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**Geoscientific Model Development**
An interactive open-access journal of the European Geosciences Union

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**Model description paper**
09 Sep 2019

**Model description paper** | 09 Sep 2019

Developing a monthly radiative kernel for surface albedo change from satellite climatologies of Earth's shortwave radiation budget: CACK v1.0

^{1}Department of Forests and Climate, Division of Forestry and Forest Resources, Norwegian Institute of Bioeconomy Research, Ås, Norway^{2}Department of Forestry and Environmental Conservation, Clemson University, Clemson, South Carolina, USA^{3}Baruch Institute of Coastal Ecology and Forest Science, Clemson University, Georgetown, South Carolina, USA

^{1}Department of Forests and Climate, Division of Forestry and Forest Resources, Norwegian Institute of Bioeconomy Research, Ås, Norway^{2}Department of Forestry and Environmental Conservation, Clemson University, Clemson, South Carolina, USA^{3}Baruch Institute of Coastal Ecology and Forest Science, Clemson University, Georgetown, South Carolina, USA

**Correspondence**: Ryan M. Bright (ryan.bright@nibio.no)

**Correspondence**: Ryan M. Bright (ryan.bright@nibio.no)

Abstract

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Due to the potential for land-use–land-cover change (LULCC) to alter
surface albedo, there is need within the LULCC science community for simple
and transparent tools for predicting radiative forcings (Δ*F*) from
surface albedo changes (Δ*α*_{s}). To that end, the radiative
kernel technique – developed by the climate modeling community to diagnose
internal feedbacks within general circulation models (GCMs) – has been
adopted by the LULCC science community as a tool to perform offline Δ*F* calculations for Δ*α*_{s}. However, the codes and data behind
the GCM kernels are not readily transparent, and the climatologies of the
atmospheric state variables used to derive them vary widely both in time
period and duration. Observation-based kernels offer an attractive
alternative to GCM-based kernels and could be updated annually at relatively
low costs. Here, we present a radiative kernel for surface albedo change
founded on a novel, simplified parameterization of shortwave radiative
transfer driven with inputs from the Clouds and the Earth's Radiant Energy
System (CERES) Energy Balance and Filled (EBAF) products. When constructed
on a 16-year climatology (2001–2016), we find that the CERES-based albedo
change kernel – or CACK – agrees remarkably well with the mean kernel of
four GCMs (rRMSE = 14 %). When the novel parameterization underlying
CACK is applied to emulate two of the GCM kernels using their own boundary
fluxes as input, we find even greater agreement (mean rRMSE = 7.4 %),
suggesting that this simple and transparent parameterization represents a
credible candidate for a satellite-based alternative to GCM kernels. We
document and compute the various sources of uncertainty underlying CACK and
include them as part of a more extensive dataset (CACK v1.0) while providing
examples showcasing its application.

How to cite

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How to cite.

Bright, R. M. and O'Halloran, T. L.: Developing a monthly radiative kernel for surface albedo change from satellite climatologies of Earth's shortwave radiation budget: CACK v1.0, Geosci. Model Dev., 12, 3975–3990, https://doi.org/10.5194/gmd-12-3975-2019, 2019.

1 Introduction

Back to toptop
Diagnosing changes to the shortwave radiation balance at the
top of the atmosphere (TOA) resulting from changes to albedo at the surface
(Δ*α*_{s}) is an important step in predicting climate change.
However, outside the climate science community, many researchers do not have
the tools to convert Δ*α* to the climate-relevant Δ*F*
measure (Bright, 2015; Jones et al., 2015), which requires a detailed
representation of the atmospheric constituents that absorb or scatter solar
radiation (e.g., cloud, aerosols, and gases) and a sophisticated radiative
transfer code. For single points in space or for small regions, these
calculations are typically performed offline – meaning without feedbacks to
the atmosphere (e.g., Randerson et al.,
2006). Large-scale investigations (e.g., Amazonian or pan-boreal land-use–land-cover change, LULCC; Bonan et al., 1992; Dickinson and Henderson-Sellers, 1988) typically
prescribe the land surface layer in a general circulation model (GCM) with initial and perturbed states,
allowing the radiative transfer code to interact with the rest of the model.
While this has the benefit of allowing interaction and feedbacks between
surface albedo and scattering or absorbing components of the model, such an
approach is computationally expensive and thereby restricts the number of
LULCC scenarios that can be investigated (Atwood et al.,
2016). Consequently, this method does not meet the needs of some modern
LULCC studies which may require millions of individual land cover
transitions to be evaluated cost effectively (Ghimire et al., 2014; Lutz
and Howarth, 2015).

Within the LULCC science community, two methods have primarily met the need
for efficient Δ*F* calculations from Δ*α*_{s}: simplified
parameterizations of atmospheric transfer of shortwave radiation (Bozzi
et al., 2015; Bright and Kvalevåg, 2013; Caiazzo et al., 2014; Carrer et
al., 2018; Cherubini et al., 2012; Muñoz et al., 2010) and radiative
kernels (Ghimire et al., 2014; O'Halloran et al., 2012; Vanderhoof et
al., 2013) derived from sophisticated radiative transfer schemes embedded in
GCMs (Block and Mauritsen, 2014; Pendergrass et al., 2018; Shell et al.,
2008; Soden et al., 2008). Simplified parameterizations of the LULCC science
community have not been evaluated comprehensively in space and time. Bright
and Kvalevåg (2013) evaluated the shortwave Δ*F*
parameterization of Cherubini et al. (2012) when applied at several
globally distributed sites on land, finding inconsistencies in performance
at individual sites despite good overall cross-site performance. Radiative
kernels (Block and Mauritsen, 2014; Pendergrass et al., 2018; Shell et
al., 2008; Soden et al., 2008) – while being based on state-of-the-art
models of radiative transfer – have the downside of being model-dependent
and not readily transparent. While the radiative transfer codes behind them
are well-documented, the scattering components (i.e., aerosols, gases, and
clouds) affecting transmission have many simplifying parameterizations, vary
widely across models, and may contain significant biases (Dolinar et al.,
2015; Wang and Su, 2013). An additional downside is that the atmospheric
state climatologies used to compute the GCM kernels vary widely in their
time periods (i.e., from the preindustrial period to the year 2007) and durations
(from 1 to 1000 years). The application of a state-dependent GCM kernel that
is outdated may be undesirable in regions undergoing rapid changes in cloud
cover or aerosol optical depth, such as in the northwest United States
(Free and Sun, 2014) and in southern (Srivastava, 2017) and eastern (Zhao et al., 2018) Asia, respectively. An albedo change
kernel based on Earth-orbiting satellite products could be updated annually
to capture changes in atmospheric state at relatively low costs.

The NASA Clouds and the Earth's Radiant Energy System (CERES) Energy Balance and Filled (EBAF) products (CERES Science Team, 2018a, b), which are based largely on satellite optical remote sensing, provide the monthly mean boundary fluxes and other atmospheric state information (e.g., cloud area fraction, cloud optical depth) that could be used to develop a more empirically based alternative to the GCM-based kernels. The latest EBAF-TOA Ed4.0 (version 4.0) products have many improvements with respect to the previous version (version 2.8; Loeb et al., 2009), including the use of advanced and more consistent input data, retrieval of cloud properties, and instrument calibration (Kato et al., 2018; Loeb et al., 2017).

Here, we present an albedo change kernel based on the CERES EBAF v4 products – or CACK. Underlying CACK is a simplified model of shortwave radiative transfer through a one-layer atmosphere. The model form (or parameterization) is selected after a two-stage performance evaluation of six model candidates: two analytical, one semiempirical, and three empirical. An initial performance screening is implemented where all six model candidates are driven with a 16-year climatology (January 2001–December 2016) of monthly all-sky boundary fluxes from CERES, with the resulting kernels benchmarked both qualitatively and quantitatively against the mean of four GCM-based kernels (Block and Mauritsen, 2014; Pendergrass et al., 2018; Shell et al., 2008; Soden et al., 2008). Top model candidates from the initial performance screening are then subjected to an additional performance evaluation where they are applied to emulate two GCM kernels using their own boundary fluxes as input, which eliminates possible biases related to differences in the GCM representation of clouds or other atmosphere state variables.

We start in Sect. 2 by providing a brief overview of existing approaches
applied in LULCC climate studies for estimating Δ*F* from Δ*α*. We then present the six model candidates in Sect. 3. Section 4
describes the model evaluation and uncertainty quantification methods, in
addition to two application examples. Results are presented in Sect. 5,
while Sect. 6 discusses the merits and uncertainties of a CERES-based
kernel relative to GCM-based kernels.

2 Review of existing approaches

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Earth's energy balance (at TOA) in an equilibrium state can be written as

$$\begin{array}{}\text{(1)}& \mathrm{0}=F={\mathrm{LW}}_{\uparrow}^{\mathrm{TOA}}-\left({\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}-{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}\right),\end{array}$$

where the equilibrium flux *F* is a balance between the net solar energy inputs
(${\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}-{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}$) and thermal energy output
(${\mathrm{LW}}_{\uparrow}^{\mathrm{TOA}}$). Perturbing this balance results in a radiative
forcing Δ*F*, while perturbing the shortwave component is referred to
as a shortwave radiative forcing and may be written as

$$\begin{array}{}\text{(2)}& \begin{array}{rl}& \mathrm{\Delta}F=\mathrm{\Delta}\left({\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}-{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}\right)\\ & =\mathrm{\Delta}{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}\left(\mathrm{1}-{\displaystyle \frac{{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}}}\right)-{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}\left(\mathrm{\Delta}{\displaystyle \frac{{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}}}\right),\end{array}\end{array}$$

where the shortwave radiative forcing results either from changes to solar energy inputs ($\mathrm{\Delta}{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}$) or from internal perturbations within the Earth system $\left(\mathrm{\Delta}\frac{{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}}\right)$. The latter can be brought about by changes to the reflective properties of Earth's surface, which is the focus of this paper.

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:

$$\begin{array}{}\text{(3)}& \begin{array}{rl}\mathrm{\Delta}{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}& ={\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}\left({\mathit{\alpha}}_{\mathrm{s}}+\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\right)-{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}\left({\mathit{\alpha}}_{\mathrm{s}}\right)\\ & ={\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\equiv {K}_{{\mathit{\alpha}}_{\mathrm{s}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}\end{array}$$

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
(Δ*F*) at TOA:

$$\begin{array}{}\text{(4)}& \begin{array}{rl}& F+\mathrm{\Delta}F={\mathrm{LW}}_{\uparrow}^{\mathrm{TOA}}-\left({\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}-{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}+{K}_{{\mathit{\alpha}}_{\mathrm{s}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\right)\\ & \mathrm{\Delta}F=-{K}_{{\mathit{\alpha}}_{\mathrm{s}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}.\end{array}\end{array}$$

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).

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.

Two simple empirical parameterizations of shortwave radiative transfer have
been widely applied within the LULCC science community for estimating
Δ*F* from Δ*α*_{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) Δ*α*_{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 Kernel model candidates

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The six candidate models (or parameterizations) for a CERES-based albedo change kernel (CACK) are presented henceforth. All requisite variables and their derivatives may be obtained directly from the CERES EBAF v4 products (at monthly and $\mathrm{1}{}^{\circ}\times \mathrm{1}{}^{\circ}$ resolution) and are presented in Table 2. To improve readability, temporal and spatial indexing is neglected and all terms presented henceforth in Sect. 3 denote the monthly pixel means.

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

$$\begin{array}{}\text{(5)}& {\mathrm{SW}}_{\uparrow ,\mathrm{SFC}}^{\mathrm{TOA}}={\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}{\mathit{\alpha}}_{\mathrm{s}}{\displaystyle \frac{{\left(\mathrm{1}-r-a\right)}^{\mathrm{2}}}{(\mathrm{1}-r{\mathit{\alpha}}_{\mathrm{s}})}},\end{array}$$

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):

$$\begin{array}{}\text{(6)}& r={\displaystyle \frac{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}{\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}-{\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}{\mathrm{SW}}_{\uparrow}^{\mathrm{SFC}}}{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}\phantom{\rule{0.125em}{0ex}}\mathrm{2}}-{\mathrm{SW}}_{\uparrow}^{\mathrm{SFC}\phantom{\rule{0.125em}{0ex}}\mathrm{2}}}},\end{array}$$

while the single-pass absorption coefficient *a* is given as

$$\begin{array}{}\text{(7)}& a=\mathrm{1}-r-T\left(\mathrm{1}-{\mathit{\alpha}}_{\mathrm{s}}r\right),\end{array}$$

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):

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{ISO}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={\displaystyle \frac{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}(\mathrm{1}-r-a{)}^{\mathrm{2}}}{(\mathrm{1}-r{\mathit{\alpha}}_{\mathrm{s}}{)}^{\mathrm{2}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}$$

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:

$$\begin{array}{}\text{(9)}& k={\displaystyle \frac{\left({T}_{\mathrm{a},\mathrm{CLR}}\right)-\left({T}_{\mathrm{a}}\right)}{\mathrm{ln}(\mathit{\tau}+\mathrm{1})}},\end{array}$$

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 ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$:

$$\begin{array}{}\text{(10)}& \begin{array}{rl}{\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}& ={K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\\ & ={\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}\left[\left({T}_{\mathrm{a}}\right)-kc\mathrm{ln}(\mathit{\tau}+\mathrm{1})\right]\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}\end{array}$$

where *c* is the cloud cover area fraction.

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:

$$\begin{array}{}\text{(11)}& {r}_{\uparrow}=\mathrm{0.05}+\mathrm{0.85}\left(\mathrm{1}-{\displaystyle \frac{{\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}}{{\mathrm{SW}}_{\downarrow ,\mathrm{CLR}}^{\mathrm{SFC}}}}\right),\end{array}$$

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:

$$\begin{array}{}\text{(12)}& {\displaystyle}{t}_{\downarrow}={\displaystyle \frac{{\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}-{r}_{\uparrow}{\mathrm{SW}}_{\uparrow}^{\mathrm{SFC}}}{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}}},\text{(13)}& {\displaystyle}{t}_{\uparrow}={T}_{\mathrm{a}}-\left[{t}_{\downarrow}-{t}_{\downarrow}(\mathrm{1}-{r}_{\uparrow}{\mathit{\alpha}}_{\mathrm{s}})\right],\end{array}$$

where *T*_{a} is the effective atmospheric transmittance (Table 2) of the
earth system.

The kernel may now be expressed as

$$\begin{array}{}\text{(14)}& {\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{ANISO}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={\displaystyle \frac{{\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}{t}_{\downarrow}{t}_{\uparrow}}{{\left(\mathrm{1}-{r}_{\uparrow}{\mathit{\alpha}}_{\mathrm{s}}\right)}^{\mathrm{2}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}$$

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

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:

$$\begin{array}{}\text{(15)}& {\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\equiv {K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{M}\mathrm{10}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={\mathrm{SW}}_{\downarrow}^{\mathrm{TOA}}{T}^{\mathrm{2}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}$$

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*:

$$\begin{array}{}\text{(16)}& {\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}\equiv {K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{C}\mathrm{12}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}k\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}$$

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).

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 1×10^{11} 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

$$\begin{array}{}\text{(17)}& {\displaystyle \frac{\partial {\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}}{\partial {\mathit{\alpha}}_{\mathrm{s}}}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}={\mathrm{SW}}_{\downarrow}^{\mathrm{SFC}}\sqrt{T}\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}}.\end{array}$$

4 Kernel model evaluation

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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

$$\begin{array}{}\text{(18)}& {\mathrm{AD}}_{m,p}^{X}=\left|{\mathrm{CERES}}_{m,p}^{X}-{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}_{m,p}\right|,\end{array}$$

where ${\mathrm{CERES}}_{m,p}^{X}$ is the kernel for CERES model candidate *x* in month
*m* and pixel *p* and ${\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}_{m,p}$ 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:

$$\begin{array}{}\text{(19)}& {\mathrm{NAD}}_{m,p}^{X}=\mathrm{1}-{\displaystyle \frac{{\mathrm{AD}}_{m,p}^{X}}{max\left({\mathrm{AD}}_{m,p}\right)}},\end{array}$$

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 ${\widehat{\mathrm{NAD}}}^{X}$:

$$\begin{array}{}\text{(20)}& {\widehat{\mathrm{NAD}}}^{X}={\displaystyle \frac{\mathrm{1}}{M}}\sum _{m=\mathrm{1}}^{M}{\displaystyle \frac{\mathrm{1}}{P}}\sum _{p=\mathrm{1}}^{P}{\mathrm{NAD}}_{m,p}^{X},\end{array}$$

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

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).

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 (*R*^{2}) 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 *x*_{1} and *x*_{2} 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 *x*_{1} and *x*_{2}
summed in quadrature (Breipohl, 1970; Clifford, 1973; Green et al.,
2017):

$$\begin{array}{}\text{(21)}& \begin{array}{rl}\mathit{\sigma}\left(y\right)\approx & \phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathrm{ME}}\left(y\right)+\sqrt{{\left({\displaystyle \frac{\partial y}{\partial {x}_{\mathrm{1}}}}\right)}^{\mathrm{2}}{\left[{\mathit{\sigma}}_{\mathrm{PV}}\left({x}_{\mathrm{1}}\right)+{\mathit{\sigma}}_{\mathrm{DU}}\left({x}_{\mathrm{1}}\right)\right]}^{\mathrm{2}}}\\ & \stackrel{\mathrm{\u203e}}{+{\left({\displaystyle \frac{\partial y}{\partial {x}_{\mathrm{2}}}}\right)}^{\mathrm{2}}{\left[{\mathit{\sigma}}_{\mathrm{PV}}\left({x}_{\mathrm{2}}\right)+{\mathit{\sigma}}_{\mathrm{DU}}\left({x}_{\mathrm{2}}\right)\right]}^{\mathrm{2}}}\\ & \stackrel{\mathrm{\u203e}}{+\sqrt{{\left(\mathrm{2}{\displaystyle \frac{\partial y}{\partial {x}_{\mathrm{1}}}}{\displaystyle \frac{\partial y}{\partial {x}_{\mathrm{2}}}}\mathit{\sigma}({x}_{\mathrm{1}},{x}_{\mathrm{2}})\right)}^{\mathrm{2}}}},\end{array}\end{array}$$

where *σ*_{PV}(*x*_{1}) and *σ*_{PV}(*x*_{2}) are the standard
deviations of the 16-year climatological record of CERES input variables
*x*_{1} and *x*_{2}, respectively, for a given grid cell and month,
*σ*_{DU} (*x*_{1}) and *σ*_{DU} (*x*_{2}) are the absolute
uncertainties of CERES input variables *x*_{1} and *x*_{2}, respectively,
for a given grid cell and month, *σ*(*x*_{1},*x*_{2}) is the covariance
within the 16-year climatological record between CERES input variables *x*_{1} and *x*_{2} 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
Δ*F* 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).

To demonstrate CACK's application when based on monthly CERES EBAF
climatology, including the handling of uncertainty, we estimate the annual
mean local Δ*F* from a Δ*α*_{s} scenario associated with
hypothetical deforestation in the tropics, where Δ*F* for a given month
is estimated as Eq. (4) where ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}$ is the 2001–2016 monthly
climatological CACK and Δ*α*_{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 Δ*F* can thus be expressed (in W m^{−2}) as

$$\begin{array}{}\text{(22)}& \begin{array}{rl}\mathit{\sigma}\left(\mathrm{\Delta}F\right)=& {\displaystyle \frac{\mathrm{1}}{\mathrm{12}}}\sum _{m=\mathrm{1}}^{\mathrm{12}}\left|\mathrm{\Delta}{F}_{m}\right|\sqrt{{\left({\displaystyle \frac{\mathit{\sigma}\left({K}_{{\mathit{\alpha}}_{\mathrm{s}},m}\right)}{{K}_{{\mathit{\alpha}}_{\mathrm{s}},m}}}\right)}^{\mathrm{2}}}\\ & \stackrel{\mathrm{\u203e}}{+{\left({\displaystyle \frac{\mathit{\sigma}\left(\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s},m}\right)}{\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s},m}}}\right)}^{\mathrm{2}}},\end{array}\end{array}$$

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 Δ*α*_{s} in month *m* defined as

$$\begin{array}{}\text{(23)}& {\displaystyle \frac{\mathit{\sigma}\left(\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s},m}\right)}{\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s},m}}}=\sqrt{{\left({\displaystyle \frac{\mathit{\sigma}\left({\mathit{\alpha}}_{\mathrm{s},m}\right)}{{\mathit{\alpha}}_{\mathrm{CRO},m}}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\mathit{\sigma}\left({\mathit{\alpha}}_{\mathrm{s},m}\right)}{{\mathit{\alpha}}_{\mathrm{EBF},m}}}\right)}^{\mathrm{2}}},\end{array}$$

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).

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 Δ*F* 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 Δ*α* incurred over the time series
together with CACK updated monthly to estimate the local annual mean
instantaneous Δ*F* at each step in the series:

$$\begin{array}{}\text{(24)}& \mathrm{\Delta}F\left(t\right)=\sum _{m=\mathrm{1}}^{m=\mathrm{12}}-{K}_{{\mathit{\alpha}}_{\mathrm{s}},m}\left(t\right)\mathrm{\Delta}{\mathit{\alpha}}_{\mathrm{s}},\end{array}$$

where ${K}_{{\mathit{\alpha}}_{\mathrm{s}},m}\left(t\right)$ is the monthly CACK in year *t* of the time
series. Δ*F* is then averaged across all grid cells in the sample, with
the results then compared to the Δ*F* 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 Δ*α*_{s} for all months and years rather than the actual
Δ*α*_{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 Δ*F*. We limit the Δ*F* uncertainty
estimate to CACK's uncertainty that includes *σ*_{DU}(*x*_{n}) and
*σ*_{ME}(*x*_{n}) but excludes *σ*_{PV}(*x*_{n}).

5 Results

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

Qualitatively, starting with December–January–February (DJF), ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ gives the best agreement with ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}$ 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
${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}$ (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 ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}$ 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.

Quantitatively, the proportion of the total variance explained by linear
regressions of monthly ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}$ on monthly
${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{CERES}}$ (i.e., “*R*^{2}”) 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
(“*B*_{0}”) 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
*R*^{2}, 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
(${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\stackrel{\mathrm{\u203e}}{\mathrm{GCM}}}$) 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).

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.

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.

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 *R*^{2}, a slope
closer to 1, and a *y* intercept closer to zero (Fig. 4). The BO18 model (or
parameterization) is therefore selected for CACK.

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).

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).

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).

When estimated with a CACK based on monthly CERES EBAF climatology, the
annual local Δ*F* from Δ*α*_{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 Δ*α*_{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 Δ*F* 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).

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 Δ*F* emerges in Fig. 7c,
where Δ*F* increases in magnitude by 0.004 W m^{−2} from 2002 to 2016.
This Δ*F* 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 Δ*α*_{s} for
2001–2002) would give slightly higher Δ*F* estimates relative to those
based on ECHAM6 and CAM5 kernels; conversely, a CACK based on 2015 CERES
EBAF inputs (applied with Δ*α*_{s} for 2015–2016) would
yield lower Δ*F* estimates relative to those based on the same two
GCM-based kernels (Fig. 7c). The use of temporally explicit CACK can therefore
capture Δ*F* trends related to a changing atmospheric state that
fixed-state GCM kernels are unable to capture.

6 Discussion

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Motivated by an increasing abundance of climate impact research focusing on land processes in recent years, we comprehensively evaluated six simplified models (or parameterizations) as candidates for an albedo change kernel based on the CERES EBAF v4 products (Kato et al., 2018; Loeb et al., 2017). Relative to albedo change kernels based on sophisticated radiative transfer schemes embedded in GCMs, a CERES-based albedo change kernel – or CACK – represents a more transparent and empirically rooted alternative that can be updated frequently at relatively low cost. This allows greater flexibility to meet the needs of research focusing on surface albedo trends within the CERES era in regions currently undergoing rapid changes to atmospheric state as it affects shortwave radiation transfer. Although some modeling groups have provided recent updates to their albedo change kernels using the latest GCM versions (e.g., Pendergrass et al., 2018), the atmospheric state conditions used to derive them may still be considered outdated or not in sync with that required for many applications (Table 1).

Based on both qualitative and quantitative benchmarking against the mean of four GCM kernels, the novel kernel parameterization obtained from machine learning ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$, together with the two (semi-)analytically derived kernels ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{QH}\mathrm{06}}$ and ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{ANISO}}$, proved far superior to the ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{ISO}}$ analytical kernel and to the two additional empirical parameterizations ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{C}\mathrm{12}}$ and ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{M}\mathrm{10}}$. When subjected to additional performance evaluation, however, we found that ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ was able to more robustly emulate two GCM kernels (ECHAM6 and CAM5) with exceptionally high agreement, suggesting that ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ could serve as a suitable candidate for CACK.

Relative to the monthly CAM5 and ECHAM6 kernels, the mean absolute monthly
emulation “error” of ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ was found to be 6.8 and
6.1 W m^{−2}, respectively – a magnitude which is only ∼20 % of the standard deviation found across four GCM kernels (annual
mean). CACK's remarkable simplicity lends support to the idea of using
machine learning to explore and detect emergent properties of radiative
transfer or other complex, interactive model outputs in future research. The
fact that the ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ parameterization emerged as the
best common solution from two independently executed machine learning
analyses each employing a random sampling unique to a specific GCM kernel
suggests that the ${K}_{{\mathit{\alpha}}_{\mathrm{s}}}^{\mathrm{BO}\mathrm{18}}$ parameterization is robust and
insensitive to the underlying GCM representation of shortwave radiative
transfer.

Despite its stronger empirical foundation over a GCM-based kernel, it is important to recognize CACK's limitations. Firstly, while CACK has a finer spatial resolution than most GCM kernels, it still represents a spatially averaged response rather than a truly local response; in other words, the state variables used to define the ${\mathrm{SW}}_{\uparrow}^{\mathrm{TOA}}$ response are averages tied to the coarse spatial (i.e., $\mathrm{1}{}^{\circ}\times \mathrm{1}{}^{\circ}$) resolution of the CERES EBAF v4 product grids. Secondly, the monthly CERES EBAF-Surface product used to define lower atmospheric boundary conditions is not strictly an observation. The spaceborne platform is not able to directly observe surface irradiances, requiring additional satellite-based estimates of cloud and aerosol properties as input to a radiative transfer model (Kato et al., 2012). Although TOA irradiances are applied to constrain the surface irradiances, they remain susceptible to errors in the radiative transfer model inputs. Regarding this error as “data uncertainty” increases CACK's overall uncertainty beyond that which is related to its underlying parameterization or “model error”. The uncertainty of CERES surface shortwave irradiances as well as extensive ground validation and testing are documented in greater detail elsewhere (Kato et al., 2013, 2018; Loeb et al., 2017, 2009) and may continue to be reduced in future EBAF-Surface versions.

To conclude, we developed, evaluated, and proposed a radiative kernel for
surface albedo change based on CERES EBAF v4 products – or CACK. Relative
to existing kernels based on GCMs, CACK provides a higher spatial-resolution, higher-transparency alternative that is more amenable to user
needs. For LULCC research of the near-past, present-day, or near-future
periods, the application of a CACK whose inputs are based on monthly
climatological means of the full CERES EBAF record can better account for
the corresponding interannual variability in Earth's atmospheric state
affecting shortwave radiative transfer. For regions undergoing changes in
atmospheric state that are detectable above the normal variability within
the CERES era, the application of a temporally explicit CACK can better account
for its influence on Δ*F* estimates from surface albedo change. CACK's
input flexibility and transparency combined with documented uncertainty make
it well-suited to be applied as part of a monitoring, reporting, and
verification (MRV) framework for biogeophysical impacts on land, analogous
to those which currently exist for land sector greenhouse gas emissions.

Code and data availability

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Code and data availability.

We make both monthly temporally explicit and monthly climatological-mean CACKs for the years 2001–2016 available as a complete data product (“CACKv1.0”; Bright and O'Halloran, 2019) that includes their respective uncertainty layers. A summary of this dataset and associated variables is provided in Table S3 of the Supplement. Octave script files for generating monthly CACK and demonstrating its application with user-specified temporal and spatial extents are bundled with the netCDF file.

CERES EBAF data are available for download at https://ceres.larc.nasa.gov/products.php?product=EBAF-TOA (last access: 5 September 2019, CERES Science Team, 2018a, b). The CAM3 kernel is available at http://people.oregonstate.edu/~shellk/kernel.html (last access: 2 September 2019, Shell, 2008). The CAM5 kernel is available at https://www.earthsystemgrid.org/ac/guest/secure/sso.html (last access: 2 September 2019, Pendergrass, 2017). The ECHAM6 kernel is available at https://swiftbrowser.dkrz.de/public/dkrz_0c07783a-0bdc-4d5e-9f3b-c1b86fac060d/Radiative_kernels/ (last access: 2 September 2019, Block and Mauritsen, 2015).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/gmd-12-3975-2019-supplement.

Author contributions

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Author contributions.

TLO conceptualized the study. RMB and TLO developed the methodology, curated the data, designed the computer programs, and carried out the formal analysis. TLO and RMB produced the figures. RMB wrote the original draft, and RMB and TLO reviewed and edited the final paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

The authors thank two anonymous reviewers for their constructive comments and feedback.

Financial support

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Financial support.

This research has been supported by the Research Council of Norway (grant nos. 244074/E20 and 250113/F20) and the USDA National Institute of Food and Agriculture (grant no. 2017-68002-26612).

Review statement

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Review statement.

This paper was edited by Rolf Sander and reviewed by two anonymous referees.

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Short summary

To determine the effects of land cover change on climate, researchers must be able to quantify the net change in energy (radiation) at the top of the atmosphere caused by changes in surface reflectance (albedo). Historically, this was done with sophisticated models that require detailed input datasets only available to specialists. Here we combine existing remotely sensed datasets and a new formulation to create a new model that is accurate, transparent, and easy to use.

To determine the effects of land cover change on climate, researchers must be able to quantify...

Geoscientific Model Development

An interactive open-access journal of the European Geosciences Union