{Benford}'s law in the natural sciences

  • Sambridge M
  • Tkalčić H
  • Jackson A
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More than 100 years ago it was predicted that the
distribution of first digits of real world observations
would not be uniform, but instead follow a trend where
measurements with lower first digit (1, 2, …{})
occur more frequently than those with higher first
digits (…{}, 8, 9). This result has long been
known but regarded largely as a mathematical curiosity
and received little attention in the natural sciences.
Here we show that the first digit rule is likely to be
a widespread phenomenon and may provide new ways to
detect anomalous signals in data. We test 15 sets of
modern observations drawn from the fields of physics,
astronomy, geophysics, chemistry, engineering and
mathematics, and show that Benford's law holds for them
all. These include geophysical observables such as the
length of time between geomagnetic reversals, depths of
earthquakes, models of Earth's gravity, geomagnetic and
seismic structure. In addition we find it also holds
for other natural science observables such as the
rotation frequencies of pulsars; green-house gas
emissions, the masses of exoplanets as well as numbers
of infectious diseases reported to the World Health
Organization. The wide range of areas where it is
manifested opens up new possibilities for exploitation.
An illustration is given of how seismic energy from an
earthquake can be detected from just the first digit
distribution of displacement counts on a seismometer,
i.e., without actually looking at the details of a
seismogram at all. This led to the first ever detection
of an earthquake using first digit information alone.

Author-supplied keywords

  • earthquakes; large data sets; observational seismo

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  • M Sambridge

  • Hrvoje Tkalčić

  • A Jackson

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