The first CERES instrument flew on the TRMM satellite, launched in November 1997, and provided data until 2000. Five CERES instruments are currently operating: two on NASA's Terra satellite (FM1 and FM2), two on NASA's Aqua satellite (FM3 and FM4), and one on the Suomi National Polar-orbiting Partnership (NPP) satellite (FM5). These CERES instruments provide radiometric measurements of the Earth's atmosphere from three broadband channels: (1) a shortwave channel to measure reflected sunlight (0.3–5 µm), (2) a long-wave channel to measure Earth-emitted thermal radiation in the window region (8–12 µm), and (3) a total channel to measure radiation from 0.3 to 200 µm.

CERES radiances are converted to TOA fluxes using ADMs. The CERES science team has an extensive database of ADMs for clear- and cloudy-sky over both land and ocean (Loeb et al., 2005). The ADMs heavily depend upon the observed scene type and are sensitive to surface characteristics as well as cloud and aerosol optical properties (Loeb et al., 2003; Zhang et al., 2005a; Patadia et al., 2011). The ADMs over ocean are dependent upon wind speed and aerosol optical thickness along with sun-satellite geometry (Zhang et al., 2005a).

The Aqua spacecraft carries two identical CERES instruments: one operates in a cross-track scan mode (FM3) and the other in a biaxial scan mode (FM4). Measurements from the biaxial scan mode were used to develop the ADMs; this provided considerable improvement over the previous generation of instruments, including the ERBE (Loeb et al., 2003, 2007).

This study uses the Single Scanner Footprint (SSF, Edition 3A) TOA SWF obtained from the Aqua CERES FM3. The SSF product is an instantaneous merge of CERES parameters with coincident cloud and aerosol parameters derived from the Aqua MODIS (Loeb et al., 2003) at the footprint level (i.e., not daily averages). The high-resolution (1 × 1 km2 at nadir) MODIS imager data are used to characterize the clear and cloudy portions of the larger CERES pixel (20 × 20 km2 at nadir). We also used broadband surface albedo over land derived using measurements from MODIS and CERES to characterize the land surface in the NN trained specifically over land.

OMI provides hyper-spectral measurements of Earth-backscattered sunlight from UV to visible wavelengths (∼ 270–500 nm) with a spectral resolution of the order of 0.5 nm (Levelt et al., 2006). Its spatial resolution is 13 × 24 km2 at nadir with a swath width of about 2600 km. Cloud, aerosol, and TCO products from OMI are used in this study. Specifically, the cloud–aerosol optical centroid pressure (OCP), effective cloud fraction (ECF; fc), Lambertian equivalent reflectivity (LER) at 354.1 nm, SZA, relative azimuth angle (RAA), and viewing zenith angle (VZA) are obtained from the OMI cloud products as detailed below, and aerosol index (AI) and TCO are obtained from the OMI-TOMS total ozone product (OMTO3, version 8.5, collection 3) (McPeters et al., 2008).

Cloud–aerosol OCP, also known as effective cloud pressure, is a measure of the reflectance-weighted pressure reached by incoming solar photons (Joiner et al., 2012). It is distinct from the cloud-top pressure (CTP). While CTP is the more important parameter needed for TOA long-wave flux, OCP is more related to atmospheric absorption in the shortwave. OCP is derived from OMI observations using two different methods (Stammes et al., 2008): (1) filling-in of solar Fraunhofer lines from rotational Raman scattering in the UV (the OMCLDRR product) (Joiner and Bhartia, 1995; Joiner et al., 2004) and (2) collision-induced oxygen absorption (O2–O2) at 477 nm (the OMCLDO2 product) (Stammes et al., 2008; Acarreta et al., 2004). Unless otherwise specified, we use the fc and OCP from OMCLDRR product here.

OMI cloud and trace-gas algorithms use a simplified mixed Lambertian cloud model to estimate observed radiances Im. In this scenario, a pixel is modeled as a having components from clear and cloudy sub-pixels weighted using an ECF fc; i.e., Im=Ig1-fc+Icfc, where Ig and Ic are the radiances computed in the Rayleigh atmosphere for Lambertian surfaces corresponding to the clear and cloudy portions of the scene, respectively; fc is defined as the fraction of the Lambertian cloud covering the pixel and is related to both the geometric cloud fraction and cloud optical thickness. It contains information similar to the LER of the scene (related to cloud and surface reflectivities). However, because it attempts to account for variations in the Earth's surface reflectivity, it is a more spectrally invariant quantity and therefore potentially more highly correlated with TOA SWF.

Formally, the ECF is wavelength dependent because it is defined by spectral quantities (Stammes et al., 2008). We conducted a simulation experiment to evaluate the wavelength dependence of fc. In this experiment, we simulate observed TOA radiances as a weighted sum of the clear-sky and cloudy radiances; i.e., Im=Ig1-fg+Icfg where Ic is the cloudy radiance computed with a plane-parallel cloud model that depends on cloud optical thickness, and fg is the geometrical cloud fraction. In our simulation, clouds have a vertically uniform distribution of the extinction coefficient and phase function. We use a cloud-top height of 5 km and a cloud layer thickness of 1 km. The assumed cloud optical depth (COD) of 20 is spectrally independent within the 320–1400 nm wavelength range. The spectrally independent optical thickness is a good approximation for clouds with sufficiently large particles (Deirmendjian, 1969). We neglect gaseous absorption in the specified spectral range. Three models of cloud phase function are used: (1) ice crystals with an effective diameter of 60 µm (Baum et al., 2014), (2) Deirmendjian's C1 model of a water cloud having droplets with an effective diameter of 12 µm (Deirmendjian, 1969), and (3) the Henyey–Greenstein model (e.g., van de Hulst and Irvine, 1963) with an asymmetry parameter of 0.85. We use a simplified model of the spectral ground reflectance: Rg= 0.05 at λ < 700 µm and Rg=0.2 at λ > 700 µm. We then calculate fc by inverting Eq. (1) assuming a Lambertian cloud with a reflectivity of 0.8; this is commonly used for trace-gas algorithms (Stammes et al., 2008).

If fc does not vary much with wavelength, then it should be highly correlated with TOA SWF and thus a good predictor in a statistical model of TOA SWF. We investigate the spectral dependence of fc in Fig. 1 where we performed calculations using fg=0.5, SZA = 45∘, and at nadir for the three phase functions assuming a cloud single-scattering albedo of unity. It can be seen that fc is nearly invariant with wavelength over a wide spectral range; it changes by only a few percent even for the steep change in the ground reflectance (simulating the so-called red edge) at 700 nm. COD of less than 20 at the same geometrical cloud fraction leads to a lower value of the ECF. However, the spectral variations of ECF remain similar to that shown in Fig. 1. The weak spectral dependence of ECF is explained by the fact that the Lambertian cloud model with cloud reflectivity of 80 % effectively accounts for Rayleigh scattering in partially cloudy scenes as it has been shown by Ahmad et al. (2004). This result holds when other input parameters in our simulation are varied. We will show the implications of the spectral invariance of fc below.

We have used the following modified cloud fraction parameter, fc_mod, as a predictor to estimate TOA SWF: fc_mod=fc×cos⁡(SZA)×1/SED2, where SED is the Sun–Earth distance. The modification accounts for variation in the incoming solar irradiance with respect to SZA and SED. Figure 2 shows a 2-D histogram for a month of collocated CERES TOA SWF and OMI fc_mod (collocation criteria described in detail below) that demonstrates the near-linear relationship between the two parameters. It also indicates that the single parameter fc_mod captures much of the variability in TOA SWF.

In addition to OMI data, a SeaWiFs-derived chlorophyll (Chl) concentration climatology is used as an input predictor when fc < 1. The precipitable water (PW) and 2 m surface wind speed (wind) are also used as predictors; these are provided in the CERES SSF data set and are taken from the Goddard Earth Observing System (GEOS) 4 reanalysis (Bloom et al., 2005).

Because the sizes of the OMI (13 km × 24 km) and CERES (20 km2) pixels are similar at nadir, we perform a simple spatial collocation by finding the closest CERES pixel corresponding to each OMI pixel. OMI and CERES collocated pixels are only included in our training and validation samples when the distance between centers of OMI and CERES pixels is less than 20 km. We do not include pixels with viewing zenith angles > 60∘. At these angles, OMI and CERES pixels become significantly larger (∼ 150 km for OMI and ∼ 200 km for CERES in the cross track direction) and therefore may contain many different scene types. We also mask OMI pixels with AI > 1 to avoid heavy absorbing aerosol loaded scenes where the fc and OCP are known to contain errors (Vasilkov et al., 2008). This does remove not non-absorbing aerosols. Non-absorbing aerosols are essentially treated as clouds within the OMI cloud data sets. We examined the frequency distribution of the distance between OMI and CERES pixels of all the collocated data sets and found that most of the collocated data, 98 and 60 % have distances less than 20 and 10 km, respectively. The quality-controlled collocated data are then averaged on equal latitude and longitude grids of 1∘ × 1∘ (unless otherwise specified) for training, testing, and validation of the neural networks.

The NN inputs (predominantly retrievals from OMI measurements) are used to train the NN to match the output (CERES-derived TOA SWF). Once the network is trained, input data sets can be used to calculate TOA SWF with characteristics similar to the CERES product. Therefore, the NN-produced TOA SWF will be referred as OMI estimated SWF throughout the paper.

The general neural network architecture has three layers of neurons: an input layer, a hidden layer, and an output layer with standard multi-layer network architecture. We use the same number of neurons in the hidden layer as in the input layer as this produced generally good results. The input layer has an identity activation function; all other layers are connected by sigmoid activation functions (Eq. 4). yx=11+e-x The network normalizes both input and output data sets with a unique linear mapping for each input and output parameter. Figure 3 provides an example of a schematic of the network used in our study. Here we used two different NNMs: one with nine nodes or parameters (NNM1) and a second with seven nodes (NNM2) in the input layer. Both of these models have one node (TOA SWF) in the output layer. Figure 3 also lists the input parameters corresponding to the NNM1 and NNM2 models. The NNM1 model is optimized for ocean cases where the OMI fc < 1.0, whereas NNM2 is optimized for cases where fc=1 (saturated cases).

NNMs require optimized training to produce accurate outputs. Here we use a standard back propagation training algorithm (Hertz et al., 1991), where inputs are iteratively sent to the neural network. In back propagation, the hidden layer weights associated with each input parameter are modified through the training process that minimizes errors between the targets and outputs (Bishop, 1995; Gardner and Dorling, 1998). After each iteration, the error is propagated backward through the network and weights are modified to bring the actual response of the network closer to the desired output in a statistical sense. The function minimized during the training is a sum of squared errors of each output for each training pattern. Once the network is trained, it can be evaluated using independent data (i.e., not used in the training data set).

Here we examine the impact of using various input parameters on the derived neural networks. This exercise is performed using data with fc < 1 with 1 month of the data over ocean (January 2007). Table 1 presents the performance of eight different NNMs, denoted models “a” through “h”, with various input parameters listed in Table 1 and described in more detail in Sect. 3.2.1–3.2.2.

We next examine the performance of the NN model h with respect to different input samples. We first examine the robustness of the NN for detection of interannual variability. In this exercise, we trained the NN with data from the first 15 days of January 2007 as above and applied it to input data from the entire months of January 2007 (Fig. 5a) and January 2006 (Fig. 5b). Figure 5a–b present 2-D histograms similar to that in Fig. 2 but here compare the TOA SWF from the NN with that from CERES for January of the 2 different years over ocean. The colors represent the 2-D histogram (or density) of coincident pairs using a bin size of 10 Wm-2. The solid black 1 to 1 line is shown with three dotted lines on both sides that represent envelopes of ±5, ±10, and ±15 % OMI–CERES differences. Figure 5a–b show that the NN performance is consistent between years. Although the number of samples in January 2006 and 2007 is a bit different, the NN model produces similar statistics.

A NN trained on 1 particular month of data is not guaranteed to perform well for a different month. In the next test, we used the NN model trained on January 2007 data with input data from July 2007. Results for July (using CERES as the benchmark) were degraded as compared with application to January data. This is due presumably to changes in observing conditions between the 2 different months (changes in solar angles).

We then trained the same NN (identical input parameters) using a subset of data from July 2007 and applied it to data from the entire month of July 2007. Results (again using CERES as the benchmark) were of similar quality to those where the NN was trained and applied to January. This exercise suggests that we may need to use different models for different months or expand our training data set for application to different months.

We next use data from the 1st day of each month of 2007 for training and data from the 16th day of each month of 2007 for evaluation. The comparisons with CERES using the training and validation data are consistent as shown in Fig. 6. The almost identical values of statistical parameters for training and validation data demonstrate that the neural network has been well trained. For example, there is a high degree of linear correlation (R=0.98 or R2=0.96) and slopes close to 1 (0.96) in both training and validation comparisons. The mean bias in both training and validation data sets is close to zero, whereas the global standard deviation remains stable and close to 30 Wm-2 in the two independent model runs. Further analysis shows that 83, 70, and 43 % of TOA SWF estimated from OMI (training and validation data combine) lie within the 15, 10, and 5 % of the CERES TOA SWF, respectively.

Further evaluation of the entire year reveals that this NN (model h) is appropriate for all months. Therefore this model will be used for subsequent analyses in this study. Creating more networks as a function of scene type or for different latitude belts or even for different months/seasons may improve results in certain regions. However, based on our results, we simplified the approach by minimizing the number of networks.

About 1–2 % of total coincident data correspond to fc=1, typical of overcast conditions with optically thick clouds. These cases were modeled using a simpler NN model with inputs of LER, SZA, VZA, RAA, OCP, O3, and PW; the surface-related parameters (surface wind speed and Chl content) do not produce a significant impact for ECF = 1 and have therefore been removed. Subsequent all-sky results shown in Fig. 6 use combined outputs from the two separate models for fc < 1 and fc=1.

Here onward, all the results presented in Figs. 7–12 and discussed are produced using model h, which is the most optimized model. Figure 7 presents the RMSE, RMSE normalized by CERES flux (NRMSE in percent), data sample (percent), and bias (percent) for 5 % ECF bins. This analysis includes both training and validation data as presented in Fig. 6. The RMSE varies between about 24 and 35 Wm-2 and continuously increases with cloud fraction (and observed TOA SWF). The NRMSE, in contrast, continuously decreases with ECF from about 18 % for 5 % ECF to ∼ 6 % for overcast conditions. The bias represents the mean error (in percent) for each ECF bin. The mean global bias shows more variability than RMSE and is highest (2.9 %) for about 10 % ECF. The bias decreases sharply from 2.9 % at fc=0.1 to about 1.2 % at fc=0.4. The bias remains low (< 1.2 %) for fc > 0.4 (usually associated with frontal or deep convective clouds). The higher biases for lower fc (usually associated with thin cirrus and broken clouds) are likely related to higher noise and uncertainties in OMI cloud parameters. For example, Joiner et al. (2012) showed that cloud OCP errors increase with decreasing fc. The biases may also be related to absorbing aerosol in the scene, particularly when it overlies clouds. This will be illustrated in more detail below as we show spatial variations in OMI–CERES differences.

In order to evaluate the NN performance at different spatial and temporal scales similar to those used by the climate community, we use data from July 2007. Figure 8 presents a comparison of daily CERES and OMI TOA SWF over ocean for six spatial scales: the OMI native pixel (13 × 24 km2 at nadir) and 0.5, 1, 2, 5, and 10∘ gridded spatial resolutions. Statistical parameters for these comparisons are reported in Table 2. As expected, the pixel level data are much noisier than the gridded data due to collocation noise in partly cloudy cases. This collocation noise averages out over larger spatial and temporal scales. Regardless of the noise, the slope (0.96) is still close to 1, and the linear correlation coefficient is 0.96 with a standard deviation of 47.7 Wm-2. Below 300 Wm-2, where the sample density is highest, the NN slightly underestimates the CERES SWF. The mean bias of the OMI-estimated SWF with respect to CERES is -1.4 Wm-2. This bias may be due to a combination of effects including uncertainties in the input parameters as well as the limitations of the NN model itself. For example, we have excluded pixels with a clear signature of absorbing aerosols (OMI-derived AI > 1) where OMI ECFs and pressures may be in error in both the training and validation data. However, in some regions where smoke and dust overlaying clouds is common (e.g., western coast of Africa), pixels with erroneous cloud fractions due to small amounts of absorbing aerosol may be present in both the training and validation data. This may produce errors in the NN model and will be examined in more detail below.

For the daily data, as the spatial averaging scales increase from 0.5 to 10∘, the OMI-estimated SWF becomes almost identical to CERES; the correlation coefficient increases from 0.97 to 0.99, and the slope increases from 0.96 to 0.97. The percentage of OMI data that fall within ±5 % of CERES increases from 37 % for 0.5∘ to 69 % for 10∘ grids. About 87 % percent of OMI-estimated 2∘ gridded daily mean TOA SWFs are within 15 % of CERES data.

Figure 9a–d show 2-D histograms of monthly mean gridded data over ocean at 0.5, 1, 2, and 5∘ spatial resolutions, respectively. The monthly intercomparisons of OMI and CERES SWF show excellent agreement at all spatial resolutions with correlation coefficients of 0.99 and slopes of 0.98 (Table 2). The global mean biases vary between -1.8 and 0.25 Wm-2. The standard deviations vary between 6.6 and 12.9 Wm-2 for the different spatial resolutions. Of monthly mean 1∘ gridded OMI estimated TOA SWFs, 97 % are within 15 % of those derived from CERES and 93 % are within 10 %.

Figure 10 presents the spatial distribution of 1∘ gridded monthly mean (July 2007) TOA SWF from CERES (Fig. 10a) and the difference with the OMI in Wm-2 (Fig. 10b) and percent difference (Fig. 10c). There are subtle differences between the NN and CERES estimates of TOA SWF as shown in Fig. 10b and c. The OMI minus CERES histograms (Fig. 10d) show that for 44 % (79 %) samples, NN fluxes are within ±2 (±5) % of CERES fluxes. About 9 % of the samples have biases of ±8 % or more. Overall, the Northern Hemisphere shows better agreement than the Southern Hemisphere during July (boreal summer). This could be due to larger errors in the OMI cloud products at higher SZAs in the Southern Hemisphere. The low biases on the western coast of Africa may be due to the presence of absorbing aerosols, particularly when they occur over clouds. The striped pattern in the Southern Hemisphere (latitudes > 40∘ S) is mainly associated with high VZAs in conjunction with high SZAs that occur on one side of the OMI swath.

Figure 11 similarly shows differences between CERES and OMI TOA SWF over ocean derived using fc and OCP from the OMI O2–O2 product in place of the OMI RRS product. Because there are slight differences in the two cloud products, we retrained the network with OMI O2–O2 cloud parameters to be consistent. The use of the O2–O2 fc and OCP improves the accuracy of the estimated TOA SWF. The regions of improvement include the western coasts of South America and southern Africa and some parts of the Indian Ocean. The O2–O2 cloud product, which uses visible wavelengths, is less affected by absorbing aerosol; this may explain the improvement in these areas where absorbing aerosol, especially over clouds, is common. However, negative biases remain over large regions off the western coast of Africa.

Figure 12a shows a time series of daily global mean values of TOA SWF over ocean from OMI and CERES for 2007. Both instruments show almost identical daily variations with differences within ±1 %. Figure 12b and c provide monthly averaged (July 2007) CERES and OMI zonal and meridional means of TOA SWF. The OMI-derived TOA SWF is able to reproduce the variability shown in the CERES data well.

We developed a similar land-only NN model that utilizes most of the input parameters from our ocean NN (e.g., OMI RRS cloud parameters). The only change is that for surface characterization we use a monthly climatology of surface broadband albedo in place of the Chl concentration and surface wind speed. The albedo product is derived using a combination of CERES and MODIS observations at 1∘ spatial resolution (Rutan et al., 2009). The land NN model is trained and validated over land only and it follows a similar approach as was used over ocean but with a slightly different set of input parameters. We have not performed extended analyses and testing over land as we have over ocean. Here, we demonstrate that the same methodology with different input parameters can reproduce CERES TOA SWF over land with similar accuracy and precision as was obtained over ocean.

Figure 13 shows results from the OMI-derived CERES-trained NN that produces TOA SWF over land. Statistical comparison with CERES over land provides results similar to those over ocean. The NN performs well over Asia and parts of Europe and the Americas. The OMI-based NN tends to underestimate TOA SWF over the high albedo desert areas of Northern Africa, Australia, and also over some regions of South America. Note that the large differences that occur in coastal regions may be due to imperfect collocations.