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76 comments on When Does Hubbert Linearization Work?

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Sam Foucher on September 29, 2005 - 11:20am Permalink | Subthread | Comments top
I'm skeptical about the use of this method to present production data because the relative error doesn't seem to be distributed uniformly. The relative error in the log domain of the vertical ordinates according to the logistic model is the following:

D(ln(aP/Q)) = Dk/k - DQ/Q x Q / (1 - Q)

where D stands for the greek symbol Delta, Dk/k and DQ/Q are the relative errors on k and Q which can be presumed constant. The error behaves has following:

  • Production start Q -> 0:
    D(ln(aP/Q)) = Dk/k
  • Production Q -> 1 (total URR has been extracted):
    D(ln(aP/Q)) = -infinity
Because we are in log domain, D(ln(aP/Q)) = -infinity means that deviation around the asymptotic line will tend toward zero! That's why, we observe these wild deviations aroud the line when production is starting whereas it seems to converge nicely when Q tend toward 1. This behavior can be misleading for an observer because it seems to reinforce that there is some inexorable mechanism at work pushing the production data around the line.
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Stuart Staniford on September 29, 2005 - 11:42am Permalink | Subthread | Parent | Comments top
It's not a log plot. It's a linear plot of P/Q versus Q.
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Sam Foucher on September 29, 2005 - 12:06pm Permalink | Subthread | Parent | Comments top
it doesn't matter! the log is applied here only to make the error analysis easier (multiplications become additions and absolute errors become reletive errors).
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Eroie Van Tanstaafl on September 29, 2005 - 2:31pm Permalink | Subthread | Parent | Comments top
From what I've gathered about the math behind these curves, I don't think there are good answers to the concerns you are raising vis error analysis.

Hubbert was not a mathematician or an econmist.  He drew his curves by hand, and calculated the areas underneath them by counting the squares on his graph paper and guessing.  He assumes logistic growth and decline, and some of the field curves look like logistic curves.

But I'm not seeing a lot of advanced statistical (or econometric) methods or models yet from the peakers.  Drawing and fitting curves is meaningless - its the equations the curves reflect and the relation of the variables in the equations to the facts in the ground that provide meaning.

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ChrisPhoenix on September 29, 2005 - 7:39pm Permalink | Subthread | Parent | Comments top
Counting squares under a curve is not "guessing," it is measuring. It's the way things were done back in the days before computers.

If this non-argument is the best you can do, maybe the Peak Oil people are right.

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Stuart Staniford on September 30, 2005 - 5:47am Permalink | Subthread | Parent | Comments top
Got it - it is certainly true that because of the Q term, the fluctuations are much smaller to the left than the right. However, that doesn't prevent the curve from diverging from the model if indeed the model is wrong. More tomorrow - I now have a couple of countries where it doesn't work that we can chew on.
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76 comments on When Does Hubbert Linearization Work?

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