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Sambridge, M, Tkalčić, H and Jackson, A (2009)

On the applicability of Benford's Law in the Geosciences

Abstract for American Geophysical Union, Fall Meeting 2009.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

Abstract: Benford's Law is a curious property of numerous datasets whereby the frequency distribution of the first digit (i.e. first non zero number from the left) follows a well defined logarithmic function, namely P_D = log_b(1+1/D), where D is the first digit and b is the base of the data. This was initially observed by Newcomb (1881) and later quantified and expanded by Benford (1938). The latter author first put forward a set of 20 distinct data sets with differing physical dimension and character which collectively obeyed this 1st digit law. The nature of each data is the most startling feature of all in that they range from physical properties of matter such as molecular weight and specific heat capacity through river areas and drainage rates to population numbers in the USA as well as American baseball league averages of 1936. A universal law of digits was proposed by Benford and in recent times has been widely accepted. Investigations into the nature and use of Benford's Law have continued in multiple fields. Mathematicians have more recently proven the correctness of this universal law of digits under general conditions and Nigrini (1992) has made use of it for uncovering anomalous data errors and fraud in accountancy practices. To date Benford's Law appears to have received no attention within the Geosciences. Here we demonstrate its widespread applicability for geophysical data sets as well as models derived from data of varying type and physical dimension. Specifically we verify Benford's Law holds for a geomagnetic Field model of the Earth (gufm1), Seismic models obtained from tomography (including mantle shear wave and regional body wave P and S models for various parts of the globe), and the GRACE gravity model up to degree 160. It would appear that Benford's Law has widespread applicability to geoscience data. Departures from Benford's Law are of interest as they seem to indicate changes in the local character of data, possibly due to fraud, error, or influence from a secondary signal. To date practical applications of Benford's Law have been of a forensic nature, i.e. the detection of subtle effects in data which are inconsistent with the global statistics. Since signal detection is a central component in many areas of the geosciences, e.g. seismology, then Benford's Law may find novel applications. We present results of experiments which show that seismic noise in general does not obey Benford's Law whereas earthquake traces do, suggesting a potential role in seismic event detection. In addition when applied to seismicity catalogs artificially assigned earthquake depths clearly show up as anomalies to the 1st digit law. Our survey of the applicability of Benford's Law to various geophysical data sets has also lead to new theoretical insight on the conditions under which it will be satisfied. We present a new generalization of Benford's Law that has broader applicability. As awareness of `digit analysis' grows we expect Benford's Law to find new applications in the Earth Sciences.

@article{, author = {Sambridge, Malcolm and Tkal{\v{c}}i{\'c}, Hrvoje and Jackson, A}, year = {2009}, month = {12}, pages = {}, title = {On the applicability of Benford's Law in the Geosciences}, journal = {AGU Fall Meeting Abstracts}, url = {}, }

Reference Type: Conference Paper

Subject Area(s): Natural Sciences