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June 14, 2006

Estimate of Uncertainty

Several readers asked for the uncertainty in my date of December 16, 2005 for the peak of world oil production. I'm increasingly aware that there are some seriously discerning readers on the other end of this web site. The answer is: from November 12, 2005 to January 7, 2006. Now, what exactly was the question?

There are two different kinds of uncertainty involved, inside and outside. The inside uncertainty accepts the Hubbert logistic-curve model and asks for the range of Hubbert-peak dates that are consistent with the observed data. The outside uncertainty asks whether some non-Hubbert model might also make an acceptable fit to the data. This note is concerned only with the inside uncertainty. If you want to go outside, you're on your own.

Of course, there has been speculation that I have issued tongue-in-cheek predictions in order to sell books. I'm really serious.

I'll try to explain exactly what I did. The evaluation proceeds in two stages: estimating the total producible oil and interpolating the calendar day when half of the oil was produced.

The initial stage uses the dates of oilfield discoveries shown on page 49 of Beyond Oil; updated to include the year 2004 and 2005 results. Those data points are in substantial agreement with an ExxonMobil graph dated 2002, as updated by the ASPO newsletter 63, page 2, March 2006 at (If you don't have a copy of Beyond Oil, just click on the title and Stephen Deffeyes will transport you magically to the land of The best-fitting straight line on page 49 has a horizontal intercept of 2.013 trillion barrels. Inserting the 2004 and 2005 exploration results does not change the 2.013 trillion barrel ultimate recovery.

I obtained the 2.013 trillion barrel estimate by making the least-squares fit of a bell-shaped logistic curve to the exploration history. The logistic equation

Annual hits = Q / (1 + exp(a (t - year)))

Q = total recoverable oil (the number we want to estimate)
a = similar to an annual interest rate (see page 38 of BO for an explanation)
t = the center year, the peak of exploration success
year = the calendar year
exp = the exponential function, the natural antilog

The actual computation was done by brute force: looping through the possible values of Q, a, and t, adding up the squared errors between the observed and computed discoveries for each year, and selecting the Q, a, and t that together gave the lowest result. (That's why it is called "least squares.") Years ago, doing it by brute force was considered inelegant and wasteful of computer time. My slow, 2.7-pound IBM X-40 laptop takes 0.17 seconds to go through all of the years for all of the combinations of Q, a, and t.

Of course, the "best" estimate is the one with the lowest sum of the squared errors. An advantage of the brute force method: I can rerun the calculation printing out all the near-miss combinations of Q, a, and t. The very best estimate, the one that gave the 2.013 result, had a sum-of-squares of 55,024. (Don't ask, I think the units of 55,024 are square barrels.) I chose to look for any other answers that came within one percent of the best, any Q, a, and t with summed square errors less than 55,574. These nearly-as-good answers have Q ranging from 2.008 to 2.016 trillion barrels. Notice that 2.013 falls, as it should, near the middle.

The interpolation to get the actual peak date uses the cumulative world production of .981324 trillion barrels through the end of 2004 and 1.007529 trillion barrels at the end of 2005. For the minimum estimate, half of 2.008 trillion barrels, the peak date would be November 12, 2005. Half of 2.016 trillion barrels gives a peak date of January 7, 2006.

If you took my advice of two years ago and celebrated the peak on the American Thanksgiving day of November 24, 2005, you were within the uncertainty window. The stronger message: January 7, 2006 is history.

To many people, and especially to Cornucopians, a range from November to January smacks of false precision. I repeat, that is the uncertainty inside the Hubbert model. Anyone is free to concoct a different quantitative model and to compute their own uncertainty.

Of course, there has been speculation that I have issued tongue-in-cheek predictions in order to sell books. I'm really serious. We need to face the problem and to take action. Meanwhile, I'm not averse to selling books; the updated paperback edition of Beyond Oil went on sale June 14, 2006.

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