A simple demonstration of climate change




I wanted to test if I can detect anthropogenic climate change from simple timeseries. To that end, I downloaded global surface temperature anomalies from NOAA [1], the good old Keeling curve from Scripps Institution of Oceanography [2] and an estimate of total solar irradiance from University of Colorado [3].

Then I was like 'hmm... how would a simple climate model look like?' I came up with the idea

dT = aS - bT^4,

dT being change of temperature, S being solar energy flux and T^4 coming from Stefan-Boltzmann's law, with a and b positive constants. (Bear with me, I know that this model is overly simplistic! The idea is what you can derive from this model, and that's quite awesome.)

Then, how could CO2 go in? I assumed

b = b0*exp(-c*CO2)

with b0 and c positive, this is a reasonable first approximation: b decreases in CO2 but convexly. If there is no relationship between CO2 and global temperature, then c=0.

Then I linearized the whole of dT around the mean of S, CO2 and T. I got the regression equation

dT = a S' + d CO2' + e T' + Intercept

where dashed variables denote centered variables (i.e., ones where the mean has been subtracted) and the signs of a, d and e follow from the physical model, so that a>0, d>0 and e<0. Simple, effective and intuitive.

It is the same as every business freshman would say: If CO2 warms the climate then I guess it has a positive regression coefficient. The same goes with the Sun. The negative coefficient of T' stems from the fact that there is a negative feedback between temperature and temperature change. (Warmer bodies emit more heat radiation.)

The regression coefficients were: Sun 0.08, with standard error 0.03; CO2 0.009, with standard error 0.001; Temperature -0.91, with standard error 0.14. This was the expected result, and highly significant. Use of heteroskedasticity and autocorrelation corrected standard errors (also known as HC and HAC) did not materially change the result.

In fact, this highly significant result is not a great wonder, looking at the image at the start of this post. The image has been drawn so that the timeseries have been "Z transformed", i.e. the mean has been subtracted from each timeseries and then each timeseries has been divided by its own standard deviation. This puts the data in the same scale but does not distort the trends, correlations etc.

It would be nice to see how the image looks with more recent data, but I could only construct a data set extending to 2018 by quick googling. Of course, the climate change is more complex than this and there are a lot of phenomena to tackle with but a simple regression analysis can demonstrate the association of CO2 and global mean temperature. Which is nice, if it can convert a single denialist.

Caveat 1: These results are obtained when dT is specifically defined as a forward difference, i.e. so that the covariates of year t affect the temperature change from year t to year t+1. If one uses backward differences instead, signs are reversed and the results are ruined. When one uses an intermediate option, i.e. so that covariates from e.g. year 2010 affect the temperature change between two full years starting from July 2009 and July 2010, the results are... intermediate. The signs are as they should but the coefficients are not significant.

Caveat 2: It is possible that the timeseries I have used have been edited, by their originators, a little bit to reflect consensus opinion, plausible values etc. That may, to some degree, reduce the reliability of the results, but I have tried to use as "raw" data as possible.

References:

[1] https://www.ncei.noaa.gov/access/monitoring/global-temperature-anomalies/anomalies

[2]  C. D. Keeling, S. C. Piper, R. B. Bacastow, M. Wahlen, T. P. Whorf, M. Heimann, and H. A. Meijer, Exchanges of atmospheric CO2 and 13CO2 with the terrestrial biosphere and  oceans from 1978 to 2000. I. Global aspects, SIO Reference Series, No. 01-06, Scripps Institution of Oceanography, San Diego, 88 pages, 2001. URL: https://scrippsco2.ucsd.edu/data/atmospheric_co2/primary_mlo_co2_record.html

[3] https://lasp.colorado.edu/lisird/data/historical_tsi




Comments

Post a Comment

Popular posts from this blog

A tale of 13 degrees

Image
  What happens if we don’t stop burning the fossil fuels? Is it the end of the world? Nobody knows certainly but I wanted to tackle this question with another back-of-the-envelope calculation. Here’s what I found: First, let’s take the same starting point as in the previous post, the climate equation dT = aS - bT^4. If the world is in a thermal equilibrium, dT = 0 and aS = bT^4. From this, we may solve how emissivity b affects the equilibrium temperature. (Remember, the effect of CO2 comes through the emissivity in this model.) We have T = (aS/b)^0.25. Thus, we may conclude that the equilibrium temperature is directly proportional to the power -0.25 of emissivity. In practice, this means that every 1% decrease in emissivity increases the equilibrium temperature by approximately 0.25%. The next question is obviously, how much unlimited burning of fossil fuels affects the emissivity. This is now hardcore speculation, but I chose to model this by using emission spectra. Above, the fir...
Read more

Staying healthy

Image
  These are some of my personal musings on the topic: 1. Above all, eat healthy and try to control your weight. What that means in practice, is a bit controversial and I will return to that below. 2. Don’t smoke. It has been anticipated since the 1930s that smoking causes cancer [1] and it has now been confirmed beyond reasonable doubt. 3. Don’t take drugs. Don’t drink too much alcohol. Take vaccines, at least the ones recommended by your government. 4. Exercise. What kind of exercise is the best? They say that to live long, you need exercise which elevates the heart rate for long periods of time, i.e. aerobic training, i.e. cycling, jogging or at the very least, walking. But I have seen several headlines which say that for old people, muscle strength predicts very strongly non-morbidity and longevity. So, it seems that to live long, you need aerobic training, but to live even longer, you need strength training, e.g. gym of crossfit. Ideally, you would combine these two types of tr...
Read more