Earth killer – composite trigonometry CO2 graph
Scientists have known for years that the amount of carbon dioxide in the atmosphere has been increasing.
The observations on the top of Hawaii’s Mauna Loa volcano have shown a disturbing rise in CO2 over the last 50 years.
Image source: National Oceanic & Atmospheric Administration
The black line on the graph represents mean data for each year (with some allowance for missing data points). The green and red oscillating lines are the result of natural “breathing” by the Earth throughout the year. In winter, when leaves drop and people burn coal, wood and oil for heating, the CO2 goes up. In summer, as the leaves reappear and there is less fossil fuel burned, the CO2 concentration drops. (This is Northern hemisphere, of course. There is less CO2 in the Southern hemisphere due to lower population, but the pattern will be similar.)
This graph reminded me of a function that I created as an example for this composite trigonometric graphs page. (It’s Exercise 1 on that page, the curve y = x2/10 − sin πx.)
So I thought it may be an interesting exercise to model NOAA’s CO2 data (1958 to 2008) and try some extrapolation to see where we’ll be in a few decades time.
A model here means an equation that connects the time variable (horizontal axis) and the concentration of CO2 (the vertical axis). Modeling is a very important concept in mathematical thinking, and unfortunately, most students never get to do any modeling, or even see it being done.
Extrapolate means that we will use the equation that we get to predict what will happen in the future (we can also extrapolate backwards in time.)
Climate modelling is probably the most important mathematics going on in the world right now.
Back to the story…
Using Excel to Model the CO2 Data
My first graph uses the full zero to 450 parts per million vertical scale, so that we can see that there is indeed a noticeable increase in CO2 concentration over the last 50 years. (In statistics, you can always exaggerate a trend by restricting the vertical scale. This is a common trick in advertising.)
For the rest of the graphs on this page, I have restricted the vertical axis scale, so that we can see more clearly how well the models work.
Linear Model
The simplest model through the given data points is a straight line. Using Excel’s “add trendline” facility, and choosing “linear”, we get the following:
Excel has given us a “line of best fit”, with the least variation from the data points. It is not a very good model. Clearly, the slope of the CO2 concentration is increasing as time goes on. If we tried to extrapolate beyond 2008, we would be under-estimating the amount of CO2.
(To see how to use Excel’s “add trendline” facility, and for another modelling example, see DJIA Model.)
We clearly have a curve, rather than a straight line, and that curve is likely to be exponential, since CO2 concentrations are related to population growth, which is also exponential.
In summary, I’m looking for a curve that passes through most of the data points and clearly follows the trend.
Exponential Model
This is the model given by Excel and it is clearly not very satisfactory. It under-estimates at the beginning and end of the data series, and doesn’t look much better than the linear model.
The problem, of course, is that Excel is making an assumption that the value at x = 0 (that is, at year 0) is a very small value (in this case, it has chosen 0.1056). However, we know (from Antarctic ice cores) that the CO2 was at a reasonably stable 280 ppm until the beginning of the Industrial Revolution.
There is no way (that i could figure out) to tell Excel to use 280 as a base line. So I subtracted 280 from each of the data points and used Excel to give me a new exponential model. Adding 280 to each of the model’s values, gave the following result.
Actually, Excel’s model was y = (1E-17)e^0.0216x which was close, but not good enough. I have tweaked this to give the graph of the model overlaid on the original data above.
Now we are getting somewhere. The model fits quite well with the data.
Here is a long-term view of the model, indicating the relatively stable CO2 levels (at 280 ppm) until around 1800 when the madness of coal burning began.
Yearly Oscillations of CO2
The yearly oscillations can be represented by a cosine curve, whose period is 1 year. The amplitude of the cosine curve is just over 3 ppm, derived from observation. There is a slight phase shift since the data actually starts in Jan 1958 and there is a lag before CO2 concentration reaches its peak for the year.
This cosine curve is simply added to the polynomial curve expression, as follows:
y = 3.07cos(2πx-1.2) + 280 + (10^-17)e^0.02181x
This graph (obtained using Scientific Notebook) looked quite close to the original NOAA data.
To check it, I resized and then overlaid the black model graph onto the original NOAA graph (green and red) and obtained:
I’m quite satisfied that the model is a good fit.
In fact, this model seemed good enough to use for extrapolation. So here is 100 years of abuse of the world’s air, from 1935 to 2035. We will have managed to increase CO2 levels by around 50% in that 100-year period, assuming the model is close. This is not a good thing.
According to this model, the CO2 concentration in 2035 will be about 470 parts per million. This assumes that the current rate of increase will continue until 2035. But with India, China and Vietnam (and many other developing countries) hell-bent on “catching up with the West”, it is likely to increase faster than this.
Why it Matters
Throughout the past 1 million years, the CO2 concentrations have ranged between 160 ppm (during ice ages) and were sitting around 280 ppm before the Industrial Revolution. (This information comes from examinations of Antarctic ice down to 3 km depth.) Current concentrations of around 380 ppm represent 869 gigatons (billion tons) of carbon in the air. [Source: The Weather Makers by Tim Flannery.]
The inevitable results of this increased carbon? More warming, more violent weather, more severe flooding and droughts, higher food prices, environmental refugees, etc.
Authentic Data
There is such a great deal of interesting authentic data out there. Why do math textbooks continue to use boring “nice” data (that is easy to plug into some formula) rather than real stuff that actually has meaning and matters?
Disclaimer
The above model is not a climate model as such. All I am doing is modelling the NOAA data as given so that I have a function that I can use for extrapolation. A real climate model will feed in all of the available data and will end up with a much more sophisticated model than this.
See also A simple climate change model.
Endpiece: Polynomial Model Limitations
My first attempts to get a good exponential model were not so successful, so I resorted to a polynomial model. Following is what it looked like.
I am usually reluctant to use a cubic polynomial model, because I find that they are often quite unrealistic either side of the data set and so cannot be used for extrapolation.
However, the following cubic polynomial model (obtained from this online regression utility) looked very promising.
The fit is very good, as you can see. Around 1990 the CO2 increase was quite rapid (probably due to Mt Pinatubo’s eruption in the Philippines) and you can see the (red) model graph sneaking through.
However, we can see the limitations of this model when we extrapolate too far beyond the 100 year period of 1935 to 2035.
The period where data exists (1958 to 2008) is indicated on the graph (in dark blue). As you can see, the model is quite unrealistic to the left of the data set.
For interest, here is the same graph with the exponential model from above (in black). It is clearly a better model than the polynomial one.
Comments
28 Responses to “Earth killer – composite trigonometry CO2 graph”
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Great post – thanks Zac.
I agree with you – we should use more ‘real’ data in our math classes.
Hi Zac,
I think that the reason the exponential models don’t do so well is because they have used only two parameters – with either y=a*b^x or y=a*exp(L*x). It is possible to get a much better fit by taking y=c+a*b^x , which actually makes sense since the exponential growth is presumably being added to a pre-human baseline which was not zero. Of course the cubic, having four parameters, can be made to fit the local data even more accurately, but, as you point out, either model can go wildly wrong if we use it for extrapolating too far from the observed interval.
Thanks for once again putting together the math and data so clearly and attractively.
cheers,
Alan
Hi Alan
In one of my attempts I used a y=c+a*b^x model but it was not an improvement on the linear or exponential ones shown. If I get a chance, I’ll have another go.
Hi again, Alan.
I re-wrote the article after finding a better exponential model. Thanks for your input.
Thanks for the analysis, Zac.
As your earlier post asked – What is the point of us? Have we just been put here to foul our own nest? if so, we’ve done a very good job.
You criticise the use of coal. But for most of the world’s poor, it is the only viable source of energy. Is it fair to deny them warmth in the winter, and cooking?
Good on you for using this ‘real-life math example’ to highlight the problem.
Greenhouse effect of CO2 is logarithmic:
Why is the greenhouse effect logarithmic?
Please use a time scale in the hundred of thousands of years.
Your data may also be in error
http://www.climateaudit.org/?p=1878
Do not scare the children.
BG
Thanks for your input, Bruce, but I suspect you have not read what I was attempting to do in this model. Let me address each of your points:
(1) Greenhouse effect is logarithmic? The article you linked to by Pilsen is talking about the effect of an increase in CO2 on temperature. My model is not talking about temperature changes or ice ages – it is simply a mathematical description of the increase in observed CO2. The increase is exponential, as shown above.
(2) Time scale: It would not be logical to use a time scale in hundreds of thousands of years for the data I am using, since people have only been measuring CO2 levels on the top of Mauna Loa for 50 years.
While there is data from Antarctic ice cores, it changes very slowly over time and is not relevant to the current exercise, except as a starting point for the model.
(3) Accuracy of data: I’m at a loss to figure out the relevance of your second link. They are not talking about the CO2 data on Mauna Loa at all and do not cast any doubt on its accuracy.
As far as I am aware, no-one has questioned the accuracy of the data I was using.
(4) Scaring the children: What I am scared about is the rantings by the anti-science lobby funded by big business, especially the oil companies, whose task it is to muddy the climate change waters.
The children should be scared.
Great analysis, zac, and great way to show how mathematics can be used with real-world data.
Bruce, your comments don’t really make sense here. The data set being modelled are CO2 over time, not temperature vs. CO2. In any case, your conclusions are not shared by the majority of climate scientists or major scientific organizations.
And please provide reasons for your requests. A longer timescale would make the graph harder to read, but would have no significant effect on the model. And I agree, children should be scared (as should we all). They’re the ones who will inherit a vastly different world; it’s silly to try to insulate them from the dangers they’ll face, especially since they have the potential to ameliorate some of them.
That Bruce guy doesn’t know what he’s talking about.
Good post Zac. I also agree the children should be scared – or better still, they should not follow in the footsteps of the adults.
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I’ve been working with the Pt Barrow CO2 data and the Arctic Polar ice extent data. I found them to be very similar in form. The differences between monthly values plotted out give a similar saw-tooth wave form that can be represented by a sine function with harmonics. Multiple regression yielded three statistically significant harmonics for both sets of data with the signs for the coefficients agreeing. The curve fit was 84% for the CO2 and 95% for the sea ice extent.
Hi Fred and thanks for your input. Have you published your results anywhere? I’d be interested to see them.
No. This is a work in progress. This last year I have been looking at all the data I can find related to climate change trying to get at the truth. I retired from EPA over 17 years ago and haven’t published in over 13 years. There is too much international politicing the issue to know what to believe. Personally I believe that any greenhouse effect of CO2 is insignificant compared with water vapor and clouds. The observed rise in background CO2 can be explained by a net increase in flux from the oceans as one would expect from a net increase in SST, It’s the old chicken or egg argument.
Hi again Fred. I’ve been thinking about your comments, especially:
There is too much international politicing the issue to know what to believe.
On the issue of “any greenhouse effect of CO2 is insignificant compared with water vapor and clouds“, it is a chicken and egg thing, but in terms of a solution, which should we kill – the chicken?
In The melting Arctic, I quoted Wadhams talking about the effect of all that reflection of heat from white snow and ice turning to absorption because of darker sea water.
And that will certainly cause another rise in sea surface temperature…
If you look at the CO2 concentrations from the ice core samples taken from Antartica, the last several hundred thousand years shows that today’s peak has not yet reached previous peak cycles. I have no doubt humans are contributing to CO2, however, we are also in the rising part of a natural cycle that has occured before. The fact that humans are also contributing during this natural peak is a coincidence but does add to the effect somewhat.
Thanks, GDI, but this BBC report talks about current CO2 being “well outside the natural range”:
However, this data from Oak Ridge National Laboratory shows concentrations of up to around 300 ppm within the last 400,000 years.
So it seems that the current 380 ppm is indeed “well outside the natural range.”
Comparing the ice core 300ppm to the background present day atmospheric measurement of 380ppm is like comparing apples to oranges. First, because the ice core data is an average value of concentrations over from about 100 to over several thousand years depending on depth, while 380ppm is an annual average. When the CO2 is deposited in the ice it’s movement, phase, and chemistry are not completely “frozen in time”. As long as the temperature is above absolute zero, there is movement in solids. Second, in making the measurement of the concentration, one must have a large enough sample or thickness of ice. Thickness, is the proxy measure of time which is the least accurate factor in ice core studies. We have a multi-variant statistic that is not so simple to treat. We should not try to compare our present day extremes to the last inter-glacial period long term averages.
Thanks again for the clarification, Fred. The implication in that ORNL article was “atmospheric concentrations of CO2″.
But seems that may not be the case.
Nice post.
The growth of CO2 in the atmosphere and the accompanying absorption of radiation is a mindbogglingly good way to secure funding, enough to keep several grad students employed (faculty can make careers with them). You need excellent skills in data and plot reading to become a successful CO2 modeler (In my humble opinion). Scientific computing is a sham (Garbage in, garbage out).
The exponential fit can be an awful optimization problem. The planet is doomed. The only future is to leave the earth, resistance is useless.
It is true that CO2 levels are increasing, but as of yet temperatures or sea levels have not reacted to it. Sea levels are increasing, they have been increasing for the last 6,000 years, but at an extremely low rate of increase. The CO2 increase has not caused the sea levels to increase their level of ascent.
I saw you said earlier that CO2 is what makes Venus uninhabitable(or something of the sort).
That is just not true. The planet just happens to be way too close to the sun! If the Earth itself was a couple of miles closer to the Sun, we would be fried! That’s what Venus’s problem is, not CO2…
Josh
You really should read more and base your ideas on scientific data, not your hopes.
According to NASA:
and also from AMES Research Center (also part of NASA):
Yes, Venus is close to the sun, but Mercury is even closer but has a lower surface temperature.
From Spectacular Conjunction, a NASA article:
So once again, Josh, Venus is hotter than Mercury. Mercury is closer to the sun but does not have a runaway CO2-induced greenhouse effect.
You are right in one respect: Earth would be hotter if we moved it closer to the sun. The other way to fry us all is to continue belching CO2 from factories, cars and by burning forests.
I’ll go with the NASA scientists, people who have studied these phenomena for years, not a climate change skeptic who is unable to produce any scientific backing for his claims.
Finally:
I’m getting more worried, now that I can see the effect of poor science education in the USA. No wonder we have problems.
Actually sir, you are outnumbered! Outside of the corrupt UN, support for man-made global warming is waning.
The IPCC report you have cited has only 2,500 scientists that support it.
31,000 scientists and counting believe that man-made global warming is a hoax.
http://www.petitionproject.org/
http://www.newsmax.com/newsfront/al_gore_global_warming/2008/05/19/97307.html
http://canadafreepress.com/index.php/article/3128
Do you need anymore?
Another thought on this:
Why is the United States being singled out? China builds new coal plants every day, and they don’t have filters on those plants like we do.
China’s too smart to be taken in…and they aren’t about to stop because Al Gore claims that Kilomanjairo is melting…
I enjoy debating you sir, but I would prefer that you not make disparaging remarks about people who disagree with you.
J.E.
God bless
Once again, Josh has chosen to ignore facts that are put in front of him.
Notice – no mention at all in his latest comment about my refuting his claim that there is no CO2 on Venus and that its heat is due to runaway greenhouse effect.
Disparaging? Where is my disparaging remark? I have been remarkably neutral in my comments, considering how I really feel. Is this what you regard as disparaging: “…who is unable to produce any scientific backing for his claims”?
Only one thing I agree with – China’s contribution to global warming is already immense and will surpass the US’s contribution in a few years. Not something to be proud of, by any means.
What I meant by “disparaging remarks” was this comment, “now that I can see the effect of poor science education in the USA”.
How come you aren’t commenting on the 31,000 scientists, 9,000 of which with PHD’s that are opposed to the idea of man-made global warming?
Now back to Venus:
Even if Venus had no CO2 in its atmosphere, it wouldn’t make any difference whatsoever in its ability to support life. More importantly, what’s the point of comparing Venus to Earth? There is not much to the planets that are similar…other than size.
Now about evidence for global warming…
What evidence? Getting a computer to run modules of climate change does not qualify as evidence in my book! We need physically verifiable evidence, not computer climate modules.
Al Gore claimed that the ice on Kilomanjaro was melting, and predicted that all ice would be gone by at least 2025, or even sooner than that.
It is a bold-faced lie!
The British government even says so!
http://www.nyunews.com/opinion/1.637337
The Artic was supposed to melting, but it has been verified that Artic ice levels are increasing!
The British government even says so!
Now for NASA
Are you aware of how much of funding NASA gets to fund climate change studies? NASA needs the money, and like someone else on here said, conducting studies on global warming is a great way to get funding. Like someone once said, “Follow the money”.
NASA needs money, because a lot of people in our Congress have cut funding to the organization(something I don’t like by the way).
That’s my view anyway, but my original point was that the so called “consensus” doesn’t really exist. Not if 31,000 scientists have anything to say about it!
God bless
Oops, the British government does not say that Artic ice levels are increasing, mix-up on my part…
I rest my case.
at http://www.kidswincom.net/climatechangepdf.pdf
Thanks, Fred for the paper. You present some interesting arguments. I’ll ponder them further…