracy and on-orbit stability to detect climate change–scale trends in cloud properties will improve confidence in the understanding of the relationship between observed climate change and cloud property trends, thus providing information to better constrain ECS and radiative forcing. This study applies a climate change uncertainty framework to quantify the impact of measurement uncertainty on trend detection times for cloud fraction, effective temperature, optical thickness, and water cloud effective radius. Although GCMs generally agree that the total cloud feedback is positive, disagreement remains on its magnitude. With the climate uncertainty framework, it is demonstrated how stringent measurement uncertainty requirements for reflected solar and infrared satellite measurements enable improved constraint of SW and LW cloud feedbacks and the ECS by significantly reducing trend uncertainties for cloud fraction, optical thickness, and effective temperature. The authors also demonstrate improved constraint on uncertainty in the aerosol indirect effect by reducing water cloud effective radius trend uncertainty.
Clouds play a significant role in Earth’s radiation budget by modulating the shortwave (SW) reflected (0.3–3.5 μm) and longwave (LW) emitted (3.5–100 μm) radiation at the top of the atmosphere (TOA) (Stephens et al. 1990; Chen et al. 2000; Stephens 2005). On a global annual scale, clouds reduce incoming SW (outgoing LW) irradiance by about 50 W m−2 (28 W m−2). Clouds, therefore, have a net cooling effect on Earth’s climate system of about 22 W m−2, according to the Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF)-TOA dataset (Loeb et al. 2009, 2012; Dolinar et al. 2015). Changes in cloud macrophysical (e.g., height, amount) and microphysical (e.g., optical thickness) properties induce positive or negative feedbacks, thus contributing to Earth’s climate system response to climate forcings and noncloud feedbacks.
Cloud response to Earth’s warming climate is one of the largest sources of uncertainty among global climate model (GCM) projections. Net cloud feedbacks in modeling experiments comprising phase 5 of the Coupled Model Intercomparison Project (CMIP5) (Taylor et al. 2012) tend to be nearly neutral or positive on average, meaning that clouds would cool the planet less as global mean surface temperature increases. Significant disagreement remains regarding the net cloud feedback magnitude among CMIP5 model output (e.g., Bony et al. 2006; Dessler and Loeb 2013; Vial et al. 2013; Webb et al. 2013; Caldwell et al. 2016). Estimating SW and LW cloud feedback from observations requires global monitoring of observed decadal changes in the SW and LW cloud radiative effect (CRE; previously cloud forcing), the difference between clear-sky and all-sky TOA irradiance (flux). Understanding the physical basis of CRE decadal trends requires a comprehensive understanding of how global cloud properties that govern trends in SW and LW CRE respond to changes in Earth’s climate. The uncertainty in CMIP5 SW cloud feedback is the largest contributor to intermodel spread in equilibrium climate sensitivity (ECS) (2.1 to 4.7 K) (Flato et al. 2013). Soden and Vecchi (2011) determined that 75% of the intermodel spread in net cloud feedback was due to low cloud, which dominates the SW cloud feedback.
In addition to the large uncertainty in cloud feedback, the aerosol indirect effect is among the greatest uncertainties in estimates of anthropogenic radiative forcing (Myhre et al. 2013). The uncertainty in the aerosol indirect effect can be better constrained by reducing uncertainty in cloud amount, cloud optical thickness, and water cloud effective radius trends. Here, we will focus on the connection between the aerosol indirect effect and water cloud effective radius. A decrease in water cloud effective radius may be indicative of an increased number of cloud condensation nuclei, which are typically dominated by aerosol particles (Twomey 1977).
To better constrain radiative forcing and cloud feedback, the tools used to observe Earth’s climate system must have sufficient accuracy and stability to detect cloud property trends on climate change–relevant scales (larger than 2000-km spatial and decadal temporal scales; Soden et al. 2008; Wielicki et al. 2013, hereafter W13). Such tools include passive remote sensing satellite measurements and associated retrieval algorithms. The accuracy and stability of both the satellite instruments and algorithms must be sufficient for unambiguous understanding of cloud response to climate change.
Climate change detection requires measurements from instruments with high accuracy and stability that provide the capability to detect what are likely to be small changes within Earth’s climate system (Goody et al. 2002; Ohring et al. 2005). W13 addressed this challenge by presenting an uncertainty framework that can be applied to a diverse group of essential climate variables (ECVs) and measurement systems to determine the necessary absolute measurement uncertainty requirements of a satellite-based observing system (Leroy et al. 2008b; Weatherhead et al. 1998).
W13 presented this uncertainty framework using, as an example, the Climate Absolute Radiance and Refractivity Observatory (CLARREO), a Tier-1 Decadal Survey–recommended climate observing mission (National Research Council 2007). The CLARREO mission concept includes reflected solar (RS) and infrared (IR) spectrometers with International System of Units (SI)-traceable on-orbit calibration designed to achieve substantially higher absolute accuracy (up to 10 times greater) than currently or previously operational Earth-observing satellite sensors (W13). These instruments will be used both for climate benchmarking and intercalibrating with instruments operational during the CLARREO lifetime. CLARREO intercalibration would include cloud imagers, such as Moderate Resolution Imaging Spectroradiometer (MODIS; King et al. 2003) and VIIRS (Visible/Infrared Imager/Radiometer Suite; Lee et al. 2006), thus enabling reduced measurement uncertainty of reflectance and brightness temperature measurements used in the corresponding retrieval algorithms. A Pathfinder to the full CLARREO mission is planned to be on the International Space Station (ISS) in the early 2020s. The CLARREO Pathfinder mission includes a RS spectrometer and has been approved for one year of operation on ISS.
The satellite sensors with which the CLARREO instruments would intercalibrate would still be essential parts of the global climate observing system. For example, cloud imagers have the spatial and temporal sampling needed for global monitoring of cloud properties, and CERES instruments have the angular sampling and broadband spectral response required to estimate TOA SW and LW irradiance. The CLARREO mission goals of unprecedented absolute accuracy and high information content for intercalibration and climate benchmarking allow for such a mission to contribute to the climate community’s needs independently and in conjunction with the other essential instruments within the climate observing system. In the studies presented here, applications of the climate change uncertainty framework are shown using the CLARREO requirements as examples of climate mission requirements.
W13 presented an uncertainty framework to quantify climate change instrument requirements based on the need to detect global mean trends in two ECVs: the SW cloud radiative effect and global mean surface temperature. However, the impact of instrument and algorithm uncertainties on delaying trend detection times in many other ECVs remains to be evaluated. This includes cloud properties which, as noted above, are a crucial but largely uncertain part of constraining the spread among climate model projections.
Other studies have applied a similar framework to study the effect of measurement errors on precipitable water vapor trend detection times (Roman et al. 2014), to compare the trend detection times between RS hyperspectral and broadband climate Observing System Simulation Experiment (OSSE) simulations (Feldman et al. 2011), and to quantify the instrument and IR spectral fingerprinting retrieval error impact on atmospheric and cloud property trend uncertainties (Leroy et al. 2008a; Kato et al. 2014; Liu et al. 2017).
In this study, we apply the principles of the W13 uncertainty framework to evaluate the impact of RS and IR instrument uncertainty requirements on trend uncertainty and trend detection times of satellite-retrieved globally averaged cloud properties. These studies are focused on instrument calibration uncertainty which, among instrument uncertainty sources (e.g., random instrument noise and orbit sampling errors), dominates on global scales (W13). This study does not address the potential impact of time-dependent retrieval algorithm biases on trend uncertainty. While such biases are currently assumed to be constant in climate change studies, future research is needed to evaluate the potential for time-dependent algorithm biases to augment the uncertainty of climate change trends (National Research Council 2015; Trenberth et al. 2013).
Current satellite retrievals of climate change from Earth-viewing reflected solar and thermal infrared spaceborne sensors rely on in-orbit estimates of instrument calibration stability along with some form of intercalibration between successively launched, temporally overlapping sensors. This is required both because the lifetime of sensors on orbit is much shorter than climate change-relevant time scales and because current Earth-viewing satellite sensors are not tied to SI-traceable standards on orbit.
Stringent characterization of the stability of instruments in orbit remains a challenge, despite the variety of techniques that have been developed to characterize instrument stability (e.g., Loeb et al. 2007). Long-term instrument degradation can have an impact on the subsequently retrieved geophysical variables, as illustrated by Lyapustin et al. (2014), which showed how the MODIS Collection 5 calibration impacted several MODIS-ST data products, including cloud properties. Several improvements have been made to the MODIS-ST Collection 6+ calibration, including a correction for the sensors’ increased sensitivity to polarization over time, that have subsequently resulted in improved geophysical property retrievals (Lyapustin et al. 2014; Platnick et al. 2017). In addition to polarization sensitivity, several other factors can impact the characterization of instrument stability such as changes in spectral response, optical surface contamination causing spectrally dependent transmission loss, and detector nonlinearity, among others.
While monitoring stability and utilizing measurement overlap are the current best practices for constructing satellite climate change records, these methods limit the confidence level in detected climate change trends. Given the large economic and societal issues at stake with climate change, a more rigorous approach to achieve high accuracy and stability levels, ideally using on-orbit SI traceability, is preferred. Current in-orbit methods to determine stability lack such SI traceability at the confidence and uncertainty levels required for climate change detection.
The present study therefore considers a rigorous and conservative approach of SI traceability to determine uncertainty requirements of orbiting sensors for observations of climate change (Anderson et al. 2004; Leroy et al. 2008b; Fox et al. 2011; W13). Such rigorous methods would achieve higher accuracy calibration in orbit for climate-related sensors. Currently, most operational sensors do not have climate change observations as part of their primary mission, nor are technologies currently available to reach the needed uncertainty levels for a large number of satellite instruments. A less costly approach is to use a few highly accurate SI-traceable orbiting reference spectrometers that span the reflected solar and infrared spectral ranges as in-orbit transfer standards, similar to those used by metrology laboratories (Fox et al. 2011; Lukashin et al. 2013; W13).
Stringent uncertainty requirements, high spectral resolution, and high spatial resolution of CLARREO-like (W13) or Traceable Radiometry Underpinning Terrestrial- and Helio- Studies (TRUTHS)-like (Fox et al. 2011) instruments could reduce measurement uncertainty in other sensors through intercalibration including cloud imagers, land imagers, and radiation budget monitoring instruments (Roithmayr et al. 2014; Wu et al. 2015). Intercalibration with an in-orbit absolute reference could then provide resilience to gaps in climate monitoring and verify instrument stability in orbit. This type of approach has been recommended by the World Meteorological Organization’s Global Space-Based Intercalibration System (WMO GSICS; Goldberg et al. 2011) as well as a joint document from the WMO, CEOS (Committee on Earth Observation Satellites), and CGMS (Coordination Group for Meteorological Satellites) called “Strategy towards an Architecture for Climate Monitoring from Space” (Dowell et al. 2013). The current study seeks to understand the calibration uncertainty required for such a designed climate monitoring system, wherein multiple, potentially nonoverlapping, instruments are needed to detect a single trend, with specific regard to detecting trends in clouds properties using cloud imagers.
The analysis described herein was conducted using cloud properties retrieved from the CERES (Wielicki et al. 1996) Cloud Property Retrieval System (CPRS) (Minnis et al. 2011), which ingests spatially subsetted MODIS reflectance and brightness temperatures. We therefore quantified the MODIS-like measurement uncertainty requirements needed to observe climate change trends in globally averaged retrieved cloud properties.
2. Climate observing system uncertainty framework
W13 demonstrated a climate observing system uncertainty framework based on earlier work by Leroy et al. (2008b) and Weatherhead et al. (1998). The uncertainty framework can be used to calculate the absolute uncertainty in the trend of a geophysical variable, , as follows:
where is the measurement record length, is the standard deviation of natural variability, is the autocorrelation time of natural variability, is the uncertainty in the retrieved geophysical variable due to the uncertainty in the instrument’s calibration, and is the calibration autocorrelation time. In the current study is the standard deviation of the variable’s global, annual mean time series. The natural variability autocorrelation time can be understood as the amount of time between independent measurements. For , we use the Weatherhead et al. (1998) definition, , where is the lag-1 autocorrelation. The calibration autocorrelation time can be understood as the time over which the calibration of the instrument can be assumed to vary within the instrument’s calibration uncertainty. Details of determining the natural variability ( and ) specific to the cloud properties examined in these studies are discussed in section 3. Phojanamongkolkij et al. (2014) found small differences in trend uncertainty estimation using the Weatherhead et al. (1998) versus Leroy et al. (2008b) definition of autocorrelation time.
Additional instrument and algorithm uncertainties can also be evaluated using Eq. (1). For retrieved geophysical variables, such as cloud properties, retrieval algorithm uncertainties may have a large impact on climate change–scale trend uncertainties. For example, the 3D optical thickness bias may not be constant on the scale of several decades as cloud type distributions change as Earth’s climate changes. Addressing the impact of algorithm uncertainties on cloud property trend uncertainties, while highly important, is planned for future study (see section 6). As discussed above in section 1, calibration uncertainty tends to dominate the trend uncertainty (among instrument noise, calibration, and sampling uncertainty) of geophysical variables on global scales (W13); therefore, we focus in this paper on calibration uncertainty for global trends of cloud properties. We also note that cloud feedbacks are determined using global mean trends (Soden et al. 2008; Dessler and Loeb 2013). Units of are dependent upon the units of the uncertainties, autocorrelation times, and record length. Consistent units should be used for natural variability and calibration uncertainty, as well as for record length and autocorrelation time.
Solving for the time series length in Eq. (1) gives an upper bound on the time it takes to begin distinguishing a trend of some magnitude from natural variability:
In Eq. (2), m is the magnitude of the trend in question, which has a relationship to the trend uncertainty, , defined by (Leroy et al. 2008b), where is the signal-to-noise ratio. In this study the trend magnitudes are stated at 95% confidence, that is, .
The trend uncertainty achieved using measurements made by a perfect instrument, , is only limited by the natural variability of the climate variable, as shown in Eq. (3) (Leroy et al. 2008b). Regardless of how flawless an instrument may be, it cannot be used to detect a secular trend in the climate system with uncertainty less than that caused by natural (internal) variability (due to, for example, El Niño or volcanic eruptions). The time for a perfect instrument to begin distinguishing a trend from natural variability, , can be calculated using Eq. (2), with the exception that the and will both be zero:
Information in Eqs. (1) and (3) can be used to determine a calibration uncertainty requirement, depending on how close to perfect it is desired for an observing system to be capable of detecting a trend, a concept that can be quantified by taking the ratio between and :
In these studies, as in W13, is assumed to be a standard expected satellite instrument lifetime of 5 years to account for different absolute instrument accuracies of multiple instruments with gaps between their operational lifetimes or uncorrected calibration drifts during their lifetime (Leroy et al. 2008b; W13). Also, , meaning that the goal for the RS and IR CLARREO instruments to enable detecting geophysical trends with trend uncertainties no more than 20% greater than those calculated using a perfect instrument limited only by natural variability.
Solving Eq. (4) for provides the required calibration uncertainty to satisfy the trend uncertainty goal, indicated by the value of Ua. The term must have the units of the retrieved geophysical variable for the Ua ratio to be unitless. To determine , the uncertainty in calibrated instrument units, we need to characterize the relationship between each cloud property and reflectance or brightness temperature in the MODIS spectral bands used to retrieve those cloud properties (section 3b). In the previous study by W13, there was a more direct relationship between the geophysical variables studied, surface temperature and CRE, and the calibrated measurements. For cloud properties, the relationship is less direct and requires the additional analysis shown in sections 3 and 4.
3. Determining requirements from uncertainty framework
a. Natural variability of CERES/MODIS cloud properties
We examine several cloud properties retrieved by the CERES (Wielicki et al. 1996) Cloud Property Retrieval System (Minnis et al. 2011): cloud fraction, cloud optical thickness (log10), liquid water cloud effective radius (log10), and cloud effective temperature. The logarithm of optical thickness was evaluated because it is approximately linearly proportional to the cloud radiative effect, and the logarithm of the water cloud effective radius was used because in section 4b we take advantage of the relationship between optical thickness and effective radius (Slingo 1989) to approximate relevant effective radius trends in the context of radiative forcing.
To estimate the natural variability of cloud properties here, we used data from operational satellites (CERES/MODIS cloud properties), combined with statistical adjustments to account for the short annual time series and any potential secular linear trends. This assumes that the anomalies in cloud properties measured from satellite adequately represent cloud property natural variability. The natural variability parameters, and , were computed from globally and annually averaged cloud property time series using 11 years of CERES/MODIS SSF1deg edition 4A cloud products (Minnis et al. 2011; CERES Science Team 2016) between July 2002 and June 2013.
These averages excluded regions poleward of 60°N/S and any 1° grid boxes containing snow or ice identified using the 1° CERES monthly compilation of snow and ice percent coverage of the National Snow and Ice Data Center’s 25-km daily coverage (Nolin et al. 1998) and the permanent snow map from the U.S. Geological Survey’s International Geosphere/Biosphere Programme (IGBP) (Loveland et al. 2000). The cloud mask algorithm operates differently when discriminating clouds from snow- or ice-covered surfaces (Trepte et al. 2003; Minnis et al. 2008), so these regions were eliminated to focus the scope of these studies. Because MODIS Terra sensor degradation has contributed to calibration-based trend artifacts in geophysical properties retrieved from the MODIS TERRA L1B data (Lyapustin et al. 2014) we used the CERES/MODIS Aqua cloud properties to compute and .
This study was conducted on global and annual scales to provide the most stringent spatial and temporal constraint on calibration uncertainty requirements. Natural variability increases at smaller zonal and regional scales compared to global and annual scales, resulting in less stringent requirements (W13). This is clearly illustrated by Eq. (4). Holding all other terms in the equation constant, if were to increase, then to maintain the same ratio, would also increase and vice versa. A second reason to use global means is that cloud feedback is most closely related to global mean changes in cloud properties (Zelinka et al. 2012b, 2013).
Global and annual scales are not the only spatial and temporal scales on which to conduct these studies. The following studies give an initial assessment of the calibration uncertainty requirements needed to evaluate cloud property trends, and evaluating instrument and retrieval algorithm requirements on regional scales and by cloud type is planned for future studies (section 6).
Using linear regression, we detrended the time series prior to calculating and to remove any linear trends, which would artificially inflate both terms. Using currently available observed time series of cloud properties to determine their natural variability results in short time series (11 yr). The of short times series tends to be underestimated. To address this, we used the Student’s t statistical distribution to scale the standard deviation using the degrees of freedom (df), rather than the value for an infinite number of samples. Rather than calculate the 95% confidence calibration uncertainty by using , we use the Student’s t value for , .
Using a short time series to estimate the autocorrelation time of natural variability contributes to uncertainty in the trend detection time (Weatherhead et al. 1998), tm [Eq. (2)]. Weatherhead et al. (1998) therefore derived the following 95% confidence interval for the trend detection time:
where tm is the trend detection time as computed in Eq. (2), and the trend uncertainty factor, B, is defined as
The natural variability parameters of the cloud properties evaluated in this study are shown in Table 1. For calculating requirements in the RS band, values were calculated relative to the 11-yr cloud property averages, also shown in Table 1.
b. Sensitivity of CPRS cloud properties to instrument changes
Using Eq. (4), (absolute and relative) was calculated for each cloud property and , shown in the last two columns of Table 1. The corresponding , the absolute calibration uncertainty requirement in calibrated measurement units (reflectance and brightness temperature) must ultimately be computed using the following relationship:
where C is the cloud property of interest (e.g., cloud fraction, optical thickness), and I is the measurement in calibrated instrument units. We used the offline CERES CPRS edition 4 with the CERES clear-sky start-up maps to calculate this sensitivity of the cloud property retrieval algorithm to small changes in reflectance and brightness temperature (BT) to the primary MODIS Aqua channels used in the daytime [solar zenith angle (SZA) < 82°], nonpolar (60°S–60°N) cloud retrievals: 0.65, 3.8, 11, and 12 μm.
The reflectance in the 0.65-μm band was perturbed by and , and the BT in the 3.8-, 11-, and 12-μm bands were each perturbed by and 1 K. Gain changes were applied in the RS band and offset changes were applied in the IR bands to emulate the type of calibration drifts expected in comparable RS and IR instruments. We calculated the absolute and relative differences between each cloud property after each individual calibration perturbation in each band and the values from the baseline run, wherein no perturbations were imposed.
As in the natural variability analysis, snow- or ice-covered pixels in nonpolar regions were excluded from this sensitivity analysis. These sensitivity studies were conducted using the highest resolution of MODIS data available at the NASA Langley Atmospheric Science Data Center (ASDC), which is subsampled to every other pixel and every other scan line from the 1-km MODIS L1B data. This results in MODIS reflectance and BT at a 1-km resolution and 2-km spatial sampling. Additionally, since MODIS is a passive instrument, only clouds with an optical thickness of at least 0.3 were included in these studies. The CLARREO RS spectrometer has been designed to match measurements with other sensors in space, time, and viewing angle (W13), meaning that its design allows for intercalibrating with a MODIS-like instrument across its full swath. We therefore evaluated cloud properties retrieved across the full MODIS swath.
Tests were conducted to determine the number of samples sufficient for robust statistics of cloud property sensitivity to reflectance and BT. Each day contains on the order of cloud pixels. Given the large number of CPRS runs needed, we determined an appropriate subset of days within a month (in our case, July 2003), such that the averaged change in each cloud property was representative of the average computed using a full month’s worth of data. We explored this using a subset of our planned CPRS sensitivity runs: the gain increases imposed upon the 0.65-μm band MODIS reflectance for the entire month of July 2003. We calculated the requirements for the 0.65-μm channel for each cloud property using differenced averages that included an increased number of days throughout the month, starting with the first day of July 2003. The final calculations for the month were differenced averages computed using the cloud data for the entire month. We found that by the 3-week mark (21 days), the requirements for each cloud property stabilized to a value that was typically 4% or less than the full month value. The only deviation we saw from this difference was a 10% relative difference from the full month value for cloud fraction. We therefore decided to use 21-day averages for the remainder of our studies.
Global, 21-day cloud property means were calculated using MODIS data from the first three weeks of July 2003. Linear regression was applied to determine the slope for each set of absolute and relative differenced averages. Because both positive and negative calibration changes were imposed, the linear parameters for both sets of changes were computed separately. This allowed examination of linearity for every band, imposed change, and cloud property across both negative and positive changes. The slopes determined from the linear regressions give the average sensitivity of each cloud property [C in Eq. (7)] to changes in MODIS reflectance or brightness temperature [I in Eq. (7)]. The standard deviations of the daily, globally averaged differences were used to determine the uncertainties in the regression slopes, allowing for estimation of the uncertainty in the sensitivities and, ultimately, the determined requirements.
Upon calculating the requirements for each cloud property and each band it was clear that certain cloud property–driven requirements served as limiting factors within each spectral band. Five of these sensitivities (slopes) are shown in Table 2 for the band(s) predominantly used to calculate each property: cloud optical thickness (0.65 μm), cloud fraction (11 and 12 μm), effective cloud temperature (11 μm), and water droplet effective radius (3.8 μm). The sensitivities shown in Table 2 are the average sensitivities determined from the linear regressions discussed above. In these cases discussed here, the relationships were linear across the increased and decreased changes.
The bands shown in Table 2 are not the only bands to which these four cloud properties were sensitive. For example, the CPRS cloud mask is determined prior to calculating cloud optical thickness using the 0.65-μm reflectance (), so although the optical thickness is predominantly sensitive to changes in the , it is also sensitive to changes in the 11- and 12-μm brightness temperatures ( and ). Information in both of those bands is used in the cloud mask, changes in which will, to some degree, impact the average magnitude of the cloud optical thickness and other subsequently retrieved cloud properties.
For simplicity and to clearly demonstrate a proof of concept for applying the climate uncertainty framework to cloud properties retrieved from cloud imagers, we have conducted these studies by considering changes in each band individually. Evaluating changes in multiple bands simultaneously remains for future study and would more realistically simulate potential changes in an operational satellite instrument.
The results from these studies are dependent on the algorithm used. Alternate results can be expected if a different algorithm (MODIS-ST cloud algorithms; Platnick et al. 2017) or cloud imager and its corresponding algorithms (e.g., VIIRS) were used to determine these sensitivities.
4. Implications for instrument requirements
a. Optical thickness, effective temperature, and cloud fraction
Combining the natural variability and sensitivity results allows for calculation of instrument requirements [Eqs. (4) and (7)]. Recall that these studies use the initial CLARREO goal of . We determined a relative for the logarithm (log10) of cloud optical thickness, log10(), of 0.621% and a of 0.85 years (Table 1). In this paper we discuss all requirements at 95% confidence (2σ); however, recall from section 3a that we use for a signal-to-noise ratio of 2 because of the tendency of shorter time series to underestimate . This resulted in a relative of 0.170% (far right column of Table 1) and a 2 of 0.379% (i.e., at 95% confidence).
To compute the 2 value, we used Eq. (7) and the relative sensitivity of the CERES/MODIS log10() to gain perturbations, which we found to be 1.38%/% (Table 2). This gives an absolute calibration requirement for the m band of 0.27%, nearly equivalent to the current CLARREO RS requirement of 0.3% (2σ) (W13).
Figure 1a shows the global optical thickness trend uncertainty, [Eq. (2), left y axis] at the 95% confidence level as a function of measurement record length . The different curves show the trend uncertainties that instruments with different 2σ calibration uncertainties, , in the 0.65-μm band could achieve (see legend, Fig. 1b). A perfect RS instrument has a value of 0%; its trend uncertainties are illustrated by the black curve (the leftmost curve) in Fig. 1a. Figure 1b shows the trend detection time delay compared to a perfect instrument, that is, . There is not a perfect curve in Fig. 1b because for a perfect instrument would be zero for all trend uncertainties. For reference a corresponding trend is shown in Fig. 1b on the right y axis, which was calculated using the relationship .
The measurement record length among different instruments spans a larger range as the required trend uncertainty approaches 0% decade−1. For example, for an optical thickness trend uncertainty of 10% decade−1 the difference in measurement record length between a perfect observing system and one with a 3.6% (2σ) uncertainty spans about a decade. However, achieving a much smaller trend uncertainty of 2% decade−1 becomes more difficult, with record length differences spanning about 25 years between a perfect observing system and one with 3.6% calibration uncertainty.
Without further information, however, the selection of the range of optical thickness trend uncertainty shown in Fig. 1 is arbitrary. This range can be placed into a climate change–relevant context by estimating the expected range of optical thickness trends that correspond to the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (AR5) equilibrium climate sensitivity (ECS) intermodel range of 2.1–4.7 K (IPCC 2013) and the corresponding SW cloud feedback. Estimating this range would help to better constrain instrument uncertainty requirements to reduce the uncertainty in detecting global trends in optical thickness.
This was estimated using two primary steps. First, we used CMIP5 climate model output and the forcing-feedback framework to estimate SW and LW cloud feedback values for a wide range of ECS. Then, we used retrieved CERES data products to estimate radiative kernels and link trends in cloud properties to trends in CRE and cloud feedback, with an assumed global surface temperature trend. We used the CERES data products for consistency with the rest of our analysis in which we use the CERES retrieved cloud properties to evaluate the natural variability and to determine the absolute calibration instrument requirements. Details of this methodology are discussed below.
We applied the forcing-feedback framework, , using the IPCC AR5 effective radiative forcing (RF) fixed sea surface temperature multimodel mean for doubled CO2, . The noncloud feedbacks were used from IPCC AR5 globally averaged model means of the Planck, water vapor, lapse rate, and surface albedo feedbacks (Flato et al. 2013), shown in Table 3.
The SW and LW cloud feedbacks used were the ensemble averages, neglecting rapid adjustments, calculated by Zelinka et al. (2013) from abrupt quadrupled CO2 simulations, in which the cloud fraction, optical thickness, and altitude contributions to the SW and LW cloud feedbacks were partitioned by isolating contributions due to changes in cloud amount, cloud optical thickness, and cloud height using output from CFMIP2/CMIP5 model simulations and CTP-τ histograms (Table 3). Using the and feedback values detailed above, we calculated an ECS of 2.53 K, which is within the AR5 intermodel range (IPCC 2013).
We used the forcing-feedback framework to calculate LW and SW cloud feedbacks solely due to changes in cloud amount, altitude, or optical thickness for a range of ECS. We describe the methodology of this process in detail using cloud optical thickness as an example. Using the AR5 for doubled CO2, feedbacks listed in Table 3, and the range of ECS considered in this analysis, , , we computed nine corresponding values of the SW cloud feedback due to changes in cloud optical thickness, with the following equation:
In Eq. (8), j indexes the number of ECS values for which we calculated , and the summation term on the right is the sum of the climate feedbacks from which the nominal shown in Table 3 is subtracted. The difference between the summation term and the nominal is equivalent to . Each computed value of was added to the nominal contributions to SW cloud feedback due to changes in cloud amount and altitude (Table 3) to compute nine values—one for each ECS evaluated. This process was repeated for each partitioned SW and LW cloud feedback.
Finally, we estimated the relationship between each partitioned SW and LW cloud feedback and their corresponding cloud property trends. We used the monthly averaged 1° gridded CERES edition 4 data products to estimate cloud radiative kernels by calculating the differences between select geophysical variables from July 2006 and July 2004 and using multiple linear regression to regress LW irradiance, SW irradiance over land, and SW irradiance over ocean (Su et al. 2015), each of which is approximately normally distributed, on those variables. Regions poleward of 60° and snow- or ice-covered nonpolar grid boxes were excluded. Using the USGS IGBP map, SW irradiance was regressed onto cloud fraction and the relative log10() separated by land and ocean surface types. The LW irradiance was regressed onto cloud fraction, effective cloud-top temperature, cloud emissivity, total column precipitable water, and surface skin temperature. The SW land and LW TOA irradiance anomalies computed with the multivariate linear regression results are each compared to their corresponding CERES-observed irradiance anomalies in Fig. 2. The regression coefficients from multivariate linear regressions were used as the estimated radiative kernels [e.g., ] in these studies and are shown in Table 4. The cloud fraction radiative kernels computed using this method were found to be comparable to those computed by Zelinka et al. (2012a).
We multiplied the cloud property-partitioned SW and LW cloud feedbacks by a global mean surface temperature trend of 0.25 K decade−1 to calculate TOA SW and LW irradiance trends (in W m−2 decade−1). Then, dividing the SW and LW irradiance trends by the radiative kernels we computed corresponding cloud property decadal trends. These analyses resulted in relationships among ECS, globally averaged cloud property trends (for cloud fraction, cloud effective temperature, and cloud optical thickness), and the SW and LW cloud feedback.
Similarly to Fig. 1, Fig. 3 shows the trend uncertainties as a function of (Fig. 3a) and the trend detection delay compared to a perfect observing system (Fig. 3b) for RS instruments with various calibration uncertainties in the 0.65-μm band. However, the Fig. 3 trend uncertainty range (left y axis) has been adjusted using the additional information relating ECS and SW cloud feedback to global decadal trends and includes the AR5 ECS intermodel range shaded in gray. The farthest right y axis shows the equivalent trend. The only difference between the left and farthest right y axes is that the trend has negative values, whereas trend uncertainty cannot be negative, so only its absolute value is shown.
From estimating this relationship we find that the globally averaged optical thickness trend range falls between −0.56% decade−1 (for 4.7 K ECS) and 0.39% decade−1 (for 2.1 K ECS) (Fig. 3, shaded). With an instrument with a 0.65-μm calibration uncertainty of 0.3% (2σ) it would take 21–27 years [with a 95% confidence interval in record length of 14–33 years, due to the uncertainty in ; Eq. (5)] to begin distinguishing trends from natural variability, depending on the magnitude of the trend, equivalent to a 2–4-yr delay compared to a perfect instrument (i.e., one limited solely by natural variability). However, continuing with calibration uncertainty levels comparable to operational cloud imagers, considering the potential for data gaps (e.g., 3.6% for MODIS/VIIRS, 2σ; Xiong et al. 2010; Cao et al. 2014), the measurement record lengths span 78 to 100 years, with the confidence interval being 54–144 years. These trend detection times give a delay compared to a perfect instrument of 60–76 years, much longer than intercalibrating such systems with a CLARREO-like RS instrument with smaller calibration uncertainty.
To demonstrate the challenge of detecting a trend of smaller absolute magnitude in we turn to the nominal ECS of 2.53 K calculated from our forcing-feedback calculation. The corresponding trend is 0.1% decade−1. It would take a perfect instrument 58 years to begin distinguishing this trend from natural variability, a feat that would take 67 years with a CLARREO-like intercalibration standard ( confidence interval 46–97 yr). With today’s absolute calibration uncertainties, we would wait over a century longer (189 yr) before beginning to distinguish this smaller trend from natural variability. Figure 3 demonstrates that observations can most quickly distinguish large absolute trends in cloud optical thickness or, equivalently, extreme values of climate sensitivity. Climate measurement records that are both longer and have smaller absolute uncertainty provide a tighter the constraint on ECS uncertainty.
The results related to the effective cloud temperature () global trend, LW cloud feedback, and ECS are shown in Fig. 4. We found a of 0.147 K and a of 0.679 years. Using the climate uncertainty framework and finding a sensitivity of to BT changes in the 11-μm band of 1.34 K K−1, we determined that for a goal of , the 11-μm band requirement is 0.06 K (2σ), which is also the current CLARREO IR uncertainty goal (W13). Applying our analysis to link the trend, LW cloud feedback (upon which cloud temperature, and therefore altitude, has a greater impact than upon SW cloud feedback), and ECS, we estimate the corresponding range of trends to be −0.036 K decade−1 (ECS of 2.1 K) to −0.33 K decade−1 (ECS of 4.7 K). This trend range, illustrated in Fig. 4 by the shaded region, is predominantly negative, indicating rising cloud heights, and is consistent with GCM simulations of cloud changes, their projections of a rising tropopause level, and their resulting calculations of positive LW cloud feedback (Zelinka et al. 2012b; Collins et al. 2013).
For the likely range of globally averaged trends, the trend detection delay compared to a perfect instrument for a cloud imager intercalibrated with a CLARREO-like spectrometer is 1–5 years. Taking into account the uncertainty in measurement record length [Eq. (5)], the confidence interval upper bound gives a maximum delay of 8 years. For today’s instruments the delay would be longer, ranging between 21 and 95 years [ confidence interval from Eq. (5) of 16–132-yr delay] for a VIIRS-like calibration uncertainty of 0.54 K (2σ) (Moeller et al. 2013) and between 26 and 117 years (considering the confidence interval, a 19–163-yr delay range) for a MODIS-like calibration uncertainty of 0.68 K (2σ) (Wenny et al. 2012).
For global cloud fraction, we found the to be 0.171%, and the to be 1.35 years. For all instances of cloud fraction–related values, except in Table 1 where the relative and are stated, cloud fraction is stated in percent cloud fraction ranging from 0% (clear) to 100% (completely overcast). The CPRS cloud mask involves several MODIS bands, depending upon the scene. Among the four primary bands investigated in this study, the globally averaged cloud fraction exhibits the most sensitivity to the 11- and 12-μm bands. We determined globally averaged sensitivities of cloud fraction to BT changes in the 11- and 12-μm bands to be −0.28% K−1 and −0.35% K−1, respectively. For these bands the calibration uncertainty requirements for are more lenient than the 0.06-K CLARREO IR requirement with 0.47 K for the 11-μm band and 0.39 K for the 12-μm band. The impact of calibration uncertainty on the trend uncertainties as a function of measurement record length and trend detection delay for both IR bands is shown in Fig. 5. Note that the current VIIRS and MODIS absolute calibration uncertainties are less lenient than both calibration uncertainty requirements shown above.
These results for cloud fraction need to be considered with some caution, however. Recall that within these studies, we have thus far evaluated the sensitivity of cloud properties to changes in four MODIS bands independently, and we have determined the impact on time to detect trends in those cloud properties based on calibration requirements in each of those bands. This should not be the only way these requirements are evaluated, however, since within the CERES/MODIS cloud mask retrieval algorithm, bands may be used individually (such as the 11-μm band, which is used to determine if the pixel is too cold to be cloud free) or the combination of information between two bands may be used together, such as the difference between the BT in the 11- and 12-μm bands. Additionally several other cloud mask tests are often applied using reflectance and brightness temperature in different wavelengths depending on the cloud type encountered. For example, there are differences in determining thin high clouds versus low thick clouds.
We have conducted preliminary investigations into the impact of these cloud types differences on the sensitivity of cloud properties to changes in the four bands considered here. In these preliminary results, we have found that the sensitivity of cloud fraction in the 11-μm band varies by cloud type. The total cloud sensitivities used in this study, however, do not necessarily sufficiently represent the variability in the sensitivity among different cloud types. Further investigation, therefore, is required that also carefully examines the natural variability of the cloud properties of different cloud types, in addition to their RS and IR instrument calibration sensitivities, the combination of which would allow for determination of calibration requirements by cloud type.
b. Water cloud effective radius
We also determined calibration uncertainty requirements for detecting global trends in effective particle size of water clouds. In the CPRS, the effective particle radius, , is retrieved primarily using the information about particle size in the 3.8-μm band. Using the method described above we determined the calibration uncertainty requirement to achieve a trend uncertainty within 20% of what a perfect instrument could achieve, which we found to be 0.01 K. Although it should be noted that the current CLARREO design does not include the 3.8-μm band, this requirement is more stringent than the uncertainty requirement for the CLARREO IR instrument (designed to span 5–50 μm).
This climate change uncertainty analysis for effective radius can be placed into a climate change–relevant context using the relationship between and the aerosol indirect effect (Twomey 1977), or as it has more recently been named, the effective radiative forcing (ERF) due to aerosol–cloud interactions (ERFaci) (IPCC 2013). Trends in the ERFaci can be linked to cloud changes in both cloud amount and optical thickness (and, therefore, effective radius); however, in the following analysis, we focused solely on the connection between the ERFaci and effective radius. A decrease in water particle size, in a cloud with constant liquid water content, increases the total water droplet cross-sectional surface area, thus increasing the cloud optical thickness. A decrease in water cloud effective particle size may indicate an increase in cloud condensation nuclei, which are typically dominated by aerosol particles. We therefore evaluated the level of instrument uncertainty required to detect global trends in to better constrain estimates of ERFaci.
Ultimately, we needed to estimate a relationship between aerosol forcing trends and effective radius trends. To quantify this relationship we used the 30-yr forcing projections from the AR5 representative concentration pathway 4.5 (RCP4.5) scenario (Collins et al. 2013). Between 2000 and 2030, the RCP4.5 total effective radiative forcing projected change is 1.31 W m−2. The total aerosol ERF (ERFari+aci) (IPCC 2013), which includes aerosol cloud interactions (aci) and aerosol radiation interactions (ari), is nearly indistinguishable among the four future scenarios, or representative concentration pathways, used in the AR5 (Cubasch et al. 2013), with the ERFari+aci becoming less negative by about 1 W m−2 during the twenty-first century. Between 2000 (−1.17 W m−2) and 2030 (−0.91 W m−2) the ERFari+aci is projected to increase by 0.26 W m−2. However, to connect the aerosol ERF to the effective radius trend, we needed to isolate the ERFaci. AR5 radiative forcing estimates for 2011 relative to 1750 show that the ERFaci and ERFari contribute 50% each to the ERFari+aci, each being about −0.45 W m−2 (Myhre et al. 2013). Assuming this ratio remains approximately constant throughout the twenty-first century, we estimate an ERFaci change between 2000 and 2030 of 0.13 W m−2 (0.043 W m−2 decade−1).
The ERFaci trend presented above () can be represented as
where the w subscript indicates water cloud, and is the SW cloud radiative effect for water cloud. The radiative kernel was computed in a manner similar to those described in the previous section and shown in Table 4. Because the focus is on liquid water clouds, these kernels were computed using one year of data to ensure a sufficient sample size. The resulting kernel value and its uncertainty is . From Eq. (9), we solve for the optical thickness trend, , and the relationship between this trend and an effective radius trend can be shown to be
From Slingo (1989), we use the parameterization that approximates the relationship between water cloud and , where C is a constant approximated by , with h being the geometric cloud height and the globally averaged liquid water path. Equation (10) simplifies to Eq. (11), considering logarithm rules and that both C and are constants, resulting in a zero trend for each. Combining Eqs. (11) and (9) provides a relationship between the ERFaci and the water cloud effective radius. In addition to the AR5 projected change of the total ERF, we modified the ERFaci to cover a range of values and computed the corresponding water cloud effective radius trend (relative trend of the logarithm of the effective radius). This relationship and the expanded analysis covering a range of potential ERFaci trends linked to corresponding trends is shown in Fig. 6.
For the specific AR5 projection discussed above for which the ERFaci trend was 0.043 W m−2 decade−1, the corresponding relative trend is 0.06% decade−1. It would take a perfect instrument 19 years to begin distinguishing this trend from natural variability. Taking the uncertainty from Eq. (5) into account, the 95% confidence interval is 13.5–27 years. For an instrument capable of detecting trends with a trend uncertainty less than 20% of the trend uncertainty possible with a perfect instrument (0.01 K, 2σ) the delay beyond a perfect instrument would be 1.5 years. With a CLARREO-like instrument, the delay would be 22 years (with the 95% confidence interval giving a delay spanning 16–31 yr). For instruments comparable to today’s operational IR cloud imagers (Wenny et al. 2012; Moeller et al. 2013), the delay in trend detection time would be over a century.
These results need to be considered with care, as we have made several assumptions within this analysis, which are stated above; however, despite the idealized context within which we obtained these results, our analysis provides important information regarding the impact of calibration uncertainty requirements on quantifying the aerosol indirect effect, which is among the greatest uncertainties in radiative forcing (Myhre et al. 2013). We have shown that with an instrument with a comparable calibration requirement to the CLARREO IR spectrometer, trends in effective radius, and therefore ERFaci could be detected eight to nine decades sooner than with existing instruments. These results illustrate, similarly to the results from W13, the importance of stringent uncertainty requirements for climate change trend detection. This study was conducted solely using the effective radius retrieved using the 3.8-μm band; however, it would also be relevant to extend this study to investigate the impact of calibration uncertainty on the time to detect trends in water cloud effective radius retrieved using reflectance in the 1.6-μm and 2.1-μm bands.
Reducing cloud property trend uncertainties using instruments with sufficient accuracy and stability for climate change detection and attribution would contribute significantly to improved understanding of climate processes. In these studies we applied a climate uncertainty framework (W13) to enable quantitatively based justification for determining what constitutes sufficient uncertainty requirements for timely globally averaged cloud property trend detection. We applied this climate uncertainty framework to quantify the impact of calibration uncertainty of RS and IR instruments on the trend uncertainties and trend detection times of cloud fraction, effective temperature, optical thickness, and water cloud effective radius retrieved by the CERES/MODIS Cloud Property Retrieval System (Wielicki et al. 1996; Minnis et al. 2011).
These studies followed the CLARREO goal for detection of climate variable trends at uncertainties no more than 20% from those determined using a perfect observing system (), limited solely by natural variability. We estimated relationships among trends in cloud properties (cloud fraction, optical thickness, and effective temperature), equilibrium climate sensitivity, and SW and LW cloud feedback. We also related trends in water cloud effective radius to trends in radiative forcing (ERFaci). These analyses provide a quantitative context within which necessary and sufficient uncertainty requirements can be defined for future climate change observing instruments to assist in reducing uncertainty climate model projections.
The CLARREO RS requirement of 0.3% (2σ) is nearly equivalent to the requirement for an instrument detecting cloud optical thickness trends within 20% of trend uncertainties that could be determined by a perfect instrument in the 0.65-μm band (), which we found to be 0.27% (2σ). The delay compared to a perfect observing system in detecting trends within the AR5 ECS intermodel range spanned about 2–7 years for such instruments to several decades for instruments with uncertainty requirements comparable to that of today’s instruments (60 years to more than a century).
The climate uncertainty framework applied to cloud effective temperature revealed a 0.06 K (2σ) requirement for the 11-μm band for an instrument with goal of , equivalent to the current CLARREO IR requirement. Detection times for instruments with calibration requirements similar to today’s instruments (0.54 K–0.68 K) span 20 years to more than a century for likely trends in Te; however, for a CLARREO-like IR instrument, detection delays are shorter, between 1 and 5 years, illustrating the substantial reduction in detection time compared to continuing with the calibration uncertainty of currently operational IR imagers.
To detect trends in cloud fraction using the goal the 11- and 12-μm band uncertainty requirements are 0.47 and 0.39 K (2σ). These requirements are less stringent than the current CLARREO design of 0.06 K (2σ) but more stringent compared to operational cloud imager absolute measurement uncertainties. A more rigorous analysis of cloud fraction by cloud type is required to determine cloud fraction–driven climate uncertainty requirements, given the complex dependence of the CPRS cloud mask for different cloud types on multiple MODIS bands. Our analyses provide the first direct link between satellite instrument absolute calibration requirements and their impact on constraints on ECS uncertainty and the detection times of climate change–scale cloud property trends.
For detecting trends in water cloud effective radius (), we determined that a requirement is much more stringent than the current CLARREO IR uncertainty requirement at 0.01 K (2σ). We linked trends in to effective radiative forcing due to aerosol–cloud interactions (ERFaci) using the aerosol indirect effect mechanism. Using information from AR5 projections revealed that detection times of could be reduced by about eight decades with an instrument calibration requirement of 0.06 K (2σ) compared to the calibration uncertainty of operational cloud imagers.
An important distinction to make is that these studies focus on absolute measurement uncertainty, not on relative stability, of passive satellite sensors. Stability is typically inferred, but not proven, by comparing operational instruments to one another. Stability intercomparisons are important, but they potentially provide a weaker test of trend uncertainty than instruments with small uncertainty levels tied to SI-traceable standards. Climate data records used to inform critical societal decisions require a high level of confidence. The present results consider tying measurements to international standards as a method to provide that higher level of confidence, also assuming from the most conservative standpoint that dependence solely on stability is insufficient.
Studies evaluating other essential climate variables with quantitative frameworks such as that demonstrated here will become increasingly important within the current U.S. and global challenge to appropriate sufficient resources for climate change monitoring. With the challenge of limited Earth science funding to develop instruments with the rigorous uncertainty requirements for climate change detection and attribution, using quantitative studies such as these can provide more rigorous justification for the design of new climate change satellite, aircraft-based, surface, and in situ sensors. A similar method for determining the required quality of climate change measurements has been demonstrated in the report entitled “Continuity of NASA Earth Observations from Space” (National Research Council 2015).
6. Future work
This study demonstrates the value of applying the climate uncertainty framework and techniques for placing the results from that framework application into a climate change–relevant context. There are, however, several areas for future studies. Although we focused on trends in individual cloud properties and connected the value of improving trend detection time to climate model projections, applying cloud fingerprints may help to detect secular trends more rapidly (e.g., Marvel et al. 2015; Roberts et al. 2014; Jin and Sun 2016). In this study, we limited our analysis to evaluating the impact of calibration requirements in individual bands on trend detection times; however, evaluating cloud property trend detection impacts of calibration requirements in multiple instrument bands simultaneously would provide a more realistic analysis. Because the CERES CPRS was used to quantify the sensitivity of cloud properties to gain and offset changes in MODIS data, the results from our study are dependent on the CERES CPRS retrieval algorithm; therefore, it would also be valuable to extend these studies to other cloud imagers (e.g., VIIRS) and retrieval algorithms (e.g., MODIS-ST).
In these studies, we focused on global trends in cloud properties for total cloud, without evaluating regional variability or individual cloud type contributions; however, climate projections have indicated that different cloud types on both a global and regional scale respond differently to and exert different feedbacks upon Earth’s changing climate. For example, there is a need for better constraint of low cloud processes to reduce uncertainty of the SW feedback and, ultimately, equilibrium climate sensitivity. It would be valuable, therefore, to expand these studies to both two-dimensional cloud type histograms and to regional scales. These analyses could then be expanded to link instrument requirements and their impact on cloud trend detection to climate model projections for those different cloud types, which would help to provide more specific constraints regarding instrument requirements.
Retrieved geophysical variable trend detection uncertainty is also dependent upon uncertainties due to assumptions and approximations made in retrieval algorithms. Time-dependent biases in retrieval algorithms could be erroneously identified as secular geophysical trends in the climate system, thereby masking or distorting the true climate change trends occurring in the climate system. In future studies, we plan to evaluate the impact of time-dependent biases (e.g., the three-dimensional bias) and uncertainties in retrieval algorithms on cloud property trend detection uncertainty in cloud properties. This is an effect that must be understood for other essential climate variables as well and, if possible, reduced.
The CERES data products were obtained from the Atmospheric Science Data Center at the NASA Langley Research Center. The authors thank David Doelling for his help with obtaining the CERES Edition 4A Cloud Property Data and the three reviewers whose comments improved the quality and presentation of this manuscript. P. Minnis and S. Sun-Mack are supported by the NASA CERES Program. Y. Shea and B. Wielicki are supported by NASA CLARREO Pre-formulation funding.
Publisher’s Note: This article was revised on 3 August 2017 to correct several acronyms when first introduced and to fix a minor typographical error, all in section 1.
Publisher’s Note: This article was revised on 25 January 2018 to include the entry for the linear regression coefficients for cloud fraction in Table 4 that was missing when originally published.