Daniel Kahneman

Theory-induced blindness and Vermeer’s and Rahmstorf’s “Global sea level linked to global temperature.”

In one of the many interesting chapters of  Thinking, Fast and Slow, Daniel Kahneman, Princeton University Emeritus Professor of Psychology and winner of the 2002 Nobel Prize in Economics discussed Daniel Bernoulli’s 250-year-old mathematical theory of risk aversion. 

Kahneman points out that “Bernoulli’s essay is a marvel of concise brilliance…

Most impressive, his analysis… has stood the test of time: it is still current in economic analysis almost 300 years later.  The longevity of the theory is all the more remarkable because it is seriously flawed.  The errors of a theory are rarely found in what it asserts explicitly; they hide in what it ignores or tacitly assumes”

Kahneman then goes on to demolish of Bernoulli’s theory.  This demolition is simple and incontrovertible, takes about one page, and is easily understood by anybody of average intelligence. Kahneman says this about the demolition…

“All this is rather obvious, isn’t it?  One could easily imagine Bernoulli himself constructing similar examples and developing a more complex theory to accommodate them; for some reason, he did not.  One could imagine colleagues of his time disagreeing with him, or later scholars objecting as they read his essay; for some reason, they did not either.

The mystery is how a conception … that is vulnerable to such obvious counterexamples survived for so long.  I can explain it only by a weakness of the scholarly mind that I have often observed in myself.  I call it theory-induced blindness: once you have accepted a theory and used it as a tool in your thinking, it is extraordinarily difficult to notice its flaws.  If you come upon an observation that does not seem to fit the model, you assume that there must be a perfectly good explanation that you are somehow missing.  You give the theory the benefit of the doubt, trusting the community of experts who have accepted it.  Many scholars have surely thought at one time or another of stories such as [the examples that Kahneman gives] and casually noted that these stories did not jibe…But they did not pursue the idea to the point of saying ‘this theory is seriously wrong because it ignores the fact[s]‘…As the psychologist Daniel Gilbert observed, disbelieving is hard work…”

What does all this have to do with ClimateSanity?  Simple – it sounds like Vermeer’s and Rahmstorf’s model linking global sea level to global temperature (“Global sea level linked to global temperature,” Proceedings of the National Academy of Science, December 22, 2009 vol. 106 no. 51 21527-21532 ).  It has been incontrovertibly demolished, but the believer’s just can’t let it go.  They must suffer theory-induced blindness.  They seem to have endless capacity to simply overlook the plethora of bizarre, improbable or impossible consequences of the Vermeer and Rahmstorf  model.

Suppose a friend of yours believes that the world is facing a dire future because of too much CO2 in the atmosphere.  The problem seems clear to him, he says, and so does the best course of action.  The fossil fuel powering of the world must come to an end.

He knows that bringing that end about is very difficult, but he perseveres.  It will be very expensive.   Many of the worlds assets will be consumed replacing a thriving fossil fuel based economy with a new “cleaner” economy.

The entire world must be convinced to cooperate, so he cajoles and he scolds.  The standard of living of the wealthy nations must be reduced.  The rest who are embracing the benefits of an energy rich society for the first time will have to wait.

Huge administrative changes will be needed, so he manuevers for control.  Great swathes of the planets landscape will have to be altered to accommodate some combination of wind power, solar power and biofuel production.

But he also has faith that the hardships of his solution will be offset by other (perhaps vague) benefits beyond the reduction of CO2.

A new idea

Now suppose along comes somebody who has another idea to solve the problem: an idea that might not result in a huge disruption of the economy. It’s an idea that might not result in all the hardships.  It is only an idea, in its infancy.  He would like your friend to take a look at it.

What does your friend say?  Does he say “Let’s see this new idea, look at the pros and cons.  Let’s run some simple tests.”  Or does he say “Go away!” and stick his fingers in his ears and say “NANANANANA - I can’t hear you – NANANANANA.”

Would you doubt your friends sincerity, or even his sanity, if he chose the latter response?

Suppose the idea was this…

The Stratospheric Particle Injection for Climate Engineering (SPICE) study, which considers the possibility spraying sulphur particles into the stratosphere, like a volcano, to force global cooling.  This is a big idea and it would be no simple feat to accomplish. What are the advantages and disadvantages?  Who knows?

The stratosphere is 20 km above the surface of the Earth.  The amount of material that would be required is enormous.  The idea is that huge balloons would pull hoses up into the stratosphere and particulate material would be pumped up from the ground.   

It is worth at least thinking about.  A small-scale project to test of the concept was planned in the UK for this fall.  A balloon would hoist a hose only one kilometer and water would be pumped into the atmosphere. 

Image is from "Good governance for geoengineering," Nature, 479, November 2011

Modern Luddites

Alas, this experiment may never take place because a lot of people like your friend may have killed it.  In effect, they said “Go away!” and stuck their fingers in their ears and chanted “NANANANANA - I can’t hear you – NANANANANA.”

According to Nature  (the journal where every environmentalist is a saint),

“the EPSRC [UK Engineering and Physical Sciences Research Council], one of the study’s main funders… received a letter and open petition, also sent to UK energy and climate-change secretary Chris Huhne and signed by more than 50 non-governmental organizations (NGOs) and civil-society organizations, demanding that the project be cancelled.”

The source of the letter and petition?  An organization called “Hands Off Mother Earth.”  The 75 or so signatory organizations to the petition include the following organizations: Amigos da Terra Brasil, Earthpeoples, Gaia Foundation, and my favorite, the Gender Climate Caucus.

Their letter states…

We believe the experiment planned to test equipment for injecting particles into the stratosphere with the aim of counteracting global warming through solar radiation management (SRM) should be cancelled…We believe that such research is a dangerous distraction from the real need: immediate and deep emissions cuts.

In other words, they do not want a simple test of an alternative idea.  I believe that makes them anti-science and anti-reason, modern luddites.

Data from the International Land Coalition, as reported in “Land Rights and the Rush for Land“  leads to this inescapable conclusion.  The Land Matrix project has recorded worldwide transitions of land ownership, leases and control.  Although they cannot record every transition, they have tried to uncover as many 200 hectare or larger changes over the last decade as possible.  In all, they have documented the transition of control and usage of over 2oo million hectares (2 million square kilometers) of land.  To put it in perspective, 200 million hectares is about the amount of cultivated land in the United States and Canada combined.

Is this land rush aimed at growing food for undernourished populations?  Not quite.  The report points out…

Cross-referenced data from the Land Matrix show that in fact the highest demand for land comes from biofuel production…The relatively high proportion of land being acquired for biofuels is particularly striking, considering the displacement of real or potential food production on these lands. This also reflects the expected profitability of biofuels.

Here are a few of their conclusions…

• Of cross-referenced deals for which the commodity is known, 78% are for agricultural production, of which three-quarters are for biofuels.
• Africa is the prime target of the land rush, accounting for 134 million hectares of reported deals.
• The best land is often being targeted for acquisition. It is often irrigable, with proximity to infrastructure, making conflict with existing land users more likely.
• Transnational and intra-national capital flows are pushing land tenure and land-based production systems in a direction that increasingly appears to be far from optimal.

If you are assuming that the folks at the International Land Coalition are a bunch of right-wing anti-environment zealots, well think again.

Ambitious Biofuel Plans in the United States

While were talking about biofuels, lets not forget the ambitious plans for the United States.   Remember the U.S. Energy Independence and Security Act requirement that the US produce 25% of its gasoline from biofuels by 2022?  This alone would require converting about 65 million hectares (equivalent to more than a third of our arable land) to biofuel production in the next 10 years.  Doesn’t make much sense, does if?

Recall the six IPCC families of temperature scenarios, summed up in the following IPCC figure.  VR2009 applied these temperature scenarios to their model to yield corresponding sea level rise rates.  Let’s consider the A1F1 and A1T temperature scenarios.

Figure 1. (top) This is figure 10.26 from the IPCC AR4 Chapter 10, "Global Climate Projections." It shows the temperature projections for each of the six IPCC SRES emission scenarios averaged for the 19 AOGCM models and 3 carbon cycle feed backs and the standard deviations. (bottom) Zoom in on A1F1 and A1t averages.

Here are the resulting VR2009 sea-level rise rates for the A1T and A1F1 scenarios…

Figure 2. Resulting sea level rise rates when the VR2009 model is applied to the A1T and A1F1 temperature scenarios.

Nothing really surprising so far. The sea level rise rates look more or less like the temperatures. 

Now consider some the following hypothetical 21st century scenarios.  Note that they can’t be considered “extreme” when compared the 21st century temperature scenarios already used by VR2009.

Figure 3. The same IPCC temperature scenarios, A1T and A1F1, as in figure 1 and three hypothetical temperature scenarios from Moriarty.

Here are the resulting sea level rise rates…

Figure 4. The sea level rise rates due to the A1T and A1F1 temperature scenarios and three the hypothetical temperature scenarios from Moriarty.

Where are the sea level rise rates for Moriarty’s hypothetical temperature scenarios?  They are perfectly hidden below the sea A1T sea level rise rate.  How can that be?  Because they were designed to be that way to make a point.  See the math here and let γ=1 in equation (VIII) and you will get the idea.  This is not some mistake in my math, but rather a direct consequence of the VR2009 and one more illustration of the bizarre consequences of their model.

From Margaret Wente, Canadian Globe and Mail…

“Although Climategate has been widely dismissed as nothing more than the usual academic sniping, it is much more than that. In some of the e-mails, scientists propose ways to massage the data to make it look better. They try to figure out how to get dissident scientists fired. Others are unhappy because they believe important information has been simplified, suppressed or misrepresented for public consumption.”

See entire article here.

Margaret Wente is one of Canada’s leading columnists, winner of the National Newspaper Award for column-writing, editor of two leading business magazines, Canadian Business and ROB Magazine, editor of the Globe’s business section, managing editor of the paper.

Please see this index of my posts concerning KMVR2011. Check back occasionally because the list of posts is slowly growing.

I will keep things almost entirely graphical this time around (no equations, YEAH!).

Figure 1. Figure 4c from KMVR2011. Global EIV land and ocean temperature and KMVR2011 equilibrium temperature.

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Figure 2. Same as figure 1 from digitized data.

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Figure 3. Same as figure 2 overlaid with GISS temperature (raw and smoothed) and with five hypothetical temperature scenarios starting around 1950

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Figure 4. Same as figure 3, zoomed in to 20th century

Consider the temperature scenarios shown in figure 4.  Which one do you think would lead to higher sea-level rise rates, γ=0.9 or γ=1.1?  Take a look at figure 5, and you may be surprised!

Figure 5. Resulting Sea-Level rise rates when the KMVR20011 model is applied to my hypothetical temperature scenarios compared to the results when the model is applied to GISS temperature.

No Mistake

This not a result of some outrageous error in my calculations.  This is a direct consequence of the KMVR2011 model.  Like VR2009, this bizarre result comes from choosing b to be negative (their choice, not mine).

Some may argue that KMVR2011 uses a wide range of values for the variables in their Bayesian updating.  True enough.  But they kept b negative.  ALL combinations of variables that they used would give qualitatively the same results that I have shown.

Please see this index of my posts concerning KMVR2011.  Check back occasionally because the list of posts is slowly growing.

T_o(t),  the “equilibrium temperature”

Recall the KMVR2011′s model includes a moving target “equilibrium temperure”, T_o(t),  given by equation Ia

The “equilibrium temperature” can be determined by inserting the temperature history or scenario into equation Ia and solving  the resulting differential equation for T_o(t).  Figure 1, below, shows an equilibrium temperature found by KMVR2011 when Mann’s Global EIV land and ocean temperature is used.

Figure 1. this is figure 4C from KMVR2011

In my previous post I laid out a formula (equation II, previous post)  for temperature vs. time that will cause the KMVR2011 model to yield an unrealistic sea level rise rate for a realistic temperature.    In this post I will take the necessary step of finding the ”equilibrium temperature” that results when my hypothetical temperature scenario is inserted into KMVR2011′s equation Ia.  In a subsequent post I will show how my hypothetical temperature scenario and its resulting equilibrium temperature affect the sea level rise rate as calculated by the KMVR2011 model.

Quick and to the point

 Here is T_o(t). 

If you are not interested in the details, you can just take my word it and stop reading here.  Otherwise, continue on the following sections.

“Reasonable” temperature scenarios

 Even the best possible model could not be expected to give reasonable results if the input is nonsensical and it would not be a fair test of the model.   That is why, for the moment, I am choosing to apply hypothetical temperatures for the past (1960 to 2000) to the KMVR2011 model.  In that way the reader can compare my temperature scenarios to the same data used by KMVR2011 for that period and decide if my scenarios are “reasonable”.  

The following graph shows five different temperature scenarios created by my temperature formula.  Each of these scenarios is identical, except for the choice of γ (gamma). 

Are these “reasonable” temperature scenarios?  Are they a fair test of the KMVR2011 model?  Let’s compare them to Hansen’s GISS instrumental temperature data and to Mann’s (Mann is the “M” in KMVR2011) own Global EIV, Land and Ocean temperature reconstruction for the same period…

T_o(t) from my hypothetical temperature scenarios

If you agree that my temperature scenarios are reasonable, then without further ado, here is the derivation of T_o(t).

Let

Inserting equation II into equation Ia gives

Letting

Then

Solving the differential equation in IIIa gives

The constant of integration, C₂, can be found by choosing a known  T_o(t)  at some time, t’…

…and solving for C₂

Now, simply substitute equation VI in equation IV for C₂

Coming Soon

Sea level rise rates from the KMVR2011 model when my simple, reasonable temperature scenarios and the corresponding KMVR2011 “equilibrium temperatures” are used.  I think you will find it interesting.

Update 11/27/11

The term (ατ + 1) were corrected to  (ατ - 1)  in equations (IV) through (VII).  This was a typographical error and all calculations had been done with the correct term.

Albert Einstein

As usual, I will refer to ”Climate related sea-level variations over the past two millennia” (Andrew C. Kemp, Benjamin P. Horton, Jeffrey P. Donnelly, Michael E. Mann, Martin Vermeer, and Stefan Rahmstorf, PNAS, 2011)  as KMVR2011.

I would like to elaborate on my previous post, in which I presented a simple temperature vs. time function that causes KMVR2011′s model relating sea level rise rate to global temperature behave in a rather peculiar manner.  I am trying to find a balance between simplicity, clarity and thoroughness.  The level of mathematical literacy of my readers may vary widely, but this time around I need to employ some calculus.  If the equations bother you,  just consider the conclusions.

Starting with the conclusions

There exists a simple class of realistic temperature vs. time functions, which when applied to KMVR2011′s model yield results that disqualify it as representing a relationship between global temperature and sea level rise rate.  This class of temperature vs. time functions gives a family of curves for which it is guaranteed that the higher the temperature the lower the sea level rise rate.  This implausible effect is so severe that if forces rejection of the KMVR2011 model.

The Math

Here is the KMVR2011 model

where

Where H is the sea level, T(t) is the global temperature, T_oo, a₁, a₂, b and τ are all constants and T_o(t) is a to-be-determined time varying function related to T(t) as defined by equation Ia.

Now, consider the following temperature evolution.  It is nearly the same as equation II from my previous post, but has an additional unitless constant, γ (a.k.a. “gamma”), in the exponential…

If equation II is inserted into equation I, then…

Rearranging terms in equation III gives…

H is the sea level.  dH/dt, the derivative of the sea level,  is the sea level rise rate.  d²H/dt², the second derivative of the sea level, is the rate at which the sea level rise rate changes.  That is, d²H/dt², is the acceleration.  If d²H/dt², is positive, the sea level rise rate is increasing.  Conversely, if d²H/dt², is negative, then the sea level rise rate is decreasing.  Taking the time derivative of equation IIIa gives…

Let’s also consider the difference in the sea level rise rates at some time, t, for different values of γ.  We can do this by analyzing the derivative of dH/dt (equation IIIa) with respect to γ.

What does the math tell us?

KMVR2011 does not conclude with specific values for their model constants and their time varying T₀(t).  Instead, they present probability density distributions for some constants, or combination of constants.  However, there are some definite constraints that can be noted about the variables and their relationships to each other.  These constraints are key to my conclusions.

Constraints:

1. a₁ + a₂ = a, where a1 and a2 are defined in KMVR2011 (see equation I, above) and a is defined in VR2009.  VR2009 found a = 5.6 mm/yr/K.
2. a₁ > 0 mm/yr/K  and a₂ > 0 mm/yr/K.  KMVR20011 states that the distribution of a₁ for their Bayesian analysis varied between 0.01 and 0.51 mm/yr/K.   Needless to say, if either of these terms were less than zero the KMVR2011 model would make even less sense that it does now.  That would be a road that the KMVR2011 authors do not want to travel.
3. b < 0 .  VR2009 found b = -49 mm/K.   KMVR2011 varied b about -49 mm/K with σ² = (10 mm/K)² for their Bayesian analysis.
4. C  > 0.  C is a unitless constant that I introduced, and for the purposes of this post I am constraining C to be greater than zero.
5. γ > 0.  γ is a unitless constant that I introduced, and for the purposes of this post I am constraining γ to be greater than zero.
6. Time, t, is restricted to about 1900 and later for my hypothetical temperature (equation II).  This insures that T(t) > T₀(t), which in turn insures that dT₀(t)/dt > 0.

The equations above, coupled with the listed constraints guarantee the signs of the derivatives shown in table 1, below.

Table 1. Derivatives of temperatures and second derivatives of sea level. Green "up arrows" indicate increasing values and red "down arrows" indicate decreasing values.

As you can see from table 1, it gets little confusing for 0<γ<1.  When a₁, a₂, b, C, and γ conform to the listed constraints, the signs of the various derivatives are known with certainty as long as…

But when …

at some point in time t- t’ will become large enough that d²H/dγdt will become positive.  When that time occurs depends on the choices of a₁, a₂, b  and γ.  If we choose a₁, a₂ and b to agree with VR2009 (recall a₁ +a₂ = a = 5.6 mm/yr/K, and b = -49 mm/K) and γ = 0.8, then d²H/dγdt will continue to be negative until t – t’ = 44 years.

The conclusion, again.

Equation 2, above, can be used to build realistic hypothetical temperature evolutions.  See figure 1, here, for some examples.  Remember, KMR2011′s model relates sea level rise to temperature, and when applied to these hypothetical temperatures it must yield realistic sea level rises.  It does not. 

Table 1 shows various aspects of temperature and sea level using my hypothetical temperature evolution and KMVR2011′s resulting sea levels.  Summed up succinctly, the table shows that with this combination the greater temperatures result in lower sea levels.  This implausible situation disqualifies KMVR2011′s model. 

Coming soon

I realize that a bunch of equations and a table do not give visceral understanding of this effect.  A graphical illustration of these points will be coming soon.

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Update (9/30/11)
Table 1 corrected.  Change makes no difference to conclusions.

As I noted in the previous post, the KMVR2011 model is the progeny Vermeer and Rahmstorf’s 2009 PNAS model and Rahmstorf’s 2007 Science model.

Here is the KMVR2011 model

where

Where H is the sea level, T(t) is the global temperature, T_oo, a₁, a₂, b and τ are all constants and T_o(t) is a to-be-determined time varying function related to T(t) as defined by equation 1a.

Now, consider a temperature evolution of the following  form, where t’ is a constant…

Note the following points about equation II…

• KMVR2001 stipulate that a₁ + a₂ = a, where a is defined in VR2009.
• VR2009 says a = 5.6 ± 0.5 mm/year/K  > 0. 
• Therefore a₁ + a₂ > 0
• b is defined in VR2009, where they claim that b =  -49 ± 10 mm/K.
• Therefore, b < 0
•  Therefore, -(a₁ + a₂)/b > 0
• Since -(a₁ + a₂)/b > 0, then the exponential in equation II increases with increasing t.
• So, if C is chosen to be positive, then T(t) is increasing with increasing t.

Does equation II present a realistic temperature evolution?  Figure 1 shows some simple examples compared to the GISS global temperature.  Figure 1 uses a =  a₁ + a₂ = 5.6 ± 0.5 mm/year/K and b =  -49 ± 10 mm/K, but it would look the same, qualitatively, for any choice of a and b used in the KMVR2011 Monte Carlo simulations used to populate their data for their ”Bayesian updating.”

Figure 1. All of these temperature models satisfy equation II.

What happens to the sea level rise rate, dH(t)/dt, when equation II is inserted into equation I?

All the terms on the right side of equation III are constants except T₀(t).    d²H(t)/dt² is the rate at which the sea level rise is increasing or decreasing.  So given a time evolution in the form of equation II…

We know that a₂ is greater than or equal to zero. (If a₂ = 0, then the KMVR2011 model becomes exactly the same at the VR2009 model.)   What about dT₀(t)/dt? (i.e. How does the equilibrium temperature change with time?) 

Consider equation Ia.  Notice that T₀(t)  is always trying to “catch up” with T(t).  That is, if T(t) > T₀(t), then T₀(t) is increasing.  Conversely, if T(t) < T₀(t), then T₀(t) is decreasing.

Since we are told that the world is now at unprecedented high temperatures compared to the last millennium, then by implication T(t) > T₀(t) for the present day.  This obvious point is confirmed for the last 100 years by KMVR2011 figures 4A & 4C.  Consequently, dT₀(t)/dt must be increasing with time for the present day and for the entire last century.

Pulling it all together

Given the KMVR2011 model described by equations 1 and 1a, and given a temperature evolution described by equation II (and as illustrated by the various model temperatures in figure 1, and “Model temperature 2″ in particular), then  d²H(t)/dt² is negative.  That is, the sea level rise rate is guaranteed to be decreasing.  This is a rather bizarre result that is a consequence solely of the design of KMVR2011′s model.  It is not some math trick or mistake.  KVMR2011 should have been able to anticipate this problem, since it parallels very closely a similar problem with VR2009.  And I know Mr. Rahmstorf was reading my blog.

KMVR2011 would likely argue that my equation II cannot represent a realistic temperature scenario, and that their model can only work for realistic temperature scenarios.   Figure 1, model temperature 2, above refutes such a claim.  We are left with the following situation: When a simple temperature scenario with a rapidly rising temperature that is similar to the last half of the 20th century is applied to KVMR2011′s model, it yields a decreasing sea level rise rate.  This point alone should be enough to raise the eyebrows of  KMVR2011′s readers. 

But there is much more to come.

I will refer to ”Climate related sea-level variations over the past two millennia” (Andrew C. Kemp, Benjamin P. Horton, Jeffrey P. Donnelly, Michael E. Mann, Martin Vermeer, and Stefan Rahmstorf, PNAS, 2011)  as KMVR2011.  This paper dishes up a third generation model relating sea level rise rate to temperature whose immediate ancestors are Rahmstorf’s 2007 model and Vermeer’s and Rahmstorf’s 2009 model.

With H being sea level and T being global temperature the models have evolved as follows.

Generation 1, form Rahmstorf’s 2007 ”A Semi-Empirical Approach to Projecting Future Sea-Level Rise“

Generation 2, from Vermeer and Rahmstorf’s 2009 ”Global sea level linked to global temperature“

And now, Generation 3, from KMVR2011

where

A cursory examination of equation I makes it plain the this new model is simply the cobbling together of  the VR2009 model (with a₁ and T_oo in this model being the same as a and T_o  respectively in VR2009) with an additional term,  a₂[T(t) - T₀(t)], taken from Jevrejeva (GRL, 37, 2010).  KMVR2011 sum up the meanings of each term in equation I as follows…

The first term captures a slow response compared to the time scale of interest (now one or two millennia, rather than one or two centuries as in [VR2009]). The second term represents intermediate time scales, where an initial linear rise gradually saturates with time scale τ as the base temperature (T0) catches up with T. In [VR2009], T0 was assumed to be constant. The third term is the immediate response term introduced by [VR2009]; it is of little consequence for the slower sea-level changes considered in this paper.

 In Rahmstorf’s 2007 model linking sea level rise rate to temperature there were only two constants (a and T_o) that needed to be determined.  The 2009 Vermeer and Rahmstorf (VR2009) model went a step further with three constants (a, T_o, and b) that needed to be determined.  The new KMVR2011 model advances the science with four constants (a₁, a₂, T_oo and b).  Count them!  But even more astonishing: this model requires not just solving for the four constants, but also a time varying function (T_o(t) )!

Back at the keyboard

I have had a leisurely summer, and have not written any blog posts for several months, but my eyes and ears have been open, and my pencil has scratched out a few equations.   This post represents the beginning of a new series on KMVR2011, which I will call the “Gordian Knot of Nonsense.”

This series will be interspersed with posts on other topics, so please check back occasionally for updates.