Cross-correlation-an objective tool to indicate induced seismicity

Differentiation between natural and induced seismicity is crucial for the ability to safely and soundly carry out various underground experiments and operations. This paper defines an objective tool for one of the criteria used to discriminate between natural and induced seismicity. The qualitative correlation between earthquake rates and the injected volume has been an established tool for investigating the possibility of induced, or triggered, seismicity. We derive mathematically, and verify using numerical examples, that the definition of normalized cross-correlation (NCC) between positive random functions exhibits high values with a limit equal to one, if these functions (such as earthquake rates and injection volumes) have a large mean and low standard deviation. In such a case, the high NCC values do not necessarily imply temporal relationship between the phenomena. Instead of positive-value time histories, the functions with their running mean subtracted should be used for cross-correlation. The NCC of such functions (called here NCCEP) may be close to zero, or may oscillate between positive and negative values in cases where seismicity is not related to injection. We apply this method for case studies of seismicity in Colorado, the United Kingdom, Switzerland and south-central Oklahoma, and show that NCCEP reliably determines induced seismicity. Finally, we introduce a geomechanical model explaining the positive cross-correlation observed in the induced seismicity data sets.