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.

Production Estimates = Demand Estimates

Finally, a brave volunteer got a oilman yo-yo to admit the sad truth that their oil supply estimates derive only from oil demand estimates. Gail got this revelation out of John Felmy, American Petroleum Institutes's Chief Economist, at a recent TOD post describing a conference call with Chevron executives.

This proves once again that big oil and EIA never intended to predict anything realistic, instead opting to project their wishes for the future. Unfortunately, that doesn't fit into any branch of scientific forecasting methods that I have ever encountered.

Update: Dissident makes a good comment that may explain the thinking as delusional economics mathematics (i.e. the dismal science)

"The strategy the economists used was as simple as it was absurd—they substituted economic variables for physical ones. Utility (a measure of economic well-being) took the place of energy; the sum of utility and expenditure replaced potential and kinetic energy. A number of well-known mathematicians and physicists told the economists that there was absolutely no basis for making these substitutions. But the economists ignored such criticisms and proceeded to claim that they had transformed their field of study into a rigorously mathematical scientific discipline."

Oh my, what medieval garbage.

So they seem to think they can make a mathematical analogy to the conservation of energy, whereby K.E. + P.E. = constant. Yes, medieval garbage based on some bizarre premise spurred by a very bad analogy.