So I have left you all in suspense as to what my big project is. This is it.
The answer appears to be yes, with difficulty. Using data from here, I identified the points at which several spikes in aerosol optical depth occurred globally since 1850. Specifically, I picked the seven largest spikes that were not before the decay of a previous eruption apparently ended. This is what the time variations of those eruptions optical depths look like relative to one another:
The red curve is the “average” eruption profile. The start dates were cross checked when possible against dates of known eruptions, but the date before the first sudden jump in AOD was chosen even if it preceded the eruption, as long as it did not do so by more than twelve months. For example, the start date for one of the above curves of December 1882 very roughly corresponds to Krakatoa-that is, the AOD increases suddenly in January of 1883. This is the beginning of the spike associated with Krakatoa…except that doesn’t quite work. There appears to have been some build-up from some other eruption or eruptions before that, since Krakatoa did not itself erupt until August of 1883. I chose to leave the start date as the month before the increase relative to the background began rather than the date of the eruption. In another case, there was a spike beginning after September of 1855, which I can’t seem to identify the associated eruption for. Sato et al. identify the spike with Cotopaxi, for what it’s worth. So these are my chosen “start” (baseline) dates:
September 1855 (Cotopaxi, others in 1856?)
December 1882 (Krakatoa, possibly some smaller eruptions earlier?)
December 1901 (Santa María, again this is months before the eruption proper, but I have chosen when the increase in Optical depth began.)
May 1912 (Novarupta, or Katmai-which seem to refer to the same volcano but different parts of it. The timing here is perfect, with the eruption happening in June, directly associated with when the spike in optical depth begins. From this point onward, I figure the dates are more reliable)
March 1963 (Agung-there was some increase in AOD from 1960 to 1961 but it essentially leveled of, I selected this date on the basis that after it the AOD jumps rapidly, and it happens to perfectly align with the timing of the eruption)
March 1982 (El Chicón, perfect timing again.)
May 1991 (Pinatubo, perfect timing again.)
Now, given these start dates, the next problem to solve becomes, how to isolate the temperature signal? The temperature record (I used HADCRUT4) contains long term trends in an apparent pattern which needs to be removed first to isolate the short term effects of volcanic eruptions from long term trends. I have previous worked on techniques to smooth short term variations out of data and identify long term signals. The technique goes like this:
First: How long is the time series? Let’s say it is n months long. At the time I originally did this that was 1958 months for the temperature data.
Second: Is n odd or even? If odd, subtract one and divide by two, if even subtract two and divided by two; call this number m-you will need it later so write it down.
Third: Take a three point centered average of the time series, such that you create a time series n-2 months long that starts at the second month and ends on the n-1 month. For those “missing” months, take an average like this: (1st month+1st month+2nd month)/3 and (nth month+nth month+n-1th month)/3 for the first and last month of the first smoothed timeseries, respectively.
Fourth: Repeat the third step m times (including the first time) treating the k-1th result as the original timeseries in the kth repetition.
EDIT: To clarify the above point, you repeat step three m times, that is, until k = m. So if you are doing this in Excel, with each smoothing column acting on the previous (that is, the kth column acting on the k-1th column) you should have m total columns (not counting the original data column 0r a column for time).
This ends up looking like this:
I was not satisfied with this as having removed the dips from volcanoes-the dips from Pinatubo and El Chicón for example appear to still be present. So I then repeating the smoothing process 9 more times, until the long term signal looked like this:
The original smooth is shown for comparison. Separating this out from the monthly data levels thus short term “noise”:
Obviously, various global “weather” is present here: ENSO etc. The data is noisy. But if we take segments of the data beginning at the dates we picked for the beginning of spikes in AOD, we get something like this:
The red line is the average temperature evolution after the volcanic eruptions. Obviously the noise in the data makes it difficult to see the signal, but averaging seems to help. Here is a plot of just the average response:
We begin to see that, not long after a volcanic eruption, the Earth’s surface temperature begins to dip somewhat, but not all that much-an average increase of about .08 in AOD seems to be associated with an average drop in temperature of less than -.15 K. We can quantify this a little better, though-but we need to determine the exact date of the minimum and estimate dip at that point that is not just the remaining noise at monthly timescales. So I use the above smoothing method on the average volcanic temperature profile (n=110 months, m=54). The result looks like this:
Note I am using this not to estimate the magnitude of the temperature dip just yet-since the smoothing probably attenuates the magnitude of temperature swings-but I am identifying the date of the temperature minimum. It appears that the minimum temperature occurs at about 25 months. Using the same smoothing technique on the AOD profile so that any bias in the date will be present in both datasets-and because there are actually two different dates of maximum in the average AOD profile-I identify the minimum in the smoothed AOD:
As occurring at 16 months after the start of the AOD spike. This amounts to a “lag” of nine months, which is only slightly longer than others seem to have found.
Now as for estimating the impact: First, I want to restore the variance loss to the temperature data. I do this by removing the means from both the raw average temperature profile and the smoothed profile, and then doing a regression where the smoothed profile is the predictor of the raw temperature profile. This yields a coefficient of approximately 1.43. I multiply the smoothed profile by that coefficient:
I then subtract the initial value of the smoothed series from both:
The minimum temperature dip is about -.11 K. The change in AOD from start to maximum is about .08. So I can estimate, linearly, using a 9 month lag and the coefficient of about -1.38, the temperature impact of volcanoes. That looks like this:
Note that the largest volcanic eruption dip, from Krakatoa, is less than -.23 K. No wonder the Wikipedia page says “citation needed” to the outlandish claim that temperature dipped by as much as 1.2 K! There is just no such evidence in the temperature data-no such large reduction in temperature could even possibly have occurred.
To see what this looks like I subtracted the above from the temperature data and then smoothed both the original and the “volcanoes linearly removed” series with 12 month running averages:
And smoothed data:
As can be seen from the above, this technique, simply linearly estimating the impact of volcanic eruptions, removes much of the signal: you can clearly see how well it removed Pinatubo from the data. But can we go further? Can we estimate the climate’s sensitivity from this data? Hm, maybe! Here is my first shot:
First, recall Equation 7 here. If we have a forcing function and a temperature function, and we can take the derivative of the temperature function, then we can use the temperature function and it’s derivative as predictor variables in a multiple regression analysis to attempt to predict the forcing function. Their coefficients will be an estimate of the inverse of the sensitivity and the time constant divided by the sensitivity, respectively. It will be easier to work with the smoothed temperature profile response than the temperatures themselves, since that is easier to estimate the derivative of. But the forcing function is just a little trickier since in order to make it directly comparable to the smoothed temperatures, it has to also be smoothed, but start at zero. So the first thing I did was take the smoothed data from Figure 8, and fit an exponential decay from the initial value to the value the smooth takes in the 61st month. I then subtracted that from the smoothed data. Let’s call that series S-E (smoothed minus exponential). The next thing I did was take the raw AOD profile from Figure 8, and subtract the initial value (that is, I made it start at zero)-let’s call that R-B (raw minus baseline). I then calculated the maximum value of R-B and of S-E and multiplied S-E by the factor that would make those values equal. Let’s called that MC-S-E (Magnitude corrected S-E) and subtracted the baseline value from the smoothed data from Figure 8-called it S-B. I then take the first 28 months of MC-S-E and the 29th month on of S-B, and combined them. The final eruption smoothed AOD profile, compared to R-B, looks like this:
The next step is to convert AOD units into radiative forcing. To get this I take the GISS forcing data, and take annual average values of AOD, and regress those values against the GISS Stratospheric Aerosol forcing. The coefficient I get is about -23.45, meaning a change of positive one tenth of a unit AOD leads to a negative forcing of -2.3 W/m^2 (compare that to 3.7 W/m^2 for a doubling of CO2) clearly large, but transient forcings come about as a result of volcanic eruptions (on average peaking out a little under -2 W/m^2).
Calculating the derivative of the temperature (dT/dt) is trickier. I’ve taken the first differences, and made a second copy of them and shifted them back one month, then averaged the two, and place zero values at the beginning and end of the dT/dt series.
Finally, doing a multiple regression of T and dT/dt onto F(t), I get the following for coefficients: inverse lambda of about 7.4, 95% confidence interval from about 5.54 to 9.26, which corresponds to a sensitivity of .500 K for a doubling of CO2, with a range from about .399 to .667 K for a doubling of CO2, and tau times inverse lambda of about 28.52 from 14.59 to 42.46, corresponding to 3.85 months, ranging from 2.63 to 4.85 months, although the fit is pretty poor due to high noise level (adjusted R squared of about .409). This provides evidence that the sensitivity is pretty small, and is in the range of my estimates from feedback fluxes, and the Faint Young Sun.
EDIT: I wondered if maybe I would get a different result if I focused on the first 25 months-that is I essentially forced the model to focus on the initial temperature drop and ignore the recovery period. It perhaps makes sense to do this because of the weird wave pattern that may be an artifact of…something. Anyway, fitting only the first 25 months results in a much better match of the initial temperature dip and large errors after that. The original 110 months is just the time between El Chicón and Pinatubo, there was nothing magically significant about try to fit nine years after an eruption. Getting the initial dip right could be seen as more important. So only doing the multiple regression on the first 25 months, I get even lower sensitives and longer response times (best fit of .369 for a doubling of CO2 and 13.67 months). The fit for those 25 months is better (adjusted R squared of about .991) but the fit over the whole dataset with those parameters is terrible. It’s interesting that to get a good fit to the initial dip required a lower sensitivity-lower than I personally believe makes sense, especially given other analyses I have done. It indicates to me that there is, realistically, not going to be a good explanation for this data with sensitivities that require even slight positive feedback.