The Spencer Identity for Global Temperature Trend Calculation

July 25th, 2014 by Roy W. Spencer, Ph. D.

After reading the comments I got on my post The Kaya Identity Crisis, I thought it would be good to follow up with a little more discussion regarding how you can create an equation with physical units that work out, but the equation is not necessarily useful.

My point was that the Kaya Identity was fine from a mathematical point of view, but I suppose I should have emphasized more that the terms in the Kaya equation (which relates total global carbon dioxide emissions to the product of population, per capita GDP, energy intensity, and CO2 emissions per unit energy used) were certainly open to debate.

So, since I’m not as knowledgeable about economics concepts as I am about global temperature trends, I thought it would be cool to have my own equation: The Spencer Identity.

I have thoroughly researched this equation for at least 15 minutes, and I believe the units balance, and it makes some physical sense (which is a good thing).

The equation doesn’t give you the real global temperature trend, which we can never know. It instead gives the consensus temperature trend, which (we have found out) is the result of political institutions and the funding they provide to researchers willing to play along:

the Spencer Identity for government-reported temperature trends:

T = s[L x D x H x W]


T = official government-reported global average temperature trend (deg. C/decade)
s = scale factor to allow easy adjustment of consensus temperature trends based upon prevailing opinion (dimensionless)
L = the number of liberal politicians in power
D = the dollars in government climate research funding per liberal politician
H = the number of hockey sticks created per research dollar spent (or maybe you could substitute mythical polar bear deaths or hidden temperature declines or number of climate models)
W = the rate of warming per hockey stick

As I said, I believe the units balance and the terms of the equation have meaning. I’m sure others can develop improved versions of the equation, so maybe this can be considered an open source project.

39 Responses to “The Spencer Identity for Global Temperature Trend Calculation”

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  1. Johan says:

    A “cool” Spencer Identity. Do I sense an agenda? 🙂

  2. Johan says:

    And another thing … so the Spencer Identy says, with absolutely no liberal politicians in power, we’d all freeze to death (officially anyway).
    A bit worrisome.

  3. MRW says:

    Haha. Good one.

  4. jimc says:

    I believe the equation should include the number of Nobel Prize nominations to be had. It has to be a whole number so as to be pointless.

  5. Greg says:

    While this seems to be an attempt to ridicule the Kaya identity, it does more to ridicule the author’s lack of basic maths skills.

    At least twice in this short text he flips back and forth between the terms equation and identity as if they were… identical.

    The two are not equivalent and interchangeable mathematical terms. Each has a separate, defined meaning.

    This same basic mistake is at the base of all the shouting about the so-called Kaya identity.

    While most of those comments are from ill-informed, uneducated people, it is disappointing to see someone with the background and standing of Dr. Spencer displaying such ignorance of basic maths.

    • Johan says:

      “At least twice in this short text he flips back and forth between the terms equation and identity as if they were identical.”

      No, he consistently calls it an equation, and simply gave this equation the name “The Spencer Identity”.
      Admittedly, it’s not very modest to name an equation after oneself, but it is mathematically correct to do so.

      To cheer you up a litlle:

      The physicist and the engineer are in a hot-air balloon. Soon, they find themselves lost in a canyon somewhere. They yell out for help: “Helllloooooo! Where are we?”
      15 minutes later, they hear an echoing voice: “Helllloooooo! You’re in a hot-air balloon!!”
      The physicist says, “That must have been a mathematician.”
      The engineer asks, “Why do you say that?”
      The physicist replied: “The answer was absolutely correct, and it was utterly useless.”

      • Massimo PORZIO says:

        Hi Hi Hi…
        Instead, I read this one about mathematicians in a book some years ago:

        There were three old English friends who were doing a trip on a train in Scotland.
        One was a butcher, the other was a physics professor and the third was a mathematician.
        At a certain time, the butcher who were reading the newspaper, closed the paper, stood up and looking through the window shouted “WOW! look there, once at home I should tell my nephews that the Scottish sheep are black!”
        In fact on the hill there was a black sheep which slowly was eating the grass.
        The physics professor which was reading a book about thermodynamics, shut up the book, stood up and approaching the window replied: “No! No! Dear friend, you cant tell your nephews that all the Scottish sheep are black, the only thing that you can tell for sure to your nephews and that Ill tell to mine, is that during this trip in Scotland we seen one black sheep on a Scottish hill, that because you cant see any other black sheep here.
        At this point, the mathematician who were reading a book about the geometries of non Euclidean universes, shut up his book, stood up and went to the window too. Looked outside the window and stated: No dear friends, you are both wrong. No ways you can state for sure that what you see here is a black sheep indeed. The only true thing that you can tell to your nephews is that one day, looking outside of the train window you seen a sheep that it was black on one side, that because you can’t tell for sure whats the true color of that sheep on the other side!

        The moral is: mathematics is the only true science; a mathematician cant rely upon any sensation; a mathematician must rely only upon what he/she can demonstrate to be absolutely true, even if the statistical probability surely would lead anybody else to believe the it was true.

        Note that I reported that tale translating from Italian and the tale was from England, so maybe I did some mistakes translating that, anyways the meaning of the tale should be clear.

        Have a great day.


        • TonyM says:

          Nice one Massimo. It came through clearly 🙂

        • JohnKl says:

          Hi Massimo Porzio,

          Your translation worked just fine. Thank you for the story/parable. However, lacking the available time I’m encouraged to find a mathematician to calculate the odds that the person I’m communicating with is in fact the REAL MASSIMO PORZIO. Lacking evidence to the contrary many will say I’m safe to do so. Of course one never knows for sure!

          Have a great day!

        • JohnKl says:

          Hi Massimo PORZIO,

          Btw, your story/parable reminds me of California impounded car auctions. If you ever attend one put on by the state or local jurisdictions you will do well to arrive early and inspect the vehicles. The side of the car viewed by bidders may look just fine but the other side of course may not. You may even find bullet riddled artistic impressions left by prior vehicular encounters with narcotic traffickers. In any case, the odds you’ll find an intact vehicle may resemble the chance of finding a dichromatic scottish sheep with each of the sheep’s sides presenting a different color. Of course, this begins to remind me of a Star Trek episode, so I think I’ll finish here.

          Have a great day!

          • Massimo PORZIO says:

            Yeah JohnKl,
            I never attended a car auction, but I can imagine what you mean.
            I’m not a mathematician, just an engineer so I can imagine without looking for any proof of what I’m imagining 😉

            Have a great day too.


            In case you didn’t find the mathematician who calculated the odds that I’m not me, I would ensure you that I’m absolutely myself, even if I know that that mathematician will conclude that there is still a little probability that I’m not me indeed!

  6. Gordon Robertson says:

    Roy…I appreciate the humour but there are inconsistencies in your formula.

    For one, I am worse than a Liberal, I am a socialist, yet I support you in your skepticism.

    For another, the government in British Columbia,Canada, is an ultra right wing government who call themselves Liberals. They stole the party from a real Liberal after their right-wing version was trounced at the polls.

    This Conservative government in Liberal clothing has had the temerity to introduce a carbon tax, and they have levied a transit tax that is aimed at forcing people onto the transit system and out of their cars. All told, we pay 25 cents a liter in carbon-like taxes per litre of gas.

    Here’s the good part. They apply the carbon tax to schools and hospitals for their use of fuel then they give the carbon tax money to a private company.

    We all know that right-wingers are slobbering all over their shoes trying to find a way to get in on the carbon tax cash cow.

    Another point is Schwarzeneggar, when he was governor of California. He was right of Attila the Hun and may even be related to him. He would fit right into your formula.

    Finally, how about Al Gore. Although he impersonated a Liberal vice president his claim to fame, with his wife Tipper, was going after rock groups who allegedly built Satanic messages into their lyrics. Not once, to my knowledge, did Gore raise his voice about global warming during his tenure as a Liberal veepee.

    Of course, like all good right-wingers he has made a fortune speaking out on climate disaster.

  7. Milton Hathaway says:

    Illuminating equation, Dr Spencer, but I can’t seem to locate the positive feedback term, which observation indicates must be present. Is it hidden in the “s” parameter?

    • Gordon Robertson says:

      @Milton Hathaway “I cant seem to locate the positive feedback term, which observation indicates must be present. Is it hidden in the s parameter?”

      Actually, observation does not require a positive feedback term it is only the invalidated climate models that are programmed with a true positive feedback that has never been observed. I say true because I seem to recall Roy explaining that some climate scientists have redefined positive feedback as a not-so-negative negative feedback.

      In Roy’s equation it could not be contained in the ‘s’ term since it is purely multiplicative. Positive feedback has to be tied directly to a gain element, as a positive fraction of the gain, so the input signal increases during each cycle.

      For example, G = gain = A (1/1-AB), where A is the amplifier gain and B is the feedback component. Note that B can never exceed A so the term AB is always 1, and the bracket expression will evaluate to a fraction (negative feedback). If the value of B is positive, the denominator of the brackets is 1, making the overall bracket expression to be > 1 (positive feedback).

      Note that the term AB can never = 1 in an electronic circuit, which would make the bracket expression undefined. In reality that cannot happen since B is defined as a fraction of A and is realized through a resistance/impedance network designed to take a fraction of the output of A and feed it back to the input of A.

      It’s important to understand that positive feedback only happens under strict conditions. It works with an electronic amplifier because the sign of the feedback controls the gain of the amplifier. It works to a degree with resonant structures where the resonance state tends to increase the overall signal. The Seattle-Tacoma Bridge collapsed due to an uncontrolled positive feedback due to resonance in the support cables of the bridge.

      In the atmosphere, there is nothing to enable positive feedback. Even if infrared from GHGs in the atmosphere are being back-radiated, there is no gain factor to amplify them. In fact, there is such a negative component due to losses that the very weak feedback could never make up the losses let alone create a positive feedback.

      Positive feedback of this type depends on amplification by an independent amplifier powered from an independent source. There is no such mechanism in the atmosphere and the 2nd Law of Thermodynamics ensures that.

      • Gordon Robertson says:

        @Gordon Robertson

        Correction: I said “Note that B can never exceed A so the term AB is always 1”

        That should read “Note that B can never exceed A so the term AB is always less than 1”

        Something got messed up between my written post and the posted version. Some information was omitted.

        I explained that in the equation G = A (1/1-AB) that the expression in the brackets evaluated to a fraction in the case of negative feedback when B is -ve. Conversely, the bracket expression evaluates to a value > 1 when B is +ve in the case of positive feedback.

        It comes down to the term 1-AB. If B is -ve, the sign changes and it becomes 1+AB, making the expression 1/1-AB a fraction.

        When B is positive, the expression remains 1-AB making the denominator a fraction and the overall expression > 1.

        • geran says:

          Gordon, Buddy, have you ever considered a therapist?

          • Gordon Robertson says:

            Enough with the ad homs. If you can’t understand the math just say so.

            My background is in electronics and positive feedback is a fundamental concept in electronics.

          • Gordon Robertson says:

            @geran “Gordon, Buddy, have you ever considered a therapist?”

            I called one just to please you and she said she knew nothing about positive feedback. Why would I consider a therapist for advice on PF or climate science?

            Maybe you need help.

          • geran says:

            If you really have a background in positive feed back, then you know you already lost this argument.

  8. Lewis Guignard says:

    Dr. Spencer,

    I believe your formula would be more accurate if it included a “C” for compliant media and a “g” for gullible people who have no interest in thinking for themselves.

    Happy Saturday

  9. Nylo says:

    Makes total sense. All other things being equal, to increase the officially reported trend in temperatures, you can either increase the number of liberals in the gvt, or increase the ammount of money spent per liberal in the gvt, or increase the number of hockey sticks published per dolar spent, or increase the amount of warming exhibited by the published hockey sticks.

    Seems to me like a perfect twin for the Kaya identity, both in its strengths (it is, mathematically, absolutely correct) and in its weaknesses (all other things will never be equal, so the identity is useless and pointless).

  10. geran says:

    The only problem with your “identity” is I get no kickback.

    Nor do you!

    After proper peer review, I strongly suggest a re-write, with suitable remunerations for all.

    (You’re welcome.)

  11. Gunga Din says:

    Love it!
    But I think,

    H = the number of hockey sticks created per research dollar spent (or maybe you could substitute mythical polar bear deaths or hidden temperature declines or number of climate models)

    Should include the BSTU value of each hockey stick. 😎

  12. David L. Hagen says:

    Your Spencer’s Identity appears to be related to SKINNER’S CONSTANT (FLANAGAN’S FINAGLING FACTOR)

    That quantity which, when multiplied by, divided by, added to, or subtracted from the answer you get, gives you the answer you should have got.

  13. JohnKl says:

    Hi Roy and everyone,

    It seems to me the equation can still be improved by including at least three additional units. The right side of the equation should include the letter “A” for the number of Adjustments made to any given data-set requisite to arrive at any given temperature result ( note this is not a scaling factor but an historical account of prior admitted adjustments- any future adjustments at this stage would be speculative, however likely!!!). “P” represents the number of Psuedo-science historical Proxies fabricated to paint some environmental image of our distant past. Finally the unit “E” signifies the quantity, nature (lower troposphere, stratosphere, mesosphere, etc.) and extent (for example regional temp projections (like U.S., Canada etc or perhaps global) of prior Erroneous temperature trends presented this will likely asymptotically approach some vertical line to infinity. The equation may or may not balance but it will nevertheless better reflect the current state of “CONSENSUS TEMPERATURE TRENDS” and the state of science in the U.S. best reflected in current zombie movies or Planet of the APES!

    Have a great day!

  14. Scott says:

    Unverified, but I think this relationship may hold up some of the time:

    (E = mc^2) ≠ (T = s[L x D x H x W])

  15. Gordon Robertson says:

    @geran …(re Gordon Robertson comment on positive feedback) “If you really have a background in positive feed back, then you know you already lost this argument.”

    There is no argument. I have presented facts about positive feedback that can be verified by anyone. The formula I presented represents feedback in a process with gain. I outlined another possibility where a resonating system can run away.

    The only other kind of true positive feedback is in servomotor control but the positive in positive feedback there describes the sign of the feedback only. There is no gain involved.

    Atmospheric positive feedback does not exist based on the definition from physics, which requires gain (an amplifier). As Roy described the situation at one time, positive feedback in climate science has been redefined as a not-so-negative negative feedback. If there is no gain in a system, the only feedback possible is negative feedback, due to losses in the system.

    Thus far, you are the one who is arguing but you have resorted to ad homs and oblique logic. If there is something wrong with the facts I have presented then lay them out so we can have a discussion. Arguments bore me.

    • Massimo PORZIO says:

      Hi Gordon,
      I’m an EE too, and I would agree with you but as yourself said the climate science redefined a little the concept of PF because of the particular context.

      You wrote:
      “If there is no gain in a system, the only feedback possible is negative feedback, due to losses in the system.”

      IMHO you fail here, because in the real world the so called “losses” are equivalent to heat transfer to the surrounding environment.
      For that, even IMHO of course, in the climate context of evaluating the changes in temperature due to atmospheric composition, the losses are a positive feedback indeed. Using an EE friendly paradigm: the more the atmospheric thermal resistance is high (that is more losses), the more the atmosphere gases heat-up.
      If (I repeat, IF, I’m no ways sure about that indeed) the atmospheric concentration of so called GHGs increases then the temperature should increase too.

      Have a great day.


      • Massimo PORZIO says:

        “If (I repeat, IF, Im no ways sure about that indeed) the atmospheric concentration of so called GHGs increases then the temperature should increase too.”

        I would write:
        “If the atmospheric concentration of so called GHGs increases then the temperature should increase too(I repeat, SHOULD, Im no ways sure about that indeed).”

        • JohnKl says:

          Hi Massimo PORZIO,

          You stated:

          If the atmospheric concentration of so called GHGs increases then the temperature should increase too(I repeat, SHOULD, Im no ways sure about that indeed).

          Thank you for stating what should be obvious to anyone, no real empirical evidence exists of actual temperature increases due to GHG’s! Note that the claim of Roy S & many others of a 30 deg Celsius increase in Earth temps due to GHG’s has been inferred from the data but not established. He makes assumptions for example about the Earth’s albedo which may or may not prove accurate. The albedo seems hardly a hard and fast number and would seem to involve some conjecture. A small change in albedo would throw off temp projections enormously. Unfortunately, I must rush and will discuss more later.

          Have a great day!

          • Massimo PORZIO says:

            Hi JohnKl,
            I’m not a climate scientist, just an electronic engineer, so I don’t believe that I’m entitled to establish what is right or what is right in this field.
            But since I’m not a mathematician I can “feel” something really bad about how some “measurements” have been done. 🙂

            By the way, here in North Italy we had probably the coldest July I can remember, but that’s weather not climate, as we know it for sure 🙂

            Have a great day too.


      • Gordon Robertson says:

        Massimo Porzio “IMHO you fail here, because in the real world the so called losses are equivalent to heat transfer to the surrounding environment”.

        Unfortunately, all we have to go on are models, according to atmospheric physicist/meteorologist, Craig Bohren. In the one model, GHGs act as a blanket to trap heat, which he claims is a metaphor at best, and at worst, plain silly.

        Heat cannot be trapped. It is not a substance, it is a description of the state of atoms/molecules based on their relative kinetic energies. Temperature is a scale of relative kinetic energies but heat is a macro description of the state of the atoms/molecules.

        The more likely model according to Bohren is two surfaces radiating against each other. One surface is the Earth’s surface which supplies the heat for the other surface which is atmospheric GHGs. So, you have a massive surface radiating an enormous IR flux against a tiny surface of GHGs that theoretically absorb the surface radiation and back-radiate it in equal amounts, according to AGW theorists.

        In such a model, there are losses on the Earth’s surface as it loses IR to the surroundings. Of course, those losses are balanced by solar energy til a stasis is formed. However, in the model, solar energy is not a factor since stasis has been achieved, therefore the surface can be regarded as operating as a surface at temperature T.

        There are also conduction losses through the medium before contact with GHG molecules. That describes only the greenhouse model.

        With the AGW model, the GHG surface is comprised only of anthropogenic CO2, which accounts for about 1/1000nds of 1% of atmospheric gases (based on 390 ppmv…IPCC). That means further losses in the back-radiation due to the very narrow bandwidth is which ACO2 can absorb and re-radiate.

        Thus far, we have talked only about IR exchange, which is not heat transfer. Heat can only be transferred in one direction, from the hotter surface to the cooler surface. Alarmists are claiming, using the 1st law, that ENERGY transferred is positive in sign therefore the 2nd law is upheld.

        That is sheer nonsense. The 2nd law was not written based on IR transfer, in fact, Clausius wrote it based purely on heat transfer. He made it clear that IR could be transferred in both directions but that heat could only be transferred from the warmer surface to the cooler surface under normal conditions.

        In other words, you would need an external power source to enable heat to be transferred from the cooler surface to the warmer surface.

        That’s where the positive feedback comes in. The theory goes that ACO2 can transfer enough energy back to the surface to warm it beyond the temperature it is warmed by solar energy. In fact, they add IR from ACO2 to solar energy to enable that theory (Rahmstorf).

        The positive feedback is described as the condition under which the super-heating of the surface by ACO2 causes extra water vapour to be boiled off which increases the overall GHG content of the atmosphere.

        That is clearly an abuse of the theory of positive feedback, which would require a mechanism with which to amplify heat.

        The only way ACO2 could affect surface temperature would be to make up for the immense losses at the surface due to surface radiation, thereby interfering with the stasis between solar energy and surface radiation. Clearly, there is not enough ACO2 to back-radiate nearly enough IR energy to replace the immense losses from every nook and cranny on the Earth’s surface.

        The weakness in your argument about resistance is the lack of quantitative data to support it. Here’s an example of what you are dealing with.

        All GHGs represent about 1% of atmospheric gases and ACO2 represents about 1/1000nds of 1% of atmospheric gases (based on 390 ppmv). Of that 1%, water vapour accounts for 96%, and of the remaining 4%, CO2 from natural sources account for most of it. ACO2 accounts for 4% of the 4%, based on 390 ppmv total CO2.

        If you have a real greenhouse with 100 panes of glass, 1 pane would represent 1% of the greenhouse glass. To get the equivalent model in the atmosphere, you would have to remove 99 panes of glass. To get the equivalent AGW model, you would have to break the remaining pane of glass into a tiny sliver.

        The problem with the AGW description of a real greenhouse is the focus on radiative energy. Glass in a real greenhouse traps air molecules which are heated through conduction as solar energy warms the soil in the greenhouse.

        Warming in a real greenhouse is due to the disruption of cooling convective currents that would normally replace heated air as it rises. AGW theory is simply wrong, from the ground up, and that should surprise no one since it was devised by mathematicians programming climate models.

        • Massimo PORZIO says:

          Hi Gordon,
          as said, I still don’t really know for sure how it works, mainly because I’m just an ignorant in climate science.

          Anyways I know what you wrote and I almost fully agree.
          If you go back in time and read my very modest contributions to this blog, you should convince yourself that I’m almost completely by your side.
          But this doesn’t mean that I know the truth in this field, I just know for sure that the current knowledge is at least insufficient to state anything.
          And when I write about “current knowledge” I refer to the current measurements done to establish a causality link between GHGs and ground temperature.
          My critic to your statement was about the negative feedback implicit in losses, which IMHO it doesn’t hold in this context, nothing less nothing more.

          My conviction is that we currently don’t have really measured the current outgoing LWIR fluxes, and the famous “bite” at 666cm-1 could be far less than the one simulated and seen at the TOA by the satellites spectrometers.

          Have a nice day.


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