Hubbert linearization

The Hubbert linearization is a way to plot production data to estimate two important parameters of a Hubbert curve, the approximated production rate of a nonrenewable resource following a logistic distribution: * the logistic growth rate and * the quantity of the resource that will be ultimately recovered. The linearization technique was introduced by Marion King Hubbert in his 1982 review paper. The Hubbert curve is the first derivative of a logistic function, which has been used for modeling the depletion of crude oil in particular, the depletion of finite mineral resources in general and also population growth patterns.

The Hubbert linearization is a way to plot production data to estimate two important parameters of a Hubbert curve, the approximated production rate of a nonrenewable resource following a logistic distribution: the logistic growth rate and the quantity of the resource that will be ultimately recovered. The linearization technique was introduced by Marion King Hubbert in his 1982 review paper. The Hubbert curve is the first derivative of a logistic function, which has been used for modeling the depletion of crude oil in particular, the depletion of finite mineral resources in general and also population growth patterns. The Norwegian Hubbert linearization estimates an URR = 30 Gb and a logistic growth rate of k = 17%. Image:HubbertLin_Norway.svg|Hubbert Linearization on Norway's oil production Image:Hubbert_Norway.svg|Hubbert curve on Norway's oil production ==Alternative techniques== ===Second Hubbert linearization=== The Hubbert linearization principle can be extended to the first derivatives of the production rate by computing the derivative of (2): The left term, the rate of change of production per current production, is often called the decline rate. The decline curve is a line that starts at +k, crosses zero at URR/2 and ends at −k. Thus, we can derive the Hubbert curve parameters: The growth parameter k is the intercept of the vertical axis. The URR value is twice the intercept of the horizontal axis. ===Hubbert parabola=== This representation was proposed by Roberto Canogar and applied to the oil depletion problem. It is equation (2) multiplied by Q. The parabola starts from the origin (0,0) and passes through (URR,0). Data points until t are used by the least squares fitting method to find an estimate for URR. ===Logit transform=== David Rutledge applied the logit transform for the analysis of coal production data, which often has a worse signal-to-noise ratio than the production data for hydrocarbons. The integrative nature of cumulation serves as a low pass, filtering noise effects. The production data is fitted to the logistic curve after transformation using e(t) as normalized exhaustion parameter going from 0 to 1. The value of URR is varied so that the linearized logit gives a best fit with a maximal coefficient of determination R^2 . ==External links== Robert Rapier: Does the Hubbert Linearization Ever Work?, The Oil Drum, 2007-03-22 David Rutledge: Energy Supplies and Climate, Caltech, 2019 - on curve fits to the production history (including excel data on historic coal production and logit fits)

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