46 comments on Links to tutorial material on Hubbert Linearization
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46 comments on Links to tutorial material on Hubbert Linearization
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Log(P) = A + B*T + C*T^2.
The constants A, B, and C are obtained fitting. Therefore, the problem reduces to the problem of fitting a quadratic rather than a straight line. This is only slightly more difficult and can be done with spreadsheet programs like Excel. The advantage is that, unlike the Hubbard Linearization, the relationship is exact. I've done this fitting myself and it works quite well. I'll see if I can paste the chart in this post:
The problem is that:
(1) The data has noise.
(2) The correct model may not be a Gaussian.
In the U.S. case, HL may have given better early predictions of the peak than the Gaussian, but that might not be true of world data. I suppose the best we can do is to try different fitting methods and get a "rough estimate" of world peak oil production. The estimate should improve over time. As you point out in your discussion, error bars and sensitivity analysis are important.
the noise is almost always data that doesn't fit the person's pre-determined opinion from what i have seen.
The same exercise for Russia showed that post-1984 cumulative Russian production was 95% of what the HL model predicted, using only production data through 1984 to generate the predicted production profile.
I proposed the Lower 48/Russian experiment to Khebab, and he chose all of the technical parameters. If you have read any of his posts, you can tell that Khebab is an objective scientist. IMO, he is a genius.
In any case, Khebab had zero preconceived expectataions of how the results would turn out. When you look at the actual 1970 and earlier Lower 48 data and actual 1984 and earlier Russian data, they both show very strong HL patterns.
"M. King Hubbert's Lower 48 Prediction Revisited" has the HL modeling of the Lower 48.
For example, back when I was a teenager I used to grind and polish astronomical mirrors (for Newtonian and Hershel-style off-axis) reflecting telescopes. The goal was to get to the Raleigh Limit--one-eighth of a wavelength of sodium light, and the figure desired was a parablola.
Guess what: for a 4 1/4 diameter F-20 mirror I figured it to a SPHERE which is well within the Raleigh limit for a parabola, even when using the off-axis style to avoid the diffraction from a Newtonian diagonal and its support.
Even at F-12 or thereabouts for a Newtonian style, a sphere is within the Raleigh Limit for a little mirror, such as 6".
BTW, for observing planets, most nights the atmosphere is too turbulent to get much if any benefit from a telescope over 6" or 8" And on many nights a 60 mm lens on a good refractor actually shows better images than you get from a big scope because of the nature of atmospheric turbulence, which is (to put it mildly) complex.
Anyway, what matters and costs the most in amateur scopes is usually the stability of the mounting rather than the quality of the lenses and mirrors.
perhaps you understand this and you are making another point when you say Hubbert Linearization is an approximation but if so, I think you may have confused some readers that are less familiar with the subject.
At the risk of teaching the many Grannies that live here to suck eggs, Hubbert proposed that cumulative production with time (Q) followed the logistic curve (or more precisely the sigmoid curve, a special case of the logistic curve). The rate of production (P) is thus the differential of this with respect to time which gives the familiar Bell shaped curve that is like a Gaussian curve but not quite the same.
Q = Qmax/(1 + exp((th - t)k)))
P = Qmax*(t/k) * ((exp((th - t)/k)
(1-exp((th - t)/k)²))Where:-
Qmax = the ultimate cumulative production
th = the time at which half of this production is extracted
k = a constant with units of time that determines the rate of depletion, the larger k the slower the extraction.
t = time
This function gives the mathematically exact relationship
P/Q = (t/k)*(1-Q/Qmax)
This shows that were production to exactly follow Hubbert's curve a plot of P/Q against Q would give a straight line slopping down to intersect the Q axis at Qmax. The line passes thrugh Qmax/2 at th and this is the time of peak production.
Plotted the first way, the majority of the curves look like this:
_files/image006.gif)
You can see the general shape of the upside-down parabola but since it is not symmetric, it can't be the logistic curve.
Makes a nice parabola and actually this looks much better than the linearization (since it's not blowing up the misfit in the tails with a 1/Q factor). The predicted URR is within error bars of what the linearization says. I only fooled around a tiny bit, but it seems that it does ok at robust prediction too:
(BTW, as a process/communication issue I would prefer that you not accuse others of acting in bad faith ("swept under the rug"), without good evidence. It's more polite to assume that the other people simply see things differently, which is usually the case, certainly around here.)
I guess I am more into the understanding rather than the predictive properties. I just don't get how this stuff works out without any kind of forcing function included. I am categorizing this set of formulations under the heading "Immaculate Conception Hubbert Peak Analyses". Without a forcing function, I might as well look into causes of Spontaneous Human Combustion.
A thought experiment for us to engage in. Say from now out, time=T, the quantity of discoveries followed as:
K/(C + (time-T))
In this case, we would still have the peak centered at the same point but the URR will blow up to infinity. Until this is discussed by someone other than me, I can say that it is "swept under the rug", which is a mildly-offensive euphemism for "ignored". And for all I know someone has discussed this, but I don't know about it.
Oops, maybe that was an insult as well. I will try to stifle myself, and just consider anything outside the bounds of decent behavior as my attempts at snark (people such as Michael Lynch, George W. Bush, and Michael Crichton excluded).
I agree, as I've said repeatedly, that we lack a theoretical understanding of why the US production curve is so Gaussian and that's unsatisfactory. However, I view that as an interesting challenge that we should try to solve rather than a reason to dismiss the fact that it has been so up to now (modulo some noise).
It was George Stigler (I think) in his first edition (late 1940s I believe) who pointed out for the first time that BECAUSE we economists do not know the empirical shape of most supply and demand curves that we should draw them as straight lines TO EMPHASIZE THAT THEY ARE ARBITRARY AND DO NOT REFER TO THE REAL WORLD. Unfortunately, after Stigler, relatively few authors make his point, thus needlessly confusing generations of miserable and bewildered and hostile students.
HOWEVER, I shout;-)
For a small change in price for a well-behaved supply or demand function a straight line is often a pretty darn good approximation to the real world.
It is almost never a decent approximation for a large (say more than 20%) change in price.
When I taught economics I explained these nuts and bolts, and guess what: Almost half my students got a fairly good understanding of supply and demand. In the typical introductory and even intermdiate microeconomics classes at U.C., Berkeley, my guess is that fewer than ten percent grasped the most basic fundamentals of supply and demand.
Now elementary need not be hard at all. No! The problem is that most teachers of lower division classes do not give a darn about teaching and could care less that 90% of the students are ignorant of fundamentals.
Grump.
The linearization talked about here concerns a technique to turn a highly non-linear function into a straight line. It has nothing to do with small perturbations affecting linearity to the first order, as a Taylor series approximation does.
That property I do believe in but that does not influence my disagreement with the original premise of using a logistic curve or gaussian curve formulation to describe the stochastic behavior.
I think (but do not know) that Stuart agrees with my line of reasoning; he has expressed it himself in somewhat different words--just a few days ago.
The key point is that Hubbert, in 1956, accurately predicted the Lower 48 peak.
What Khebab and I attempted to address was how good the HL model was at predicting post-peak cumulative production, using only Lower 48 production data through 1970. The answer was that actual cumulative Lower 48 production was 99% of what the HL model predicted that it would be.
Assuming that Deffeyes is right that we are past the peak of conventional crude + condensate production, the HL model should therefore offer us a very accurate prediction for post-peak world production.