## Rahmstorf vs. Rahmstorf

March 5, 2012

Oh, what a tangled web we create, when first we practice to exaggerate.

ClimateSanity
with apologies to Sir Walter Scott

Intrepid mathematician Stefan Rahmstorf has calculated the global temperature increase rate for the last 31 years.  (Global temperature evolution 1979–2010, Foster and Rahmstorf, Environ. Res. Lett. 6, 2011) For the fun of it, lets take him at his word.  The problem is that when his temperatures from this new paper are inserted into his sea level rise rate formula from one of his earlier papers (Global sea level linked to global temperature, Vermeer and Rahmstorf, PNAS, 2009), the calculated sea level rise rate isn’t anywhere close to reality.

These papers can’t both be correct.  My guess is that neither of them are.

In the 2011 paper he starts with five different global temperature records and adds his version of corrections for volcanoes, el Nino and solar variations.  He then calculates the temperature rate of change per decade for each of the five temperature records.  The five ranged between 0.141 °C/decade to 0.175 ° C/decade, but the average was 0.163 °C/decade as shown in figure 1,  below.

He also calculated the temperature rise rate acceleration, and found none.  In his own words

“To look for changes in the warming rates over time, we computed the rate in adjusted data sets for different time intervals, for all start years from 1979 to 2005 and ending with the present. The results show no sign of a change in the warming rate during the period of common coverage.”

Figure 1 Rahmstorf's version of global temperature for 1979 to 2010. This is figure 4 and table 1 from Foster and Rahmstorf. Trendline, based on the average of table 1, added by ClimateSanity

You know what higher temperatures mean: higher sea level rise rates.  Nobody knows this better than Herr Rahmstorf, who has spent the better part of his career making the point.  He has even provided a formula in his 2009 paper to translate the global temperature to the sea level rise rate.

## Some easy math

Assuming his calculated temperature increase rates for the last three decades are correct, what does his sea level rise rate formula tell us?  In Rahmstorf’s parlance H is the sea level and dH/dt is the sea level rise rate.  His formula, from which sprang the famous 1.8 meter sea level rise for the 21st century meme, looks like this…

From Rahmstorf’s graph of global temperature from 1979 to 2010 (figure 1, above), we see that his temperature and the rate of temperature change are given by …

Substituting equations II & III into equation I and gathering terms reveals

While equation IV won’t tell us the exact sea level rise rate for a particular year, it will tell use how much the sea level rise rate changes between two years.  That is

Let’s say that Rahmstorf’s temperature data from the his 2011 Environmental Research Letters paper is correct and his formula relating sea level rise rate from his 2009 PNAS paper is correct.  And let’s say that we wanted to know how  much the sea level rise rate had increased between (oh, I don’t know – how about) 1993 and 2010. Then equation V would tell us that the sea level rise rate should have increased by 1.55 mm/year (0.09128 mm/year X (2010-1993)).

## Comparing to reality

Lucky for us, we have measured sea level data to compare the calculated value to.  As figure 2, below makes abundantly clear, the sea leve rise rate has been about 3.1 mm/year over this time period.   The combination of Rahmstorf’s 2009 PNAS paper and 2011 Environmental Research Letters paper indicate that it should have increased by 1.55 mm/year (an additional 50%).
How can this discrepancy be explained?  Oh yeah, I almost forgot, we already know the Rahmstorf’s formula relating sea level rise rate to global temperature is totally bogus.

## Sea level data set to music. Yeah, that’s right.

January 29, 2012

Vermeer’s and Rahmstorf’s “Global sea level linked to global temperature” (PNAS, 2009) relied on Church’s and White’s “A 20th century acceleration in global sea-level rise” (GEOPHYSICAL RESEARCH LETTERS, VOL. 33,) for their sea level data.  Church and White built their sea level time series from the Permanent Service for Mean Sea Level (PSMSL) tide gauge data.

The following video shows all the PSMSL tide gauge data so you can search for a sea level rise acceleration.  Or you can dance or sing along!

There is no attempt to analyse the data here, but I have started that process and will report on it later.  The first two minutes may be a little boring, but please read along.  It livens up later.   For now, sit back and enjoy.

Update, 3/11/12: My original videos have been banned by Youtube for violating music licenses.  They contained music by REM (The End of the World As We Know It), Johnny Cash (How High’s the Water, Mama) and James Taylor (the traditional “The Water is Wide”).

I have replaced the music with Creative Commons licensed music.

## “Disbelieving is hard work”

January 19, 2012

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.

## Rahmstorf (2009): Off the mark again (part 13), 21st century projections with gamma = 1

December 16, 2011

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 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.

## Gordian Knot of Nonsense – Part 5. Resulting sea-level rise rates

November 20, 2011

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.

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.

## Gordian Knot of Nonsense – Part 4. Solving for To(t) using my hypothetical temperature scenarios

October 17, 2011

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.

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

### To(t),  the “equilibrium temperature”

Recall the KMVR2011’s model includes a moving target “equilibrium temperure”, To(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 To(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 To(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…

### To(t) from my hypothetical temperature scenarios

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

## Gordian Knot of Nonsense – Part 3. More Math (Sorry about that.)

September 22, 2011

“Make everything as simple as possible, but not simpler”

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, Too, a1, a2, b and τ are all constants and To(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 > 0γ 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 a+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.