Guest Post by Willis Eschenbach [SEE UPDATE AT END]
In my most recent post, called “Where Is The Top Of The Atmosphere“, I used what is called “Ordinary Least Squares” (OLS) linear regression. This is the standard kind of linear regression that gives you the trend of a variable. For example, here’s the OLS linear regression trend of the CERES surface temperature from March 2000 to February 2021.
Figure 1. OLS regression, temperature (vertical or “Y” axis) versus time (horizontal or “X” axis). Red circles mark the ends of the correct regression trend line.
However, there’s an important caveat about OLS linear regression that I was unaware of. Thanks to a statistics-savvy commenter on my last post, I found out that there is something that must always be considered regarding the use of OLS linear regression.
It only gives the correct answer when there is no error in the data shown on the X-axis.
Now, if you’re looking at some variable on the Y-axis versus time on the X-axis, this isn’t a problem. Although there is usually some uncertainty in the values of a variable such as the global average temperature shown in Figure 1, in general we know the time of the observations quite accurately.
But suppose, using the exact same data, we put time on the Y-axis and the temperature on the X-axis, and use OLS regression to get the trend. Here’s that result.
Figure 2. OLS regression, time (vertical or “Y” axis) versus temperature (horizontal or “X” axis). As in Figure 1, red circles mark the ends of the correct regression trend line.
YIKES! That is way, way wrong. It greatly underestimates the true trend.
Fortunately, there is a solution. It’s called “Deming regression”, and it requires that you know the errors in both the X and Y-axis variables. Here’s Figure 2, with the Deming regression trend line shown in red.
Figure 3. OLS and Deming regression, time (vertical or “Y” axis) versus temperature (horizontal or “X” axis). As in Figure 1, red circles mark the ends of the correct regression trend line.
As you can see, the Deming regression gives the correct answer.
And this can be very important. For example, in my last post, I used OLS regression in a scatterplot comparing top-of-atmosphere (TOA) upwelling longwave (Y-axis) with surface temperature (X-axis). The problem is that both the TOA upwelling LW and the temperature data contain errors. Here’s that plot:
Figure 4. Scatterplot, monthly top-of-atmosphere upwelling longwave (TOA LW) versus surface temperature. The blue line is the incorrect OLS regression trend line.
But that’s not correct, because of the error in the X-axis. Once the commenter pointed out the problem, I replaced it with the correct Deming regression trend line.
Figure 5. Scatterplot, monthly top-of-atmosphere upwelling longwave (TOA LW) versus surface temperature. The yellow line is the correct Deming regression trend line.
And this is quite important. Using the incorrect trend shown by the blue line in Figure 4, I incorrectly calculated the equilibrium climate sensitivity as being 1°C for a doubling of CO2.
But using the correct trend shown by the blue line in Figure 5, I calculate the equilibrium climate sensitivity as being 0.6 °C for a doubling of CO2 … a significant difference.
I do love writing for the web. No matter what subject I pick to write about, I can guarantee that there are people reading my posts who know much more than I do about the subject in question … and as a result, I’m constantly learning new things. It’s the world’s best peer-review.
[UPDATE] My friend Rud said in the comments below:
First, CERES is too short a data set to estimate ECS.
I replied that climate sensitivity depends on the idea that temperature must increase to offset the loss of upwelling TOA LW. What I’ve done is measure the relationship between temperature and TOA LW. I asked him to please present evidence that that relationship has changed over time … because if it has not, why would a longer dataset help us?
Of course, me being me, I then had to go take a look at a longer dataset. NOAA has records of upwelling TOA longwave since 1979, and Berkeley Earth has global gridded temperatures since 1850. So I looked at the period of overlap between the two, which is January 1979 to December 2020. Here’s that graph.
Figure 6. Scatterplot, NOAA monthly top-of-atmosphere upwelling longwave (TOA LW) versus Berkeley Earth surface temperature. The yellow line is the correct Deming regression trend line.
Would you look at that. Instead of using CERES data for the graph, I’ve used two completely different datasets—upwelling TOA longwave from NOAA and global gridded temperature data from Berkeley Earth. And despite that, I get the exact same answer to the nearest tenth of a watt per square meter— 3.0 W/m2 per °C.
My thanks to the commenter who put me on the right path, and my best regards to all,
w.
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IMPORTANT NOTE: I’ve updated the head post, doing the same analysis for a longer time period (1979-2020) and using totally different data (NOAA upwelling TOA LW and Berkeley Earth surface temperature).
TL;DR: The results agree to within a tenth of a W/m2 per °C.
w.
These results are close to Kaplan’s 1960 paper on the subject. He refuted Plass’s modeled CO2 numbers as being 2-3x too high. Plass is the basis of all of Gavin Schmidt’s work.
Willis: Very interesting analysis. I’m ashamed to say that I have a degree in math with and emphasis on statistics and yet was not familiar with the Deming regression. In my defense, I did look through my college textbooks and found no mention of it. Of course that was long ago and perhaps before Deming was recognized for is brilliance by US professors. I did read Deming and Juran later in life, but I don’t recall seeing this discussed. Anyway, I did often analyze correlation data by regressing both X vs Y and Y vs X and when the two results agreed concluded that more confidence was warranted. Disagreement to me meant the data was suspect.
One question though. How does the R-squared value or the correlation coefficient compare between the OLS and Deming regressions?
Thank you for an illuminating discussion. We all can still learn a lot.
According to the Stefan-Boltzmann law a 1 degree increase in temperature from 287K to 288K would result in a 5.4 W/m^2 increase in radiation. The fact that Willis got a value of
3W/m^2 means that he is effectively calculating the emissivity of the earth using two different data sets and it should be no surprise that he got the same answer each time.
Emitted by the ground at 288 K is 390 W/m^2 IR, while TOA is 240 W/m^2… the difference being what most people refer to the “green house effect”….Willis’ calcs are TOA…
Willis: A commenter pointed out Deming regression to you June 13, 2012 in your post entitled “Observations on TOA Forcing vs Temperature”. Deja vu all over again? Here’s a bit of his comment:
Willis: The slope of the linear regression of Y on X is simply the reciprocal of the slope of the linear regression of X on Y.
Commenter:
that’s seldom true (that is, the probability of it being true is 0.) If you perform the linear regression of y on x and call the slope estimate b(Y|X), and do the regression of x on y and call the slope estimate b(X|Y), then b(Y|X) =/= 1/b(X|Y).
You can look this up on Wikipedia under the topic “Deming Regression”. I have found Wikipedia entries on statistical topics to be quite good. You can also look it up on Mathematica.
I’ve left comments regarding Deming Regression in a different context before. Here’s a comment I left on an analysis (not yours) that just threw out data because it was higher error than other data:
meab:
It’s not proper to eliminate data because of perceived high error. You can weight the data, but if you eliminate it you’ve biased the result. By doing this, they booted their analysis. Look up the Deming method for properly weighting data by its associated error. And yes, it’s the same Deming who radically improved Japanese production quality.
One of my degrees is in Mathematics with an emphasis on statistics – an area in which many alarmist analyses are demonstrably lacking.
Meab, thanks for mentioning that this was from The Deming. He singlehandedly saved Japanese industry which forced the American auto industry to eventually use his SPC methods for quality control.
The one good recommendation from the climatgate enquiries was for a person skilled in stats to be attached to each climate research team.
I understand this has not been implemented.
From AR6
https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_SPM_final.pdf
My instinct tells me the ECS is less than the radiative forcing alone (ie less than 1.1C) because CO2 is passive and feedbacks will be overall negative in the long term.
There is a hilarious list of contributing factors “factors du jour” in the IPCC report.
Danny Braswell and I demonstrated the importance of this issue in our 2011 Remote Sensing paper, where we showed that a mixture of radiative forcing (mostly uncorrelated with temperature because of the climate system’s heat capacity) and radiative feedback (highly correlated with temperature), leads to scatter plots of temperature versus radiation that produce OLS regression estimates of the feedback parameter which are too low (biased toward high climate sensitivity). We further harped on this issue in the following years in other papers. The Remote Sensing paper is the one where the editor resigned the following day following publication after Trenberth criticized him for allowing publication. https://www.mdpi.com/2072-4292/3/8/1603
Yeah, it was pretty sad how he could harp on about how your simplified model was too simple (eg no ENSO) when the simulations of ENSO at the time were biased and poor. Apparently its better to take into account an effect that’s modelled badly than to simplify it.
Thanks, Dr. Roy, I’ll take a close read of the paper.
And thanks as well for all of your contributions to the climate discussion and to climate science over lo these many years.
w.
In the discussion of errors here, it is not entirely clear what kind of errors we are talking about. Regression (curve fitting) is a process of reducing the “error” between some data and a mathematical model which is intended to show a relationship between some independent variable or variables and a dependent variable. The data may have errors due to the fact that we simply lack the ability to measure or know exactly what the value of something is. The curve fitting error can result from the fact that we simply have not accounted for all the independent variables, or it can be due to the fact that there are errors in the determination of the values of both the dependent and independent variables. Can someone talk about this please.
Hallo Willis
I finally figured it out. The “global’ warming…
https://breadonthewater.co.za/2022/01/10/global-warming-due-to-ehhh-global-greening/
Let me know what you think.
Linear regression only works if ALL of the variables are taken into account and ALL of the regression equations solved simultaneously. This includes the regression of the dependent variable on its own previous values. The ARX method, for “auto-regression with exogenous variable”, is described here: https://blackjay.net.au/measuring-climate-change/ . It gives both Impulse Response and Sensitivity.
Willis,
The paper Ramanathan, V., and A. Inamdar, 2006: The radiative forcing due to clouds and water vapor. In J. Kiehl & V. Ramanathan (Eds.), Frontiers of Climate Modeling (pp. 119-151). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511535857.006 gives a value for your quantity which is similar, but not the same to 1 decimal place, and is:
3.53 Wm-2K-1
I am writing a paper on climate feedback which uses this value.
HTH, Rich.
A lot of scatter.
What are the 1sd error bars on the 0.6C per doubling?
Good question. However, it’s hard to say, because the MODTRAN values have no error bar. Nor does the 3.7 W/m2. So let me make a rough calculation.
The value I’ve used is 0.525 for the TOA value as a fraction of the 3.7 W/m2 Myhre troposphere value. I’d estimate the error on that to be ±0.025. That’s ± 4.76%
The 3.7 W/m2 likely has an error on the order of ±0.1 W/m2. That’s ± 2.7%.
The slope of the trendline between temperature and upwelling TOA LW is 3.03 ± 0.21. That’s ± 7.23%.
For multiplication and division, the percentage errors add in quadrature, so the total error is sqrt(4.76^2 + 2.7^2 + 7.23^2) = ± 9.09%.
So the final estimate would be .525 * 3.7 / 3.03 = 0.64 ± 9.09%
= 0.64 ± 0.06 °C per doubling of CO2
Best regards.
w.
Willis,
Please pardon the length and lateness of this. Geoff S
……………………
A crucial, but neglected duty of the senior scientist of today is to examine the “perceived wisdom” to see if it is really “deliberate deceit”; to publicise examples of the latter, to try to change guesses into reliable, derived values that can be reproduced.
Willis, you wrote above “… the standard error of the mean of the 64,800 individual gridcells that are averaged to give each month’s value. For the temperature, this is about 0.06°C”.
The question arises, “Is this the appropriate error to use to support the work?” You used the Ceres satellite measurement of surface temperature as well as the Berkeley Earth temperature product.
In the case of measurement of water temperature, one can draw on experience of measurement performance by experts under top, controlled conditions. A few years ago I asked the Brits at their National Physics Laboratory how good they were at measuring water temperatures.
My question was –
“Does NPL have a publication that gives numbers in degrees for the accuracy and precision for the temperature measurement of quite pure water under NPL controlled laboratory conditions?
At how many degrees of total error would NPL consider improvement impossible with present state-of-art equipment?”
Part of their answer was –
“NPL has a water bath in which the temperature is controlled to ~0.001 °C, and our measurement capability for calibrations in the bath in the range up to 100 °C is 0.005 °C. However, measurement precision is significantly better than this. The limit of what is technically possible would depend on the circumstances and what exactly is wanted.”
Australia’s National Measurement Institute answered “The selection of a suitable temperature sensor and its readout is mostly based on the overall uncertainty, the physical constraint (contact/immersion), manual or auto-logging, available budget… The most accurate (most expensive) sensor is a standard platinum resistance thermometer at mK level uncertainty.” (A mK is 0.001 Kelvin).
……………………….
Moving from optimised specialist laboratories to the real world of ocean T measurement by buoys, we see claims like this – ” The temperatures in the Argo profiles are accurate to ± 0.002°C”
https://argo.ucsd.edu/data/data-faq/#accurate
This claim is laughable. It is an example of “deliberate deceit” to claim that they can do as well as the top measurement labs under laboratory conditions.
Next. we have the Ceres claim of the standard error of the mean of about 0.06 deg C. However, for your application, Willis, you do not need the standard error of the means of the gridcells – you need the absolute error involved over the whole measurement process.
Finally to Berkeley Earth. I searched the Net for about 20 minutes looking for a mathematical error figure for ocean temperatures and managed to find one. In the following reference, we have “Figure 2. Component uncertainties for the ocean average of HadSST v3 and the corresponding transformed forms of those components after the application of the interpolation scheme described in the text. All uncertainties are expressed as appropriate for 95 % confidence intervals on annual ocean averages.” Their graph of total uncertainty varies over the decades from about 0.05 to 0.2 deg C.
https://essd.copernicus.org/articles/12/3469/2020/essd-12-3469-2020.pdf
……………………………
In summary, many modern authors are wary about quoting any numbers for accuracy, error or uncertainty of ocean temperature measurements. Most do not even know the difference between these 3 terms. There is a preference to carry on as if these are matters, like the content of sausages, about which Mark Twain said with the law in mind “People who love sausage and respect the law should never watch either one being made.”
Thanks, Jeff, most interesting.
Some experimentation shows that the Deming regression is sensitive, not to the absolute values of the errors, but to the difference between the errors. In other words, if I double the estimate of both the temperature and the upwelling TOA LW errors, the slope of the Deming regression stays the same.
However, if I double one or the other of them the slope changes. And in neither case does the resulting trend give anything near an eyeball fit to the data.
This is good news, since the CERES EBAF dataset is likely to have at least somewhat similar errors throughout.
My conclusions are strengthened by the fact that using the Berkeley Earth temperature and the NOAA TOA upwelling LW gives me the same answer (3.03 CERES vs 2.96 Berkeley/NOAA).
I’ve given a first cut at an error estimate immediately above … let me know what you think.
w.
Willis, you usually say ECS but don’t you really mean TCR? The immediate change rather than the one that takes place years or centuries later? If you’ve answered this a hundred times before, please ignore!
The relationship between surface temperature and TOA upwelling LW has no lag—radiation moves at the speed of light.
So whether it takes one year or 1000 years for the surface temperature to warm by 0.6°C, that’s how much it has to warm to re-establish the TOA radiation balance. So we’re talking about the ECS.
w.
It takes awhile for the surface temperature to respond to an energy imbalance though. There’s currently a +0.8 W/m2 imbalance which will take at least a couple of decades to increase the temperature enough to restore a balance even if the forcing that caused imbalance drops to zero.
No, ECS is after the oceans have warmed, so you are not dealing with ECS.
You guys are missing the point. The relationship between TOA LW and surface temperature is as shown above. That measures how much the surface temperature will have to rise to offset a given change, not how long it will take to rise that much.
So it doesn’t matter if the ocean has equilibrated or not, and it doesn’t matter how long it takes the temperature to rise that much.
w.
No, you are missing the point. ECS is not defined that way. You have invented your own ECS, the Eschenbach Climate Sensitivity.
What is your definition of the ECS? My understanding has always been that the ECS is the rise in temperature necessary to offset the effect of a doubling of CO2 … which is what I calculated.
What’s your definition??
w.
But the effect isn’t immediate. If so you could have calculated the sensitivity from the daily solar input variation. And it’s not just the oceans. It takes time to warm the atmosphere too, where most of the radiation originates, 2-3 months judging by the surface-UAH/RSS lag.
https://en.wikipedia.org/wiki/Climate_sensitivity
“It is a prediction of the new global mean near-surface air temperature once the CO2 concentration has stopped increasing, and most of the feedbacks have had time to have their full effect”
Why didn’t you calculate the sensitivity using the post-1979 data?
Interesting that this number is very similar to this sensitivity calculation.
https://www.academia.edu/39277492/Challenging_the_Greenhouse_Effect_Specification_and_the_Climate_Sensitivity_of_the_IPCC?email_work_card=title
“According to the two methods of this study, the climate sensitivity parameter λ is 0.27 K/(Wm-2 ). It is about half of the λ value 0.5 K/(Wm-2) applied by the IPCC and the reason is in water feedback. Based on these two findings, the TCS is only 0.6°C.”
I believe what is actually being calculated here is the amount of energy required to raise the temperature within our atmosphere. It has nothing to do with CO2. Downwelling radiation is an irrelevancy. Temperature change is based on energy in – energy out.
The only way to change the the temperature is to increase energy in or decrease energy out. While it may look like blocking some outgoing radiation could reduce energy out, all it does is activate other energy transport mechanisms within the atmosphere. The same amount of energy still gets radiated to space.
What really happens with increased CO2 is you get a little bit more activity within the atmosphere which helps spread out the energy a little more evenly. This will warm colder areas and cool warmer areas.
So, why have we warmed? More energy is coming in due to a reduction in cloud reflectivity.
Most of the radiation to space comes from the atmosphere,
The temperature of the atmosphere or surface or both will have to increase to restore the balance, and it’s of course both. The surface is warmer than the atmosphere so there is no reason to believe the temperature of the surface will increase less than the temperature of the atmosphere.
???—The change in temperature in the two regressions (Deming and OLS) are identical—–the angle of the two regression lines are of course different as the dependent variable and independent variable were switched—-but the two are exactly the same—-it is impossible for them not to be the same.
I just checked this. RSS data into excel and added trends. The ‘normal’ way the trend is 0.021 K/yr. Switching the variables the trend becomes 34.47 yr/K. 1/34.47=0.031 K/yr, not 0.021 so Willis is right.