## Friday, February 19, 2010

### CO2 Derivatives (Not Al Gore's Kind Of Derivatives)

A few days ago an article appeared on What's Up With That that claimed to have disproven AGW using statistics. That claim seems rather bold to me, but there were aspects of the paper I found interesting.

The paper talked about the "1st differences" and "2nd differences" of CO2. By this, the author meant the 1st and 2nd derivatives of CO2. Here I mean derivative as the term is used in Calculus, not the CO2 financial derivatives Al Gore wants to sell you on his carbon stock exchange.

This post shows how to calculate the 1st and 2nd derivatives of a variable in the Climate Scientist Starter Kit.

Calculating Derivatives In The Climate Scientist Starter Kit
It turns out it's pretty easy to calculate 1st and 2nd derivatives in the Climate Scientist Starter Kit. A derivative is calculated as dy/dx. In English, this means the change in y divided by the change in x. In this case, the term "y" refers to the y axis and "x" refers to the x axis. So you can calculate a derivative with the following spreadsheet formula:

`=(Y_Cell - Previous_Y_Cell) / (X_Cell - Previous_X_Cell)`

But its actually even easier than that. Because the X axis represents a series of months, the difference between any two adjoining X values is always 1. This simplifies the formula to:

`=(Y_Cell - Previous_Y_Cell) / 1`

And because the result of any number divided by one is that number, the formula simplifies even further:

`=(Y_Cell - Previous_Y_Cell)`

And there's our derivative!

Calculating the 2nd derivative is just the process of getting the derivative of the first derivative. In other words, it's just the same (Y_CELL - Previous_Y_Cell) run against the 1st derivative rather than the original data series.

The 1st And 2nd Derivative Of CO2
The graphs below show the CO2 data from the Climate Scientist Starter Kit, and the 1st and 2nd derivatives of that data.

CO2 Data

CO2 1st Derivative

CO2 2nd Derivative

The 1st derivative tells us the rate of change in the amount of CO2. The 2nd derivative tells us the rate of change in the rate of change in the amount of CO2.

Conclusion
Originally, I had planned to show how the 2nd derivative of CO2 has a good match with changes in cosmic rays. To do this you just lay the normalized cosmic rays on the graph with the 2nd derivative of CO2.

I had done this very quickly with a couple of decades of data when I first read the article I mentioned above. The match was very good. The 2nd derivative of CO2 and cosmic rays changed in lockstep with one another. Unfortunately, when I extended the analysis to the full range of data for the purposes of writing this post, the new range didn't have that nice correlation.

So I have no cool correlation to show you, but now you know how to calculate 1st and 2nd derivatives of data in the Climate Scientist Starter Kit. Well, ok, I do have one correlation to show you. It's between the 1st derivative of CO2 and the Solar Ephemeris. A similar correlation also exists with the 2nd derivative of CO2.

References:
New paper on mathematical analysis of GHG
Climate Scientist Starter Kit v1.5