Kevin Kilty
Early last summer, during some internet search, Google’s AI Overview told me that Earth’s energy imbalance (EEI), that is the difference between incoming solar irradiance and outgoing solar plus long wave infrared, is 0.5 ± 0.47 W/m2. Like most facts that AI Overview offers, I gave it little attention at the time, but filed it away in my memory.
Later, I began to ponder the number. Several things occurred to me. First, the uncertainty seemed to me an attempt to avoid writing the value as 0.5 ± 0.5 and thus potentially including zero in whatever confidence interval was being expressed. After all, it is difficult to speak of the world starting to cook, when a confidence interval on the source of heat includes zero – i.e. no heat source at all. Second, the number is very small considering that at any given time in the temperate zones incoming solar irradiance might be over 1,100 W/m2 and is highly variable to boot. I wondered “How does one measure this imbalance number and how well can we characterize its uncertainty?” What I expected to learn was that EEI is a very small number when compared to the variability of climate, and that its magnitude is greatly uncertain.
It’s difficult to get AI Overview to repeat things at a later time, and there was no use searching for the original source of the 0.5 ± 0.47 W/m2 figure using it. I eventually located the origin of this value but learned that it referred not to the EEI per se. Rather, it was a measure of decade to decade change in EEI. Along the search, however, I found an array of EEI values at different places stated in different ways.[1] Estimates ranged from 0.47 W/m2 to 1.0 W/m2.
Some of this dispersion apparent in all these values can be explained as figures covering different time periods. For example, figures of 0.47 W/m2 and 0.76 W/m2 quoted by Von Stuckmann et al [2] are for the time periods 1971-2018 and 2010-2018 respectively. Nonetheless, this is a rather large dispersion of estimates for energy imbalance considering that one person described it as “ the most fundamental metric that the scientific community and public must be aware of as the measure of how well the world is doing in the task of bringing climate change under control…”
In addition, the values pertaining to uncertainty for this very important number are unclear. Are they one-standard deviation values, 95% confidence intervals, 90% values or something else? It turns out they are a mixture. Since a number of people argue that the EEI is increasing with time, are these mean values over some time period plus uncertainty that are detrended or are they instantaneous at some particular time? Many questions to answer.
Let’s see what we may learn in some selected literature research.
Energy accounting
Earth’s energy accounting is not terribly different from double-entry financial accounting. There is a revenue account (incoming solar irradiance) and the expenses of outgoing longwave and reflected solar irradiance. Meanwhile in our energy ledger we have an assets account where I would place any energy additions to doing useful things like warming the planet or greening it, but which alarmists would call the liabilities account, perhaps, and a Earth’s energy equity account.
However we might name these accounts, our double entry system must balance, and this should provide two independent ways of determining energy balance or lack thereof. In one way we look at the flows of energy crossing a boundary at satellite level. In the other method we look at all the places where energy appears to have accumulated. In principle the two ought to produce the same numbers to within the uncertainty of each.
Magnitude and Uncertainty
Since energy imbalance is one of many metrics being employed to argue for changes to livestyles, products, and the economy as a whole, it had better be proven to be a highly certain menace. At a minimum this means being accurately measured.
Uncertainty of a measurement is not merely a matter of the randomness of a process being measured or randomness in sampling of a population. It has to include all aspects of a measurement, from the inherent randomness of the process being measured, through uncertainties of instrument construction, calibration, drift, installation (think of the surface stations project here), algorithms, and data reduction. If a stationary process is sampled in a truly random fashion, then the dispersion of measurements might encompass all of the components of uncertainty.[3] Exceptions to this include systematic errors and bias. Errors and bias are a particular concern when specialized or unique instruments and platforms are involved, because this equipment, consisting maybe of only a few units, will not produce a statistical ensemble, and the biases are unlikely to be recognized in the measurement statistics alone.
In particular, Henrion and Fischoff found that measurements of physical constants tended to group near one another even when later, and better, measurements showed this grouping to be extremely unlikely.[4] They suggested this to be an artifact of the sociology of physical science. In one sense there was a conflict between a desire to produce the definitive work (small uncertainty) and the surety of being correct (larger confidence bounds) with the former winning out often. Another finding was that scientists did a poor job of evaluating bias and that supposedly independent research groups could be nudged toward a value found by the most influential group among them (Bandwagon effect). In climate science a similar nudge is produced by a desire to produce shocking results – an effect one might call this the “it’s worse than we thought” bandwagon.
Direct measurements of energy flow
A direct measurement of energy imbalance is available from satellite measurements. Nasa’s Clouds and the Earth’s Radiant Energy System experiment (Ceres) involved seven different radiometers flying on five different satellites beginning in 1997. These radiometers enabled direct measurement of incoming solar radiation, outgoing solar radiation, and outgoing LWIR. Two of these satellites had an end of operational life in 2023, but continued to operate afterward. Terra and Aqua are both now experiencing fuel and power limitations. A new addition to this experiment, Libera, is scheduled to fly early in 2028. This program has discovered some interesting things over the years.
In a September 2022 conference, Norman G. Loeb summarized the CERES mission, its findings, and some consequences of these findings.[5] According to this presentation, Earth’s energy imbalance has doubled from 0.5 ± 0.2 W/m2 during the first 10 years of this century to 1.0 ± 0.2 W/m2 recently. The increase is the result of a 0.9 ± 0.3 W/m2 increase absorbed solar radiation that is partially offset by a 0.4 ± 0.25 W/m2 increase in outgoing longwave radiation.
What do these uncertainty estimates mean and how credible are they? To answer the “what” part of this question, slide number 20 in the powerpoint is reproduced in Figure 1.
Figure 1. From Loeb, N. 2022 [5].
This shows a comparison of direct measures of in and out radiation against planetary heat uptake, which I will address next, but which amounts to an independent, indirect measure of radiance imbalance. The two measures trend closely together at 0.5 ± 0.47 W/m2 per decade. This slide reveals the source of AI Overview’s summary! The uncertainty is identified as a confidence interval (5%, 95%); so, ±0.47 translates to ±0.29 as a standard deviation. What looks good in this graph is the agreement in what we suppose to be two independent estimates of energy imbalance.
Further research shows this agreement to not be all it seems. Whatever coverage factor these ± figures refer to (𝞼, 2𝞼, 90% CI, 95% CI, etc), a number built from change in solar energy minus outgoing longwave would calculate to 0.5 ± 0.39 W/m2; this combined with 0.5 ± 0.2 W/m2 of a previous decade would equate to 1.0 ± 0.44 W/m2, not 1.0 ± 0.2 W/m2. Unless of course there is some negative correlation between and among them which is never explained. I have had real trouble making the figures I read consistent with one another. There’s more.
I began my research on uncertainty of the energy imbalance by referral back to a special volume of JGR from May 1986 dealing with the instrumentation for the predecessor of CERES which was the Earth Radiation Budget Experiment (ERBE).[6] There we find
“These instruments are designed to provide radiant flux and radiant exitance measurements to better than 1% measurement uncertainty over a 2-year orbital lifetime.”[7]
This source of uncertainty does not cover the total that is possible, but what is important is to compare this source of uncertainty with attempts to rectify CERES satellite estimates of EEI with ones based on heat uptake. In a paper the title of which explains a lot,”Toward Optimal Closure of the Earth’s Top-of-Atmosphere Radiation Budget,” we learn that calibrations of the instruments involved are accurate to 1-2%.[8] The underlying uncertainty of the instruments, themselves, doesn’t seem to have improved from ERBE to CERES.
There are two tables in this paper that are pertinent to our question of uncertainty. Table 1 from this paper shows the very large variations in imbalance of tens of W/m2 that are due to internal climate variability – variations which are far larger than credible EEI due to forcing. Table 2 from this paper shows a list of known and unknown biases. The known ones run the range from -2 to +7 W/m2, and that the EEI obtained from satellite measurements is too large by a factor of perhaps five to be credible.
While it seems that the intrinsic uncertainty of the instruments has not improved significantly from ERBE to CERES, there are improvements to the measurements of the energy budget that have come from correlations between CERES instruments, and other space platforms and ground truth. Irrespective of the cleverness of these schemes they bring with them uncertainties from other instruments, platforms and algorithms which should be propagated into results. I wished to see a full analysis of uncertainty somewhere, but I did not find one.
More concerning, though, is the discussion on algorithms to improve this circumstance for purposes of climate modeling by, in one case, making adjustments to LWIR radiance measurement and also different adjustments to SW radiance measurements in order to produce an EEI compatible with estimates of planetary heat uptake. The adjustments made at this time were not small but rather a factor of two larger than EEI itself.[9] In other words, estimates of heat uptake in the oceans and atmosphere have nudged the satellite measurements in a preferred direction. The two methods of estimating EEI are not independent of one another. We don’t really have a double entry accounting system.
Planetary Heat Uptake
Those places on Earth where excess energy might be retained are as follows: 1) Temperature change of atmosphere below tropopause, 2) Temperature change of oceans above 2000m depth, 3) Temperature of ground surface/subsurface and 4) Change to cryosphere temperature and phase change. A recent review of all this data and results can be found in von Schuckmann et al (2023).[2] Some estimates are truly remarkable with a claimed relative uncertainty below 10% (uncertainty divided by expected value) considering that 2) through 4) are places hidden from direct measurement.
However, the authors appear to recognize their uncertainty estimates are not so rosy. They say
“The ensemble spread gives an indication of the agreement among products and can be used as a proxy for uncertainty. The basic assumption for the error distribution is Gaussian with a mean of zero, which can be approximated by an ensemble of various products. However, it does not account for systematic errors that may result in biases across the ensemble and does not represent the full uncertainty.”
Nonetheless, we are told in perhaps the most optimistic case that the present rate of heat uptake is 0.76 ± 0.1 W/m2 [2] or perhaps as good as 0.77 ± 0.06 W/m2[10]. These figures come from adding up the totality of contributions 1) through 4) in our list; though 89% is from the ocean alone. The indicated uncertainty should come at least from adding the respective uncertainties of the contributors in quadrature, just as though these data are uncorrelated, independent and unbiased estimates of the same thing.
Figure 2. Storage of energy imbalance in the ocean. From K. von Schuckmann et al. (2023) Heat stored in the Earth system 1960–2020, © Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License.
The gigantic effort of accounting for where all the energy imbalance goes done by von Struckmann et al [2] which involves 68 co-authors at 61 institutions is an avalanche of data stated in various units, over different time periods, and coming from numerous sources and subsets of sources. It is extremely difficult to sort out. Figure 2 shows just one example of what I see as discrepancies in the compilation. Compare, for instance, heat stored in the ocean from 1960 to 2020 to heat stored from 1993 to 2020. The depth range 0-2000m surely is made from the sum of 0-700m and 700-2000m as the sums in each case clearly show. Yet the uncertainty figures are impossible to reconcile. In one case uncertainties of 0.1 and 0.04 combine to 0.1 and in the other case the same combine to 0.2. Maybe there is covariance between the two depth ranges; though Figure 2 in Von Schuckmann et al [2] shows obvious positive correlation in all depth ranges – something that should amplify uncertainty not reduce it.
Acceleration of EEI
One prominent theme of all this work on energy imbalance is that the heat uptake is accelerating. Loeb et al [10] suggest that satellite observations show energy imbalance has doubled from 0.5±0.2 W/m2 during the 2000 to 2010 time period to 1.0±0.2 W/m2 during the 2010 to 2020 period – what I’d call decadal estimates. The increase is the result of a 0.9±0.3 W/m2 more absorbed solar radiation, partially offset by 0.4±0.25 W/m2 increase in outgoing longwave radiation.
Once again I face some troubles making all of this consistent. One would infer the result of increasing solar radiation and also increasing outgoing LWIR, here, by the rules of adding uncorrelated estimates, to be 0.5±0.4 W/m2, but which Loeb [5] characterizes as 0.5±0.47 W/m2, with a (5% – 95%) confidence interval. It’s puzzling.
Von Struckmann et al [2] suggest the equivalent decadal numbers to be 0.48±0.1 W/m2 and 0.76±0.2 W/m2 (which uncertainties they call a 90% confidence interval, 5%-95%), which are, if the stated uncertainties are to be believed, far smaller than what Loeb et al [10] calculates.
Neglecting all the differences, however, it is apparent all groups are presenting evidence of accelerated imbalance. Is this well established, though?
I decided to perform a little experiment with the uahncdn_lt_6.0 dataset (University of Alabama, Huntsville). I used an ordinary least squares linear regression. The month to month linear coefficient of increase is 0.0013C, and the 95% confidence interval runs (0.0012 to 0.0014 C/month). Figure 4 shows this. As Dr. Spencer has written on his blog, this 0.02C change per year is what results from increased CO2 forcing and anything departing from it is some influence of climate variability.
A close inspection of Figure 3 indicates the data are not randomly scattered about the best fit line. Their trend looks rather flat near the origin and at several places there is a bit more data below the line than above. Some extreme values occur at the end of the time series. This shape suggests some curvature like a squared term.
Figure 3.
So, I fit a quadratic function only to the data (small diagram in the upper left of Figure 3), and got what seems to be a remarkably good fitting curve that has zero slope at the origin and rises monotonically to the end. Ah! Evidence of an accelerating trend?
Not quite.
The fraction of variation explained by a line is 51% and by a quadratic is 53% – an insignificant difference. The standard error of residuals is no different (0.198 versus 0.195), and one suspects that the data don’t quite meet the requirement of being independent and identically distributed across its domain. In fact, it is simple to show there is correlation in this data across multiple time scales, and the correlation alone might fool a person into seeing a trend not there. This tendency toward a flattening followed by a brief but sharp rise appears in other data covering the entire past century as well, say 1920-1945. These temperature data are much too variable to establish a preference for the quadratic trend over the linear one.
Conclusions
When I began this effort I expected to discover that my search for energy imbalance would find a very small number with a very large uncertainty. This is exactly what I found. In addition, I was surprised to see that estimates of heat uptake provided a large nudge to satellite data, which leaves the estimates not independent. I also should have figured that whatever value of EEI that resulted would serve some purpose in promoting worry over climate change.
Something that all investigators in this EEI community do is tie energy imbalance to a litany of problems in the present world. To them a warming planet is nothing but trouble; there seems no upside at all. Von Schuckmann et al [2] even go so far as to calculate, from their estimates of imbalance, and even considering the acknowledged unknown uncertainties and assumptions involved, how much CO2 would have to be removed from the atmosphere to bring us back to year 1988, or to keep the Earth below 2C temperature change from the late 1800s. Why these two goals are important isn’t clear but we are told that they somehow will prevent climate problems.
But issues that cause panic are not related to puny temperature change. Heat waves do not arise from a 1C or even 1.5C background temperature rise. They arise from UHI approaching 10C and resulting from the way we have constructed our mega-cities, which are full of energy dissipating equipment and cut off from countryside breezes. Sea level rise, being puny, does not cause flooding – failures of civil engineering and people building in places they shouldn’t are what cause flooding. Knowing the EEI, which might be interesting in its own right, will aid in preventing none of this.
References:
1- See for example: Trenberth, et al, Earth’s Energy Imbalance, Journal of Climate, 27, 9, p.3129-3144, 2014 DOI: https://doi.org/10.1175/JCLI-D-13-00294.1; V. Schuckmann, et al, Heat stored in the Earth system: where does the energy go?, Earth Syst. Sci. Data, 12, 2013–2041, https://doi.org/10.5194/essd-12-2013-2020, 2020; Loeb, N.G., et al. Observational Assessment of Changes in Earth’s Energy Imbalance Since 2000. Surv Geophys (2024). https://doi.org/10.1007/s10712-024-09838-8;
2-v Schuckmann, et al, 2023, Heat stored in the Earth system 1960–2020: where does the energy go? System Science Data, 15(4), 1675-1709. https://doi.org/10.5194/essd-15-1675-2023
3 -Guide to Uncertainty in Measurement. Available online at https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf
4-Max Henrion and Baruch Fischoff, Am J. Physics. 54,791, 1989. See also Science. 289, 2260-2262, 29 September 2000.
5-Loeb, Norman, Trends in Earth’s Energy Imbalance, ISSI GEWEX Workshop, September 26-30, 2022, Bern, Switzerland. Powerpoint online at: https://ntrs.nasa.gov/api/citations/20220010511/downloads/Loeb_2022_compressed.pptx.pdf
6-B. Barkstrom, and G. Smith, The Earth Radiation Budget Experiment’ Science and Implementation, Reviews of Geophysics, v. 24, n 2, p. 379-390, May 1986
7-Kopia, L. Earth Radiation Budget Experiment Scanner Instrument, Reviews of Geophysics, v. 24, n 2, p. 400-406, May 1986
8-Loeb et al, 2018, Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) Top-of-Atmosphere (TOA) Edition-4.0 Data Product, Journal of Climate, DOI: 10.1175/JCLI-D-17-0208.1 “With the most recent CERES edition-4 instrument calibration improvements, the net imbalance from the standard CERES data products is approximately 4.3 , much larger than the expected EEI.“
9-Loeb, et al, 2009, Toward Optimal Closure of the Earth’s Top-of-Atmosphere Radiation Budget. J. Climate, 22, 748–766, https://doi.org/10.1175/2008JCLI2637.1.
10-Loeb, et al, 2024, Observational Assessment of Changes in Earth’s Energy Imbalance Since 2000. Surveys in Geophysics; https://doi.org/10.1007/s10712-024-09838-8
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