General Dispersive Discovery and The Laplace Transform
Posted by Khebab on October 3, 2008 - 9:15am
Topic: Supply/Production
Tags: discovery, original, shock model [list all tags]
This is a guest post by WebHubbleTelescope.
I find it interesting that much of the mathematics of depletion modeling arises from considerations of basic time-series analysis coupled with useful transforms from signal processing. As a case in point, Khebab has postulated how the idea of loglet theory fits into multi-peak production profiles, which have a close relationship to the practical wavelet theory of signal processing. Similarly, the Oil Shock Model uses the convolution of simple data flow transfer functions that we can also express as cascading infinite impulse response filters acting on a stimulated discovery profile. This enables one to use basic time series techniques to potentially extrapolate future oil production levels, in particular using reserve growth models ala Khebab's HSM or the maturation phase DD. [1]
In keeping with this tradition, it turns out that the generalized Dispersive Discovery model fits into a classic canonical mathematical form that makes it very accessible to all sorts of additional time-series and spatial analysis. Actually the transform has existed for a very long while -- just ask the guy to the right.
Much of the basis of this formulation came from a comment discussion originally started by Vitalis and an observation by Khebab (scroll down if curious). I mentioned in the comments that the canonical end result turns into the Laplace transform 
of the underling container volume density; this becomes the aforementioned classic form familiar to many an engineer and scientist. The various densities include an exponential damping (e.g. more finds near the surface), a point value (corresponding to a seam at a finite depth), a uniform density abruptly ending at a fixed depth, and combinations of the above.
The following derivation goes through the steps in casting the dispersive discovery equations into a Laplace transform. The s variable in Laplace parlance takes the form of the reciprocal of the dispersed depth, 1/lambda.
What does the term lambda really signify? A fairly good analogy, although not perfect, comes from the dynamics of an endurance race consisting of thousands of competitors of hugely varying skill or with different handicaps. If one considers that at the start of the race, the basic extent of the mob has a fairly narrow spread, roughly equal to the distance traveled. The value of lambda over distance traveled describes this dispersion. I postulate that the dispersion of the mob increases with the average distance that the center of gravity of the mob has traveled. Overall, we empirically observe enough stragglers that the standard deviation of the dispersive spread may to first-order match this average distance. The analogy comes about when we equate the endurance racers to a large group of oil prospectors seeking oil discoveries in different regions of the world. The dispersion term lambda signifies that the same spread in skills (or conversely the difficulty in prospecting equating with certain competitors having to run through mud or while wearing cement boots) would occur in the discovery cycle just like it does in an endurance race. The more varied the difficulties that we as competitors get faced with, the greater the dispersion will become and a significant number of stragglers will always remain. The notion of stragglers then directly corresponds to the downside of a discovery profile -- we will always have discovery stragglers exploring the nooks and crannies of inaccessible parts of the world for oil.
The basic idea behind dispersive discovery assumes that we search through the probability space of container densities, and accumulate discoveries proportional to the total size searched (see the equation derivation in Figure 1) . The search depths themselves get dispersed so that values exceeding the cross-section of the container density random variable x with the largest of the search variables h getting weighted as a potential find. In terms of the math, this shows up as a conditional probability in the 3rd equation, and due to the simplification of the inner integral, it turns into a Laplace transform as shown in the 4th equation.
Figure 1: Fundamental equations describing generalized Dispersive Discovery
The fun starts when we realize that the container function f(x) becomes the target of the Laplace transform. Hence, for any f(x) that we can dream up, we can short-circuit much of the additional heavy-duty math derivation by checking first to see if we can find an entry in any of the commonly available Laplace transform tables.
In the square bracketed terms shown after the derivation, I provided a few selected transforms giving a range of shapes for the cumulative discovery function, U-bar. Remember that we still need to substitute the lambda term with a realistic time dependent form. In the case of substituting an exponential growth term for an exponentially distributed container, lambda ~ exp(kt), the first example turns directly into the legendary Logistic sigmoid function that we derived and demonstrated previously.
The second example provides some needed intuition how this all works out. A point container describes something akin to a seam of oil found at a finite depth L0 below the surface.[2] Note that it takes much longer for the dispersive search to probabilistically "reach" this quantity of oil as illustrated in the following figure. Only an infinitesimal fraction of the fast dispersive searches will reach this point initially as it takes a while for the bulk of the searches to approach the average depth of the seam. I find it fascinating how the math reveals the probability aspects so clearly while we need much hand-waving and subjective reasoning to convince a lay-person that this type of behavior could actually occur.
Figure 2: Cumulative discoveries for different container density distributions analytically calculated from their corresponding Laplace transforms. The curves as plotted assume a constant search rate. An accelerating search rate will make each of the curves more closely resemble the classic S-shaped cumulative growth curve. For an exponentially increasing average search rate, the curve in red (labeled exponential) will actually transform directly into the Logistic Sigmoid curve -- in other words, the classic Hubbert curve.
The 3rd example describes the original motivator for the Dispersive Discovery model, that of a rectangular or uniform density. I used the classical engineering unit-step impulse function u(x) to describe the rectangular density. As a sanity check, the lookup in the Laplace transform table matches exactly what I derived previously in a non-generalized form, i.e. without the benefit of the transform.
Khebab also suggests that an oil window "sweet spot" likely exists in the real world, which would correspond to a container density function somewhere in between the "seam" container and the other two examples. I suggest two alternatives that would work (and would conveniently provide straightforward analytical Laplace transforms). The first would involve a more narrow uniform distribution that would look similar to the 3rd transform. The second would use a higher order exponential, such as a gamma density that would appear similar to the 1st transform example (see table entry 2d in the Wikipedia Laplace transform table):
Interestingly, this function, under an exponentially increasing search rate will look like a Logistic sigmoid cumulative raised to the nth power, where n is the order of the gamma density! (Have any oil depletion analysts have ever empirically observed such a shape?)
The following figures represent some substantiation for the "sweet spot" theory as it plots Hubbert's original discovery versus cumulative footage chart against one possible distribution -- essentially the Laplace Transform of a Gamma of order-2.
 |
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| Figure 3: Derivative of the oil window "sweet spot" Laplace transform. |
Figure 4: Eyeball fit to Hubbert's cumulative footage data. |
The following scatter plots (Figures 5 to 8) demonstrate how we can visualize the potential discovery densities. Each one of the densities gets represented by a Monte Carlo simulation of randomized discovery locations. Each dot represents a
Figure 5: A uniform density of potential discoveries over a finite volume gives a normalized average value of 0.5. This distribution was the impetus for theoriginal Dispersive Discovery model.
Figure 6: A damped exponential density of potential discoveries over a finite volume gives a normalized average value of 0.5. When combined with an exponentially accelerating dispersive search rate, this will result in the Logistic Sigmoid curve.
Figure 7: A gamma order-5 density of potential discoveries over a finite volume narrows the spread around 0.5
Figure 8: A gamma order-10 density of potential discoveries over a finite volume further narrows the spread around 0.5. At the limit of even high orders, the density approaches that of the "seam" shown as the solid line drawn at 0.5.
I discovered an interesting side result independent of the use of any of the distributions. It turns out that the tails of the instantaneous discovery rates (i.e. the first derivative of the cumulative discovery) essentially converge to the same asymptote as shown in Figure 9. This has to do more with the much stronger dispersion effect than that of the particular container density function.
Figure 9: The set of first derivatives of the Laplace Transforms for various container density functions. Note that for larger dispersed depths (or volumes) that the tails tend to converge to a common asymptote. This implies that the backsides of the peak will generally look alike for a given accelerating search function.
In summary, using the Laplace Transform technique for analyzing the Dispersive Discovery model works in much the same way as it does in other engineering fields. It essentially provides a widely used toolbox that simplifies much of the heavy-lifting analytical work. It also provides some insight to those analysts that can think in terms of the forbidding and mind-altering reciprocal space. Indeed, if one ponders why this particular model has take this long to emerge (recall that it does derive the Hubbert Logistic model from first principles and it also explains the enigma of reserve growth exceedingly well), you can almost infer that it probably has to do with the left-field mathematical foundation it stems from. After all, I don't think that even the legendary King Hubbert contemplated that a Laplace Transform could describe peak oil ...
Footnotes
[1] I recently posted here how the Oil Shock Model gets represented
as a statistical set of "shocklets" to aid in unifying with the loglet
and HSM and DD approaches
[2] 
I use depth and volume interchangeably for describing the spatial density. Instead of using depth with a one-dimensional search space, essentially the same result applies if we consider a container volume with the search space emanating in 3 dimensions (see figure to the right). The extra 2 dimensions essentially reinforce the dispersion effects, so that the qualitative and quantitative results remain the same with the appropriate scaling effects. I fall back on the traditional "group theory" argument at this stage to avoid unnecessarily complicating the derivation.
Web References
- http://mobjectivist.blogspot.com
- http://graphoilogy.blogspot.com
- Finding Needles in a Haystack
- Application of the Dispersive Discovery Model
- The Shock Model (A Review) : Part I
- The Shock Model : Part II
- The Derivation of "Logistic-shaped" Discovery
- Solving the "Enigma" of Reserve Growth
TheOilDrum.com
Huh?
Seriously, this is what I like about TOD; it forces me to learn. Thanks for the link to the "enigma of reserve growth." That was a huge help.
For further analogies, see the next comment. The math is pretty straightforward but inverted in the sense of no one thinks in terms of rates. Once you do that, it may help in understanding.
The "enigma" arises because no one ever thought to come up with a workable model.
A couple of examples of dispersion that I have analyzed since writing this post:
The dispersion in finish times among marathon runners
http://mobjectivist.blogspot.com/2008/09/marathon-dispersion.html
The dispersion in network latencies between two interconnected nodes
http://mobjectivist.blogspot.com/2008/09/network-dispersion.html
These give a bit of context into other domains. The math is the same but the perspective differs. In all the cases the goal is to reach the finish line first.
I need to digest this more. I've just been playing around with iterated sigmoid curves and they get more pronounced with a steeper inflection; maybe that's the discovery sweet spot.
On dispersive discovery generally I could point out that Fosset's plane was outside the 'container'. Also what is happening now with oil, gas and metals is technology enabling of already discovered reserves, some of the acronyms being EOR, ISL and UCG.
With the damped exponential or gamma forms, there is no strict "container", it just becomes more and more improbable as you go along the profile.
So if you are referring to Steve Fosset, the analogy is that there is a slight probability that they could have found his plane all the way on Mars -- it is just a very, very, very slight possibility using an exponential distribution. Yet the fact that it took this long to discover it makes sense to me, as the rate is effectively very slow in exploring outside of the zone they initially started searching. As you radiate outward, you effectively go that much more slowly as you have more ground to cover.
... EOR, ISL and UCG.
Sounds good. To first order, at least some of these may be the equivalent of the long tails in the dispersive profile. They don't get reported until now so have gotten deferred from the reserve accounting for quite some time. Yes, it could be that one of these technologies could turn into a Black Swan event. Yet, in general, you should make the new enabled regions exponentially more difficult or slower to process since that is a historically accurate observation.
Very nice. Not sure I get it all, but helpful none the less. I know simple is better, but is there some way to introduce a new component (multiplier) applied over the cumulative length of hole drilled? That number would actually expand Easy Access Volume, Difficult Access Volume and More Difficult Access by virtue of improved exploration/exploitationist efficiency and product price as the mean explorer matures? Not simply shift the population betwixt and between.
Could that component be modeled by inference by curve fitting the known increased reserves in existing fields over time? That retroactive reserve growth rate does extend to improvements in techniques in new efforts as well. Sort of an "attack effectiveness" rate coupled with a change of perception as to what is an ore (economic), not just a mineral (fossil). It might better suggest behavior of the "tail".
Also do the shifting sands of political barriers alter the relative availability of the access volumes or just move forward or push out discoveries as a function of hole drilled?
To some degree, that's in there already. The multiplier could be an exponentially increasing search speed. If you add this to the exponential Laplace profile you get the classic Logistic Hubbert curve!
1/(1+1/x) => x ~ K*exp(at) => 1/(1+1/(K*exp(at))The key thing to remember is that dispersion and the average speed (aka multiplier) are distinct characteristics. Dispersion is like the variance and the multiplier is the mean.
So with an inbetween search speed we would get something between the two?
Yes, you certainly can vary the speed along the path, and this will get you something in between the formulas to the left and right.
For example, here are a couple of sets of graphs demonstrating exponential growth and power-law growth over different container distributions. The first curve shown is the classic Logistic Hubbert curve.
As you can see, the power-law growth starts off faster than exponential but then slows down relative to the exponential. This gives you a much broader peak as it takes longer to sweep through the container volume along the critical interval.
Also what I find interesting is that the tails of the Hubbert peak have nothing to do with the underlying distribution of potential finds, but everything to do with the dispersion of search rates.
"but everything to do with the dispersion of search rates"
Applying the map to the territory, does it suggest that the various dispersed searchers in the various dispersed geologically and constrained politically basins will broaden the peak but not so greatly affect the shape or area under the tail?
yes. First, if you look at all those curves, the area underneath each curve is identical. (This is not always obvious on a semi-logarithmic plot, but trust me)
The speed in the search will effect the profile like squeezing a water-filled balloon. If you slow it down one place, it will pop up in another part of the curve
The speed also has an effect on the tail but apparently only in its slope. Once the fast dispersions sweep through their search space container and the slow ones start to approach the boundaries of the container, the actual distribution of discoveries ceases to become important. It essentially amounts to picking up stragglers from all over the search space. This is basically an entropy argument, as all the details in the original distribution (seam, uniform, gamma, etc) get washed out with dispersion.
Ugh, this brought back several nightmare visions for the second time today. This morning, a guy at work who is taking Calc III, mentioned using a cross product on a test this week on vectors.
Ditto, thanks for reawakening the PTSD of Convolution Theory. Wrote the major mid term with a hangover only a 22-year old could survive and somehow managed to pass the course.
Then we go down diffy-calc memory lane - ugh! (Well basic calculus really). However, the salient point to take home from this, and I believe HubbleT, may have academically alluded to, is the analysis lies in the frequency domain - not in the time domain.
In fact, the revelation and acceptance of frequency domain as the dominant mode of analysis is considered an informal right of passage in electrical engineering. With a novice's understanding of many other technically based professions, I think this same condition holds throughout. The rookie financial traders work in the time domain (daily, quarterly, etc), while the pro's work in the frequency domain; i.e. Buffet's latest cash coup by putting cash into certain positions. He saw the cycles, not the year over year quarterly earnings.
And to echo what has been said on this thread, this is why I like TOD. Where else can you read volunteer posts of this caliber that can stand the scrutiny of any peer reviewed journal? And, sometimes a joke is thrown in for levity!
Why do you need to remind me about Buffet? I took a huge (HUGE) bath on stupid mutual funds the last two weeks, and now realize that perhaps I should have been applying my "expertise" in signal processing to personal finances instead of trying to educate people on when we might deplete the world's oil supply.
Khebab, on the other hand, is probably a stock market multi-millionaire with his skill set.
My mutuals are tanking too. Sorry about the Buffet reminder. A young engineer in our office circulated a question about opinions or knowledge regarding the retirement mutual funds a couple of weeks ago. I responded with, "Who told you you get to retire? Didn't you get the memo? Retirement will be allotted by discretion with a minimum age of 82."
They took me seriously and I just had to laugh. However the last laugh may be on me if this does in fact become truth 15 years from now.
Mine are too. Collectively (PSPFX, OEPIX) around 50% down from their peak at the beginning of July.
Daily reading the dour news on TOD, as well as seeing ever declining inventories, I figured it would rebound any day.
Certainly I am not the only one who considers our energy resources - not finance, not real estate, not gold -to be the main support on which our civilization survives.
All the other resources we have are recyclable, all the gold that has ever been on Earth is still here, same with all metals, elements, whatever. ( with very minor corrections for that launched or thermonuclearly changed ). Our hydrocarbon energy resources are the sole exception to this. There is no such thing as recycled fuel.
What I am seeing confounds me to no end. Its like one knows the well is the only source of water - yet is selling it. The well would be the LAST thing I would sell. Take my gold, take my horses, but once I lose that well and what it provides, my lifetime can be measured in hours.
I think you're on the wrong track.
You have volume/dispersive depth L-lambda but L-lambda(transform 2d) in the really represents a median/standard deviation of a variant of the standard binomial probability function a.k.a
the Poisson probability function
Pr=e^-L * L^n/n!
gives the probability of independent events like find/no find occuring with an empirical mean of L(lambda) which is also equal to . The gamma probability function F, just gives the waiting time to the nth successful event is just 1 - cummulative poisson probability and the the exponential function is the gamma function for the first successful event.
These events are 'random' and mutually exclusive. They have little to do with finite volumes or finite depths.
Bayes Theorem which partitions events by conditional probabilities(infinitely?) would tend to prove that human judgement rather than a specific physical mechanism might be behind the logistic function.
http://en.wikipedia.org/wiki/Thomas_Bayes
There is a difference between what you use to describe a probability density function (PDF) and the application you use it in.
So for example, if you keep increasing the gamma order, you can transform it into a seam delta. It makes for an understandable mutable parameter. The mean and variance for each order of the gamma are well characterized which makes it a good statistical metric as well.
And plus I don't buy into that the only way you can understand the gamma is via Poisson statistics. That is a Stat 101 point-of-view. You can also get an order N=2 by convolving 2 damped exponential PDF's. This may be the result of a probability that a layer of oil gets diffused to the surface convolved with another exponential that says it starts at a given depth.
I would like to see the math behind the Bayes/human judgement approach. If it is worth bringing up, it might be worth demonstrating.
What your saying would be on top of if you will the geologic constraints or physical constraints of the problem. The probability of rolling a 12 with a normal die is zero. Its outside of the physical constraints. What WHT is proposing is the constraint under which you could apply Bayes Theorem.
I'd say the biggest problem is that a worse distortion using the runner analog is to consider the case of a distribution of sprinters and long distance runners with a small number of super athletes if you will that can run 20 times as long as the rest.
What happens with you introduce this inhomogeneous population vs assuming equality is interesting. Other constraints such as your suggesting would act to effectively change the population. For example methods that extract oil faster could be modeled using the runner analogue as adding more sprinters. Export land can be treated as removing some of the runners etc. Bayes would be another mutation.
In a perfect world production would be and exact mirror of discovery. Because of above ground factors this is not true so we know real distortions can be caused by economic and political events. And we know from the production vs sizes of fields vs quality etc that our actual oil production is heavily weighted to a small number of fields.
So then the issue becomes how important is the base discovery curve in predicting future production even if its on a sound mathematical basis.
You can certainly add in distortions but then how do you prove you have accounted for all of them ? I'd use export land as and example its significance was discovered very late. And Bayesian is great but how to you weight ? You can quickly create any answer you want even given that your also probably correct that Bayesian stats play a role. All that really means is that underneath the covers the system is non-homogeneous or its weighted. And we know thats true.
What really happens is it seems you lose all predictability to a large extent. However almost all the probable distortions serve the reduce the total amount of oil produced and steepen the declines that you pretty sure that what ever the model you use the real answer is probably lower.
And now to go a bit further a field this is actually the result that I seem to get as you attempt to add more and more shocks and feedbacks and other effects the real solution i.e how much oil will be produced in the future goes to zero. The only thing that keeps the curve from being a simple cliff is that the distortions themselves take time to build up and interact.
Again using export land will drive total exports to zero well before we run out of oil but it has its own doubling time and growth rate.
So the real answer for future production is it will go to zero well before we run low on oil.
How soon depends on the external factors and how you model them. Thus although I suspect your right and the better model is Bayesian our ability to actually create the right weighting scheme is limited. This is actually a problem that plagues neural nets and other AI methods.
Its not the algorithms thats the problems but the inability to either quantify effects or even show that you have "thought of everything" that hobbles these sort of holistic approaches.
The human brain gets around this by simply inventing and answer then using facts and fabrications to justify it. The art of the WAG.
Sorry for this tangent but my point is that if you start down this path of trying to add more and more distortions or shocks or what ever you want to call them you end up with two problems.
To many subjective weighting decisions and and obvious trend that the solution drops to zero much faster than the base equation suggests with the error making it harder and harder to predict when. Better maybe to be simple and wrong but realize it.
I also think a very interesting case is the one where no dispersion exists at all. This I believe happens in a few special cases. That of extinction, see the famous figure =>
And what happens in gold rush towns, where everyone uses the same technology, the number of prospectors accelerate quickly, and the collapse occurs suddenly. People jump out of this situation the minute they see the peak hit the cliff edge. Unfortunately, with dispersion in technology and geography, this isn't always possible to detect, and we get into the situation akin to the frog being slowly boiled alive. We don't know what is happening until it is much too late, and get lulled into a false sense of security by the slowly uncovered new discoveries. Gold rush towns know immediately when they hit the boundary, yet we as oil consumers don't and keep on saying "Drill, Baby, Drill", not knowing what diminishing returns lay ahead.
I call this "The Curse of Dispersion"
http://mobjectivist.blogspot.com/2008/09/curse-of-dispersion.html
Thanks WHT. I sleep much better at night knowing that there are people like you amongst us who have a far greater intellectual capacity than I can aspire to.
It reminds me that as much as we think we understand here how the oil gets from the well to our tank, there is so much more complexity, which is so remote from the consumers experience, that is built into the system that keeps the whole industrial-consumer economy going.
While I didn't understand anything much of what you said after "I find it interesting..." what I did take away from it is that we focus on a peak of production that is built on a very broad and complex production system that is made up of thousands of much smaller components. All that we can ever do is look at small samples of those components and try to get them to reveal their secrets through mathematical modelling.
Perhaps if there was a healthy dose of energy modelling put into financial modelling, we wouldn't have the mess and panic that is happening today.
Perhaps if there was a healthy dose of energy modelling put into financial modelling, we wouldn't have the mess and panic that is happening today.
A bit harder, but indeed just as vital. I think what makes it more difficult is the number of feedback loops built into the financial system.
O.k., after reading the comments by the mathematicians in the group, I am convinced the Tower of Babel was really the Mathematics depart of a Babylonian university.
I believe that even HL was mathematical overkill. All this much more so. I believe that what can be predicted about peaking, can be predicted using a simple graph. It will be somewhat wrong, but no more wrong than something more sophisticated. There are areas where mathematics is very powerful, and areas where it is totally useless. This is in between. MRoMI is low here.
Excellent, well said. This is verifiable as well. A few hours researching stories in Bloomberg about oil producing nations and a few more hours looking at the numbers on oil production will give you as good an idea about what to expect from production as all these models and formulas.
I disagree actually see my long drawn out post.
I think WHT is 100% right. And it serves as a baseline to work from. The time shifted discovery curve is for sure the best case model. We cannot pump oil we have not discovered. Next his methodology is spot on in my opinion.
You will not get this out of looking at production since production is not a predictor of discovery.
However the problem you run into is fairly simple. Lets take our economy as and example its based on the concept of infinite growth as long as we continued to increase our resources it was a pretty sure bet we would continue to grow. Thus the discovery curve is a excellent predictor for production during the growth phase. For the most part this is int the past.
However it becomes and upper bound for long term production. And looking at oil production or bottom up approaches become short term because they don't take into account depletion.
The problem is that when complex systems begin to decline it seems at least as far as I can tell we don't have the mathematical tools to handle them. Whats interesting is I ran into a similar problem when I was working on chaos theory. We have basically zero tools to predict the dynamics of chaotic systems. So it turns out that real post peak production is a similar problem. Or global economy is actually entering into this contraction phase for example and the dance if you will between global contraction and oil production is as far as I can tell unpredictable because of the nature of the system itself at least with any tools I'm aware of.
This failure of mathematics or at least the mathematics we know is what I find fascinating.
I've gotten interested in three problems chaos, wave mechanics, and complex systems. All three end in a sort of complexity breakdown and it becomes impossible to use math as a predictive tool that guides experiments we are back it seems to the place where experiment will have to lead math. Given that one of the problems is how our society will evolve over the near future this is not comforting.
Neither approach takes into account external factors or distortions thus both will be very poor predictors of future production. And worse both approaches are optimistic.
But you can't begin to get this out of looking at production the models play and important role esp the discovery model since any real model for production has to be based of a good understanding of discovery. Heuristics like HL are powerful but are not based on what we call first principals. Given a model built on a firm foundation one expects that it will produce a result that is similar to HL but HL in and of itself is not a "real" model of oil production.
And finally the real problem at hand is oil production over the next 5-10 years. Most of our models show that decline will be obvious within ten years and taking into account export land export decline will almost certainly be obvious within 5 years. This means that from five years into the future forward the world will be forced to deal with peak oil and what ever the real decline rate is. If we are lucky the models we have developed will be accurate enough to help policy makers decide how to deal with peak oil.
However its the next five years that are going to be the interesting part to say the least and in my opinion nothing I've seen is capable of predicting when and how we slide down the peak oil slope.
However given the above base models some exploration into extraction rates and recently some research into the financial problems and the impact they will have on oil production we do have some important results. One of them is that we will not see all the production come back online in the GOM after the hurricanes. Exactly how much depends on the details.
Your not going to get a prediction like this looking at production numbers and I've not seen it made any where else but the oildrum. And it would have been impossible to make without basic models like what WHT works on.
This failure of mathematics or at least the mathematics we know is what I find fascinating.
This is deep, Memmel. If this is the Holy Grail that you have been seeking out all this time, you have a kindred spirit here.
Mathematics allows us to probe into areas that our pea-brains struggle to comprehend. And when the math starts breaking down, I figure that we are completely lost in the wilderness. So, if we do enter into an oil/financial "collapse", and we have no understanding of what is going on, predictions will become purely arbitrary.
First, I just want to thank both you and the TOD editors. You for having put together this post and the TOD editors for being willing to host it and expose us all to alternate perspectives.
With regard to the remarks by both you and Memmel on collapse, would it be fair to restate this as the occurrence of a set of black swan events which take place on the boundary, or outside of, current knowledge domains?
The second aspect of this collapse would appear to be that it occurs too quickly for us to gain insight and understanding of it. In essence, real world events "speed up" and the dynamic exceeds our ability to interpret, rationalize, and react to it. Would you concur with that observation?
Tainter appears to provide the current accepted interpretation of civilizational collapse. I am not comfortable with Tainter but have yet to devote the time required for a through critique (the collapse of the Roman Empire was a positive event. Had it not collapsed our own civilization may not have had the opportunity to occur and Londinium would still be a small peripheral market town). Do you have any comment on your insights into collapse and those of Tainter?
Apologies for moving the goalposts on the discussion. It's what one does when one is mathematically challenged :-)
One telling feature that I think is vitally important to this discussion is that the search speed in the Dispersive Discovery model has to continue to exponentially increase to match the Logistic Hubbert curve profile. More directly, it has to exponentially increase on the downslope of the Logistic. As many a number-cruncher knows, when you take a large number and multiply it by a small number (the search spots remaining), the result is very sensitive to values in each if it isn't compensated by a constant additive factor.
You gave me a good idea. I will try to look at the noise characteristics on each side of the Logistic Dispersive Discovery curve after placing fluctuations on the search speed. This will likely be very revealing, and it may tell us about chaotic effects on the backside that Memmel is hinting at.
Thanks Web. I've been wandering the wilderness so to speak for a while seeing if we could come up with something tangible to convince people we are facing a serious problem.
The result is that mathematics seems to ensure that the crystal ball so to speak is cloudly.
We can do what your doing with confidence but as we step outside of this like you have to some extent with your shock model we suffer parameter explosion but worse a massive increase in uncertainty. As the complexity increases we seems to be forced into having to have better and better measurements. This leads eventually to the quantum limit. This opens up the reverse question which is is quantum mechanics itself the result of some underlying complexity ?
Everything I've seen about quantum gravity hints at a complexity explosion also happening in the small.
However
And I'll use this as and example.
http://www.iop.org/EJ/article/1538-3881/118/3/1177/990128.text.html
Regular orbits are dense in phase space. Indeed semi-classical approaches use the fact that quantum probability densities are highest around normal orbits to derive quantum analogs of complex systems.
Again we can reverse the problem and ask ourselves if this is actually a property of complexity and not just a property of quantum mechanics ? So we make the assertion that complex systems exhibit this property and even though the complexity seems to hide the right answer whats really happening is the system is concentrating around one of its stable regions or orbits.
In fact we have knowledge of complex systems that indicate that they indeed are capable of picking out stability.
http://en.wikipedia.org/wiki/Stochastic_resonance
This leads to a conjecture that complex systems tend to act as some sort of signal filter/generator etc they begin to calculate something.
Suddenly the problem at least for me becomes obvious as systems become more complex they start creating their own equations and laws if you will out of the basic laws of physics. You run into the problem of a turing machine trying to deduce the state of another turing machine.
Small wonder your seeing signal processing concepts start to emerge. Signal processing is simply a symptom of a calculation taking place.
Now its no wonder that Bayesian concepts can work and also that fairly simple dispersive theory works. In the particular case of discovery it seems that the complex system is working to determine a regular equation.
And I'll end this with and observation Wolfram was throughly discredited with his finite automata approach and in my opinion for good reason since he was unable to show how it worked.
But in my opinion he was both right and wrong. He was right in that complex systems exhibit the ability to calculate he was wrong in restricting the model to just finite automata.
The right way is the approach your taking and starting with signal processing concepts which can generate both regular and simple results and also effectively opaque engines or signal processors which work in non obvious ways.
And back to the top we know from both chaos theory and quantum mechanics that simple regular solutions are both dense and preferred.
The invention of classical physics was in itself dependent on this as is your result.
For complex systems the problem is we don't know the virtual coordinates if you will that the system begins to operate under. It works the same way but it synthesis its own coordinate space and the regular orbits are expressed in this unknown coordinate system. This can be seen with empirical methods such as HL we don't actually know what the "real" coordinates or laws are which underly the success of the empirical method. We don't know what its mixing together and what equations are actually in use. Its opaque and impossible to derive backwards if you will from HL to the real underlying dynamics.
This paper does a really good job of exposing the deus ex machina.
On a bigger scale we can be safe in assuming that history is and excellent guide to the evolution of our society as oil runs low. Left undisturbed the equation will almost certainly settle into one of the known modes common when societies collapse. We can given the above assert this with a very high confidence interval. And we also know from history that almost all of these modes are considered undesirable by the majority.
And last but not least we can even explain why we can't predict the future easily. I.e we cannot prove bad things will happen or the exact course the system will take before it stabilizes around one of the possible outcomes.
We can show that the regular solutions are all bad however.
I think that at some point in the future assuming we keep our technical civilization we will develop methods that allow us to detect the fingerprint if you will of complex systems and deduce what they are calculating. In fact signal processing itself ends with image recognition as its limit. Eventually we will have the math skills to see the image thats being generated by complex systems and thus quickly put them into a category.
Whats really funny is this is the reason I left graduate school I wanted to approach chaos theory from the approach of doing image processing and recognition or signal processing. In short I wanted to develop a way to see chaos.
I knew it was a signal processing problem and twenty years later you offer proof.
As and example Nate Hagens posted a funny correlation between music quality and I believe oil supply. I forget since it was just a funny graph. However the point is complex system always form patterns no matter how stupid its intrinsic. The formation of patterns is a signal and a sign of some sort of processing. Most pseudo science is actually based on playing tricks with this property of complex systems so real scientist tend to dismiss it. Its been hiding in plain sight for a long time. The properties are intrinsic. At one extreme we have these nonsense variables that people use to "prove" all kinds of stupid stuff. And at the other we have the classic equations. In between we have a grey area thats not been explored.
These are sensible and convoluted mixing of real variables in a complex manner.
Anyway good luck finding the signal.
And as I finish this it looks like our fate is sealed just as we begin to understand the problems we face it looks like the world has chosen the wrong answer.