blue_cd

REPLY: Well if you’ve proven it to be a non proof by those methods, then show you work. - Anthony

Here you are.

The following is in response to the conclusions drawn from data presented in Figure 3 and Figure 6 of Dr. Spencer's guest article "Spencer Part 2: More CO2 Peculiarities - The C13/C12 Isotope Ratio. By the time I had written this up, the thread was closed for comments. Please note, that since that posting I have registered at wordpress and had to change my moniker from "blue" to "bluegrue" in the process. You can find this entire post in my livejournal entry, too. I recommend to read it there, because of the better layout.

Notation
Some of the expressions in the following proof can get quite long, so I am using a very compact notation, suitable for a text blog:

1. Variables are one letter only and case-sensitive.
2. x(t) denotes a timeseries, i.e. a collection of n tuples (t_i,x_i) where i = 1 .. n
3. I'll need a lot of sums over all elements of a timeseries and elementwise sums or products of these timeseries. Multiplication signs are dropped. I'll use square brackets to denote them. Let [x] be the sum over all terms of the indicated timeseries. Examples:
   • [x] = x₁ + x₂ + ... + x_n
   • [xy] = (x₁*y₁) + (x₂*y₂) + ... + (x_n*y_n)
   • The mean value of a timeseries in this notation is [x]/n
   • Caveat:
     [1] = 1 + 1 + .... + 1 = n

Regression Coefficient
We will need the slope m of the linear regression line of y(x), where x(t) and y(t) are series with n members:

     m = (n[xy]-[x][y]) / (n[xx]-[x][x])

You can verify that in literature or see e.g. here at MathWorld equation 15. If requested, I can give you the derivation for this from least square principles.

Now on to the proof
Let u(t) and w(t) be two time series with n elements each, sampled at equidistant timesteps (i.e. daily, monthly or similar data).

Create two new time series, by linearly detrending the above time series u(t) and w(t), where e,f,g and h are constants (Dr. Spencer used the regression lines for this, I'm proving the more general case) :

     U(t) :=  u(t) - et - f
     W(t) :=  w(t) - gt - h

Now take the derivatives with respect to time

     x(t) := du(t)/dt
     y(t) := dw(t)/dt
     X(t) := dU(t)/dt = d(u(t) - et - f)/dt = x(t) - e
     Y(t) := dW(t)/dt = d(w(t) - gt - h)/dt = y(t) - g

We can observe, that only the constants g and e survive, when taking the derivatives.

x(t) and y(t) are what Dr. Spencer called the "Trend Signal".
X(t) and Y(t) are, what Dr. Spencer called the "Interannual Signal".

The correlation coefficient of the "Trend Signal" is just

  m = (n[xy]-[x][y])   / (n[xx]-[x][x])

Now let us calculate the correlation coefficient of the "Interannual Signal":

  M = (n[XY]-[X][Y])   / (n[XX]-[X][X])
    = (n[(x-e)(y-g)]-[(x-e)][(y-g)])   / (n[(x-e)(x-e)]-[(x-e)][(x-e)])
    = (n[xy-gx-ey+eg]-([x]-ne)([y]-ng)) / (n[xx-2ex+ee]-([x]-ne)([x]-ne))
    =   (n[xy]-ng[x]-ne[y]+nneg - [x][y]+ng[x]+ne[y]-nneg)
      / (n[xx]-ne[x]-ne[x]+nnee - [x][x]+ne[x]+ne[x]-nnee)
    =   (n[xy]-[x][y])/(n[xx]-[x][x])
    = m 

So the correlation coefficient of Dr. Spencers "Trend Signal" and "Interannual Signal" of any equidistantly spaced time series are identical by mathematical neccessity. Arguing "m equals M, therefore A holds true" is equivalent to saying "4 = 4, therefore A holds true"

Anthony, feel free to copy this post to where ever you feel appropriate.

Taking up the discussion, which ensued on Met Office tipping point blog post on WUWT

DAV wrote

bluegrue (03:20:38) : Satisfied, DAV?

Not quite but maybe getting there. The question was: what has convinced you of anthropomorphic climate changes in the latter part of the past century? IOW: I was asking you to make a case for the anthropogenic climate change charge. Perhaps you misunderstood as you haven’t really answered the question but instead listed a number of largely irrelevant points. I am not particularly interested in what you have read nor your background nor your opinions on other arguments. I would like you to explicitly state what you perceive is the evidence for the previous weather patterns having an anthropomorphic cause.

Can you do this or not?

In short: The increase in atmospheric CO2 from pre-industrial levels is anthropogenic, the attribution studies show that the GHGs dominate the changes in the second half of the 20th century. Hence, AGW.

Longer:
1) As a background, all the above points you brush aside.
2) We know the time series of CO2 and the other greenhouse gases from direct measurements and ice core data. We have increased considerably from pre-industrial levels
3) We know pretty well how much fossil fuel (the major part of our CO2 contribution) we have burned in the past. The increase in CO2 in the atmosphere, the drop in O2 in the atmosphere and the decrease of oceanic pH are consistent with our burning of fossil fuels in the past. The changes of isotopic composition of atmospheric CO2 confirms this picture. I have yet to see anyone to explain the above effects, if it was not us, and if so where our CO2 has gone instead.
4) Investigations like those by Solanki and others, which show that it was not TSI. Other solar effects have yet to be proven, if they do exist they are so small that they are non-obvious and hence can not explain the obvious changes in climate in the second half of the 20th century. I’ll defer you to Leif Svalgaard, he knows the solar stuff much better than I do.

What convinced you the climate change in the 20th century is not anthropogenic?
BTW, will you take up my little challenge? Just curious.

[Edit]

DAV (09:26:54) :

bluegrue (08:47:33) : let’s take our off-topic discussion from this page, we are distracting others.

I’m not so sure it is off topic. I’d rather keep it here but OK. Tomorrow perhaps. I’ve spent enough time on this tody.

My move was prompted by Anthony's announcement "Wasted Effort" on the main page and the comment by AKD (19:55:05) on the tipping-point thread. We really were not talking about the Met Office any longer. ;-)

 
A little challenge. Your mission, should you decide to accept it, is to locate one primary data source of a webpage.

There is this page by Monte Hieb, which is listed in WUWT's resource section. It has been repeatedly been waved around in the comment sections [1] to prove that the influence of anthropogenic CO2 in the atmosphere is negligible. It is here:

http://www.geocraft.com/WVFossils/greenhouse_data.html

Monte Hieb lists in Table 1 amongst others two columns:
- "man-made additions"
- "natural additions"
The caption asserts, that this is data from the DOE. If you follow the link given in the footnote, you end up here:
http://cdiac.esd.ornl.gov/pns/current_ghg.html
Funny thing, this page only covers the other columns, but neither the "man-made additions" nor the "natural additions". Does not now, never has (courtesy of Archive.Org) So the caption is bunk already. But hey, hidden in the footnote there is another source: The IEA Greenhouse Gas R&D Programme. But alas, data is "(data now available only to "members")".
How convenient: The data that is so crucial to Hieb's argument, namely that only 1/7 of the CO2 increase in "man-made", is not accessible.
How inconvenient: The IEA Greenhouse Gas R&D Programme points its readers to the IPCC reports for details
http://www.ieagreen.org.uk/climate.html

Now my challenge: go to Archive.Org, find me the data for
- "man-made additions"
- "natural additions"
either on any incarnation of
http://cdiac.esd.ornl.gov/pns/current_ghg.html
or
http://www.ieagreen.org.uk
At least one goes to pretty much functional incarnation

If you want to save yourself some time, Hieb states on his page that it was last revised January 10, 2003.

Please report your findings, be it success or failure

[1] by Ed Scott, Harold Pierce Jr, Evan Jones, Steve S, KuhnKat, HasItBeen4YearsYet?, Tom in Florida

In response to
Just something to keep in mind. While Mr. Milbank did write this satirical account of the hearing, his position on Climate Change becomes clear in this take on Senator Inhofe:
http://www.danamilbank.com/inhofe.html

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In response to a comment by Smokey

Smokey (13:22:32)

The planet isn’t getting warmer. In fact, lately it has been getting cooler: click

You put a lot of trust into a 6th degree polynomial trendline. I have taken the liberty of updating the data up to and including December 2008, that's all the data available now (the grey data in the plot).

t2lt data and trend speculation:


You will notice, that as of December 2008 the anomaly has rebounded to the level observed in the period 2002 to 2007. The plot contains also a speculative scenario. Let us assume that the anomalies in 2009 will be an exact copy of the data in 2005. How does this affect the trendline? It would flatline with a small upward bend. A single year of data could overthrow your hailed "trend".

My conclusion? A 6th degree polynomial "trend line" is anything but. It may highlight weather, but it is useless as a predictor for climate trends.

This is just a place to anchor my moniker in debates about climate change.

I set this up after getting into debates about identity here:
http://wattsupwiththat.com/2008/01/28/spencer-pt2-more-co2-peculiarities-the-c13c12-isotope-ratio/
Another occasion where I commented substantially:
http://wattsupwiththat.com/2008/08/08/this-year-there-was-an-exceptional-amount-of-ice/

Here's the personal information I am willing to divulge. I am:

• German
• a physicist
• not related to climate research (just a hobbyist taking part in the debate on blogs)

Moniker is "blue", "_cd" is for ClimateDebate, as the undecorated name was taken already.