The Oil Drum

22 comments on Linearizing a Gaussian

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nero on January 11, 2006 - 5:33pm

I can make a fair argument for some sort of "S" type curve but it is only so much hand waving. My criticism was about the argument that the logistics equation ought to be used over some other equally apropriate curve.

The logistics curve has a logical mechanical reason for applying to bacterial growth where in an unconstrained situation the rate of increase is proportional to the current bacteria population and the availability of the constraining resource.

However for oil production it isn't the past cumulative historical production (Q=sum(P)) that determines the rate of production growth but the current size of the exploration industry (I). Here is an alternative simple model that has to my mind some more relevant significance to the terms.

dI/dt = (h(Qt-Q)-j)I,
dP/dt= r(Qt-Q)I - nP,

where:
r is a parameter related to the exploratory success rate
n is the average infield decline rate
j is a depreciation factor
h is a parameter associated with the insentives to increase exploratory effort.

If we add some apropriate parameters this also forms a nice bell curve. Is this a better model than the logistic curve? I couldn't say, but it does have the advantage that it makes some mechanistic sense.

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WebHubbleTelescope on January 12, 2006 - 12:08am

Nero is darn near close to what I advocate, as far as I can tell:

I(t) = Discoveries,
dQ/dt = I(t) - n Q(t),
P(t) = a Q

I go through a few more 1st-order transforms because you have to consider latencies corresponding to fallow periods, construction periods, and maturation periods. I have the math all worked out, have the source code to do the numerical integration, and it basically looks like this if you assume that the Discoveries curve follows a quadratic growth curve for the USA (peaking after 1930)

That blue curve comes out to a quadratic (i.e. peaked parabola) convolved with a 4-th order gamma curve (i.e. 4 exponentials of the same rate convolved sequentially). It accurately maps over a range 5 orders of magnitude.

http://mobjectivist.blogspot.com/2006/01/would-you-believe.html