Posts Tagged ‘Climate related sea-level variations over the past two millennia’

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Rahmstorf (2011): Robust or Just Busted (Part 4): First results from new code

September 14, 2012

This is part 4 of a multi-part series about “Testing the robustness of semi-empirical sea level projections,” Rahmstorf, et. al., Climate Dynamics, 2011. You can see an index of all parts here. I frequently refer to this paper as R2011.

I will refer to Stefan Rahmstorf’s ”Testing the robustness of semi-empirical sea level projections”  as R2011 [1].

The new code for consistent processing of temperature and sea level data according to the predominant Vermeer and Rahmstorf 2009 model (VR2009)[2] is complete.

It is written LabView V7.1.  There have been several upgrades to LabView since V7.1, but I believe my code will open in any of them.  I prefer this older version of LabView for a variety of reasons that I will not go into here.  But one advantage is that anyone who is interested in running this code can find a used student version of LabView on Ebay at a very reasonable cost.

My code can be downloaded here.

VR2009 input the GISS temperature, Church’s and White’s 2006 sea level data, and modified the sea level data with a correction for reservoir storage from Chao and determined the fit parameters, a, b, and To  for their model…

Rahmstorf and company figured that once a, b, and To were found they could insert hypothesized temperature scenarios for the 21st century into equation 1 and calculate the resulting sea levels.  I have provided a long list of criticisms of their logic.  One of the most devastating observations is that their own source of 20th century sea level data(Church and White, 2006[3]) had revised their data, and the new version of data (Church and White 2009[4] or Church and White 2011[5]) resulted in much lower sea levels by the end of the 21st century when inserted in to equation 1.

Two years ago I reproduced the VR2009 fit parameters, a, b, and To, to demonstrate that I could accurately reproduce their model.

In R2011 Rahmstorf re-works the numbers with the same inputs used in VR2009, and I have reworked the numbers with this new code.  And for the same inputs used back on VR2009, everything lines up within Rahmstorf’s stated uncertainties.  But that is a minor point.  Rahmstorf’s primary objective in R2011 is to defuse my observation that Church’s and White’s newer, more accurate sea level data causes Rahmstorf’s model to yield much lower sea level projections for the 21st century.  Plenty of time to deal with that issue later.

But for now and for the record: in VR2009 Vermeer and Rahmstorf found

a = 5.6 ± 0.5 mm/year/K

b= -49 ± 10 mm/K

To = -0.41 ± 0.03 K

In 2010, using my implementation of their model, I found

a = 5.6  mm/year/K

b= -52 mm/K

To = -0.42 K

In R2011 Rahmstorf presents slightly different numbers than he did in VR2009 for the same input conditions.  Similarly, with my new code I now get slightly different numbers for the same input conditions.

With the new code I found

a = 5.8  mm/year/K

b= -54 mm/K

To = -0.41 K

Presentation of my results

In R2011 Rahmstorf makes some claims based the same model as equation 1, but with various combinations of temperature and sea level data from different sources.  His claim is that he gets essentially the same results – no matter what inputs he uses – indicting that his model is “robust.”

I will also be presenting a lot of results for different possible inputs in the days to come.   But my results will be very detailed, complete, and entirely open for your examination.  You also have access to my complete code.

My code will always generate four files for any set of inputs.  Three of those files are images of: graphs of the input data;  graphs of the model fits to the input data (used to derive a, b, and To); and graphs of sea level projections based on various temperature scenarios for the 21st century, including the SRES emission scenarios used in VR2009 and the RCP45 and RCP85 scenarios used in R2011.  The fourth file is a tab delimited text file with all setup parameters, fit plots and results, and projections.

Note that the graph images of the 21st century sea level projections will not be autoscaled.  That is, the Y axis of the projection graphs will all have the same scaling.  This will make many of the graphs look crowded, but it will also be easy to make a qualitative comparison of the projections from different input data.   You can always open the tab delimited text file in the spreadsheet of your choice and replot the data as you see fit.

Below you can see an example of the graph images and the corresponding tab delimited text file that is generated by my code with the same input data used to find the model fit parameters listed above.  That is, I will use the  GISS temperature, Church and White’s 2006 sea level data and the Chao reservoir correction, which result in my values of a, b, and To, shown above.

The tab delimited text file is shown below.  I have truncated the columns of data (which could be thousands of rows long).   The headers and columns would line up better if you opened the file in a spreadsheet.

INPUTS
Temperature filename: T GISS Land Ocean.txt
Original source: http://data.giss.nasa.gov/gistemp/graphs_v3/Fig.A2.txt		

http://data.giss.nasa.gov/gistemp/graphs_v3/

Sea level filename: SL CW06.txt
Original source: http://www.psmsl.org/products/reconstructions/church_white_grl_gmsl.lis

Modifier filename: RS Chao 2008.txt
Original source: “Impact of Artificial Reservoir Water Impoundment on Global Sea Level”		
Chao, et al., Science 320, 212 (2008)

SETUP PARAMETERS
Minimizing residual: dH/dt
Extension (years): 15.0
Smoothing Gaussian FWHM (years): 15.0
input years used: 1880.0 - 2000.0

FIT PARAMETERS
a: 5.8
b: -54
To: -0.41
H mse: 1.986
dH/dt mse: 0.250

FIT CURVES
date	model H (mm)	data H (mm)	H residuals (mm)	model dH/dt (mm/year)	data dH/dt (mm/year)	dH/dt residuals (mm/year)
1880.050000	-76.997238	-76.648275	0.348963	1.252341	0.699570	-0.552771
1880.150000	-76.873236	-76.577572	0.295664	1.240020	0.714500	-0.525521
1880.250000	-76.750402	-76.505711	0.244692	1.228336	0.722720	-0.505615
    |               |                |              |              |                |                |      
    |               |                |              |              |                |                |    
PROJECTIONS
year	RCP45	RCP85	A1B max	A1B mid	A1B min	A1F1 max	A1F1 mid	A1F1 min	A1T max	A1T mid	A1T min	A2 max	A2 mid	A2 min	B1 max	B1 mid	B1 min	B2 max	B2 mid	B2 min
2000.500000	3.564485	3.462285	4.177685	4.330985	4.330985	4.841985	4.688685	4.586485	4.279885	4.228785	4.688685	4.126585	4.382085	4.790885	4.126585	4.841985	4.688685	4.841985	4.841985	4.790885
2001.500000	7.325070	7.132270	8.226370	8.413370	8.668870	8.815270	8.679370	8.997570	7.908170	8.169470	8.730470	8.181070	8.458670	9.178770	8.181070	9.019670	9.037070	8.917470	8.917470	8.923270
2002.500000	11.429255	11.515155	12.424755	12.588555	13.019455	12.938255	12.819755	13.511955	11.681455	12.169255	12.916155	12.283055	12.628055	13.567755	12.334155	13.170555	13.392355	12.568955	12.875555	13.085755
   |               |                |              |              |                |                |      |               |                |              |              |                |                |

Tab delimited text: VR summary 120913-212735.doc

The three associated graph images…

Input data image: http://climatesanity.files.wordpress.com/2012/09/vr-input-image-120913-212735.png

Fit image: http://climatesanity.files.wordpress.com/2012/09/vr-fit-image-120913-212735.png

projections image: http://climatesanity.files.wordpress.com/2012/09/vr-projections-image-120913-212735.png

___________________________________

[1]  Rahmstorf, S., Perrette, M., and Vermeer, M., “Testing the robustness of semi-empirical sea level projections” Climate Dynamics, 2011

[2] Vermeer, M., Rahmstorf, S., “Global sea level linked to global temperature,” PNAS, 2009

[3] Church, J. A., and N. J. White, “A 20th century acceleration in global sea-level rise“,  Geophys. Res. Lett., 33, 2006

[4] www.psmsl.org/products/reconstructions/church_white_new_gmsl.lis

[5] Church, J. A. and N.J. White, “Sea-level rise from the late 19th to  the early 21st Century“, Surveys in Geophysics, 2011

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Gordian Knot of Nonsense – Part 6. Irrelevance of Baysian Analysis

May 28, 2012

It has been a while since I wrote about ”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), which I will refer to as KMVR2011.

Please see this index of my posts concerning KMVR2011.

I want to sew up one loose end here.  Last time around I showed that this latest incarnation of the Rahmstorf model relating sea level to temperature was just as bogus at the previous versions. But I did not talk about one of their interesting (but ultimately irrelevant) new twists. Another layer of complexity was added by the application of Bayesian analysis, or in KMVR2011 nomenclature: “Bayesian multiple change-point regression.”

Bayesian analysis is a useful, but often counter intuitive, statistical method to tease out an underlying distribution from an observed distribution. That being said, the KMVR2011 application of Bayesian analysis starts out with a bogus model, which has been demonstrated ad nauseam. (See here and here.)  This added layer of complexity simply obfuscates the failures of the starting model, rather that addressing those failures.

My next series of posts will move on to another recent outing by Rahmstorf and company – Testing the robustness of semi-empirical sea level projections  (Rahmstrof, et. al., Climate Dynamics, November 2011)

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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.

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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.

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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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