Posted: Sat Jan 07, 2006 10:44 pm Post subject: How Reliable is the Hubbert Linearization Method?
Motivation:
The Hubbert linearization is a simple method to estimate the parameters (URR and growth rate K) of the logistic model. It consists in plotting the ratio of Production on cumulative production versus cumulative production (P/Q vs Q). This representation is often used as a predictive tool, see for instance the excellent threads posted by Stuart staniford on TOD:
This method gives impressive results on mature countries. For instance Norway:
I'm trying to answer the following questions:
1- is the Hubbert linearization a good approach in the estimation of the URR and production growth rate?
2- what confidence levels can I have on these estimates?
3- is it a good predictive tool on immature data (far from the production midpoint)?
I'm not discussing here the validity of the logistic model for the depletion modeling of oil fields. I'm assuming that it is a valid model.
Methodology:
The proposed approach consists in generating a pool of random datasets from a known logistic curve on which we apply the Hubbert linearization method to estimate the URR and K. The true parameter values are the following (Hubbert model for the US production):
Code:
URR= 222.34 Gb
K= 6.08 %
I consider 7 levels of maturity (20%, 30%, 40%, 50%, 60%, 70%, 80%). 50% corresponding to the production midpoint and below this value the dataset is considered immature. For each level, I simulate 1,000 datasets constructed from the true logistic curve on which I add a gaussian random noise of standard-deviaiton equals to 0.12 Gb (Note: this value is derived from the residuals of the logistic model applied on the US production data).
Example of a random data set: simulated production on the left and hubbert linearization on the right. The red curve corresponds to the true logistic curve. The blue points are the data used for the Hubbert Linearization.
Results:
The distribution of the relative error is derived from all the estimates from which we produce the following curves:
Observed precision for the URR parameter estimation for different levels of maturity.
We define Precision as the maximum absolute relative error wanted on the parameter. The confidence is the percentage of estimates that have reached a particular precision.
Observed precision for the K estimation for different levels of maturity.
We define Precision as the maximum absolute relative error wanted on the parameter. The confidence is the percentage of estimates that have reached a particular precision.
We can observe that the estimation is much more precise for K than for the URR. If we decide to tolerate a maximum of relative error below 10%, we have the following probabilities to be sucessful:
Discussion:
The Hubbert linearization seems to be a good tool to estimate the growth rate even on immature production data (we have 90% of chance to estimate K with less than 10% of error when only 30% of the URR has been produced). However, for the URR estimation we need to reach the production midpoint to have the same confidence level.
Regarding the Hubbert Linearization applied to the world production, there are a lot of evidence that we are now near 45% of the URR which means that we can be about 80% confident that the URR is around 2,350 Gb+/- 10% which is the value resulting from the Hubbert linearization (see figure below).
Posted: Sun Jan 08, 2006 6:49 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
nice graphs khebab
here´s a little more inspiration to the linearization process
important: cumulative production graphs are analog to overshoot graphs, hence, they are squished beetween two exponentials, consequently they
linearize very fast
it´s easy to "cook" the angular factor of the linearized graph and a small rotation in the line produces a wide result in the crossing beetween the linearized line and the X axis, but when extraction data is true, the linearization allows predictions like hubbert´s one, 14 years (-+)1year.
when you deal with random data squished beetween two exponentials (narrowing or widening along time), it´s possible to fit the data using black and scholes model which is being done for triggering decision making in stock options.
and the decision process is done using this kind of graphic:
Posted: Sun Jan 08, 2006 7:52 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
Quote:
Regarding the Hubbert Linearization applied to the world production, there are a lot of evidence that we are now near 45% of the URR which means that we can be about 80% confident that the URR is around 2,350 Gb+/- 10% which is the value resulting from the Hubbert linearization (see figure below).
Khebab:
Congratulations on an excellent post!
I think the key point is the one above: accuracy of the method improves the farther to the right on the extraction curve you go.
If Spike were to review this, he would say that at present levels of apparent extraction, there is still a possibility of new discoveries, etc. that would extend the oil supply sufficiently to give us more time before the peak. What you are saying is that there is indeed this possibility, and it's about 10% that the URR will be greater than 2580 gb (2350+10%). It's also 10% possible that URR will be less than about 2000 gb.
This is just about enough chance to give the cornucopians some hope, but allow the doomers to continue to make their survival plans. Ironically, even at 90% extraction, there is still some uncertainty, which says that the cornucopians will be able to hang on to some hope right up until the end.
Posted: Sun Jan 08, 2006 9:32 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
bantri wrote:
here´s a little more inspiration to the linearization process
important: cumulative production graphs are analog to overshoot graphs, hence, they are squished beetween two exponentials, consequently they
linearize very fast
Thanks for the comments, very interesting! It seems that the properties of the Hubbert linearization are quite general and are independent of the function considered as long as it is a positive function. For instance, If I strongly pertubate the curve with a periodic signal (cosinus), I still have a good fit:
Curiously, there is no litterature on that subject.
pup55 wrote:
If Spike were to review this, he would say that at present levels of apparent extraction, there is still a possibility of new discoveries, etc. that would extend the oil supply sufficiently to give us more time before the peak.
In a next post, I wish to explore the link between the discovery pattern (i.e. the amount of yet to be find discoveries) and the result of Hubbert linearization but it's a complex problem.
pup55 wrote:
What you are saying is that there is indeed this possibility, and it's about 10% that the URR will be greater than 2580 gb (2350+10%). It's also 10% possible that URR will be less than about 2000 gb.
Yes, you're right, the estimate could be lower but the error is not spread uniformly and estimates for the URR have a tendency to be underestimated before maturity. I will try to make another figure about that. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Sun Jan 08, 2006 10:15 pm Post subject: Re: How Reliable is the Hubbert Linearization Method?
Actually that brings up a very good point. The peak region is the only meaty part of the fit. Any curve that has a parabolic fit around the peak will give a good Hubbert Linearization. And since every symmetric peak is approximated by an upside-down parabola (i.e. quadratic) due to Taylor-series, that means that just about any peaked curve will work.
Posted: Mon Jan 09, 2006 2:22 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
Good post.
Your calcuations show good precision of hubbert prediction, but one could object than you take as guaranted that production is generally hubbert-shaped, with some random variations around the base hubbert curve.
There are countries where production is not Hubbert-shaped at all, for one of the following reasons :
# Production quotas (opec countries)
# disruption by war, embargo (Iraq), political collapse (FSU)
# several distinct production zones with large time lag, then production can be modelled as the sum of two or more hubbert cycles.
# production is constrainbed by pipeline capacity (Chad, Ecuador...).
Now, according to your results, if the production is "free" (no constraint => hubbert shape), we can have a reasonnably good estimation of URR even with only the 20% first percents of it gone. Then perharps we could get some rough estimation of Middle East's ultimate using figures up to 1980, before the quota were established ?
Also, any model is only as good as th input data you throw in. For historical figure, I currently use BP statistical review, but it has some flaws.
It only starts in 1965. For some countries, I'd like to have figures going somewhat further in the past.
Also, it includes refinry gain. Refinery gain has nothing to do with production, it can be done from imported crude, and depends of the market structure (it is higher when refiners maximize gasoline output, like in the us).
Where could I find historical figures excludin refinery gain? It would be even better to have a split into crude, NGL's, and non-conventionnal production.
Posted: Mon Jan 09, 2006 10:24 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
Quote:
Curiously, there is no litterature on that subject.
To my current knowledge there´s none.
(just sharing a little 2+2=4 here:)
Just assume that separate areas of knowledge (financial, physical and energy related, in this case) connect at some point, and keep researching.
Extra Hint:
The simulation using the sum of a hubbert curve and a constant amplitude cosine wave is useful to reach the conclusion that the linearization process narrows along time, but, when comparing with real models, it´s possible to conclude that this simulation doesn´t narrow as fast as the real ones.
i suggest a simulation using a DAMPED cosine wave, where it´s possible to adjust the damping factor to fit the curve better for comparison with true and real measurements (like Norway´s) without "human adjusted" factors.
more on this later....
Posted: Mon Jan 09, 2006 11:02 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
WebHubbleTelescope wrote:
Such is the power of integration.
Exactly
WebHubbleTelescope wrote:
Actually that brings up a very good point. The peak region is the only meaty part of the fit. Any curve that has a parabolic fit around the peak will give a good Hubbert Linearization. And since every symmetric peak is approximated by an upside-down parabola (i.e. quadratic) due to Taylor-series, that means that just about any peaked curve will work.
You're right, for a small time step d around the production peak at tmid, Q is a straight line and P a quadratic:
P(tmid+d)= P(tmid) + P''(tmid).d2/2+O(d3)
Q(tmid+d)= Q(tmid) + P(tmid).d+O(d3)
Therefore, a first order approximation of P/Q around the peak should be a decreasing straight line:
(P/Q)(tmid+d)~ (P/Q)(tmid)(1-P(tmid).d)+O(d2) _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Mon Jan 09, 2006 11:11 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
bantri wrote:
To my current knowledge there´s none.
(just sharing a little 2+2=4 here:)
Just assume that separate areas of knowledge (financial, physical and energy related, in this case) connect at some point, and keep researching.
Thanks for the hints, I know that P/Q vs Q have been used aslo for population study. By the way, do you have a link for the first image you posted before.
bantri wrote:
Extra Hint:
The simulation using the sum of a hubbert curve and a constant amplitude cosine wave is useful to reach the conclusion that the linearization process narrows along time, but, when comparing with real models, it´s possible to conclude that this simulation doesn´t narrow as fast as the real ones.
i suggest a simulation using a DAMPED cosine wave, where it´s possible to adjust the damping factor to fit the curve better for comparison with true and real measurements (like Norway´s) without "human adjusted" factors.
more on this later....
Ok but it seems that the damping occurs naturally in P/Q because of the division by an increasing Q. I'm not sure I fully understand your comment. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Mon Jan 09, 2006 11:23 am Post subject: Re: How Reliable is the Hubbert Linearization Method?
Raminagrobis wrote:
but one could object than you take as guaranted that production is generally hubbert-shaped, with some random variations around the base hubbert curve.
Yes it is the main assumption here.
Raminagrobis wrote:
There are countries where production is not Hubbert-shaped at all, for one of the following reasons :
# Production quotas (opec countries)
# disruption by war, embargo (Iraq), political collapse (FSU)
# several distinct production zones with large time lag, then production can be modelled as the sum of two or more hubbert cycles.
# production is constrainbed by pipeline capacity (Chad, Ecuador...).
Agreed, production modeling is really difficult for some countries especially when production is immature and the YTF is still important (ex: Nigeria) with possibly multiple Hubbert cycles involved.
Raminagrobis wrote:
Now, according to your results, if the production is "free" (no constraint => hubbert shape), we can have a reasonnably good estimation of URR even with only the 20% first percents of it gone. Then perharps we could get some rough estimation of Middle East's ultimate using figures up to 1980, before the quota were established ?
hmm... the confidence level at 20% of Qinf is rather low (~20%).
Raminagrobis wrote:
Where could I find historical figures excludin refinery gain? It would be even better to have a split into crude, NGL's, and non-conventionnal production.
Finding good and reliable data is a big issue, some private database (ex: IHS) have probably this level of details. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Mon Jan 09, 2006 4:11 pm Post subject: Re: How Reliable is the Hubbert Linearization Method?
WebHubbleTelescope wrote:
Actually that brings up a very good point. The peak region is the only meaty part of the fit. Any curve that has a parabolic fit around the peak will give a good Hubbert Linearization. And since every symmetric peak is approximated by an upside-down parabola (i.e. quadratic) due to Taylor-series, that means that just about any peaked curve will work.
Does this suggest that the linearization will not be very
good at the tail end in case the whole curve is not quadratic?
In that case, the estimate of the URR based on the linearization
may be quite a bit off. It also explains the poor approximation
at the start of the curve.
Posted: Mon Jan 09, 2006 9:38 pm Post subject: Re: How Reliable is the Hubbert Linearization Method?
rrb wrote:
WebHubbleTelescope wrote:
Actually that brings up a very good point. The peak region is the only meaty part of the fit. Any curve that has a parabolic fit around the peak will give a good Hubbert Linearization. And since every symmetric peak is approximated by an upside-down parabola (i.e. quadratic) due to Taylor-series, that means that just about any peaked curve will work.
Does this suggest that the linearization will not be very
good at the tail end in case the whole curve is not quadratic?
In that case, the estimate of the URR based on the linearization
may be quite a bit off. It also explains the poor approximation
at the start of the curve.
Actually, it works perfectly over the whole range only for the logistic curve. That is the interesting property of the logistic curve. How this came about is through a sequence of steps:
1. Somebody notices the production curve kind of fits a logistics curve
2. Somebody notices the logistic curve linearizes nicely when plotted a certain way.
3. Lots of people start plotting production curves this way.
4. Everyone nods their head in agreement, ignoring the fact that it actually doesn't fit over the entire range anyways
This next step hasn't happened yet.
5. Other models will work just as well over a certain range, thus making the logistic curve just another interesting empirical relationship.
Ok but it seems that the damping occurs naturally in P/Q because of the division by an increasing Q. I'm not sure I fully understand your comment.
indeed it happens mathematically, but try consider also a physical point of view, exponential damping also happens in decaying energy systems.
if you put a resistor R in an LC circuit, and start to dissipate the initial energy given inside the system to the outside, in the form of heat, the oscilations may be random but they are contained inside a narrowing exponential funnel.
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