i think it´s also worth noting that the graph of your first stage resembles a mirror of another one i´ve seen before which adds the concept of diminishing eroei, just pick a bitmap editor and flip left and right sides and watch the net energy curve.



to insert diminishing eroei (net energy) into the system you should use a different logistic step function.
one way to build it is to consider logarithmic accelerating time axis for a default logistic step, or to consider the first half of the frequency response of a butterworth bandpass filter.

It's not a unit step function. If it was, the output would asymptotically reach a constant value. This would be the analogue of an infinite supply of oil.

The stimulus on the input is actually decaying exponential (in this particular example). The area under the curve of the stimulus is equivalent to the area under the curve of the response curves. Put simply, volume of discoveries = volume of production. The stages in between can be thought as virtual volumes that we never see in the abstract sense. They correspond to transition from the discovery state to the fallow state to the construction state to the maturation state and finally ending up in the "concrete" production state.

oooops! my fault



It´s possible to conclude that when you look for a circuit that started with zero energy, and the graph shows that in some point in the future it will return to zero, it was obviously stimulated by something that started in zero and will end in zero energy, like a decaying pulse that instantly jumps from zero to some constant and decays back to zero at infinity.

by considering the unit step function, i´ve just integrated my interpretation, because when i look at energy consumption, i like to watch and account the total accumulated energy consumed.

your idea of creating a physical electrical model to do some analogy to whats going on in the "macroscopic" level is a fantastic tool for doing some predictions, let´s discuss a few improvements, because when it starts to improve, the graphs obtained in the linearization process, if the model is well calibrated, should reflect exactly the real life graphs, which is good, because is gives ahead information sharing for us, here at peakoil.

IMO i think that is better to create a model on the integration side and watch for the zero and first derivative graphs, instead of looking only at the derivative side and integrating the results.
this will be surely is more difficult but the precision of the results are worth the work involved.
IMO the initial idea is to conceive a circuit that starts with C1 (which can be zero or not) initial energy and by the flip of a switch it jumps to C1+K level of energy.
the behaviour of this circuit (which i have no clue on how to make) at the input will be obviously a step function, but the objective at the output is to obtain a logistic simmetrical step output at D0 and a hubbert curve at D1.
now that the purpose of this black box is know, what are some useful parameters to it? how many of them?

now comes a very interesting part, which is to conceive an "eroei" stage.(which i have much less clue than the first stage)
the eroei stage states that if you supply a hubbert curve on the input side, the result on the output stage is a mirrored capacitor discharge curve like the one that appears in your first line of graph of in the net energy my net energy graph (already mirrored).
if you reflect for some time about this, then you can ask yourself.
how is it possible to conceive a circuit that behaves exactly like a mirrored capacitor charge discharge graph? especially when this graph appears as the result of the input of an exponential decay pulse?
because, at a first glance, to obtain the mirror for this graph would mean to assume "negative" linear time, which is not possible right? this is aparently the only way to mirror this curve....
i´ll leave a few private conclusions for the next post, to allow some reflection on this puzzle, but i´m curious to know what are your ideas on how to deal with the implementation of a circuit for this task.



do you mean by this, that all the areas under the curves of all stages are the same?

All this is really just a mathematical equivalence. If you want to see how the exact same equations apply, its better to go to the source for my model:
http://mobjectivist.blogspot.com/2005/1 ... posts.html

If you see more electrical circuit analogies, I would certainly consider adding them to the model.

good page,
lots of inspiration and excelent graphs, thanks

i´ve never tried spice modeling, so, for the moment, i´ll pass. (but is surely a good investment and useful for discussing and exchanging circuit stages)

indeed, by using the mathematical simulation process, it´s much easier to obtain the tasks mentioned above.

i just caught myself thinking, why i didn´t think this before, so i´ll just have to think more about it.

The simulations at the the beginning of this thread were implicitly assuming an i.i.d. white noise for the residuals. This is of course not true because residuals are clearly realizations of a correlated random process:

[align=center]
Residuals of the logistic fit for the US production and auto-correlation function (the two dotted lines are the limit for an i.i.d. process).
[/align]

So, I decided to redo the simulations assuming an AR (auto-regressive) random process:

where n is a independent Gaussian noise.

Below are the resulting confidence intervals:

[align=center][/align]

We can see that the estimator is really well behaved because confidence intervals are nested and the median estimate is almost a straight line.

When Hubbert made his famous prediction (1956), the production was about 25% mature which is very early in production and he had 10% chance to estimate the URR within a 10% error margin! the 90% confidence intervals was then [-44.04, 446] Gb.

Now, we are past 80% of maturity and we have 80% of chance to have an URR estimate within a 10% error margin (the 90% confidence interval is [215.53, 255.56] Gb).

Motivation This thread is a continuation of the very popular two previous threads:

How Reliable is the Hubbert Linearization Method?
Bootstrapping Technique Applied to the Hubbert Linearization The objective is still the same: try to put some confidence intervals around the URR and K but this time I look at the world production.

Methodology I assume a particular logistic model which gives a production maximum in 2009 and an URR of 2,450 Gb (all liquids).

[align=center][/align]

I model the residuals using a 4th-order AR random process:

the AR model replicates the observed first- order (variance) and second order (correlation) residual statistics:
[align=center][/align]
Then, for each production maturity levels (20% to 80%) , I generate 1,000 random realizations of the residuals which I add to the "true logistic model" before using the Hubbert linearization to estimate the URR and K.

Results

[align=center][/align]

The distribution of the sample estimates for the URR and K at 50% of maturity are the following:
[align=center]
Distribution of the 1,000 samples' estimates at 50% of maturity. The full red line is the median value and the two dotted red lines are the limits of the 80% confidence interval.[/align]

Discussion

1- K estimation is more reliable than the URR
2- if we are near 50% of maturity (peak production) the uncertainty on the URR estimation is quite large and the 90% confidence interval is [1.551, 3.854] Tb with a median estimate around 2.335 Tb which covers the ASPO and USGS lower estimates.
3- Assuming a given URR we have about 30% chance to be wrong by more than 10% (higher or lower) if we are near peak production.
4- The Hubbert linearization seems to be fairly reliable even if residuals are strongly correlated

I suppose we should be thankful that the probability of a surprise on the upside in the direction of a greater-than-expected URR is greater than the probability of a surprise on the downside (in other words, the probability distribution is asymmetrical and probably gaussian or something.)

The function becomes more symmetrical the more mature the production gets, which is logical. The farther to the right you go, the more certain you are that you did not get lucky and find more oil than you expected.

thanks for your comment pu55!
The distribution shape looks like more like a gamma or a chi2 function. Note that the maximum is around 2,000 Gb which means that the URR is probably lower than we think.

Correct.

Does anyone know if President Jimmy Carter read or was familiar with M. King Hubbert's work before his fireside chat.
I believe he spoke those words soon after the US peak. Did the two ever met? Was Carter referring to the future world peak?

Good question.

It may not have been necessary for Carter to have spoken to Hubbert. The ideas of energy depletion were widespread in the culture, with a high level of analysis.

For example, the April 19, 1974 issue of SCIENCE was devoted to the energy crisis.
(The articles are online at http://www.sciencemag.org/feature/data/ ... y-1974.dtl )

- Bart

Thanks for the Science Article. It could have been written today. Canadian Tar sands were not yet making money and Colorado Shale was just around the corner.Even 'measured' feel-good optimism should be called out. It is much more dangerous than healthy scepticism.

Thanks for the articles Bart. This'll be a great rebuttal to the 'somebody predicted depletion way back when...' argument. "Oh yeah? Well here's what people thought was going to save us..."

"The only thing new in the world is the history you don’t know" - Harry S Truman

This report from CIBC World Markets (investment division of one of Canada's largest banks) called Hubbert's Peak in hindsight for 2004, and sees conventional oil supply decreasing again next year, with the 3.6 million barrels coming onstream in 2006 offsetting losses in other fields. All the growth hereafter is from non-conventional sources such as deepwater & oil sands. They're calling for over $70/bbl this year, and 100/bbl by the end of next year.

Another interesting fact from the report - much of the oil that isn't depleting is either in state control (such as Saudi) or been renationalized (such as Venezuela) - the result is about 80% of the world's reserves remain off-limits to private investment. Not looking good for the major oil companies.