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:
.
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.

No Sudden Noises

Apropos of nothing, I heard Matt Drudge on his radio show last night refer to scientists as witch doctors and what they practice as witch-craft. Apparently too much talk of global warming set him off. He then proceeded to crow tremendously about his WiFi skills. In my pagan view, deconstructing borderline bipolar tendencies can entertain as well as keep the mind sharp and engaged.

With that, I present a bit more witch-craft.

Rembrandt completed his 3 part series on reserve growth at TOD. He has zeroed in on the enigma of reserve growth that I have blogged on recently:

The problem with this method is the way how crude oil reserves are reported in the US which has been described in detail in part 2. Because of the practice of reporting only proven reserves, the amount of reserve growth is very high in the US when compared to other regions. In addition several heavy/extra heavy oil fields such as the kern river oil field are included in the assessment, which showed huge reserve growth due to the advancement in steam technology necessary to dilute the oil to produce it in the middle of the 20th century. It is erroneous to apply reserve growth in such heavy/extra heavy oil fields with medium and light crude oil fields.

This fits in with the censored data issue I raised. So the historically censored data can either arise from shut-in fields or can get a spurt of new technology such as what happened with heavy oil in California (the Kern River oil field as Rembrandt mentioned). It could also stem from earlier quota restrictions or lower throughput during some stretches of time. In general, any throughput lower than extrapolated would lead to speculative predictions, which would imply the reserve growth predictions would be remain low during those times. This makes sense, since reserve estimates proportionally track the production rates, as the US oil companies have historically used this mapping as the proverbial "carrot before the horse" to lure the investment community.

I could find a bunch more analogies to this misuse of censored data. A fairly intriguing one that overlaps pretty nicely with a 15 year time frame, concerns the similarity to investment return rate prior to the year 2000. For example, say you didn't know how much an original stock investment was from the early part of last century. But you had the data that showed significant growth rates from the period from mid-1980's through the 1990's. This period in fact showed huge growth rates in investments, usually above 20% per year. If you would have used A&R's technique of not considering censored data, you would have heaped an incredible investment growth from investments originally made in 1900 (much, much more than the actual growth and much greater than reserve growth of oil, AAMOF). If then you hypothetically worked as an investment advisor and tried to pull this creative bookkeeping using the A&R technique on your clients, the SEC would put your sorry ass in jail in no time. This eerily parallels the equivalent of the standard financial disclaimer of "past performance is no guarantee on future returns", yet it looks as if A&R got away with it! They basically say instead that "current performance is indicative of past growth rates". Do you realize what kind of suckers that makes us?

Not showing the pessimistic growth rates also makes the USGS look really bad, and probably speculatively fraudulent if the SEC decided to look into it. Unfortunately, the USGS does not exist as a corporation and therefore immune to prosecution. The shills have participated as useful tools of the oil industry's game of 3-card Monte. And we fell for a sucker bet.

And the 3-card geologist at peakoil.com said essentially the same thing, not realizing that he buttressed up my argument.

And of course better extraction techniques have an effect on reserve growth, the problem with R&A's method is WHERE those changes fall, based primarily on the age of the fields where better extraction techniques have had the biggest effect ( heavy oil in California springs to mind ).

(my emphasis) Thank you, now get off my street corner.

Censored Data

The lower 48 reserve growth analysis from Attanasi & Root likely suffers from the curse of censored data. If you look at the data posted on TOD recently, you can see the window gets clipped in a ~15 year timeframe from 1977 to 1991. Yet the authors decide to show the data as if the growth looks continuous up to 90 years.

This leads to charts like the following where the authors deem it alright to extrapolate backwards to a much larger window:

Now, most analysts admit that many of the fields included in the analysis become "shut-in" or dormant or plugged during their lifetime, and then when the economics become reasonable, the field can produce oil with a renewed vigor. Unfortunately this means that reserve growth has to drop somewhat reasonably to a near-to-zero value during the dormancy stage. The field remains essentially abandoned so you would think that this approximation would make some sense. Yet you would never notice this from a cursory glance of the sliding data of the first chart transforming into the larger aperture of the second chart. Properly supplying reasonable censored data for a much-larger-than-15 year window would in fact suppress the slope of the profile in the second chart considerably.

Doing censored data correctly falls into the class of statistical problems called survivability analysis. Analyze censored data incorrectly and you will face results fraught with errors. A classical case frequently referenced concerns survival studies of tobacco smokers. These studies would go on for 10 years or so, and the statisticians would collect enough data to project death rates in the future. The results become problematic when the extrapolator assumes a linear projection whereas in reality the actual data could vary non-linearly. Without the uncensored data available, i.e. a longer time scale, one could come to the wrong conclusion. Big Tobacco has used this statistical sleight-of-hand in the past to justify any conclusion they wanted to, preferably to stress that tobacco does not cause premature death (or addiction). I fear that Big Oil can practice the same sleight-of-hand until people start to catch on.

The censoring in the particular case of reserve growth matches the typical tobacco survival in a general fashion only; for reserve growth the censoring of data happens in the previous years to the 1977-1991 window. Let me give you a better analogy as to what this means. Say, instead of oil, you looked at growth data of people (e.g. say their height) placed in suspended animation at age 15, and you added that to data 100 years from now after reviving the subjects from their hibernating stupor. Assuming that suspended animation prevented aging, you would have a completely misguided view of how people grew as a function of time. Without understanding that cold storage prevented growth, you would think that humans continue to increase in height well past their 90's. To top it off, these humans would also have to consist of a range of sizes from tiny dwarfs to giants so as to match the range in oil field sizes. That basically amounts to what R&A did with their data. They completely neglected any part of the data that didn't show an increase in value. A conservative read of the data would only justify looking at the first 15 years of the second chart.

To summarize, for all we know, many of those regions outside the 15 year data collection window became shut-in or dormant which meant they had zero reserve growth for a long period. But then to resurrect the data and show growth 90 years after discovery displays an incredible amount of deception or rank ignorance on how censored data works. If Attanasi & Root had an agenda of deception, then I consider them equally as evil as the tobacco "scientists" who jiggered the survival statistics over the years leading to the premature deaths of millions of people worldwide.

If on the other hand, A&R simply chose to bathe in their own mathematical ignorance, these two should forever inhabit a special state of suspended animation, and we shouldn't wake them up to do more harm.

It took me about a year to actually understand what this weird reserve growth meant. Now that the extra insight hit me like a ton of bricks, I have become more than a bit peaved. No wonder they call it censored data, Big Oil doesn't want the truth to come out. You'd think a company would fire somebody who makes this bad a mistake, but think about the agenda of the employer and they would deem the dynamic duo of A&R as vital to their long term best interests. Exactly. They want to present this optimistic reserve growth to the public and if they can find tools (i.e. useful idiots) to keep parading around their bad statistics, they would just as soon keep said employee on staff than to fire his sorry behind. Welcome to how the Tobacco industry worked in the 20th century and the way Big Oil works today.

Russian Reserve Growth

Spurred by a post on TOD, I tried to fit the reserve growth from oil fields of the West Siberian Basin.

This uses the self-limiting parabolic growth law from here, with an a/C(0) ratio of 1/0.07.

Original data: