Characterization of noise in airborne transient electromagnetic data using Benford's law

摘要

Given any data set, extract the first significant digit of every number, and count how often whole numbers 1 to 9 appear as the first significant digit in the data set. Benford's law states that the frequency distribution of first digits 1 to 9 obeys a particular probability for many naturally occurring data, especially those spanning many orders of magnitude in a log-uniform distribution. I find that airborne transient EM data obey Benford's law, whereas a random noise with a zero-mean normal distribution does not. So the relative magnitude of such noise can be characterized by the deviation of the first-digit frequency from Benford's law. A high noise level can result in a large Benford deviation. This approach is first demonstrated by a simple mathematical example using a synthetic data set contaminated by Gaussian noises in different strengths. Then I apply Benford's law to three real airborne transient EM data sets, including MegaTEM (2006), HeliGEOTEM (2008), and HeliTEM (2012). The HeliTEM data set has shown to be in great conformity to Benford's law, and it is inferred to have the lowest relative noise level of the three. Application of Benford's law does not involve sophisticated statistics, data preparation and visualization, because it only counts the frequencies of the first digits. Its simplicity makes it a practically useful tool in the first-pass noise characterization of large airborne EM data sets.