Graphing Out Loud: ups and downs

A while back, we started looking at a poorly thought-out article from the website C3Headlines. C3 is starting to make a name for itself as a goldmine of climate comedy- their claims have recently been addressed at Tamino and SkepticalScience.

We’re going to keep digging into C3‘s claim that carbon dioxide concentrations have been increasing linearly over the 20th century. They seemed to draw this claim by eyeballing the graph of CO2 concentrations and qualitatively describing them as linear, apparently using the inset in their first figure to compare linear, quadratic, and exponential trends. This is a faulty method: it’s an elementary fact of calculus that ANY smooth curve, when viewed appropriately, will appear linear. The point has already been made but it’s worthwhile to keep looking because there are some interesting graphical follies at play; examining them further might help us understand how and why graphs are misunderstood.

Figure 1: From C3Headlines' article on "The Left/Liberal Bizarro Anti-Science Hyperbole", which claims that CO2 concentrations are increasing linearly. Click to read it, if you dare...

C3‘s second graph in this article measures the change in atmospheric CO2 by calculating a month-to-month percentage change. It’s not entirely clear why they are using a percent change, rather than the standard practice of expressing rate of change as concentration change per year (like the source of their data uses). Whereas ppm/year is an absolute measure, each datum generated by the percentage-change method depends strongly upon the value of the previous month. As a measure of long-term rate of change, it is a bit questionable.

My primary concern, though, is with their use of monthly data in the first place. In my last article, we noted that, without explanation, C3 confined their focus to January CO2 concentrations. Were they consistent, they’d also look at January rates of change – of course, doing so might lead to unacceptable conclusions.

Figure 2. Rates of CO2 accumulation have been calculated for the month of January, consistent with earlier investigation of January CO2 concentration. Over the period of observation, rates have increased at a significant (P~0.0005) acceleration of 0.11 ppm/year^2. Monthly rates throughout this article have been calculated by considering the change in CO2 between adjacent months, and assuming that a month is 1/12 of a year. Interpolated values of CO2 were used to avoid annoying data holes early in the record.

Instead, they look at the rate of change for every single month on record. Why do I find that problematic? Well, let’s look at the full record, with monthly resolution:

Figure 3. The Mauna Loa CO2 record, at monthly resolution. Imagine the green line was a roller coaster - would you notice the slow climb uphill indicated by the red line? Or would you be too busy holding onto your dentures due to the quick up and down?

As you can see, superimposed on the long-term trend, there’s an annual oscillation. This is the result of annual cycles of photosynthesis. During the summer, plants store carbon in solid carbohydrates, removing it from the air. During the winter, there is little photosynthesis, and CO2 released from decaying leaves builds up in the atmosphere. The seasons are out of phase in the northern and southern hemispheres, but there is more landmass in the north, so its contribution dominates.*

This annual cycle means that, when you calculate the rate of change on a monthly time scale, it will be positive during the winter and negative during the summer. Look back at the CO2 concentrations plotted in Figure 3. The rate of change at each point on the graph is the slope of the graph at that point. Imagine the graph was the track of a roller coaster – every summer the car slides downwards, and every winter it climbs upwards. This means that over the course of a year, the rate of change in CO2 will fluctuate dramatically, from strongly negative to strongly positive values. When I calculate the monthly rates of change, their magnitude comes out much, much larger than the annual average rate:

[FIG]

Figure 4. Rates of change in atmospheric CO2. The red line is the monthly rate as measured at MLO; the black line is a linear regression to these data. The blue dots, included for reference, are the annual rates of CO2 accumulation, which have a significant increase over time. If it's not obvious that they are increasing at this level of magnification, why should we believe that the monthly MLO data *aren't* increasing?

Along with the monthly rates, I’ve plotted the annual rates. Although we already have seen a significant increase in the annual rates, the wild variations at the monthly time scale act to hide the incline. A linear regression to the monthly data agrees relatively well with the annual trend (actually, the regression to the monthly data gives an acceleration in CO2 concentrations about 1.5 times as large as the annual data), but the enormous variability reduces its significance. It’s a little like driving a car up a rocky mountain road: the trip may well be too bumpy, moment to moment, for you to notice that the road is slowly sloping uphill.

A graph of monthly rates obscures more than it reveals. So what can the monthly data tell us?

Figure 5. Acceleration in atmospheric CO2, by month. Asterisks denote significant (p<0.05) trends.

Here, I’ve calculated the long-term trend in the CO2 rates for each month; this tells how the CO2 is accelerating in each month. Between May and August, the acceleration is negative – this corresponds to the time of the year when CO2 is removed. The biosphere is taking deeper and deeper inhalations as CO2 levels rise. During the rest of the year, the acceleration is positive. Not only that, looking at the September-April months, it’s clear that there is more positive acceleration than there is negative acceleration – exhalations (which include fossil fuel emissions) are growing faster than inhalations. This means that, taken over the whole year, there is a net acceleration in CO2 concentrations. A linear trend would, by definition, show no acceleration, so the data that C3 presents once again undermines their thesis.

Figure 5 also gives some insight into a popular talking point, that increased carbon dioxide will be better for plants, and that it will be balanced by greater plant growth. Although it is true that the inhalations are getting deeper during the summer months, perhaps from CO2 fertilization, longer growing seasons, or other effects, we can see that they don’t balance out fossil fuel emissions.

In fact, while increased carbon dioxide can help plants to a certain extent, that extent has been over-estimated in the past. For example, many early measurements of CO2 fertilization were done in greenhouses or carefully controlled chambers. Such an experimental setup might be fine as a first approximation, but as (Long et al, 2006) point out,

“no agrochemical or plant-breeding company would base its business plan for a new chemical or variety solely on greenhouse studies without
rigorous field trials.”

Long et al. compare the results of greenhouse studies to those done under the more realistic Free Air Concentration Enrichment (FACE) experiments, in which plants are grown like any other crop, with sprayers elevating local CO2. What did they find?

“In [FACE] trials, elevated CO2 enhanced yield by ~50% less than in enclosure studies. This casts serious doubt on projections that rising CO2 will fully offset losses due to climate change. ”

It doesn’t take much creativity to imagine what would have happened if the results had come out the other way around, if early estimates of CO2 fertilization had been a factor of 2 too low, rather than too high. The c3Headlines might read: “Important Climate Figures Incorrect; Earth Scientists Don’t Know That CO2 Is Plant Food.”

All the elements are there – new developments overturning older accepted ‘truths’, major flaws in IPCC models – and yet, I’ve never seen the skuptix mention (Long et al. 2006).

I wonder why?

~~~

* There is an excellent description of the annual CO2 cycle in this video near minute 40. It includes some very cool visualizations of data and animations of atmospheric models. I’ve also made a graph of the annual cycle at MLO by detrending the monthly data with the quadratic model we built last time. You can see it here.

Monthly CO2 Data and Annual Growth Rate Data from MLO

Long, S., Elizabeth A. Ainsworth, Andrew D. B. Leakey, Josef Nosberger, & Donald R. Ort (2006). Food for Thought: Lower-Than-Expected Crop Yield Stimulation with Rising CO2 Concentrations Science, 312 (5782), 1918-1921 DOI: 10.1126/science.1114722

Graphing Out Loud: curves and lines

I love graphs – my eyes quickly glaze over at a table of numeric data, but a graph, used correctly, can quickly and easily tell the whole story.

‘Used correctly’ is the key phrase – for all their power, graphs are infamously easy to bungle, and when used incorrectly they can misinform – or lie outright.

I’m going to look at an example that touches on a few graphical and statistical concepts near and dear to my heart, as well as carbon geochemistry.

Fig. 1: An image from C3Headlines; the 3 C's are "Climate, Conservative, Consumer". Oh, and the article is titled "The Left/Liberal Bizarro, Anti-Science Hyperbole Continues". It sure would be tragic if they made obvious n00b mistakes after using such language. Click for link!

Coming from an article on the website C3Headlines, this image claims that carbon dioxide concentrations have ‘Linear, Not Exponential Growth’. thereby ‘expos[ing] the lunacy of typical left/liberal/progressive/Democrat anti-science’, The author has reached this conclusion by graphing January CO2 levels* and fitting a linear trendline to them.

Already this is a warning sign – the comparisons the author makes are entirely qualitative, apparently  based up on eyeballing the graph. However, trend lines are created by a statistical process called a linear regression, which comes with a caveat: it will fit a trend line to ANY data given to it, linear or nonlinear. Fortunately, there are also ways of evaluating how good a trend line is.

temperature aNOMalies

If you are new to climate science, you might be wondering what, exactly, this ‘temperature anomaly’ thing is that you keep hearing about. I know I was a bit confused at first! This post explains the concept, using a real-world example.

Absolute temperatures (yearly averaged) from two sites in the UK: one urban (St. James Park, green) and one rural (Rothamsted, red). Although the urban site is consistently warmer, the two sites show the same warming trend. But is there a way to compare them directly? Data from Jones et al. 2008, kindly provided by Dr. Jones.

Cities tend to be warmer than their surrounding countrysides, a fact known as the urban heat island effect (UHI). This occasionally is offered as an alternative explanation for greenhouse warming, but it fails on closer inspection. We can use data from Jones et al. (2008) [PDF] to see one reason UHI can’t explain observed warming. One time series is from St. James Park, in the city of London; the other is from nearby Rothamsted, a rural site some tens of miles away. As you can see, the urban location is consistently about 2 C warmer; however, the warming is nearly identical at both sites (a strongly significant 0.03 deg C/year). Jones et al. note:

“… the evolution of the time series is almost identical. As for trends since 1961 all sites give similar values …  in terms of anomalies from a common base period, all sites would give similar values.”

This gives us a hint about what a temperature anomaly is:

cnfusin rained and chas

Last time, we looked at a very simple atmospheric model known as the Lorenz equations, and saw it exhibit the ‘Butterfly Effect,’ in which even very small changes in initial conditions can dramatically effect which path the system takes. However, we also saw that the initial condition had a relatively small impact on the statistical properties of the system. Because climate is a statistical property of the earth system, asking

“How can we expect to predict future climate when we can’t predict the weather?”

is a lot like asking

“How can we claim to know the half-life of a radioactive element when we can’t predict when a given atom will decay?”

To those familiar with chaos, this shouldn’t come as a surprise. Lorenz didn’t just discover apparent disorder in his model, but a deeper, eerie structure lurking in the noise.

The Lorenz Attractor: wibbly-wobbly mess of the millenium. Three simulation runs (red, green, blue) are shown; they start close together but quickly spin off on different trajectories, demonstrating sensitivity to initial conditions. Nonetheless, the trajectories quickly converge on an intricate structure in the phase space, called an 'attractor'. The attractor doesn't vary with initial conditions, but is instead a feature of the Lorenz equations themselves. Image generated with code from TitanLab - click to check them out

You may remember that the Lorenz equations relate three variables (X, Y, Z), which vary over time. In the above image, I’ve plotted the evolution of three runs of the Lorenz model by putting a dot at each (X(t), Y(t), Z(t)) coordinate, at every time t in the given interval. The three runs start very close together in this three-dimensional ‘phase space’, but quickly diverge.

However, despite their different individual behaviors, these runs are confined to a structure in phase space, known as the Lorenz attractor – an attractor, because all trajectories converge on it, regardless of their initial conditions. If you perturb the system by bouncing it off the attractor, it quickly settles back into the same loops through phase space. Lorenz (1963) described it:

CO2 Trouble: Ocean Acidification, Dr. Everett, and Congressional Science Standards

A companion article at ArkFab shares my thoughts on peer review in regards to this project and DIY/community/citizen science in general. 

At long last, the much-anticipated booklet, “CO2 Trouble: Ocean Acidification, Dr. Everett, and Congressional Science Standards” is available and approved for human consumption! Download and share HERE (or at Scribd HERE).

In this document, I have bundled, updated, and expanded my series of essays debunking the congressional testimony of Dr. John Everett regarding the environmental chemistry of carbon dioxide.

It has been designed to be a fairly short (less than 30 pages, including images, appendicies, etc.) and accessible read. It has been challenging but fun to write; I have had to learn a lot about GIMP, Python, Scribus, social networking, and of course ocean acidification to get to this point.

It was also very useful for me as an opportunity to go back through my earlier remarks and double-check my work. For example, I later realized that the documentation which Dr. Everett provides for his CO2 data in part two is ambiguous: Although the citation for the rate data is referred to as “Recent Global CO2”, the URL provided links to the longer record as measured at Mauna Loa Observatory. This confusion had led me in the past to make incorrect claims about some of the figures he presents. Ultimately it was inconsequential to my argument, but it was frustrating to have to deal with such ambiguities. On the other hand, this led me into comparing the Mauna Loa record with the global record (Appendix B) which was an interesting exercise.

In researching this project, I also came across new phenomena I wasn’t previously aware of. For example, while I was calculating historical rates of CO2 change, I ran though the 1000-year Law Dome record and saw this:

Did chaos theory kill the climatology star?

Last time, we saw that some mathematical systems are so sensitive to initial conditions that even very small uncertainties in their initial state can snowball, causing even very similar states to evolve very differently. The equations describing fluid turbulence are examples of such a system; Lorenz’s discovery of extreme sensitivity to initial conditions ended hopes for long term weather forecasting. Because the state of the weather can only be known so well, the small errors and uncertainties will quickly build up, rendering weather simulations useless for looking more than a few days ahead of time.

But Lorenz’s discovery doesn’t have much impact on climate modelling, contrary to the claims of some climate skuptix. Climate is not weather, and modelling is not forecasting.

Weather refers to the state of the atmosphere at a particular time and place: What temperature is it? Is it raining? How hard is the wind blowing, and in which direction? Climate, on the other hand, is defined in terms of the statistical behavior of these quantities:

“Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. [...] Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer. ” IPCC

Many climate skuptik talking points derive from confusing these two quantities, in much the same way that a gambler might win a few hands of poker and decide that they are on a roll.

Although it is generally not possible to predict a specific future state of the weather (there is no telling what temperature it will be in Oregon on December 21 2012), it is still possible to make statistical claims about the climate (it is very likely that Oregon’s December 2012 temperatures will be colder than its July 2012 temperatures). It is very likely that the reverse will be true in New Zealand. It is safe to conclude that precipitation will be more frequent in the Amazon than in the Sahara, even if you can’t tell exactly when and where that rain will fall.

In fact, Lorenz’s groundbreaking paper, ‘Deterministic Nonperiodic Flow’, would seem to endorse this sort of statistical approach to understanding fluid dynamics:

“Because instantaneous turbulent flow patterns are so irregular, attention is often confined to the statistics of turbulence, which, in contrast to the details of turbulence, often behave in a regular well-organized manner.” (Lorenz 1963)

Let’s take a closer look.

Fig. 1. Three solutions of the Lorenz equations, starting at virtually identical points. Although the solutions are similar at first, they rapidly decouple around T=12.

The Lorenz equations consist of three variables describing turbulent fluid flow (X,Y, and Z), and three controlling parameters (r, b, and s). The equations are differential equations, meaning that a variable is described in terms of how it changes over time- saying ‘Johnny is driving west at 60 miles per hour’ is a simple differential equation. In order to solve a DiffEq, you need an initial condition – “Johnny started in Chicago” is an initial condition; without knowing that, you can’t say where she will be after driving for three hours.

You can’t predict the weather

Regarding climate models, physician and science fiction writer Michael Crichton had this to say:

“Since climate may be a chaotic system—no one is sure—these predictions are inherently doubtful, to be polite.” (Aliens Cause Global Warming)

What does he mean when he says that climate may be chaotic, and what impact does this have on climate modelling?

Flash back to the early 1960s. Meteorologist Edward Lorenz was studying a bare-bones weather model, consisting of three differential equations. Give the model an initial state and the differential equations would describe how the state changes over time, in much the same way that you can predict where Johnny will be in three hours’ time, given that he starts in Chicago and is driving west at 60 miles per hour. The hope was that with a big enough computer, a powerful enough model, and an accurately measured state of the atmosphere, the weather could one day be predicted far in advance.

Lorenz, the story goes, found a run of the model which interested him, and sat down to replay the simulation. He entered the initial conditions and set the model in motion, only to watch in bewilderment as the replay rapidly diverged from the original simulation.

"From nearly the same starting point, Edward Lorenz saw his computer weather produce patterns that grew farther and farther apart until all resemblance disappeared" (Image and caption from Chaos: Making a New Science, by James Gleick, 1987, p.17)

Lorenz tore his code apart looking for the error, only to realise that the error had been in his assumptions. In a distinctly Crichtonesque twist, the computer worked with numbers to six decimal places (0.123456) but only printed out values to three decimal places (0.123) in order to save space. It was these shortened number which Lorenz entered as the initial conditions for his model. Surely those last digits were inconsequential; after all, they were but a few hundred parts per million, comparable to the atmospheric concentration of the trace gas carbon dioxide.

Oh, but the consequences! Its roots stretched back to earlier anomalies and the term ‘chaos’ would not be introduced for another decade, but it was Lorenz’s observation which heralded the beginnings of chaos theory.

Lorenz had discovered that even very small changes in the state of a chaotic system can quickly and radically change the way that the system develops over time. This property is known as extreme sensitivity to initial conditions, also called the ‘Butterfly Effect’ because it suggested neglecting an event as small as the flapping of a butterfly’s wings could be enough to derail a weather forecast. There is more to chaotic systems than the Butterfly Effect, but this characteristic is one of their best know properties. Lorenz’s work put and end to hopes of long-term weather forecasting. The state of the atmosphere could only be known so well, and even the smallest of imprecisions would lead the simulations to catastrophic failure.

‘Nobody believes a weather prediction twelve hours ahead. Now we’re asked to believe a prediction that goes out 100 years into the future? And make financial investments based on that prediction? Has everybody lost their minds?’ – Crichton

But does chaos theory signal doom for climate modelling? Stay tuned for part II….

John Everett, part IV.0: Things and Things’ Changing, dva

A part of my John Everett series – read more: 0/I - II.0 - II.5 - II.75 -  III.0 - III.3 - IV.0 - IV.4 - IV.8 - V - VII - VIII - Full Report 

People who minimize or deny the threat of climate change (or ocean acidification, as in part IV of Dr. Everett’s testimony) will often demand that the change be “unprecedented” – that nothing like it has ever happened before in Earth history. (eg, here) The reasoning seems to be that if there have been ecological events like anthropogenic climate change in the past, then current events must not be alarming, since life on earth has each time survived and recovered:

“We know that the Earth has seen these conditions before, and that all the same types of animals and plants of the oceans successfully made it through far more extreme conditions. ” – Everertt (part V)

 

This has always seemed to me like it’s setting the bar a bit low: Do we only become alarmed when faced with the possibility of sterilizing the planet? And considering the amount of violence which earth life has withstood over the ages, it doesn’t seem a very strong statement that human impact is unlikely to wipe it out.

John Everett, part II.75: Carbon Pie

A part of my John Everett series – read more: 0/I - II.0 - II.5 - II.75 -  III.0 - III.3 - IV.0 - IV.4 - IV.8 - V - VII - VIII - Full Report 

The last couple posts looked at Dr. Everett’s discussion of the growth rate of carbon dioxide. There’s one other claim in this section which warrants inspection: that a constant airborne fraction is a challenge to projected acidification.

I got the bright idea to sudo rm -rf in my /etc/ and now GIMP is broken. So none of my sweet graphics this episode. Instead here's a diagram of the carbon cycle, courtesy of NASA (click for sauce.) It's just as well. The coolest thing I could think to draw was some pictures of pie. Mmmm pie.

Here’s what he has to say:

The meaning of this information [the supposed leveling off of CO2 growth rate] (and the future of all climate models[)] became VERY cloudy on 31 December 2009 with the ScienceDaily acknowledgment of a paper published by American Geophysical Union and authored by Wolfgang Knorr that shows “No Rise of Atmospheric Carbon Dioxide Fraction in Past 160 Years”, despite the predictions of carbon cycle/climate models3. The implications of this have yet to be assimilated by the modeling community. This does not mean that CO2 proportion is not rising but rather that the proportion not being assimilated has not changed since 1850. Importantly, it means that the rate of CO2 cycling increases as it becomes more concentrated, and does not decrease as assumed in climate models. The rate of projected growth in CO2 appears to be greatly exaggerated.

John Everett, part II.5: Significance Junkies

A part of my John Everett series – read more: 0/I - II.0 - II.5 - II.75 -  III.0 - III.3 - IV.0 - IV.4 - IV.8 - V - VII - VIII - Full Report 

The CO2 scenarios are literally falling flat and need revision. The observational trend line shows monotonic growth – pretty much a straight line as in the chart below of global marine CO2 measurements (NOAA data)4, while the IPCC scenarios used in most research rely on an accelerating growth. Certainly the predicted rapid acceleration of the IS92a model (see solid black line in middle of the figure on the right) is missing from the NOAA data plotted below. In fact, if the last 8 or 12 years are representative of the future, we might imagine a downward slope in the growth rate.

Last time, we looked at one claim Dr. Everett makes in this paragraph: that the measured rate of change in atmospheric carbon dioxide is inconsistent with the emissions scenarios used to predict future ocean acidification. To do this, he plays fast and loose with quantities and their derivatives (the rates at which they change.) The imprecision extends even to his quoted numbers: in pidginthe previous paragraph, he gives a growth rate as “3.05 ppm”. That’s a not a growth rate; it’s a concentration. He means 3.05 ppm per year. His projection is an extrapolation of “the average rate of increase for the past 10 years (1.87/year)…” 1.87 WHAT per year? I know that he means 1.87 ppm/year, but a lot of people wouldn’t, and I shouldn’t have to make assumptions. If Everett is being sloppy with his units, he’s being sloppy with his science.

The other claim that Dr. Everett draws from the rate of change in CO2 is that “the growth rate seems to be leveling off, if not declining [...] In fact, if the last 8 or 12 years are representative of the future, we might imagine a downward slope in the growth rate. ” Look at the graph of the growth rate again. It goes up and down- a lot.

The record of changes in atmospheric carbon dioxide since 1980. It's got it's ups and downs. Click to see the full record back to 1959.