Gold Rush dynamics, the Dispersive Discovery sanity check

Dynamics such as those that lead to extinction events and of boom-bust periods first motivated me to generalize discovery dynamics in terms of dispersive effects.

If we look into an extinction event such as passenger pigeons in the 1800's, we find a steadily accelerating harvest per year until culling hit a critical point and then fell precipitously. The harvests went spectacularly to zero and so, unfortunately, did the pigeon population.

I can say the same for boom-bust cycles, such as happened during the gold-rush days of the 1800's. In most cases, a boom occurred on the onset of an isolated discovery as many prospectors joined the search, enough time passed to enable the building of a huge infrastructure and then suddenly everything dried up with the infrastructure left standing in place.

But that hasn't happened with the discoveries of fossil fuel around the world. Although discoveries did increase at an accelerating pace until about the mid-part of the 20^the century, reaching a peak a little after 1960, many discoveries continue to occur and the bottom did not fall out, unlike the cases of extinction and nini-boom-busts. We explain this by considering the role of dispersion in the discoveries. The following figure shows a non-dispersed discovery function, which reaches a sharp peak and then drops to zero as prospectors finish searching an isolated volume of potential finds.



This basically happens when a highly localized search takes place, as with the case of the blanket coverage of passenger pigeon flyways with an efficient army of hunters (often equipped with explosives!). Its also happens with prospectors sifting everything with the equivalent of a fine-tooth comb in some localized gold strike area.

But the discovery of oil differs as dispersion in the rates of discovery in various parts of the world lead to a broad smearing of the bust peak. In fact, the effective bust peak (equal integrated volume) only lines up on the backside of the dispersed profile. This all makes consistent sense and provides a further argument against the use of the Logistic function to model any of these kinds of search processes, dispersed or not.

In other words someone has to explain why a symmetric Logistic function (ala the classic Hubbert curve) does not explain the steep drop-off displayed in many culling-forced extinction examples and of the bust drop-off in gold-rush cases.

Of course, this all gets the hand-wave treatment by the classically trained Hubbert modelers that use the Logistic function. Which I find really and truly odd as the Verhulst birth-death equations theoretically apply most effectively in localized Petri dish style experiments. Translation: analysis by Logistic approaches does not meet yet another sanity check and only serves as a cheap heuristic.

The Sigmoid Fraud

Unlike Sigmund Freud, I don't do psychology as a career but I do see something seriously disturbing about the fact that a majority of depletion analysts view the Logistic Function as something that contains some deep and significant meaning.

On the contrary, the Sigmoid curve --as the simplest manifestation of the Logistic-- remains a cheap empirical relationship that describes a value that increases and then saturates below some constrained limit. It indeed does follow from the solution of a non-linear differential equation, but this equation describes the temporal dynamics of a simplistic birth-death model used to describe interacting entities. One can choose populations of biological creatures or concentrations of chemical reagents to plug in to the equation. But you don't insert oil molecules into the equation and expect it to make any sense.

Take a look at this post at TOD on whale oil harvesting in the 1800's. Although the original poster does not bring up the Logistic to describe the saturation, plenty of commenters do. Fair enough, whales do fall into a biological classification, and they do give birth and die. But whale oil harvesting never tracked a population rise in whales themselves. It actually tracked the reverse. So, instead of calling it a "birth-death" model we should refer to it as a "death-birth" model. The parameter "death" represents the culling of the whale population for oil and any residual "birth" comes about because the whales can reproduce themselves based on the size of their population. Then as an exercise for the reader, one can plug some values into the birth-death equations as described here: Derivation of Logistic Growth.

But then we get to the real twist. Since whales do reproduce, if we play our cards right, then the amount of whale oil that we can harvest has no limit! The URR of whale oil essentially becomes infinite since the cumulative never abates. And unless we harvest the whales to extinction, the Logistic Function will fail miserably in describing whale oil production. (In actuality, cumulative whale oil production likely saturated because crude oil replaced whale oil as a harvestable resource.) See passenger pigeons if you want to get closer to a saturated harvest driven to extinction.

This whole analysis when incorrectly applied to oil exploration and production can induce early psychosis. On the one hand, oil does not reproduce like a biological entity nor does it act like a chemical reagent. So the equations themselves make no sense. But since oil only gets consumed and obeys the rules of a finite resource (abiotic-oil-mental-midgets notwithstanding), it will eventually saturate. So the Sigmoid falls into our lap in spite of itself. The fraud survives in effect only because it looks like an S-curve !

To avoid this mental anguish, I prefer to use the Dispersive Discovery formulation for discoveries and the Oil Shock model for extraction/production dynamics. This approach makes intuitive sense, the math falls out naturally, and you don't have to continue to psychoanalyze insane ramblings of people that live in some freakish world where square pegs fit into round holes and empiricism has the dynamic range of a stupid heuristic. The rise and fall of the oil culture deserves a better understanding than the Logistic can ever offer.

End promised rant;

Same: More Of

Kevin Phillips

What more can I say but that the inflation numbers and the dynamics therein remind me a lot about the essentially corporate and government cover-up behind the truth in oil reserves. We can easily figure this out if we had more committed and numerate people.

Unless you believe that growth consists in how many songs you can put into an mp3 player or the accelerating absurdity of Grand Theft Auto, money doesn't buy as much as it used to.

Scaling and the Dispersive Discovery growth function

The search growth function I use for the Dispersive Discovery model follows a T⁶ time dependence. The derivation comes from a quadratic growth term on top of a single dimension of volume. When the quadratic gets multiplied along the three dimensions of volume, the T⁶ dependence results.

High-order growth terms such as T⁶ have some similarity to exponential growth terms as a particular order in the Taylor's series polynomial expansion dominates over a certain interval. The following chart shows the cumulative dispersive discovery using T⁶ plotted alongside an e^kT growth term inserted into the Dispersive Discovery equation. I normalized the two curves via an affine transformation so they intersect at T=1.

Note that the doubling time for the exponential is about 10% of T at T=1, which roughly coincides to the doubling time for the T⁶ growth.

For world crude oil discoveries, the T=1 time point scales to approximately 100 years (the time period from 1858 to the early 1960's when we observed a global peak). This means that the discovery growth doubling time equated to roughly 10 years in historical terms -- premised on that you believe the Dispersive Discovery model applies. If you look closely at the two curves beyond T=1, the exponential reaches the asymptote much more quickly than the T⁶ growth curve. This makes perfect sense as the higher order polynomial terms in the Taylor's expansion of the exponential take over, and push to the asymptote more quickly, and thus minimizing the effects of dispersion.

Some might find the exponential growth model more understandable or intuitive, as this emulates technological advances such as those described by Moore's law (i.e. which shows doubling of microprocessor speed every two years), or approximates population growth and the demand and acceleration in prospecting effort that this implies.

Whether the exponential growth actually provides a more realistic picture of the dynamics, I can't say but know for certain that it requires a much stronger growth stimulus -- thus implying that a doubling of search effort must occur every 10 years for the foreseeable future. On the other hand, a high-order function such as T⁶, though it continues to accelerate, will show progressively longer doubling periods as T increases.

We know that Moore's Law has recently shown signs of abating. This could result from an abatement of technological progress as researches start to give up on scaling techniques¹, which in the past has guaranteed speed increases as long as the research fabs could continue to reduce circuit dimensions. Or it could stem from a hard limit on the scaling itself, due to parasitics and losses as the electrical properties encounter quantum limits. I have a feeling that something similar to a "dispersive discovery" in the research growth advances will allow Moore's Law to continue to limp along, as researchers will continue to find niches and corners in the ultimately constrained and finite "volume" of semiconductor combinations available to us.

So what happens to oil prospecting effort as we start hitting the walls remains unknown. We may want to pay close attention to how Moore's Law shakes out just out of curiosity and to see how a "smooth landing" applies in that technology area². In any event, it definitely will pay to start using the exponential growth model in conjunction with the T⁶ growth term as the two complementary cumulative dispersive discovery curves don't show a significant amount of difference, and moreover demonstrates that the underlying model shows a certain amount of robustness in terms of parametric variation. In particular, the exponential provides a good way of calculating differential margins should we want to assume a stronger post-peak discovery search pressure.

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¹ Years ago, I sat in an adjacent office to Robert Dennard, a really nice guy by the way. The scaling theory that he formulated, along with his invention of DRAM, had a lot to do with the correctness of Gordon Moore's predictions. I would find it fascinating if I could get Dennard's opinion (or Moore's for that matter) on how the Dispersive Discovery "scaling" theory could apply in a macro sense. I bet they would both admit that the endless doubling would not continue indefinitely, both in classical semiconductor scaling and likely in oil discoveries as well.

² The key area of research interest looks like a focus on multi-threading and concurrent functionality. Building more parallelism into microprocessors allows them to continue on an upward performance path, even though the speed improvement turns into a "virtual" or ephemeral achievement. And that assumes that we can get our arms around creating algorithms that take advantage of multi-threading -- not the easiest or most amenable idioms to formal techniques that programmers would prefer to encounter. But some researchers do have grand hopes; in an EE Times article titled "Berkeley researcher describes parallel path", one professor thinks he has discovered unity energy savings on this path:

Energy and the environmental issues are also driving work in ubiquitous computing, said S. Shankar Sastry, dean of engineering at Berkeley.

"We need to think about a building OS that handles all the heating and cooling systems and controls elevators," he said, describing work that could make these large energy consumers into generators. "We need to create buildings that not only consume zero net energy but have zero net cost," he added.

Incredible.


Next: Stay tuned for a final skewering of the Logistic production model.

Idiot Wind

Bush Press Conference, April 29, 2009

"You know, I just told you that there's about 27 million gallons of diesel and gasoline that could be from domestically produced crude oil that's not being utilized.

And not only that, we can explore in environmentally friendly ways. New technologies enables for — to be able to drill like we've never been able to do so before; slant-hole technologies and the capacity to use a drill site — a single drill site to be able to explore a field in a way that doesn't damage the environment."

Mr. President, do you consider us morons?

BUSH: "... just cut off your mike. You can't. No."

Seriously, no one can explain what any of this means, as it leaves Bush's lips. (Mike Malloy rebroadcast this excerpt and I still couldn't figure it out)

QUESTION: Mr. President, you have spoken today about opening ANWR for drilling and also refineries. But these are clearly long-term solutions to the problem of rising gas prices. What can you tell Americans about what your administration is going in the short term?

And, secondly, have you been briefed on tomorrow's GDP numbers...

BUSH: No, I haven't.

QUESTION: ... and are you concerned -- OK. OK, and are you concerned that they will show us to officially be in a recession?

BUSH: I think they'll show that it's a very slow economy. I can't guess what the number will be.

And I haven't been shown truly.

And, by the way, opening up ANWR is not long term. It's intermediate term. But it sends a clear signal, is what it does. It sends a signal to the markets that the United States is not going to restrict exploration; the United States is going to encourage exploration.

And in the meantime, we have done -- increasing CAFTA, for example. But the market's going to, you know, do as much for encouraging conservation as anything else is now.

And so I firmly believe that, you know, if there was a magic wand to wave, I'd be waving it, of course. It's -- you know, I strongly believe it's in our interest that we reduce gas prices -- gasoline price.

It'd be like a major tax cut for people.

QUESTION: (OFF-MIKE)

BUSH: Let me finish, please, Cheryl (ph). Strike one on the exclusive.

Excuse me, strike two.

That -- made me lose my train of thought. Of course, maybe that's what you're attempting to do.

No, I think that if there was a magic wand to say, OK, drop price, I'd do that.

And so part of this is to set the psychology right that says to the world, We're not going to become more beholden on your oil, we're going to open up and be aggressive and have an aggressive energy policy.

Secondly, we're going to be sending a signal we're going to be building new refineries.

But there is no magic wand to wave right now. It took us a while to get to this fix. That's why I told you that if Congress had responded -- matter of fact, Congress did pass ANWR in the late 1900s -- 1990s -- end of the 1900s -- 1990s, but it didn't go forward.

And, you know, it's my considered judgment, given the technological advances, to say, This is -- you know, will destroy the environment, is just -- I don't think it's an accurate statement.

And so I think it's very important, Cheryl (ph), for Congress.

The other thing Congress can do, if you want to send a good signal during these uncertain times, is make the tax cuts permanent, is to let people -- send the signal that people are going to be able to keep their money. And I think that'll help the psychology of the country.

Update: I heard the talk again today on Olbermann and Bush said clearly 27 million gallons a day. So, the transcript was wrong. But he still sounds lik an idiot.

Adversaries: Earth

Was this a throwaway comment?

"We're more likely to see other companies as collaborators rather than adversaries ... We aren't so much competing with other as we are competing with the Earth. And maybe that's a healthy way to look at it." -- George Kirkland, Chairman and Managing Director Nigeria Chevron Limited (as quoted in the Goodman's "Th Exception to The Rulers"

I have always used this as a first-order rule when analyzing the conditional premises surrounding the dynamics of oil depletion. Unrelenting greed becomes the overriding factor in the stimulus. Model it as a process of exploit, exploit, exploit, and then exploit some more, and you have captured the oil grab mentality. I really don't believe economics plays much of a role in the driving stimulus as technology and human consumption turn it into a monotonically increasing function. Which makes it a good first-order rule.

I did happen to see Amy Goodman and David Goodman speak today, whence she paraphrased the above quote. That bit of tacit knowledge, perhaps inadvertently spoken by a oil honcho, basically outlines the entire premise of dispersive discovery and the oil shock model. It basically says find as much as you can while you can, and turn on the taps as much as they can handle.

I learned that David Goodman has also written quite a bit on backcountry, downhill, and x-c skiing in New England. He joins Bill McKibben with his X-C manifesto Endurance on my short list of New England-based progressive environmental/journalists who want to save the planet, and who also enjoy a most obscure recreational pasttime.

While on the subject of lists, actor Matthew Modine has organized www.bicycleforaday.org slated for later this year. He joins fellow costar of "Married to the Mob", Michelle Pfeiffer, in a list of actors from the greatest Mafia movie of all time, who have advanced the state of bicycle ridership. She, according to urban legend once said "I relax by taking my bicycle apart and putting it back together again." (google it) Could have fooled me, but someone should ask Modine to get the real scoop. He'll need all the help he can get to make that day a success. I chipped in and bought a card from him that says "card-carrying liberal".

Splainations

Fafblog has returned, in the nick of time to splain everythin.