1. Introduction
Understanding how clouds and atmospheric properties change with time under radiative forcing is necessary to understand feedback (e.g., Knutti and Hegerl 2008). Generally, global clouds and atmospheric properties are retrieved from satellite-based instruments. Subsequently, retrieved values from an instrument’s field of view are averaged and the time rate of change of cloud or atmospheric properties can be inferred from averaged properties. This is simple in concept but identifying artifacts of the retrieval is difficult in practice. Even if the same algorithm is used for the retrieval throughout the period, instrument calibration shift and changes of inputs used for the retrieval (e.g., source of temperature or water vapor profiles) cause artifacts. In addition, the complexity of retrieval algorithms makes the tracing of error propagations and the error estimate in retrieved properties difficult, especially if the error is time-dependent. An alternative way to derive a trend of cloud and atmospheric properties is tying their property change directly to the observed radiance change. This average-then-retrieve approach carries observed radiances as far as possible and retrieved values are directly tied to instrument stability (i.e., repeatability and reproducibility; Taylor and Kuyatt 1994). However, it requires separating cloud and atmospheric property changes contributing to the highly spatially and temporally averaged observed radiance change. In addition, the average-then-retrieve approach can retrieve anomalies or changes of cloud and atmospheric properties occurring over a region and a time period directly. Although mean values are not derived, there is an advantage in deriving anomalies. If the error is time-dependent (e.g., aliasing seasonal temperature or water vapor variability), the error in mean values affecting anomalies is difficult to eliminate. Because anomalies are much smaller than their means, a small time-dependent error affects anomalies significantly. In addition, although unknown bias errors in the calibration affect both approaches, once observed radiances are averaged, both instrument noise and natural variability are reduced—the larger the time and space averaging, the larger the reduction. These advantages of the average-then-retrieve approach might be utilized in detecting trends and investigating the variability of cloud and atmospheric properties. A possibility of such detections is investigated in earlier studies using longwave spectral radiances by Leroy et al. (2008a), Huang et al. (2010b), and Kato et al. (2011) and shortwave spectral radiances by Feldman et al. (2011), Jin et al. (2011), and longwave spectral radiances combined with radio occultation data by Huang et al. (2010a). However, the error in the trend estimated from derived properties using the average-then-retrieve approach is unknown.
Assumptions used in retrievals from highly spatially and temporally averaged spectral radiances are as follow: 1) top-of-atmosphere (TOA) spectral radiance change can be expressed as a linear combination of spectral radiance changes caused by cloud and atmospheric properties; 2) the magnitude of spectral radiance changes linearly corresponding to a small perturbation of cloud or atmospheric property, at least in the relevant parts of the spectrum; and 3) changes of cloud and atmospheric properties provide unique spectral radiances that can be separated by a linear regression. These assumptions are tested in an earlier study (Kato et al. 2011) and proved to be valid assumptions at least for annual and monthly and 10° latitude zonal scales.
To extend the study by Kato et al. (2011), this study has two objectives. Earlier studies (Huang et al. 2010a,b; Kato et al. 2011) show that the error in retrieved cloud properties from the difference of spectral radiance averaged over two time periods is rather large especially for low-level clouds. The scene identification over an instrument footprint often reduces the retrieval error from instantaneous radiances (e.g., clouds affecting temperature retrieval by the retrieve-then-average approach). This paper, therefore, investigates whether or not separating clear-sky from cloudy-sky also reduces the retrieval error when highly averaged spectral radiances are used. The second objective of this paper is to estimate the error in a trend estimated from properties derived from highly averaged spectral radiances. Given results of earlier studies that show a large error in retrieved cloud properties (Huang et al. 2010b; Kato et al. 2011), the motivation of the second objective and utility of the result seem to be questionable. If the retrieval needs to separate the observed spectral radiance trend into trend contributions by cloud and atmospheric properties, then two necessary requirements to obtain a small error in the retrieved trend, based on studies by Weatherhead et al. (1998), Leroy et al. (2008b), and Wielicki et al. (2013), seem to be 1) any retrieval bias error must be stable at a level much smaller than that of the trend of interest, and 2) any random error must be sufficiently small with sufficient independent samples to allow accurate detection of the trend.
Section 2 explains the source of the error in the simulation, section 3 describes the method and data used in this study, and section 4 describes the results. Section 5 describes why the error in the trend estimate is small even when the error in the retrieved anomalies is relatively large.
2. Retrieving atmospheric and cloud property anomalies from spatially and temporally averaged spectral radiance
When 
Spectral radiance change computed by perturbing the monthly 10° latitude zonal mean values indicated at the top of each panel and averaging over a year (blue line). The red line indicates the spectral radiance change computed by perturbing monthly 10° latitude zonal value and averaging over a year minus the change computed by perturbing corresponding instantaneous values sampled by a 90° inclined polar orbit. (top) For the latitude between 30° and 40°S and (bottom) for the latitude between 40° and 50°N. Spectral radiance changes are computed for year 1990.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
An earlier study by Kato et al. (2011) defines the spectral radiance difference 
3. Method
We model spectral radiances using cloud and atmospheric properties taken from three data files (tavg1_2d_slv_Nx, tavg1_2d_rad_Nx, and inst6_3d_ana_Np) of the Modern-Era Retrospective Analysis for Research and Applications (MERRA; Rienecker et al. 2011). MERRA outputs temperature, humidity, and ozone concentrations every 6 h in a 0.66° longitude by 0.5° latitude grid. Near surface (2 m above the ground) air temperature and humidity are given hourly. MERRA also outputs high, mid, and low cloud properties every hour. Because MERRA produces an abundance of optically thin clouds, we exclude clouds with a visible optical thickness less than 0.3. The 0.3 threshold is equivalent to the cloud sensitivity threshold of a passive sensor such as the Moderate Resolution Imaging Spectroradiometer (MODIS; Minnis et al. 2008a,b) and about the emissivity threshold of 0.15 in the 11-μm window region (Minnis et al. 1993). The resulting zonal mean clear fraction (1 minus cloud fraction) is shown in Fig. 2. The zonal clear-sky occurrence is similar to that derived from the Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP; Winker et al. 2010) and CloudSat (Stephens et al. 2008). Note that the clear-sky fraction from CALIOP and CloudSat shown in Fig. 2 is also derived by screening clouds with optical thickness less than 0.3 using the extinction coefficient vertical profile derived from CALIOP. We use a temperature threshold to determine the cloud phase (−13°C; e.g., Cheng et al. 2012) and cloud properties given by Yang et al. (2001) to compute phase function and single scattering albedo. All clouds are classified as either high-, mid-, or low-level clouds. Clouds with a top pressure smaller than 400 hPa and greater than 700 hPa are classified, respectively, as high- and low-level clouds, and clouds with the top pressure in between are midlevel clouds. When more than one cloud type exists in a grid, the cloud fraction exposed to space is computed by the cloud type with a no overlap assumption (Fig. 3). The spectral radiance with clouds exposed to space is then computed for all cloud types. For example, when all high-, mid-, and low-level clouds exposed to space are present in a grid, the spectral radiance is computed for single-layer high-, mid-, and low-level clouds as well as clear sky. The results are area weighted and averaged. Because subgrid-scale information is not available from MERRA, it is assumed that clouds are horizontally uniformly distributed within a 0.66° longitude by a 0.5° latitude grid box. One simplification is applied to cloud height variability. Because the set of MERRA products used for this study does not provide complete vertical cloud profile information in a grid box, only the cloud-top height for the highest cloud type in an hourly grid box varies while the cloud-top height for other lower cloud types stays constant (Fig. 3a). The cloud-top height of low-level clouds, therefore, varies when there is no higher cloud in the hourly grid box (Fig. 3b). The cloud-top height of lower clouds is set at 500 hPa for midlevel clouds and surface pressure minus 100 hPa for low-level clouds when upper-layer clouds are in an hourly grid box.
Clear-sky fraction (1 − cloud fraction) derived from MERRA (blue solid line) and CALIOP and CloudSat (red dashed line) as a function of latitude; 28 years of MERRA and 3 years (2007–09) of CALIOP and CloudSat data are used. Cloud layers with visible optical thickness less than 0.3 are excluded from both MERRA and CALIOP–CloudSat data. Error bars of MERRA clear fraction indicate the standard deviation of annual mean clear fraction over 28 years. The error bars of CALIOP–CloudSat clear fraction indicate the maximum and minimum values of three annual means.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
Simple schematic of cloud overlap assumptions used in the simulation. MERRA provides high-, mid-, and low-level clouds fraction assuming random overlap and the highest cloud top height in 0.66° × 0.5° grids every hour. Shown are (a) a 0.66° × 0.5° grid box having all three cloud types and (b) a 0.66° × 0.5° grid box that has only one cloud type. The cloud fraction exposed to space is computed from MERRA clouds and used in our simulation with no overlap. Clouds exposed to space are separated by cloud type with the independent column approximation. The spectral radiance is computed with single layer high-, mid-, and low-level clouds in addition to clear sky and averaged weighed by area. When higher-level clouds are present, mid- and low-level cloud-top heights are fixed, respectively, at 500 hPa and surface pressure minus 100 hPa. Note that the effective cloud-top height for all cloud types can vary because of the variability of the optical thickness of each cloud type.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
We compute spectral radiance from 50 to 2760 cm−1 with a 0.5 cm−1 resolution using the Principal Component–based Radiative Transfer Model (PCRTM; Liu et al. 2006). MERRA outputs of 28 yr from 1983 through 2010 are used. In the computations, water vapor and ozone concentrations vary according to MERRA outputs but the concentrations of CO2, CH4, N2O, and CO are fixed to their climatological values. No aerosols are included in this simulation. We use a 90° inclined polar orbit to simulate a satellite orbit and sample every 30 s from the closest MERRA grid and hour box. The 90° inclined orbit assures full diurnal cycle sampling; the sampling error by one instrument on a 90° polar orbit is 25% of the sampling error from sun-synchronous orbits (Kirk-Davidoff et al. 2005). As a consequence, the sampling noise of nadir view only observations from a single 90° inclined polar orbit is negligible to detect changes occurring at a large scale. Sampling from a single 90° inclined polar orbit increases the variance of global annual mean top-of-atmosphere broadband window irradiance less than 5% (Wielicki et al. 2013). Nadir-view spectral radiances are computed for every field of view to simulate instantaneous spectral radiances. We treated inputs sampled from the 90° inclined polar orbit and used them in the spectral radiance computations as truth in evaluating retrieved values.
There are four steps involved in computing annual mean spectral radiative kernels. First, monthly 10° latitude zonal mean cloud and atmospheric properties are computed by averaging those properties sampled from the 90° inclined polar orbit over the course of a month. Second, we perturb monthly 10° latitude zonal mean cloud and atmospheric properties one at a time by the amount listed in Table 1. The perturbation magnitudes Δx* are equivalent to either expected decadal change or the standard deviation of 10° latitude zonal deseasonalized anomalies (Kato et al. 2011). As discussed in Kato et al. (2011), we also perturb all variables together by Δx* to crudely account for the spectral radiance change caused by more than one variable perturbed together. Third, the spectral radiance computed with the unperturbed cloud and atmospheric properties (i.e., using monthly 10° latitude zonal mean values) is subtracted from the perturbed result. Fourth, monthly spectral radiance changes are averaged over the entire 28-yr period (i.e., 12 times 28 spectral radiance differences in a 10° latitude zone) to compute annual 10° latitude zonal mean spectral radiative kernels. The resulting annual 10° latitude zonal mean spectral radiance changes are used in S. When atmospheric and cloud property anomalies are retrieved from the 10° latitude zonal annual spectral radiance anomalies using Eq. (10), we have, therefore, 28 retrieved values for a given property in a 10° latitude zone. Note that the exact value of of Δx* is not critical to the retrieval results. However, it needs to be in a range of annual zonal variability of cloud and atmospheric properties to make retrieved values close to 1.
Perturbed values in building spectral radiative kernels.
4. Results
Figure 4 shows the retrieved annual 10° latitude zonal air and surface temperature and water vapor amount anomalies between 30° and 40°S as a function of year. These are derived from all-sky spectral radiance anomalies. Even though clear-sky scene spectral radiances are not separated from cloudy scene spectral radiance anomalies, temperature and water vapor amount anomalies are retrieved relatively well compared to cloud property anomalies for the 10° latitude zone where the cloud fraction is about 80% (Fig. 2). Three reasons for this are 1) spectral features of radiance change caused by temperature and water vapor perturbations are sufficiently different from each other and from spectral features caused by cloud property changes, especially for those perturbed at high altitudes; 2) the sum of all individual radiance changes caused by cloud and atmospheric property changes is nearly equal to the total radiance change; and 3) radiance anomalies caused by cloud and atmospheric property anomalies are small so that the radiance changes linearly (Kato et al. 2011).
(left) Retrieved (red dots) and true (blue circles) temperature anomalies as a function of year. From top to bottom, 100–10 hPa, 200–100 hPa, 500–200 hPa, 850–500 hPa, 850 hPa–surface, and surface (near-surface air and skin) temperature. A 10° latitude zone of 40°S–30°S is used for the plot. (right) As at left, but for relative water vapor amount anomalies for, from top to bottom, 500–200 hPa, 850–500 hPa, and 850 hPa–surface layers. The vertical bars indicate the uncertainty of the retrieved values computed by the method described by Eq. (B7) in Kato et al. (2011). Note that two terms, the spectral radiance difference of control runs computed using instantaneous and monthly mean properties and the standard deviation of annual signature matrices, are considered in computing the error bars.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
To understand whether or not separating clear-sky scenes from all-sky scenes improves the retrieval result, we also retrieve temperature and water vapor amount anomalies using clear-sky scenes only. On the one hand, separating clear-sky scenes avoids the influence of the cloud property anomaly retrieval error on retrieved clear-sky property anomalies. On the other hand, clear-sky temperature and water vapor amount and spectral radiances from clear-sky scenes can be significantly different from those under all-sky conditions (Cess and Potter 1987; Sohn et al. 2010; Kato et al. 2013), even though the difference of means does not necessarily lead to different anomalies between under clear-sky and all-sky conditions. The separation of clear-sky from all-sky for the temperature and water vapor anomaly retrieval, therefore, does not necessarily improve retrieved values but can lead to a bias error over regions where those anomalies under all-sky conditions are significantly different from those under clear-sky conditions. In the following section, we test the effect of cloud property anomaly retrieval errors on the temperature and humidity anomaly retrieval.
Effect of scene identification
To test the effect of scene identification on the retrieval, we use two different methods to compute the annual 10° latitude zonal mean clear-sky spectral radiances. The first mean clear-sky spectral radiance is computed by removing clouds (hereinafter, cloud removed clear-sky mean spectral radiance), hence clear-sky scenes happen everywhere all the time regardless of the presence of clouds. The second mean is computed by averaging radiances weighted by the clear-sky fraction (hereinafter clear-sky weighted mean spectral radiance) over the MERRA grid. Note that if observations are used, retrieval from clear-sky weighted mean spectral radiances is the only way to retrieve clear-sky property anomalies. Because clear-sky temperature and water vapor amount might be different from those under all-sky conditions, and the presence of clouds affects the retrieval, the retrieval error of atmospheric temperature and water vapor amount anomalies derived from the cloud-removed clear-sky mean spectral radiance is smaller than the error that can be achieved using observations. Therefore, the retrieval error from cloud-removed clear-sky spectral radiance anomaly is a lower limit and can be used to assess the impact of cloud retrieval error. If the retrieval error using all-sky mean spectral radiance anomalies is equivalent to that using cloud-removed clear-sky mean spectral radiance anomalies, we can conclude that the presence of clouds does not affect the accuracy of retrieved temperature and water vapor amount anomalies.
The left column of Fig. 5 shows the root-mean-square (RMS) difference between the retrieved surface temperature using Eq. (10) and truth as a function of latitude. The RMS difference is computed from 28 retrieved anomalies from 10° latitude zonal annual spectral radiance anomalies. The RMS difference of retrieved values from all-sky spectral radiance anomalies is equivalent to the RMS difference computed with cloud-removed clear-sky spectral radiance over the tropics and midlatitude. This indicates that the presence of clouds has a small influence on the retrieval. The RMS difference of surface temperature anomalies retrieved from clear-sky weighted spectral radiance anomalies is significantly larger over regions where cloud fraction is larger, such as the 10° latitude zones between 50° and 70°S. In addition, using the clear-sky fraction weighted mean spectral radiance introduces biases in the retrieved temperature and causes a larger RMS difference than the other two cases for layers even for the altitude above the height of clouds (e.g., over midlatitude at 100–10 hPa). Note that the RMS difference increases toward polar regions, probably because the magnitude of the temperature anomaly increases toward polar regions. (Kato 2009).
(left) RMS difference and (right) correlation coefficient between retrieved and true temperature annual anomalies. Temperature anomalies for the layer indicated at the top of each panel are retrieved from 10° latitude zonal annual mean spectral radiance anomalies and compared with true anomalies used for the simulation. Anomalies are defined as the deviation from the 28-yr mean (deseasonalized anomalies). Blue and red bars indicate the result of retrievals, respectively, using cloud-removed clear-sky spectral radiance anomalies and using clear-sky fraction-weighted clear-sky spectral radiance anomalies. Green bars indicate the retrieval result using all-sky spectral radiance anomalies. Horizontal dashed lines indicate the correlation coefficient that is significant at a 95% confidence level.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
A negligible impact of the presence of clouds on retrieved temperature anomalies from all-sky spectral radiance anomalies is also apparent in the correlation coefficient (Fig. 5, right). The correlation coefficients between true temperature anomalies and temperature anomalies retrieved from all-sky spectral radiance anomalies are equivalent to the correlation coefficient computed using cloud-removed clear-sky for most regions and pressure levels, except for the surface to 850-hPa layer. Note that the correlation coefficient at a 95% significance level with 28 degrees of freedom is 0.37. Larger RMS differences and smaller correlation coefficients over the tropics at the 200–100-hPa layer are probably caused by the presence of the tropopause in the layer so that the temperature does not decrease linearly with height within this layer. As a consequence, the temperature within the layer probably does not uniformly change. Using a spectral radiative kernel computed by perturbing the layer temperature uniformly, therefore, leads to a larger error in the retrieved temperature anomaly for the 200–100-hPa layer. Similarly, a poor correlation for the layer from the surface to 850 hPa at subsidence regions suggests that the annual mean temperature in the layer may not change uniformly because of a temperature inversion at the top of the boundary layer. This result suggests that a uniform temperature change within a layer is a requirement in discretizing the atmospheric layer. We expect that separating these layer into multiple layers or using a climatological boundary layer height as a boundary could mitigate the problem but the study is left for the future.
Figure 6 shows the RMS difference and correlation coefficient for water vapor amount relative anomalies. The relative anomaly is defined as the water vapor amount deviation from the 28-yr mean divided by the 28-yr mean value. The result is similar to that of the temperature anomaly retrieval; the RMS difference of retrieved values from all-sky scenes is equivalent or smaller than that for retrieved values from cloud-removed clear-sky mean spectral radiance. Note that the smaller RMS is due to the smoothing metrics 
(left) RMS difference and (right) correlation coefficient between retrieved and true water vapor amount relative annual anomalies. Water vapor relative anomalies for the layer indicated at the top of each panel are retrieved from 10° latitude zonal annual mean spectral radiance anomalies and compared with true relative anomalies used for simulation. Relative anomalies are defined as the deviation from the 28-yr mean divided by the 28-yr mean value. Blue and red bars indicate the result of retrievals, respectively, using cloud-removed clear-sky spectral radiance anomalies and using clear-sky fraction-weighted clear-sky spectral radiance anomalies. Green bars indicate the retrieval result using all-sky spectral radiance anomalies. Horizontal dashed lines indicate the correlation coefficient that is significant at a 95% confidence level.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
(top left) RMS difference and (top right) correlation coefficient between retrieved and true cloud fraction annual anomalies for low- (blue), mid- (red), and high- (green) level clouds. (bottom) As at top, but for cloud height annual anomalies. Cloud fraction and height anomalies are retrieved from all-sky spectral radiance anomalies. The RMS and correlation coefficients are based on 28 anomalies. Horizontal dashed lines indicate the correlation coefficient that is significant at a 95% confidence level.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
5. Impact of retrieval errors on trends
Retrieved (red dots) low-level cloud fraction annual anomalies with true anomalies from MERRA (blue circles) for the latitude between 30° and 40°S. Red and blue dashed lines are linear regression lines computed with, respectively, retrieved and true anomalies.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
To investigate whether or not the result shown in Fig. 8 is shared by other latitudes and other retrieved properties, we plot the slope of the regression line derived from retrieved and true values in Figs. 9, 10, and 11. The error bars indicate a 95% confidence interval of the slope derived from the linear regression. The true temperature trend is within the 95% confidence interval estimated from retrieved temperature anomalies except for the near-surface temperature in polar regions (Fig. 9). Among five layers and surface temperatures over 18 zones, approximately 60% true trends are within the 95% confidence interval of the retrieved trends. The trend of water vapor amount in the lower atmosphere is similar to the result of the temperature trend (Fig. 10). While a larger error occurs in layers near the surface in polar regions among three layers of water vapor amounts over 18 zones, approximately 60% true trends are within a 95% confidence interval of retrieved trends. In addition, 40% of 10° latitude zone trends of cloud fraction estimated from the truth also fall within a 95% confidence interval estimated from retrieved cloud fraction anomalies (Fig. 10). In contrast, the trend of cloud height estimated from retrieved cloud height anomalies has a large error (not shown) and needs a significant improvement. Only 20% of true 10° latitude zonal trends are within the 95% confidence interval of retrieved trends. Retrieving cloud-top effective temperature anomaly instead of cloud-top height anomaly might improve the cloud height trend estimate.
Temperature trends derived from retrieved annual anomalies (red dots) for the atmospheric layer indicated at the top of the panels. Blue circles indicate trends derived from true anomalies. Error bars indicate a 95% confidence interval of the trend.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
The trend of relative water vapor amount derived from retrieved annual relative anomalies (red dots) for the atmospheric layer indicated at the top of the panels. Blue circles indicate trends derived from true anomalies. Error bars indicate a 95% confidence interval of the trend.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
Cloud fraction trend derived from retrieved annual anomalies (red dots) for the cloud type indicated at the top of the panels. Blue circles indicate trends derived from true anomalies. Error bars indicate a 95% confidence interval of the trend.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
The 95% confidence interval computed with true values is caused only by natural variability. The size of the 95% confidence interval for the trend derived from retrieved values is determined by two factors, natural variability and the retrieval error. The 95% confidence intervals for the retrieved trend, however, can be smaller than the confidence interval derived from true values because the variability of retrieved values is controlled by the smoothing matrix 
Ratio of the standard deviation of retrieved annual 10° latitude zonal anomalies (σret) to the standard deviation of true annual 10° latitude zonal anomalies (σn) as a function of latitude. (top) Values are derived by using an identity matrix for 
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
Fraction of the true trends that are in the 95% confidence interval of retrieved trends as a function of ratio of standard deviation of retrieved anomalies σret and true anomalies σn less than or equal to the value indicated on the x axis. Temperature (all three layers and surface), water vapor (all three layers), cloud height (three cloud types), and cloud fractions (three cloud types) over all eighteen 10° latitude zones are considered.
Citation: Journal of Climate 27, 12; 10.1175/JCLI-D-13-00566.1
One caveat of this simulation result is that cloud and atmospheric properties used for simulating observed spectral radiances are also used in computing spectral radiative kernels. In practice, cloud and atmospheric properties are unknown so that the error in climatological mean cloud and atmospheric properties also affects spectral radiative kernels. Therefore, the error in the trend derived from retrieved properties estimated in this study excludes the error in the climatological mean. The climatological mean can be obtained from other retrievals such as those using instantaneous satellite observations. It can also be derived from the difference between modeled and observed spectral radiances averaged over the time period using Eq. (10). In this climatological mean retrieval, 
In summary, the result of this study shows that if the retrieval error is nearly random and linear, an observed spectral radiance trend can be separated into the contribution of cloud and atmospheric property anomalies. Using spectral radiative kernels averaged over the entire observational period and retrieving anomalies every year reduces the error in the trend estimate. This contrasts from the retrieval using the difference of observed spectral radiances from two periods to retrieve the cloud and atmospheric property difference once. The result of this study suggests the possibility of using such algorithms in climate change signal detection. Possible improvements of the algorithm are further discussed in the appendix.
6. Summary and conclusions
We used 28 years of atmospheric and cloud properties from MERRA and simulated spectral nadir-view instantaneous radiances observed from a 90° inclined polar orbit. We also used the same gridded atmospheric and cloud properties but averaged 10° latitude zonal over a month to compute spectral radiative kernels by perturbing cloud and atmospheric properties. The resulting spectral radiative kernels are averaged over the entire 28 years. We then retrieved annual 10° latitude zonal cloud and atmospheric property anomalies using mean radiative kernels. Because earlier studies indicate that the errors in retrieved properties are large especially for cloud properties, one of two purposes of the simulation is to test whether or not separating clear sky from cloudy sky reduces the retrieval error. The results suggest that the presence of clouds does not influence the retrieval of temperature and humidity anomalies when a simple linear regression is used. When temperature and humidity anomalies are derived from clear-sky fraction-weighted spectral radiances, the RMS difference tends to be larger because temperature and humidity profiles under clear-sky conditions are different from those under all-sky conditions. The result does not rule out the possibility of reducing the retrieval error by separating clear sky from cloudy sky when a more sophisticated algorithm is used, especially for retrieving cloud property anomalies. If the clear-sky retrieval is separated from all-sky retrieval, however, one needs to contemplate at least the following two points. 1) The clear sky needs to be well defined because the sensitivity of cloud detection varies with instrument type and the resolution. When the clear-sky scene is defined, the effect of the scene identification error on the retrieval needs to be small. 2) Even if clear-sky retrieval is separated from all sky with a minimum error, temperature and humidity anomalies need to be retrieved from cloudy or overcast spectral radiances because those under all-sky or overcast conditions might be different from those under clear-sky conditions. The separation of clear sky from cloudy sky has to bring additional constraints to compensate the error introduced in identifying clear-sky scenes.
The second objective is to estimate the error in the trend derived from retrieved properties derived from highly averaged spectral radiances. The simulation in this study demonstrates that cloud and atmospheric property changes over time can be extracted from highly averaged spectral radiances. As mentioned, the requirements are 1) any retrieval bias error must be stable at a level much smaller than that of the trend of interest, and 2) any random error must be sufficiently small with sufficient independent samples to allow accurate detection of the trend. The results of this study also suggest that the atmospheric layer needs to be discretized so that the temperature or humidity in the layer changes uniformly.
Acknowledgments
We thank Drs. Stephen Leroy, Xianglei Huang, Oleg Dubovic, Norman Loeb, and Steven Platnick for helpful discussions and suggestions and Ms. Amber Richards for proofreading the manuscript. The work was supported by the NASA Science Directorate through the CLARREO project.
APPENDIX
Retrieval with Unequal Weights to Cloud and Atmospheric Properties
This appendix provides possible ways to improve retrieval results in the future by following the numerical inversion technique discussed in Twomey (1977) and Dubovik (2004).
Number of 10° latitude zones that meet two conditions: 1) the true trend is within the 95% confidence interval of trend derived from retrieved anomalies, and 2) the ratio of standard deviation from retrieved anomalies to the standard deviation of true anomalies is between 0.9 and 1.1 (second column), between 0.8 and 1.2 (third column), between 0.7 and 1.3 (fourth column), between 0.6 and 1.4 (fifth column), and between 0.5 and 1.5 (sixth column). An identity matrix for 
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