Abscissa-Transforming Second-Order Polynomial Functions to Approximate the Unknown Historic Production of Non-renewable Resources

For many non-renewable resources, reliable production data are only available from a certain point in time but not from the beginning of production periods. In order to constrain the unknown historic production of such resources for those ancient times for which no reliable annual production data are available we present a novel mathematical technique, based on Verhulst’s logistic function. The method is validated by the United States’ crude oil production for which the complete production cycle, starting in 1859, is well documented. Assuming that the oil production in the USA between 1859 and 1929 is unknown, our method yields values for this period of 16.0 gigabarrels (Gb) based on a second-order polynomial fit and 13.5 Gb based on a third-order polynomial fit of post-1929 production data, respectively. Especially the latter amount compares well with the actual value of 12.1 Gb, thus illustrating the strength of the method. For global gold (Au) production, our method yields an ancient production up to the year 1850, when official and reliable production statistics began, of approximately 10,000 metric tons (t) based on a second-order polynomial fit. For mercury (Hg) a production of 64,000 t was determined for the time up to the year 1900, when annual production figures started to become available, again using a second-order polynomial fit. While the results obtained by the application of second-order polynomial functions could be confirmed by higher-order polynomial functions in the cases of both USA oil as well as global Au production, this was not possible in the case of global Hg production because of a highly irregular production curve.