Logistic Curve

In this paper we consider the parameter estimation (PE) problem for the logistic function-model in case when it is not possible to measure its values. We show that the PE problem for the logistic function can be reduced to the PE problem for its derivative known as the Hubbert function. Our proposed method is based on finite differences and the total least squares method.
Given the data (pi, ti, yi), i = 1, …, m, m > 3, we give necessary and sufficient conditions which guarantee the existence of the total least squares estimate of parameters for the Hubbert function, suggest a choice of a good initial approximation and give some numerical examples.

Published in: Proceedings of the Conference on Applied Mathematics and Scientific Computing 2005, Part II, Pages 217-234
Available from: SpringerLink

A quantitative analytical method, using a spreadsheet, has been developed which allows the determination of values of the three parameters that characterize the Hubbert-style Gaussian error curve that best fits the conventional oil production data both for the U.S. and the world. The three parameters are: the total area under the Gaussian which represents the estimated ultimate (oil) recovery ( EUR), the date of the maximum of the curve, and the half-width of the curve. The "best fit" is determined by adjusting the values of the three parameters to minimize the root-mean-square deviation ( RMSD ) between the data and the Gaussian. The sensitivity of the fit to changes in values of the parameters is indicated by an exploration of the rate at which the RMSD increases as values of the three parameters are varied from the values that give the best fit. The results of the analysis are:

1) the size of the U.S. estimated ultimate recovery ( EUR ) of oil is suggested to be 2.22 x 1011 barrels (0.222 trillion bbl) of which approximately three-fourths appears to have been produced through 1995;

2) if the world EUR is 2.0 x 1012 bbl, (2.0 trillion bbl) a little less than half of this oil has been produced through 1995, and the maximum of world oil production is indicated to be in 2004;

3) each increase of one billion barrels in the size of the world EUR beyond the value of 2.0 x 1012 bbl can be expected to result in a delay of approximately 5.5 days in the date of maximum production;

4) alternate production scenarios are presented for EURs of 3.0 and 4.0 trillion bbl.

Published in: Mathematical Geology, Volume 32, Issue 1
Available from: Al Bartlett.org or
Hubbertpeak.com