Immaculate Conception Peak Oil Models
Staniford revisited his Hubbert Linearization (HL) formulation in some recent TOD posts. In the latter comment, he makes a mention of the good fit over a range of data of US oil production using a Gaussian curve.
I don't recall pointing this out in anything more than a passing comment, but one can represent a Gaussian curve in terms of a rate equation. The familiar bell-shaped curve of a Gaussian follows from this partial differential equation:
dP(t)/P(t) = K(T1-t)where P = production, t = time, T1 = the peak date, and K parameterizes the width of the Gaussian. I suppose one can read that as exponentially growing production with the throttle linearly pulled back through to where it switches sign at the peak. However, the T1 number looks suspiciously preordained as opposed to coming out of some intuitive process.
Although Staniford has some good empirical fits using Hubbert Linearization, he still doesn't know why they work out so well:
However, we don't, at the moment, have a very good theoretical understanding of exactly why, so there remains room for doubt about whether it will continue to work as well in the future.and
Where HL works well (eg the US), it seems to be because the production curve is close to Gaussian. Presumably there is some kind of central limit theorem "adding lots of random variables together" or "random walk through oil exploration space" kind of reason for this. If so, the asymmetry of individual field profiles may not necessarily give rise to an asymmetric overall shape. However, since we lack a clear and persuasive account of why the Gaussian shape arises, it's hard to say.I will start to label this style of analysis under the category of Immaculate Conception Peak Oil Models. The empiricism leads straight to the fact that no one has ever suggested a forcing function to cause the temporal behavior in any of these models, including the logistic curve. They remain at best a set of heuristics that tend to at least shadow the production data. Without a clearly identified forcing function, I half-jokingly suggest we cook up analogies to Spontaneous Human Combustion to get better insight.
For a more practical model, go here.
On the other hand, I could side with Staniford and look at the Gaussian from a law-of-large-numbers/ central-limit-theorem-flavored approach, which would mean for me to give up and punt away my understanding, while waving my hands wildly. A hand-waving heretic? Not me. An Immaculate Conception believer? Not me. An agnostic function forcer? That's the ticket.