Fitting Multiple Bell Curves Stably and Accurately to a Time Series as Applied to Hubbert Cycles or Other Phenomena
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Bell curves are applicable to understanding many observations and measurements across the sciences. Relating Gaussian curves to data is a common because of its relation to both the Central Limit Theorem and to random error. Similarly, fitting logistic derivatives to oil or other non-renewable resource production is common practice. Fitting bell curves to a time series is an inherently non-linear problem requiring initial estimates of the parameters describing the bell-curves. Poor estimates lead to instability and divergent solutions. Fitting to a cumulative curve improves stability, but at the expense of accuracy of the final solution. Jointly fitting multiple bell curves is superior to extraction of curves one at a time, but further exacerbates the non-linearity. Including both the cumulative data and the bell-curve data within the inversion, can exploit the greater stability of the cumulative fit and the greater accuracy of a direct fit. The algorithm presented here inverts for multiple bells by combining cumulative and direct fits to exploit the best features of both. The versatility and accuracy of the algorithm are demonstrated using two different Earth Science examples: a seismo-volcanic sequence recorded by a hydrophone array moored to the seafloor and US coal production. The MatLab function used here for joint curve determination is included in the online manuscript complementary material.
KeywordsGaussian Hubbert multicycles Curve-fitting
I thank Del Bohnenstiehl for the seismo-volcanic azimuthal detection analysis. I thank Ken Anderson whose interest in Hubbert cycles first prompted my thoughts of the problem of simultaneously fitting multiple bell curves to complex systems. This work was improved by the thoughtful comments and suggestions of two anonymous reviewers as well as the journal editor, Roussos Dimitrakopoulos. This work was partially supported by National Science Foundation grant OCE-0825424.
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