a. Ice clouds

Cirrus clouds have a profound impact on the radiation balance of the earth–atmosphere system by virtue of their reflective properties at solar wavelengths and their absorptive properties at infrared wavelengths characteristic of terrestrial radiation (Stephens et al. 1981, 1990; Liou 1986; Sassen 2002; Stephens 2002). The relative magnitudes of these competing effects and their spatial and temporal variability represents a crucial component modulating global atmospheric circulations and must, therefore, be accounted for in any energy budget model. To this end, an algorithm is applied to infrared radiances from the Visible and Infrared Sounder (VIRS) instrument to detect ice clouds and provide a rough estimate of their optical properties.

The cirrus cloud algorithm employed here is that of Cooper et al. (2003), which can be considered as an advanced version of the split-window technique (Inoue 1985; Prabhakara et al. 1988). The technique finds its roots in the fact that ice particles on the order of 30 μm or smaller more efficiently absorb and emit radiation at a wavelength of 12 μm than at 10.8 μm. As a result, cirrus clouds composed of small ice particles appear “colder” at 12 μm than at 10.8 μm. The combination of radiances at these two channels, therefore, provides insight into both the optical depth of the cloud and the size of its constituent particles. The optimal estimation formulation employed by Cooper et al. (2003) is well suited to the radiative budget applications in this work for two reasons. The first is the inclusion of explicit cloud boundary information in the retrieval, which is found to significantly reduce the biases inherent in traditional implementations of the approach and provides a means for determining the vertical placement of the cloud in subsequent radiative transfer calculations. Furthermore, the optimal estimation framework provides error diagnostics for all retrieved parameters often lacking in other retrievals. Such estimates are critical for assessing uncertainties in the radiative fluxes that will be derived below.

The dominant sources of uncertainty in the algorithm are due to a strong dependence on ice crystal habit, which is generally unknown, and simplifying approximations required to increase the speed of radiative transfer calculations. Expected uncertainties in each of the algorithm products—optical depth, effective radius, and cloud thermodynamic temperature—are summarized in Fig. 1. Errors in retrieved optical depth range from 20% to 40% while uncertainties in effective radius estimates typically lie between 60% and 100%. As noted in Cooper et al. (2003), the fact that optical depth is retrieved to much higher precision than effective radius is a consequence of the sensitivity of the forward model to these parameters. Estimates of cloud thermodynamic temperature are generally accurate to ±7 K but improve substantially for colder clouds as a result of their greater impact on upwelling infrared radiances from below. These estimates are employed in a series of sensitivity studies to determine error bounds on the radiative components of the tropical energy budget in section 4a.

b. Nonprecipitating liquid clouds

In addition to the well-documented impact of ice clouds on the ERB, there have been a number of studies which suggest that water clouds have a significant impact on the global climate through strong radiative interactions (e.g., Paltridge 1980; Somerville and Remer 1984). Furthermore, using data from the Nimbus-7 and ERBE satellites, Stephens and Greenwald (1991b) demonstrated that the response of the ERB to changes in cloud liquid water can vary substantially from region to region resulting from “gross macrophysical differences” between the clouds in disparate locations. The impact of liquid clouds on outgoing longwave radiation at the TOA is generally small (but not negligible) since they reside in the lowest levels of the atmosphere at warm temperatures relative to ice clouds and, therefore, have emission characteristics similar to the earth's surface. These clouds are, however, highly reflective and strongly influence the shortwave (SW) components of the radiation budget, particularly in the absence of high cloud. In light of these observations, we employ crude estimates of the location and amount of cloud liquid water in the Tropics from the passive microwave measurements available from the TRMM Microwave Imager (TMI). The resulting liquid water path (LWP) estimates and a climatologically based estimate of effective radius are then employed in the radiative transfer calculations which follow.

The method adopted for retrieving nonprecipitating liquid clouds parallels that of Greenwald et al. (1993) and is based on both horizontally and vertically polarized brightness temperatures at 19 and 37 GHz. While originally applied to retrievals from the Special Sensor Microwave Imager (SSM/I), this technique is readily adapted to the TMI as both instruments possess channels at the required frequencies. The algorithm is based on the fact that the degree of polarization of emitted microwave radiation from the ocean's surface is reduced by the absorption and reemission by water molecules and quasi-spherical liquid cloud droplets in the atmosphere before reaching the satellite. Given a model of the emission properties of a wind-roughened surface, then, one can estimate the amount of water vapor and liquid water present in the atmospheric column provided measurements of two independent polarizations at two different frequencies are available.

The liquid water path retrieval algorithm employed in the present study differs from that outlined in Greenwald et al. (1993) in two ways. First, Greenwald et al. (1993) adjust the 37-GHz observations from the SSM/I to correct a constant offset observed in water vapor optical depth comparisons made in that study. No such correction is employed here since considerable effort has been made by the TRMM science team to ensure that the TMI radiances are well calibrated. Second, in order to determine the surface emission coefficients, ϵ^H,V, Greenwald et al. (1993) adopt the empirical relation of Goodberlet et al. (1989) for surface wind speed as a function of 22.235-GHz brightness temperature observed by the SSM/I instrument. Since the water vapor channel on the TMI has been shifted off the center of the water vapor band to 21.3 GHz, the corresponding fit parameters are likely to be different. Rather than reparameterize this relationship using collocated TMI and buoy data, we chose to use the water vapor product of Remote Sensing Systems (RSS). [These data are obtained through the Remote Sensing Systems web site www.ssmi.com. For details of the algorithm see Wentz (1997) and Wentz et al. (2000).] These data provide high-resolution, 0.25° × 0.25°, estimates of surface wind speeds derived from the TMI observations as well as an independent estimate of column water vapor, which was used to test the performance of the present algorithm. The emissivity of the wind-roughened surface is then calculated based on the model developed by Petty (1990) as in Greenwald et al. (1993).

The accuracy of retrieved liquid water paths was assessed using a series of sensitivity studies in which the two dominant factors governing the results, sea surface temperature (SST) and surface wind speed, were perturbed by amounts consistent with expected uncertainties in the RSS data. For a given set of brightness temperatures, increasing either SST or surface wind speed generally increases the estimated LWP while decreasing these quantities leads to a decrease in LWP. Average differences between baseline retrievals and LWP retrieved after both augmenting and decreasing SST by 2 K and V_av by 2.5 m s⁻¹ establish upper and lower bounds, respectively, under the assumption that the retrieval model itself is perfect. Ultimately the uncertainty in retrieved LWP is a combination of sensitivity to SST and wind speed errors and some inherent model error. To account for this, the mean of the forward model uncertainty quoted in Greenwald et al. (1993) will be added to the component owing to uncertainties in SST and V_av determined previously. This approach ensures that uncertainty estimates employed in the radiative transfer calculations provide a maximum error bound on the results.

Resulting uncertainties in LWP are presented in Fig. 2. The solid lines represent the uncertainty in the LWP estimates that will be assumed in the radiative transfer calculations to follow while the error bars indicate the values one would obtain using the minimum and maximum model uncertainty quoted in Greenwald et al. (1993), 25% and 40%, respectively. The results suggest that the retrieved LWP is most sensitive to the external input at low LWP. Physically this seems reasonable since one expects the contribution of surface emission, and therefore SST and surface wind speed, to the radiation observed by the TMI to decrease as LWP increases.

A disadvantage to using passive microwave data to infer LWP is that cloud droplets are typically much smaller than the wavelength of the radiation used, so the measurements provide little information regarding particle size. In the present study, an effective radius of 11 μm will be adopted consistent with the mean value presented in Table 1 of Miles et al. (2000), who compiled a database of in situ observations of low-level stratiform clouds made between 1972 and 1995. These values are also consistent with those quoted in Han et al. (1998). Liquid cloud vertical boundaries will be assumed to be 0.5 and 1.5 km, also based on Miles et al. (2000). A detailed analysis of the radiative implications of these assumptions, the uncertainties presented in Fig. 2, and errors in cloud detection are discussed in section 4a.

c. Precipitation

Considerable effort has been made in the past two decades to refine and improve satellite-based rainfall retrieval algorithms. The algorithms developed for TRMM represent some of the most advanced in operation, offering both estimates of surface rainfall rate as well as some information regarding the distribution of liquid and frozen hydrometeors in the entire atmospheric column. For this reason we make use of data from the TMI-based Goddard profiling algorithm (GPROF) that employs a Bayesian Monte Carlo inversion method and a database of precipitation realizations from cloud resolving model (CRM) simulations to estimate profiles of cloud and rainfall using the full complement of TMI radiances.

The GPROF algorithm, outlined in Kummerow and Giglio (1994), is performed in three steps. First, a number of cloud profiles are simulated using CRMs initialized with soundings representative of a variety of meteorological conditions. These profiles are then combined with a corresponding set of simulated TMI observations to form a very large database of scenes, which the TRMM satellite may observe as it orbits the globe. The retrieval proceeds by assigning a weight to each profile in the database depending on how closely the simulated measurements match those observed by the satellite at a given location. The final retrieved profile is a weighted sum of all the profiles in the database and consists of a surface rain rate as well as profiles of both liquid and frozen hydrometeors in 14 atmospheric layers between 0.5 and 18 km.

L'Ecuyer and Stephens (2002) have developed a model for propagating uncertainties in retrieval parameters through Bayesian retrieval algorithms and have applied it to estimate fractional uncertainties in the products of the GPROF algorithm. Omitting the mathematical details of both the algorithm and the error model for brevity, the results are summarized in Fig. 3. Uncertainties in instantaneous surface rain-rate estimates are found to range from 30% to 60% in rainfall up to 18 mm h⁻¹. In heavier rain, uncertainties rapidly increase due to the fact that all TMI channels become saturated requiring surface rainfall to be inferred solely from the nonunique relationship between surface rain rate and the ice scattering signature in the strongly coupled 37- and 85-GHz brightness temperatures.

Errors in GPROF estimates of frozen hydrometeor concentrations are on the order of 100% while those in liquid hydrometeor concentrations generally lie between 40 and 60%. As a result, we make use of infrared radiances from the Visible and Infrared Scanner (VIRS) aboard TRMM to establish cloud-top height in precipitating regions. Assuming that precipitating clouds are optically thick, the 10.8-μm brightness temperature from VIRS is taken to be equal to the cloud-top temperature. When compared with an appropriate temperature profile it is possible to infer an effective cloud-top height above which hydrometeor water contents in all GPROF classes are set to zero regardless of that which is retrieved.

d. Background atmosphere and surface properties

In addition to clouds and precipitation, radiative flux calculations are sensitive to clear-sky atmospheric properties such as water vapor, temperature, and aerosols as well as those of the underlying surface, such as SST and surface wind speed. Since many of these properties exhibit significant spatial and temporal variability, accurate estimates of each are critical to model radiative fluxes. Profiles of temperature and moisture in the background atmosphere are obtained by matching forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF) to each TRMM overpass and interpolating the results to the vertical resolution of the GPROF rainfall profiles. Based on sensitivity studies of Eyre (1990) and Eyre et al. (1993), we anticipate uncertainties in layer temperatures and specific humidities to be approximately 2 K and 30%, respectively. The concentrations of less variable gases such as carbon dioxide and ozone are assumed to follow the McClatchey et al. (1972) tropical atmosphere. For consistency with the other TRMM datasets used, SSTs and surface wind speeds are specified from the TMI-based RSS products. The estimated accuracies of these products are ∼1.5 K and 2.5 m s⁻¹, respectively. (These estimates were obtained from the RSS web site www.ssmi.com/tmi/tmi_validation.html.)

Modeling the global distribution of aerosol optical properties presents a difficult challenge since both their concentrations and composition are highly variable in space and time. Satellite-based global measurements (see King et al. 1999 and references therein) are susceptible to uncertainties introduced by assumptions regarding their shape and composition, while the detailed ground-based observations of aerosol composition required to characterize their optical properties are restricted to localized regional experiments. In the present study a crude model of the global distribution of aerosols is developed from a combination of ground-based observations of their optical properties and satellite-derived estimates of their concentration. The spectral variations of aerosol extinction, asymmetry parameter, and single-scatter albedo and their vertical distributions are modeled after Toon and Pollack (1976). The integrated optical depth from this model is then scaled to agree with the mean aerosol optical depth at 0.55 μm from the Global Aerosol Climatology Project (GACP) from February 1985 through October 1988. The mean aerosol optical depth over this period is presented in Fig. 4 (top) for the oceanic regions observed by TRMM. Figure 4 (bottom) presents standard deviations over the same period providing a rough estimate of uncertainties due to temporal variability in aerosol concentrations.

It should be emphasized that this aerosol model is approximate and the error model derives simply from the statistical properties over the 4-yr period of the dataset not accounting for variability in chemical and optical properties of the aerosols at any given location. While we expect the model to be representative of average conditions, local variability in the sources and composition of aerosol particles and extreme events such as volcanic eruptions or strong dust storms are neglected. The development of accurate global aerosol models remains an open area of research so detailed analysis of such events and of the sensitivity of the results to the choice of aerosol optical properties will be left as a topic for future study.

a. Radiative transfer

Up- and downwelling broadband radiative fluxes are computed for each pixel within the cloud system making use of an implementation of the δ-two-stream approximation to the radiative transfer equation (Stephens and Webster 1979; Ritter and Geleyn 1992; Stephens et al. 2001) over 6 shortwave and 12 longwave spectral bands. The model, known as BUGSrad, is similar to that currently implemented in the Colorado State University (CSU) General Circulation Model (GCM) as well as the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM). The implementation of BUGSrad used here has been modified to account for the effects of scattering, absorption, and emission from all four hydrometeor classes retrieved by GPROF using the parameterization of cloud optical properties introduced in Stephens et al. (1990). Each hydrometeor class is assumed to follow a modified gamma distribution

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and the asymmetry parameter, single scatter albedo, and volume absorption and extinction coefficients are parameterized as a function of the distribution width parameter, p, and the characteristic radius, r_m. Liquid and ice clouds are assumed to follow a modified gamma distribution with p = 2 and characteristic radii defined by retrievals in the case of ice clouds and climatological values for liquid clouds. For consistency with the CRM simulations used in the creation of the GPROF databases, liquid and frozen precipitation hydrometeors are assumed to follow a Marshall–Palmer (Marshall and Palmer 1948) distribution with p = 1 and characteristic radius defined according to r_m = 3(LWP/8πρ)^0.25 while liquid and ice cloud hydrometeors in precipitating pixels are taken to be monodispersed with effective radii of 100 μm. The total volume concentration, N₀, derives from the liquid or ice water content in any given layer. Gaseous absorption is treated using the correlated k-distribution formulation of Fu and Liou (1992) and surface albedo is fixed at 0.12, consistent with the ERBE-derived mean cited in Harrison et al. (1990). A vertical resolution of 0.5 km is adopted consistent with the lowest levels of the GPROF products and the independent pixel approximation (IPA) is invoked in which each 0.25° × 0.25° pixel is treated as radiatively isolated from its neighbors. This approximation neglects the the effects of horizontal inhomogeneity within clouds and precipitation but such information has already been lost as a result of the footprint size of the instruments and the retrieval methods employed in the previous section. A series of sensitivity studies in which fluxes are recomputed calculations with various input parameters perturbed by an amount consistent with their accuracy provide error bounds on the resulting fluxes.

b. Latent heating

The cooling of the atmosphere through radiative processes is largely balanced by the release of latent heat by precipitation. This is particularly true in the Tropics, which accounts for two-thirds of global precipitation (Simpson et al. 1996). Thus some estimate of latent heating due to precipitation is required to fully characterize the impact of precipitating clouds in the tropical atmosphere. When averaged over a suitably large horizontal domain, small-scale eddies in the horizontal wind components can be neglected, and the “apparent” atmospheric heat source can be written as the sum of vertical eddy flux convergence of heat, latent heating from condensation, evaporation, and sublimation, and radiative cooling (Yanai et al. 1973; Tao et al. 1993; Olson et al. 1999).

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where c, e, f, m, d, and s represent the rates of condensation, evaporation, freezing, melting, deposition, and sublimation per unit mass of air. Here, L_υ, L_f, and L_s denote the latent heats of vaporization, fusion, and sublimation; π = (p/1000 mb)^R/c_p, R is the dry gas constant, c_p is the specific heat of air at constant pressure, θ is the potential temperature, w the vertical velocity, and p is pressure. The overbar denotes horizontal averages while the primes denote deviations from these means.

Provided a spatial and temporal domain large enough to encompass the geographic evolution and lifecycle of the cloud complex is considered, the latent heating (LH) terms can be vertically integrated from the surface to the top of the cloud to yield

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where R is the surface precipitation rate. Thus the column-integrated LH is, in a space–time average sense, related to the total condensate removed from the system by precipitation processes since a net exchange of energy between the surface and atmosphere requires that water leave the earth's surface in the vapor phase through evaporation and reenter it in the liquid or solid phase through precipitation. Clouds that never lead to precipitation at the surface must reevaporate in the atmosphere, a process that ultimately cancels the latent heat released during their formation. This is supported by the results of Tao et al. (1993) who find that vertically integrated condensate and storm-total precipitation vary by less than 10% for modeled squall lines. Fractional uncertainties in column-integrated latent heating estimates determined in this fashion are identical to the errors in the surface rainfall estimates themselves. The most pessimistic estimate is ∼50% assuming that the surface rainfall errors are all systematic and are not reduced through averaging over large spatial and temporal domains. A more realistic estimate, however, is 20% based on comparisons of the monthly mean rainfall products from the TMI and precipitation radar (PR) products (Kummerow et al. 2000).

To characterize the vertical structure of diabatic heating in the atmosphere more completely, one would prefer to include profiles of latent heating to complement the radiative flux profiles provided by BUGSrad. Profiles of LH, however, not only depend on profiles of condensate and water vapor but also on the dynamics of the local environment and are considerably more challenging to estimate as a result (Olson et al. 1999). The present study is, therefore, restricted to column-integrated estimates of latent heating, but a number of techniques are currently being developed for inferring latent heating profiles from TRMM observations. These methods will be incorporated into the algorithm as soon as they become available (e.g., Tao et al. 1993; Olson et al. 1999; Yang and Smith 1999a,b; 2000; Tao et al. 2001).

c. February 1998

Figure 6 illustrates mean surface rain rate from the GPROF algorithm and the fraction of TRMM observations for which the cloud retrievals indicated the presence of high and low clouds over the month of February 1998. As one would expect the largest concentrations of rainfall and cirrus clouds are associated with deep convection in the intertropical convergence zone (ITCZ) and South Pacific convergence zone (SPCZ). High clouds are also evident associated with storm tracks north of 30°, which also give rise to regions of low-level liquid cloud. Finally, persistent regions of capped stratocumulus clouds associated with subtropical highs are evident to the east of Central and South America and Africa.

These clouds and precipitation provide two important mechanisms for the exchange of energy between the surface and atmosphere. The first of these, latent heating due to all phase changes in the development of precipitation in the atmosphere from water vapor that originally evaporated at the earth's surface, is presented in Fig. 7 (top) for the month of February 1998. Since this estimate is derived following Eq. (3), the structure mimics that of the surface rainfall presented in Fig. 6 (top) tracing out the convergence zones and storm tracks. In addition to latent heating through phase changes, clouds and precipitation also modify the relative amounts of absorption and emission in the atmosphere and thus influence its total radiative heating. Figure 7 (bottom) presents the net change in column-integrated radiative heating induced by the presence of clouds and precipitation defined as the difference in net flux between the surface and TOA for all conditions minus the corresponding gflux difference in the absence of clouds and precipitation:

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The subscript all-sky refers to the observed atmosphere while clear denotes that from which all clouds and precipitation have been removed. The cloud and precipitation components of atmospheric radiative heating is strongest in regions of prevalent high cirrus, again associated with storm tracks and convergence zones. Conversely, the capped stratocumulus clouds associated with subtropical highs have a cooling effect on the atmosphere in three large regions off the west coasts of North and South America and southern Africa. While the exception of these regions, there is a strong correlation between radiative and latent heating impacts of clouds and precipitation in the tropical atmosphere. It is important to note, however, that the physical mechanisms for radiative and latent heating from a precipitating cloud system often operate at different altitudes, thus more complete examination of their vertical distribution is required before more substantive conclusions can be drawn. This will be considered in the companion paper (L'Ecuyer and Stephens 2003, unpublished manuscript).

a. Sensitivity studies

Results of sensitivity studies detailing the impact of uncertainties in a wide variety of input parameters on longwave and shortwave fluxes at TOA and the surface are summarized in Tables 1 and 2, respectively. The first two columns in each table describe the parameter studied and the magnitude by which it is perturbed. The remaining columns summarize the impact of each perturbation on daily mean fluxes relative to the set of assumptions used in nominal implementation of the algorithm. Quoted are the deviations in mean fluxes averaged over the oceanic domain between 40°N and 40°S, while the quantities in parentheses correspond to root-mean-square (rms) differences in individual pixels at 1.0° resolution. Since a majority of the pixels sampled by TRMM are observed only 1 time per day, the rms differences isolate the average contribution of each input parameter to the uncertainty in flux estimates at pixel resolution, while the tropic-mean differences indicate the expected order of magnitude of biases that could result in daily radiation budget estimates due to systematic errors in the input parameters. Note that downwelling fluxes at TOA are omitted from these tables as they are not sensitive to any of the input parameters. We assume that solar insolation is known to accuracies of ∼0.5 W m⁻² from measurements by active cavity radiometers [e.g., the Active Cavity Radiometer Irradiance Monitor (ACRIM) instrument described in Willson (1984)] while incoming longwave radiation is negligible.

Table 1 suggests that longwave fluxes are most susceptible to errors in profiles of temperature and humidity by virtue of their role in both clear and cloudy pixels. Systematic perturbation of either field leads to errors of 5 W m⁻² in outgoing longwave radiation (OLR). Interestingly, the impact of water vapor on OLR is equally distributed between that above and below 500 hPa despite the fact that less than 10% of the total atmospheric water vapor mass resides above 500 hPa. The other important source of uncertainty in OLR is the distribution of ice cloud microphysical properties. At the individual pixel level errors in retrieved effective radii and thermodynamic temperatures of ice clouds can lead to errors of as much as 90 W m⁻² in estimates of OLR but typically result in biases of 2–3 W m⁻² when averaged with all noncloudy pixels over the entire TRMM domain. Downwelling longwave fluxes at the earth's surface are most sensitive to errors in water vapor at levels below 500 hPa that introduce uncertainties of up to 13 W m⁻². A secondary source of error is the uncertainty in sea surface temperature that can lead to a bias of as much as 6 W m⁻² when integrated over the whole domain. The sole contribution to uncertainties in upwelling longwave fluxes from the surface arises from errors in specified SSTs, which lead to errors of ∼10 W m⁻².

Table 2 demonstrates that uncertainties in shortwave fluxes are dominated by errors in surface albedo and in the retrieved distribution of liquid clouds and their microphysical properties. In individual pixels, uncertainties in LWP and effective radius often lead to errors in outgoing shortwave radiation (OSR) and downwelling shortwave fluxes at the surface on the order of 100 W m⁻². Averaging with surrounding clear-sky pixels reduces their impact to 4–8 W m⁻². Systematic errors in the assumed surface albedo lead to errors of 9 W m⁻² in the reflected solar radiation from the earth's surface. After some attenuation these errors lead to a 7 W m⁻² error in OSR. Finally, as a result of the absorption of shortwave radiation by water vapor, errors in the specification of water vapor concentrations below 500 hPa can also lead to errors of up to 4 W m⁻² on the downwelling shortwave radiation at the surface.

Perhaps surprisingly, the impact of uncertainties in precipitating regions on tropical mean fluxes is negligible. This is primarily a result of the fact that, on average, raining pixels account for less than 10% of the domain considered and thus do not impact the averages much. It should be noted that at the pixel level, however, errors in shortwave fluxes often exceed 50 W m⁻². Conversely, errors in longwave fluxes seldom exceed 10 W m⁻² due to the fact that the clouds associated with precipitation are optically thick and their impact on longwave fluxes has saturated in many cases. In fact, rms errors in shortwave fluxes at the pixel level (in parentheses in Tables 1 and 2) are generally much larger than errors determined by comparing tropical means in all cases involving cloud and precipitation properties. Many of the dominant sources of uncertainty in longwave fluxes, such as atmospheric temperature and water vapor, have a more or less uniform effect from pixel to pixel. As a result, pixel-scale rms errors in longwave flux estimates, on the other hand, are generally very similar to those determined from the tropical means. The dominant sources of error in shortwave fluxes, such as liquid water path and effective radius, are strongly dependent on the characteristics of the scene being viewed. This, coupled with the diurnal cycle of solar radiation at the top of the atmosphere leads to greater pixel-to-pixel variability in the resulting errors. Thus, at the individual pixel level we anticipate larger uncertainties in shortwave than longwave fluxes but comparable uncertainties in TRMM domain-average fluxes.

In addition to the sources described earlier, it is important to characterize the impact of errors in cloud detection, a problem encountered in most passive remote-sensing techniques. This has been modeled by varying the LWP and brightness temperature thresholds by which liquid and ice clouds are identified. The results demonstrate that accurate detection of liquid clouds is extremely important for modeling shortwave fluxes in the atmosphere. Failure to model a liquid cloud can lead to errors in OSR and downwelling shortwave fluxes at the surface exceeding 200 W m⁻². Equivalently, false detection of a cloud leads to uncertainties of equal magnitude but in the opposite direction. When averaged over the TRMM domain, systematic errors in liquid cloud detection lead to uncertainties of 10–15 W m⁻² in shortwave fluxes and represent the largest single error source on these estimates. A solution to this problem may come in the form of a multisensor technique that combines visible radiances from VIRS with the TMI microwave radiances such as that introduced by Masunaga et al. (2002). While this approach is only applicable during the daytime, it may be supplemented with the microwave-only technique at night since longwave fluxes are much less sensitive to errors in cloud detection.

Errors introduced by misrepresentation of ice clouds is small by comparision primarily due to the fact that all errors are restricted to the identification of extremely thin cirrus clouds (IR optical depths less than 0.2). At the individual pixel level these clouds can lead to errors of as much as 50 W m⁻² in OLR but when integrated over the entire oceanic TRMM sampling domain, their effect is on the order of 2 W m⁻². This result is consistent with the fact that the algorithm used to infer cirrus cloud microphysical properties is explicitly designed to detect thin cirrus clouds (Inoue 1985; Prabhakara et al. 1988).

It should be noted that the error estimates provided in Tables 1 and 2 apply only to fluxes at atmospheric boundaries. Uncertainties in corresponding vertical flux profiles are likely to be much larger due to the inherent difficulties in extracting cloud boundary information from passive observations. Analyses of the vertical structure of radiative heating in the atmosphere using this technique should, therefore, be restricted to a few coarse atmospheric layers even though flux profiles are computed at a vertical resolution of 0.5 km. Furthermore, there exist a number of additional sources of uncertainty that have not been addressed here. These include errors due to temporal and spatial sampling, the sensitivity of the results to subpixel variability in cloud and precipitation fields, and errors due to the radiative transfer model itself such as parameterizations of cloud optical properties the aerosol model adopted. Evaluation of the impact of these error sources requires a more rigorous treatment and are topics of ongoing research.

b. External validation

In climate study applications involving long-term and wide-area means or in variational data assimilation where accurate observational covariance matrices are required, random errors must be distinguished from biases. While the results of the previous section demonstrate the sensitivity of the algorithm to errors in the input, it is not possible to separate random and systematic contributions to the resulting uncertainty estimates. Comparisons with observations of TOA fluxes from the CERES instrument are particularly well suited for this purpose since CERES also flies aboard the TRMM satellite and therefore provides a dataset that is temporally and spatially matched to the other observations used in the present study [for more information on the objectives and characteristics of the CERES instrument see Wielicki et al. (1996)].

Estimates of OLR at 1.0° resolution for February 1998 are compared with CERES observations over the same period in Fig. 8. The CERES data are obtained by averaging data from the CERES ERBE-like Monthly Regional Averages (ES-9) product¹ over the entire month to avoid misinterpreting the effects of diurnal models employed in producing CERES monthly mean products as errors. Differences between our results and those from CERES may be loosely interpreted as “error estimates” with the caveat that the CERES data are not perfect themselves. All error estimates made in this way are therefore approximate but are assumed to provide a ballpark and independent assessment of the accuracy of our TOA results.

The correlation coefficient, root-mean-square difference, and bias between the two datasets is provided in the first row of Table 3. Given the large uncertainties expected in ice cloud boundaries, agreement is very good. The data correlate better than 96% and exhibit a bias of less than 1 W m⁻². Figure 8 indicates a slight positive bias in the present study relative to CERES at low OLR (<235 W m⁻²) suggesting that cloud in the vicinity of convection may be too low or optically too thin, consistent with the inherent lack of information regarding cloud-sized ice hydrometeors in the GPROF retrievals. On the tropic-wide mean, this warm bias is, however, cancelled by a small negative bias in clear-sky regions (OLR > 275 W m⁻²).

A similar comparison of outgoing shortwave radiation estimates from the present study and CERES is presented in Fig. 9 and the corresponding correlation coefficient, rms difference, and bias are summarized in the second row of Table 3. Agreement in shortwave flux estimates is not as good as in their longwave counterparts. The correlation between the shortwave datasets is less than 80% while rms differences are ∼23 W m⁻². Note, however, that the relative bias between the two datasets is less than 3 W m⁻² due to a cancellation of overestimated reflection from nonprecipitating liquid clouds (OSR < 100 W m⁻²) and underestimated reflection from precipitation (OSR > 100 W m⁻²). The former may result from the albedo parameterizations used in BUGSrad overestimating the reflection due to nonprecipitating cirrus and stratus clouds or from systematic errors in LWP estimates. The latter can easily be explained by the fact that ice particles at the top of thick raining clouds are assumed to follow a monodispersed distribution with radius 100 μm consistent with assumptions in the GPROF database. In nature, particles at the tops of these clouds are much smaller resulting in significantly more reflection than is modeled in the present study.

For completeness, a comparison with the National Oceanic and Atmospheric Administration (NOAA) interpolated OLR product is provided in the last row of Table 3. (These data were provided by the NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado, from their web site at http://www.cdc.noaa.gov.) Due to differences in the sampling of the Advanced Very High Resolution Radiometer (AVHRR) on the Geostationary Operational Environmental Satellites (GOES) from which the NOAA OLR product is derived and the TRMM satellite, both the correlation and rms differences are substantially worse relative to the comparison with CERES data. In fact, correlations between the NOAA and CERES products themselves are less than 70%. It is encouraging to note, however, that biases remain small (∼10 W m⁻²) suggesting that under-sampling by the TRMM satellite is substantially reduced when data are averaged to monthly timescales.

These results indicate that the bias component of the uncertainty is substantially less than the “worst case” total indicated by the sensitivity studies in the previous section. This is likely a result of both cancellation of competing sources of uncertainty and temporal averaging.

CERES data have also been used to probe the breakdown of the uncertainties in the present model into random and systematic components. Estimates of OLR and OSR are compared to CERES observations on a variety of different temporal scales in Fig. 10. On short timescales, these differences provide an estimate of the sum of random and bias errors while on longer scales, random errors, by definition, diminish through the averaging process leaving only the component owing to systematic biases. Uncertainties in instantaneous fluxes, represented by the 1-day averages in Fig. 10, are 27 and 80 W m⁻² in OLR and OSR, respectively. At the other end of the spectrum, uncertainties in long-term means are 8 and 21 W m⁻². These results imply that random errors account for between 75% and 85% of the uncertainty in instantaneous fluxes, while the remainder must be due to systematic differences between the present work and the CERES product. Differences in 5- and 10-day, and monthly averages are highlighted for convenience. Random errors are also reduced through spatial averaging. Rms differences in 5° OLR and OSR estimates, however, are found to improve by only 10% relative to those at 1° resolution presented in Fig. 10.

In principle, the strong sensitivity of the current model to assumptions regarding cloud height, detection, and particle size provide a means for constraining the retrieval using broadband flux observations such as those made by CERES. While tempting, the possibility of calibrating the model by tuning parameters for better agreement with CERES is avoided for three reasons: 1) it eliminates CERES data as an independent source of information for comparison purposes, 2) while we can make general arguments regarding the dominant sources of the errors in the technique, not all sources are known nor are their relative importance in the big picture so it is difficult to favor tuning one parameter over another, and (3) the tuning process may put too much confidence in the CERES estimates, which are themselves imperfect and could introduce additional biases in the results. As a result, no effort is made to force agreement between the two datasets at this time. To the contrary, these differences provide a useful source of additional information, which may prove invaluable in future applications to study cloud–radiation–climate interactions.