Posts Tagged ‘PNAS’

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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.

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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.

******************

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

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Gordian Knot of Nonsense – Part 2. A simple hypothetical temperature model

September 5, 2011

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.

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, 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 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.

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Gordian Knot of Nonsense – Part 1. Rahmstorf and company strike again.

August 28, 2011

Rahmstorf and friends are at it again, but this time they have signed on a bigger fish: Michael Mann of hockey stick infamy.  Somehow it does not surprise me that this new serving of dribble comes to us via the Proceedings of the National Academy of Sciences.  Frankly, it grieves me to know that this is the state of the scientific culture in the US. 123

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 Too in this model being the same as a and To  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 To) that needed to be determined.  The 2009 Vermeer and Rahmstorf (VR2009) model went a step further with three constants (a, To, and b) that needed to be determined.  The new KMVR2011 model advances the science with four constants (a1, a2, Too and b).  Count them!  But even more astonishing: this model requires not just solving for the four constants, but also a time varying function (To(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.

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Rahmstorf (2009): Off the mark again (Part 12). A mathematical comedy

February 13, 2011

Here is one more post about the laughably bad PNAS “Global Sea level linked to global temperature” by Vermeer and Rahmstorf.  Will this  fount of absurdity never run dry?

Much has been said about Rahmstorf’s data smoothing techniques.  But the little gem you are about read may make your head spin.

Remember the Chao reservoir correction?  This was the correction that VR2009 applied to the Church and White sea level data to compensate for water that has been impounded in man-made reservoirs.  Never mind the fact that VR2009 paid lip service to, but did not include, a counter-correction for water that has been pumped from the aquifers and has artificially added to the sea level.  Let’s look at some details of how VR2009 handled this correction.

Here is something amazing…

VR2009 had the 2006 Church and White sea level data, which is rather noisy.  They also had the Chao reservoir correction data, which is also noisy.  They correctly saw the need to smooth the noisy data.  It seems that they could have done it one of  two ways: smooth each set separately, then  add the smoothed Chao data to the smoothed Church and White data, or add the unsmoothed Chao data to the unsmoothed Church and White data and then smooth the result.

When I reproduced VR2009’s basic algorithm, I choose the first method.  But VR2009 doubled up on smoothing the Choa reservoir correction.  They smoothed the Chao data, added it to the unsmoothed Church and White data, then smoothed the sum again.  So, the Chao data was effectively smoothed twice.

But here is the really amazing thing:  Look at the overlay of Chao’s data, VR2009’s smooth for the Chao data, and my smooth for the Chao data…

Figure 1. Chao correction to sea level rise rate with VR2009 smooth and Moriarty smooth

Wow! All I can say is “Wow!”  Can you believe how terrible the VR2009 fit for the additional sea level rise rate is?  It’s just amazingly bad! 

How did VR2009 come up with this bizarre data smooth?

In the Matlab program file that VR2009 uses to find the relationship between sea level and temperature (sealevel2.m, get copy here) they first import the unsmoothed Church and White data (church_13221.txt, get copy here) with the following code…

% load the church & white sea level data
load church_13221.txt;
seayear = church_13221(:,1);
sealevel = church_13221(:,2)/10;

Two arrays are created, one with the year, one with the sea level.  The “/10″ in the last line of code converts the sea level data from mm to cm.

Then they apply their Chao reservoir correction.  Instead of importing a time series with the Chao data, they apply a function…

% Apply Chao et al (2008) reservoir correction:
if chao == ‘y’
     sealevel = sealevel + 1.65 + (3.7/3.1415)*atan2(seayear-1978,13);
end

So, VR2009 claims the term “1.65 + (3.7/3.1415)*atan2(seayear-1978,13)” is a representation of the Chao reservoir correction.  Figure 1, above shows the derivative of the Chao reservoir correction (which you can see as figure 3 in Chao’s Science paper).  So the derivative of VR2009’s Chao correction term should at least be close to the derivative provided in Chao’s paper.  Alas, instead it looks like the blue peak in figure 1, above. 

How did VR2009 come up with this strange correction that “fits” the Chao reservoir correction to an inverse tangent (atan2) function?  VR2009 claims to use sophisticated single spectrum analysis (SSA) to smooth its sea level and temperature data.  But their SSA code yields a numerical result, not an analytic one (that is, a time series of numbers, not a formula).  So SSA was NOT used to generate VR2009’s Chao correction term.

If you use my smooth of the Chao data as a baseline, then the VR2009 fit is about 0.2 mm too low around 1960 and about 0.3 mm too high by 1980.  By using their fit to the Chao reservoir sea level rise rate correction, they have effectively increased the sea level rise rate from 1960 to 1980 by an additional 0.5 mm per year.  They have pushed the Chao sea leve rise rate correction to later in the century which, of course, fits their general theme.

The following plot shows the 2006 Church and White sea level data with the questionable VR2009 version of the  Chao reservoir correction data and my version of the Chao reservoir correction.  At first they do not look much different.  But consider this: The VR2009 version causes the average sea level rise rate from 1950 to 1970 to be 1.66 mm/year, and for 1970 to 1990 to be  1.99 mm/year.  That’s a 16% increase.  If my version is used there is an average DECREASE in sea level rise rate, from 1.87 mm/year to 1.78 mm/year.  That is a 5% drop.  Look at figure 1, above, and ask yourself “Whose smooth of the Chao data is better?”

I will not attempt to assign motivation for this laughably bad smooth of the Chao reservoir correction data.  Suffice it to say that it is just one more in long series of blunders and bizarre consequences for VR2009.

Read more about the comedy known as the PNAS “Global Sea level linked to global temperature” by Vermeer and Rahmstorf.

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Rahmstorf (2009): Off the mark again (part 11). VR2009 Matlab code

February 6, 2011

I have written much about Vermeer and Rahmstorf’s 2009 Proceedings of the National Academy of sciences paper “Global sea level linked to global temperature“  (referred to as “VR2009″ in my series of posts).  I have reproduced their algorithm with my own code. 

I thought it would be useful to provide any interested readers easy access to VR2009’s code.  The PDF version of their paper  had a link (which has been long broken) to The National Academy of Sciences website that was supposed to provide the VR2009 code in a zipped file format.  After the original link was broken Kay McLaughlin (PNAS Editorial Staff) kindly sent me this new link.  Just in case that link  breaks, I have archived the zipped file here.  If you unpack the zipped file you will get 24 files: some Matlab code files, some text files, and some .dat files.  Or you can simply look at the list of 24 links at the bottom of this post to download any one of the 24 files individually.

I am not a Matlab programmer, nor was I willing to pay several thousand dollars to buy Matlab.  However, there is a free set of software call GNU Octave, which is highly compatible with Matlab.  With some minor modifications I was able to get VR2009’s “sealevel2.m” file to execute.  The fly in the ointment is that VR2009’s “Sealevel2.m” calls another Matlab file called ‘ssatrend.m,” which is supposed to do single spectrum analysis to smooth the input sea level and temperature data.  VR2009 did not write “ssatrend.m,” but rather reference Aslak Grinsted for this code.  I could not get this version of “ssatrent.m” to work with VR2009’s “sealevel2.m.” 

I had gone down a rabbit hole when I tried to reproduce Rahmstorf’s 2007 results, and I was not going to make the same mistake twice.  So, I simply used LabVIEW to reproduce VR2009’s basic algorithm using my own preferred smoothing method, which I have written about extensively.

It is interesting to note that Rahmstorf’s 2007 science paper also used “ssatrend.m.”  Nicolas Nierenberg at Neirenberg’s Climate Musings pointed out that…

I wrote Dr. Grinsted who wrote me back very promptly, and sent me the source code to ssatrend.m. He also commented that he had no idea how Dr. Rahmstorf had gotten a copy of it, and that he had never meant for it to be distributed. I think that he was just concerned about it being unsupported. My own view is that it is pretty strange to use some random piece of code in a published paper without making the code your own.

Maybe by 2009 Rahmstorf or Vermeer had made contact with Aslak Grinsted and made more conventional arrangements to use his code for VR2009, but I don’s know.

For your reading pleasure, here is an unzipped version of VR2009’s code and results. 
eta_1000y_10be.dat
ipcc_scenarios.dat
sat_1000y_10be.dat
sat_1000y_14c.dat
sat_1750_2100_SRES.dat
ssh_steric_1750_2100_SRES.dat
eta_1000y_14c.dat
chao.m
jack.m
millennium.m
RECE.m
sealevel_predict.m
sealevel2.m
delay_cont.m
autocorr.m
magicc_scenarios.mat
church_13221.txt
giss_landocean.txt
README.TXT
resleft.txt
resright.txt
sl.txt
resid_ab.txt
resid_a.txt

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What is RealClimate afraid of?

December 10, 2010

I left a comment over at RealClimate on December 4th 6th and they deleted it.   I expected them to delete it, since that is what they have done before.  I had the foresight to take a screen shot of their page with the comment and you can read it by clicking on the following image.  Yes it was off-topic, but they don’t seem to delete other off-topic (sycophantic) comments.   You can make your own judgement about why they deleted it.   

My comment dealt with a very serious issue that needs to be addressed by Stefan Rahmstorf – he can only ignore it for so long.

The issues pointed out in the comment are covered in more depth here, here, and here.

If you have read the three above links, then please answer the following poll…

I’m sorry that I spelled “Rahmstorf” incorrectly in the salutation of my comment.  My name is also frequently spelled wrong, but I’m used to it.

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