I have now gone through the calculations for predicting the Hubbert's Peak based on the logistics curve. Here is the differential equation.
and Q is the cumulative production, Q' its time derivative (this can be taken to be the yearly production), Q_o is the ultimate endowment, and 'a' is the slope of the straight line in Q'/Q vs. Q/Q_0 coordinates. (or when it is multiplied by 100 it is the annual percentage growth of the normalized cumulatative production). This is the figure on page 157 of Deffeyes's book.
A couple of comments:
1. I took the production data from Campbell's book for the years 1930-1960 and his data for how much had been produced before 1930.
2. The data after 1930 is from Energy Information Services and it seems to check Campbell's data.
3. The curve fit is based on putting a least squares line through the last few years of the data plotted as mentioned above. >From this one can get the ultimate production, the slope "a" and then the Hubbert's Peak.
4. The estimate is sensitive which years are included in the least squares fit. If one starts in successively from years 1990, 1991, 1992 etc and includes data through year 2000, you get different results. Hence, as is implicit in Hubbert's work, and this was mentioned by Deffeyes, one must know by independent analysis what the ultimate endowment is. The work of Jean Laherrere on parabolic fractals and the creaming curve are very important in establishing a good guess of the ultimate by a study of all the oil provinces and then estimating how much is yet-to-find.
5. After fiddling around for a while, I came to the conclusion that using the data from 1990-2000 gives the best fit and a peak in 2004-2005. (one should not use the earlier data).
It was quite bold for Hubbert to suggest a peak in 1970 for the US based on data from prior to 1956. Deffeyes says that he based his analysis on the logistics curve and the data shows that the "straight line" behavior sets in only quite late. I have new appreciation for how much other work went to the analysis of the production and depletion, but as we are so close to the Hubbert's Peak, the prediction becomes more and more certain.
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