Well, could we linearize this, then?
Following our discussion of Hubbert Linearization of the Middle East, I was inspired to take a quick look at Romania. Romania is similar to the US in that production began very early (in fact the world's first commercial-scale oil production occurred in Romania in 1857), and peaked in the 1970s (1976 was the year of peak production). However it differs in that the amount of production has never been that large - peaking at just over 300,000 barrels a day, an amount that a single large modern offshore platform might handle. It's also different in that the production profile (above) has been interrupted on a huge scale, especially by WWI and WWII, both of which caused an approximate 50% reduction in production.
Note that the scale on the bottom totals 160 years. The peak looks sharp, but really the peak is 30 years ago.
Thus the profile is at least as choppy as the Middle East. But it turns out that this does not render the curve-fitting methods useless, as you might imagine. Firstly, if one supposed that production followed some underlying dynamic that led to a Gaussian shape, then the log of production would follow an upturned parabola (quadratic):
So it seems like the quadratic does about as good a job of modeling this as any smooth curve is likely to do with such a choppy time series. But could it have predicted anything, or does the noise just destabilize things too much? Well, I didn't have time to do a thorough investigation, but I checked at the peak year of 1976, and the Gaussian knew very well it was at the peak:
But maybe it would think that at previous points too? How about the pre-WWII peak in 1936? Well, it wasn't fooled. It knew perfectly well that wasn't the real peak, though it had a very poor idea at that stage of when the peak would be:
Ok, so how about the linearization? Well, as Laharrere has shown before, this is about the best poster child there is:
Not only would the linearization have given you about the right answer (or at least about the same answer it would give you today) at the peak in 1976, it would have been telling you pretty much the same answer from some point in the early 1960s on (I hedge a little since that falls in my missing data gap that I linearly interpolated). Also, this is the most mature country profile in the world, both because it started so early and also because the peak is narrower than the US peak (since K is around 6.7% versus the 5.6% for the US).
Next, I looked at the idea we explored the other day at the suggestion of WebHubbleTelescope of plotting P (annual production) against Q (cumulative production). Since the logistic differential equation is P = dQ/dt = KQ(1-Q), a plot of P versus Q should be a quadratic if the production was exactly Hubbertian. Again, this is very roughly true in Romania, though the production profile is so noisy that the fit is not as pretty as in the US case:
However, it predicts the same URR as the linearization. But it turns out that back at the peak it would have given you a much less clear idea of what was going on than either the linearization or the Gaussian fit to logP:
It's 50% higher on the URR estimate than a modern estimate would be, and also thinks the peak is still a decade away when in fact it has arrived.
So in Romania, linearization works despite a very choppy profile, and again it seems to be because the profile is roughly Gaussian (decorated with major ugly noise) and the linearization technique serves as a useful robust predictor in that situation. However, directly fitting the Gaussian in logP versus t space works quite well here also. Fitting the differential equation in P versus Q space works eventually but is less stable to noise early on.
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