World Forecast Update

I applied the discovery model of quadratic growth with negative feedback to the world oil shock model in the clickable chart below:
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This model reduces to three input parameters:

1. The acceleration term for quadratic discovery growth = c
2. The feedback term for discovery growth damping = a
3. Parameter to the shock model for each of the lag terms: fallow, construction, maturation, and extraction. I set each of these to the same value of 12.5 years (i.e. a stochastic rate of 0.08/year). So you can see the phases progress in increasingly darker shades as the lags accumulate.

I kept the same starting point for discoveries of 1859 as the USA model, but slightly reduced the feedback term and more than doubled the acceleration term to account for a greater worldwide interest in finding oil; in other words, a greater population presents a higher acceleration of human resources devoted to discovery. However, the world feedback term should not change too much from the USA model as it presents the same damping of enthusiasm as discoveries top off.

This formulation presents a slightly different tact than previous attempts with estimated backdated discoveries, as the model has become completely analytic (though I still solve it numerically) with the only adjustable parameters provided by the physically based and potentially measurable rate terms. I really believe that each of these input terms have significance beyond that of the typical Hubbert heuristics -- definitely not of the inscrutable Logistic model variety.

After the solution of the differential equations, the result gives P(t), the yearly world-wide production of oil assuming an initially finite resource and impending collapse.

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I post this as I listen to author Dilip Hiro discuss his latest book ("Blood of the Earth: The Battle for the World's Vanishing Oil Resources") on Laura Flanders' Air America radio show (here). I really could not follow too much of what he said because of a hyper-speedy Indian accent (somewhere in there I heard a mention of "Hubert's (sic) curve"), so I suppose I shouldn't feel bad if I lose somebody due to too much math in my own posts, ha ha. Must ... try ... to ... concentrate. Apparently Chomsky likes the book.

Debate Tactics

Roger Cohen, who spent more than 25 years with ExxonMobil Corp said:

As for a peak and then abrupt decline in world oil production, Cohen called such predictions "an incoherent set of anecdotes." The peak theory is based on a model developed in 1956 by Marion King Hubbert, a Shell Oil Co. geophysicist who predicted that U.S. oil production would peak in 1970.

I long figured that the local right-wing blogosphere, consisting of such pseudo-luminaries as the Powerline lawyers and Lileks, act and write more like former (perhaps frustrated) high-school debaters or college-bowl quizzters than genuinely concerned citizenry. They all seem to display a competitive urge to argue a side, in which they don't actually have to believe in, in hopes that they continue to "win". What exactly they want to win I haven't a clue, but it seems like a good theory. So I knew that the NoDak educational system crowned Jim Lileks state debate champion long ago, but now I find out that 2/3 of the Powerline team have decided to share their master debating years with their readers:

That does it. If now everyone would just realize that all these right-wing ramblings constitute nothing more than reliving past faded glories, we could all go on and basically let them argue amongst themselves.

And the talking point about referring to a Democrat as belonging to the Democrat Party? It really amounts to nothing more than right-wing encoding of a swear word. The naughty wingnuts remain too civil to engage in curse words so basically use the secretly encrypted code to vent and whine to the other kids in the clubhouse, the student council, and their now adult debate team-mates.

Liquidation model?

Al Gore provides a potential name for the latest model sprung forth from the fevered imaginings of this here blog. Thus,

civilization is "operating planet earth like a business in liquidation"

I thought "Resource collapse" sounded pretty good but "Liquidation" about nails it.

The Missing Link

As far as I can tell, few models exist for the discovery dynamics of valuable yet finite resources. From a previous post, I introduced the concept of quadratic growth. This kind of growth has an underlying mechanism of a constant acceleration term -- in other words the rate of growth itself increases linearly with time. To first order, this explains scenarios that involve a rapidly increasing uptake of resources, and particularly those that spread by word of mouth. The growth of wiki-words in Wikipedia provides the best current-day example of quadratic growth. Unfortunately wiki-words grow out of an almost endless supply of alpha-numeric strings, which shows no signs of declining. However, for non-infinite resources we all know that growth ultimately abates and (quite frequently) suddenly. I first review two prime examples of this kind of dynamics: the old-fashioned gold rush and the extinction of species.

First, if we consider the gold rushes that occurred in places like California and Alaska during the 1800's, we invariably witnessed an accelerated search for the mineral as prospectors swarmed to a region. This accelerating growth in claims never lasted for long though -- within a few years, the region became scoured clean and history usually records a decline typically more spectacular than the original rise. We pretty much have to agree that finite resources played an important role in this behavior, and numerous ghost towns remain the only concrete evidence that any activity even occurred.

The passenger pigeon extinction provides an even more dramatic example of accelerating growth followed by sudden decline. From historical accounts of the colonial days of the new country, a few settlers started realizing that pigeon populations provided an easy source of food. More important, other settlers joined in and discovered increasingly lethal ways of decimating the bird population. So this perhaps century long accelerating increase in harvest numbers formed a framework for a precipitous decline in pigeon population within a few decades, ultimately followed by extinction of the species. The pigeon population essentially became a finite resource as reproduction dynamics could not overcome decimation by the numbers (e.g. through creative uses of dynamite). Although I have found few reliable estimates of the numbers (here), no one argues that wild pigeons essentially became extinct within the span of a few years from the late 1800's into the first few years of the 20th century.

So I ask the question: can we create a model of this "gold-rush"-like discovery of resources which effectively matches those of gold, passenger pigeons, and ultimately oil? I call this a missing link, because although I have a fairly good understanding of oil depletion post-discovery, the oil shock model, I don't yet have a good handle on the dynamics of discovery. For example, I hacked on a contrived decline term to my previous model of quadratic growth.

Clearly, this artificial break in quadratic growth does not follow from any physical process and I did this mainly to match empirical observations.

I assert that the key to modeling a finite resource limited decline lies in combining a quadratic growth term with a first-order feedback term describing the constraint. The latter term adds a variant of exponential growth/decline to the quadratic term.

Quadratic Growth : d2Q(t)/dt2 = k
Exponential Growth : dQ(t)/dt = aQ(t)

The combined real equation looks like the following:

d2Discovery(t)/dt2 = c - a3*Integral(Discovery(t))

Which basically says that the acceleration in discoveries is proportional to a constant suppressed by a drag factor that increases as discoveries accumulate. The drag term essentially describes the finite resource. If, on the other hand, I switch the sign on the drag factor, it becomes an exponential growth term which eventually dominates the quadratic term, forming a type of positive feedback (which models population dynamics). However, for negative feedback, the acceleration eventually becomes negative and the discoveries get driven into the ground. You can see the behavior in the following figure, where I plot the square root so you can see the divergence from pure quadratic growth depending on the feedback sign:

We can use calculus and Laplace transforms¹ to come up with the solution to the quadratic/feedback differential equation:

Discovery(t) = InitialValue +
   c*(e-at-eat/2(cos(K*a*t/2)-K*sin(K*a*t/2))/(K*a)2

where K = square root of 3. The term a acts like a characteristic value to the solution of a third order differential equation, while the value for c sets the amplitude.

I decided fit the model to historical estimates of oil discoveries, based on seeing the following kind of data reported:

Note that this figure shows a histogram of numbers of world discoveries which does not include the individual size of the discoveries; I consider this reasonable as the size forms a stochastically independent variable to the number of discoveries and random fluctuations would certainly modulate this profile -- but only in a statistical sense.

Initially, I decided to look at USA data, as the discovery estimates provided by Laherrere and the production numbers by Staniford at TOD generate a good dynamic range (for links see here).


I used the same oil shock model from the earlier post, and plotted the results for the USA below. It easily betters the Gaussian model as it accounts for the causality in the initial discoveries by Drake in Titusville, PA in 1859 (i.e. something has to start out the "gold rush"). It also models the out-years quite effectively, as the decline will become much less steep than a Gaussian will predict, something that Staniford would also likely admit to if he extended the profile much beyond the year 2010.


Also notice when I plot the model on a linear scale (see below), it becomes nearly as symmetric as a Gaussian! I can easily explain this as the right-heavy asymmetry of the quadratic feedback model gets balanced by the left-heavy asymmetry of the gamma distributions that form the basis of the oil shock model. The convolution of the two models effectively cancels out the left/right asymmetries and a fairly symmetric model results. With this revelation, I suggest we should seriously think about throwing out the Logistic and Gaussian formulations of the Hubbert peak models. We finally have a combined model which springs forth from first principles -- something that neither the logistic nor the Gaussian formulations can deliver. In other words, with the missing link uncovered, we have very little need for unfounded heuristics in modeling oil depletion or any kind of finite resource depletion (barring the role of economics of course).



Since the quadratic/feedback formulation shows self-scaling similarities similar to that of trig functions (i.e. period and amplitude), we can describe certain characteristics which depend on the a and c constants.


So we can easily estimate the terms for quadratic/feedback growth by simply overlaying the scaled profile on potential discovery curves. The figure to the right should match world discoveries to first order.

I will definitely pursue this analysis further, as the quadratic/feedback formulation looks promising. What to call it though; I have a feeling the particular equation I use exists with a fancy name somewhere in the literature. In the meantime, perhaps the "resource collapse" model would provide a fitting name?

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¹ The Laplace transform of the quadratic/feedback differential equation:
P(s) = c/(s³ + a³)

Twilight in the Mind?

A leading independent British book publisher, Profile Books, has released a cornucopian treatise called "The Battle For Barrels: Peak Oil Myths & World Oil Futures" by Duncan Clarke. If nothing else, the book will provide good target practice. At last we get a live one, not some fake little vanity press book (see Corsi, et al).

Check out the last blurb from the marketing page:

The essential crisis inside Peak Oil reflects a condition alike some Twilight In The Mind.

I haven't an idea what this truly means, but such lofty, pretentious pronouncements tend to resonate among the legions of the right-wing, who like to project their own inadequacy on others (e.g. "morbid outlook").

Unqualified

I find it entertaining to discover some unqualified dude in the administration making up strategy with respect to the Iraqi occupation effort. And as a commenter on DailyKos notes, the dude looks like Newman! from Seinfeld. Big daddy Donald allowed the dude to get a PhD in 3 years.

Bush looked about ready to cry on 60 Minutes today. No wonder.

Thought Dynamics

Dave Roberts, who usually posts at the Grist.org, wrote an intriguing piece on the position of moderation regarding global warming. In Roberts' opinion, moderation does not sit MOR, right square between believers and deniers, but shifted toward the believers, who nonetheless attack the populist alarmists. The reason that some pundits drum up such a middle position? They consider them separate from the left-leaning alarmists who consist largely of the rabble (i.e. the dirty hippies) leftover from the 60's, and thus can remain civil in their discourse by appealing to such "moderate" sensibilities.

To get back to the topic at hand: The last thing the global warming debate needs is for this kind of dynamic to develop. Really. If for no other reason than my head would explode.

This argument also works for the Iraq war (pro & anti & "hippie"). I find it unnerving that if we replace "global warming debate" with the "peak oil debate", we might enter another round of bifurcation in opinions. I honestly don't think this will happen though, because enough people think the peak oil alarmists have an inside track with the oil industry as much as vice versa. With the peak oil debate, most people don't know which side the "dirty hippies" belong to. The minute it looks as if the debate swings to a conservation-minded ethic, someone will claim that Big Oil essentially dreamed up the oil depletion issue as a pricing mechanism. I don't see a middle ground arising anytime soon, as it remains a no-man's zone of opinion, not attached to either side and no one to really appeal to.

From a comment on the same HuffPo blog posting, conspiracy journalist Wayne Madsen theorizes the NYC smell emanates from warming methane deposits off the east coast, essentially dragging along H2S as it rises from the ocean floor.

A lot of things have started to make circumstantial sense whether anyone came prove the logic or not.