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## Sunday, September 16, 2007

### Logistic Model for HL purely a Birth Model

TOD analyst Khebab reminds us that:
There are two contexts for the use of a Hubbert/Logistic curve:
1. logistic demographic modeling as initially proposed by Pierre François Verhulst.
2. curve fitting as Hubbert did.

WHT's arguments are dealing with the first context whereas you are putting yourself in the second context. The power of (1) is that you are trying to physically model the observed phenomenon using a differential equation. In the case of a logistic curve, the differential equation is modeling a birth-death process as explained by WHT
I responded to his comment:
Khebab nailed it by placing it in the 2 contexts. As I said before, you cannot ascribe any physical meaning to the parameters if it is simply a curve-fitting exercise. And in the other context, we just can't make any sense of the model in terms of oil production.

Let me present another perplexing situation. If we actually consider the full birth-death model:
`dP/dt = (B0-B1*P-D0-D1*P)*P = (B0-D0)*P - (B1+D1)*P2`
you notice that in terms of birth and death, this gives us asymptotic P as the current carrying capacity. The equivalence between carrying capacity and URR makes absolutely no sense if we keep the death terms in the equation. Remember that deaths essentially knock out entities from the current population and the number of entities that have actually existed over all time would be infinite! But we know that URR << infinity.

What this means for the oil analogy is that we would have to present the Birth/Death model as purely a Birth model. In other words, births don't give rise to deaths but they do add to the cumulative growth. Otherwise, we would significantly undercount the URR as deaths would invisibly remove "entities" from the cumulative count.

So the next time someone talks about oil fields dying and trying to relate that to Logistic/HL modeling, point them here.

As Khebab said, no problem if you use HL as simply a curve fitting exercise, but you cannot ascribe any further meaning to it.
So now we have a logical proof that the Logistic model has absolutely nothing to do with "deaths" of oil wells or oil regions. I know this won't prevent TOD commenters from continuing to use improper analogies to "explain" what the Logistic (and HL) means but we have to draw a line in the sand.

As an obvious alternative to the HL in its only legitimate curve-fitting role, I recommended the First-Derivative Linearization (FDL):
If we want to go with something simple, then go with something simple. For instance, why the heck don't we just plot the first derivative of yearly production with respect to time and then plot that? Around the peak, this will turn into a straight line with negative slope and you should be able to discern the peak by where it goes through zero (i.e. the slope flips sign around the peak).

This is based purely on calculus and the Taylor series approximation that every quasi-symmetrically peaked curve has major terms like A-B*(t-t0)2 around the peak. Take the derivative of that curve and you get 2B*(t0-t) which you and I and everyone else can easily understand as a negatively sloped straight line which crosses the axis at peak.

I know that this completely obscures the subject of URR, but URR is not even important here, based on the same assumption of a quasi-symmetric peak. As much of the production will appear on one side of the peak as the other for a more-or-less symmetric curve, so just ignore the URR.

So I see it that we have two routes to take:
1) Go for the trivial analysis as above (therefore undermining HL, which has proven to be a perfect example of a concocted and contrived analysis)
2) Go for a real model of oil discovery and depletion

I'm all for (2) but if we really want to do (1) then let's really agree to finding the faults with HL.

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