We conduct a theoretical investigation into whether changes in the outgoing longwave radiation (OLR) spectrum can be used to constrain longwave greenhouse-gas forcing and climate feedbacks, with a focus on isolating and quantifying their contributions to the total OLR change in all-sky conditions. First, we numerically compute the spectral signals of CO2 forcing and feedbacks of temperature, water vapor, and cloud. Then, we investigate whether we can separate these signals from the total change in the OLR spectrum through an optimal detection method. Uncertainty in optimal detection arises from the uncertainty in the shape of the spectral fingerprints, the natural variability of the OLR spectrum, and a nonlinearity effect due to the cross-correlation of different climate responses. We find that the uncertainties in optimally detected greenhouse-gas forcing, water vapor, and temperature feedbacks are substantially less than their overall magnitudes in a double-CO2 experiment, and thus the detection results are robust. The accuracy in surface temperature and cloud feedbacks, however, is limited by the ambiguity in their fingerprints. Combining ambiguous feedback signals reduces the uncertainty in the combined signal. Auxiliary data are required to fully resolve the difficulty.
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 As reviewed by Bony et al.  and the Intergovernmental Panel on Climate Change (IPCC) , the differences between model projections of future climate change can be largely attributed to the uncertainties in the climate feedbacks of atmospheric temperature, water vapor, clouds, and so on. An improved quantification of these feedbacks is imperative for a better understanding of climate sensitivity.
 Our climate feedback analysis is referenced to the top-of-atmosphere (TOA) radiation energy budget [Ramaswamy et al., 2001], and a feedback effect is quantified by its individual contribution to the total change in TOA radiation energy flux. Two conventional methods exist for separating the individual contributions: the partial radiative perturbation (PRP) method [e.g., Wetherald and Manabe, 1988] and the radiative kernel–based method [e.g., Huang et al., 2007a; Shell et al., 2008; Soden et al., 2008; Lu and Cai, 2009]. These methods facilitate intercomparison between different climate models but are not intended for quantifying individual feedbacks from observational data.
 TOA broadband radiation flux measurements provide key observational constraints for understanding the energy balance of the climate system, but it is difficult to diagnose the individual climate effects from such measurements because the scalar observable always represents collective impact. In contrast, the impacts of different controlling factors of the radiation energy budget are characterized by the longwave spectral changes they induce [Kiehl, 1983; Slingo and Webb, 1997; Harries et al., 2001; Huang and Ramaswamy, 2008, 2009]. On the basis of such recognizable spectral fingerprints, Leroy et al.  proposed a method that separates individual effects by applying an optimal detection technique to the outgoing longwave radiation (OLR) spectra that can be measured with ensured accuracy in space [Anderson et al., 2004; Dykema and Anderson, 2006]. Extending the clear-sky work of Leroy et al. , we investigate the application of the optimal detection approach to global all-sky measurements of OLR spectra.
 Eventually the climate research community will obtain a time series of well-calibrated OLR spectra from which climate variations can be analyzed. In this study, however, we use a double-CO2 numeric experiment to provide a “climate change” context for examining the optimal detection technique. This experiment and the all-sky radiance spectral calculation are described in section 2. The optimal detection results are presented and analyzed in section 3. In section 4, we discuss the key issues to consider when applying the method to real data.
2. Double-CO2 Experiment and Spectral Radiance Calculation
 We use the difference between the beginning and ending steady states in a double-CO2 (280–560 ppmv) experiment from the Cloud Feedback Model Intercomparison Project (CFMIP) [Williams and Webb, 2009] as a surrogate for climate change. In this experiment, a model is first integrated to a steady state, then the CO2 concentration prescribed to the model is doubled as a step function of time, and then the model is integrated to another steady state. Among the models submitted to CFMIP, the Canadian Centre for Climate Modelling and Analysis (CCCMA) model is chosen for our study because it is one of few models that archives the information on clouds needed to simulate radiance spectra. The model itself consists of an Atmospheric General Circulation Model (GCM) coupled to a slab ocean model. In the 60-year CCCMA integration archived by CFMIP, CO2 is doubled after year 20, and the model reequilibrates in about 15 years. We have used years 1–20 and 46–60 as the beginning and ending steady states, respectively.
 We calculate the OLR spectral radiance (R, in W m−2 cm sr−1) with a moderate-resolution radiative transfer model, MODTRAN 4 [Bernstein et al., 1996], at 1 cm−1 resolution. For the atmospheric and surface conditions, we use the monthly mean geophysical variable fields generated by CCCMA. These fields, including temperature, specific humidity, cloud fraction, liquid water, and ice concentrations, are stored at 17 fixed pressure levels (from 1000 to 10 hPa) with a horizontal grid of 2.5° by 2.5°. Following Huang et al. [2007b], 5 additional layers of standard atmosphere are patched to the top of the 17-layer profiles in order to properly represent the stratospheric absorbers. Two spectra, one clear-sky and one all-sky, are computed at each GCM grid point for each monthly time step. For the all-sky computation, clouds at different vertical levels are assumed to overlap randomly, adapting the method of Huang et al. [2007b]; the clear-sky spectrum is calculated with the same atmospheric temperature (T) and moisture profiles (q) but without cloud condensates (C).
 In feedback analyses, climate feedbacks are usually measured by TOA radiation energy flux change divided by surface temperature change (in W m−2 K−1). However, as the goal of optimal detection is to separate the total change in radiation energy flux (W m−2) into individual contributions, in the following discussions we focus on this radiation aspect.
 To provide a “truth” (reference calculation) to which the optimal detection results may be compared, changes in the OLR spectrum (δR) due to the radiative forcing and associated feedback processes are obtained by using the PRP method [Wetherald and Manabe, 1988]. We first compute the mean OLR spectra for the postdoubling steady period, then perform a series of suppression simulations based on this period. In each of these suppression simulations, the change in a geophysical variable is replaced by the climatological mean calculated from the beginning period prior to doubling CO2, the OLR spectra are recalculated, and the spectral fingerprint of the suppressed variable is computed as the difference of postdoubling spectra less the suppression spectra. We have obtained the spectral fingerprints of CO2 forcing and responses of surface temperature, tropospheric and stratospheric temperature, tropospheric and stratospheric water vapor, and lower, middle, and upper clouds (Table 1). Both clear- and all-sky fingerprints are obtained for each suppression. The boundary between troposphere and stratosphere is set to 200 hPa. In the categorization of clouds the International Satellite Cloud Climatology Project definition is adopted, with the low cloud being below 700 hPa, the middle cloud being between 400 and 700 hPa, and the high cloud being above 400 hPa. We also conduct a simulation for which the changes of all the above variables are suppressed simultaneously and the overall OLR spectral change compared to the prior-to-doubling climatology is thus obtained.
Table 1. Suppression Experiments and Spectral Signals, δRXia
Spectral Radiance Change
R(…, Xi, …) represents the OLR radiance spectrum at the ending steady state in the double-CO2 run, and R(…, i, …) represents the radiance spectrum computed while suppressing the change in Xi. For instance, the spectral change induced by surface temperature change is computed as δRTs = R(rco2, TS, Ttrop, Tstrat, qtrop, qstrat, Clow, Cmid, Chgh) − R(rco2, , Ttrop, Tstrat, qtrop, qstrat, Clow, Cmid, Chgh), where TS is the surface temperature at the steady state after doubling CO2 and S is the climatological mean at the steady state prior to CO2 doubling.
Figure 1 illustrates the global distributions of total change in OLR broadband flux (δOLR, W m−2) in the CCCMA double-CO2 experiment and the PRP-computed individual contributions (δOLRXi) to the total change due to different physical causes. The broadband flux values are approximated by integration of the radiance changes δR (W m−2 cm sr−1) over the spectral domain and multiplied by π to integrate over solid angle. The all-sky global mean δOLRXi values are summarized in the first column of Table 2.
Table 2. Global Mean All-Sky Outgoing Longwave Radiation Change due to Different Causesa
Global Mean OLR Change
Partial Radiative Perturbation
Units of δOLR: W m−2. In case 1, optimal detection is applied to the global mean δRtotal, with the uncertainty covariance matrix computed as Σ = Σv + Σu. Case 2 is the same as case 1, but with Σ = Σv + Σu + Σnl. Case 3 is the same as case 2, but signals with ambiguous fingerprint shapes are combined. The detection results are presented together with marginal uncertainties.
−2.52 ± 0.05
−2.72 ± 0.12
−2.72 ± 0.12
−0.47 ± 0.15
−0.46 ± 0.19
−0.46 ± 0.17
−4.57 ± 0.53
−4.94 ± 0.77
−4.94 ± 0.70
−0.26 ± 0.11
−0.26 ± 0.13
−0.26 ± 0.12
2.89 ± 2.36
3.36 ± 2.47
3.42 ± 0.78
−0.25 ± 2.29
0.04 ± 2.35
0.19 ± 0.17
0.08 ± 0.19
8.70 ± 0.68
−1.00 ± 1.21
−1.19 ± 1.49
10.06 ± 0.98
9.83 ± 1.11
 We define a normalized radiance change δRN = , which represents the radiance change per unit broadband flux change. Although the signs of the OLR broadband flux change δOLR and the radiance spectrum change δR are highly geographically variable and can differ from model to model for the same region the normalization tends to yield a uniform spectral shape of δRN. The global mean δRN spectra due to different physical causes as listed in Table 1 are shown in Figure 2; these δRN spectra are used as “fingerprints” to identify the forcing and feedbacks as detailed below. Also shown in Figure 2 is the standard deviation of the δRN spectra across all the CCCMA grid points globally.
 Comparing the all-sky and clear-sky results in Figure 2, it is evident that the noncloud spectral fingerprints as identified by Leroy et al.  are generally unchanged in the all-sky case. Inspection of the means (blue) compared to the standard deviations (red) in Figure 2 shows that the CO2-forcing fingerprint has the highest signal-to-noise ratio and little similarity to the rest of the fingerprints, which suggests that the greenhouse forcing is the most unambiguously identifiable signal. While no two fingerprints are identical, some pairs of fingerprints are noticeably more similar. These pairs are surface temperature-low cloud, tropospheric temperature-tropospheric humidity, and tropospheric temperature-high cloud. Cloud fingerprints tend to have large standard deviations because of strong geographical variations in model-predicted cloud properties and also in background atmospheric profiles. Note that even for a uniformly prescribed change in cloud properties, cloud longwave radiative impact is highly dependent on the background atmospheric profile. The large variability in cloud spectral fingerprints as displayed in Figure 2 is representative of the uncertainty that would be measured by the large intermodel difference in cloud feedback strength [IPCC, 2007].
 By changing one variable at a time, the PRP method estimates the partial OLR change due solely to each of the variables listed in Table 1. The OLR changes thus obtained approximate the linear terms in a Taylor expansion of δOLR (see equation (1) of Huang et al. [2007a] for an example). However, because these variables are not independent of each other, part of δOLR is also due to the cross-correlation between the variables, so that the sum of the linear terms does not exactly reproduce the total OLR change that is obtained with all the variables changing simultaneously. Our results from PRP computations show that this nonlinearity effect is more significant when cloud perturbations are included than in the clear-sky case. Figure 3 shows a comparison between the all-sky total change in OLR spectrum and the sum of the individual contributions. Between 400 and 1400 cm−1, where the OLR spectrum shows most prominent changes, the summed signals agree with the total change to within 15% at most frequencies and 5% in the window region (833–1250 cm−1). The nonlinearity-induced spectral residual, although it is relatively small, does add to other noise sources to impact the optimal detection of the forcing and feedbacks. We show below that by properly taking into account this residual signal in the optimal detection (OD) process, we can limit its impact on the accuracy of the detection results.
3. Optimal Detection
 Attributing measurable OLR spectral change to different physical causes can be formulated as a multivariate linear regression problem:
where the column vector y is the total change of the OLR spectrum with a dimension of [nw × 1] (nw is the number of frequency intervals), the matrix S [nw × nf] represents the spectral fingerprints of nf causes being studied, the column vector a [nf × 1] is the solution to the problem we seek, that is, the amplitude of each of the nf causes, and r is the residual in the total change that cannot be explained by the linear combination of the fingerprints. The residual arises from sources such as natural variability, uncertainty in the fingerprints, and the nonlinearity effect discussed above.
where the contravariant spectral fingerprint F is constructed as
Σ is the covariance matrix of r, the superscript T denotes transpose, and the superscript −1 denotes a matrix pseudo-inversion. In practice, we compute the first ne empirical orthogonal functions (EOFs) (e [nw × ne]) and accompanying eigenvalues (λ [ne × ne]) of Σ and then compute the pseudo-inverse matrix Σ−1 as eλ−1eT. Note nw > ne > nf. Also note that ΣΣ−1 is not the identity matrix but a tensor (eeT) that describes the spectral subspace spanned by the eigenvectors e.
 The posterior fitting error is
which has a chi-square distribution with nf degrees of freedom. The marginal uncertainties of individual elements of a are the square roots of the diagonal elements of (STΣ−1S)−1.
 The spectral fingerprint of a physical cause can be obtained by numerical computation of the radiance spectral change with the PRP method. We use the normalized radiance spectral changes (δRXiN, as illustrated in Figure 2) as fingerprints and investigate how the total change δRtotal can be explained by linear combination of the δRXiN. Internal variabilities and the fingerprint shape uncertainty discussed by Leroy et al.  contribute to the postfit residuals. Here, we also take into account an additional contributor, the nonlinearity effect, that is, the sum of δRXi being not exactly equal to the total change δRtotal. The uncertainty covariance associated with all the postfit residuals becomes
where the natural variability covariance, Σv, and the fingerprint shape uncertainty matrix, Σu, are defined by Leroy et al. :
in which 〈〉 denotes an unperturbed time series (20 years in this study) of the global annual mean OLR spectra, and
where δRXiN and δOLRXi are normalized spectral changes and integrated broadband flux changes at each CCCMA grid point and the angle brackets indicate a global average. Ideally, the angle brackets should be averages over an ensemble of many different climate models, but no such ensemble of model simulations with appropriate outputs needed for all-sky infrared radiance computation is available. Instead, we use spatial variability of the fingerprints to represent the fingerprint shape uncertainty. Fingerprint shape uncertainty measured in this way depends on the location and extent of the spatial regions used to define the mean fingerprint. Evaluating the matrix with all the grid points globally, we include as much spatial variability as allowed by the double-CO2 experiment to define the fingerprint shape uncertainty.
 The nonlinearity uncertainty covariance Σnl is computed from
where δRtotal and δRXi are spectral radiance changes at each grid point computed with the PRP method. By introducing Σnl, we treat the residual signal r caused by the nonlinearity effect in a similar way to other noise sources. This component of the total uncertainty matrix makes the linear decomposition in OD emphasize the spectral regions that are less subject to the nonlinearity effect.
3.2. Detection Results
 Tests show that including Σnl reduces the biases in OD results incurred when not accounting for the nonlinearity impact. Comparing cases 1 and 2 in Table 2, we find that after including Σnl, optimal detection accuracy improves substantially at the cost of some loss of precision (measured by marginal uncertainty). Although the precision loss is inevitable because Σnl increases the magnitude and thus the eigenvalues of Σ, this loss is small compared to the improvement in accuracy. Hence, the practice of limiting the nonlinearity impact by inclusion of Σnl is proven to be sound.
 When, instead of using a set of globally or regionally averaged fingerprints at all grid points, we use one set of error-free fingerprints locally evaluated at each point, we find that optimal detection reproduces with great fidelity the linear portion of the OLR change due to CO2 forcing and each feedback as determined by the PRP method. The global distribution of detection errors obtained with error-free fingerprints is shown in Figure 4. Compared to the magnitudes of forcing and feedbacks shown in Figure 1, the errors are generally smaller than 10% and decrease when the number of EOFs retained in the inversion of Σ increases. When keeping the first 50 EOFs, the clear-sky detection errors are mostly less than 0.1 W m−2, except for tropospheric water vapor. The larger error in this case is due to the larger uncertainty in its fingerprint shape (see Figure 2). This error can be reduced if more EOFs are retained.
 Uncertainty in spectral fingerprints arises due to incomplete knowledge of the changes in the surface and atmospheric states as well as of their present mean states. This uncertainty limits the overall accuracy in the OD results, which is made evident by the trial cases in Table 3. Here, we divide the globe into regions. In each region, we compute a set of regional mean fingerprints and apply this single set of fingerprints to detect forcing and feedbacks at all the grid points within this region. We then compute an RMS error with all the grid points globally and use it to assess the accuracy in the OD results. As shown by the numbers in Table 3, as the size of the regions decreases, the regional mean fingerprints differ less from the error-free fingerprints at each grid point, and thus the RMS error diminishes. This means that detection results greatly improve as fingerprints become more accurate. In the worst-case scenario (one single set of globally averaged fingerprints), the RMS errors of CO2 forcing, tropospheric water vapor, and temperature feedbacks are still less by a few factors than the magnitudes of these forcing and feedbacks (Table 2), whereas the detectability of surface, cloud, and stratospheric signals is more impaired.
Table 3. Global RMS Error in Optimally Detected All-Sky Outgoing Longwave Radiation Changesa
δOLR (W m−2)
30° × 60°
10° × 20°
Pointwise (3.75° × 3.75°)
Optimal detection errors generally decrease as the extent of regions used to define the fingerprints is reduced from global to 3.75° × 3.75° grid boxes.
 Besides the uncertainty in fingerprint shape, joint detection of multiple signals is also impaired when there exist strong similarities in the shape of some spectral fingerprints. One recognized problem in the clear-sky case is the ambiguity between atmospheric temperature and water vapor. In the all-sky case, ambiguity may also exist between low cloud and surface temperature, between upper clouds and tropospheric temperature, and between clouds of adjacent levels. This is reflected by the relatively large, sometimes strongly anticorrelated detection errors in these signal pairs in Figure 4b: most noticeable are the compensating errors in the surface temperature-low cloud pair. The ellipses in Figure 5 illustrate the marginal uncertainty in joint detection, that is, the joint probability distribution function of (δOLRXi, δOLRXj). The major axes of the uncertainty ellipses have slopes very close to −1, indicating high probability for completely offsetting errors when surface temperature and low cloud, or when tropospheric temperature and high cloud, are detected jointly. However, the problem can be removed by treating the ambiguous signals as a single unit. As demonstrated by case 3 compared to case 2 in Table 2, the marginal uncertainty of the detection is greatly reduced when the ambiguous signals are combined and detected as one signal.
4. Discussion and Conclusion
 The uncertainty in the optimal detection results may arise from multiple sources. In addition to the internal variability and the uncertainty in the fingerprint shapes, this study shows that it is important to take into account the nonlinearity effect due to the cross-correlation of geophysical variables in order to obtain accurate detection results. We note that the relative importance of the three sources may vary in different problems. In assessing the climate change between two steady states before and after doubling CO2, the fingerprint shape uncertainty and nonlinearity effect dominate the internal variability. If, however, a short-term transient climate change is studied, the internal variability may dominate because its impact scales inversely with time.
 Some signals cannot be unambiguously detected because the distinctions in their fingerprints are obscured by the uncertainties. This ambiguity issue exists to different extents for the signals of low cloud and surface temperature, of tropospheric temperature and high cloud, of tropospheric temperature and water vapor, and of clouds at adjacent levels. Greenhouse-gas forcing and tropospheric water vapor and temperature feedbacks in the double-CO2 experiment can be unambiguously detected with marginal uncertainties substantially less than their magnitudes. Surface temperature response and cloud feedbacks, however, are subject to larger errors that may yield null detection results if the fingerprint shape uncertainty is not adequately constrained. Because the accuracy in fingerprints limits the accuracy in OD results, in applying OD to real observational data, it is advisable to minimize the uncertainty in spectral fingerprints. For example, for detecting regional forcing and feedbacks, fingerprints should be computed from local profiles.
 Signal ambiguity means that it is difficult to disentangle some signals from longwave spectral measurements alone. However, such ambiguity can be largely removed by incorporating other measurements. For instance, reflected solar measurements may aid in distinguishing low cloud signal from surface temperature signal due to their distinctive optical properties in this spectral region. GPS radio occultation measurements may help distinguish atmospheric temperature from high cloud because microwave radio signals are sensitive to temperature but insensitive to clouds. We plan to focus on such investigations in future research.
 There are a few caveats about this study. First, the simulated climate change in OLR spectrum may differ from the actual climate change signal for several reasons: the change in atmospheric and surface states simulated by CCCMA may differ to some unknown extent from the actual climate change, the all-sky radiance simulated from monthly mean profiles with a coarse vertical resolution (17 layers) may be biased, and some forcing species such as O3 and CH4 that have noticeable impacts on the longwave spectrum and interact with other geophysical variables (e.g., atmospheric temperature) are omitted. However, the OLR spectral change forced by doubling CO2 as simulated in this study does closely resemble that of Huang and Ramaswamy  forced by multiple greenhouse gases and aerosols and using a different climate model; the major differences lie in the absorption bands of those greenhouse gases not included in this study, that is, O3, CH4, N2O, and CFCs. Huang and Ramaswamy's  simulations prescribed with different forcing combinations also show that the surface and atmospheric responses to different well-mixed greenhouse gases are similar in spatial pattern and in OLR spectral characteristics. Nevertheless, attention should be paid to these details in future simulation experiments and eventually when seeking to explain real data using OD. Second, there are also limitations in the computation of spectral fingerprints and quantification of their uncertainties. Due to the approximations used in our spectral simulations, the distinctiveness between some spectral fingerprints may have been underestimated. For instance, the cloud emissivity in this study is determined from an empirical approximation of coarse spectral resolution while Earth's surface is assumed to be a blackbody. If more realistic schemes are used, the low cloud and surface ambiguity may be reduced. In addition, representing fingerprint shape uncertainty by the spatial variability of fingerprints may lead to an overestimate of the uncertainty. Setting the tropopause to 200 hPa globally as a matter of convenience may artificially alias some tropospheric signals into stratospheric signals or vice versa. This may obscure the distinctiveness of stratospheric fingerprints. Last, although determination of surface temperature change is necessary for quantifying the conventionally defined climate feedbacks, benchmark measurements of surface temperature are beyond the scope of this study, so we have focused on the attribution of OLR changes, which is a key aspect in the longwave feedback analysis.
 In conclusion, based on a GCM simulation of climate change forced by doubling of CO2 in the atmosphere, we have theoretically investigated the separation of longwave climate feedbacks from spectral OLR observations in the all-sky condition. Spectral fingerprints of greenhouse-gas forcing and different feedbacks, namely atmospheric temperature, water vapor, and cloud feedbacks, are shown to exhibit different radiance spectral features that allow the detection and separation of individual signals from the overall change in the OLR spectrum. This study establishes that the optimal detection method can be directly applied to all-sky satellite spectral radiance measurement and that longwave climate forcing and feedback amplitudes can be acquired from observational data. This study also provides an understanding of the strengths as well as limitations of this method.
 We thank Richard Goody and three anonymous reviewers for their thoughtful comments, which improved the quality of this paper. Y.H. is supported by a NOAA Climate and Global Change postdoctoral fellowship.