A new measurement of Equilibrium Climate Sensitivity (ECS)

A dynamic analysis of global temperature data  gives a value of ECS = 2.5C ± 0.5C . Values above 3.0C or below 2.0C are ruled out. This analysis is based on two assumptions: 1) That net climate forcing follows that used in CMIP5  (ref 1).  2) That climate equilibrium is reached with an e-folding time of 15 years (derived from GISS Model-II).

Detailed comparison of Hadcrut4.6 with calculated temperatures for different ECS values.

Analysis Method

Equilibrium Climate Change or ECS is defined as the increase in global temperatures following a doubling of CO2, once the climate system has stabilised. Models can calculate ECS by running a step function for CO2 concentrations from say 280ppm to 560ppm and then plotting how the temperature responds with time. Each  model gives a different value for ECS, and the spread in values represents in AR5 as an estimate of the uncertainty. I am going to use one of the simplest models, GISS Model II to investigate this lag effect of climate stabilisation which is mainly caused by the heat inertia response of the oceans to increased forcing.

Fig 1: Response temperature curve from a pulse doubling of CO2 in 1958 and fit described i the text

Fig 1: Response temperature curve for GISS Model 2 following a sudden doubling of CO2 in 1958. and fit described in the text. The e-folding time is 15 years.

After roughly 100 years the climate reaches a new stable state, and shows that GISS Model II gives a value for ECS of 4.4 C. The red curve is a fit to the temperature response curve which can be written in terms of temperature anomalies in the general form

\Delta{T} = ECS \times (1-e^\frac{-t}{15})

In reality CO2 levels in the atmosphere have been slowly growing over the last 200 years by annual increments as recorded since 1950 by the Mauna Loa data. The direct radiative forcing from increased CO2 has been calculated by radiative transfer codes. My derivation of this formula is described here.  A more precise parameterisation of that forcing is the well known formula

DS = 5.34 \times \ln{C/C_0}

where C_0 is the initial CO2 concentration and C is the incremental value. A doubling of CO2 alone give a forcing of ~3.7 W/m2 which at equilibrium is balanced by a surface temperature rise of 1.1C by applying Stefan Boltzmann’s law.

S = \sigma \epsilon \times T^4 \Longrightarrow DS = 4 \times \sigma \epsilon T^3 DT

Ignoring feedbacks and using \epsilon = 0.6 results in ECS being ~ 1.1C. Higher values of ECS are due to net positive climate feedbacks, mainly from increased H2O. CMIP5 models give a large spread in predicted ECS values due to the different ways H2O and cloud feedbacks are handled. Can we measure ECS directly from the data ?

The problem with measuring  ECS from the temperature data is that net forcing is increasing every year so we can never wait long enough for the climate to reach an equilibrium state. Given these constraints I adopt a different approach.

We treat the temperature record at any time as the response to the sum of previous discrete annual pulses of forcing. Each pulse causes a time dependent temperature response as shown in Figure 1. The resultant annual temperature for year n is then the integral of all previous responses up to that year.

Each pulse response is tracked through time and integrated to yield the overall instantaneous temperature at year N:

\Delta{T}(N) = \sum_{k=1}^N (\Delta{T_0}(1 - e^\frac{(N-k)}{15}))    – Equation 1.

This procedure can then be repeated for various possible values of ECS and compared directly to the temperature data. Rather than using the CO2 forcing directly we use the ‘Total Anthropogenic’ AR5 forcing data as shown below, which turn out to be almost the same thing.



The actual forcing data used in this analysis which is almost identical to CMIP5 net Anthropogenic.

The equilibrium temperature response \Delta{T}_{0} to an incremental forcing DS is  \frac{DS}{3.5-f} , where f is calculated from each possible value of ECS by using:

ECS = \frac{3.7}{3.5-f}

where 3.7 is the direct forcing due to a doubling of CO2 (calculated from 5.34 \ln{2} ) and f is the feedback parameter. This then allows  to calculate the feedback parameter f corresponding to a particular value of ECS, and then use f to to calculate the impulse forcing response. The resultant values of f are as follows.

1.5 1.03
2.0 1.65
2.5 2.02
3.0 2.26
3.5 2.44
4.0 2.57

A perl script was written to integrate forward past temperature responses  into a predicted annual temperature for various values of ECS by applying equation 1. The results are compared to the annual Hadcrut4.6 values.

Comparison of Hadcrut4.6 annual temperature anomalies with predicted temperatures for different values of ECS.

It is instructive to look in more detail at the recent data as it then becomes obvious that high values and very low values of ECS are ruled out.

Detailed comparison of Hadcrut4.6 from 1940 to 2017 with predicted temperatures assuming different values for ECS. Error bars are ±0.05C

The best fit to the observed temperature distribution using this method is ECS = 2.5C. High values above 3.oC  and very low values below 2.0C are ruled out. So my best estimate is

ECS = 2.5 ± 0.5C    (95% probability)

The error is based on post 2000 temperature values. ECS=2.0 falls within just 12% of data point errors (0.05C) while ECS = 3.0 falls within 24%.  This is to be compared with ECS=2.5C which falls within 84%. Both ECS=2.0 and ECS=3.0 are about 2 sigma from the mean average shown in black.  By 2017 ECS=2.0 lies 0.15C below the mean and ECS=3.0 lies 0.26C above the mean. Therefore I estimate a 95% probability that ECS lies within this range.

If climate sensitivity is 2.5C then  global temperatures can never rise more that 2.5C above pre-industrial levels so long as CO2 levels are kept below 560 ppm. This is a far more achievable goal than many activists are calling for since it requires only gradual reductions in CO2 emissions by 2100. This then gives us time to develop realistic alternatives, which I am convinced must have a strong nuclear base.


Forster et al. Evaluating adjusted forcing and model spread for historical and future scenarios in the CMIP5 generation of climate models, J. Geophys. Res.,118 1139-1150, 2013

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36 Responses to A new measurement of Equilibrium Climate Sensitivity (ECS)

  1. J Martin says:

    Clive Best just joined the alarmist camp. Didn’t Spencer & Christie get an ecs of 1.1 ?

    • Clive Best says:

      I am being generous and assuming that all warming is due to human activity, that models apply the correct forcing and that the globally temperature data is correct. Even then climate sensitivity is not that alarming.

      Temperatures change by 4C between an ice age and an interglacial. At worst CO2 emissions will be about half that.

  2. J Martin says:

    The third assumption that has been made is that all warming is caused by co2

    • Clive Best says:

      That is true. If there are natural cycles lasting many decades then that changes the results. However my point is that even the worst case scenario is not really that alarming.

    • A C Osborn says:

      J Martin, the first mistake is to assume that there is such a thing as ECS in the first place.
      I could understand someone thinking it might apply to H2O, but not CO2.

  3. Nick Stokes says:

    There has been a lot of experimentation with this over the years. ATTP gives a summary here. You are using a one box model – just an exponential decay, one time constant. Tho box models consider two time constants. ATTP got an ECS 2.6, Tamino gets 2.5. So a lot of consistency there.

    I think the way to think about it is this. T is a linear functional of past F, so it is a convolution with some (unknown) impulse response w. You can get some handle on w by trying various F’s. The standard ECS uses a step F and gets T at infinity. That gives the integral of w, or zeroth moment, which is the ECS. TCS is an experiment with a linear ramp F, etc. You are adding the assumption that w is exponential decay with time constant 15 years. That means that the zeroth moment fixes w. You could deduce TCR etc.

    • Clive Best says:


      Yes that’s right. The problem is that all we have to pin down climate sensitivity is the long term temperature record which is one dimensional, so far as ‘global’ warming goes. I don’t think adding a 2-Box Ocean & Land makes a lot of difference, unless you compare ocean and land separately.

      Since we all get an ECS value of 2.5C (mine includes data up to 2017), why is there still so much IPCC uncertainty ? Is it not perhaps to keep the modellers happy ? Those models with high sensitivity are clearly wrong(IMHO) and their continued impact is being exploited by green activists. The usual argument they use is the precautionary principal. i.e. we should plan for the worst case scenario in case it turns out to be true. The problem with that argument is that we only get one shot at this, and if we back the wrong technology by acting too soon, modern society will collapse.

      For TCR I get 1.6C

      I think ECS is the crucial value to get right, and so too is really understanding how the Carbon Cycle works. If emissions stop, how will CO2 levels really decay back to natural levels. I don’t think the carbon cycle is really understood. The Bern model is just a parameterisation.

      • Nick Stokes says:

        “why is there still so much IPCC uncertainty ?”
        Because there are much longer time scales that your simple exponential model does not take account of. You are using GISS Model II, which is a 1983 effort. It was not coupled air/ocean. It was air only, which only a very simple modelling of the SST interface. It did not capture the long time scales of diffusion into the ocean.

        ECS is the integral of the impulse response function. Long timescale processes (tails) don’t interfere much with a fit on a 15 year scale, but can add hugely to that integral.

        • Clive Best says:

          So are you not really saying that the uncertainty lies in ESMs and our understanding of the carbon cycle ?

          • Christian says:


            The uncertainty comes also from forcing so as from observation. If you play out them, you get nearly the same range as in IPCC-Reports e.g which is done(partly) in Haunstein et al 2017 as a result of, human induced warming could less/more then observed warming.

            Your approach is assume, that warming is all induced by forcing and forcing also observation is correct, but you dont know for sure, its also possible, that natural variability has warmed/cooled, early observation is incorrect or forcing is over/undereastimate. Another point is, you strictly assume that forcing and warming is linear, if in real climate system are none linearities, your calculation becomes wrong.

            In other words, such approaches can mean all and also nothing, only time will tell

            You say:

            ” However my point is that even the worst case scenario is not really that alarming.”

            Isnt really true, you showing the more likely scenario, worst case would mean, overestimated forcing, underestimated warming in observation and and cooling in observation due natural factors, them you get fast ECS arround 5K or higher.

          • No one seems to understand diffusion. The tails are not damped exponential, but 1/t or 1/sqrt(t) depending on how much dispersion there is.

          • Clive Best says:

            Presumably you mean heat diffusion in the oceans. Even if the diffusion time s longer I would assume the normalised temperature profile remains the same before and after a doubling of CO2. Furthermore this is something that should be reflected in seasonal temperature change profiles which is of the order of 10C for SST.

  4. MarkR says:

    If you apply these same calculations to CMIP5 models, what do you get? How does it compare to the known model ECS values?

    • Clive Best says:


      Here is a comparison of CMIP5 models to data. You see again that models with high sensitivity cannot reproduce the measured temperature record.

      The best fit is GIS-E2-R which has ECS ~ 2.3C This was from mid 2015.

      • Sid Wit says:

        I think Mark means something different. Assume you take the temperature time series of a CMIP5 model, treat is as the observed temperature and infer ECS. Then compare it to the actual ECS of that particular model.
        How well do the two numbers agree? The result will probably that you strongly underestimate the uncertainty in your results above.

        • Clive Best says:

          I am pretty sure that the numbers would agree almost exactly. In other words the theoretical curve for that ECS would pass straight through the model points. However that does assume that the relaxation time for each model follows approximately the same exponential curve.

          The interesting question is: Do models behave differently to each other following a step doubling of CO2 . If model A needs 1000 years to reach equilibrium whereas model B needs just 50 years then there is a serious problem – not only with my simple analysis but with CMIP5 in general.

          Does anyone know the answer?

          • MarkR says:

            From 21 abrupt4xCO2 simulations median e-folding time is 13.4 years and 5–95 % range is 7.1–23.1 years.

            Inferred ECS is 1.9 (1.1–2.4) C versus truth of 3.2 (2.1–4.2) C.

            I think we understand the differences thanks to e.g. Zhou, Armour, Knutti, Rugenstein… have you read any of their papers from the last few years? Also doi: 10.1038/nclimate3066

      • MarkR says:

        That looks like modelled global air temperature versus non-global, air/water temperature with HadCRUT4’s sea-ice quirk. Sampling the models like the observations should give Supplementary Figure 5 of: http://dx.doi.org/10.1038/nclimate3066

        For 1861-1880 to 2007-2016 I now get HadCRUT4 lying near the 50th percentile of the models. Same for Berkeley Earth vs global model air/water temperatures.

        In my first comment, I mean did you try calculating the ECS of individual models using their historical runs with your technique? Does your calculated ECS match the known value for a given model?

        If your results matched then it would be interesting to see why. It would be exciting if a technique that uses the temperature series favours a global-air ECS around 2.5 C, given that modelled temperature change basically matches the observations over 1861-2017!

  5. Windchaser says:

    Did you spatially mask the model results to be the same as the HadCRUT coverage before comparing them? (HadCRUT doesn’t cover the whole globe; the models do).

    Same question re: ocean temperatures and air-just-above-the-ocean temperatures.

    Last, did you use the real-world forcings in your model, before comparing its results to the real-world temperatures?

    Make sure you’re comparing apples-to-apples.

    • Clive Best says:

      If you are asking whether HadCRUT4.6 represents the best estimate of the global temperature, then probably no because there is some coverage bias. Cowtan and Way correct somewhat for the lack of Arctic coverage. Nick Stokes and myself use 3D spherical triangulation to extend coverage over the poles, and essentially get the same result. http://clivebest.com/blog/?p=7886

      However here I am doing something much simpler. I take the global scalar temperature and compare it to an integral of simplified global forcing model based on the AR5 model forcings. As such there is no need to mask land/ocean etc.

      So yes I am comparing real world (global) temperatures to real world (global) forcing. The assumptions are

      1) a 15 year e-folding stabilisation time for each annual increment in forcing
      2) an equilibrium temperature response DT = \frac{DS}{(3.5-f)} where f is adjusted so that ECS equals the chosen value

  6. Ron Graf says:

    Clive, it’s always good to see new approaches to reckon ECS. I am thinking that one could look for effects of AGW independent of GMST to help constrain ECS. The GHE not only raises GMST it also dampens diurnal and seasonal variation. The Moon’s surface gives us data for what the surface temp response would be without an atmosphere. The GHE essentially is adding atmosphere and thus we can simply extrapolate what that would do.

    1) The peaks and valleys in seasonal temp should be slightly delayed and dampened in amplitude. (Of course as GMST rises.)

    2) The peaks and valleys in diurnal temp should also delay and dampen. A shrinking of diurnal temperature range over the last century is documented for land in Vose (2003), Thorne (2016).

    3) Lastly, we should see land warming faster than SST. Although this technically is being observed it is only due to proportionally more land in the upper northern hemisphere. A great apples to apples test is to look at island stations compared to non-marine climate land stations. I did this quickly at Nick’s site last month and discussed the result with Nick at his blog here. Still waiting if he has a resolution. Maybe you could take a look at this Clive. Surprisingly the islands show the same response as inner continental stations.

    • Ron Graf says:

      I should disclaim that non-GMST responses are directly related to ECS since they do not likely correlate with feedbacks. However, it would be nice to have independent indications of AGW forcing to constrain it from natural variability and non-climate effects, (recording biases like UHI).

    • Clive Best says:

      Yes nights should warm slightly as IR loss from the surface gets slightly dampened. This is very difficult to separate from UHI in urban areas. Likewise winters should loose slightly less heat to space.

      The land ocean temperature comparison is a bit strange as land seems to warm more only in the late 20th century onwards. In reality the ocean data is sparse in the time of bucket measurements on ships. This was acknowledged by Tim Osborne.

      Tim Osborn?
      Following Following @TimOsbornClim
      Replying to @plazaeme @clivehbest
      Tim Osborn Retweeted Zeke Hausfather
      Perhaps the early 20thC SST warming is overestimated in our dataset? Intriguing possibility that has just been published https://twitter.com/hausfath/status/947910844976742401
      If true, that would give land/ocean warming contrast for this early period too

      Tim Osborn added,

      Zeke Hausfather

      We are closer to NOAA’s record after WW2, showing a bit cooler temperaturatures from 1940 through 1979. We also show warmer temperature in the early 1900s, eliminating the cold excursion around 1910 found in both NOAA and Hadley.

      • Ron Graf says:

        The ship bucket and engine intake temperature measurements have been adjusted many times, each with a pretext but also without a scientific basis, like in situ testing. I suspect that the problem of island and coastal anomalies not matching the adjacent seas was known but ignored. I’m hoping that it will not be used now simply as another degree of freedom to adjust past wrongs with the pretext of new insight. Call me cynical, but it seems like the adjustments always to bias recent warming up and fix old adjustments back to where they needed to be, or to smooth out variability. The beauty of Karl15 was that they had an opportunity to bias the past temps (or un-bias them) and never need to deal with the ship adjustments coming back to haunt them again since they were forever being replaced by buoys and Argo. They could take a sweet parting shot.

        If Zeke is looking at the island and coastal stations he must see that their warming is much faster than the seas around them but no faster than non-coastal. My suspicion would be this is evidence of non-climate effects in land stations. Zeke’s conclusion is likely that the SST warming is not properly being captured. But in all cases if the island and coastal trends are matching the non-coastal we have a puzzle because there is assumed to be a steadily growing and quite sizable radiative imbalance. And, marine stations and non marine stations should be recording divergent trends.

        Clive, you agreed that night and winter cooling should be dampened from EGHE. You were silent on my proposal that seasons and diurnal radiative responses per se should be flattened temporally as well.

        By the way, I meant Vose 2005, not 2003 studied global diurnal temperature range. Both Vose (2205) and Thorne (2016) agreed that the DTR narrowed significantly between 1950-1980 but pretty much flat lined from 1980 to present. This does not follow an expected fingerprint of EGHE from AGW. I suppose if it did there would be big news and there would be more than two papers in 50 years on the topic.

        • Clive Best says:


          Yes DTR should decrease because minimum temperatures rise faster than maximum temperatures.

          Islands will show much the same trend, because land based diurnal temperatures are much greater than sea surface temperature ranges. For this reason I don’t think Zeke can really use them to correct the SST data. Of course their real problem is to match energy balance of oceans and land. Oceans have to be cooler to match that increase in Ocean Heat Content !

          • Ron Graf says:

            Clive, there are dozens of 50-year-plus land stations that are less than 300 yards from the ocean. How much is their expected profile supposed to match land rather than sea temp behavior? There must be a spectrum from buoy to tiny island to large island to coastal mainland to non-coastal mainland. My point is that if a coastal station correlates 60% with the sea in its daily and seasonal temperature profile but has zero correlation in its anomaly trend then either the sea measurement is biased or the land is. Zeke is likely ready to assume the sea is biased due to less rigid measurements. Why? Because systematic non-climate effects occurring at land stations coastal stations is an unthinkable scenario for Zeke.

          • Ron Graf says:

            In fact island and coastal stations should be ideal for quantitative analysis of non-climate effects since they have the perfect control comparison: a temperature buoy in the vicinity which can be assumed to have zero non-climate effects.

            Typically non-climate effects like UHI and micro-site raise the daily Tmin. Thus if the warmer station trends are due primarily to increasing Tmin that would be very strong evidence the stations were biased. And if non-climate effect bias was found in coastal stations which are classified as rural, (assumed pristine,) that would have radical implications for the entire land record.

          • MarkR says:

            Hey Ron,

            Have you read the coastal stations paper? Have you got any analysis that supports your claims?

            I can’t make any sense of your comments.

          • Clive Best says:


            I hope to get round to looking at the coastal stations in the next week or so. I agree that they should follow the same rend as surrounding ocean even if their absolute temperatures are higher.

          • Ron Graf says:

            Mark R, I had not been aware of Cowtan, Rhode and Hausfather (2018) [full version] until you pointed it out.

            My thoughts on using island and coastal stations pre-dates the publishing this paper. As I mentioned I brought the topic to Nick Stokes here a month ago after bringing the topic up here at Clive’s months before that.

            Now that I have read Zeke’s paper I see it’s exactly as I had imagined. They found exactly what they were looking for: underestimated warming in SST and smoothing of the record to look more like the models (getting rid of unexplained variability). And, they overlooked what they could have found: an independent tool to clean out non-climate effects in the land stations.

            As the paper points out in its conclusion, there is no inherent way to know that the the coastal stations are more reliable than the SST.

            The differences between the coastal hybrid and existing sea surface temperature reconstructions are not necessarily indicative
            of problems in the existing records, although divergence between the existing records means that both cannot be correct.

            However, it would be safe to assume that the SST and marine air temp (MAT) are free from micro-site and UHI. So it would make sense to use them as a tool to get a ground truth on the coastal land stations. This could be done using the same type of analysis used in the paper, having SST and MAT as the origin on all marine climate effects in comparison to inland. Then if the temperature anomaly forms a step from SST to coastal station and all the other effects are a continuum relative to proximity then the step represents non-climate effects in the stations.

            There would likely be no conflict in the ERRST and COBE and HADSST if the focus had been finding the truth rather than finding innovative ways to adjust, warming as much as could be done with a straight face.

  7. dpy6629 says:

    Hi Clive, How does your analysis differ from that in Lewis and Curry?

    • Clive Best says:

      I think Lewis and Curry is based on a pure energy balance model. Something like

      ECS = \frac{F_{2x}\Delta{T}}{\Delta{F}-\Delta{Q}}

      TCR = \frac{F_{2x}\Delta{T}}{\Delta{F}}

      where F is the forcing at doubling CO2 and Q Ocean heat uptake.

      I am trying to fit the current global temperature curve with an integral over annual increments in forcing assuming a 15-y exponential equilibration. In effect the sum of the non-equilibrated parts in the annual exponential curves is energy imbalance or ocean heat uptake.

  8. robertok06 says:

    Hello Clive:

    on the same subject, a different estimate of the ECS:



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