‘Benford’s Law’ or ‘Newcomb-Benford Law’, or the ‘First Digit Law’ describes the distribution of first digits in observational data (Newcomb, 1881; Benford, 1938) — in a distribution of random real world observations, the frequency of the first digit of the observation value (1,2,….9) will be non-uniform: i.e., Lower digits (1,2) occur more frequently.

A key use of Benford’s Law is to assess whether data is anomalous (fraudulent or incomplete). I have found a few Earth science examples: Nigrini and Miller (2007) looked at hydrological datasets; Sambridge et al. (2010) investigated a wide range of Earth Science datasets, and Geyer and Martí (2012) assessed datasets from volcanology. More recently Joannes-Boyau et al. (2015) recently used Benford’s Law to investigate a dataset of tropical cyclone tracks.

Beyond assessing the completeness and ‘Benford-ness’ of observational datasets is the idea of assessing models using Benford’s Law. I first came across this idea in Tolle et al. (2000). If observations of a system conform to Benford’s law, then perhaps a method to assess a model of the system is to make sure the model also conforms to Benford’s Law.

Because numerical models have broad application in geomorphology and many different (competing) models are developed to simulate the same processes, I wonder if Benford’s Law could be a useful discriminatory test? What if observations conform to Benford’s Law and a model fits the observations but is not Benford? Does this suggest that something is missing from the model? Or maybe conformity/nonconformity to Benford’s law means nothing for model assessment?

Radiolab did an episode about Benford’s law.

**For further reading, two books have been released in the past year:**

- Berger and Hill, 2015, An Introduction to Benford’s Law. Princepton University Press.(see the DSWeb review here and the MAA review here)
- Miller, S.J., ed., (2015), Benford’s Law: Theory and Applications, Princepton University Press.

**References:**

- Benford, F. (1938). The law of anomalous numbers.
*Proceedings of the American Philosophical Society*, 551-572. - Geyer, A., and Martí, J. (2012). Applying Benford’s law to volcanology.
*Geology*,*40*(4), 327-330. - Joannes-Boyau, R., Bodin, T., Scheffers, A., Sambridge, M., & May, S. M. (2015). Using Benford’s law to investigate Natural Hazard dataset homogeneity.
*Scientific reports*,*5*. - Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers.
*American Journal of Mathematics*,*4*(1), 39-40. - Nigrini, M. J., & Miller, S. J. (2007). Benford’s law applied to hydrology data—results and relevance to other geophysical data.
*Mathematical Geology*,*39*(5), 469-490. - Sambridge, M., Tkalčić, H., & Jackson, A. (2010). Benford’s law in the natural sciences.
*Geophysical research letters*,*37*(22). - Tolle, C. R., Budzien, J. L., & LaViolette, R. A. (2000). Do dynamical systems follow Benford’s law?.
*Chaos: An Interdisciplinary Journal of Nonlinear Science*,*10*(2), 331-336.