2.1 GCM-based radiative kernels

The radiative kernel technique was developed as a way to assess various climate feedbacks from climate change simulations across multiple climate models in a computationally efficient manner (Shell et al., 2008; Soden et al., 2008). A radiative kernel is defined as the differential response of an outgoing radiation flux at TOA to an incremental change in some climate state variable – such as water vapor, air temperature, or surface albedo (Soden et al., 2008). To generate a radiative kernel for a change in surface albedo with a GCM, the prescribed surface albedo change is perturbed incrementally by 1 %, and the response by the outgoing shortwave radiation flux at TOA is recorded:

where ${\mathrm{SW}}_{\uparrow }^{\mathrm{TOA}}$ is the outgoing shortwave flux at TOA and $K_{{\mathit{\alpha }}_{\mathrm{s}}}$ is the radiative kernel (in W m⁻²), which can then be used with Eq. (1) to estimate an instantaneous shortwave radiative forcing (ΔF) at TOA:

To the best of our knowledge, four albedo change kernels have been developed based on the following GCMs: the Community Atmosphere Model version 3, or CAM3 (Shell et al., 2008), the Community Atmosphere Model version 5, or CAM5 (Pendergrass et al., 2018), the European Center and Hamburg model version 6, or ECHAM6 (Block and Mauritsen, 2014), and the Geophysical Fluid Dynamics Laboratory model version AM2p12b, or GFDL (Soden et al., 2008). These four GCM kernels vary in their vertical and horizontal resolutions, their parameterizations of shortwave radiative transfer, and their prescribed atmospheric state climatologies. These differences are summarized in Table 1. Apart from differences in their prescribed atmospheric background states and radiative transfer schemes, a major source of uncertainty in GCM-based kernels is related to the GCM representation of atmospheric liquid water or ice associated with convective clouds; of the four aforementioned GCMs, only CAM5 and GFDL attempt to model the effects of convective core ice and liquid in their radiation calculations (Li et al., 2013).

Table 1Attributes of existing GCM kernels, all of which having a monthly temporal resolution.

^∗ Atmospheric CO₂ concentration = 284.7 ppmv; exact time period unknown.

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2.2 Single-layer atmosphere models of shortwave radiation transfer

Within the atmospheric science community, various simplified analytical or semiempirical modeling frameworks have been developed, either to diagnose effective surface and atmospheric optical properties from climate model outputs or to study the relative contributions of changes to these properties on shortwave flux changes at the top and bottom of the atmosphere (Atwood et al., 2016; Donohoe and Battisti, 2011; Kashimura et al., 2017; Qu and Hall, 2006; Rasool and Schneider, 1971; Taylor et al., 2007; Winton, 2005, 2006). While these frameworks all treat the atmosphere as a single layer, they differ by whether or not the reflection and transmission properties of this layer are assumed to have a directional dependency (Stephens et al., 2015) and by whether or not inputs other than those derived from the boundary fluxes are required (e.g., cloud properties; Qu and Hall, 2006).

Winton (2005) presented a semiempirical four-parameter optical model to account for the directional dependency of up- and downwelling shortwave fluxes through the one-layer atmosphere and found good agreement (rRMSE <2 % globally) when this was benchmarked to online radiative transfer calculations. Also considering a directional dependency of the atmospheric optical properties, Taylor et al. (2007) presented a two-parameter analytical model where atmospheric absorption was assumed to occur at a level above atmospheric reflection. The analytical model of Donohoe and Battisti (2011) subsequently relaxed the directional dependency assumption and found the atmospheric attenuation of the surface albedo contribution to planetary albedo to be 8 % higher than the model of Taylor et al. (2007). Elsewhere, Qu and Hall (2006) developed an analytical framework making use of additional atmospheric properties such as cloud cover fraction, cloud optical thickness, and the clear-sky planetary albedo, which proved highly accurate when model estimates of planetary albedo were evaluated against climate models and satellite-based datasets.

2.3 Simple empirical parameterizations of the LULCC science community

Two simple empirical parameterizations of shortwave radiative transfer have been widely applied within the LULCC science community for estimating ΔF from Δα_s (Bozzi et al., 2015; Caiazzo et al., 2014; Carrer et al., 2018; Cherubini et al., 2012; Lutz et al., 2015; Muñoz et al., 2010). While these parameterizations are also based on a single-layer atmosphere model of shortwave radiative transfer, at the core of these parameterizations is the fundamental assumption that radiative transfer is wholly independent of (or unaffected by) Δα_s. In other words, they neglect the change in the attenuating effect of multiple reflections between the surface and the atmosphere that accompanies a change in the surface albedo. Nevertheless, due to their simplicity and ease of application they continue to be widely employed in climate research.

3.1 Analytical kernels

The first kernel candidate may be analytically derived from the CERES EBAF all-sky boundary fluxes and their derivatives. The surface contribution to the outgoing shortwave flux at TOA ${\mathrm{SW}}_{\uparrow ,\mathrm{SFC}}^{\mathrm{TOA}}$ can be expressed (Donohoe and Battisti, 2011; Stephens et al., 2015; Winton, 2005) as

where r is a single-pass atmospheric reflection coefficient, a is a single-pass atmospheric absorption coefficient, ${\mathrm{SW}}_{\downarrow }^{\mathrm{TOA}}$ is the extraterrestrial (downwelling) shortwave flux at TOA, and α_s is the surface albedo (defined in Table 2). The expression in the denominator of the right-hand term represents a fraction attenuated by multiple reflections between the surface and the atmosphere. This model assumes that the atmospheric optical properties r and a are insensitive to the origin and direction of shortwave fluxes or – in other words – that they are isotropic.

The single-pass reflectance coefficient is calculated from the system boundary fluxes (Table 2) following Winton (2005) and Kashimura et al. (2017):

while the single-pass absorption coefficient a is given as

where T is the clearness index (defined in Table 2). Our interest is in quantifying the ${\mathrm{SW}}_{\uparrow ,\mathrm{SFC}}^{\mathrm{TOA}}$ response to an albedo perturbation at the surface – or the partial derivative of ${\mathrm{SW}}_{\uparrow ,\mathrm{SFC}}^{\mathrm{TOA}}$ with respect to α in Eq. (5):

where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ISO}}$ is referred to henceforth as the isotropic kernel.

The second analytical kernel is based on the model of Qu and Hall (2006) which makes use of auxiliary cloud property information commonly provided in satellite-based products of Earth's radiation budget – including CERES EBAF – such as cloud cover area fraction, cloud visible optical depth, and clear-sky planetary albedo. This model links all-sky and clear-sky effective atmospheric transmissivities of the earth system through a linear coefficient k relating the logarithm of cloud visible optical depth to the effective all-sky atmospheric transmissivity:

where T_a,CLR is the clear-sky effective system transmissivity, T_a is the all-sky effective system transmissivity, and τ is the cloud visible optical depth. This linear coefficient can then be used together with the cloud cover area fraction to derive a shortwave kernel based on the model of Qu and Hall (2006) – or $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$:

where c is the cloud cover area fraction.

3.2 Semiempirical kernel

The third kernel makes use of three directionally dependent (anisotropic) bulk optical properties r_↑, t_↑, and t_↓, where the first is the atmospheric reflectivity to upwelling shortwave radiation and the latter two are the atmospheric transmission coefficients for upwelling and downwelling shortwave radiation, respectively (Winton, 2005). It is not possible to derive r_↑ analytically from the all-sky boundary fluxes; however, Winton (2005) provides an empirical formula relating upwelling reflectivity r_↑ to the ratio of all-sky to clear-sky fluxes incident at the surface:

where ${\mathrm{SW}}_{\downarrow ,\mathrm{CLR}}^{\mathrm{SFC}}$ is the clear-sky shortwave flux incident at the surface.

Knowing r_↑, we can then solve for the two remaining optical parameters needed to obtain our kernel:

where T_a is the effective atmospheric transmittance (Table 2) of the earth system.

The kernel may now be expressed as

where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ANISO}}$ is henceforth referred to as the anisotropic kernel.

3.3 Existing empirical parameterizations

Although not referred to as “kernels” in the literature per se, we present the simple empirical parameterizations as such to ensure consistency with previously described notation and terminology.

The first candidate parameterization, originally presented in Muñoz et al. (2010), makes use of a local two-way transmittance factor based on the local clearness index:

where ${\mathrm{SW}}_{\downarrow }^{\mathrm{TOA}}$ is the local incoming solar flux at TOA, T is the local clearness index, and $\partial {\mathrm{SW}}_{\uparrow }^{\mathrm{TOA}}/\partial {\mathit{\alpha }}_{\mathrm{s}}$ is the approximated change in the upwelling shortwave flux at TOA due to a change in the surface albedo.

The second candidate parameterization, originally proposed in Cherubini et al. (2012), makes direct use of the solar flux incident at the surface ${\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}$ combined with a one-way transmission constant k:

where k is based on the global annual mean share of surface reflected shortwave radiation exiting a clear sky (Lacis and Hansen, 1974; Lenton and Vaughan, 2009) and is hence temporally and spatially invariant. This value – or 0.85 – is similar to the global mean ratio of forward-to-total shortwave scattering reported in Iqbal (1983). Bright and Kvalevåg (2013) evaluated Eq. (16) at several global locations and found large biases for some regions and months, despite good overall performance globally (rRMSE = 7 %; n=120 months).

3.4 Proposed empirical parameterization

To determine whether the GCM-based kernels could be approximated with sufficient fidelity using other simpler model formulations based on their own boundary data, we applied machine learning to identify potential model forms using GCM shortwave boundary fluxes as input. For the two GCMs kernels in which the GCM's own shortwave boundary fluxes are also made available (CAM5 and ECHAM6), we used machine learning to minimize the sum of squared residuals between the four shortwave boundary fluxes (i.e., ${\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}$, ${\mathrm{SW}}_{\downarrow }^{\mathrm{TOA}}$, ${\mathrm{SW}}_{\uparrow }^{\mathrm{SFC}}$, ${\mathrm{SW}}_{\uparrow }^{\mathrm{TOA}}$) and the GCM kernel at the monthly time step. The reference dataset consisted of a random global sample of 200 000 monthly kernel grid cells at a native model resolution (97 % and 32 % of all cells for ECHAM6 and CAM5, respectively), of which 50 % were used for training and 50 % for validation. Models were identified using a form of genetic programming known as symbolic regression (Eureqa^®; Nutonian Inc.; Schmidt and Lipson, 2009, 2010), which searches a wide space of model structures as constrained by user input. In our case, we allowed the model to include the operators (i.e., addition, subtraction, multiplication, division, sine, cosine, tangent, exponential, natural logarithm, factorial, power, square root), but numerical coefficients were forbidden. The model search was allowed to continue until the percent convergence and maturity metrics exceeded 98 % and 50 %, respectively, at which point more than 1×10¹¹ formulae had been evaluated. A parsimonious solution was chosen by minimizing the error metric and model complexity using the Pareto front (Fig. S1 of the Supplement) (Smits and Kotanchek, 2005). Between CAM5 and ECHAM6, four common model solutions were found (Table S1 of the Supplement). The best of these common solutions is subsequently referred to as $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ and is given as

4.1 Initial candidate screening

The four GCM kernels presented in Sect. 2.1 are employed as benchmarks to initially screen the six simple model candidates introduced from Sect. 3.2 to 3.4. We compute a skill metric analogous to the “relative error” metric used to evaluate GCMs by Anav et al. (2013) that takes into account error in the spatial pattern between a model and an observation. Because we have no true observational reference, our evaluation instead focuses on the disagreement or deviation between CERES and GCM kernels at the monthly time step. Given interannual climate variability in the earth system, the challenge of comparing the multiyear CERES kernel to a single-year GCM kernel can be partially overcome by averaging the four GCM kernels.

Using the multi-GCM mean as the reference, we first compute the absolute deviation ${\mathrm{AD}}_{m,p}^X$ as

where ${\mathrm{CERES}}_{m,p}^X$ is the kernel for CERES model candidate x in month m and pixel p and ${\stackrel{\mathrm{‾}}{\mathrm{GCM}}}_{m,p}$ is the multi-GCM mean of the same pixel and month. ${\mathrm{AD}}_{m,p}^X$is then normalized to the maximum absolute deviation of all six CERES kernels for the same pixel and month to obtain a normalized absolute deviation, ${\mathrm{NAD}}_{m,p}^X$, which is analogous to the relative error metric of Anav et al. (2013), having values ranging between 0 and 1:

where $max({\mathrm{AD}}_{m,p}$) is the maximum absolute deviation of all six CERES kernels at pixel p and month m.

CERES kernel ranking is based on the mean relative absolute deviation in both space and time – or ${\widehat{\mathrm{NAD}}}^X$:

where M is the total number of months (i.e., 12) and P is the total number of grid cells.

4.2 GCM kernel emulation

In order to eliminate any bias related to differences in the atmospheric state embedded in the GCM kernel input climatologies, we emulate them by applying the top candidate models (as identified from the initial performance screening described in Sect. 4.1) using the original GCM boundary fluxes as input. Emulation is only done for two of the GCM-based kernels since only two of them have provided the accompanying boundary fluxes needed to do so: ECHAM6 (Block and Mauritsen, 2014) and CAM5 (Pendergrass et al., 2018). Emulation enables a more critical evaluation of the functional form of the candidate models in relation to the more sophisticated radiative transfer schemes employed by ECHAM6 (Stevens et al., 2013) and CAM5 (Hurrell et al., 2013).

4.3 CACK model uncertainty

Following emulation, monthly GCM kernels are then regressed on the monthly kernels emulated with the leading model candidates. The model that best emulates both GCM kernels – as measured in terms of the mean coefficient of determination (R²) and mean RMSE – is chosen to represent CACK.

Three sources of uncertainty are considered for CACK when based on the CERES boundary flux climatology (i.e., 2001–2016 monthly means): (1) physical variability, (2) data uncertainty, and (3) model error (Mahadevan and Sarkar, 2009). The first is related to the interannual variability of Earth's atmospheric state and boundary radiative fluxes. The second is related to the uncertainty of the CERES EBAF v4 variables used as input to CACK (including measurement error). The third source of uncertainty is the error related to CACK's model form. CACK's combined uncertainty for any given pixel and month is estimated as follows, where if CACK or y is some nonlinear function of the CERES boundary inputs x₁ and x₂ that covary in time and space, then the combined uncertainty of y – or σ(y) – may be expressed as the sum of the model error plus the combined physical variability and data uncertainty associated with x₁ and x₂ summed in quadrature (Breipohl, 1970; Clifford, 1973; Green et al., 2017):

where σ_PV(x₁) and σ_PV(x₂) are the standard deviations of the 16-year climatological record of CERES input variables x₁ and x₂, respectively, for a given grid cell and month, σ_DU (x₁) and σ_DU (x₂) are the absolute uncertainties of CERES input variables x₁ and x₂, respectively, for a given grid cell and month, σ(x₁,x₂) is the covariance within the 16-year climatological record between CERES input variables x₁ and x₂ for a given month and grid cell, and σ_ME is the monthly grid cell model error. Model error (σ_ME(y)) and data uncertainties (σ_DU(x_n)) for any given grid cell and month are based on the relative RMSE (Supplement) and relative uncertainties of CERES boundary terms reported in Kato et al. (2018) (cf. Table 8, “Monthly gridded, Ocean + Land”) and Loeb et al. (2017) (cf. Table 8, “All-sky, Terra-Aqua period”). For the model error, we take the relative RMSE of the machine learning model solutions for ECHAM5 and CAM5. For the relative uncertainty of the incoming solar flux at TOA (${\mathrm{SW}}_{\downarrow }^{\mathrm{TOA}}$), we use the 1 % “calibration uncertainty” reported in Loeb et al. (2017).

If CACK's intended application is to estimate a temporally explicit ΔF within the CERES era (i.e., if temporally explicit rather than the climatological-mean CERES boundary fluxes are desired to compute CACK), the uncertainty related to physical variability (σ_PV(x_n)) can be dropped from Eq. (21).

4.4 Climatological CACK application example

To demonstrate CACK's application when based on monthly CERES EBAF climatology, including the handling of uncertainty, we estimate the annual mean local ΔF from a Δα_s scenario associated with hypothetical deforestation in the tropics, where ΔF for a given month is estimated as Eq. (4) where $K_{{\mathit{\alpha }}_{\mathrm{s}}}$ is the 2001–2016 monthly climatological CACK and Δα_s is the difference in the 2001–2011 monthly climatological-mean white-sky surface albedo between “croplands” (CRO) and “evergreen broadleaved forests” (EBF) taken from Gao et al. (2014), which is based on International Geosphere-Biosphere Program definitions of land cover classification.

The monthly climatological albedo lookup maps of Gao et al. (2014) contain their own uncertainties, which we take as the mean absolute difference between the monthly albedos reconstructed using their lookup model and the monthly MODIS retrieval record (cf. Table 3 in Gao et al., 2014).

The total estimated uncertainty linked to the annual local (i.e., grid cell) instantaneous ΔF can thus be expressed (in W m⁻²) as

where $\mathit{\sigma }\left(K_{{\mathit{\alpha }}_{\mathrm{s}},m}\right)/K_{{\mathit{\alpha }}_{\mathrm{s},m}}$ is the relative grid cell uncertainty of CACK and $\mathit{\sigma }(\mathrm{\Delta }{\mathit{\alpha }}_{\mathrm{s}},m)/\mathrm{\Delta }{\mathit{\alpha }}_{\mathrm{s},m}$ is the relative uncertainty of Δα_s in month m defined as

where σ(α_s,m) is the monthly absolute uncertainty of the climatological-mean surface albedo (i.e., of the Gao et al., 2014 product).

4.5 Temporally explicit CACK application example

Use of a temporally explicit CACK may be desirable for time-sensitive applications within the CERES era. This is particularly true for regions experiencing significant changes to the atmospheric state affecting shortwave radiation transfer. A good example is in southern Amazonia where tropical deforestation has been linked to changes in cloud cover (Durieux et al., 2003; Lawrence and Vandecar, 2014; Wright et al., 2017). To exemplify this, we estimate the annual mean instantaneous ΔF for CERES grid cells in the region having experienced both significant positive trends in surface albedo and negative trends in cloud area fraction during the 2001–2016 period. Grid cell trends in surface albedo and cloud area fraction are deemed significant if the slopes of linear fits obtained from local (i.e., grid cell) ordinary least squares regressions have p values ≤0.05. We then apply the slope of the surface albedo trend to represent the monthly mean interannual Δα incurred over the time series together with CACK updated monthly to estimate the local annual mean instantaneous ΔF at each step in the series:

where $K_{{\mathit{\alpha }}_{\mathrm{s}},m}\left(t\right)$ is the monthly CACK in year t of the time series. ΔF is then averaged across all grid cells in the sample, with the results then compared to the ΔF that is computed for the same grid sample using the time-insensitive CAM5 and ECHAM6 kernels (i.e., $K_{{\mathit{\alpha }}_{\mathrm{s}},m}\ne f\left(t\right)$). Using the slope of the surface albedo trend as the Δα_s for all months and years rather than the actual Δα_s,m(t) (i.e., $\mathrm{\Delta }{\mathit{\alpha }}_{\mathrm{s},m}\left(t\right)={\mathit{\alpha }}_{\mathrm{s},m,t}-{\mathit{\alpha }}_{\mathrm{s},m,t-\mathrm{1}}$) yields the same result when averaged over the full time period but allows us to isolate the effect of the changing atmospheric state on calculations of ΔF. We limit the ΔF uncertainty estimate to CACK's uncertainty that includes σ_DU(x_n) and σ_ME(x_n) but excludes σ_PV(x_n).

5.1 Initial performance screening

Seasonally, differences in latitude band means between the CERES kernel candidates and the multi-GCM mean kernels are shown in Fig. 1.

Figure 1Latitudinal (1^∘) and seasonal means of the multi-GCM mean ($K_{\mathit{\alpha }}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$) and CACK model candidates for (a) December–January–February (DJF); (b) March–April–May (MAM); (c) June–July–August (JJA); (d) September–October–November (SON). CACK model candidates refer to those presented in Sect. 3 and not to those of the model selection phase of the machine learning algorithm.

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Qualitatively, starting with December–January–February (DJF), $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ gives the best agreement with $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$ with the exception of the zone around 55–65^∘ S (−55 to −65^∘), where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$ gives slightly better agreement (Fig. 1a). In March–April–May (MAM), $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ appears to give the best overall agreement with the exception of the high Arctic, where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ANISO}}$ and $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{C}\mathrm{12}}$ give better agreement, and with the exception of the zone around 60–65^∘ S (−60 to −65^∘), where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$, $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ANISO}}$, and $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{C}\mathrm{12}}$ agree best with $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$ (Fig. 1b). The largest spread in disagreement across all six CERES kernels is found in June–July–August (JJA; Fig. 1c) at northern high latitudes. $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ appears to agree best both here and elsewhere with the exception of the zone between ∼20–35^∘ N, where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$ gives slightly better agreement.

In September–October–November (SON), $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ agrees best with $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$ at all latitudes except the zone between 10–25^∘ N and 55–65^∘ S, where $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$agrees slightly better.

Figure 2(a–f) Scatter–density regressions of global monthly mean $K_{\mathit{\alpha }}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$ (y axis) and $K_{\mathit{\alpha }}^{\mathrm{CERES}}$ (x axis), with the CERES kernel identifier shown at the top of each subpanel. “m”: slope; “B₀”: y intercept. The color scale indicates the percentage of regression points that fall within an averaging bin, where the x axis and y axis have been gridded into 100×100 equally spaced bins to help illustrate the density of overlapping points.

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Quantitatively, the proportion of the total variance explained by linear regressions of monthly $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$ on monthly $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{CERES}}$ (i.e., “R²”) is highest and equal for the CERES kernels based on the ANISO, QH06, and BO18 models (Fig. 2b, c, d). Of these three, $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$ has a y intercept (“B₀”) closest to 0 and a slope (“m”) of 1, although the root mean squared error (“RMSE”) – an accuracy measure – is slightly better (lower) for $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$. The two CERES kernels with the lowest R², highest slopes (negative deviations), highest RMSEs, and y intercepts with the largest absolute difference from zero – or the worst performing candidates – are those based on the ISO and M10 models (Fig. 2a, e).

Although the y intercept deviation from 0 for $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{C}\mathrm{12}}$ is relatively low, its RMSE is ∼50 % higher than that of $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$, $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$, and $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ANISO}}$ and leads to notable positive deviation from the multi-GCM mean ($K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\stackrel{\mathrm{‾}}{\mathrm{GCM}}}$) judging by its slope of 0.92.

Globally, $\widehat{\mathrm{NAD}}$ for the QH06, ANISO, and BO18 kernels is far superior to the ISO, M10, and C12 kernels (Table 3).

Table 3Normalized absolute deviation and CERES kernel model candidate ranking.

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After filtering to remove grid cells for oceans and other water bodies, $\widehat{\mathrm{NAD}}$ scores for these three kernels decreased; the decrease was smallest for $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ (−0.03) and largest for $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$ (−0.06). Despite constraining the analysis to land surfaces only, the rank order remained unchanged (Table 3), and $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$, $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$, and $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{ANISO}}$ are subjected to further evaluation.

5.2 GCM kernel emulation and additional performance evaluation

Because the QH06 model ($K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$) required auxiliary inputs for cloud cover area fraction and cloud optical depth – two atmospheric state variables not provided with the ECHAM6 and CAM5 kernel datasets – it was not possible to emulate these two GCM kernels with $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$. Additional performance evaluation through GCM kernel emulation is therefore restricted to the ANISO and BO18 models.

Figure 3(a) Mean annual bias of the CAM5 albedo change kernel emulated with the ANISO semiempirical model; (b) mean annual bias of the CAM5 albedo change kernel emulated with the BO18 parameterization; (c) mean annual bias of the ECHAM6 albedo change kernel emulated with the ANISO semiempirical model; (d) mean annual bias of the ECHAM6 albedo change kernel emulated with the BO18 parameterization.

Globally, the kernel based on the ANISO model displays larger annual mean bias relative to BO18 when compared to both ECHAM6 and CAM5 kernels (Fig. 3). Notable positive biases over land with respect to both ECHAM6 and CAM5 kernels are evident in the northern Andes region of South America, the Tibetan Plateau, and the tropical island region comprising Indonesia, Malaysia, and Papua New Guinea (Fig. 3a, c). Notable negative biases over land with respect to both ECHAM6 and CAM5 kernels are evident over Greenland, Antarctica, northeastern Africa, and the Arabian Peninsula (Fig. 3a, c).

Globally, annual biases for BO18 are generally found to be lower than for ANISO and are mostly non-existent in extra-tropical ocean regions (Fig. 3b, d). Patterns in biases over land are mostly negative with the exception of Saharan Africa where the annual mean bias with respect to both GCMs is positive. For BO18, systematic positive biases – or biases evident with respect to both GCM kernels – appear over eastern tropical and subtropical marine coastal upwelling zones where marine stratocumulus cloud dynamics are difficult for GCMs to resolve (Bretherton et al., 2004; Richter, 2015).

Regression statistics (Fig. 4) indicate a greater overall performance for BO18 than for ANISO. RMSEs for monthly kernels emulated with BO18 are 9.0 and 8.2 W m⁻² for CAM5 and ECHAM6, respectively – which is ∼50 %–60 % of the RMSEs emulated with the ANISO model. Relative to ANISO, the BO18 model also gives a higher R², a slope closer to 1, and a y intercept closer to zero (Fig. 4). The BO18 model (or parameterization) is therefore selected for CACK.

Figure 4(a–d) Scatter–density regressions of $K_{\mathit{\alpha }}^{\mathrm{GCM}}$ (y axis) and $K_{\mathit{\alpha }}^{\mathrm{GCM}}$ emulated with the ANISO semiempirical model and BO18 parameterization using the GCM's own inputs (x axis); “m”: slope; “B₀”: y intercept. See Fig. 2 caption for a description of the color scale.

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Focusing only on the GCM kernels emulated with $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ henceforth, global mean negative biases are evident in all months (Table 4), with the largest biases (in magnitude) appearing in May (−4.4 W m⁻²) and November (−2.5 W m⁻²) for CAM5 and ECHAM6, respectively. In absolute terms, the largest biases of 8.6 and 6.8 W m⁻² appear in June for CAM5 and ECHAM6, respectively. Annually, the mean absolute bias for CAM5 and ECHAM6 is 6.8 and 6.1 W m⁻², respectively – a magnitude which seems remarkably low if one compares this to the annual mean disagreement (standard deviation) of 33 W m⁻² across all four GCM kernels (not shown; for seasonal mean standard deviations, see Fig. 1).

Table 4Global monthly mean bias (MB) and mean absolute bias (MAB) for $K_{\mathit{\alpha }}^{\mathrm{BO}\mathrm{18}}$ emulated with T and ${\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}$ from ECHAM6 and CAM5. For reference, the global mean value of $K_{\mathit{\alpha }}^{\mathrm{BO}\mathrm{18}}$ is 133 W m⁻².

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5.3 CACK uncertainty

For a kernel based on 2001–2016 monthly mean CERES EBAF climatology, Fig. 5 illustrates the contribution of the absolute error related to $K_{{\mathit{\alpha }}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$'s model form (Fig. 5a, annual mean) relative to CACK's total absolute uncertainty (Fig. 5c, annual mean), which includes the uncertainty surrounding CERES EBAF v4 input variables ${\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}$ and ${\mathrm{SW}}_{\downarrow }^{\mathrm{TOA}}$ and their interannual variability (Fig. 5b, annual mean).

Figure 5Annual uncertainty of a CACK based on 2001–2016 monthly mean CERES EBAF v4 climatology: (a) the absolute uncertainty related to model error (i.e., the parameterization); (b) the total propagated absolute uncertainty related to physical variability and data uncertainty of CACK input variables; (c) total absolute uncertainty; (d) total relative uncertainty.

Total propagated σ_PV and σ_DU far exceeds σ_ME, is dominated by ${\mathit{\sigma }}_{\mathrm{DU}}\left({\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}\right)$ and ${\mathit{\sigma }}_{\mathrm{PV}}({\mathrm{SW}}_{\downarrow }^{\mathrm{SFC}}$), and is largest in the Pacific region to the south of the intertropical convergence zone (ITCZ). Over land, the annual σ_PV and σ_DU as well as the annual σ_total are generally largest in arid or high-altitude regions (Fig. 5b). However, annual CACK values are also large in these regions, reducing the relative uncertainty (Fig. 5d). The largest relative uncertainties over land (on an annual basis) – which can approach 50 % – are found over central Europe, northwestern Asia, southeastern China, Andean Chile, and northwestern North America (Fig. 5d).

5.4 Climatological CACK application

When estimated with a CACK based on monthly CERES EBAF climatology, the annual local ΔF from Δα_s linked to hypothetical deforestation in the tropics is negative in most regions, approaching −20 W m⁻² locally in some regions of the Brazilian Cerrado and south of the Sahel region in Africa (Fig. 6b). The combined CACK and Δα_s uncertainty for these regions can approach ±5 W m⁻² annually (Fig. 6c) in regions like the Brazilian Cerrado and sub-Sahel Africa. Relative to the ΔF magnitude, however, the largest uncertainties (annual) may be found in the subtropical regions of Central America, southern Brazil, southern Asia, and northern Australia, where they can approach 30 %–40 % (Fig. 6d).

Figure 6Example application of a CACK based on the 2001–2016 monthly mean CERES EBAF v4 climatology to estimate the local annual mean ΔF from a hypothetical land cover change within a CERES grid cell. (a) Annual mean of the climatological (i.e., 2001–2011) monthly mean difference in white-sky surface albedo between croplands and evergreen broadleaved forests (Δα_s) based on the 1^∘ product of Gao et al. (2014); (b) annual mean local (i.e., within grid cell) instantaneous radiative forcing (ΔF) of monthly mean Δα_s estimated with CACK; (c) absolute uncertainty (annual mean) of the CACK-basedΔF estimate, including the uncertainty of Δα_s; (d) relative uncertainty (annual mean) of the CACK-based ΔF estimate.

5.5 Temporally explicit CACK application

The effect of a decreasing cloud cover and increasing surface albedo trend in southern Amazonia (Fig. 7b) on shortwave radiative transfer and thus a CACK-based estimate of regional mean annual ΔF emerges in Fig. 7c, where ΔF increases in magnitude by 0.004 W m⁻² from 2002 to 2016. This ΔF trend would otherwise go undetected if a GCM-based kernel were applied to the same surface albedo trend – that is, to a sustained positive interannual monthly albedo change “pulse”. Alternatively, a CACK based on 2001 CERES EBAF inputs (applied with Δα_s for 2001–2002) would give slightly higher ΔF estimates relative to those based on ECHAM6 and CAM5 kernels; conversely, a CACK based on 2015 CERES EBAF inputs (applied with Δα_s for 2015–2016) would yield lower ΔF estimates relative to those based on the same two GCM-based kernels (Fig. 7c). The use of temporally explicit CACK can therefore capture ΔF trends related to a changing atmospheric state that fixed-state GCM kernels are unable to capture.

Figure 7Example application of a temporally explicit CACK; (a) 2001–2016 statistically significant positive trends in all-sky surface albedo derived from CERES EBAF-Surface v4; (b) 2001–2016 statistically significant negative trends in cloud area derived from CERES EBAF-TOA v4; (c) mean ΔF from Δα_s when estimated with the CACK, ECHAM6, and CAM5 surface albedo change kernels. ΔF is the mean of all grid cells plotted in panel (a). The 1σ confidence interval (“CI”) shown for CACK excludes the uncertainty component related to physical variability.