Open Mind

Multiple Regression

April 12, 2009 · 92 Comments

Not too long ago I got an email inquiry from Barton Paul Levenson. He had done a multiple linear regression of annual average temperature against ln(CO2) (the logarithm of CO2 concentration, proportional to the climate forcing from CO2), DVI (the “dust veil index,” a measure of the cooling effect of volcanic eruptions), TSI (total solar irradiance), and CH4 (methane concentration). The temperature data were from GISS, based on meteorological stations only (rather than the more familiar land-ocean index). To his surprise, the coefficient for CH4 came out negative. CH4 is a potent greenhouse gas, more so than CO2 on a per-molecule basis, although CH4 concentration is far less than that of CO2 (CO2 concentration is usually measured in ppm, or parts per million, while CH4 concentration is generally expressed as ppb, or parts per billion). But according to the regression coefficients, more CH4 meant cooler temperatures. How could this be?

It’s not because of autocorrelation in the data; the temperature data are annual averages, and the autocorrelation for annual data is not very large (although it’s not zero either). The answer lies in the fact that the results of multiple linear regression can be misleading when two or more of the predictor variables are strongly correlated. In this case, ln(CO2) and CH4 are very strongly correlated, as is evident from a graph of their normalized values:


The correlation between the two variables is a whopping 0.9846, so their time evolution is nearly identical when used as predictor variables for global temperature. We can remove the effect they have in common, by modelling CH4 concentration as a function of ln(CO2), then taking residuals; that leaves this:


This is what “counts” when using CH4 as a predictor variable in addition to ln(CO2). Now we can see why adding CH4 into the mix of predictor variables gives it a negative coefficient. CH4 concentration — relative to ln(CO2) — rises rapidly from about 1940 to 1970, when temperature levels off. Then it falls rapidly from 1980 onwards, just as temperature rises rapidly. Hence the correlation between temperature and what’s left of CH4 concentration after removing what it has in common with ln(CO2) turns out to be negative.

If you just look at the significance level of the coefficients (using standard tests) from the multiple regression, the CH4 coefficient appears to be highly significant (99.5% confidence). However, the high significance level is an artifact of the strong correlation between CH4 and ln(CO2). If, instead, you compare two competing models, one using CH4 as one of the predictors and the other not, you’ll find that the influence of CH4 concentration is definitely not significant. Both AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) indicate that the model without CH4 is better. In this case, we don’t even have to resort to information criteria, because the without-CH4 model is a subset of the with-CH4 model. Hence we can directly apply an F-test to see whether or not there’s a significant influence, giving an F-statistic of 0.2257 on 1 and 123 degrees of freedom — not even close to being significant.

Therefore, for these data, the “model of choice” is one which omits the influence of CH4. This doesn’t mean that CH4 hasn’t had an effect; surely it has, but its influence isn’t strong enough (due to the low atmospheric concentration) to be verifiable with these data.

The “take-home” point is that multiple regression can be a tricky animal, especially when two or more of the predictor variables are strongly correlated. And when they’re very strongly correlated, you might even get the wrong sign for the influence, indicating CH4 has a cooling effect when surely, as a potent greenhouse gas, it has a warming effect.

Categories: Global Warming

92 responses so far ↓

  • TCO // April 12, 2009 at 9:51 pm | Reply

    I took a junior in college level survey class in Design of Experiments. Pretty simple stuff. This aspect was already covered in that course. This was way far from some super Mannian linear algebra course. And it was still in there. We get into these problems because we have all these amateurs running around who don’t even have a copy of Box Hunter Hunter (and have not internalized the concepts)…

  • David B. Benson // April 12, 2009 at 9:54 pm | Reply

    Well done again, Tamino!

    Pleased to see you back in form and hoping you’ll soon start a thread on AIC…

  • Deep Climate // April 12, 2009 at 11:41 pm | Reply

    Another way around this problem would be to model all significant GHGs together as one independent “CO2 equivalent” variable.

    Having said that, it appears it would not have made much difference as CO2 would far outweigh the other GHGs.

  • Saltator // April 13, 2009 at 3:14 am | Reply

    Why not factor in water vapour which is the most prevalent greenhouse gas?

    [Response: Since water vapor concentration is controlled by climate, it becomes a feedback, not a forcing.]

  • Chad // April 13, 2009 at 3:17 am | Reply

    I had been experimenting with a similar regression analysis using CO2, TSI, aerosols and MEI. I would never have thought such a complication would arise due to using two strongly correlated variables. Guess I’ve got a lot more to learn. Great post.

  • TCO // April 13, 2009 at 3:29 am | Reply

    I would think things like variable energy storage with the ocean via different weather patterns ala el Nino, would make it hard to do much with this sort of regression analysis.

  • Barton Paul Levenson // April 13, 2009 at 10:29 am | Reply


    Hey, thanks for that analysis! I appreciate it. The fact that I was getting a significant negative coefficient for methane, which I knew to be a powerful greenhouse gas, was driving me crazy. I’ve got to study more about multiple regression. This stuff fascinates me.

  • Soil Creep // April 13, 2009 at 11:31 am | Reply

    I believe this issue is referred to as multicollinearity:

  • Philippe Chantreau // April 13, 2009 at 3:28 pm | Reply

    Very interesting. Excellent post once again Tamino.

  • Chad // April 13, 2009 at 4:57 pm | Reply

    Hey Bart,
    I looked up the dust veil index and was displeased to find it stops in 1983. I found a similar time series covering 1890-1999.

    And there’s also this file on radiative forcing for things other than aerosols (GHG, H2O, etc) from GISS. (1880-2003)

  • Chad // April 13, 2009 at 5:44 pm | Reply

    correction: I meant to write “Hey Barton”.

  • Barton Paul Levenson // April 13, 2009 at 6:39 pm | Reply

    Soil Creep,

    My first thought was multicolinearity. But in classic multicolinearity, the regression coefficients are of low statistical signficance. This wasn’t the case here, even though the dependent variables were highly correlated.

    Tamino, is there another term for this kind of effect? If some particular economist or statistician studied it, we could then call it “Jones’s Multicolinearity,” or something of the sort.


    Michael Mann extended the DVI to 1995 to account for Pinatubo; that was the series I used. But the lack of figures from 1984 up (or 1996 up) is extremely annoying. I’d love to find a global aerosol time series that covered more years. Also, the Ammann et al. data is a little hard to use for me since it is given by month and for 64 latitude regions. I don’t mind averaging but with a data set this size it can take a long time.


    Wasn’t I using the land-surface figures? I thought I was. I’ll bet I picked the wrong table again. Darn it, I keep doing that!

    [Response: I think you're right; it's my mistake.]

  • Chad // April 13, 2009 at 7:48 pm | Reply

    I too found that series by Amman. I wrote a script to calculate the monthly global averages. I could also calculate annual averages if that’s what you need. Let me know, I’ll send it to you.

  • george // April 14, 2009 at 12:45 am | Reply

    I know it probably does not change anything substantially, but the radiative forcing for methane is actually approximately proportional to the square root of the concentration.

    So wouldn’t you want to regress that (rather than simple concentration) against temperature? And likewise, for your analysis above, consider “square root of CH4 concentration as a function of ln(CO2)” rather than “CH4 concentration as a function of ln(CO2)”?

    [Response: Yes on both counts. It does make more sense to regress against sqrt(CH4), and it doesn't change anything substantially, the correlation with ln(CO2) is still strong enough to give an unreliable coefficient for the CH4 effect.]

  • Kipp Alpert // April 14, 2009 at 2:50 am | Reply

    Tamino:Chris Colose has said and written that CH4 and CO2 were comparable greenhouse gases, but since there is much less CH4 absorbed, it would have a larger effect, in the bands where it absorbs. He even said that CO2,
    because of it’s symmetry, was a better absorber, and would start moving out to it’s wings, as it has absorbed mostly in the middle part of it’s bands.
    Could you clarify this,thanks,kipp

    [Response: I'm no expert on the absorption of radiation by molecular species. I do know that the higher concentration of CO2 means that its absorption bands are approaching saturation, so increasing concentration mainly increases absorption in the wings of the absorption bands. I believe that's one of the reasons that at present concentrations, CO2 climate forcing is proportional to the logarithm of concentration while CH4 forcing is proportional to the square root of concentration.

    But as I say, I'm no expert on this. If the Rabett is about, he may be able to enlighten us.]

  • Kipp Alpert // April 14, 2009 at 3:10 am | Reply

    Tamino;After learning all of this new information, 1 month of deep learning from (Kipp for Physics) you are getting me all flustered and I will become a denier if you do not answer this,please.Direct from Chris Colose part2

    The gases degree of “saturation” is also relevant; it is often remarked that methane is more powerful than CO2, but this is not due to some intrinsic property of the gas, but precisely because methane exists in lower concentrations and so has yet to fill its primary bands. If methane existed in higher concentrations than CO2, the reverse would be true: CO2 would be more powerful on a molecule-by-molecule basis. Adding a gas against lower background concentrations will have more of an effect than adding it against higher background concentrations as the absorption spreads away from the peak.

    [Response: Didn't I say something similar? I certainly don't think Chris and I are in disagreement.]

  • Chad // April 14, 2009 at 3:41 am | Reply

    I suppose one could also use the simplified radiative forcing expressions found on this page:

  • Phil. // April 14, 2009 at 7:04 am | Reply

    Weakly absorbing lines give a linear response, as the lines saturate at the line centers and the increase in broadening gives rise to ~log dependence further broadening leads to square root dependence.

  • Timothy Chase // April 14, 2009 at 7:41 am | Reply

    Tamino wrote inline:

    Response: I’m no expert on the absorption of radiation by molecular species. I do know that the higher concentration of CO2 means that its absorption bands are approaching saturation, so increasing concentration mainly increases absorption in the wings of the absorption bands. I believe that’s one of the reasons that at present concentrations, CO2 climate forcing is proportional to the logarithm of concentration while CH4 forcing is proportional to the square root of concentration.

    That is how I understand it. In any case I found this bit of text:

    For gases such as halocarbons, where the naturally occurring concentrations are zero or very small, their forcing is close to linear for present-day concentrations. Gases such as methane and nitrous oxide are present in such quantities that significant absorption is already occurring, and it is found that their forcing is approximately proportional to the square root of their concentration. For carbon dioxide, parts of the spectrum are already so opaque that additional molecules are almost ineffective; the forcing is found to be only logarithmic in concentration.

    It would appear that at very small concentrations nearly all of the marginal absorption (borrowing a term from economics, but please read “additional” absorption as concentration increases) is taking place at the peak. At moderate concentrations marginal absorption is already beginning to spread out, although some is still taking place in the peak. Then at higher concentrations essentially all marginal absorption is taking place in the wings.

  • Timothy Chase // April 14, 2009 at 8:10 am | Reply

    Humorous sidenote…

    Incidentally, when we had over at Real Climate the climate skeptic Frank James Cripwell (he called himself “Jim Cripwell” there), I looked him up, found out that he had gone to Cavendish Laboratories, had a background in radiation physics, was living in Canada, had taken up quilting when his wife died, etc. But I had also found where as a skeptical argument, he had argued that if absorption due to carbon dioxide was a logarithmic function of concentration, then the absorption at a concentration of zero would be nonsensical (negative infinity).

    It was at that point that I decided I needed to know how absorption behaved at low concentrations — just in case he tried to make the argument at Real Climate. And then I found out the three regimes, and how the first regime was linear due to it all taking place in the peak, whereas the absorption became a logarithmic function of concentration as higher concentrations — when any additional absorption had to take place in the wings. Nothing technical — text about as informative as the link. But it would have been enough.

    And I believe all of this was before “A Saturated Gassy Argument,” etc.

  • jyyh // April 14, 2009 at 11:56 am | Reply

    I’ll try to answer Kipp…

    Bonds on CO2 are doubled in the opposite directions and they may resonate to withstand a higher energy level, if I’ve understood that correctly in the Phys Chem. -courses. CH4 is also symmetric, pyramidal, but in this case addition of more energy to the system will produce distorted shapes that will either break the bond or relax on a low frequency radiation that’s not readily re-absorbed. Please correct if I’m totally out in the forest on this one.

  • jyyh // April 14, 2009 at 1:38 pm | Reply

    Oops, …re-absorbed by anything else but water… , that is, of course. Ehh, I keep on forgetting H2O (gas).

  • Ray Ladbury // April 14, 2009 at 1:38 pm | Reply

    Tamino and Kipp, I imagine the Rabett is tired out from hippety-hopping around for Easter.
    If you’ve seen the shape of an absorption curve, it looks like a normal distribution in the middle, but with very broad wings/tails. It looks kind of like a Cauchy distribution. Imagine a spectrum of IR photons moving up from a source (e.g. Earth) through an atmosphere with a greenhouse gas. If you have only a few ghg molecules and lots of photons, you will preferentially absorb photons in the peak of the distribution. However, there are lots of photons, so if you put more ghg into the atmosphere, those too will absorb photons–giving rise to a nearly linear dependence on absorption and ghg concentration.

    As you keep increasing the ghg concentration, there are as many photons in the peak of the distribution for the new molecules to absorb, so the relation between increasing concentration and absorption will be less than linear. This is the region where you get power law dependence–the photon flux in the peak is following some extinction function and absorption in the tails is also starting to be an important contributor to the incremental increase in absorption.

    As you increase the concentration still further, you are absorbing almost all the photons in the peak of the distribution, so most of the incremental absorption is of photons in the wings of the absorption distribution. However, as we said above, the absorption distribution looks like a Cauchy distribution, so no matter how far our you go into the tails, the probability of absorption remains nonzero. This is where you get logarithmic dependence between energy absorbed and increased concentration.

    I don’t know if this helps you that much, Kipp, but it’s at least a starting point for you to ask more questions.

  • Saltator // April 14, 2009 at 1:43 pm | Reply


    “Response: Since water vapor concentration is controlled by climate, it becomes a feedback, not a forcing.”

    I think that clouds and water vapor are also radiative forcing agents.


  • Saltator // April 14, 2009 at 1:47 pm | Reply

    What skeptical science says:

    “What the science says…

    Water vapour is indeed the most dominant greenhouse gas. The radiative forcing for water is around 75 W/m2 while carbon dioxide contributes 32 W/m2 (Kiehl 1997). Water vapour is also the dominant positive feedback in our climate system and a major reason why temperature is so sensitive to changes in CO2.”

  • Demesure // April 14, 2009 at 2:23 pm | Reply

    “Water vapour is also the dominant positive feedback in our climate system and a major reason why temperature is so sensitive to changes in CO2.”
    If temperature is “so sensitive to changes in CO2″, why can’t it be “so sensitive” to changes in water vapor ?
    If an increase in WV is triggered by an increase in GHG (CO2, methane, CFC…) , why can’t it be triggered by and increase in the readily available WV ?

    What keeps the water vapor forcing at “around 75W/m2″ and what prevent the positive feedback loop from increasing indefinitely the WV concentration in the atmosphere (warmer => more WV => warmer => more WV…) ?

    [Response: Very basic physics. Water is condensible, so at earth-like conditions (where water is abundant at the surface) its concentration is mainly determined by temperature. If you could magically decrease the water vapor content of the atmosphere, it would quickly (with a few weeks) return to previous concetrations due to evaporation. Likewise, if you magically increase water vapor content it will quickly return to previous concentrations due to precipitation.]

  • Ray Ladbury // April 14, 2009 at 2:31 pm | Reply

    Saltator, since temperature determines H2O vapor concentration, and since feedbacks are also temperature driven, it makes more sense to treat H2O vapor as a feedback rather than an independent forcing. The thing to remember is that H2O is not an independent variable that can be changed–it’s fixed by temperature and the C-C equation.

    Clouds are another matter.

  • Deep Climate // April 14, 2009 at 2:41 pm | Reply

    Ray Ladbury’s explanation hierarchy of the relationship between saturation and marginal absorption is elaborated in a couple of sources:

    SF6 – linear 350 ppm

    CH4 also has a small negative term to account for overlap with NO2 in the more rigourous GHG accounting .

    See Chris Hope, “Marginal Impacts of Co2, Ch4 and SF6 emissions), (PAGE 2002 Model): (p. 7)

    A good discussion of GHGs and various absorption rates at RealClimate:

    Obviously I have to retract my “CO2 equivalent” comment above, as that describes global warming “potential” over a period of time and wouldn’t apply to an annual time series regression.

  • Deep Climate // April 14, 2009 at 2:44 pm | Reply

    Mangled again! Please delete previous post.

    Ray Ladbury’s explanation hierarchy of the relationship between saturation and marginal absorption is elaborated in a couple of sources:

    SF6 – lt 1 ppb – linear
    CH4 about 1 ppm – sqrt
    Co2 – gt 350 ppm – ln

    CH4 also has a small negative term to account for overlap with NO2 in the more rigourous GHG accounting .

    See Chris Hope, “Marginal Impacts of Co2, Ch4 and SF6 emissions), (PAGE 2002 Model): (p. 7)

    A good discussion of GHGs and various absorption rates at RealClimate:

    Obviously I have to retract my “CO2 equivalent” comment above, as that describes global warming “potential” over a period of time and wouldn’t apply to an annual time series regression.

  • Demesure // April 14, 2009 at 5:12 pm | Reply

    “Likewise, if you magically increase water vapor content it will quickly return to previous concentrations due to precipitation.]”
    If water vapor is increased, GHG effect will increase. Then temperature will increase (according to the GHG theory). Then more WP retention capacity. Then more water vapor… hence the positive feedback.
    The mean relative humidity of the Earth is around 60% so there is ample margin for WV to increase and an illimited reservoir (oceans) for candidate WV before precipitation. What makes precipitations occur to keep the WV content stable “within a few week” ? Why should it be stable on such timescales ? What breaks the positive feedback loop ?
    Why should WV be more sensitive to CO2 (through GHG forcing hence temperature) than to WV itself, both are GH gases, aren’t they ?

    [Response: You haven't thought this through very carefully. If you can work out the mathematics of this "feedback loop," you'll see the ultimate result is no change in final water vapor content.]

  • Kipp Alpert // April 14, 2009 at 5:56 pm | Reply

    Chad,Phil,JyyhTimothy Chase,Deep Climate,Ray Ladbury(Nano Man)
    I think I got it for now.Thanks.So I won’t become a Septic. KIPP

  • Kipp Alpert // April 14, 2009 at 6:03 pm | Reply

    Tamino:Thnaks your answer was spot as far as I know I just didn’t understand what you meant .thanks,Kipp

  • Barton Paul Levenson // April 14, 2009 at 6:10 pm | Reply


    Yes, please! I love annual time series data. [Austin Powers voice:] It’s my thing, man.

  • David B. Benson // April 14, 2009 at 8:00 pm | Reply

    Demesure // April 14, 2009 at 2:23 pm — What changes with increased air temperature is the absolute humidity, but relative humidity remains about the same for the reasons Tamino gave.

    A good paper, relevant here, is Ray Pierrehumbert’s (with co-author) CalTechWater.pdf, available from his publications page.

  • Demesure // April 14, 2009 at 9:11 pm | Reply

    @David B. Benson,
    Which reasons Tamino gave for RH to remain constant please ? I don’t see any, accept “work out the mathematics of the thing”.

    Suppose RH should remain constant (for whatever reason) , it means that the GHG concentration increases with temperature which increases GHG concentration and so on. I still don’t see when the feedback would stop.

    With a +0.7°C warming, the concentration of WV has increased, right (since you assume RH is constant)? So this added WV produces a GHG forcing. How is it counted in the total GHG forcing? If zero, why and if not zero, how much in W/m2 ?

    [Response: The reason you don't see when the feedback would stop is that you don't understand the mathematics of the process. But you're still willing to pontificate.

    Lose the attitude, or get lost.]

  • David B. Benson // April 14, 2009 at 10:53 pm | Reply

    Demesure // April 14, 2009 at 9:11 pm — If RH goes to 1, clouds form and precipitation likely; RH goes down. If RH is low, no clouds, more evaporation, so RH goes up. Indeed, global average RH is about constant.

    Indeed, more WV total (in conditions of near equilibrium) warms the air, which warms the sea; the sea then expresses CO2 producting further warming… This is a positive feedback, f less than 1.

    Suppose it is 1/2 for purposes of illustration only. Then one has
    1 + 1/2 + 1/4 + 1/8 + … = 2
    as the new equilibrium in the limit.

    SO WV is a feedback, not a forcing. It is part of the equilibrium climate sensitivity of close to 3 K from a doubling of CO2 concentrations.

    Here are two chapters from a good book:
    with on-line models

  • Ray Ladbury // April 15, 2009 at 1:08 am | Reply

    Demesure, Look up infinite series. This looked to be a fairly good treatment:

    In any case, you know that infinite series can be finite–or else integral calculus wouldn’t work.

  • george // April 15, 2009 at 2:52 am | Reply


    Tamino himself wrote an excellent post on feedback Even called it that and talked about water vapor.

    In general, for 0 < f < 1

    1+ f + f^2 + f^3 + f^4 + … = 1/(1-f)

    see, for example

    Geometric series

  • J. Lowe // April 15, 2009 at 2:56 am | Reply

    What did the regression show?

    It is noted that the formula for CO2 alone should give:

    Temperature C Change = 0.75 * 5.35*ln(387/280) = 1.3C

    Did the regression return a lower coefficient for CO2?

  • Kipp Alpert // April 15, 2009 at 3:18 am | Reply

    Ray Ladbury:There are different measurements not necessarily logical to me.CH4 is PPM,co2 is ppm,linear,square root,logarithmic eTC. What is the quauntifying reason for these differences.

  • Kipp Alpert // April 15, 2009 at 3:18 am | Reply


  • Demesure // April 15, 2009 at 7:18 am | Reply

    George, Ray & David,
    Sorry, but feedback is not represented by an arithmetic but geometric series and the way Tamino explains feedback (a infinite sum so the total gain is finite) is not how feedbacks work : see for example wiki or in any textbook
    The way to represent feedback by a infinite sum is plucked out of the air.

    Besides, according to Tamino’s link, the moderating factor for the (computed) WV feedback is the lapse rate which is also computed by models so it’s not a scientific evidence.

    Anyway, the total “feedback strength”, according to the link’s graphic, is around 2W/m2, that is huge compared to a CO2 forcing of 1.6 W/m2 (2.6 W/m2 for doubling).
    A system with positive feedback (let alone “large” positive feedback) can’t be stable. Period. If someone says it is, then either he uses the term positive feedback improperly, or there is no “large” positive feedback.

    And it still does not respond to my question: if WV is feedback for CO2, why can’t it be feedback for WV ?

    [Response: My characterization of feedback is correct. This is also discussed in a RealClimate post, and if you had bothered to follow the links you'd have found a nice exposition in Roe and Baker (2007).

    Yet, just as I said earlier, you're willing to pontificate in spite of being in the grip of the most profound ignorance. Until you realize how wrong you are, any attempt to enlighten you is a waste of time.

    Further comments claiming that right is wrong will be deleted. While your idiocy has been somewhat entertaining, it interferes with worthwhile efforts.]

  • Barton Paul Levenson // April 15, 2009 at 11:05 am | Reply


    It’s a converging series, not a diverging one. Not all infinite series diverge. Take, for example, the series 1 + 1/2 + 1/4 + 1/8… That never reaches 2 unless you use an infinite number of terms, and even then it never passes 2. Same with water vapor-temperature feedback.

  • Barton Paul Levenson // April 15, 2009 at 11:13 am | Reply


    The regression equation of anomaly on ln CO2 and DVI for 1880-2008 is:

    Anom = -1882.34802 + 327.1476513 ln CO2 – 0.041756985 DVI
    (-21.36) (15.30) (-2.88)

    R2 = 78% N = 129

    (t-statistics in parentheses).

    This indicates that doubling CO2 from 280 ppmv to 560 ppmv would raise the surface temperature 2.27 K on average, which is well within the IPCC estimated range of 2-4.5 K.

  • Ray Ladbury // April 15, 2009 at 12:06 pm | Reply

    Kipp, the reason for the units is merely convenience. 385 ppmv is easier to write than 0.0385% by volume. And since the argument of the function is a ratio of the current concentration to the starting concentration, the units don’t matter.
    The logarithmic vs. power law approximations have to do with the slope of the absorption curve as one moves into the wings.
    Did you take calculus? It’s a little like determining how many terms of the Taylor series you have to keep go get good enough agreement.

  • Uli // April 15, 2009 at 12:44 pm | Reply

    Barton Paul Levenson,
    have you considered the ocean heat uptake?

  • george // April 15, 2009 at 2:04 pm | Reply

    Demesure (Damnsure?)

    George, Ray & David,
    Sorry, but feedback is not represented by an arithmetic but geometric series… “

    Had you actually read what I posted, you would have seen the geometric series that I used as an example — and seen (and followed) the link to “geometric series” that I gave.

    Tamino’s post also gives a geometric series for feedback.

    And your absolute claim that

    “A system with positive feedback (let alone “large” positive feedback) can’t be stable. Period “

    is just nonsense.

    People have tried to help you understand but I have to say that there’s really no point in further responding to someone who apparently is more interested in making a point (however mistaken) than actually understanding — especially to someone won’t even read the comments he/she is supposedly responding to.

  • Philippe Chantreau // April 15, 2009 at 6:10 pm | Reply

    Demesure, all manners of atmosheric lapse rates have been measured empirically for a long time. Models just use standardized or averaged values for them. Going down the road, “it’s a model so it’s not scientific evidence” won’t lead you to better understanding.

  • george // April 15, 2009 at 7:52 pm | Reply

    It’s more than a little ironic (and humorous) that Demesure quoted wikipedia on feedback above to argue against the possibility of “stable positive feedback”:

    see for example wiki or in any textbook
    The way to represent feedback by a infinite sum is plucked out of the air.

    When I just did a google search on “positive feedback”the link to the wikipedia article “positive feedback” was ranked first and had this to say

    Consider a linear amplifier with linear feedback. As long as the loop gain, i.e. the forward gain multiplied with the feedback gain, is lower than 1 the result is a stable (convergent) output. This is of course always true for a negative feedback but also for lower positive feedbacks. In electronic amplifiers the normal case is that the forward gain is quite high and the amplifier becomes unstable for quite small positive feedbacks.

    of course, the “loop gain less than 1″ is precisely analogous to “f < 1″ for the convergent geometric series that I gave above (and that Tamino gave in his post)

    And, as a reminder: Demesure was arguing against even the possibility o f “stable positive feedback”:

    “A system with positive feedback (let alone “large” positive feedback) can’t be stable. Period. “

    Finally, quite frankly, I’m left scratching my head over Demesure’s comment that

    the total “feedback strength”, according to the link’s graphic, is around 2W/m2, that is huge compared to a CO2 forcing of 1.6 W/m2 (2.6 W/m2 for doubling).

    It looks like he is mixing up apples and oranges ( and throwing in wrong values to boot) and I know I already said there was no point in responding further…

    …but I would simply note a few things more:

    First, the radiative forcing given by IPCC for doubling of CO2 is 3.7 W/m^2 which (in the absence of feedbacks) would lead to about 1 C increase in temperature,

    “Without any feedbacks, a doubling of CO2 (which amounts to a forcing of 3.7 W/m2) would result in 1°C global warming, which is easy to calculate and is undisputed. The remaining uncertainty is due entirely to feedbacks in the system, namely, the water vapor feedback, the ice-albedo feedback, the cloud feedback, and the lapse rate feedback.”[1]

    The normally quoted 3C for doubling of CO2 already includes feedbacks, so really, the sensitivity to CO2 alone is about 1K/3.7 W/m^2 = .27K/W/m^2

    But there are feedbacks and they are positive (notwithstanding the claims of some).

    From the graphic that Tamino shows on his “feedback” post, we can estimate the (positive) radiative feedback due to an initial 1K change caused by the CO2 doubling to be about 2 W/m^2, which would lead to an additional temperature increase (due to the feedback) of about 0.27*2 = 0.54K.

    So, our “feedback value” (in response to the initial 1K change due to CO2 doubling) is really about 0.54K.

    But the feedback is not yet “complete”.

    That was just the first step, which in turn, would lead (at the “next step” of the geometric progression) to an additional temperature increase of (.54) ^2, which would in turn lead to a yet smaller (.54) ^3 increase and on down the geometric progression line.

    The cumulative effect of the feedback would be approximately 1/(1 – 0.54) ~= 2C

    So, the total temperature increase due to (CO2 doubling + feedback) is 1C + 2C ~= 3C

    in other words, overall increase (including feedback) due to CO2 doubling is 3C (the “best” value given by IPCC)

    Note: I realize the numbers I use above are not precise, but they are probably not too far off and do nonetheless (I think) give an idea of what is going on.

  • george // April 15, 2009 at 8:28 pm | Reply

    I just realized I double counted the contribution from CO2 in the sum of the series above.

    The total sum should be 2C (not 3C).

    This statement is correct but applies to the total effect:
    The cumulative effect of the feedback would be approximately 1/(1 – 0.54) ~= 2C

    So, the total temperature increase due to (CO2 doubling + feedback) is ~= 2C

  • infernojones // April 15, 2009 at 8:37 pm | Reply

    Where is the sun in all of this?

    [Response: In BPL's analysis, the solar influence is included through "TSI" = total solar irradiance.

    I hope you can tell how laughably ridiculous the link you've provided is. If not...]

  • dko // April 15, 2009 at 10:09 pm | Reply

    How does BPL’s analysis account for the lag in time between the variables and resulting temperature? We can’t “stop” the experiment and wait several decades for equilibrium and better readings before moving on to the next GHG level. IOW, todays temperature anomaly does not fully reflect today’s CO2 concentration — but maybe that from the 1980s.

    So is there a way to incorp0rate the fact that many forcings are only partially realized?

  • dhogaza // April 15, 2009 at 10:13 pm | Reply

    You know, that denialdepot blog might be a spoof.

    Sample post:

    “If Global Warming is true, why are there still snowstorms?”

    similar to the creationist screech “if people evolved from apes, why are there still apes?”.

    At least, I hope the site’s a farce … but I’m probably wrong.

  • dhogaza // April 15, 2009 at 10:17 pm | Reply

    Yeah, there are only two posts there, and the “about” section is hilarious. Maybe you shouldn’t approve my post above or this one – don’t want to be responsible for spoiling the blogger’s fun.

    With luck it will grow in popularity until it rivals WUWT.

  • luminous beauty // April 15, 2009 at 11:08 pm | Reply

    “Nothing is more fairly distributed than common sense: no one thinks he needs more of it than he already has.”

    –Rene Descartes

  • Deep Climate // April 15, 2009 at 11:17 pm | Reply

    I hope you can tell how laughably ridiculous the link you’ve provided is. If not…

    But there’s a reference… to the Weblog Science Blog of the Year, no less!


  • Mike Bantom // April 15, 2009 at 11:30 pm | Reply

    Oh my lord.

    That has to set some new record for idiotic arguments. That’s like this guy I know who argued that the planes that hit the World Trade Centers couldn’t have damaged them because the buildings weighed over a thousand times more. I asked him how much a bullet weighs compared to a human body.

  • ChuckG // April 15, 2009 at 11:53 pm | Reply

    infernojones probably is the “owner” of:

    that_is_nonsense said…
    What is this rubbish??? Snowstorms disprove global warming??? is this April 1st still? Give up on your blog, you are hopelessly arrogant AND ignorant and I dont care if you delete this comment I just couldnt let this go unmentioned.
    14 April 2009 14:29

    that_is_nonsense said…
    oh and stop spamming everywhere with your blog link
    14 April 2009 14:30

    Inferno said…
    No “mr nonsense” I won’t delete your post. I will leave it as an example of how alarmists resort to name-calling because they have no science
    15 April 2009 11:44

    bolds added

  • Hank Roberts // April 16, 2009 at 12:02 am | Reply

    > denialdepot

    Dogbert: “… a big round number. It’s B-I-I-I-G and R-O-O-UND . . .”
    Dilbert: “Stop it!”

  • David B. Benson // April 16, 2009 at 1:58 am | Reply

    dko // April 15, 2009 at 10:09 pm — Yes thre is. Tamino hass an earlier thread on this topic of fast vs slow feedbacks.

  • Kipp Alpert // April 16, 2009 at 2:49 am | Reply

    Ray Ladbury:When you derive from any volume you might want to use square root,or watts per meter squared to explain proportion and linier is more like 380ppm.Is this right.

  • Kipp Alpert // April 16, 2009 at 3:01 am | Reply

    Ray Ladbury:I’ve Always have been a photographer, and you do what you do best because your not only a natural at it, it’s like something that you always new and studying was easy, right. I’m not dumb, but as I know what global warming is and how it works, than I am honest. This stuff does not come easy for me.The best thing I ever did in science was to show a rose dying over three days once an hour with an old twins lens Rollei. So it’s like you guy’s are in the coolest club and I’ll just read on the sidelines and soak up whatever a well meaning, scientifically challenged person can do. Listen and learn. Maybe I.ll do a group Portrait someday. Kipp

  • Eric // April 16, 2009 at 6:58 am | Reply

    That denialdepot site is obviously a spoof site. I mean, it reads as follows at one point, about exponential notation:
    ‘Exponentials are often used as shorthand by people who don’t want to bother writing all the zeros out. That’s fine if you are just keeping score in a very long game of football! But if you are an engineer designing a saftey system, or a scientist conducting an experiment, you should really take the care to accurately write out whole numbers.’
    Note that it says exponential notation is useful for ‘long game[s] of football’ but that it is better to ‘write out whole numbers’ for doing science. If that’s not a giveaway that the whole thing is a joke, I don’t know what is.
    It’s a pity that it actually looks enough like a *real* denialist to provoke people into making serious replies. It shows the low standard of debate in this field.

  • Geckko // April 16, 2009 at 10:15 am | Reply

    What are the results of Dickey Fuller tests on the resultant regressions?

  • George Darroch // April 16, 2009 at 10:28 am | Reply

    A serious question…

    Water vapor is the dominant greenhouse gas, and is almost entirely non-anthropogenic. It also circulates rapidly.

    However, almost entirely non-anthropogenic is not the same as completely non-anthropogenic. Coal and nuclear power-stations use water for cooling and emit large quantities of heated water vapor into the air. Additionally, this figure is increasing as developed and particularly developing countries increase their coal power output.

    Other human influences may decrease atmospheric uptake of water vapor.

    Has any work quantified the amount of water vapor that is emitted as a direct result of human activity and what the GWP of these water vapor emissions are?

    Water vapor is an area of serious climate research, and I don’t feel qualified to assess any of it. I do note that of course water vapor is a feedback, but looking over the research I haven’t found discussion of what contribution humans have created, except as a feedback of other greenhouse gases. Which is indeed an important area of research.

  • Barton Paul Levenson // April 16, 2009 at 12:13 pm | Reply


    Clearly my regression is only measuring short-term effects. Long-term sensitivity would involve the ocean-atmosphere feedbacks and heat storage.

  • Ray Ladbury // April 16, 2009 at 12:56 pm | Reply

    Kipp, Learning is what it is about. WRT the issue of dimensions. Generally speaking, the argument of a mathematical function has to be dimensionless. Otherwise, you could change the result by changing the units, and that wouldn’t make sense. In physics, this gets handled in a couple of ways:

    1)If the phenomenon depends on changes in a quantity, you can work with ratios–e.g. 380 ppmv/280 ppmv ~1.36. It doesn’t matter then whether you are using percent volume or ppmv.

    2)If there is a coefficient, the units (or “dimensions”)used in the coefficient have to match those of the variable, so you get the appropriate cancellations and the quantity has the right units. Lev Landau (of Landau and Lifshitz fame) was the master at looking at a problem and solving it up to a constant based solely on “dimensional analysis”.

    Keep on it. The science is difficult, but you are making good progress. Just for fun–to remind you of why you are trying to learn all this physics–you might want to pick up a copy of Jearl Walker’s “Flying Circus of Physics”. A great book you can have fun with, and which is invaluable when kids start asking questions like “Why is the sky blue?”

  • Ray Ladbury // April 16, 2009 at 2:01 pm | Reply

    Eric says: “That denialdepot site is obviously a spoof site. ”

    Is it really that obvious? I mean we couldn’t tell if G&T was a spoof, and a lot of what Roy Spencer has done lately would be hard to satirize. Maybe what we have is an indication that Poe’s law applies to climate denialists:

    Without a winking smiley or other blatant display of humor, it is impossible to create a parody of Fundamentalism that SOMEONE won’t mistake for the real thing.–Nathan Poe

  • Igor Samoylenko // April 16, 2009 at 2:31 pm | Reply

    From the About section at DenialDepot:

    “When so-called “experts” in their “peer reviewed journals” say one thing, we dare the impossible and find imaginative ways to believe something else entirely.”


    It has got to be a spoof, surely?

    But then I have seen even more ridiculous claims made by real deniers/contrarians/sceptics in forums…

    This quote would make a great forum signature.

  • Philippe Chantreau // April 16, 2009 at 3:12 pm | Reply

    Eric, it’s hard to tell really. I wouldn’t be surprised if the guy was actually serious. Recently, Watts (science blog of the year, mind you) posted a thread on a paper presenting evidence that GCRs could promote Ozone breakdown by halogens. However, the title of Watts’ post suggested that the paper showed GCRs, not CFCs were responsible for ozone loss. It was as if he did not even bother reading so much as the abstract, which was perfectly clear. Then an endless succession of posts by Watts’ hoi polloi elaborates on how the “greenies” were such a pain for making everyone change what they used for their cars’ AC. It was obvious none of them read the abstract either, until finally someone pointed that the paper actually said something different. Yet the hoi polloi continues on its merry way, even indirectly trashing Rowland/Molina/Crutzen.
    When Dhogaza asked Watts why he devised the title of the thread the way he did, Watts eluded the question. The hoi polloi continued on. So really, the depot thingy might be for real. It is that bad.

  • dhogaza // April 16, 2009 at 5:22 pm | Reply

    Maybe what we have is an indication that Poe’s law applies to climate denialists

    Indeed, it does. When I first quickly skimmed the two posts at Denial Depot I assumed it was real, by someone perhaps 10% more whacked out than Watts.

    Then I read the “about” blurb, part of which has been posted above by Igor, re-read the “snow disproves global warming post” more closely and realized the poster wasn’t saying “late snow this year”, but rather the existence of snow at all, and had a fit of the giggles.

    It’s a pity we’ve outed the owner so quickly. I think the site has real potential if the typical poster over at WUWT can be lured over :)

  • Hank Roberts // April 16, 2009 at 5:55 pm | Reply

    > Has any work quantified the amount of water
    > vapor that is emitted as a direct result of
    > human activity

    George, tha question could mean many things.
    Emitted — it goes up
    Residence time — it stays up
    Rain — it comes back down

    Global warming potential depends on residence time — how long something affects climate.
    Water? Not long:

    Human activity from fountains and firehoses, showers and steam, and H2O from burning the hydrogen in hydrocarbons, adds a little bit of water to the atmosphere. You can find numbers on it if you search. It’s tiny, like comparing the amount of heat added to the climate from human fuel use, to the amount added from sunshine.

    Water rains out (”residence time”) quickly so doesn’t make much difference compared to the natural background humidity. It’s possible warming may raise the background humidity enough to make a difference, e.g.

    Robust responses of the hydrological cycle to global warming
    IM Held, BJ Soden – Journal of Climate, 2006
    Cited by 126:

  • Gavin's Pussycat // April 16, 2009 at 7:27 pm | Reply

    George Darroch, water vapour is on¶ly a feedback. It rains out on an e-folding time scale of ten days.
    The amounts humans put directly into the atmosphere by power plants etc. are completely negligible compared to natural processes — back-of-the-envelope already tells so.
    What apparently is a significant human contribution, at least regionally, is the change in natural evapotranspiration due to changes in vegetation cover. Worth studying. This affects not only specific humidity but also cloud cover.

  • David B. Benson // April 16, 2009 at 8:52 pm | Reply

    George Darroch // April 16, 2009 at 10:28 am — The direct amounts of water vapor are insignificantly small. But the indirect affects from land use cahnge might well matter; everyhting from impounded water to agriculture to deforestation. That said, note that the majority of the surface of the globe is water.

  • George Darroch // April 17, 2009 at 12:21 am | Reply

    Thanks Hank, David, Gavin’s Pussycat.

    “George, that question could mean many things.”

    In my experience, when you know how to ask a question well enough to get the information you want, you already know the subject pretty well =) Half the problem is knowing what questions to ask – which is why the contrarians do so badly, asking questions which have already been answered, and acting like they’ve uncovered something spectacular.

    I don’t doubt that human influence on water vapor is largely insignificant, was just interested as it is such a potent (natural) gas and any non-natural or non-feedback increase or decrease would have some impact.

    Having looked (in a non comprehensive manner) at the literature I’m not sure that the effect of land use changes has been quantified – and I imagine that doing so would be particularly difficult, but that land use satellite data might be employed for the job.

  • Joel Shore // April 17, 2009 at 12:34 am | Reply

    It is scary that things have gotten so bad that we can’t tell if is a parody or not. Still, I am going to go out on a limb and say that I am >99% sure it is.

    On the other hand, I also said that G&T would never see the light of day in any real physics journal, so I’ve had to eat some crow lately. The fact that IJMPB actually published G&T makes me wonder why Sokal ( ) perpetrated his hoax on a social science journal…Clearly, he set his sites too low as one can find physics journals these days who will publish pseudoscientific gibberish masquerading as physics. Urgh…I’m embarrassed for my profession!!!

  • Geckko // April 17, 2009 at 10:49 am | Reply

    Seems to have been mised. What are the results of Dickey Fuller tests on the resultant regressions?

  • Deep Climate // April 17, 2009 at 7:37 pm | Reply

    DenialDepot – yep, it’s a joke. I should have read a little more carefully, instead of just being drawn to the WattsUp link.

    Then again … have you seen Heliogenic Climate Change? That’s pretty funny too (presumably unintentional).

    “The Sun, not a harmless essential trace gas, drives climate change”

    Some, um, compelling discussion and comments about Steig et al study, including Richard Courtney talking about the wrong IPCC graph.

  • Kipp Alpert // April 19, 2009 at 2:19 am | Reply

    Dhogaza:It’s a farce that Denial Depot, Or they are crazy. The picture of the sun and the snow is to red and too blue, like a windows Vista version of Photo shop, contrast dialed way up, for effect. Is this the parody to Mike Moron’s new blog, Climate Depot. They did mention Poe, so it is some of you funny, to smart for your own good warmers. kipp

  • Kipp Alpert // April 19, 2009 at 2:28 am | Reply

    Ray Ladbury:I just purchased the book you recommended at Amazon. Looks pretty good, and a good orientation, which is a start. I have stopped over at Accuweather, they are like trilobites, and nasty to boot. Screw them. I have better things to do. Thanks, Kipp

  • dhogaza // April 19, 2009 at 5:42 pm | Reply

    Yes, it’s a farce, Kipp, but you’ve made an observation that I didn’t make:

    Is this the parody to Mike Moron’s new blog, Climate Depot.

    I’m sure you’re right … “Denial Depot” … “Climate Depot”.

    So, again, too bad we “outed” the site as a parody, but then again a lot of people don’t read this blog, so the owner might have success, yet!

  • Kipp Alpert // April 20, 2009 at 3:35 am | Reply

    Dhog:We must start one of our own, like global warming “The Onion” style.
    A little humor?
    President Obama has begun building the world’s biggest air conditioner.New Ice Age hits Florida nudest colony. Dr.James Hansen siezes’Trump Headquaters for new world Order.ETC.San Francsco earthquake deniers are smoking mad.

  • JH // April 20, 2009 at 10:50 pm | Reply

    Looking at your residual graph, I wonder if you have checked diagnostics, such as residual plot, probability plot and VIFs, to see if the assumptions of homoscedasticity and linearity are satisfied.

    One can also try to center the variables when there is multicollinearity (i.e., when two or more predictor variables in a multiple regression model are highly correlated). It’s also possible that there is a nonlinear relationship between temperature and say, NH4.

    My 2 cents.

  • Richard Steckis // April 21, 2009 at 5:26 am | Reply

    “It’s also possible that there is a nonlinear relationship between temperature and say, NH4″

    That is almost certain. All gases conform to Beer-Lambert’s Law (even water vapour). Therefore, as the concentration of the gas rises , the ability of each extra tonne of gas to absorb radiation diminishes proportionally. ie. the relationship is logarithmic.

  • Saltator // April 21, 2009 at 3:02 pm | Reply


    Don’t you mean CH4? NH4 is ammonium.

  • Kipp Alpert // April 22, 2009 at 3:02 am | Reply

    Richard Steckis: Thanks for your note over at my site, that was really nice. You know the picture I like, so how much is a copy. If you want to leave a note over there, please do. Montage of two pictures Ala Photo shop.

  • Barton Paul Levenson // April 23, 2009 at 3:20 pm | Reply

    michel writes:

    I also thoroughly approve of Lucia’s efforts. Whether her analysis is right or wrong, it is quality work technically

    No, it isn’t. It’s incompetent work, technically. She’s using too-short periods of analysis, ignoring error bars, and generally making a flaming ass of herself — more so because she refuses to take any correction on the matter.

  • Barton Paul Levenson // April 23, 2009 at 3:21 pm | Reply

    Sorry, that was supposed to go into the open thread, but I hit some wrong keys.

  • walter crain // April 25, 2009 at 3:11 am | Reply

    sorry if this is totally off topic and in poor form etc…
    i loved your analysis of the central england temp record a while back. i am having this discussion with “jbob” over at realclimate. he is a real bona fide scientist (”physical scientist”, i believe) who has done an analysis of the central england temperature record. apparently, he has shown that you guys have this global warming thing all wrong…

    (knowing that central england does not represent global temperatures) what can you say about the graphs he as produced using the central england temp record? see this link:

  • Dan Satterfield // April 26, 2009 at 3:58 am | Reply

    Residual issues have come up in my past readings and had always confused me.
    NO MORE!

    Thank you very much for writing this.

  • Johan // May 7, 2009 at 3:30 am | Reply

    As you seem to be a stats savy groupd I have an issue with multi-colinearity in multiple regression. I’m looking at what factors contribute to water clarity as measured by Kd-par (light extinction coefficient of photosynthetically active radiation) in a freshwater delta environment. My potential independent variables being water column Chlorphyll a, dissolved organic carbon (N.S.) and total suspended solids (TSS: not used) and its subcomponents; innorganic suspended solids (ISS: clay and silt) and organic suspended solids (OSS: detritus really). As I have river (n=33) and lakes (n=98) sample data I thought I would analyse the 2 groups seperately, and while I get verry similar coefficents for all included variables, the odd bit is OSS takes on a negative coefficent, ie. more organic suspended matter the clearer the water which seems silly. Oddly the coefficient is very similar for both river and lake data sets. Now no surprise there is a strong collinearity between ISS and OSS. Additional complication is that the method of estimating OSS it is possible to get some degree of overestimation of OSS when ISS is very high, and as my ISS data covers 2 orders of magnitude that may be a contributing issue.

    So while ISS, OSS, Chl a (N.S. in rivers) gives me a good R2 the negative cofficent of OSS bugs me. And yes individually OSS is a possitively correlated with Kdpar (decreasing water clarity).
    So while I could just use the TSS value and not use the ISS and OSS data this seems unsatisfactory as 1) the R2 is not as good and 2) while OSS and ISS are correlated there are trends in the ratio between them such that in clear water, total TSS is about 40-60% OSS, while in turbid water OSS is 5-10% of TSS.

    If your curious my work mostly focusses on trends in physical/chemical/biological differences along hydrological connectivity gradients as that is the major factor controling delta floodplain systems.

    So basically any advice or insights would be welcome.



    [Response: It's hard to speculate without actual data, but here's a thought. Of course the R2 value will be better when you use both ISS and OSS rather than just TSS, it's unavoidable that using the extra information will reduce the residual variance whether the extra information is relevant or not. If you added beer sales data for Osaka, Japan that too would improve R2, in spite of its obviously being irrelevant. You need to know whether the improvement in R2 is enough to be called significant; for that purpose you can use an F-test, or an information criterion like AIC or BIC. It's very possible that the improvement in R2 is entirely accidental, and has no statistically meaningful value.

    And don't forget the oft-repeated adage, "correlation is not causation." Could there be some other variable/process which affects both clarity and OSS, causing then to vary in opposite directions and leading to an apparent (but unreal) causal relationship between the two?

    Instead of using ISS and OSS, try using TSS(=ISS+OSS) and (ISS-OSS). It's even better if you use TSS and whatever linear combination of ISS and OSS which is *orthogonal* to TSS. There's no guarantee, but the result may be enlightening.

    Bottom line: if you can find a professional statistician to help you in your analysis, perhaps the situation will become as clear as the best of your water samples.]

  • Chris S. // May 7, 2009 at 12:46 pm | Reply


    Just a thought – you state “more organic suspended matter the clearer the water “. Could it be the other way round? i.e. The clearer the water the better conditions for organisms (algae, plankton, fish etc.) thus the more organic detritus floating around.

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