# Roles of Earth’s Albedo Variations and Top-of-the-Atmosphere Energy Imbalance in Recent Warming: New Insights from Satellite and Surface Observations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{−2}increase in solar-energy uptake by the planet from 2000 to 2020 and its effect on GSAT. Even more surprisingly, Section 7.2.2.1 of the IPCC WG1 Contribution features two graphs in their Figure 7.3 (on p. 936) showing a positive trend in the Earth’s reflected solar radiation and a negative trend in the outgoing thermal flux since 2000 that are supposedly based on CERES data. However, these trends are the opposite of what CERES has actually measured and contradict prior published results.

## 2. Data and Methods

#### 2.1. Satellite and Surface Datasets

^{−2}) represent only a fraction of the observed changes in the absorbed shortwave flux depicted in Figure 1b. According to CERES observations, the Earth’s all-sky albedo has declined by approximately 0.79% since 2000 causing an increase of planetary shortwave radiation absorption of ≈2.7 W m

^{−2}. For comparison, the IPCC AR6 estimated a total anthropogenic forcing of 2.72 W m

^{−2}driving climate change from 1750 to 2019 [2] (Section 7.3.5.2). Thus, the solar forcing measured over the past 2.4 decades has the same magnitude as the total anthropogenic forcing estimated by models for the past 27 decades. This fact further stresses the urgency to quantify the contribution of observed albedo changes to the recent planetary warming.

#### 2.2. Modeling the Response of Global Surface Air Temperature to Solar Forcing

^{−2}) and the mean surface total atmospheric pressure ($P$, Pa), i.e.:

^{−2})

^{0.25}inferred from a generic analytical formula for calculating the average surface temperature of airless planetary bodies derived by Volokin and ReLlez [13] via spherical integration of the Stephan–Boltzmann radiation law, and ${E}_{a}\left(P\right)$ is the Relative Atmospheric Thermal Effect/Enhancement (RATE), which is a dimensionless, nonlinear function of $P$. The product $A{S}^{0.25}$ defines the global surface temperature of a planetary body (K) in the absence of atmosphere, while ${E}_{a}\left(P\right)$ accounts for the adiabatic enhancement of the no-atmosphere temperature due to the force of air pressure. The dependence of RATE on pressure shown in Figure 6 was empirically quantified using non-linear regression analysis of measured NASA data from 6 rocky planets and moons spanning a broad range of physical conditions in the Solar System. We will not discuss the functional form of ${E}_{a}\left(P\right)$ here, since it is not relevant to the present analysis and has been explained elsewhere [12]. Although technically speaking $A$ is not a constant (as it depends on the magnitude of a planet’s average geothermal flux heating the surface), it is nevertheless a conservative quantity with a narrow range of variation in the Solar System ($32.44\le A\le 33.68$). For Earth, $A=32.51$ K/(W m

^{−2})

^{0.25}.

^{−2}) and $\mathsf{\Delta}s$ is the TSI departure (anomaly) from ${S}_{b}$. The solution to Equation (6) is:

^{−2}) to TSI and its Bond albedo ($\alpha $), i.e.:

^{−2}as means for the Bond albedo and TSI, respectively, during the period March 2000–December 2023. For comparison, the latest record of the Active Cavity Radiometer Irradiance Monitor (ACRIM) suggests an average TSI of 1361.26 W m

^{−2}over the same period [14]. ACRIM’s mean is 0.41 W m

^{−2}higher than the CERES value, which is an inconsequential difference for the numerical analysis presented here. Thus, in our calculations, we adopted the ${S}_{b}$ value inferred from CERES measurements. The Earth’s average absolute GSAT (${T}_{b}$) during the CERES observational period was computed from the global temperature anomalies shown in Figure 5a and an estimate by Jones and Harpham [15] that the Earth’s average absolute GSAT between 1981 and 2010 was 14.05 ± 0.15 °C (287.2 ± 0.15 K). This yielded ${T}_{b}=287.51$ K for the period March 2000–December 2023.

## 3. Results

#### 3.1. Drivers of Recent Warming

^{−2})/decade) produces an empirical estimate of the Earth’s climate sensitivity to absorbed shortwave radiation: 0.288 K/(W m

^{−2}). For comparison, Equation (15b), which solely quantifies the effect of absorbed solar fluxes on GSAT, yields a climate sensitivity of 0.297 K/(W m

^{−2}). The close agreement between these two estimates, along with the slightly higher value of the modeled sensitivity, indicates that the observed warming during the past 24 years was likely caused by a planet-wide increase of shortwave-radiation absorption resulting from a decrease of Earth’s albedo.

^{−2}), which is approximately 38% lower than the Earth’s overall climate sensitivity estimated from our model and the data series shown in Figure 7. The lower climate sensitivity of the upper ocean is explained by a higher heat capacity of sea water compared to the average heat capacity of Earth’s surface.

^{−2}[2]. Forster et al. [19] updated this estimate to 2.91 (2.19–3.63) W m

^{−2}for the period 1750–2022. The extra absorption of solar energy by Earth over the past 24 years alone was 2.7 W m

^{−2}(Figure 1b). Also, between 2011 and 2022, TAF rose 0.61 W m

^{−2}according to climate-model calculations discussed by Forster et al. [19], while the uptake of shortwave energy measured by CERES increased by 1.13 W m

^{−2}, i.e., nearly twice as much as TAF. These facts suggest that the measured solar forcing is much stronger than the modeled TAF. If the 1.17 W m

^{−2}increase of TAF estimated by IPCC AR6 from 2000 to 2022 [14] was as effective at heating the planet as the observed 1.98 W m

^{−2}rise of Earth’s shortwave absorption (Figure 1b), then we should have seen an additional warming of at least 0.34 K over this 23-year period, assuming a climate sensitivity of 0.29 K/(W m

^{−2}). If we employ a climate sensitivity of 0.48 K/(W m

^{−2}) calculated from the GSAT record in Figure 5 and TAF data for 2011 and 2022 presented by Forster et al. [19] in their Table 3, then the additional warming should have been 0.56 K. This 21st-century transient climate sensitivity is almost identical to the long-term sensitivity of 0.47 K/(W m

^{−2}) calculated from the modeled TAF value of 2.72 W m

^{−2}and its 1.29 K contribution to the GSAT increase estimated by the IPCC AR6 from 1750 to 2019 [2] (Sections 7.3.5.2 and 7.3.5.3, respectively). Finally, if we use the IPCC’s Equilibrium Climate Sensitivity (ECS) of 0.76 K/(W m

^{−2}) calculated from the projected average global warming of 3.0 K by climate models in response to an effective radiative forcing of 3.93 W m

^{−2}due to a CO

_{2}doubling [2], then we should have witnessed an extra warming of 0.89 K over the period 2000–2023.

#### 3.2. Contributions of TSI and Albedo Variations to Recent Warming

^{−2}) to the absorbed solar flux (Equation (15b)), but only 0.053 K/(W m

^{−2}) to TSI (Equation (8)). The GSAT sensitivity to an albedo change is −1.02 K per 1% increase of albedo (Equation (15a)). Thus, the Earth’s climate is 5.6 times more sensitive to changes in sunlight absorption than to TSI variations. This estimate agrees well with empirical results reported by Shaviv [22] regarding the effect of TSI cycles on several climatic variables. This author found that “the total radiative forcing associated with solar cycles variations is about 5 to 7 times larger than just those associated with the TSI variations”. Shaviv did not propose a specific TSI amplification mechanism, but our analysis indicates that this is likely the cloud albedo-mediated change of Earth’s shortwave-radiation absorption.

## 4. Discussion

#### 4.1. The Earth Energy Imbalance

^{−2})/decade from Mar. 2000 through Dec. 2023 due to the fact that Earth’s rising shortwave absorption has been outpacing the rate of the planet’s infrared cooling to Space. Loeb et al. [8] reported the same EEI trend for the period mid-2005–2019. The average EEI over the CERES monitoring period is 0.81 W m

^{−2}, which is in good agreement with the IPCC AR6 estimate of 0.79 (0.52–1.06) W m

^{−2}for the period 2006–2018 [2] (Section 7.2.2.2), the 0.87 ± 0.12 W m

^{−2}estimate by von Schuckmann et al. [32] for the period 2010–2018, and the most recent estimate of 0.76 ± 0.2 W m

^{−2}by von Schuckmann et al. [33] for the period 2006–2020.

_{2}and methane based on the a priori assumption that these gases retain heat in the troposphere by impeding the Earth’s infrared cooling to Space [2,33]. Therefore, a positive EEI has been interpreted as evidence of heat accumulation in the climate system that commits Earth to future warming for decades and even centuries to come [32,33]. This belief has prompted climate scientists to start integrating the EEI estimates over space and time for different periods in order to come up with total heat/energy gain by the Earth system measured in Zetta Joules (1 ZJ = 10

^{21}Joule). For example, the IPCC AR6 states that, due to heat trapping by greenhouse gases, the Earth had a net energy gain of 289.2 ZJ over the period 1993–2018 and 152.4 ZJ over the period 2006–2018 [2] (Section 7.2.2.2). As a result of such an interpretation, EEI is currently considered “the most fundamental indicator for climate change” and was proposed by an international team of 68 research collaborators to be implemented in the Paris Agreement’s Global Stocktake as a science-based measure of the World’s progress toward “bringing anthropogenic climate change under control” [33]. The fundamental assumption behind this proposal is that a reduction of the atmospheric CO

_{2}concentration to 353 ppm via the limiting of anthropogenic carbon emissions would eliminate the 0.87 W m

^{−2}EEI (estimated for the period 2010–2018) by increasing thermal radiation to Space, thus restoring Earth’s energy balance and stabilizing the climate [32]. Interestingly, however, only 1–2% of the Earth’s heat gain attributed to EEI is estimated to have accumulated in the atmosphere, causing the observed surface global warming for the last 50 years. The remaining 98–99% of the estimated energy gain is believed to have been stored under the surface (i.e., in the oceans, land masses, and glaciers) [2,33]. However, the published studies do not explain how 98% of the heat supposedly trapped in the troposphere by increasing greenhouse gases is transported below the Earth surface, when it is well-known that net fluxes of turbulent and radiative heat exchange flow upward and away from the surface on average.

#### 4.2. Physical Nature of the Earth Energy Imbalance

^{−2}) to semantically distinguish it from $\mathsf{\Delta}{s}_{a}$, although they are physically equivalent. From Equation (15b), we derive the following expression for $\mathsf{\Delta}{F}_{S}$:

^{−2}to the 13-month running average TSI anomaly depicted in Figure 2. Note that $\mathsf{\Delta}{F}_{S}$ and $\mathsf{\Delta}{s}_{a}$ are physically equivalent quantities, since the absorbed shortwave anomalies are fully converted into surface energy fluxes once GSAT responds to the solar forcing.

^{2}> 0.8) and exhibit clear positive trends over the 24-year period, as expected from a warming environment and the 2nd Law of Thermodynamics. However, the surface energy flux increases 2.45 times faster than the TOA LW flux (Figure 14a). This is because the magnitude of the measured TOA LW anomalies ($\mathsf{\Delta}{F}_{TOA}$, W m

^{−2}) is on average less than half the size of surface fluxes (Figure 14b). As shown in Figure 14b, the relationship between $\mathsf{\Delta}{F}_{S}$ and $\mathsf{\Delta}{F}_{TOA}$ is $\mathsf{\Delta}{F}_{TOA}=0.479\mathsf{\Delta}{F}_{S}$. In other words, only 47.9% of the energy flux emitted by the surface reaches the TOA and is detected as an outgoing LW radiation to Space. This tropospheric flux attenuation is key to understanding the observed EEI.

^{−2}, calculated as a sum of the upwelling LW radiation (398 W m

^{−2}), sensible heat flux (21 W m

^{−2}), and total evaporation or latent heat flux (82 W m

^{−2}). The TOA outgoing thermal flux is shown as 239 W m

^{−2}. This yields a Tropospheric Energy-Flux Attenuation Coefficient (TEFAC) of 239/501 = 0.477. The CERES data suggest an average TOA thermal upwelling flux of 240.1 W m

^{−2}during the first 4 years of the 21st Century. Using this slightly higher measured value exactly reproduces the TEFAC derived from our model and shown in Figure 14b, i.e., 240.1/501 = 0.479. The fact that both the flux anomalies and the Earth’s steady-state global energy budget reveal the same TEFAC is not a coincidence, but it does indicate a common underlying physical mechanism.

#### The Earth Energy Imbalance Explained by Thermodynamic Theory

^{−3}) at altitude $z$, and $g$ is the gravitational acceleration (m s

^{−2}). According to the Ideal Gas Law, density is a function of pressure (P, Pa), temperature (T, K), atmospheric molar mass (M, kg/mol), and the universal gas constant (R = 8.314 J mol

^{−1}K

^{−1}), i.e., ρ = PM/(RT). Also, the temperature at an altitude z can be described as a function of the surface temperature ${T}_{o}$ (K) and a lapse rate $L$ (K/m), i.e., ${T=T}_{o}+Lz$. Replacing the air density and temperature at altitude z in Equation (18a) with their equivalents followed by a separation of variables and integration of both sides over levels of pressure and altitude yields the following integral equation:

^{3}), $n$ is the polytropic index, and $C$ is a constant. Equation (19) implies that the product $P{V}^{n}$ is invariant with altitude in the troposphere.

_{p}) and at constant volume (v), respectively. The ratio ${\gamma =C}_{p}/{C}_{v}$ is known as the adiabatic index, since it is key in describing a standard adiabatic process. For the Earth’s atmosphere, ${C}_{p}$ = 1005 J kg

^{−1}K

^{−1}and ${C}_{v}$ = 718 J kg

^{−1}K

^{−1}, which numerically restricts the polytropic index to the range $1.0<n<1.4$. This implies that the thermodynamic state of the troposphere lies somewhere between isothermal ($n=1.0$) and isentropic ($n=1.4$). The term “isentropic” means “constant entropy” and refers to an adiabatic process that is fully reversable. In an isentropic process, the thermal kinetic energy and temperature of an air parcel only change as a result of work done on or by the parcel (e.g., compression or expansion) without any heat exchange with the surrounding environment. In the real atmosphere, however, air parcels gain and lose heat due to water-vapor condensation and the vaporization of water droplets, respectively. Evaporation typically cools the near-surface environment, while condensation of water vapor at higher altitudes warms the upper troposphere through the release of latent heat, which reduces the absolute temperature lapse rate below its dry adiabatic value of 9.8 K/km. In addition, the atmosphere directly intercepts approximately a third of the total solar radiation absorbed by the planet while emitting LW radiation to Space. Nevertheless, the heat exchange between convective air parcels and the surrounding environment is not efficient enough to create an isothermal troposphere. As a result, the thermodynamics of the troposphere becomes quasi-adiabatic, where typically $1.1<n<1.4$. For the Earth’s standard atmosphere with an average tropospheric lapse rate of −6.5 K/km, the polytropic index is $n\approx 1.235$.

^{−1}K

^{−1}). The exponent ${R}_{a}/{C}_{p}$ in Poisson’s formula can be derived from the power term $(n-1)/n$ in Equation (22) by replacing the polytropic index with the specific heat ratio ${C}_{p}/{C}_{v}$ and applying Mayer’s relation for ideal gases, ${R}_{a}={C}_{p}-{C}_{v}$.

^{−2}, the above thermodynamic equation can be applied to compute TEFAC by specifying an effective altitude ${Z}_{e}$, where the Earth’s thermal radiation escapes to Space. It is known that the Earth’s effective emission height is between 5000 and 5700 m asl, depending on the lapse rate [38,39]. Equation (25) reproduces the TEFAC value of 0.479 inferred from our model (see Figure 14b) at altitude ${Z}_{e}$ = 5250 m if $n=1.235$, corresponding to $L$ = −0.0065 K/m (−6.5 K/km). Between 5300 m and 5000 m altitude, Equation (25) yields $0.477{\le \mathsf{\Phi}}_{a}\le 0.491$, which covers the range of empirical TEFAC estimates. For $n=1.213$ corresponding to $L$ = −0.006 K/m (−6.0 K/km), Equation (25) predicts ${\mathsf{\Phi}}_{a}\left({Z}_{e}\right)=0.478$ at altitude ${Z}_{e}$ = 5700 m. Hence, this thermodynamic formula explains the observed attenuation of thermal energy in the troposphere as a function of decreasing atmospheric pressure with height.

#### 4.3. Recapitulation of Findings about the Earth Energy Imbalance

## 5. Conclusions

^{2}= 0.8) between the shortwave radiation uptake and the mean annual temperature anomaly of the 0–100 m global oceanic layer (Figure 8). These results suggest a lack of physical reality to both the anthropogenic radiative forcing attributed to rising greenhouse gases and the positive (amplifying) feedbacks hypothesized by the greenhouse theory and simulated by climate models. This is because any real forcing (or amplifying feedback) outside of the increased planetary uptake of solar radiation would have produced additional warming above and beyond the amount explained by changes in the planetary albedo and TSI. However, no such extra warming is observed in the available temperature records. Hence, the anthropogenic radiative forcing and associated positive feedbacks are likely model artifacts rather than real phenomena. The empirical data and model calculations analyzed in our study also indicate that the Earth’s climate sensitivity to radiative forcing is only 0.29–0.30 K/(W m

^{−2}). Therefore, the greenhouse theory overestimates this parameter by 56–158%.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Monthly radiative anomalies derived from the CERES EBAF 4.2 dataset: (

**a**) Earth’s global albedo calculated via dividing the reflected all-sky shortwave anomaly by the globally averaged incident solar flux at the TOA (i.e., the global insolation) and multiplying the resulting fraction by 100 to convert to a percent; (

**b**) Earth’s absorbed solar flux calculated via multiplying the CERES reflected all-sky shortwave anomaly by −1 based on the fact that radiation absorption is opposite (and complimentary) to reflection.

**Figure 2.**Deseasonalized monthly anomalies of the Total Solar Irradiance (TSI) calculated from CERES observations via multiplying the reported TOA global shortwave isolation anomalies by 4.0.

**Figure 3.**GSAT 13-month running mean anomalies from 7 datasets. Each time series is referenced to its respective mean over the period March 2000–December 2023. The satellite-based datasets report temperature changes in the lower troposphere (TLT) measured by Microwave Sounding Units (MSU), while surface-based databases are derived from thermometer readings on the ground.

**Figure 4.**Linear trends of 7 global surface and lower-troposphere temperature datasets over the period March 2000–December 2023. UAH was excluded from our analysis, because it showed an anomalously low rate of warming during the 21st Century (i.e., less than 0.18 °C/decade) compared to the other datasets.

**Figure 5.**GSAT monthly anomalies calculated by averaging of 6 global datasets (i.e., HadCRUT5, GISTEMP4, NOAA GlobalTemp, BEST, RSS, and NOAA STAR) and a 13-month running mean used to smooth the seasonal variability: (

**a**) Over the period 1981–2023; (

**b**) Over the CERES observational period (March 2000–December 2023).

**Figure 6.**The Relative Atmospheric Thermal Effect/Enhancement (RATE) as a function of the mean surface atmospheric pressure across 6 well-studied planetary bodies with rocky surfaces in the Solar System. RATE is the ratio of a planet’s observed long-term GSAT under an atmosphere (${T}_{b}$) to the planet’s estimated global surface temperature without an atmosphere (${T}_{na}$). The red curve is the result of non-linear regression analysis of NASA planetary data. See Nikolov & Zeller [12] for details.

**Figure 7.**Comparison between observed GSAT anomalies and CERES-reported changes in the Earth’s absorbed solar flux. The two data series, representing 13-month running means, are highly correlated with the absorbed SW flux, explaining 78% of the GSAT variation (R

^{2}= 0.78). Also, GSAT lags the absorbed shortwave radiation between 0 and 9 months, which indicates that GSAT is controlled by changes in sunlight absorption.

**Figure 8.**Comparison between observed temperature anomalies of the upper 100 m global-ocean layer [11] and changes in the Earth’s absorbed solar flux reported by CERES. Both time series represent 13-month running averages. Although completely independent of each other, the data series are highly correlated with the absorbed SW flux, explaining 80% of the interannual ocean temperature variability (R

^{2}= 0.80). Ocean temperatures are from the IAP 4.0 dataset (see Table 1) and were provided by Prof. Lijing Cheng at the Institute of Atmospheric Physics (IAP) of the Chinese Academy of Sciences.

**Figure 9.**Comparison of modeled and observed GSAT anomalies over the CERES monitoring period. The modeled curve is generated by Equation (16) using CERES-reported anomalies of albedo (Figure 1a) and TSI (Figure 2) as input. The observed GSAT lags the modeled temperature between 0 and 9 months, because GSAT lags the absorbed solar flux: (

**a**) Time series of observed and modeled GSAT anomalies. The modeled temperature series has been shifted 4 months forward after June of 2007 to partially compensate for the lag in the observed GSAT series; (

**b**) Scatter plot of observed vs. modeled GSAT anomalies. The dashed blue line is a linear regression. The solar forcing explains 100% of the multidecadal warming trend and 83% of the GSAT interannual variability after partially accounting for the lag of the observed GSAT, or 79% of the GSAT variability without accounting for lags (R

^{2}= Correlation Coefficient; S = slope of the linear relationship).

**Figure 10.**Illustrating the cause of the 2023 global heat anomaly: (

**a**) Observed GSAT in relation to changes of TSI and the absorbed shortwave radiation measured by CERES; (

**b**) Observed GSAT in relation to the modeled global temperature response to solar forcing (TSI and albedo combined) according to Equation (16). The observed GSAT lags the modeled GSAT because global temperature lags changes in the absorbed solar flux.

**Figure 11.**Global surface temperature changes predicted in response to observed variations of TOA TSI (Equation (8)) and the planetary albedo (Equation (15a)) over the CERES monitoring period. These are compared to measured GSAT anomalies. The observed multi-decadal warming trend and the El Niño-Southern Oscillation (ENSO) cycles are almost entirely explained by changes in the Earth’s albedo, while TSI only has a small, almost inconsequential contribution.

**Figure 12.**Estimated contributions of TSI and albedo variations to the evolution of Earth’s GSAT since 1981. The TSI-induced temperature change is calculated by Equation (8) using ACRIM measurements of TSI [14]. The albedo-induced GSAT anomalies are estimated by subtracting the TSI contribution from the GSAT record shown in Figure 5a. The assumption that the cloud albedo has been decreasing since the mid-1980s is based on an observed reduction of the global cloud-cover fraction reported by the ISCCP-FH dataset [21].

**Figure 13.**Relationship between Earth’s Energy Imbalance (EEI) measured by CERES and GSAT anomalies based on 6 global datasets: (

**a**) Comparison of GSAT and EEI time series representing 13-month running means; (

**b**) Linear correlation between EEI and GSAT. The dashed red line is a linear regression. EEI only explains 36% of the GSAT interannual variability, while the solar forcing explains 83% of it (Figure 9).

**Figure 14.**Observed outgoing LW flux at the TOA by CERES and modeled total energy flux at the surface by Equation (17) using the GSAT record in Figure 5b as input: (

**a**) Time series and trends of the observed and modeled fluxes; (

**b**) Relationship between the modeled surface total energy-flux anomalies and the observed TOA upwelling thermal-flux anomalies. The dashed magenta line is a linear regression. The magnitude of the TOA flux anomalies is, on average, 47.9% the size of surface fluxes.

**Figure 15.**The global mean energy budget of Earth according to IPCC AR6 [2] (Section 7.2.1), their Figure 7.2. Numbers indicate the best estimates for magnitudes of the globally averaged energy balance components in W m

^{−2}, together with their uncertainty ranges in parentheses (5–95% confidence range), representing all-sky climate conditions at the beginning of the 21st century. According to this diagram, the Tropospheric Energy-Flux Attenuation Coefficient (TEFAC) is 239/(398 + 82 + 21) = 0.477.

**Figure 16.**Atmospheric pressure as a function of altitude according to the barometric formula (Equation (18c)) for three different lapse rates: dry adiabatic (L = −0.0098 K/m), humid environmental (L = −0.0065 K/m), and moist adiabatic (L = −0.005 K/m). Note that variations of the lapse rate only have a minor impact on the decrease of pressure with altitude.

**Figure 17.**Decrease of the thermal kinetic energy in the troposphere relative to the thermal kinetic energy of air at the surface for a standard atmosphere under 3 scenarios defined by different polytropic indices and temperature lapse rates applicable to dry, humid, or moist environments, respectively. According to the Gas Law, the thermal kinetic energy of air is defined by the product PV.

**Figure 18.**Comparison of observed and modeled outgoing thermal radiative fluxes at the TOA. The observed anomalies are from the CERES EBAF 4.2 dataset. The modeled time series is produced by multiplying the surface total-energy fluxes calculated by Equation (17) and shown in Figure 14a by 0.477 to account for the quasi-adiabatic energy dissipation in the troposphere. This correction makes the modeled timeseries agree almost perfectly with the observed timeseries.

**Figure 19.**CERES TOA outgoing LW fluxes compared to adiabatically adjusted CERES anomalies of the absorbed solar flux. The adiabatic adjustment consists of multiplying the original shortwave (SW) data series in Figure 1b by TEFAC = 0.477, inferred from the IPCC energy-budget diagram in Figure 15. This makes the absorbed solar flux closely match the trend and interannual variability of the outgoing LW radiation flux, which indicates that the Earth’s Energy Imbalance (EEI), quantified as a difference between absorbed shortwave and outgoing LW radiation at the TOA, is an artifact of a quasi-adiabatic tropospheric cooling driven by a decreasing pressure with altitude.

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**MDPI and ACS Style**

Nikolov, N.; Zeller, K.F.
Roles of Earth’s Albedo Variations and Top-of-the-Atmosphere Energy Imbalance in Recent Warming: New Insights from Satellite and Surface Observations. *Geomatics* **2024**, *4*, 311-341.
https://doi.org/10.3390/geomatics4030017

**AMA Style**

Nikolov N, Zeller KF.
Roles of Earth’s Albedo Variations and Top-of-the-Atmosphere Energy Imbalance in Recent Warming: New Insights from Satellite and Surface Observations. *Geomatics*. 2024; 4(3):311-341.
https://doi.org/10.3390/geomatics4030017

**Chicago/Turabian Style**

Nikolov, Ned, and Karl F. Zeller.
2024. "Roles of Earth’s Albedo Variations and Top-of-the-Atmosphere Energy Imbalance in Recent Warming: New Insights from Satellite and Surface Observations" *Geomatics* 4, no. 3: 311-341.
https://doi.org/10.3390/geomatics4030017