2011-12-01 14:02:22Understanding Solar Forcing
Jsquared

jfjanak@myfairpoint...
97.83.150.37

http://www.skepticalscience.com/Understanding_Solar_Forcing.html

NOTE:  This was originally published in the chat section here, which also has some commentary on the post.


In recent work on solar forcing, several different authors get very different results for the size of the effect. This is an attempt to try to understand those differences.

The Problem

Solar forcing is the response of the average global temperature to fluctuations in the solar output, δT/δI, where I is the insolation at the top of the atmosphere and T is the globally-averaged temperature.  A simple estimate given at the end of this post leads to the value δT/δI = 0.053 K-m2/W.  The problem is that the temperature data contain responses to several forcings - solar, volcanism, sea-surface temperatures, for example - and one needs a reliable way to extract the solar component.  The problem is made more difficult by the fact that the fluctuations in solar input, due mostly to the 11-year solar cycle, are not large, typically less than 1 W/m2.  Methods that have been tried include extracting the solar part of the temperature by examining latitudinal variations, then using linear regression to fing the best value of δT/δI, and multiple regression using time series for several inputs - solar, volcanic, etc, to extract the solar component.

Regression

The calculations described below use regression techniques. For readers who are unfamiliar with the procedure, the idea is to find the best fit to some function of time - it could be a series of data points rather than a continuous function - as a sum of one or more other functions. The basic procedure is to minimize the squared error between the function and the sum. This is perhaps most familiar in the form of finding the best-fit straight line to a time series. If the function to be fitted is δT, one forms Σ (δT - at - b)2, where the symbol Σ denotes the sum over all the (time) data points. Then one chooses a and b to minimize the sum (for which there are standard formulas). More generally, one could use some other function (say, δI) to fit δT by forming Σ(δT - k δI)2 and finding k by minimizing. Still more generally, one could use several functions ("multiple regression") by forming Σ(δT - k 1 δI - k 2 δV - k 3 δS)2, where δI, δV, and δS are known functions of time, and minimizing to find the values of the k's. The set of "basis" functions that are used is usually physically motivated - the functions are those that are expected to "control" or "force" the behavior of δT, and hopefully all the important ones have been included.

The Calculations

Camp and Tung (2008) removed the linear (global-warming) part of the surface temperature, then extracted a solar component by a spatial (latitudinal) filtering method. They did a linear regression of the result against total solar insolation, and got δT/δI = 0.18 K-m2/W. A graph of their temperature and TSI data appears below:

Douglass et al (2004) did a multiple regression of satellite lower-atmosphere temperature data against sea-surface temperatures, volcanism, solar input, and a linear term. They found δT/δI = 0.1 K-m2/W. Their temperature residual (after regression) with everything but the solar component removed is compared to the solar irradiance data in this graph:

Click to enlarge

The correlation is (to my eye, anyway) a lot cleaner in their results.

Benestad and Schmidt (2009) also performed a multiple regression of the temperature data against the GISS forcings, obtaining δT/δI = 0.09 K-m2/W. They also did several numerical experiments on regression techniques, concluding that linear regression can overestimate the size of the effect. They also indicate that regression techniques in general can produce spurious results if the signals are noisy enough.

Discussion

One way to think about multiple regression is as a decomposition of a vector in a multidimensional vector space, where the basis vectors aren't necessarily orthogonal. If the basis is incomplete, or if two of the basis vectors are almost parallel, the decomposition can become problematic.  These vectors could be functions of space and time, so the decomposition becomes a functional expansion. For the solar forcing problem, if the basis functions are incomplete, as for linear regression (where there is only one basis function), indirect solar forcing (say, for example, in sea-surface temperatures), can show up in the direct component, making it larger than it should be. Similarly, if the signals are dominated by noise, regression techniques might start projecting noise onto noise instead of signal onto signal. It follows that the basis used for multiple regression really needs to be chosen very carefully to get acceptable results. But it is encouraging that the two calculations using multiple regression get nearly the same aswer.

Regarding the factor-of-2 difference between the simple estimate of δT/δI = 0.05 K-m2/W given below and the actual results, there are several possible causes. First, the estimate didn't include water-vapor feedback (just the existing greenhouse effect, not the enhancement due to increased evaporation at the higher temperature). A factor 2 is not out of the question, but seems a bit large compared, for example, to the value of 1.6 inferred from studies of atmospheric cooling following the Mount Pinatubo eruption ( Soden et al, 2002). The IPCC climate models suggest a value between 1.5 and 2.5.

Another potential source of the difference is indirect solar effects. For example, the variations in solar output over a solar cycle are an order of magnitude larger in the UV than in the visible (1% compared to 0.07%; see, eg, Gray et al (2010) and references therein). This implies increased stratospheric heating, and there could be turbulent mixing and heat transfer between the stratosphere and the upper troposphere, eventually leading to increased heating of the lower atmosphere. This is likely to be very hard to model.

Still another channel is through cosmic-ray modulation of cloud cover. Most authors consider this effect to be too small to be observable. It is in phase with the direct effect of the solar cycle, and thus very hard to separate from the direct effect. But it is in the empirical temperature and not included in the simple model below.

Conclusions

Despite the difficulties inherent in trying to extract the size of the effect of direct solar forcing on the global temperature, calculations by several authors seem to be converging on the value δT/δI ≈ 0.1 K-m2/W for the total response of surface temperature to changes in top-of-atmosphere insolation. This number is within the range to be expected from theory including water-vapor feedback, and shows it is very unlikely that the observed 0.5 oC warming since about 1960 is due to solar effects: the average output of the sun has not changed by 5 W/m2 in that time.

Quantifying the Effect

One can get a quick estimate of how large the effect is from global energy balance, which requires that σT4 = (1-α)I/4(1-f), where σ is the Stephan-Boltzmann constant, σ = 5.67x10-8 W/m2-K4, T is the absolute temperature, α ≈ 0.3 is the albedo, I is the insolation at the top of the atmosphere, I ≈ 1365 W/m2, the factor 4 accounts for angular and day-night averaging, and 1/(1-f) represents the greenhouse effect of the atmosphere.  If there were no atmosphere (f = 0), T ≈ 255K; with an atmosphere, T ≈ 290K. so 1/(1-f) = (290/255)4 = 1.69, so f = 0.41.

Now consider small changes: 4σT3δT = (1-α)δI/4(1-f).  At 290K, 4σT3 =5.53 W/m2-K, so δT/δI =0.053 K-m2/W.

2011-12-01 19:54:33
Rob Painting
Rob
paintingskeri@vodafone.co...
118.93.31.116

I have no idea what you're talking about.

2011-12-01 22:22:25
Jsquared

jfjanak@myfairpoint...
71.173.83.223

Rob Painting, what can I do to make it clearer?  I probably ought to move the mathematics down to the end as an appendix, so people don't have to read it if they don't want to, but what else?

2011-12-01 22:52:47
Jsquared

jfjanak@myfairpoint...
71.173.83.223

I've cleaned up the post a little.  The modified post is at the link at the top.  I think it's too long to keep plugging in here

2011-12-01 23:32:15
Daniel Bailey
Daniel Bailey
yooper49855@hotmail...
97.83.150.37

How I do it is to over-write the initial version posted above with the latest version.

2011-12-02 00:43:51
Jsquared

jfjanak@myfairpoint...
71.173.83.223

Daniel Bailey: Do you have permissions to manipulate existing posts that the rest of us don't? I'm unable to delete the old one or paste into it.  Am I missing something?

2011-12-02 00:53:26
Daniel Bailey
Daniel Bailey
yooper49855@hotmail...
24.213.18.68

"I'm unable to delete the old one or paste into it."

Try Step 3 below:

 


 

1.  To access your Author Admin section, go here:
     http://www.skepticalscience.com/admin_author.php

2.  Then under the Author Admin heading, click on the Blogs link:
     http://www.skepticalscience.com/admin_author.php?Action=Blogs

3.  From there, select the link to your blog in question:  Understanding Solar Forcing
     http://www.skepticalscience.com/admin_author.php?Action=EditBlogForm&BlogId=1143

4.  That will bring up your blog post in the WYSIWYG editor.  Same interface as the Forum. 
     When done editing, click on the Save Blog Post at the very bottom left (save your work often).

5.  From there, you can click on the Preview blog post link for a preview (http://www.skepticalscience.com/Understanding_Solar_Forcing.html).
     To fix anything you spy at this point, go up to the link in Step 3 above.

 


 

If you refer to the comment at the top, look to the very lower right corner of the comment; there should be the word Edit that is hyperlinked.  Clicking on that will open up that comment at the very bottom in the editor window.  You have the ability to edit any comment you make here in the Forum.

 


 

Senior Admin's such as dana1981, John Cook and myself have the access to do this for you (should you need help).

2011-12-02 01:00:09
Alex C

coultera@umich...
67.194.33.211

If I understand this correctly, the first authors removed the linear trend and performed a linear regression of the data (which I guess consists of plotting each value at a given time from each dataset against each other and finding a linear trend through them? I understand little of statistics), and got a high value for dI/dT.  Removing further signals (multiple regression?) such as from volcanic activity resulted in a lower trend.

Is the temperature plot shown from Douglass et al the result of filtering of noise?

Whether I am right or not about how multiple regression is done, that does neatly show the problem with the article: it is still very technical (with respect to what a layman might know).

2011-12-02 03:18:12
Jsquared

jfjanak@myfairpoint...
71.173.83.223

Daniel Bailey: Thanks - I didn't notice the 'edit' on the lower right.

Alex C: What the first authors did was to pull a latitude-dependent 'trick' on the temperature data to extract just the solar part, then did a 'linear regression': construct Σ(δT - k δI)2 , where  the sum is over all the (time) data points.  Then you choos the best value of k by minimizing.  'Multiple regression' is doing something like Σ(δT - k1δI -k2δV -k3δS)2, where δI is a time series for solar input , δV is for volcanic aerosols, δS is sea-surface temperatures, and you put in all the things you think drive the temperature. Then find all the k's by minimizing.  The basic idea is that the temperature fluctuations are driven by those forces you put in, and you're finding the best fit under that assumption.  If you choose a pooor set of forcings, you get a lousy fit.

The temperature curve from Douglass et al is what you get after you've done the multiple regression, found the k's, then formed δT -k2δV -k3δS - all but the δI term.  A comparison of that 'residual' temperature to δI gives you some feeling for how good the fit actually is.

It's kind of hard to describe all this without the mathematics, which I was trying to avoid.  I'll try to make the explanation of regression clearer - maybe that will help.  

2011-12-02 04:58:38
Alex C

coultera@umich...
67.194.11.74

So linear regression essentially is just finding a line of best fit through a data set (the slope; scaling factor) - with any given dataset the line will be where the sum of the squared residuals of all points is minimized.  The math makes sense for the linear regression comparing two data sets, I think that it is indeed the case that if you plotted I and T on different axes then you would be able to obtain the K you are talking about(?).

The multiple regression makes sense too, it's essentially just removing other signals that would otherwise contribute to a change in T that is not due to I.

Is that a correct interpretation?

 

>>>A comparison of that 'residual' temperature to δI gives you some feeling for how good the fit actually is.

Small residuals = good fit.  Probably just a relatively constant residual would be adequate too I would presume.

2011-12-02 05:51:59
Jsquared

jfjanak@myfairpoint...
71.173.83.223

Alex C: the word "linear" really applies to the linearity in the coefficients, not to a straight line.  That is, the multiple regression I mentioned is still linear in this sense (linear in the k's).  If I use angle brackets to indicate sums, e.g. <δT> = Σ(δT), then expanding Σ(δT - k δI)2 = <δT2> -2k<δT δI> + k2 <δI2>.  The value of k that minimizes this is k = <δTδI>/<δI2>.  The idea is that's the best you are going to do with one function (it doesn't have to be a straight line).

Multiple regression gives an improvement provided the functions used are "independent" - if you think of them as vectors, none of them is nearly parallel to any other.  You won't do very well if the functions "overlap" a lot - as couold happen if there's a lot of noise.  But noise filtering is  a separate topic.

All I have to do now is to make all this clearer in the original post. :)

2011-12-02 05:57:11
Jsquared

jfjanak@myfairpoint...
71.173.83.223

AlexC: PS I think the formula I quoted for k = <δT δI>/<δI2> is exactly what you get for the best fit straight line when you plot δT against δI.  That's a cute way of looking at it - thanks.   

2011-12-02 06:26:32
KR

k-ryan@comcast...
216.185.0.2

Jsquared - you might also want to look at Taminos post on this:

http://tamino.wordpress.com/2011/01/20/how-fast-is-earth-warming/

He does a nice regression on multiple factors, although he doesn't spend much time on explaining multiple linear regression. His modelling is a combination of weighting and delays for each basis factor (ENSO/MEI, insolation via sunspot numbers, volcanic eruptions, the yearly cycle, and a linear trend), solved for each simultaneously.The various component graphs are quite interesting.

In particular he finds that satellite tropospheric records are more sensitive to these basis functions than surface temperature; 1.5x for volcanic activity and ~2x for ENSO and solar influences.

I believe he has an article in review or press on the topic right now - I eagerly await publication. 

---

In regard to selecting basis factors, you need to put in some effort to ensure that they are reasonably orthogonal (i.e., not directly correlated), and examine your residuals to see if you are missing something - high residuals means a poor model. If your basis functions are poorly cross-correlated and your model shows a small residual, you are likely to have chosen good basis functions.

Note that a chosen basis function that does not result in a reasonable coefficient (the regression shows it not to contribute significantly) indicates that the basis function in question doesn't relate to the data very much - it's worth trying the regression again without it to see if that has a positive or negative influence on the residuals.

2011-12-02 06:28:24
Alex C

coultera@umich...
67.194.11.74

Well glad I could help simplify the concept in some way that's easier to visualize.  I think I understand better too what you mean with the linearity of the K's, at least I understand the lack of necessity fo best linear fit.

2011-12-02 06:32:25
Jsquared

jfjanak@myfairpoint...
71.173.83.223

AlexC: I just added a short section on regression, which I would appreciate your critiquing.

KR: Thanks for the pointer to tamino.  I'll try to add a few comments on optimum choice of basis.

2011-12-02 06:50:11
Alex C

coultera@umich...
67.194.11.74

Looks good Jsquared, it makes sense to me.  Perhaps a short statement on why the squares have to be minimized, and not just the errors themselves (which I personally understand, but maybe not some others).

I think that the explanation of the basic reasoning behind the math, is very useful, especially since the post actually goes into the math.

I still pose one of my earlier questions, the predicted value from the equilibrium forcing equation is after lag has been taken into account (rather, the answer is what you would expect after equilibrium, but not in transit).  It's not clear that the papers take into account an assumed temporal lag between the forcing components and temperature, which I think would change the basic equation from

∑(dT - k*dI)^2

to

∑(dT_ti - k* dI_tj)^2

where ti > tj (starting the time series at t = 0).

(I think.)

I don't know how much of a discussion of this is necessary.  I remember reading Tamino's post myself, and I think he gives certain lag assumptions to each variable.

2011-12-02 06:51:36
Alex C

coultera@umich...
67.194.11.74

Oh, I actually hadn't asked that question before...

2011-12-02 07:32:26
KR

k-ryan@comcast...
216.185.0.2

Jsquared - Reading the post, I believe that you should replace the global mean data from Tung with (or supplemented by) the "zonal mean, annual mean spatial pattern" in the next figure. The latitude dependencies are a major part of that paper, and show much better correlation than the global observations in your shown graph.

Also, from Tung, "On 11-year time scale of about dS~1 watts m-2 . dQ=dS/4~0.25 watts m-2. 30% of that is reflected back to space by clouds and snow and surface."

70% of the 0.25 W/m^2 is 0.175 W/m^2. Are you looking at an apples/oranges comparison with the 0.1 and 0.18 values, when it should be 0.175 vs. 0.18? Note that I could be completely blowing smoke here, but I think it's worth checking...

discrepancy
2011-12-02 07:32:39
Jsquared

jfjanak@myfairpoint...
71.173.83.223

AlexC: Thanks for looking at it.  I wish I could find a way to say it without all that math. 

You don't have to lag the target function, just the basis functions - and you'd expect them to have different lags, in general.  The Douglass paper didn't use lags, and Benestad and Schmidt did. The lags are usually taken as constant - ti = tj + T in your notation, where the T's would be different for each basis function, but are otherwise constant.  You could treat them as additional variables to be determined by minimization - which makes for a messy numerical problem - or just choose some reasonable fixed number.  In some cases - volcanic aerosols - it's probably already in the data.  If I read the Benestad paper correctly, they didn't find a big difference. 

2011-12-02 07:41:17
Alex C

coultera@umich...
67.194.11.74

KR - the 0.18 came from Tung, the 0.1 from Douglass.  I think you're confusing the two data points, 0.175 rounds to 0.18.  So, you wouldn't replace Douglass' figure with the calculated from Tung's.  In other words, comparing 0.175 to 0.18 is, if I'm not mistaken, comparing apples to... well, the same apples.

(Sorry, reading the Tung link I see there's more to where you're getting the 0.1.  I'll leave you guys to this.)

Jsquared - OK; if there was no big difference then perhaps there's not much merit it bringing it up.

2011-12-02 07:45:34
Jsquared

jfjanak@myfairpoint...
71.173.83.223

KR: you are right about zonal means in Camp and Tung.  I'll fix that up.  I'l also have to go back and look at their paper again - I thought they were saying δT/δI = 0.18, where δI is TOA (around 1 W/m2).  The forcing would be δF = 0.175, so they would have λ = δT/δF  ~ 1 K-m2/W. compared to the "Planck" value of 0.3. 

2011-12-02 08:16:14
Jsquared

jfjanak@myfairpoint...
71.173.83.223

KR: bottom of p.3 of Camp and Tung: "This value of k is about 50–70% higher than the regression coefficients of temperature against irradiance variability previously deduced [Douglass and Clader,2002;Lean, 2005; Scafetta and West, 2005], of 0.1 K global- mean surface warming attributable to the solar cycles." And the graph is pretty clearly TSI at top of atmosphere (1 W/m2).

2011-12-04 12:13:21
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.137.110.252

It's an interesting post, but at the end I'm left wondering what it all means.  Is the suggestion that models are understimating the solar forcing by a factor of two?

2011-12-05 02:12:42
Jsquared

jfjanak@myfairpoint...
71.173.66.209

dana1981: the simplest model (Stephan-Boltzmann , or Planck, or whatever), underestimates by about a factor of 2.  Some of the sources for corrections (maybe I need to amlify this discussion a little? - you call) are water-vapor feedback, not included in the simple model, which does include feedback, but assumes a fixed atmosphere; effects of the sun on the stratosphere - because of the order-of-magnitude larger fluctuations in solar output in the UV, but not very well understood; and the (presumably very small or almost nonexistent) effect of cosmic rays on clouds.  Sheer guessing on my part, but I expect feedback to be around 1.5, and the other 0.5 to be stratospheric. 

I didn't talk about lags in the response, and maybe I should.  One of the really interesting questions to my mind is whether the stratospheric effect needs a lag compared to the direct effect.  But then I don't know how to model the effect of stratospheric heating on the lower troposphere in the first place.   

2011-12-05 03:21:44
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.137.110.252
I would suggest clarifying where those feedbacks are and aren't included. For example, they're not in the simplest mode, but are they in global climate models?
2011-12-05 05:09:12
Jsquared

jfjanak@myfairpoint...
71.173.66.209

dana1981: I added a sentence and a couple of references on feedbacks.  How does it look to you now? 

2011-12-05 05:30:03
nealjking

nealjking@gmail...
81.253.24.1

I believe the vast majority of our audience will not understand this at all.

2011-12-05 05:37:53
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.137.110.252

It's a bit better, but I still think the end is missing a take-home message.  What are we supposed to conclude from this discussion?

I agree most people won't be able to follow this, but not every post needs to be simple.  It's okay to throw in a complex one from time to time, but as I said, it's important to boil it down to a take-home message at the end.

2011-12-05 06:27:22
Jsquared

jfjanak@myfairpoint...
71.173.66.209

nealjking and dana1981: I added a paragraph on conclusions at the end.  You are much better judges of the audience than I am at this point.  Perhaps SkS is the wrong place for a moderately technical discussion (not that I'm an expert).  If so, I apologize for wasting people's time. I agree it shouldn't go public unless you regulars see some value in it.

2011-12-05 06:41:21
Rob Painting
Rob
paintingskeri@vodafone.co...
118.93.11.191

Jsquared - don't be put off by the criticism. Can you distill down into a paragraph (no math) what the message is that you are trying to get across? It needs to be comprehensible to a layperson. 

2011-12-05 07:06:53
Jsquared

jfjanak@myfairpoint...
71.173.66.209

Rob Painting - not put off at all.  The criticisms have all been well-founded and constructive.  Maybe I should amplify a little at the beginning, and I will try to formulate a words-only description of multiple regression, and move the math to the end as an appendix.  What I'm trying to get across is (1) regression is not arcane; (2) it's commonly used, so one really needs to know how it works; (3) it's got a number of pitfalls that can fool you if you're not aware of them.  The solar forcing problem is a nice example. 

2011-12-05 09:46:44
nealjking

nealjking@gmail...
81.253.10.63

J2:

Think about this message: "What I'm trying to get across is (1) regression is not arcane; (2) it's commonly used, so one really needs to know how it works; (3) it's got a number of pitfalls that can fool you if you're not aware of them.  The solar forcing problem is a nice example."

Who is it for? Who can understand it?

What is the overlap with the SkS target audience?

As I see it, the typical layperson, concerned with climate issues because he lives on Earth:

1) Will find regression horribly arcane

2a) Will think, "Commonly used by whom?"

2b) Will wonder, "Why should I know how it works? I'm not a statistician or a climate scientist. I'm an inhabitant of this planet."

3) & 4) "Pitfalls? Nice example?"

=> "Beam me up!"

SkS is not about teaching people how to use statistics; for that they can take a course in statistics.

The language level of this article is way too high, and the mathematical level is many many times too high.

2011-12-06 12:11:18feedback bigger than expected
Larry Wade

lwade@caltech...
137.78.4.246

Hi Jsquared,


Feedback between non-condensible greenhouse gases and water appears to be much larger than you estimated. Lacis et al, Science V330, 356 (2010) found that the dependent increase in water vapor was 3 times larger than the direct CO2 effect.

It's also worth noting that solar irradiance is typically over-estimated (Kopp and Lean, Geophys Res Lett, 38, L01706 (2011)), meaning that atmospheric insulation is even more important than typically realized. 

What exaclty is your goal? Are you trying to separate solar variability from other changes, and thereby demonstrate that solar variability is not a significant contributor to global warming??  If so look at Huber and Knutti, Nat Geosci, DOI:10.1038 for a nice job of separating these effects.

Solar forcings of a part of the atmosphere kind of misses the point.  Surface temperatures are the correct metric to use as all of the atmospheric chemistry, mixing, and variation of transmissivity as a function of wavelegth are captured.  It even captures the increase in the depth of the troposphere as a result for the increased atmospheric density due to anthropogenic emissions.  More importantly it captures water.  Water is where 90% of the energy absorbed by the earth (including the atmosphere) is captured (Murphy et al., J Geophys Res, D17107 (2009)).  The atmosphere matters because that it is where the spectral filtering occurs, but the impact is most importantly measured in the ocean.

Best regards,

Larry

 

 

 

 

 

2011-12-06 22:52:01
Jsquared

jfjanak@myfairpoint...
71.173.78.151

Larry: enough people have said the post is too technical for SkS that I've given up on it.  Hopefully it will die a quiet death. 

However, what I was trying to do was examine the reliability of those calculations that try to extract the effect of solar forcing on surface temperature data, essentially by comparing two measured time series, and if there's anything left that can't be explained by water vapor feedback or other omitted effects like stratospheric warming.    

2011-12-07 08:12:40taking a harder look is a good idea
Larry Wade

lwade@caltech...
137.78.4.246

Hi Jsquared,
That's always a good idea.  Being a newbie here, I'm also not sure how to best to handle the math within the Sks forum.  The truth is that one can only wave hands without it.  With it you can at least point while waving hands;)

Take a hard look at the Knutti paper. If I understand you correctly, they took an approach similar to the one you wanted to pursue (use the heat balance/conservation of energy to constrain forcing terms and the residual uncertainties).  By taking advantage of their existing model infrastructure they also had the tools to do, and present, a nicer-than-normal error analysis. 

Larry

 

2011-12-07 08:31:37
Jsquared

jfjanak@myfairpoint...
71.173.93.84

Larry,

Do you have a link to a .pdf for that paper?  All I can get to is the abstract.

Jsquared

 

2011-12-07 08:58:01
Rob Painting
Rob
paintingskeri@vodafone.co...
118.93.136.112

Jsquared - scroll to the bottom of this page. That the paper you're after?

2011-12-07 10:51:22
Jsquared

jfjanak@myfairpoint...
71.173.71.72

Rob Painting - That's the one.  Thanks.