The Seventh Fold

Disclaimer

I am a firm believer in peak oil theory, and there is compelling evidence that we have already reached a jagged plateau which may (or may not) prove to be the ultimate peak. But because oil plays so many important roles in the global economy, oil production forecasting models deserve critical evaluation. As you read this post, take care not to confuse the model (Hubbert linearization) with the theory. While all forecast techniques are flawed, pointing out these shortcomings does not discredit the theory of peak oil.

Preamble

Oil literally and figuratively fuels the global economy. Without abundant oil resources, the global fleets of ships, planes, trucks, cars, and trains would slow to a crawl, and so too would trade in goods and services. Market areas would necessarily shrink and economies of scale would evaporate. Without access to goods and services produced in distant markets, regions would be forced to diversify their economic base or simply do without, and the benefits to regional specialization would disappear.

Oil is, quite simply, the master resource. Yet we know oil to be finite and non-growing, and we can therefore assert without reservation or qualification that production rates cannot grow forever or even be sustained at current levels indefinitely. Rather, production rates “will rise, pass through one or several maxima, and then decline asymptotically to zero.” (Hubbert, 1959)

Given the importance of oil to the global economy, four questions are of immediate concern.

1. How much oil is left?
2. How fast can it be produced?
3. How close are we to reaching the ultimate peak?
4. After reaching the peak, how fast will production subsequently decline?

Perhaps a fifth question is of equal importance. Will we know when we’ve reached the ultimate peak, and if so, how?

Scientists, engineers, and economists have for decades sought the answers to these questions and have created parsimonious models to support their claims.

Much to their collective chagrin, however, a reliable model of medium or long term global oil production remains an elusive target because oil production is influenced by diverse, complex, and interrelated forces – from the geophysical to the economic and from the political to the technological.

Rather than attempting to simulate all the complex contingencies which characterize the real world, the preferred solution has been to abstract and distill simple models from what is a much more complex reality. Using Okham’s metaphorical razor, all non-essential forces which act on oil production are trimmed away from the analyses.

This process, however, opens the door to criticism that rather than pairing away the inconsequential, these models in fact leave critical and influential factors on the cutting room floor. As all forecast models abstract from the essential, none provide definitive answers to such important questions as, “When will production peak?” and “How fast will production decline?”

While this is an important and valid criticism, by no means does it invalidate the theory of peak oil. It is a mistake to conflate or confuse production models with peak oil theory. Regardless of whether or not we accurately predict the year when production will peak or the precise rate of the subsequent decline, oil production will at some point peak and decline.

At the same time these models should not be abandoned because they provide valuable insights into the forces which act upon production. Hence, it is in our best interest to understand the strengths and limitations of production models, and there is no better place to start than with the most widely known and accessible technique: Hubbert Linearization.

Hubbert Linearization

The term Hubbert Linearization (HL) was coined by Dr. Stuart Staniford. It is in fact, an accurate description of the methodology, which is a simplified linearization technique applied to Dr. M. King Hubbert’s original methodology which he used in 1956 to correctly predict that U.S. oil production would peak in 1970. This particular method could just have accurately been called Deffeyes Linearization after Dr. Kenneth Deffeyes, who transformed pages and pages of Hubbert’s differential equations into a much simplified, three-step procedure.

Whereas Hubbert’s original methodology requires an exogenous estimate of ultimately recoverable reserves (URR), in Deffeyes’ reformulation, the model itself predicts not only yearly production rates, but the URR as well. Despite this difference and the fact that the Hubbert linearization technique represents a greatly simplified version of Hubbert’s own work, Deffeyes assures his followers that his model is “exactly identical to Hubbert’s… [it] does not cheat, nothing is left out, and is the full Hubbert methodology.” (Deffeyes, p. 36)

The Three Steps of Hubbert Linearization

1) The first step in Hubbert linearization is to set up a simple linear regression model where cumulative production (Q) is the independent variable and the ratio of yearly production to cumulative production (P/Q) is the dependent variable.

2) In the second step, yearly production (P) and the ratio of yearly production to cumulative production (P/Q) are substituted for the ‘x’ and ‘y’ terms in the equation which defines the regression line, and the equation is re-solved for ‘P’. The result is a logistic function, which when graphed produces a symmetrical, bell-shaped curve with a peak at the midpoint which is when exactly half of all recoverable oil has been produced.

3) The final step in the Hubbert Linearization technique is to backcast production and compare predicted values to observed production rates in order to verify the model.

Critique

As we can see, Deffeyes’ model is driven by two (related) variables: yearly production (P) and cumulative production to date (Q). Deffeyes asserts that, “our ability to produce [oil] is linearly dependent on the fraction of oil that remains.” (Deffeyes, p. 39) And that’s it. No factor other than the amount of recoverable oil that has yet to be produced significantly influences oil production. Politics, economics, technical innovations, natural disasters, and wars are inconsequential. As Deffeyes puts it, “What infuriates Cornucopians is Hubbert’s implication that nothing else matters, only the undiscovered fraction.” (Deffeyes, p. 39)

The key to Hubbert Linearization – and hence the key to Hubbert’s original model – hinges on whether the data literally fall into line when graphed. Perhaps realizing this limitation, Deffeyes preemptively warns his readers about this first step. As Deffeyes puts it, “I warn you, when you let me draw that straight line, you will have bought into the whole Hubbert story lock, stock, and barrels.” (Deffeyes, p. 36)

While Deffeyes is content with a visual examination of the data, the gravity of the matter calls for more rigorous tests of linearity (not to mention normality and homoscedasticity). Upon passing these test, we may indeed conclude that “our ability to produce oil depends entirely, and linearly, on the unproduced fraction.” (Deffeyes, p. 39) But, if the data used in the regression do not meet these criteria, we must conclude that the results are not generalizable.

Turning to Deffeyes’ scatterplot of cumulative production in the U.S. (Q) to the ratio of yearly production to cumulative production (P/Q) we see quite a bit of what Deffeyes refers to as ‘noise’ in the data associated with the early years of the ‘oil age’. This noise diminishes over time, and looking only at the last few decades of production, it does appear that the relationship is linear (at least so long as the range of values on the y-axis are ‘zoomed out’ which effectively minimizes the appearance of variance).

By allowing Deffeyes to draw a line of best fit which excludes data points before 1958 (in the U.S. case) and 1983 (in the world case), we are tacitly accepting that there is either a qualitative difference between the data before and after those dates, or that the early data points are unsystematic outliers. These are the only defensible reasons for ignoring 100+ years of data.

There may or may not be a qualitative difference between the data which is excluded and the data which is considered for the regression, but if there is a qualitative difference, Deffeyes does not does not let his readers in on the secret. Rather, he explains that the ‘noise’ experienced in the early decades is due to the fact that in the early years, “cumulative production (in the denominator) was low.” (Deffeyes, p. 36)

Expanding on this facile explanation, we may conclude that because the denominator grows more rapidly than the numerator, any volatility in the numerator (yearly production) is magnified in the early years of the oil age, and de-emphasized in later years when the denominator has grown orders of magnitude larger. Hence we are more likely to see volatility (noise) in the earlier data points.

There are two essential problems with this justification for the exclusion of earlier data. First, these are not random outliers. They are, instead, the systematic result of the measurement technique. They result from the fact that yearly production – the subject of interest – dominates the P/Q ratio in the early years of the oil age, while cumulative production dominates the ratio in the later years.

Second, even when the so-called noise is taken out, the relationship still appears to be curvilinear – the slope of the line is not constant, but rather decreases at a decreasing rate. Hence the assumption of linearity appears (at least to me) to have been violated even when only the most recent data is included in the model.

If Hubbert linearization is a valid technique for forecasting future production, it should work equally well when all data is considered. And before we allow 100+ years of production data to be deemed inconsequential and excluded from the analysis, we must insist on a robust explanation of why our ability to produce oil was not linearly dependent on the fraction of oil yet to be produced during this earlier period in time.

Even if the exclusion of this data is justified, the data considered for the regression should be tested to ensure that it meets assumptions of linearity, normality, and homoscedasticity. If any of these three assumptions are not met, the results of the regression model are not generalizable.

Thought Experiment: Hubbert Linearization Applied to a Renewable Resource

Before testing the oil production data, let’s take a look at two hypothetical scenarios which highlight some important shortcomings of the Hubbert linearization technique. In the first scenario, we will examine the case where a renewable resource is extracted at a constant and sustainable rate. In the second scenario, extraction of the same renewable resource starts at a level far below the maximum sustainable rate of extraction (MSR), and grows at a modest and variable annual rate until reaching the MSR.

Scenario 1: Trees Harvested at a Constant Rate

The nice thing about hypothetical examples is that we can choose to work with round numbers. Imagine that trees are harvested from a large plot of land. Before harvesting starts, the timber company that owns the land determines that the renewable rate of extraction is 10,000 trees per year. Let’s imagine that the company has all of the equipment required to produce 10,000 trees during their first year of operation.

In the first year of production, then, the timber company harvests 10,000 trees, and cumulative production is also 10,000. In the second year, 10,000 trees are again harvested, and cumulative production climbs to 20,000. In the third year, 10,000 trees are again harvested, and cumulative production climbs to 30,000… and on and on we go. Eventually, this constant rate of sustainable extraction will produce the graph below. Note that the relationship is not linear. The slope is not constant but rather starts off quite steep, and it decreases at a decreasing rate. Also note that when viewed at this scale, the later data appear to be linear.

Now if we were to apply the Hubbert linearization technique to this hypothetical model of timber production, we would begin by arbitrarily picking a critical date and discarding all earlier observations. In this example, only the last 30 years of timber production are included in the regression analysis. Using only the 30 most recent observations, a straight regression line can be drawn, and the coefficient of determination is very high (R² = 0.99). Note in the scatterplot below that when the regression line is drawn, and the y-axis is re-scaled, that the data no longer appear to be linear.

The high coefficient of determination is presented as evidence that the straight line is a good fit – that it has high predictive capabilities. And while the regression line predicts most of the variance in the observations, we know that the linear function will systematically under-predict future production rates, and the error will grow with time because the data is non-linear.

One simple test of linearity is to compare the coefficient of determination (r-squared) from the linear regression to the coefficients of determination associated with logistic, exponential, and power functions (all of which are curvilinear functions).

Even when only the last 30 years of data are considered, we see that both the logistic and exponential regressions produce higher coefficients of determination. And when we use a power function, the coefficient of determination grows to 1.0, meaning that the model predicts observations with 100% accuracy. The fact that higher coefficients of determination are associated with curvilinear regression lines is strong evidence that the relationship is non-linear.

A second, more sophisticated test of linearity is accomplished through the visual examination of a residuals plot. In this case, predicted values are arranged in rising order on the x-axis, and standardized residuals – the difference between observed values and predicted values – are on the y-axis. The residuals measure the error between observed and predicted values. If the relationship is linear, the residuals will not follow a pattern.

In cases where the residual is equal to zero, we know that the model accurately predicted the dependent variable. In cases where the model over-predicted the dependent variable, the residual will be negative, and in cases where the model under-predicted the dependent variable, the residual will be positive.

When examining a residuals plot, nonlinearity is indicated by a curved pattern to the points (Norusis, 1998). In the residuals plot below it is clear that the data are non-linear… despite the fact that the linear regression is highly predictive.

At this point, the HL analysis should be abandoned because the data fail the test of linearity, but there is value in thinking through the final step: model verification through backcasting.

Let’s ignore this issue for a moment and continue on with the Hubbert linearization of timber production. According to the HL model, total cumulative timber production when the last tree is felled will equal 2.39 million. This is the value of x when y equals zero (i.e. when the last tree is chopped down). This value is also equal to the area under the production curve.

Despite the fact that production can be sustained at the maximum sustainable rate indefinitely, the Hubbert linearization technique insists that cumulative production has an upper bound. Again, this inaccurate prediction of cumulative production is rooted in the non-linearity of the data. If we kept plugging 10,000 trees per year into the spreadsheet, we would see that the curve would asymptotically approach, but never reach zero (as the earlier scatterplot suggests).

In order to test the model, the model is used to backcast historical production. And Hubberterians insist that if the model ‘fits’ then it is in fact robust. But when Q is substituted for x and P/Q for y in the regression line and the function describing the regression line is rearranged and solved for P, a bell-shaped logistic curve is produced. This points to an obvious problem: production was constant and sustainable (10,000 trees per year), yet the model suggests that production grew, reached a peak, then declined to zero.

Failing this test of model verification, we must conclude that the Hubbert linearization technique is not applicable to this scenario. The model should be abandoned, but not because the regression line had a low coefficient of determination (it was very high), and not because the data was shown to be non-linear (linearity tests are not explicitly stated as a step in the HL technique) – but because the model fails to correctly backcast historical production rates.

Scenario 2: Trees harvested at an increasing rate until reaching the maximum sustainable rate of extraction

It is possible, however, to create a hypothetical scenario in which production of a renewable resource can be modeled using the Hubbert linearization technique, and still ace the backcasting test despite the fact that production can in fact be sustained indefinitely.

In this alternative scenario, the primary parameter remains unchanged: the maximum sustainable rate of harvest is 10,000 trees per year. Instead of assuming that the maximum sustainable rate of harvest is reached in the first year, however, let us instead assume that production starts slowly (320 trees in the first year) and grows at a declining and variable compound rate until reaching the MSR. In this example the average rate of expansion is 3.5% per year. Upon reaching the MSR, production ceases to grow, but does not decline.

Just as in the previous example, the scatterplot of cumulative production (Q) to the ratio of yearly production to cumulative production (P/Q) eventually flattens enough that a regression line can be drawn through the later data points such that the data almost appear to be linear. Again, the data appear to be linear because the denominator has grown so large that the ratio P/Q barely changes year over year (though zooming in on the last 20 years of production and rescaling the y-axis makes the curvature of the data more obvious).

We might even be so bold as to claim that the harvest rate of trees depends entirely, and linearly, on the unproduced fraction. We may draw a regression line and claim that URR is equal to the x-intercept. We might go so far as to rearrange the variables into the form of a logistic equation, identify a date for the peak, and the rate of subsequent decline.

In this second scenario, backcasting would appear to validate the model. In the graph below, we see that through 2010, the model does a very good job of predicting historical production (though there is some systematic bias). If we didn’t know that production was going to level off at 10,000 trees per year in 2011, we might find this model to be quite persuasive. Of course, we know that timber production could be sustained indefinitely at 10,000 trees per year, so when we look at the forecast, we see the truth in the errors!

So what are we to make of the Hubbert linearization technique now that we’ve seen its vulnerabilities? More precisely, why did the model lead us to conclude that tree production would peak and decline when in fact tree harvesting can continue indefinitely at the MSR? The answer is that the relationship between cumulative production and the ratio of yearly production to cumulative production is non-linear.

Back to the oil patch

Turning back to the oil production data, we see that over time, the volatility of production declines, but even more importantly, the total quantity of oil produced to date grows at a rate which far outpaces growth in yearly production.

Mathematically, the growth of the denominator minimizes the influence of the numerator (production) in the ratio of P/Q. As a consequence, the dependent variable (P/Q) becomes much more predictable over time. But – and this is important - the increase in predictability has less to do with the predictability of production (the numerator – P) than it does with the magnitude of cumulative production (the denominator – Q).

A scatterplot of cumulative production (Q) and the ratio of production to cumulative production (P/Q), shows what appears to be a non-linear relationship between the two variables, similar to what we saw in the two hypothetical timber harvest scenarios.

Deffeyes conveniently excludes earlier data, building the regression using only post-1983 data. To the naked eye, it appears that these data points fall in line, and it is for this reason that Deffeyes (incorrectly) concludes that production “is linearly dependent on the fraction of oil that remains.” (Deffeyes, p. 39) But when inferring causality, it is not possible to conclude with certainty that the predictability of P/Q is due to the predictability of production, and it is rather a stretch to understand why this might be the case from a purely theoretical perspective.

It seems just as plausible that the strength of the relationship results from the fact that cumulative production is being used to predict a ratio in which cumulative production dominates the value. After all, if Deffeyes’ own argument holds water – that early volatility was due to cumulative production being low – would it not follow that the lack of volatility in more recent data is caused by cumulative production being high?

This is a serious question that deserves careful consideration. Deffeyes implies that the earlier data can be excluded on the grounds that volatility is due to the denominator being low, but we are only justified in excluding early observations if there is nothing systematic about them. There is, in fact, a systematic bias which is inherent in the calculation of the variable itself, and excluding systematic (non-random) outliers compromises the model and the generalizability of the results.

Moving beyond the issue of unfounded data exclusion, we must still ask whether the relationship between Q and P/Q is, in fact, linear. While a regression line with a high coefficient of determination can be derived from the data points, this alone is not sufficient evidence of linearity.

Using yearly production data from the BP Annual Statistical Review (2010), we can reproduce Deffeyes’ analysis and test the data to see whether it meets the assumption of linearity (a requirement for conducting a linear regression). The BP data only extends back to 1965, and at the time of writing, the most recent data was from 2009.

Granting Deffeyes the benefit of the doubt, I will follow his lead and exclude all data points before 1983. While Deffeyes’ data set only extends to 2003, the following analysis includes data through 2009. If Deffeyes’ assertion that production depends linearly on the proportion yet to be produced, I see no reason to exclude these later data points from the analysis. If anything Deffeyes should welcome their inclusion as more data should add further support in favor of his claims.

In the scatterplot below, we see that the coefficient of correlation derived from a linear regression is high. In fact, we can say that over 95% of the variation in the P/Q ratio (the dependent variable) is explained by the variation in cumulative production (the independent variable).

Unfortunately for Deffeyes – and for proponents of Hubbert linearization more generally – there is ample evidence indicating that the data is not linear. When an exponential regression line is drawn the coefficient of determination increases to 97.6%, and a logarithmic regression line fits the data even more precisely (R² = .986). If the relationship was linear, then curvilinear regressions would not fit as well as a linear regression, but the opposite is in fact the case.

Even stronger evidence is seen in the plot of standardized residuals. Recall that when the relationship is linear, the points cluster around zero. In the residuals plot below, the residuals are in fact organized systematically. Turning to a residuals table organized in chronological order, we see that the linear regression model consistently under-predicts all pre-1990 and post-2004 observations while consistently over-predicting all observations between 1991 and 2003. This is exactly the pattern that we would expect to see when examining a non-linear relationship.

Because the relationship is non-linear – that it is best represented by an upwardly concave power function – we should in fact expect the URR to be greater, perhaps much greater, than the linear regression would suggest. Even more importantly, because the relationship is not well represented by a linear function, the bell-shaped logistic curve that results from substituting variables and rearranging the linear equation will not likely reflect reality. After all, there is no way to derive a logistic function from the actual line of best fit (y=385994x^0.5935).

So what are we to think? Some so-labeled ‘Cornucopians’ argue that the problem with the model is that it grossly underestimates the URR (ultimately recoverable reserves). They assert that new technologies and the rising price of crude will bring previously unrecoverable reserves to production. They are correct. This will undoubtedly be the case. But, this underestimation of URR does not necessarily mean that the predicted date of peak production is in err. It is quite possible that the predicted date and magnitude of the peak is ‘right for the wrong reasons’, but that the decline rate is largely overestimated. If this ends up being the case, it will be welcome news! Alternatively, the predicted date of the peak may be early. BUT… and this is important… The further the peak is pushed into the future, the steeper will be the rate of decline!

In a proactive defense against this ‘Cornucopian’ logic, Deffeyes argues that “improved technologies and incentives have been appearing all along, and there seems to be no abrupt dramatic improvement that will put an immediate bend in the straight line.” (Deffeyes, p. 39) To this, I say: 1) there does not need to be an abrupt jump in observed yearly production for the impact on the rate of future production to be profound, and 2) there was in fact a fairly abrupt bump between 2002 and 2004.

But even more importantly, the relationship between the independent and dependent variables is not linear – not even when only recent production is considered. While the ‘Cornucopians’ argument that rising prices and emerging technologies will bring more oil into production, the problem with Hubbert linearization is far more fundamental than such an attack on the assumptions implies. The problem is that cumulative production – the independent variable – comes to dominate the ratio of yearly production to cumulative production such that only a very specific set of ephemeral circumstances would ever result in the relationship being linear.

Does this critique mean that production will not follow a bell-shaped curve? Not necessarily. My interpretation is that the systematic under-estimation will result in a bell-shaped curve with a longer than predicted plateau and a fatter than predicted tail. Of course this can only be the case if the peak occurs before half of the world’s recoverable oil is produced.

Does this critique of Hubbert linearization mean that the predicted date of peak and the level of production at the time of peak are necessarily wrong? No. It is quite possible that production will peak sometime between 2005 and 2015 (which is roughly the range of values that various Hubbert linearization techniques predict). I believe that observed production history suggests that the world in fact reached a plateau in 2005, so perhaps the predictions are correct. But if they are correct, though, they are correct for the wrong reasons.

Does this critique mean that peak oil theory has been debunked? Absolutely not. Oil is a finite and essentially non-growing resource, therefore production rates will grow, reach a peak or plateau, and decline to zero, and it is likely that the ride will be quite a bit more bumpy than the logistic function implies.