Posts Tagged ‘michael mann’

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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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Amazing multiplying hockey stick proxies

February 3, 2010

In my previous post I wrote about the five super-simple steps for building a hockey stick:   

Step 1. Gather time series.
Step 2. Select those time series that fit the instrumental (measured) temperature record of choice. Assume that since these time series match the measured temperature in some way, then they are, in fact, temperature proxies.
Step 3. Combine the chosen proxies in some fashion and note, not surprisingly, that the combined proxies match the temperature record. (duh) Call this your temperature reconstruction.
Step 4. Call this thing made from the combined proxies your temperature reconstruction, and therefore assume that the combined proxies are also a match for the temperature that occurred prior to the temperature measurement records.
Step 5. Note that the reconstruction shows that the temperature prior to the instrumental data is relatively flat, and conclude that the temperature prior to the instrumental record changed very little.   

This post is about a little subplot in gathering of time series for the Michael Mann’s 2008 version of the hockey stick (Proxy based reconstructions of hemispheric and global surface temperature variations over the past two millenia, PNAS, 2008)   

Mann used 1209 proxies for this reconstruction.  He explains the breakdown as follows…   

We made use of a multiple proxy (‘‘multiproxy’’) database consisting of a diverse (1,209) set of annually(1,158) and decadally (51) resolved proxy series … including tree-ring, marine sediment, speleothem, lacustrine, ice core, coral, and historical documentary series. All 1,209 series were available back to at least A.D. 1800, 460 extend back to A.D. 1600, 177 back to A.D. 1400, 59 back to A.D. 1000, 36 back to A.D. 500, and 25 back to year ‘‘0’’ (i.e., 1 B.C.).   

Figure 1. Northern Hemisphere proxies in alphabetical order

 

Mann split his analysis between the Northern and Southern hemispheres.  I am going to talk about the 1,036 of the 1,209 proxies that applied to the North.  The following two images show the plots of the these 1,036 proxies, just click on them to enlarge.  The file sizes are less than a megabyte each and should open quickly in your browser.  Figure 1 is the plots arranged in alphabetical order.  If you scroll through this image you will see a lot of proxies that don’t look much like a hockey stick, and a few scattered here and there that do.  However, there is a series of 71 proxies named lutannt1 through lutannt71 that look very much like hockey sticks.    

These lutannt# proxies are from Pauling A Luterbacher, the researcher who “provided” them.  More on this important point later   

Figure 2. All Northern Hemisphere proxies in order of correlation with Northern Hemisphere instrumental temperature record.

 

As explained in the five easy steps for hockey stick construction, the proxies that look much like a hockey stick are likely to be weighted heavily in the final hockey stick construction.  If all the 1,036 proxies are correlated (For the math inclined: see correlation formula below) with the northern hemisphere instrumental temperature record, and the plots laid out from the worst correlation to the best, it will look like figure 2.  Scroll through this figure from top to bottom.  You will see the worst correlations at the top and the best on the bottom.  Note that the Luterbacher proxies are among the best correlated, and show up near the bottom.   

Figure 3. All Northern Hemisphere proxies, except Luterbacher proxies, in order of correlation with Northern Hemisphere instrumental temperature record.

 

Figure 3 is the same as figure 2, but with the Luterbacher proxies removed.  Scroll through, and it is quite clear that there are far fewer hockey stick-like proxies now.   

The Amazing Multiplying Proxies

Remember, the point of a hockey stick is not that it goes up in the 20th century – this is a given because the hockey stick is deliberately constructed from proxies that go up in the 20th century.  The real point is that it is more or less flat prior to the 20th century. (See step 5 of the super-simple steps for building a hockey stick.)  The 71 Luterbacher time series are tailor-made for this purpose, because they tend to show temperature rising in the 20th century but flat prior to that.  The problem with the 71 Luterbacher proxies is that they are actually not 71 separate proxies at all.    

Luterbacher, et.al., (European Seasonal and Annual Temperature Variability, Trends, and Extremes Since 1500, Science, 2004) used about 150 “predictors” spread out over Europe to reconstruct European surface temperature fields.  These predictors consisted of “instrumental temperature and pressure data and documentary proxy evidence.”    Figure 4, taken from Luterbacher’s  supplemental material, shows the geographical distribution of these predictors.   

Figure 4. Luterbacher's original caption: (A) station pressure locations (red triangles) and surface temperature sites (B, red circles) used to reconstruct the monthly European temperature fields (25°W-40°E; 35°N-70°N given by the rectangular blue box). Blue circles indicate documentary monthly-resolved data, blue dots represent documentary information with seasonal resolution back to 1500. Green dots stand for seasonally resolved temperature proxy reconstructions from tree-ring and ice core evidence.

 

 Lutenbacher used combinations of the predictors to interpolate the data to…   

“a new gridded (0.5° x 0.5° resolution) reconstruction of monthly (back to 1659) and seasonal (from 1500 to 1658) temperature fields for European land areas (25°W to 40°E and 35°N to 70°N).”    

Each of these grid points in the reconstruction is like one of the lutannt# graphs that show up in the list of proxies for Mann’s 2008 version of the hockey stick.  Mann ends up with 71 lutannt# “proxies” by simply taking 71 points using 5° x 5° resolution from Luterbacher’s temperature field reconstruction.   

Here’s  the rub: Not all the predictors used to make Luterbacher’s temprature field reconstruction go all the way back to 1500.  In fact, prior to about 1760 only about 10 of the total 150 predictors are used, and these predictors are primarily “documentary information.”  Prior to about 1660, only about 7 are used.   Figure 5, which also comes from Luterbacher’s supplementary material, shows the number of predictors used for each year to reconstruct his surface temperature fields.   

Figure 5. Luterbacher's original caption: Number of predictors through time.

 

Figure 6 shows the location of Mann’s 71 selected “proxies” and the location of the “documentary information” sources.  Not the best match in the world, is it?  Amazingly, the construction of some of the proxies prior to 1750 is based on data from sources over 1000 kilometers away!  

Figure 6. Blue dot show the location of Mann lutannt# "proxies." Red dots show the location of Luterbachers early "documentary information" sources.

 

 The important point is that all the data for Mann’s 71 lutannt# “proxies” prior to about 1760 is made up of some combination of the same 10 or so “documentary information” predictors.  This short list of predictors are the “Amazing Multiplying Hockey Stick Proxies.”  These 10 predictors are multiplied into 71 proxies, and these proxies all rank high for correlation to the instrumental temperature record from 1850 to the present.  Consequently, these 71 “proxies” likely weigh heavily in Mann’s 2008 hockey stick, and these 10 “documentary information” predictors, sometimes folded into “proxies” over a thousand kilometers away, have an undeserved multiplied effect in making the flat part of the hockey stick prior to the instrumental temperature record. 

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correlation coefficient: 

 

where P is the proxy, and Pi is the ith year of the proxy
T is the temperature, and Ti is the ith year of the temperature