## 1. Introduction

The Clouds and the Earth’s Radiant Energy System (CERES) instruments (Wielicki et al. 1996) that measure broadband shortwave (reflected) and longwave (emitted) radiance at the top of the atmosphere (TOA) fly on board the National Aeronautics and Space Administration (NASA) Tropical Rainfall Measurement Mission (TRMM), Earth Observing System (EOS) *Terra* and *Aqua*, and the Suomi National Polar-Orbiting Partnership (Suomi NPP) satellites. The level 2 CERES data product, the clouds and radiative swath (CRS) (Charlock et al. 1997, 2006), contains modeled irradiances computed by a two-stream radiative transfer model for nearly all CERES footprints. Because estimated global surface irradiance often relies on satellite observations, computations test the accuracy of modeled irradiance by such a radiative transfer model (Wielicki et al. 1995). The CERES project derives TOA irradiances from observed radiances using angular distribution models (Loeb et al. 2005). Irradiances computed for a CERES footprint do not necessarily agree with the CERES-derived irradiance. This difference is caused by many reasons, including assumptions used in the radiative transfer model, errors in inputs such as satellite-derived cloud and aerosol properties, as well as erroneous temperature and humidity profiles. If model inputs are adjusted within estimated uncertainties and constrained by CERES-derived TOA irradiances, then the accuracy of computed irradiance at the surface might improve. A basic assumption of this constrainment method is that the difference of computed and CERES-derived irradiances is determined by the error in input used in irradiance computations.

When CERES data products were formulated in the early 1990s, it was decided that in addition to top-of-atmosphere radiance observations and irradiances derived from them, the team would release additional cloud and radiative transfer model results as well. The desire was to model both surface and in-atmosphere irradiances following two paths to the surface. The first path would be to include parameterized models that derive surface downwelling and net irradiances directly from TOA CERES observations. These values are included in the CERES single satellite footprint (SSF) product and use algorithms developed by several groups (Gupta et al. 2001, 2010; Inamdar and Ramanathan 1997; Zhou et al. 2007). Known as surface-only flux algorithms (SOFA), they were assessed in several papers (Gupta et al. 2004; Kratz et al. 2010) and are available in the SSF product. Because of the uncertainty implicit in TOA to surface parameterizations, a second path was also chosen. This was to directly calculate the radiative irradiance profile using a fast radiative transfer model with high vertical resolution. These results would be put into a separate data product, the CRS, that would contain both the SSF record as well as output from the full radiative transfer code at a subset of the original vertical resolution. Several models were considered at that time, with the group settling on the model developed by Qiang Fu and Kuo-nan Liou (Fu and Liou 1993). Given the improvements in the then-new CERES instruments over the Earth Radiation Budget Experiment (ERBE) instruments and the limited knowledge of the uncertainties in TOA radiative transfer estimates, it was decided to include an algorithm that would utilize CERES TOA irradiance as input to a constrainment algorithm and adjust model inputs to better match the TOA observations and hence potentially improve the in-atmosphere irradiance profile (including surface downwelling irradiance) and/or the model inputs. Such an algorithm would necessarily take into consideration the uncertainty of model inputs, the model itself, and the uncertainty of CERES-derived TOA irradiance for an individual footprint to which the model is being matched [now known to be approximately 5% (Loeb et al. 2007)]. The purpose of this paper is, therefore, 1) to describe the CRS Edition 2 algorithm to constrain computed irradiances by CERES-derived TOA irradiances and 2) to assess whether the constraint applied to instantaneous irradiances improves the bias and/or standard deviations between computed irradiances and observations at both the TOA and surface. Our conclusions are that the algorithm does improve the model results at the TOA when compared to CERES-derived irradiance. The algorithm, however, does not improve model-calculated downwelling irradiance compared to observations at the surface. Results comparing model versus observed irradiance within the atmospheric column await the development of an observation system capable of such measurements.

The paper is outlined as follows. Section 1 briefly describes the CERES instruments and the radiative transfer model used in CRS calculations. Section 2 details the use of a Lagrangian multiplier minimization technique for our purposes. Section 3 discusses the results of initial and constrained calculations at TOA and shows the improvement to model results with respect to TOA observations. We also discuss the impact of tuning on surface calculations, which is somewhat detrimental to shortwave (SW) and negligible on longwave (LW) irradiances, though overall it does not significantly affect our model comparisons to surface flux observations.

## 2. CERES CRS product description

### a. CERES instruments

CERES instruments measure broadband radiance in three channels: SW reflected (mostly from 0.2 to 4.0 μm), “total broadband” (mostly 0.2–100.0 μm), and in the window range of ~8.0–12.0 μm. LW-emitted radiance (mostly 4.0–100.0 μm) is inferred as the difference between the total and SW channels. CERES’s preflight model (PFM) operated for 8 months on board the TRMM satellite in 1998. Five other CERES instruments are currently on board NASA EOS satellites: *Terra* flight models 1 and 2 (FM1 and FM2, respectively), *Aqua* flight models 3 and 4 (FM3 and FM4, respectively), and the Suomi NPP satellite (FM5). An unfiltering process described in Loeb et al. (2001) takes into account CERES instrument filtering functions and produces shortwave and longwave unfiltered radiances. The unfiltered shortwave irradiance includes reflected radiance by the earth for all wavelengths. The unfiltered longwave irradiance includes emitted radiance by the earth for all wavelengths.

Because direct measurements of irradiance upwelling to a satellite can represent the earth radiation budget (ERB) accurately at only very coarse time and space resolutions (Smith and Green 1981), it is preferable to scan for radiance and infer irradiance by applying scene-dependent, empirically based statistical models (Suttles et al. 1988; Loeb et al. 2005). CERES surpasses ERBE in this regard, both by inferring the scene from high-resolution Moderate Resolution Imaging Spectroradiometer (MODIS) data (Minnis et al. 2011a,b) and by applying fairly robust angular distribution models (ADMs; Loeb et al. 2005) founded on the complex CERES rotating azimuth plane scan (RAPS) mode to establish bidirectional reflectance distribution functions (BDRF) on the scale of ~30-km broadband footprints; no other EOS instrument can employ RAPS to buttress the products of its routine cross-track scan. The detectors themselves and the conversion of radiance to irradiance each inherently introduce uncertainty into the CERES archived irradiance values. For this reason our algorithm does not attempt to match CERES irradiance values exactly but instead use that uncertainty as a constraint within the adjustment procedure.

### b. Unconstrained irradiance computations in CRS

One task of the CERES project is to calculate vertical profiles of heating rates, globally and continuously beneath CERES footprint observations of TOA irradiances. (Charlock et al. 1997, 2006) This is accomplished using a fast radiative transfer code originally developed by Qiang Fu and Kuo-Nan Liou (Fu and Liou 1993) and subsequently modified by the CERES team and known colloquially as “Langley, Fu, and Liou 2005” (LFL05) (Fu and Liou 1993; Fu et al. 1998; Kratz and Rose 1999; Kato et al. 1999, 2005).

The SW portion of the LFL05 model is a delta-two-stream radiation transfer code with 15 spectral bands from 0.175 to 4.0 *μ*m. Cloud optical properties within the CERES field of view are defined separately in the CERES SSF data product. In the SSF, MODIS imager pixels are collocated within the larger CERES footprint and weighted by the energy distribution of the CERES instrument (Green and Wielicki 1997). From MODIS spectral radiances, cloud properties, including fractional cloud cover, cloud optical depth, and particle size and phase, are derived and used as input for the radiative transfer model (Minnis et al. 1995, 2011a,b). Operationally, the LFL05 code is run for up to three conditions for a given footprint: a clear-sky portion and clouds at two levels, a low and a high cloud. Low and high clouds are defined relative to each other and do not have any cloud height bounds for each cloud. Cloud optical thickness is expressed as a gamma distribution estimated from the linear and logarithmic cloud optical thickness means (Barker 1996; Oreopoulos and Barker 1999; Kato et al. 2005). Once the distribution of cloud optical thickness is estimated for each cloud type, the gamma-weighted two-stream radiative transfer model (Kato et al. 2005) is used to compute the shortwave irradiance for each cloud.

Aerosol optical depths are input from MODIS (MOD04) product (Remer et al. 2005). When lacking the instantaneous MOD04 product in a CERES footprint, the first alternate is the daily, gridded MOD08; and the second is the Model for Atmospheric Transport and Chemistry (MATCH) (Collins et al. 2001), supplied by Dr. David Fillmore in Boulder, Colorado. MATCH is a global aerosol model that assimilates MODIS retrievals of aerosol optical depth (AOD). The MATCH is also used to define aerosol constituents using Optical Properties of Aerosols and Clouds (OPAC) from Hess et al. (1998) and updated Tegen and Lacis (1996) desert dust properties supplied courtesy of Andrew Lacis of the Goddard Institute for Space Studies (GISS). MATCH is also used to place aerosols throughout the vertical atmospheric column. Pressure, temperature, and water vapor profiles are specified from Goddard Earth Observing System version 4 (GEOS-4) (Bloom et al. 2005), a 1° gridded reanalysis data product from the Goddard Modeling and Assimilation Office. Ozone comes from the National Centers for Environmental Prediction (NCEP)’s Stratosphere Monitoring Ozone Blended Analysis (SMOBA) product (Yang et al. 1998).

The LW portion of the code uses an innovative two-/four-stream method. The two-/four-stream version (Fu et al. 1998) uses the faster two-stream LW source function, but places it in the four-stream framework for accuracy comparable to straight four-stream calculations. The code accounts for the radiative effects of H_{2}O, CO_{2}, O_{3}, O_{2}, CH_{4}, and N_{2}O; Rayleigh scattering; aerosols; liquid cloud droplets; hexagonal ice crystals (random orientation); and spectrally dependent surface reflectivity. Twelve spectral intervals are used in the LW (0–2850 cm^{−1}, or 3.5 to >100 *μ*m). The water vapor continuum currently used is the Clough–Kneizys–Davies (CKD) model version 2.4. The uniform mixing ratios for CO_{2}, CH_{4}, and N_{2}O are 360, 1.6, and 0.28 ppmv, respectively. In the Edition 2 CRS data product well-mixed gases are left constant throughout the time series. Sensitivity studies show that the effect of increased CO_{2} has less than 1 W m^{−2} effect across the 10 years of the data product. It is anticipated that these gases will be variable in time for an Edition 4 of the product when it is run. For the principal atmospheric gases, the Fu and Liou (1993) code matches a line-by-line (LBL) simulation of irradiance to within 0.05% for SW, 0.2% for LW excepting O_{3}, and ~2% for LW fluxes due to O_{3}. An updated line database is not expected to change results substantially. It should be noted, however, that there are significant uncertainties relating to the treatment of the H_{2}O continuum (i.e., Clough et al. 1992) in radiative transfer codes generally. The LFL05 model is typically run operationally using up to 36 atmospheric levels for over 2 million CERES footprints every day, and the CRS data product is now available for over 10 years of CERES observations.

Each CRS file contains 1 h of CERES footprint data and includes CERES TOA–observed radiances and irradiances and an analysis of those MODIS pixels collocated within the larger CERES footprint, which supply cloud properties inside the footprint. Along with CERES TOA irradiance and cloud analyses, CRS data files include column fluxes at five atmospheric pressure levels (surface: 500, 200, 70 hPa; and TOA: 0.1 hPa). These irradiances are a subset of the radiation transfer calculations, which were executed at higher vertical resolutions and under various atmospheric conditions to study aerosol and cloud effects on column radiative transfer. These conditions include pristine (no aerosol and no cloud), clear sky (no clouds), with clouds (no aerosol), and complete atmospheric conditions (all sky). Results for all of these separate runs are included in the CRS data product. An important point for understanding the CRS data product is that the irradiances provided at the five atmospheric pressure levels are irradiances produced from a second set of computations after the initial model inputs have been processed through a mathematical algorithm (constrainment or “tuning”) process. In short, LFL05 is run for a footprint; computation results at TOA are compared to CERES observations; and a tuning algorithm processes this information, changing certain model input variables (depending on atmospheric state), and then LFL05 is rerun with the new inputs. It is the results of this second computation that are archived on the CRS along with the amount by which the initial computed irradiance was changed. CRS also archives the initial and tuned values of the model input parameters.

### c. Constrained irradiance computations

We develop a constrainment algorithm that minimizes the difference between computed and CERES-derived irradiances using Lagrangian multipliers. The irradiance adjustment is based on the relation among the differences of computed and CERES-derived TOA SW, LW, window (WN) irradiances and TOA WN and LW radiances; a set of appropriate model input uncertainties; and the sensitivity of irradiances to all inputs. The steps in the process to adjust irradiances are as follows:

- Define the range of likely uncertainty for all radiative transfer model inputs and the CERES-observed irradiances.
- Precalculate tables of derivatives that encompass the expected effect of changing input variables on TOA computed irradiances.
- Select a subset of input variables (call them tuning variables) appropriate for the atmospheric state.
- Determine the magnitude of the adjustment for each tuning variable so as to better match the observed TOA fluxes.
- Ensure that no variable is overadjusted relative to its predefined uncertainty.
- Rerun the radiative transfer model using the adjusted model inputs to create a “tuned” computed irradiance.

An additional requirement for the process is that it also needs to be computationally inexpensive, because this process is used in an operational code and run over 2 million times to process one day of data. Because of this requirement, we limit the number of variables adjusted in step 3. We can reduce computation time by selecting variables that either likely contain a large uncertainty or to which TOA irradiances are relatively sensitive when adjusted. Because the uncertainty in imager-derived cloud properties and the sensitivity of TOA irradiances changes with atmospheric conditions, the selection of variables to be adjusted must be a function of the surface/atmospheric state. For example, TOA outgoing longwave radiation (OLR) is more sensitive to surface temperature when the atmosphere is dryer. For cloud-free dry conditions, therefore, skin temperature error is more likely to cause OLR error than atmospheric temperature error. Hence, the skin temperature is adjusted more favorably than the atmospheric temperature. However, in a warm moist atmosphere under clear-sky conditions, OLR is less sensitive to the surface temperature, so that atmospheric temperature or humidity would be adjusted more favorably. In a moist and cloudy situation, the humidity profile and cloud properties, respectively, are primarily adjusted because of their larger uncertainties than temperature uncertainties. Similar selections are done for shortwave irradiance-constrained calculations. The terms “constrain” and “tune” are used interchangeably throughout the text.

In practice, longwave and shortwave irradiances are constrained together by the CERES-derived longwave and shortwave irradiances. By adjusting longwave and shortwave irradiances together, crude spectral information is used to select variables to be adjusted. Adjustments of inputs must be done objectively using knowledge of the input uncertainties of temperature and humidity profiles as well as imager-derived cloud and aerosol properties. In addition, resulting input variables must be physically plausible, such as surface albedos less than 1.0.

*F*represent Δ

_{k}*F*, Δ

_{s}*F*, and Δ

_{L}*F*, which are the differences between the observed and computed upward SW, LW, and CERES window irradiances at the TOA, respectively. We assume that these differences can be expressed according to the following relationship:The variables in Eq. (1) are defined as follows:

_{W}*i*—cloud conditions, current maximum of three, clear, and two cloud levels within any footprint (i.e.,

*i*≤ 3);

*j—*the number of input variables to be adjusted (

*j*varies as a function of scene as described above);

*k—*TOA flux variables; currently three: upward SW, LW, and CERES window flux;

*C*—the fraction of footprint covered with cloud condition

_{i}*I*;

*υ*—model input variable changed to better match model to observed irradiance (see Table 1);

_{j}*F*—irradiance at TOA defined by subscript

*k*;

*δC*—the difference between initial and new tuned (to be determined) cloud condition;

_{i}*δυ*—the difference between initial and new tuned (to be determined) input variable;

_{i}*δF*—the maximum allowed absolute difference in TOA irradiance

_{k}*k*; and Δ

*F*—the difference between initial computed and observed TOA irradiance

_{k}*k*.

Sigma uncertainties for TOA irradiance and radiance observations.

Equation (1) simply states that the differences between computed and CERES-derived TOA irradiances Δ*F _{k}* are a result of differences between the true atmospheric state and the initial atmospheric state specified by the inputs (

*δυ*, temperature, humidity, etc.), the cloud condition derived by the CERES cloud algorithm

_{i}*δC*, and the uncertainty in the CERES-derived irradiance

_{i}*δF*. To control the amount by which any of the variables might be adjusted, we introduce an uncertainty parameter σ for each input variable in Eq. (1) that we allow to be tuned. Derivation and the values of σ parameters are discussed in more detail in section 2d. For brevity we continue with a matrix formulation of the equations and give a simple example for clarity in appendix A.

_{k}*c*is the cloud condition adjustment normalized by the uncertainty parameter of cloud condition; δ

_{i}*i*th cloud, and the column vectors

_{i}is made up of the partial derivatives of irradiances with respect to each adjustment variable. These partial derivatives are calculated in advance and maintained as tables in the code. Certain variables are transformed into log space before derivatives are calculated to minimize the nonlinear effects of rapidly changing derivative functions. They are labeled in Tables 2 and 3 with the term “log derivative.” Irradiance is computed separately for each cloud condition (up to two cloud types plus the clear area in a CERES footprint) and is accounted for by the cloud and clear area fraction

*C*. Atmospheric-state variables used for computations are the same for all cloud types and the clear-sky area within a CERES footprint and so are adjusted the same amount for all cloud types. The additional constraint of ∑

_{i}*δC*= 0 is used so that the total cloud/clear fraction remains equal to one. The goal of the constrainment algorithm is to minimize changes to model input variables while simultaneously reducing Δ

_{i}**F**determined by 1D radiative transfer theory provided the uncertainty of input variables and TOA irradiances. To do this we minimize the sum of the squares of the pertinent variables

*Z*as defined byconstrained by the set of equations defined by Eq. (2). Equation (3) results from the assumption that the error space in which each variable exists is normally distributed and independent of the other tuning variables (see appendix A). This independence is an underlying assumption of the algorithm, should be noted, and is a potential source of error for the algorithm.

Sigma uncertainties for ocean/atmosphere parameters.

Sigma uncertainties for land/atmosphere parameters.

*Z*, given the difference between computed and CERES-derived TOA irradiances (i.e., Δ

*F*). Letting

_{k}*X*= ∑δ

*C*we set up Eq. (4) aswhere

_{i}**L**

^{T}= [

*λ*

_{c}λ_{1}

*λ*

_{2}], and λ

_{0}are Lagrangian multipliers. We take the derivative of

*Y*with respect to each Lagrangian multiplier and with respect to each element of δ

*c*, δ

_{i}_{0}and

**L**

^{T}, as well as δ

*c*, δ

_{i}*F*though only at the TOA and surface. Adjusted amounts of the three midatmospheric levels are not included to decrease the total data volume on the CRS product. To recover the untuned TOA and surface irradiances, one simply calculates

### d. Input variable uncertainties

Tables 1 through 3 show estimates for the *Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations* (*CALIPSO*) and *CloudSat* observations provide an opportunity to assess retrieved cloud properties from passive instruments. We will incorporate such information into the future version. Table 1 gives estimates of the uncertainty in the CERES-observed TOA irradiances, which come primarily from the inversion of observed radiance into irradiance using the CERES ADMs. These models and uncertainties are described in Loeb et al. (2005, 2007), where they show that the uncertainty of instantaneous irradiance is about 5%, while the uncertainty decreases when irradiances derived from a wide range of viewing and relative azimuth angles are averaged. Uncertainties in CERES-derived irradiance are better defined in an aggregate sense depending on region, scene type, season, and viewing zenith angle of CERES observations (Loeb et al. 2007). Distinguishing error as a function of scene type within the constrainment algorithm, however, would add a complication to the method. Therefore, we assume 1% for TOA SW and LW irradiance uncertainty to represent the entire globe regardless of scene viewed for this study. Scene- and regional-type-dependent uncertainty will be treated in the future version of CRS.

In Tables 2 and 3 we separate the *Journal of Geophysical Research**,* 1992, Vol. 97, Issue D17). The latter is due to the existence of a temperature gradient in a vegetation canopy, with different view angles exposing different temperatures (Minnis et al. 2004). The land–sea contrast is also apparent in the

Uncertainties in AOD are more complex, as they are a function of the magnitude of the total AOD. For example, Ignatov and Stowe (2000) gave the uncertainty of a single-channel Visible and Infrared Scanner (VIRS on TRMM) as *δτ =* ±(0.04–0.08) ± 0.35*τ*_{1} over open ocean and possibly larger over coastlines. More recent work (Remer et al. 2005) summarizes the uncertainties found by multichannel approaches using the MODIS instrument. These studies point to significant improvements in AOD retrieval uncertainties, giving *δτ* = ±0.03 ± 0.05*τ* and *δτ* = ±0.05 ± 0.15*τ* over ocean and land, respectively. Given these numbers the sigma uncertainty of 50% for AOD over ocean may seem surprising. A larger sigma allows the adjustment to be larger. Over the ocean we have more confidence in the surface albedo from the COART model and so we keep the sigma very low, 0.002 (in albedo units, so ~3.3% for an ocean albedo of 0.06), thus not allowing much change in the ocean albedo. Given that small constraint on ocean albedo then, the algorithm must be allowed to change other variables to match the CERES TOA irradiance. So, even though the 50% AOD is greater than known uncertainties in MODIS-based estimates, we allow the AOD to vary the most, as we believe it is the most uncertain parameter within the subset of variables used when tuning the radiative transfer equation over clear-sky ocean scenes.

Cloud uncertainties are estimated from knowledge of the derivation of cloud properties using MODIS imager pixels collocated within CERES footprints. Edition 2 cloud algorithms were developed by the CERES “cloud” group led by Dr. Pat Minnis. Descriptions of these algorithms and their uncertainties are found in Minnis et al. (2011a,b). Uncertainties of cloud properties listed in Table 3 are comparable to the difference of cloud properties derived from a cloud algorithm using passive and active sensors and an algorithm using a passive sensor only discussed in Kato et al. (2011).

## 3. Impact of tuning algorithm on TOA and surface irradiance computation, results compared to observations

### a. Comparison at TOA

In this section we present operational CRS computation results, both tuned and untuned, compared to observations by the CERES instruments at the TOA and to surface observations at a selected number of surface sites. Representing typical results we show in Fig. 1 approximately 2.2 million footprints processed for one day of CERES/CRS processing from the *Terra* FM1 instrument on 1 May 2006. Top and bottom plots are, respectively, observed shortwave and longwave irradiance compared to the difference of computed-minus-observed CERES irradiance. Initial computations are on the left and TOA irradiance after execution of the constrainment code to determine adjusted input variables is on the right. One can easily see the reduction in variance after the tuning model is run using the constrained variables, and the reduction of bias and standard deviation are verified in the statistics. The TOA SW irradiance bias is reduced by almost a factor of 4 with the standard deviation being halved. TOA LW bias is reduced by a factor of 2, and variance is nearly halved. This is what the algorithm was designed to do and it does, in fact, work rather well.

However, breaking the statistics down as a function of scene type reveals improvement is not universal. Results are shown in Tables 4 and 5 for the TOA SW and TOA LW comparisons, respectively. The largest improvements in SW bias occur for footprints whose cloud condition is defined by either overcast ice clouds, overcast water clouds, or cloud-free land. Since the first two scene types dominate the sample, overall reduction in bias and standard deviation resembles these scenes. For the cryosphere scenes, the bias increases slightly and is of opposite sign, allowing for some cancellation of errors in the aggregate. Standard deviation is reduced by nearly half in all scene types except clear cryosphere, where it is reduced by about 33%.

TOA upward SW irradiance comparison (computed − observed for 1 May 2006, all in W m^{−2} except for footprint count in column 2).

As in Table 4, but for TOA upward LW irradiance (W m^{−2} except in column 2).

The algorithm is, of course, sensitive to which variables can be adjusted. Hence, though we know we tend to underestimate clear-sky snow surface albedo over clear cryosphere scenes, there is often neither much water vapor nor aerosol for adjustment purposes and so the algorithm has less ability to improve the standard deviation of TOA SW irradiance for those footprints. What improvement does arise comes primarily from changes to the surface albedo.

Model calculations compared to CERES observations for TOA outgoing longwave radiation tells a similar story (Table 5). Again, standard deviation is reduced for all scene types and changes to bias depending on the ability to adjust input variables. Note that here the reduction for the cryosphere bias is slightly better for LW irradiance because small total PW over Antarctica combined with adjustments to surface skin temperature can have a larger impact on TOA LW irradiance than over lower latitudes where total PW is much higher.

Tables 4 and 5 give results for a single day. To ensure these results are valid for longer time periods, Tables 6 and 7 show the SW and LW TOA bias and standard deviations for several different scene types for the entire year of 2007. While the biases change slightly, the standard deviation and its reduction by the constrainment algorithm remain remarkably stable.

TOA upward SW irradiance comparison (computed − observed for 2007, all in W m^{−2} except footprint count in column 2).

TOA upward LW irradiance comparison (computed − observed for 2007, all in W m^{−2} except for footprint count in column 2).

The adjustments by the tuning algorithm to the model inputs are also of interest. Figure 2 shows the effect of the constrainment algorithm on the adjustable cloud variables: cloud fraction (Fig. 2a), cloud optical depth (Fig. 2b), and cloud effective temperature (Fig. 2c). The red bars indicate the initial distribution of the cloud variables and the blue line shows the distribution after adjustments are made. Recall that these adjustments are solved for simultaneously, so that for any given footprint all three variables might be adjusted. The cloud fraction σ is only 2.5% and so is not adjusted much compared to other cloud variables. The optical thickness distribution of occurrence increased for the thinner clouds and decreased for thicker clouds. The cloud effective temperature tends to decrease for higher-temperature clouds and increase for lower-temperature clouds. The mean results for daytime water clouds are summarized in Table 8. Overall, the algorithm tends to decrease the cloud amount slightly and reduce the optical thickness of the clouds by about 7% while at the same time “raising” the clouds in the vertical by reducing their mean effective temperature by approximately 0.3%.

Initial mean value and bias/standard deviation after constrainment algorithm is run for cloud variables for one day of processing, 1 May 2006.

### b. Comparison at the surface

The constrainment algorithm is designed to reduce the variance of model calculations with TOA CERES observations. It does not take the resultant surface irradiance calculations into account in any way. This allows the use of surface observations as an independent measure as to the robustness on the adjustments made to the input variables. To see the impact of the adjusted variables on calculated surface irradiance, we collect observed surface irradiance from a number of sources around the globe and extract a validation dataset of model calculations from the CRS data product for all CERES footprints near these surface sites. Surface site locations are detailed in appendix B. Instantaneous comparisons of model calculations to surface observations can lead to large variances, especially in the SW, simply because of differences in what a satellite is viewing by “looking down” while a surface radiometer is “looking up.” One can easily imagine a situation where the scene viewed by the satellite indicates broken clouds, but one of those clouds is shadowing the pyrheliometer viewing the direct solar radiation. This can lead to instantaneous differences on the order of hundreds of watts per square meter. Nor do we address issues related to absolute uncertainties of surface instrumentation. For instance, it has been shown that pyranometers can vary by up to 2% (high or low as a function of instrument manufacture) when set side by side under ideal conditions (Michalsky et al. 2005). As we are less interested in these potentially large variances caused by mismatched views or smaller uncertainties due to instrumentation and more interested in the effect of the constrainment algorithm upon the computations, for comparisons at the surface we convert bias and standard deviations to percentages for comparison in Tables 9 through 12. Note also, where at the TOA we compared nearly every footprint across 1 day and 1 year, for the surface comparisons we collect data across almost 10 years since the sampling is significantly less, as the sun-synchronous orbit limits the number of observations at surface sites.

Monthly-mean surface downward all-sky shortwave initial and constrained calculations compared (computed − observed) to surface observations* for different surface scene types over 10 yr (March 2000–February 2010).

Individual surface irradiance calculations from the CRS product are compared to observed surface irradiance in the following way. CERES footprints used are those that fall within 15 km in space and 7 min in time to a surface observation site where the surface observations have been averaged over 15 min. To account for slight changes in solar zenith angle within those 7 min, SW calculations are adjusted by the ratio of the solar zenith angle at the CERES observation time to the solar zenith angle at the surface observation time. These footprint comparisons are then averaged over a month to provide a monthly-mean value from which the statistics in Tables 9 through 12 are derived. Note that these are monthly averages for the times of the *Terra* satellite overpass of the surface sites, not true diurnal monthly-mean values. The *N* in the second column of each table indicates the number of months available for comparison from all the sites indicated in column 1. Comparisons run across 10 years of CERES CRS data, from March 2000 through February 2010. The term “clear sky” is a function of the cloud condition identified in the CERES observation, where all MODIS pixels collocated inside the CERES footprint are identified by the CERES cloud algorithm as clear. It does not guarantee that the sky observed by the surface radiometer is 100% clear. Figure 3 shows the distribution of the monthly-mean bias for SW insolation for 10 land surface site locations listed in appendix B. One immediately finds the tuned distribution shifts relative to untuned with a commensurate change in bias and standard deviation. Tables 9 through 12 give statistics quantifying the effect of the constrainment algorithm on the surface irradiance calculations. Tables 9 and 10 show the difference between observed and computed SW irradiances (computed − observed) at the surface at 24 locations grouped, generally, by surface scene type. For each group in all-sky conditions (Table 9), we find an increase in bias ranging from a low of 0.1% over deserts to a high of 1.7% over the island sites and increases in standard deviation with a low again of 0.1% over the desert sites and a high of 2% over polar sites. It is encouraging that, except for the polar locations, standard deviations are increasing by only a fraction of a percent.

As in Table 9, but for CERES clear-sky footprints.

The clear-sky surface shortwave comparisons in Table 10 show less effect of the tuning algorithm. This is primarily due to a decrease in the complexity of the radiation transfer calculations once clouds are removed from the equation. There is virtually no change, neither in bias nor standard deviation regardless of surface conditions.

Results for monthly-mean LW surface observations compared to model calculations are shown in Tables 11 and 12, again for all-sky and clear sky CERES footprints, respectively. The impact of the constrainment algorithm on the downward longwave flux is less than that for surface SW insolation. For all scenes and sky conditions, the bias either remains the same or slightly decreases, again except for the polar sites, while the standard deviation increases by a maximum of 0.3% for those same polar locations. The downward longwave irradiance is largely driven by near-surface atmospheric and cloud properties, while the TOA longwave irradiance is more sensitive to upper-troposphere atmospheric and cloud properties. Therefore, the difference between computed and observed TOA irradiance does not provide much information that directly impacts the surface LW irradiance. Hence, adjusting cloud condition and upper-troposphere atmospheric properties, to which the TOA longwave irradiance is more sensitive, has much less impact on the longwave surface irradiance.

Monthly-mean surface downward all-sky longwave initial and constrained calculations compared to surface observations for different surface scene types over 10 yr (Mar 2000 through Feb 2010.)

As in Table 11, but for CERES clear-sky footprints.

## 4. Discussion

The tuning algorithm, as it is currently implemented, in the Edition 2 CERES CRS data product is designed to reduce the computed variance at the TOA compared with observations. The intent of the algorithm was that if one could decrease the uncertainty of the calculations at the TOA, then one might also decrease the uncertainty through the atmospheric column (including the surface) by adjusting inputs in a quantifiable way, such that TOA model results better matched values observed by the CERES instruments at the TOA. While the agreement at TOA with CERES-derived irradiance improves significantly after the tuning is applied, the comparison with surface observations shows that neither bias nor variance decrease substantially with the constraint. There are several reasons for this result. First, surface irradiances computed by a two-stream radiative transfer model using satellite-derived cloud and aerosol properties and atmospheric properties from reanalysis agree with CERES-derived TOA irradiance and observed surface irradiances relatively well. The bias (standard deviation) of TOA instantaneous irradiance of 2.8 (15.7) W m^{−2} for shortwave and 0.7 (6.2) W m^{−2} for longwave is better than the bias of 11 (7) W m^{−2} for shortwave and −1 (4) W m^{−2} for longwave reported by Rossow and Zhang (1995), who used monthly-mean irradiances to compare ERBE-derived TOA irradiances. The bias (standard deviation) for the surface downward shortwave irradiance is 11 (22) W m^{−2} and for the downward longwave irradiance it is −6 (7) W m^{−2}, while values reported by Rossow and Zhang are 10–20 W m^{−2} for shortwave and less than 15 W m^{−2} for longwave. Monthly-mean irradiance differences given by Rossow and Zhang (1995) for their comparison are equivalent or larger than our differences computed with instantaneous irradiances at TOA and the surface. Improved gaseous absorption treatment in the radiative transfer model, more accurate auxiliary data such as surface albedo and emissivity maps and snow ice maps, and improved retrievals of cloud and aerosol properties from MODIS are the probable reasons for the improvement.

Other potential reasons for a lack of improvement of surface irradiance are in regard to how the constraint algorithm is applied. 1) The constraint algorithm is applied to instantaneous irradiance computations. Given that the uncertainty in instantaneous CERES-derived irradiances, and inputs used to compute instantaneous irradiances are large so that the constraint is not effective at the surface. Using temporally and spatially averaged TOA irradiances or radiances allows us to use smaller uncertainties, which subsequently allow for tighter constraints. The CERES team is experimenting with such constraints. A constraint by monthly-mean and regional mean TOA irradiance is, therefore, expected to provide a better result. 2) Uniform uncertainty values of surface, cloud, and atmospheric variables for the entire globe give unrealistic uncertainty estimate in some regions. 3) Bias error of input variables are not taken out before the Lagrangian multiplier procedure is applied. While many studies of cloud retrieval errors using 1D radiative transfer theory are performed, observational estimates of retrieved cloud fraction and cloud height only become possible after *CALIPSO* and *CloudSat* data are made available. *CALIPSO* and *CloudSat* and data from the A-Train constellation will help to drive temporal- and regional-dependent uncertainty of surface, cloud, and atmospheric properties. For example, Kato et al. (2011) quantitatively showed that the cloud-base height can be lower if we consider multilayer clouds, which in turn affects the downward longwave irradiance at the surface. In addition, absorption of shortwave irradiance can increase by changing from a single-layer cloud to multilayer clouds. Similarly, our knowledge of aerosols and problems with its derivation has increased over the past several years. For example, Marshak et al. (2008) show MODIS aerosols can be too large near the edges of clouds due to three-dimensional effects. This knowledge will be incorporated into future constraint algorithms.

## 5. Conclusions

A methodology has been outlined for constraining radiative transfer calculations to instantaneous observations of irradiance (and radiance) from the CERES instruments on board the NASA TRMM, *Terra*, *Aqua*, and the Suomi NPP satellites. The algorithm minimizes the change of the calculated irradiance relative to the observations by calculating changes to input variables subject to predetermined allowable variances in those inputs. We also account for known uncertainty in observed TOA irradiance by not requiring that the constrainment algorithm determine inputs to the radiative transfer model that will exactly match observations. This helps guarantee the maximum amount of change with respect to the observations with a minimal amount of change to the inputs. The algorithm as developed reduces both bias and the standard deviation of the calculated TOA irradiance relative to CERES-observed irradiance. For all footprints (scene types), the mean bias is reduced by over 50% in both LW and SW TOA irradiance due to significant improvements in partly cloudy and overcast scenes. Improvement in bias is not uniform, however, with some scene types having very slight increases in bias. Improvements in standard deviation between calculated and observed irradiance is uniform across all scene types for both TOA SW and LW irradiance as found in Tables 4 and 5. On average the improvement in calculate TOA SW and LW standard deviation is 40% and 54%, respectively.

The algorithm adjusts radiative transfer inputs based on comparisons to observed irradiance at the TOA only. This allows observations of irradiance at the earth’s surface to be used as a robust tool for measuring the impact of the constrainment algorithm on the radiative transfer through the atmospheric column when collocated with operational CRS calculations. Comparison of monthly-mean (at satellite overpass time) calculations minus observations shows that for the majority of sites considered, there is little change in the bias or variance due to the constrainment algorithm for most surface locations except in the cryosphere. The algorithm negatively impacts the downward SW irradiance more than the downward LW irradiance at the surface. We find the largest increase in either bias or variance in the all-sky surface SW irradiance was primarily due to clouds.

Reasons for the algorithm impacting computed surface flux negatively are 1) that constraint is applied to instantaneous irradiance of which uncertainty might be large, 2) that globally uniform uncertainty is applied, and 3) that bias error is not taken out before the Lagrangian multiplier is applied. Furthermore, untuned irradiances agree with CERES-derived TOA irradiance fairly well, so that a relatively large uncertainty associated with CERES-derived TOA irradiance does not constrain computed irradiance very much. For this reason we recommend that if a data product user is comparing the CRS model results to their own calculations at the TOA or surface, that they calculate the unconstrained irradiance values for comparison purposes. This is easily done using Eq. (6) above. Because the uncertainty of CERES-derived irradiance decreases by averaging over time and a region, we expect that the algorithm is more effective in constraining monthly-mean and regional mean computed irradiance. In addition to improving the instantaneous irradiance computation, we will investigate constraining mean irradiances in the future.

## Acknowledgments

CERES data are supplied from the NASA Langley Research Center’s Atmospheric Sciences Data Center. ARM data are made available through the U.S. Department of Energy as part of the Atmospheric Radiation Measurement Program. NOAA Global Monitoring Division (GMD) data are made available through the NOAA/GMD Solar and Thermal Radiation (STAR) group. SURFRAD data are made available through NOAA’s Air Resources Laboratory/Surface Radiation Research Branch. World Climate Research Programme (WCRP) Baseline Surface Radiation Network (BSRN) data are distributed from the BSRN website (http://www.bsrn.awi.de/). All the surface observations used in this study (after averaging to 15 min) can be found at the CERES/ARM Validation Experiment website (http://www-cave.larc.nasa.gov/cave). The authors would also like to thank the three anonymous reviewers for their insightful comments and questions, which encouraged significant improvements to the original manuscript.

## APPENDIX A

### Simple Example of Constrainment Algorithm

*w*and surface temperature

*T*for clear-sky OLR at night (thus excluding the SW variables as well). Equation (1) then simplifies toIf we assume that the errors in the water vapor and temperature follow a normal bivariate distribution, thenwhereThus, to maximize the probability we need to minimize

*Z*constrained by Eq. (A1), which is illustrated in Fig. A1. This figure shows, for example purposes, a bivariate normal distribution with zero means, σ

_{ΔT}= 1.5, σ

_{Δw}= 1.0, and

*ρ*= 0. The solid line represents the constraint of Eq. (A1) with the circle the point on the probability distribution of the temperature and water vapor that will be solved for using the method of Lagrangian multiplier to minimize

*Z*. Once Δ

*T*and Δ

*w*are known, we use Eq. (5) to solve for the required input adjustment to

*T*and

*w*.

## APPENDIX B

### Surface Observation Network Information

The grouping of surface observation locations is intended to sample the downwelling irradiance as a function of general surface scene type. These groupings are listed in Table B1, which details the number of sites used in each group, their programmatic source, and their location. The primary sources of data are the Baseline Surface Radiation Network, Ohmura et al. (1998), NOAA’s Global Monitoring Division, Augustine et al. (2000), and the U.S. Department of Energy (DOE)’s Atmospheric Radiation Measurement Program (ARM). Much of these data have been collected and collated with other surface observation at the CERES/ARM Validation Experiment (CAVE) website (http://www-cave.larc.nasa.gov/cave) (Rutan et al. 2001).

Site group, number, observation program, and list of surface sites used in surface validation of CRS data product.

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